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Physics Letters B 762 (2016) 535–542

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Fubini instantons in Dilatonic Einstein–Gauss–Bonnet theory of

gravitation

Wonwoo Lee

, Bum-Hoon Lee

a,b,c

, Daeho Ro

a,∗

Asia Paciﬁc Center for Theoretical Physics, Pohang 37673, Republic of Korea

Center for Quantum Spacetime, Sogang University, Seoul 04107, Republic of Korea

Department of Physics, Sogang University, Seoul 04107, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:

Received

13 July 2016

Received

in revised form 19 August 2016

Accepted

9 September 2016

Available

online 14 September 2016

Editor: M.

Cveti

Keywords:

Fubini

instanton

DEGB

theory

Tunneling

without barrier

In this paper, we investigate various types of Fubini instantons in the Dilatonic Einstein–Gauss–Bonnet

theory of gravitation which describes the decay of the vacuum state at a hilltop potential through

tunneling without barrier. It is shown that the vacuum states are modiﬁed by the non-minimally coupled

higher-curvature term. Accordingly, we present the new solutions which describe the tunneling from

new vacuum states in anti-de Sitter and de Sitter backgrounds. The decay probabilities of the vacuum

states are also inﬂuenced. We thus show that the semiclassical exponents can be decreased for speciﬁc

parameter ranges, thereby increasing tunneling probability.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

If an initial state of a potential is a false vacuum or metastable

vacuum, the vacuum state can decay through tunneling [1–3] or

rolling as a consequence of quantum and classical mechanics, re-

spectively.

The bounce solution describes the quantum tunneling

process in Euclidean space, which has a negative mode [4]. The ef-

fect

is more subtle in the presence of gravitation [5,6] The bounce

solution determines the semi-classical exponent of the decay rate

of an initial state. The decay rate per unit time per unit volume

given by /V = Ae

−B

, where the pre-exponential factor A is a

functional determinant around the classical solution and the semi-

classical

exponent B is the Euclidean action difference between

the bounce solution and the background. The mechanism of the

false vacuum decay is later developed with gravitation [7,8] and

expanded upon in more detail [9].

the quantum tunneling process, a particle can penetrate a

potential barrier to a region which is classically considered forbid-

den.

The phenomenon of tunneling without barrier can also occur

and it is very peculiar, because the tunneling occurs in a classically

allowed region. If the time-scale of tunneling on the potential is

Corresponding author.

E-mail

addresses: warrior@sogang.ac.kr (W. Lee), bhl@sogang.ac.kr (B.-H. Lee),

daeho.ro@apctp.org (D. Ro).

shorter than that of rolling, the tunneling phenomenon can occure

despite

the lack of a potential barrier. This phenomenon was stud-

ied

ﬁrst as a Fubini instanton [10,11], a bounce solution on the

hilltop potential consisting of a negative quartic scalar ﬁeld. The

solution is used in the case of tunneling without a potential barrier

between the top of a potential and an arbitrary state. The explicit

form of the Fubini instanton was obtained through the conformal

invariance of scalar ﬁeld theory which allows the existence of a

one-parameter family of a Euclidean solution of an arbitrary size

but with the same probability. The conformal invariance can be

broken by introducing a mass term. This makes the solution disap-

pear

[12]. There have been intensive studies on tunneling without

barrier in the absence of gravitation [13] and in the presence of

gravitation [14] which are expanded upon later [15–18].

The

inﬂationary universe scenario [19–22] was proposed to

solve a number of questions in the standard model of hot big-

bang

cosmology and it is now well established as a leading theory

describing the early universe. Inﬂationary models with many dif-

ferent

types of a potential have been studied and some of them

have continued to stay relevant even after precision observation.

Recently, it has been found that Planck data [23] prefers hilltop

and plateau inﬂationary models [24–29]. In addition, the instan-

ton

solutions in those potentials have renewed interest in the

AdS/CFT correspondence [30,31]. In line with these scenarios, it is

http://dx.doi.org/10.1016/j.physletb.2016.09.013

0370-2693/

SCOAP

536 W. Lee et al. / Physics Letters B 762 (2016) 535–542

worthwhile to observe quantum tunneling in a model with a hill-

top

potential as a case of Fubini instanton.

one considers the very early universe of the Plank scale, the

effects of the spacetime curvature become signiﬁcant such that the

early universe was in the quantum gravity regime. For this rea-

son,

we consider the higher-curvature Gauss–Bonnet (GB) term as

a correction, in which the GB coeﬃcient α is a dimensionless pa-

rameter.

In four-dimensional spacetime, the theory using the GB

term does not have a ghost particle or negative energy states [32].

It also does not affect the equations of motion and the solutions.

In order to introduce the contribution from the GB term, the term

is coupled with a dilaton ﬁeld with the coupling constant γ which

has a length dimension [33–35]. The nonminimally coupled higher-

curvature

term appears in the ﬁrst order α



-correction (16ακ in

the present paper) of the string effective action [36]. This is called

the Dilatonic Einstein–Gauss–Bonnet theory of gravitation (DEGB

theory). It may provide a chance to avoid the initial singularity

of the universe [37–39]. When taking into account the model of

the Fubini instanton in the DEGB theory, the shape of the effec-

tive

potential may change signiﬁcantly depending on the signs of

α and γ .

The

paper is organized as follows: in the next section, we set

up the basic framework with the action and the equations of mo-

tion

for the DEGB theory. In section 3, we analyze how the vacuum

states are modiﬁed by the dilaton coupling with higher-order cur-

vature

terms. In section 4, we present various types of solutions in

DEGB theory. Finally in section 5, we summarize our results and

discuss their implication for cosmology.

2. Set up

Let us consider the action that the scalar ﬁeld is interacting

with the Gauss–Bonnet (GB) term as follows:

S =



√

−g



2κ

−

∂

φ∂

φ − U (φ) + f (φ)R



YGH

, (1)

where g = det g

μν

with the signs (−, +, +, +), κ = 8π G, R is the

scalar curvature of a space-time manifold M, U (φ) is a scalar

ﬁeld potential, and GB term is given by R

= R

μνρσ

−

μν

+ R

. The function f (φ) is a coupling function be-

tween

the scalar ﬁeld and GB term. S

YGH

is the generalized York–

Gibbons–Hawking

boundary term [40–43].

adopting the analytic continuation from Lorentzian to Eu-

clidean,

the action is changed with omitting boundary term as

follows:



√



−

2κ

∂

φ∂

φ + U (φ) − f (φ)R



(2)

where the signs of metric are now (+, +, +, +) and the boundary

term will be canceled in the calculation of the difference between

the action of the solution and background. The equation of motion

for scalar ﬁeld and Einstein’s equation are

0 =∇

φ − U



(φ) + f



(φ)R

, (3)

2κ



μν

−

μν



−

∂

φ∂

μν

∂

φ∂

φ +

μν

U (φ) + (GB)

μν

, (4)

where we use the primed notation for the derivative with respect

to the scalar ﬁeld φ. The last term of second equation is obtained

by GB term variation that in four-dimensional space is

(GB)

μν

=−2(∇

∇

f (φ))R +2g

μν

(∇

f (φ))R

+4(∇

∇

f (φ))R

+4(∇

∇

f (φ))R

−4(∇

f (φ))R

μν

−4g

μν

(∇

∇

f (φ))R

ρσ

+4(∇

∇

f (φ))R

μρ ν σ

. (5)

There were the terms linear in f (φ) but those terms are canceled

each

other especially in four-dimensional space [44]. Thus, (GB)

μν

can be zero when the coupling function f (φ) is given by a con-

stant

which means that the scalar ﬁeld and GB term are minimally

coupled each other.

consider Euclidean O (4) symmetry for the dominant con-

tribution

to the decay probability [45]. Then, the ﬁeld φ as well as

ρ depends only on η which is the radial coordinate of Euclidean

space. The geometry is written as

=dη

+ρ(η)



dχ

+sin

χ(dθ

+sin

θdψ

)



. (6)

The scalar curvature and GB term turn out to be

R =−6

(

−1 + ρ

ρ)

, and R

= 24

ρ(

−1)

, (7)

where the dotted notation denotes the derivative with respect to η.

The equations of motion for φ and (η, η), (χ, χ) components of

Einstein’s equation are obtained from plugging Eq. (6) into Eqs. (3)

and (4) as

follows:

0 =

φ + 3

φ −



(φ) +24 f



(φ)

ρ(

−1)

, (8)

2κ

(

−1)

−

(φ) − 12

f (φ)

ρ(

−1)

, (9)

−1 + 2ρ

2κ

(φ) − 8

f (φ)

−4

f (φ)(

−1). (10)

The last equation is different from the result obtained in Ref. [46].

order to solve the equations of motion, we should impose

appropriate boundary conditions. In the presence of gravity, there

need boundary conditions for not only φ but also ρ. These bound-

ary

conditions are divided into two types depending on the sign

of U (φ

). The maximum value of η is η

max

=∞for AdS and ﬂat

backgrounds, while it is ﬁnite for dS background which satisﬁes

ρ(η

max

) = 0.

For

the ﬂat and AdS space, we can impose the boundary condi-

tions

as follows:

ρ(0) = 0,

ρ(0) = 1,

φ(0) = 0, and φ(η

max

) = φ

, (11)

where the ﬁrst condition is for a geodesically complete space and

second condition comes from the Eq. (10) and φ

is the vacuum

value of scalar ﬁeld which was zero in Einstein’s theory of gravita-

tion

(Einstein theory). For dS space, we can impose the boundary

conditions as follows:

ρ(0) = 0, ρ(η

max

) = 0,

φ(0) = 0, and

φ(

max

) = 0. (12)

We consider the potential

U (φ) =−

+ U

, (13)

where λ is a positive constant [10] and U

is a vacuum value of

potential in Einstein theory. It plays a roll of cosmological constant

as  = κ U

[16] while it is modiﬁed as  = κU (φ

) in DEGB the-

ory.

The coupling function is

f (φ) = αe

−γ φ

, (14)

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