# Constraint on the Higgs-Dilaton potential via Warm inflation in Two-Time
Physics

Vo Quoc Phonga,b<EMAIL_ADDRESS>a Department of Theoretical Physics,
University of Science, Ho Chi Minh City 700000, Vietnam
b Vietnam National University, Ho Chi Minh City 700000, Vietnam  Ngo Phuc Duc
Loca,b<EMAIL_ADDRESS>a Department of Theoretical Physics, University of
Science, Ho Chi Minh City 700000, Vietnam
b Vietnam National University, Ho Chi Minh City 700000, Vietnam

###### Abstract

Within the $SP(2,R)$ symmetry, the Two-time model (2T model) has six
dimensions with two dimensions of time and the dilaton field that can be
identified as inflaton in a warm inflation scenario with potential of the form
$\sim\phi^{4}$. From that consideration, we derive the range of parameters for
the Higgs-Dilaton potential, the coupling constant between Higgs and Dialton
($\alpha$) is larger than $0.0053$ and the mass of Dilaton is smaller than
$10^{-7}$ GeV. Therefore, the 2T-model indirectly suggests that extra-
dimension can also be a source of inflation.

###### pacs:

11.15.Ex, 12.60.Fr, 98.80.Cq

Keywords: Spontaneous breaking of gauge symmetries, Extensions of electroweak
Higgs sector, Particle-theory models (Early Universe)

##  I Introduction

From quantum mechanics, there is a symmetry between coordinates and momentums
that can be described by a symplectic group. When we combine this spatial
symmetry and Poincaré symmetry, it can be generally described by the $Sp(2,R)$
group or leading to the space-time must be larger than the ordinary one, there
is exactly an extra dimension of space and time bars1999 ; bars2000a . From
there, Two-time physics (2T Physics) was formed. The 2T physics provides a
better understanding of symmetries as well as the concept of time.

The 2T model bars1999 ; bars2000a ; bars2001 ; survey ; bars2006 ; kuo2006
could be a well known choice. The model presents us with an unusual vision of
time that advances on a plane, i.e., it has two dimensions of time. The 2T SM
(The Standard Model in 2T Physics) has better features than regular SM in 1T
physics. The strong CP problem of QCD has been solved without axion. It shows
us the 1T SM as the ”shadow” of 2T framework bars2006 ; kuo2006 . The concepts
of force, particles and their interaction in 2T SM are reduced to 1T Physics
through space-time contractions which are examples that clearly show 1T SM
(The Standard Model in 1T Physics) fully embedded in 2T SM bars2006 ; kuo2006
. The 2T concept can be extended in gravity survey . This theory has proposed
expanding the number of spacetime dimensions to represent the whole cosmos, as
well as the introduction of dilaton and its properties, for more than a decade
(since 2008). The remarkable new quantity in the 2T SM model is the Higgs-
Dilaton potential which gives us a new type of symmetry breaking bars2006 ;
kuo2006 . Importantly, the Dilaton potential in this model is open, so it
provides us with new bases for solving other dilemmas. In addition, the
development directions as well as the success of 2T Physcis can be detailed in
Ref. bars2001 .

After Alan Guth introduced the concept of inflation in 1981 8ag , the nature
of the inflaton field responsible for inflation remained elusive. For obvious
economical reasons, researchers usually consider Higgs boson as a promising
candidate higgsinf1 ; higgsinf2 ; higgsinf3 . Currently, there are many
inflation scenarios. The basic scenario is the slow-roll approximation in
which the potential changes slowly with respect to the field. We therefore
examine this basic approximation in the 2T model through the Higgs-dilaton
coupling.

We recognize that the inflation problem has a deep connection with the number
of dimensions of space-time. The 2T model therefore has both an extra-
dimensional effect and exotic particles but the inflation is not yet analyzed
in detail, i.e, extra dimensions and dilaton boson can be new materials for
inflation. Therefore, the 2T SM or specifically the Dilaton potential in the
2T SM is chosen to analyze the inflation problem or determine the parameters
of this model when considering Dialton as an Inflaton.

This article is organized as follows. In section II, we give a brief
introduction to the 2T model and introduce to a new Dilaton potential. In
section III, we introduce some salient features of warm inflation and show
that the dilaton can be identified as inflaton in the warm inflation scenario
and get the parameter domain of the Higgs-Dilaton potential. Finally, we
summarize and make outlooks in section IV.

##  II A brief introduction to 2T model and a new Dilaton potential

###  II.1 The $Sp(2,R)$ symmetry in the 2T model

The main new feature of 2T-physics is a new gauge symmetry called $Sp(2,R)$
that works in phase space $(X^{M},P^{M})$ (not just in spacetime like
1T-physics). We will show that Sp(2,R gauge symmetry) necessarily requires 2
timelike dimensions.

In 2T-physics, spacetime coordinates and energy-momentum are united in a
doublet $(X_{1}^{M},X_{2}^{M})$. $X_{1}^{M}=X^{M}$ is spacetime coordinates in
2T-physics and $X_{2}^{M}=P^{M}$ is energy-momentum in 2T-physics. In short,
we can label them as $X_{i}^{M}$ with i=1,2. The symplectic transformation
will act on this doublet and therefore, spacetime and energy-momentum can
transform to each other. This is a brand new property of 2T-physics. In
1T-physics, we just have transformations between spacetime coordinates, for
example, the Lorentz transformation or even in gauge theory, the local gauge
transformation $e^{i\theta(x^{\mu})}$ just involve spacetime coordinates. But
in 2T-physics, we treat spacetime coordinates and energy-momentum equally and
they can transform to each other.

The local symplectic transformation is defined as 18

$\delta_{\omega}X_{i}^{M}(\tau)=\epsilon_{ik}\omega^{kl}(\tau)X_{l}^{M}(\tau),$
(1)

where $\tau$ is the proper time, $\omega^{ij}(\tau)=\omega^{ji}(\tau)$ are
parameters of Sp(2,R) and $\epsilon^{ij}$ is tensor Levi-Cavita. To have a
connection with the concept of Sp(2,R) group in section 3.1.1, we can write it
out explicitly like this

$\begin{cases}\delta_{\omega}X^{M}(\tau)=\omega^{21}(\tau)X^{M}(\tau)+\omega^{22}(\tau)P^{M}(\tau)\\\
\delta_{\omega}P^{M}(\tau)=-\omega^{11}(\tau)X^{M}(\tau)-\omega^{12}(\tau)P^{M}(\tau).\end{cases}$
(2)

Because $X^{\prime M}=X^{M}+\delta_{\omega}X^{M}$, $P^{\prime
M}=P^{M}+\delta_{\omega}P^{M}$ so in the matrix form

$\begin{pmatrix}X^{\prime M}\\\ P^{\prime
M}\end{pmatrix}=\begin{pmatrix}1+\omega^{21}&\omega^{22}\\\
-\omega^{11}&1-\omega^{12}\end{pmatrix}\begin{pmatrix}X^{M}\\\
P^{M}\end{pmatrix}=S\begin{pmatrix}X^{M}\\\ P^{M}\end{pmatrix}.$ (3)

So S should be an element of the symplectic group

$\displaystyle\begin{split}S^{T}JS&=\begin{pmatrix}1+\omega^{21}&-\omega^{11}\\\
\omega^{22}&1-\omega^{12}\end{pmatrix}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}1+\omega^{21}&\omega^{22}\\\
-\omega^{11}&1-\omega^{12}\end{pmatrix}\\\
&=\begin{pmatrix}0&1-(\omega^{12})^{2}+\omega^{11}\omega^{22}\\\
-[1-(\omega^{12})^{2}+\omega^{11}\omega^{22}]&0\end{pmatrix}.\end{split}$ (4)

Therefore, we found the constraints of symplectic parameters
$(\omega^{12})^{2}=\omega^{11}\omega^{22}$ or

$det\begin{pmatrix}\omega^{11}&\omega^{12}\\\
\omega^{21}&\omega^{22}\end{pmatrix}=0.$ (5)

And remember that $\omega^{12}=\omega^{21}$ by definition. In that case, S is
indeed a symplectic element. Meanwhile, the local symplectic transformations
for the gauge fields is

$\delta_{\omega}A^{ij}=\partial_{\tau}\omega^{ij}+\omega^{ik}\epsilon_{kl}A^{lj}+\omega^{jk}\epsilon_{kl}A^{il},$
(6)

for the gauge fields $A^{ij}=A^{ji}$. The covariant derivative is

$D_{\tau}X_{i}^{M}=\partial_{\tau}X_{i}^{M}-\epsilon_{ik}A^{kl}X_{l}^{M}.$ (7)

The gauge fields here are not normal gauge fields corresponding to internal
transformation of the field. In 1T-physics, in order to make the Lagrangian
invariant under local gauge transformations of the scalar field, or spinor,…
(we call this type of transformations internal transformations, they transform
the fields, not coordinates) we have to replace the normal derivative by the
covariant derivative that contains the gauge field and add the kinetic term of
the gauge fields (we can not add the mass term since it will not be invariant
under the gauge transformation of the gauge field, therefore we need the Higgs
mechanism to generate the mass of the gauge fields). In 2T-physics, however,
we also have the gauge fields corresponding to the transformations of the
coordinates (we call this type of transformations external transformations).
This is a new property that we do not have in 1T-physics.

The Lagrangian that is invariant under this local symplectic transformation
is18

$\displaystyle\begin{split}S_{0}&=\frac{1}{2}\int
d\tau(D_{\tau}X_{i}^{M})\epsilon^{ij}X_{j}^{N}\eta_{MN}\\\ &=\frac{1}{2}\int
d\tau(\partial_{\tau}X_{i}^{M}-\epsilon_{ik}A^{kl}X_{l}^{M})\epsilon^{ij}X_{j}^{N}\eta_{MN}\\\
&=\int
d\tau\left(\dot{X}^{M}P_{M}-\frac{1}{2}A^{ij}X_{i}^{M}X_{j}^{N}\eta_{MN}\right).\end{split}$
(8)

The dot denotes the time derivative with respect to $\tau$. $A_{ij}$ is the
$Sp(2,R)$ gauge potential. The equation of motion of this potential goes to
the constraints, the target spacetime (1T) Ref. survey

We will take the variation of this action to get the equation of motion for
the gauge fields and for X, P (Note that:
$\delta(X^{2})=\delta(X^{M}X_{M})=X_{M}\delta X^{M}+X^{M}\delta
X_{M}=2X^{M}\delta X_{M}$, $\delta(X.P)=\delta(X^{M}p_{M})=P_{M}\delta
X^{M}+X^{M}\delta P_{M}=P^{M}\delta X_{M}+X^{M}\delta P_{M}$).

$\displaystyle\begin{split}\delta S&=\int
d\tau\begin{pmatrix}P_{M}\delta\dot{X}^{M}+\dot{X}^{M}\delta
P_{M}-\frac{1}{2}X^{2}\delta A^{11}-\frac{1}{2}X.P\delta
A^{12}-\frac{1}{2}P.X\delta A^{21}-\frac{1}{2}P^{2}\delta A^{22}\\\
-\frac{1}{2}A^{11}\delta(X^{2})-\frac{1}{2}A^{12}\delta(X.P)-\frac{1}{2}A^{21}\delta(P.X)-\frac{1}{2}A^{22}\delta(P^{2})\end{pmatrix}\\\
&=\int d\tau\begin{pmatrix}P_{M}\delta\dot{X}^{M}+\dot{X}^{M}\delta
P_{M}-\frac{1}{2}X^{2}\delta A^{11}-\frac{1}{2}X.P\delta
A^{12}-\frac{1}{2}P.X\delta A^{21}-\frac{1}{2}P^{2}\delta A^{22}\\\
-A^{11}X^{M}\delta X_{M}-A^{12}P^{M}\delta X_{M}-A^{21}X^{M}\delta
P_{M}-A^{22}P^{M}\delta P_{M}\end{pmatrix}\\\ &=\int
d\tau(-\frac{1}{2}X^{2}\delta A^{11}-\frac{1}{2}(X.P+P.X)\delta
A^{12}-\frac{1}{2}P^{2}\delta A^{22})+\int d\tau\frac{d}{d\tau}(\delta
X^{M}P_{M})\\\ &+\int d\tau(\dot{X}^{M}\delta P_{M}-\dot{P}_{M}\delta
X^{M})+\int d\tau(-A^{11}X^{M}\delta X_{M}-A^{12}P^{M}\delta
X_{M}-A^{21}X^{M}\delta P_{M}-A^{22}P^{M}\delta P_{M})\\\ &=\int
d\tau(-\frac{1}{2}X^{2}\delta A^{11}-\frac{1}{2}(X.P+P.X)\delta
A^{12}-\frac{1}{2}P^{2}\delta A^{22})+\int d\tau\frac{d}{d\tau}(\delta
X^{M}P_{M})\\\ &+\int d\tau(\dot{X}^{M}-A^{12}X^{M}-A^{22}P^{M})\delta
P_{M}-\int d\tau(\dot{P}^{M}+A^{11}X^{M}+A^{12}P^{M})\delta X_{M}.\end{split}$
(9)

Set $\delta S=0$, we have the equation of motion for the gauge field
$A^{11},A^{12},A^{22}$ as

$X^{2}=P^{2}=X.P+P.X=0.$ (10)

These are the constraints for Sp(2,R) generators (The generators of Sp(2,R) is
in general a function of X and P, $Q_{ij}(X,P)$. In flat spacetime with no
background field, they are $Q_{11}=X^{2}$, $Q_{12}=Q_{21}=X.P+P.X$ and
$Q_{22}=P^{2}$). The above equations have nontrivial solutions only if there
are two times, with fewer times these conditions collapse the system to
triviality. To see why, consider at first with no time dimension, and ask if
there are solutions to equation such as $X^{2}+Y^{2}+Z^{2}+W^{2}=0$ when there
are no minus sign for time. Clearly, this equation just has trivial solution
$X^{M}=P^{M}=0$ which has no physical content. Next consider the case with one
time dimension, we see that the only solution is that $X^{M}\sim P^{M}$ must
be parallel light-like (null-like) vectors proportional to each other. The
angular momentum in defined as

$L^{MN}=X^{M}P^{N}-X^{N}P^{M},$ (11)

will vanish, and hence this is a trivial solution that does not represent even
free motion in 1T-physics. In the case of two time dimensions, these equations
permit an infinite number of non-trivial solutions. Evidently, the extra time
is not introduced ”by hand” but the symmetry demands its existence. The number
of spatial dimensions is not limited, so in general, the spacetime of
2T-physics is $d+2$ dimensions where d is the number of spatial dimensions.
Meanwhile, the equations of motion for X,P are

$\begin{cases}\dot{X}^{M}-A^{12}X^{M}-A^{22}P^{M}=0\\\
\dot{P}^{M}+A^{11}X^{M}+A^{12}P^{M}=0.\end{cases}$ (12)

The particle model in 2T is still like the standard model bars2006 , but the
space-time is larger than the standard model bars2006 . Another important
thing, the particle model in 2T includes an extra Dilaton in the Higgs
potential bars2006 .

Dilaton is a scalar field which lives in 4+2 dimensions of spacetime. In the
standard model of 2T-physics bars2006 , in order to initiate the electroweak
phase transition that explains the source of mass for all massive matter, the
4 + 2 standard model requires the presence of a dilaton field coupled to the
Higgs field. The dilaton is required also by the 2T formulation of general
relativity in 4 + 2 dimensions. More details about dilaton in 2T-physics can
be found in Ref. bars2006 .

###  II.2 The Higgs-Dilaton potential and gauge fixing technique

The Higgs-dilaton potential has the following form which is as a result of
b-gauge symmetry, the inspiration of this symmetry comes from BRST formalism
bars2006 ; kuo2006  but it ultimately comes from the underlying Sp(2,R),

$V(\Phi,H)=\lambda\left(H^{\dagger}H-\alpha^{2}\Phi^{2}\right)^{2}+V(\Phi)\,,$

where $\lambda,\alpha$ are dimensionless couplings. $H$ and $\Phi$ are the
SU(2) Higgs and dilaton doublet respectively. $V(\Phi)$ is the dilaton
potential and plays a role in construction of the effective potential.

The more complicated work will be done with the Higgs-dilaton Lagrangian,

$L(A,H,\Phi)=\dfrac{1}{2}\Phi\partial^{2}\Phi+\dfrac{1}{2}\Big{(}H^{\dagger}D^{2}H+(D^{2}H)^{\dagger}H\Big{)}-\lambda\left(H^{\dagger}H-\alpha^{2}\Phi^{2}\right)^{2}-V(\Phi)\,.$

It is convenient if one chooses a lightcone basis in 4+2 dimensions written as

$\displaystyle
X^{\pm^{\prime}}_{i}=\dfrac{1}{\sqrt{2}}\left(X^{0^{\prime}}_{i}\pm
X^{1^{\prime}}_{i}\right),\text{where
}X_{i}^{M}=(X^{M}_{1},X^{M}_{2})\equiv(X^{M},P^{M})$ (13)
$\displaystyle\Rightarrow$
$\displaystyle\begin{cases}X^{2}=-2X^{+^{\prime}}X^{-^{\prime}}+X^{\mu}X_{\mu},\\\
X^{M}P_{M}=-X^{+^{\prime}}P^{-^{\prime}}-X^{-^{\prime}}P^{+^{\prime}}+X^{\mu}P_{\mu}\,.\end{cases}$
(14)

We choose the Bars’s parametrization bars2000a  for the components of $X^{M}$
as follows

$\displaystyle
X^{+^{\prime}}=\kappa,X^{-^{\prime}}=\kappa\Lambda,X^{\mu}=\kappa x^{\mu}$
(15)
$\displaystyle\Rightarrow\kappa=X^{+^{\prime}},\Lambda=\dfrac{X^{-^{\prime}}}{X^{+^{\prime}}},x^{\mu}=\dfrac{X^{\mu}}{X^{+^{\prime}}}\,,$
(16)

from this choose (the gauge fixing technology), we can reduce the 2T metric to
1T metric (Minkowski metric).

Scalar fields will be reduced to the following:

$\displaystyle\begin{cases}\Phi(X)\longrightarrow\dfrac{1}{\kappa}\phi(x)\quad;\quad
H(X)\longrightarrow\dfrac{1}{\kappa}h(x),\\\
\partial^{2}\Phi(X)=\partial^{M}\partial_{M}\Phi(X)\longrightarrow\dfrac{1}{\kappa^{3}}\dfrac{\partial^{2}\phi(x)}{\partial
x^{\mu}\partial x_{\mu}}\,,\\\
D^{2}H(X)=D^{M}D_{M}H(X)\longrightarrow\dfrac{1}{\kappa^{3}}D^{\mu}D_{\mu}h(x)\,.\end{cases}$
(17)

With the above gauge fixing, one would be able to derive the following
reduction

$\displaystyle L(A,H,\Phi)\longrightarrow\ $
$\displaystyle\dfrac{1}{2\kappa^{4}}\phi\dfrac{\partial^{2}\phi}{\partial
x^{\mu}\partial
x_{\mu}}+\dfrac{1}{2\kappa^{4}}\left[h^{\dagger}D_{\mu}D^{\mu}h+(D^{\mu}D_{\mu}h)^{\dagger}h\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\dfrac{\lambda}{\kappa^{4}}\left(h^{\dagger}h-\alpha^{2}\phi^{2}\right)^{2}-V(\phi)\,.$
(18)

Note that all the $\kappa$ coefficients will be simplified by the scaling
invariant of the actions through $\delta(X^{2})$ when reducing from 2T to 1T
bars2006 ; kuo2006 . The Higgs and Dilaton have redefined bars2006 ; kuo2006
as

$h(x)=\dfrac{1}{\sqrt{2}}\begin{pmatrix}0\\\ v+\sigma(x)\end{pmatrix};\langle
h(x)\rangle=v\,;\phi(x)=\dfrac{1}{\alpha\sqrt{2}}(v+\alpha
d(x));\langle\phi(x)\rangle=\dfrac{v}{\alpha}\,.$ (19)

As discussed in Refs. bars2006 ; kuo2006 , there are some reasons to expect
that the new dilaton potential only takes the form $V(\Phi)=\rho\Phi^{4}$ and
after gauge fixing to 1T-physics it takes the form $V(\phi)=\rho\phi^{4}$. We
assume that the resulting scalar field after reducing the dilaton field from
2T-physics to 1T-physics is indeed the inflaton field, and we will use this
quartic form of the potential

$V(\phi)=\rho\phi^{4}$ (20)

to study warm inflation in the following sections and notice that $\rho$ has a
very small value, which will be explained in a later section.

The minimum process of this full Higgs-Dilaton potential can be considered as
the minimum process of the potential when not $V(\phi)$. Because $\rho$ is
very small. The minimum process is detailed in Ref.bars2006 .

The full Higgs-Dilaton potential is expanded as follows

$\displaystyle V(h,\phi)$
$\displaystyle=\dfrac{\lambda}{4}\left[(v+h)^{2}-(v+\alpha
d)^{2}\right]^{2}+\dfrac{\rho}{4\alpha^{4}}(v+\alpha d)^{4}$
$\displaystyle=\dfrac{\lambda}{4}(h-\alpha d)^{2}(2v+h+\alpha d)^{2}$
$\displaystyle\qquad+\dfrac{\rho}{4\alpha^{4}}(v^{4}+4\alpha
v^{3}d+6\alpha^{2}v^{2}d^{2}+4\alpha^{3}vd^{3}+\alpha^{4}d^{4})$
$\displaystyle=\dfrac{\lambda}{4}(4v^{2}h^{2}+4\alpha^{2}v^{2}d^{2}-8\alpha
v^{2}hd)$ $\displaystyle\qquad+\dfrac{\rho}{4\alpha^{4}}(v^{4}+4\alpha
v^{3}d+6\alpha^{2}v^{2}d^{2})+\text{3,4-fields interaction terms}$ (21)
$\displaystyle=\dfrac{\lambda}{4}(4v^{2}h^{2}+4\alpha^{2}v^{2}d^{2}-8\alpha
v^{2}hd)+\frac{3\rho}{2\alpha^{2}}v^{2}d^{2}$
$\displaystyle\qquad+\dfrac{\rho}{4\alpha^{4}}(v^{4}+4\alpha
v^{3}d)+\text{3,4-fields interaction terms}$ (22) $\displaystyle=v^{2}(h\quad
d).\begin{pmatrix}\lambda&-\alpha\lambda\\\
-\alpha\lambda&\lambda\alpha^{2}+\frac{3\rho}{2\alpha^{2}}\end{pmatrix}.\begin{pmatrix}h\\\
d\end{pmatrix}+\dfrac{\rho}{4\alpha^{4}}(v^{4}+4\alpha
v^{3}d)+\text{3,4-fields terms}$ (23)
$\displaystyle\approx\frac{1}{2}(h^{\prime}\quad
d^{\prime}).\begin{pmatrix}m^{2}_{h^{\prime}}&0\\\
0&m^{2}_{d^{\prime}}\end{pmatrix}.\begin{pmatrix}h^{\prime}\\\
d^{\prime}\end{pmatrix}+const+\text{3,4-fields terms},$ (24)

in which

$\displaystyle
m^{2}_{d^{\prime}}=\frac{v^{2}\left(2\alpha^{4}\lambda+2\alpha^{2}\lambda-\sqrt{\left(2\alpha^{4}\lambda+2\alpha^{2}\lambda+3\rho\right)^{2}-24\alpha^{2}\lambda\rho}+3\rho\right)}{\alpha^{2}},$
(25) $\displaystyle
m^{2}_{h^{\prime}}=\frac{v^{2}\left(2\alpha^{4}\lambda+2\alpha^{2}\lambda+\sqrt{\left(2\alpha^{4}\lambda+2\alpha^{2}\lambda+3\rho\right)^{2}-24\alpha^{2}\lambda\rho}+3\rho\right)}{\alpha^{2}}.$
(26)

The physical fields are

$\displaystyle\begin{pmatrix}h^{\prime}\\\
d^{\prime}\end{pmatrix}=\left(\begin{array}[]{cc}A&B\\\ 1&1\\\
\end{array}\right).\begin{pmatrix}h\\\ d\end{pmatrix},$ (29) $\displaystyle
A=\frac{2\alpha^{4}\lambda-2\alpha^{2}\lambda+\sqrt{\left(2\alpha^{4}\lambda+2\alpha^{2}\lambda+3\rho\right)^{2}-24\alpha^{2}\lambda\rho}+3\rho}{4\alpha^{3}\lambda},$
(30) $\displaystyle
B=-\frac{-2\alpha^{4}\lambda+2\alpha^{2}\lambda+\sqrt{\left(2\alpha^{4}\lambda+2\alpha^{2}\lambda+3\rho\right)^{2}-24\alpha^{2}\lambda\rho}-3\rho}{4\alpha^{3}\lambda}.$
(31)

To do the above minima and approximation, we have to remove the term
$\dfrac{\rho v^{3}d}{\alpha^{3}}$ or $\dfrac{\rho v^{3}}{\alpha^{3}}\ll 1$. In
addition, the angle of mixing between $h$ and $\phi$ is proportional to
$\alpha$, which is significant in an inflationary scenarino. With large
$\alpha$, $\phi$ couples to the standard model particles as large as $h$ does.
The signal strength that produces $\phi$ at the LHC, for example by gluon
fusion, is comparable to that of the Higgs field them1 ; them2 . Therefore,
the large $\alpha$ situation must be omitted.

From Eqs. 25 and 26, when $\rho$ goes to zero, we derive the massless Dilaton,
this is consistent with the discussion in Ref. bars2006 ; kuo2006 .

##  III Dilaton as inflaton in the warm $\phi^{4}$ inflation

Inflation is a very short period of exponential expansion in the early
Universe. This era is assumed to happen in order to explain some problems of
the standard Big Bang cosmology including the horizon problem, the flatness
problem, and the seeds of cosmic structures. The observational evidences
strongly support the general ideas of inflation, but the details of inflation
remain elusive.

In standard inflation, there are two evident ways to calculate the inflaton
mass. The first one is using the quadratic inflaton potential
$V=\frac{1}{2}m^{2}\varphi^{2}$, where $m$ is the inflaton mass. The second
one is using the exponential potential $V=e^{-\beta\varphi}$. In the latter
case, we expand the Taylor series of the potential and the term in front of
$\varphi^{2}$ is the inflaton mass.

Warm inflation is different to the standard inflation, is a model of inflation
in which, apart from the quantum fluctuations of the inflaton field, the
inflaton field is also subject to thermal fluctuations due to the interaction
with the background thermal bath. This idea was first initiated by Berrera in
1995 Ber95 . Intuitively, we expect that in this model the inflationary
gravitational waves amplitudes must be smaller than that in the standard
”cold” inflation. The reason is the following. The kinetic energy of the
inflaton field will be dissipated due to the additional thermal fluctuations
and hence its roll is slowed down further. This means that the inflaton field
excursion can be small and yet still produces enough the number of e-folds to
resolve the horizon or flatness problems. In turn, this implies that the
tensor-to-scalar ratio is also small due to the Lyth bound. Indeed, we will
see that this expectation turns out to be true. The $\phi^{4}$ potential in
the standard ”cold” inflation is ruled out since it predicts a too large
tensor-to-scalar ratio; on the other hand, the $\phi^{4}$ potential in warm
inflation works perfectly well since the predicted tensor-to-scalar ratio in
this model is much smaller.

However, both of them use the slow rolling approximation (Slow-roll
approximation states that $\frac{1}{2}\dot{\varphi}^{2}\ll V$, and therefore
$\epsilon,|\eta|\ll 1$. Slow-roll approximation is the must for inflation to
happen. There are two formalisms of slow-roll approximation which are
potential slow-roll approximation (PSRA) and Hubble slow-roll approximation
(HSRA). In this paper, we will always use the former). The slow-roll
parameters are defined as 39

$\epsilon=\frac{M_{p}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\hskip
8.5359pt;\hskip 8.5359pt\eta=M_{p}^{2}\frac{V^{\prime\prime}}{V}\,,$

where $M_{p}$ is 4D Planck scale and primes denote derivatives with respect to
$\varphi$.

In warm inflation, we have two cases which are the weak and the strong
dissipative regimes defined as $R\ll 1$ and $R\gg 1$, respectively:

$R=\frac{\Gamma}{3H},$ (32)

where $H$ is the Hubble parameter and $\Gamma$ is the inflaton decay rate. For
the warm inflation, in addition to the two usual slow-roll parameters, we have
the following two parameters wr1 ; wr2 ; wr3 ; phi4a ; phi4b ; phi4c ,

$\beta=M_{p}^{2}\frac{\Gamma^{\prime}V^{\prime}}{\Gamma
V},\sigma=M_{p}^{2}\frac{V^{\prime}}{\varphi V}.$ (33)

The slow-roll conditions can be expressed as wr1 ; wr2 ; wr3 ; phi4a ; phi4b ;
phi4c

$\epsilon,|\eta|,|\beta|,|\sigma|\ll 1+R.$ (34)

In order to make sure that the potential $V\sim\varphi^{4}$ is still reliable,
we need to calculate two important quantities, which are the scalar spectral
index $n_{s}$ and the tensor-to-scalar ratio $r$. To do that, we first need to
mention an important quantity in inflation which is the e-folding number $N$
representing the amount of inflation. In the context of potential slow-roll
approximation, the e-folding number is

$N\equiv
ln\frac{a(t_{f})}{a(t_{i})}\simeq-\frac{8\pi}{M_{4}^{2}}\int_{\varphi_{i}}^{\varphi_{f}}\frac{V(1+R)}{V^{\prime}}d\varphi\,.$
(35)

From the condition of inflation to end $\epsilon\sim 1+R$, we get the final
value of the inflaton field $\varphi_{f}$. Then, using Eq. (35) to calculate
the initial value of the inflaton field $\varphi_{i}$ when the perturbations
exit the horizon. Therefore, $n_{s}$ and $r$ can be expressed in terms of the
slow-roll parameters and assuming an inflaton decay rate $\Gamma=aT$. Finally,
with the inflation at 60 e-folds, one gets $10^{-15}<\rho<10^{-13}$ in all
case wr1 ; wr2 ; wr3 ; phi4a ; phi4b ; phi4c .

Considering our assumption in (20) to fit the $SP(2,R)$ symmetry, the Dilaton
potential has the quartic form. Thus, this form is compatible with a warm
inflation scenario. A quartic inflaton potential in the warm inflation
scenario has also been studied in wr1 ; wr2 ; wr3 ; phi4a ; phi4b ; phi4c .

Furthermore, the determination of the value of dissipative coefficient
$\Gamma=aT$ (or $a$) currently only stops at the estimate in the example
models. Because the determination of the dissipative components of Inflaton is
not specific. This can go beyond the standard model and SM-like versions as 2T
model. Calculations in some specific cases and the expansion of $\Gamma$ can
be found in Refs. wr1 ; 22 ; 23 ; 24 ; 25 ; 27 ; 28 . However, we can roughly
outline that $a$ is a function of coupling constants of the dissipative
channels of Inflaton.

Based on Ref. phi4b , with strong and weak dissipative regime, the important
quantities are summarized in Table 1. Especially the values of $\rho$ (or
$\lambda$) and $a$ is derived when combined with Planck data 30planck .

Quantity Regime | Weak dissipative regime | Strong dissipative regime
---|---|---
The power spectrum:$P_{R}^{1/2}\sim some\times 10^{-5}$ | $\left(\frac{\lambda^{\prime}\sqrt{a}N^{3}}{6\sqrt{70}\pi^{3}}\right)^{1/3}$ | $\left[\frac{4N^{3}\lambda^{\prime}}{125\pi^{8/3}}\left(\frac{2}{315}\right)^{1/3}\right]^{3/8}$
The tensor-to-scalar ratio: $r(n_{s})$ | $\frac{4\sqrt{14}}{625\sqrt{5}\,a^{1/2}}(1-n_{s})$ | $8.5\times 10^{-9}\frac{\,\pi^{10/3}}{a^{4}}(1-n_{s})$
$\rho=\frac{\lambda^{\prime}}{4}$ | $0.25\times 10^{-15}<\rho<0.25\times 10^{-13}$ | $some\times 10^{-15}$
$a$ | $6.5\times 10^{-5}<a<3.4\times 10^{-2}$ | $3.4\times 10^{-2}<a$
Table 1: The values of quantities are calculated to give the result $r(n_{s})$
that satisfy the Planck data 30planck . $n_{s}$ is the scalar spectral index.

In the Table 1, to satisfy all cases, we choose $\rho=10^{-14}$. Thus, in the
weak dissipative regime, $a=5\times 10^{-4}$. In the strong dissipative
regime, we choose $a=5\times 10^{-2}$.

With $m_{h^{\prime}}=125$ GeV, $v=246$ GeV and $\rho=10^{-14}$, we can deduce
the following relationship

$\displaystyle\lambda=\frac{3.05176\times 10^{21}\alpha^{2}-7.09172\times
10^{8}}{4.72781\times 10^{22}\alpha^{4}+4.72781\times
10^{22}\alpha^{2}-1.09866\times 10^{10}}.$ (36)

Furthermore, we have the condition,

$\displaystyle\frac{\rho v^{3}}{\alpha^{3}}=\frac{10^{-14}\times
246^{3}}{\alpha^{3}}\ll 1\Longrightarrow\alpha\gg 0.0053.$ (37)

We then deduce the mass of dilaton in Fig. 1. Because $\alpha$ is small, the
mass of Dilaton decreases, must be smaller than $10^{-7}$ GeV when
$0.1<\alpha<1$. So Fig. 2 is plotted, to show how $r(n_{s})$ is compatible
with the Planck data 30planck .

Figure 1: The mass of Dilaton with $\rho=10^{-14}$. Figure 2: The function $r$
depends on $n_{s}$. $\rho=10^{-14}$, $0.1<\alpha<1$.

We see that, the larger the $\alpha$, the smaller the mass of the dilaton.
Therefore, the signal of Dilaton will be very small even though the coupling
constant with Higgs is large.

Finally, when combined with the Planck data 30planck , in all regimes we get
$\rho=10^{-14}$, from which the coupling constant ($\alpha$) between Higgs and
Dilaton can be deduced.

##  IV Conclusion and discussion

We showed that Dilaton can be a candidate of Inflaton in the warm inflation
scenario. Accordingly, the parameter domain of the Higgs-Dilaton potential is
retrieved. This indirectly shows that extra dimensions can play an important
role in the inflation problem.

The Higgs-Dilaton potential can be added $\phi^{2}$ that is the dominant
component of the standard inflation, thus a dimensional mass parameter can be
added along with $\phi^{2}$. But it breaks the basic $Sp(2;R)$ symmetry. This
parameter can be derived from the interaction of Dilaton with $S(x)$ which is
an additional scalar field, can be seen as Inflaton 1bars ; 2bars ; 2barsb .
This has been done consistently with the scale invariance required by
2T-Physics Ref. 3bars .

According to Refs. wr1 ; 22 ; 23 ; 24 ; 25 ; 27 ; 28 , to be able to find more
dissipative components in 2T physics, we need to extend this model in
combination with supersymmetry or extending the Higgs-Dilaton potential one
more time with exotic particles but must conform to the $Sp(2,R)$ symmetry.
Also, because in the 2T model, Dilaton only interacts directly with Higgs.
Therefore, there are not enough dissipative components to have a suitable
value of $a$ or $r$ for the Planck data 30planck . This is an interesting work
after this article.

There are the other studies in Cosmology, or example, the cosmological
constant and inflation 42a ; 42b . But in this model, the inflation and
reheating period have not been analyzed in detail. We considered that Dilaton
is Inflaton in the warm scenario. This presents another opportunity to study
extra dimensions as well as other inflationary scenarios.

The extra dimensional phenomena can give many prediction for experiments. Our
next work focuses on extra dimensions, in order to describe their properties
in the baryogenesis or leptongenesis scenario and cosmology problems.

The 2T model is a hologram of string theory and has a connection with SM.
Therefore, analyzing and refining this model is a ”road” to create a good
bridge between two theories, in which the problem of inflation and the matter-
antimatter asymmetry are significant problems.

New physical effects from the interaction between Higgs and Dilaton may change
the electroweak, but this model still fits into the standard model through the
Bars transformation that has been mentioned in section II.2. Thus, these
changes (such as the Higgs couplings to fermions) will indeed provide new
explanations for difficult problems (such as Baryogenesis, etc. ) without
changing the electroweak nature.

In a context with too many free parameters, the first step in our paper shows
that the small Dilaton potential can be combined with the warm inflation
problem, and then derive the parameter domain $\alpha$. If further extended to
other problems, the range of values $\alpha$ that is derived in this paper,
need to be confirmed more fully.

## ACKNOWLEDGMENTS

In memory of the day our venerable teacher, Dr. Vo Thanh Van, passed away on 5
July 2016, this series of our papers highlights the continuation of what he
has done through many years in the Department of Theoretical Physics.

## References

  * (1) I. Bars, C. Deliduman and D. Mimic, Phys. Rev. D 59, 125004 (1999).
  * (2) I. Bars, Phys. Rev. D 62, 046007 (2000).
  * (3) I. Bars, AIP conference proceedings 607, 17, Fukuoka (Japan) (2002).
  * (4) I. Bars, Class. and Quantum Grav. 18, 3113-3130 (2001).
  * (5) I. Bars, Phys. Rev. D 74, 085019 (2006).
  * (6) I. Bars and Y-C. Kuo, Phys. Rev. D 74 085020 (2006).
  * (7) Alan H. Guth, Phys. Rev. D 23, 347 (1981).
  * (8) F.L. Bezrukov, M.E. Shaposhnikov, Phys. Lett. B 659: 703-706 (2008).
  * (9) Seiga Sato and Kei-ichi Maeda, Phys. Rev. D 97, 083512 (2018).
  * (10) Ryusuke Jinno, Kunio Kaneta, and Kin-ya Oda, Phys. Rev. D 97, 023523 (2018).
  * (11) I. Bars, C. Deliduman, O. Andreev, Phys. Rev. D 58, 066004 (1998).
  * (12) CMS Collaboration, Phys. Lett. B 767, 147 (2017).
  * (13) Giorgio Arcadi, Farinaldo S. Queiroz, Clarissa Siqueira, Phys. Lett. B 775, 105 (2017).
  * (14) Arjun Berera, Phys. Rev. Lett. 75, 3218 (1995).
  * (15) J. E. Lidsey, A. R. Liddle, E. W. Kolb, E. J. Copeland, T. Barreiro, M. Abney, Rev. Mod. Phys.69:373-410 (1997).
  * (16) M. Bastero-Gil, A. Berera, R.O. Ramos, JCAP 1107, 030 (2011).
  * (17) S. Bartrum, M. Bastero-Gil, A. Berera, R. Cerezo, R.O. Ramos, J.G. Rosa, Phys. Lett. B 732, 116 (2014).
  * (18) M. Bastero-Gil, A. Berera, I.G. Moss, R.O. Ramos, JCAP 1405, 004 (2014).
  * (19) Erandy Ramirez and Dominik J. Schwarz, Phys.Rev.D 80, 023525 (2009).
  * (20) Grigorios Panotopoulos, Nelson Videla, Eur. Phys. J. C 75, 525 (2015).
  * (21) V. V. Kiselev, S. A. Timofeev, arXiv:0801.2453v3 [gr-qc] (2008).
  * (22) I.G. Moss, C. Xiong, arXiv:hep-ph/0603266
  * (23) A. Berera, M. Gleiser, R.O. Ramos, Phys. Rev. D 58, 123508 (1998)
  * (24) A. Berera, R.O. Ramos, Phys. Rev. D 63, 103509 (2001)
  * (25) Y. Zhang, JCAP 0903, 023 (2009)
  * (26) M. Bastero-Gil, A. Berera, R.O. Ramos, J.G. Rosa, JCAP 1301, 016 (2013)
  * (27) M. Bastero-Gil, A. Berera, R.O. Ramos, J.G. Rosa, JCAP 1410, 053 (2014)
  * (28) P.A.R. Ade et al. [Planck Collaboration], arXiv:1502.02114.
  * (29) I. Bars, P. Steinhardt, Neil Turok, Phys.Rev.D 89, 043515 (2014).
  * (30) I. Bars, Phys.Rev.D 77, 125027 (2008).
  * (31) I. Bars, S-H Chen, Phys. Rev. D 79 085021 (2009).
  * (32) I. Bars, Phys.Rev.D 82, 125025 (2010).
  * (33) Y. Fujii, Prog. Theor. Phys.110, 433-439 (2003).
  * (34) S.Casas, M. Pauly, J.Rubio, Phys. Rev. D 97, 043520 (2018).