Instanton solutions mediating tunneling between the degenerate vacua in curved space
aa r X i v : . [ h e p - t h ] J u l CQUeST-2009-0297
Instanton solutions mediating tunneling between thedegenerate vacua in curved space
Bum − Hoon Lee † Chul H . Lee § Wonwoo Lee † Changheon Oh §‡ † Department of Physics and BK21 Division, and Center for Quantum Spacetime, Sogang University,Seoul 121-742, Korea § Department of Physics, Hanyang University, Seoul 133-791, Korea ‡ Principal Researcher Center, Technovation Partners, Seoul 135-824, Korea
Abstract
We investigate the instanton solution between the degenerate vacua incurved space. We show that there exist O (4)-symmetric solutions notonly in de Sitter but also in both flat and anti-de Sitter space. Thegeometry of the new type of solutions is finite and preserves the Z symmetry. The nontrivial solution corresponding to the tunneling ispossible only if gravity is taken into account. The numerical solutionsas well as the analytic computations using the thin-wall approximationare presented. We expect that these solutions do not have any negativemode as in the instanton solution. PACS numbers: 04.62.+v, 98.80.Cq email:[email protected] email:[email protected] email:[email protected] email:[email protected] Introduction
The instanton solution with O (4) symmetry exists between the degenerate vacua in de Sitter(dS)space. This can be understood in the particle analogy picture due to the change of the dampingterm into the accelerating term. The numerical solution of the scalar field Φ was obtained inRef. [1]. The analytic computation and meanings of this solution were further studied in Ref.[2]. Can we also obtain the instanton solution between the degenerate vacua in both flat andanti-de Sitter(AdS) space? The aim of this paper is to obtain solutions in these backgrounds.We study boundary conditions needed for these solutions not only in dS but also in both flatand AdS space. These boundary conditions respect the Z symmetry, different from thosestudied in bounce solutions [3, 4, 5, 6]. We will show that there exist a new type of solutionsgiving rise to the finite geometry with the Z symmetry.An instanton solution is a solution to the equation of motion of the classical field theory inEuclidean space satisfying appropriate boundary conditions. Yang-Mills instantons give finiteaction, and have been extensively studied in gauge theories [7] as well as in string theory [8](for a review, see Ref. [9]). In gravitational theory, there are several kinds of instantons [10].A recent summary is in Ref. [11] and references therein. Some of those have finite action, andothers an infinite one [12]. There is also an issue on the sign of the action for the solutionsdescribing quantum creation of the inflationary universe [13]. Some of the authors in Ref. [13]considered the counterclockwise Euclidean rotation rather than the clockwise rotation as theanalytic continuation in the complex t plane to suppress the probability with negative action.This is related to the fact that the Euclidean Einstein action is not positive definite, which isknown as the conformal factor problem in Euclidean quantum gravity [14]. However, this issueis not central to the problem and therefore we will not discuss the issue any more in the presentpaper.Quantum mechanically, the tunneling process in a symmetric double-well potential is de-scribed by the instanton solution. This tunneling shifts the ground state energy of the classicalvacuum due to the presence of an additional potential well lifting the classical degeneracy. Thesymmetric ground state wave function describes that there is a higher probability of findingit somewhere between the two classical vacua. In the semiclassical approximation, the actionis dominated by the instanton configuration. The instanton solution can be interpreted as aparticle motion in the inverted potential starting from one vacuum state at the minus infiniteEuclidean time and reaching the other one at the plus infinite time. We may consider themulti-instanton solutions as describing the tunneling back and forth between the two vacua.For field theoretical solutions with O (4) symmetry, the equation of motion has an additionalterm, which can be interpreted as a friction term in the inverted potential. The solution forthe potential with degenerate minima is known for the dS case. However, there are no studiesof the solution in flat or AdS space.The tunneling solution in the field theory with an asymmetric potential is called a bouncesolution. The solution is related to the nucleation of a true (false) vacuum bubble describingthe decay of a metastable state. There have been many studies on the decay of a metastable1acuum. The process was first investigated in Ref. [3] and developed in both flat [4] and curvedspacetime [5, 6]. Another type of transition describing the scalar field jumping simultaneouslyat the top of the potential barrier was investigated by Hawking and Moss [15] and later in Ref.[16]. Further recent studies can be found in Refs. [17, 18].The mechanism for the nucleation of a false vacuum bubble within the true vacuum back-ground has also been studied. The nucleation process of the large false vacuum bubble in dSspace was originally obtained in Ref. [19], and the case with a global monopole in Ref. [20]. Thesmall false vacuum bubble in the Einstein theory of gravity with a nonminimally coupled scalarfield was explored using a mechanical analogy in Ref. [21], and the case with the Gauss-Bonnetterm in Ref. [22]. The nucleation rate as well as classification depending on the size of the falsevacuum bubble in dS space was obtained in Ref. [2] in the Einstein theory of gravity. The clas-sification of true and false vacuum bubbles in the dS background space depends on the value ofthe cosmological constant. Large bubbles have large values of the cosmological constant, whilesmall bubbles have small values of the cosmological constant. These processes may provide analternative paradigm in the string theory landscape [23] or eternal inflation [24].The natural question is on the instanton solution between the degenerate vacua and itsrelation to the bubble. The clue of these solutions could have been seen in Refs. [1, 2, 18]. Theinstanton solution in the potential with degenerate vacua in de Sitter background was obtainednumerically [1]. It was also shown that the solution can be analyzed as the limiting case ofthe large true vacuum bubble or large false vacuum bubble [2]. In this paper, we will showthat there also exist solutions between the degenerate vacua in both flat and AdS backgrounds.These solutions can not be obtained as the limiting cases of the previously known vacuumbubble solutions.In flat spacetime the bounce solution has exactly one negative mode [25]. When gravityis taken into account, it is a more involved problem [26]. On the other hand, our solutionsdescribe quantum mechanical mixing between the degenerate vacua and we expect that thesolutions do not have any negative mode.The time evolution of the solution after its materialization can be studied by the analyticcontinuation to Lorentzian spacetime. We will study the dynamics of the wall following theintroductory work of Coleman [4]. The wall has a trajectory of the hyperboloid in Minkowskispacetime. The method in curved spacetime was also studied [27]. The metric junction condi-tion [28] can also be employed for the evolution of the wall (for recent works, see Ref. [29] andreferences therein). The evolution is simply classified into two types from observer’s point ofview on the wall: One is shrinking while the other is expanding. For the type of the shrinkingwall, there are two cases. One is related to the creation of a child universe. The other is relatedto the black hole formation or nothing. For the type of the expanding wall, there are also twocases. One is the bubble eating up the original background. The other is the case with theexpanding inside region as well as the outside region, which will be related to the present work.The outline of this paper is as follows: In Sec. 2 we investigate the instanton solutionwith O (4) symmetry between the degenerate vacua in not only de Sitter space but also flatand AdS space. We discuss the new type of boundary conditions with Z symmetry. Under2hese boundary conditions, we explore the possibility for the existence of the instanton solutionqualitatively. From this perspective, one of the central questions that we intend to is to obtainthe condition for the existence of the solution. The condition will depend on the local maximumvalue of the potential. In Sec. 3 we employ the metric junction condition to get the dynamicsof the wall. In Sec. 4 we summarize and discuss our results. We consider the following action S = Z M √− gd x (cid:20) R κ − ∇ α Φ ∇ α Φ − U (Φ) (cid:21) + I ∂ M √− hd x Kκ , (1)where κ ≡ πG , g ≡ detg µν , and the second term on the right-hand side is the boundary term[30].The vacuum-to-vacuum transition amplitude can be semiclassically represented as Ae − ∆ S ,where the exponent ∆ S is the difference between the Euclidean action corresponding to aninstanton solution and the background action itself. This exponent is also related to the splittingof the energy levels for the symmetric double-well potential. The prefactor A comes from thefirst order quantum correction.If we consider the asymmetric double-well potential, the metastable state will tunnel intothe ground state through the bubble nucleation process. The action corresponding to thetunneling process will then have an imaginary part. The decay rate of the background vacuumis described by a bounce solution [3, 4, 5, 25, 31].Our main concern in this paper is the case of the symmetric double-well potential. Thescalar potential U (Φ) in Eq. (1) has two degenerate minima U (Φ) = λ (cid:18) Φ − µ λ (cid:19) + U o . (2)The space will be dS, flat, or AdS depending on whether U o > U o = 0, or U o <
0. Wewant to obtain instanton solutions describing the tunneling between the degenerate vacua inthe present work.We consider O (4)-symmetric ansatz for both the geometry and the scalar field ds = dη + ρ ( η )[ dχ + sin χ ( dθ + sin θdφ )] . (3)Then, Φ and ρ depend only on η and the Euclidean field equations for them have the formΦ ′′ + 3 ρ ′ ρ Φ ′ = dUd Φ and ρ ′′ = − κ ρ (Φ ′ + U ) , (4)3espectively and the Hamiltonian constraint is given by ρ ′ − − κρ (cid:18)
12 Φ ′ − U (cid:19) = 0 . (5)We now have to impose the boundary conditions to solve Eqs. (4) and (5). In this work,we have to consider 4 conditions for two equations of 2nd order. They can be the values ofthe fields Φ and ρ , or their derivatives Φ ′ and ρ ′ at either η = 0 or at η = η max . For example,in flat spacetime without gravity where ρ is identified with η , Φ at η max , and Φ ′ at η = 0 wasimposed as the boundary conditions for Φ in Ref. [4]. The condition Φ ′ = 0 is a natural choiceto have a nonsingular solution. In the case with gravity, we need further conditions on ρ . Onemay choose the initial condition type of boundary conditions ρ = 0 and ρ ′ = const at η = 0.These boundary conditions will give small bubbles within large backgrounds regardless ofthe background geometry [2]. On the other hand, the theory has a wide variety of solutiontypes if the effect of gravity becomes more significant. Large bubbles have large values of thecosmological constant, while small bubbles have small values of the cosmological constant [2].In the case of Euclidean de Sitter background with compact geometry, the following choiceof boundary conditions is also possible: Φ ′ = 0 and ρ = 0 at η = 0 and η max = 0 [1]. Thisboundary condition has Z symmetry under the exchange of two points corresponding to η = 0and η max = 0. In general, for a potential with degenerate vacua, we can choose the boundarycondition with Z symmetry as follows: d Φ dη (cid:12)(cid:12)(cid:12) η =0 = 0 , d Φ dη (cid:12)(cid:12)(cid:12) η = max = 0 , ρ | η =0 = 0 , and ρ | η = η max = ˜ ρ ( η max ) = 0 , (6)where η max will have a finite value in the cases we will consider. We adopt this boundarycondition for our instanton solutions. Φ( η max ) is exponentially approaching to but not reachingΦ vf . The above boundary conditions have been used in dS background space [1] as alreadymentioned. We will also choose ˜ ρ ( η max ) = 0 for both flat and AdS space to impose Z symmetry.This will allow new solutions with Z symmetry even both flat and AdS space. Because of Z symmetry about the wall, the inside geometry will be identical to the outside one. As a result,the whole geometry is finite. This is very different from the known solutions with infinitegeometry [4, 5, 6, 32].In what follows, we explore in more detail the possibility for the existence of the instan-ton solution. To understand solutions qualitatively, we rearrange the terms in Eq. (4) aftermultiplying by d Φ dη Z η max d (cid:20)
12 Φ ′ − U (cid:21) = − Z η max dη ρ ′ ρ Φ ′ . (7)The quantity in the square brackets here can be interpreted as the total energy of the particlewith the potential energy − U . The term on the right-hand side can be considered as thedissipation rate of the total energy as long as ρ ′ >
0. However, the role of the term can bechanged from damping to acceleration if ρ ′ changes sign during the transition. For the tunneling4etween the degenerate vacua the region for ρ ′ > ρ ′ < Z symmetry, hence the term on the right-hand side will be vanished. Thus, the total energy atboth ends, η = 0 and η = η max , is conserved as we can see from Eq. (7) and the third columnin Fig. 1. The condition for the change of sign depends on the maximum value of the potential.To allow the change during the transition, the local maximum value of the potential U (0) mustbe positive. This can be seen from Eq. (5). That is to say, U (0) = 3 κρ + 12 Φ ′ | > . In other words, the sign of the second term of the first equation in Eq. (4) can be changedduring the transition for − µ / λ < U o .The numerical solutions for the equations are illustrated in Fig. 1. In this work, we employthe following dimensionless variables [21]:ˆ U ( ˆΦ) = λU (Φ) µ , ˆΦ = λ Φ µ , ˆ η = µη, ˆ ρ = µρ, ˆ κ = µ κλ . (8)These variables give ˆ U ( ˆΦ) = 18 (cid:16) ˆΦ − (cid:17) + ˆ U o , (9)and the Euclidean field equations for Φ and ρ becomeˆΦ ′′ + 3 ˆ ρ ′ ˆ ρ ˆΦ ′ = d ˆ Ud ˆΦ and ˆ ρ ′′ = − ˆ κ ρ ( ˆΦ ′ + ˆ U ) , (10)respectively and the Hamiltonian constraint is given byˆ ρ ′ − − ˆ κ ˆ ρ (cid:18)
12 ˆΦ ′ − ˆ U (cid:19) = 0 . (11)We use relaxation methods to solve Eqs. (4) and (5) with boundary conditions (6). For dSspace, this solution was already studied in Refs. [1, 2]. In Fig. 1, the first row (A) illustratesthe transition between the degenerate vacua in dS space, U o >
0. The second row (B) illustratesthe transition between the degenerate vacua in flat space. The third row (C) illustrates thetransition between the degenerate vacua in AdS space. The first column corresponds to thesolution of ρ . The second column corresponds to the solution of Φ. The third column illustratesthe evolution of the second term on the left side of Eq. (4). In the first column in Fig. 1, wecan see that the role of the second term on the left side of Eq. (4) changes from damping ρ ′ > ρ ′ < S , ρ starts from zero at η = 0,reaches the maximum size at η = η max /
2, and becomes zero again at η = η max . Obviously, thegeometry is finite and Z symmetry. Note that ˜ ρ ( η max ) goes to zero unlike the bounce solutionswhere ˜ ρ ( η max ) goes to infinity for both flat and AdS space.5
10 20 30 40 50 60 7005101520253035 (A) Instanton solution in de Sitter space (B) Instanton solution in flat space (C) Instanton solution in anti-de Sitter spaceFigure 1:
The first and second columns show the numerical solutions on ρ and Φ, respectively. The thirdcolumn shows the evolution of the second term in the left one of Eq. (4). Row (A) illustrates the transitionbetween the degenerate vacua in dS space, ˆ U o = 0 .
01 and ˆ κ ≃ . U o = 0 and ˆ κ ≃ . U o = − .
01 and ˆ κ ≃ .
6e must be careful in evaluating the action difference between the action of the solutionand the background action itself. In previous works [4, 5, 6], the outside contribution to theaction of the solution is equal to that from outside of the background. In our solution, theoutside geometry of the solution is quite different from that of the background. We can notemploy Eq. (3.8) in Ref. [5] directly because the role of the second term ρ ′ is changed fromthe initial role of damping to the finial role of acceleration. When we do integration by partsin an analytic computation we cannot simply drop out the surface term due to our boundaryconditions Eq. (6).In the thin-wall approximation scheme, the Euclidean action can be divided into three parts:∆ S = ∆ S in + ∆ S wall + ∆ S out , where ∆ S = S E ( solution ) − S E ( background ). For flat and AdScases the inside part ∆ S in gives no contribution because of unchanged inside geometry in thetunneling solution and background. Here, we will send the contribution of the inside part dueto the diminished inside bulk shape to the contribution of the outside part for simplicity. Thus,we only need to consider the contribution of the wall and the outside part. The contributionof the wall is given by ∆ S wall = 2 π ¯ ρ S o , where the surface tension of the wall S o (= 2 µ / λ )is a constant [5]. We note that it corresponds to the energy of a kink in one-dimensional spacein the form of the potential without U o in Eq. (2). The contribution from the outside part isgiven by ∆ S out = S E ( solution ) out − S E ( background ) out .In the thin-wall approximation scheme, the relation between dρ and dη can be seen by therelation dρ = ± dη (cid:20) − κρ U (cid:21) / , (12)where + is for 0 ≤ η < η max /
2, 0 is for η = η max /
2, and − is for η max / < η ≤ η max [2].Outside geometry after tunneling will be finite as that of dS space. Therefore we will considerthe initial background space having the size ˜ ρ imax ( η imax ). η imax denotes η max in the initial space.The finite size in the initial space will go to infinity as ˜ ρ imax ( η imax ) goes to infinity except for dSspace. We also study the effect as the size increases. For both flat and AdS space, we have tocarry out integration by parts carefully. The contribution from integration by parts Eq. (3.8)in Ref. [5] gives as follows S E ( background ) out = 4 π Z η imax ¯ η (cid:18) ρ U − ρκ (cid:19) dη − ∆ S ibp , (13)where the second term on the right-hand side of the above equation is from the effect of the sur-face term at ˜ ρ imax and ∆ S ibp = − (6 π /κ ) ˜ ρ i max (1 − κ ˜ ρ i max U o / / . For dS space, ρ imax ( η max ) = 0and ∆ S ibp = 0.The analytic computation in dS space was obtained [2] as follows:∆ S bulk = ∆ S in + ∆ S out = 12 π κ U o (cid:16) − κ U o ¯ ρ (cid:17) / + 12 π κ U o (cid:16) − κ U o ¯ ρ (cid:17) / . (14)The first term is from the contribution of the inside bulk part due to the diminished inside bulkshape and the second term is from the outside bulk part due to the same reason. Then, the7oefficient ∆ S and the critical radius of the wall are evaluated to be¯ ρ = 2 κ q S o + κ U o and ∆ S = 12 π S o κ U o q S o + κ U o , where Eq. (12) is used.The final form from the contribution of the outside part in both flat and AdS space isevaluated to be∆ S out = 12 π κ U o (cid:20) (cid:16) − κ ρ U o (cid:17) / − − (cid:16) − κ ρ i max U o (cid:17) / (cid:21) + ∆ S ibp . (15)The minus sign in front of the third term in the square parentheses is owing to the integrationof the outside part in flat and AdS space. For dS space, the minus sign is changed into theplus. This is because Eq. (12) is used for both the tunneling solution and the background inevaluating ∆ S out , while Eq. (12) is used only for the tunneling solution in both flat and AdScases. Equation (15) with the plus sign and ˜ ρ imax ( η imax ) = 0 will reproduce the result for dS inEq. (14).The position of the wall obtained by extremizing ∆ S is given by¯ ρ = 2 κ q S o + κ U o . (16)This form has the same one as the tunneling in dS space [2] except that U o <
0. The transitionrate is evaluated to be∆ S = 12 π κ U o S o q S o + κ U o − − (cid:16) − κ ρ i max U o (cid:17) / + ∆ S ibp . (17)We first consider the case for 1 < κ ˜ ρ i max | U o | . As the initial value of ˜ ρ imax goes to infinite,the exponent becomes ∆ S ∼ − π r | U o | κ ˜ ρ i max . As the initial space goes to infinite, the transition rate has an exponentially growing value.We now consider the case for 1 > κ ˜ ρ i max | U o | . Then the final form of ∆ S is evaluated to be∆ S ∼ − π κ S o − π ˜ ρ i max | U o | , where | U o | is equal to zero for the case of flat space. The exponent ∆ S has the negativeminimum value at ρ = ¯ ρ . The action of the solution is less than the action of the background8or the cases in both flat and AdS space. For the case in dS space, the action difference has thepositive value, ∆ S >
E <
In this section, we study the growth of the wall after its materialization. To obtain the dynamicsof the solutions in Lorentzian spacetime, we now apply the analytic continuation as in Ref. [27] χ = iξ + π/ , (18)where π/ − , + , + , +). The only difference is thatone has to continue the scalar field as well as O (4)-invariant Euclidean space into an O (3 , ds = dη + η [ − dξ + cosh ξ ( dθ + sin θdφ )] for flat , (19) ds = dη + 3Λ sinh r Λ3 η [ − dξ + cosh ξ ( dθ + sin θdφ )] for AdS , (20) ds = dη + 3Λ sin r Λ3 η [ − dξ + cosh ξ ( dθ + sin θdφ )] for dS , (21)where Λ = κ | U o | and ξ becomes the timelike coordinate. In Minkowski spacetime [see Eq. (19)],the inside geometry has the relation η = − t + r for the range of 0 ≤ η < η max /
2. On the otherhand, the outside geometry has the relation ˜ η = − t + r for the range of η max / < η ≤ η max ,where we defined ˜ η = η max − η . The geometry can be constructed by joining two spacetimesalong the wall, ¯ η = η max /
2. As a result, the spacetime has Z symmetry. At the initialLorentzian time ( t = 0), ¯ η is equal to R o as shown in the left figure in Fig. 2. The methodobtaining the left figure in Fig. 2 was studied in Ref. [4].The geometry and properties of our solutions as compared with the domain wall [34, 35]were studied in Ref. [2]. Domain walls can form in any model having a spontaneously brokendiscrete symmetry. An inertial observer sees the domain wall accelerating away with a specificacceleration. The domain wall has repulsive gravitational fields [34, 35]. On the other hand,our solutions are different from the formation process of a domain wall. The spacetime in thepresent work depends only on the cosmological constant and also has the spherical Rindler-typemetric. 9 ’r outin R o R o
00 tt Identify r o rV eff Figure 2:
The figures illustrate the time evolution after the nucleation. The left figure represents the evolutionof the wall under the inside and outside observers’s point of view. The right figure represents the effectivepotential in the junction equation.
Now we will employ the Israel junction condition. In the thin-wall limit, all the energymomentum is localized on the wall and therefore the thin wall has a surface stress-energytensor. Thus, it becomes a kind of surface layer [28]. In this framework, only the vacuumEinstein equations should be solved on either side of the wall. We use another coordinatetransformation [2, 27] for O (3) invariant configurations: R = η cosh ξ, T = η sinh ξ, for flat R = r
3Λ sinh r Λ3 η cosh ξ, T = r
3Λ tan − sinh ξ tanh r Λ3 η ! , for AdS (22) R = r
3Λ sin r Λ3 η cosh ξ, T = 12 r
3Λ ln cos p Λ / η + sin p Λ / η sinh ξ cos p Λ / η − sin p Λ / η sinh ξ , for dS . Because of the exact Z symmetry of the whole spacetime, both inside and outside have thesame behavior. The metric in Eqs. (19, 20, 21) becomes ds = − (cid:18) ± Λ3 R (cid:19) dT + dR (cid:0) ± Λ3 R (cid:1) + R d Ω . (23)We introduce the Gaussian normal coordinate system near the wall dS = − dτ + dη + ¯ r ( τ, η ) d Ω , (24)where g ττ = − r ( τ, ¯ η ) = r ( τ ). It must agree with the coordinate R of the left and rightregions. The angle variables can be taken to be invariant in all regions. The induced metric on10he wall is given by dS = − dτ + r ( τ ) d Ω , (25)where τ is the proper time measured by an observer at rest with respect to the wall and r ( τ ) isthe proper radius of Σ. The metrics in both sides of the wall give the same induced metric onthe wall. The following condition should be satisfied: 1 = (1 ± Λ R /
3) ˙ T − (1 ± Λ R / − ˙ R ,where the dot is a derivation with respect to τ . Because of the spherical symmetry, the extrinsiccurvature has only two components, K θθ ≡ K φφ and K ττ . The junction equation is related to K θθ and the covariant acceleration in the normal direction is related to K ττ . The junction conditionthen becomes r ± Λ3 r + ˙ r + r ± Λ3 r + ˙ r = κ σr, (26)where σ is the surface-energy density and equivalent to the surface tension of the wall S o . Aftersquaring, we obtain the metric junction equation. The effective potential has the form V eff = 12 (cid:20) − κ σ ± Λ3 (cid:21) r + 12 . (27)The condition for the existence of the expanding solution is Λ < κ σ /
16 for AdS spacetime,while the expanding solution is always possible for dS spacetime.The left figure in Fig. 2 represents the time evolution of the inside and outside spacetimefrom the inside and outside observers’s point of view, respectively. The location of walls isdenoted by R o . Because the configuration of the whole spacetime has Z symmetry the wallsmay be identified. Then both observers in the left and right sides of the wall feel that they areinside of the wall. The right figure represents the effective potential describing the dynamicsof the wall under the metric junction equation. We can read off directly the properties ofthe trajectory of the wall from the shape of the effective potential. The location of r o where V eff = 0 in the right figure in Fig. 2 is given by r o = 2 κ q σ ∓ κ . (28)This r o is the same as the position of the wall in Eq. (16). After the tunneling transition, thewall can expand without eating up bulk (inside and outside or left and right) spacetime. In this paper, we have studied instanton solutions with O (4) symmetry between the degeneratevacua in curved space. The boundary conditions we imposed have Z symmetry, which aredifferent from the usual ones. The solution also has the exact Z symmetry and gives riseto the geometry of a finite size. We obtained the numerical solutions as well as performedthe analytic computations using the thin-wall approximation. These nontrivial solutions are11ossible only if gravity is taken into account. The solutions which represent the tunnelingin the opposite direction, anti-instanton, are easily obtained by η → η max − η . The leadingsemiclassical exponent ∆ S is given by the instanton solution. To evaluate ∆ S we consideredthe initial background space with the finite size ˜ ρ imax cutoff. The finite size in the initial spacewill go to infinity as ˜ ρ imax goes to infinity except for dS space. We studied the effect as the sizeincreases. In the thin-wall approximation, the solution describing tunneling in flat space hasthe negative minimum value of the exponent ∆ S . This will be the dominant contribution tothe Euclidean path integral. For AdS space, the transition rate has a negative exponentiallygrowing value due to the diverging background subtraction. For dS space, the coefficient ∆ S has a positive value. The difference among the dS, flat, and AdS space is caused by the initialsize of the background Euclidean space or the diverging background subtraction dependingon the cosmological constant. The instanton solution in dS space can be interpreted as theordinary formation process of a kink or domain wall. On the other hand, the solution in bothflat and AdS space can be interpreted as the mixed state of the two degenerate vacuum states.We expect that this phenomenon may be interpreted as the analog of the energy difference,∆ E <
0, between the energy of the mixed state and the harmonic oscillator in the left or rightpotential. It is not clear how to interpret the physical meaning of the negative value of ∆ S inthe present work. However, the exponent ∆ S can have a finite value if the tunneling describingour instanton solutions occurs in the initial background with finite size.In these solutions mediating tunneling between the degenerate vacua, the observers’s point ofview is different from the ordinary formation process of the vacuum bubble or domain wall. Ourinstanton solutions have specific dynamics in Lorentzian spacetime. In this formation, there isonly one point of view as inside. For example, every observer sees themselves surrounded by thewall after the tunneling in dS space. In other words, the wall is located at R o (= ρ max ( η max / r = 0 and r ′ = 0, respectively. After theanalytic continuation from Euclidean to Lorentzian the observer is inside of the wall and theother is also inside of the wall over the dS horizon. Thus, every observer remains inside of thewall. The situation will be similar in both flat and AdS space although they do not have a realhorizon. The wall formed when the particle rolls down and up the inverted potential in themechanical analogy. The middle of the wall has the energy density corresponding to the top ofthe potential, which is equivalent to the vacuum energy of dS space. Thus, the wall somehowplays the role of the dS horizon, but not the real horizon. The other observer exists over thewall, which is located at r ′ = 0. It seems that the solutions represent 2-brane formation with Z symmetry. Thus, the topology of the initial spacetime could be changed under the influenceof this solution.In order to get the analytic computation on the action one may consider carefully thecontribution from the Gibbons-Hawking term. This is because the geometry of the outsidewall of our solutions is different from that of the background. This is not needed for the usualbubble solutions where the outside geometry does not change. We expect that the point ˜ η = 0after the tunneling process is smooth due to Z symmetry.We propose the solutions with exact Z symmetry can represent the nucleation of the12raneworld-like object if the mechanism is applied in higher-dimensional theory. The braneworldhaving the finite size with the exact Z symmetry can be nucleated not only in dS but also inflat and AdS bulk spacetime, and then expand, seen from an observer’s point of view on thewall, without eating up bulk (inside and outside) spacetime.It will be interesting to see if this type of solution can be extended to the theories withgauge fields or in various dimensions. These solutions are interpreted as an instanton solutionrather than a bounce solution in not only dS but also flat and AdS space. The study on theinstanton solution in curved space may be important for the understanding of the propertiesof the vacuum structure in these theories as well as to see if the processes may happen in theearly universe. Can we obtain the situation describing each observer as an outside one? It maybe related to the wormhole. The study including a wormhole solution will also be interesting. Acknowledgements
We would like to thank E. J. Weinberg, M. B. Paranjape, Sang-Jin Sin, and Hongsu Kim forhelpful discussions and comments. We would like to thank Kei-ichi Maeda for helpful discussionsat the 18th Workshop on General Relativity and Gravitation in Japan (JGRG18). This workwas supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by theKorea government(MEST) through the Center for Quantum Spacetime(CQUeST) of SogangUniversity with grant number R11 - 2005 - 021. W.L. was supported by the Korea ResearchFoundation Grant funded by the Korean Government(MOEHRD) (KRF-2007-355-C00014).13 eferences [1] J. C. Hackworth and E. J. Weinberg, Phys. Rev. D , 044014 (2005).[2] B.-H. Lee and W. Lee, Classical Quantum Gravity , 225002 (2009).[3] M. B. Voloshin, I. Yu. Kobzarev, and L. B. Okun, Yad. Fiz. , 1229 (1974)[Sov. J. Nucl.Phys. , 644 (1975)].[4] S. Coleman, Phys. Rev. D , 2929 (1977); ibid. D , 1248(E) (1977).[5] S. Coleman and F. De Luccia, Phys. Rev. D , 3305 (1980).[6] S. Parke, Phys. Lett. , 313 (1983).[7] A. Belavin, A Polyakov, A Schwartz, and Y. Tyupkin, Phys. Lett. , 85 (1975); G.’tHooft, Phys. Rev. D , 3432 (1976); Phys. Rev. Lett. , 8 (1976).[8] G. Gibbons, M. Green, and M. Perry, Phys. Lett. B , 37 (1996); A. V. Belitsky, S.Vandoren, and P. van Nieuwenhuizen, Classical Quantum Gravity , 3521 (2000).[9] S. Vandoren and P. van Nieuwenhuizen, arXiv:0802.1862.[10] S. W. Hawking, Phys. Lett. , 81 (1977); G. W. Gibbons and S. W. Hawking, Phys.Lett. , 430 (1978); G. ’tHooft, Nucl. Phys. B315 , 517 (1989).[11] D. N. Page, arXiv:0912.4922.[12] E. Baum, Phys. Lett. , 185 (1983); S. W. Hawking, Phys. Lett. , 403 (1984);A. D. Linde, JETP , 211 (1984); A. Vilenkin, Phys. Rev. D , 509 (1984).[13] S. W. Hawking and N. Turok, Phys. Lett. B , 25 (1998); ibid. B , 271 (1998); A.Vilenkin, Phys. Rev. D , R7069 (1998); A. Linde, Phys. Rev. D , 083514 (1998).[14] G. W. Gibbons, S. W. Hawking, and M. J. Perry, Nucl. Phys. B138 , 141 (1978).[15] S. W. Hawking and I. G. Moss, Phys. Lett. , 35 (1982).[16] L. G. Jensen and P. H. Steinhardt, Nucl. Phys.
B237 , 176 (1984); ibid.
B317 , 693 (1989); J.Garriga and A. Vilenkin, Phys. Rev. D , 2230 (1998); T. Banks, arXiv:hep-th/0211160.[17] U. Gen and M. Sasaki, Phys. Rev. D , 103508 (2000); K. Marvel and N. Turok,arXiv:0712.2719; A. R. Brown and E. J. Weinberg, Phys. Rev. D , 064003 (2007);E. J. Weinberg, Phys. Rev. Lett. , 251303 (2007); S.-H. Henry Tye, D. Wohns, and Y.Zhang, Int. J. Mod. Phys. A , 1019 (2010); K. Marvel and D. Wesley, J. High EnergyPhys. 12 (2008) 034; S.-S. Xue, J. Korean Phys. Soc. , 759 (2006); Int. J. Mod. Phys. A , 3865 (2009). 1418] T. Banks and M. Johnson, hep-th/0512141, A. Aguirre, T. Banks, and M. Johnson, J.High Energy Phys. 08 (2006) 065; R. Bousso, B. Freivogel, and M. Lippert, Phys. Rev. D , 046008 (2006).[19] K. Lee and E. J. Weinberg, Phys. Rev. D , 1088 (1987).[20] Y. Kim, K. Maeda, and N. Sakai, Nucl. Phys. B481 , 453 (1996); Y. Kim, S. J. Lee, K.Maeda, and N. Sakai, Phys. Lett. B , 214 (1999).[21] W. Lee, B.-H. Lee, C. H. Lee, and C. Park, Phys. Rev. D , 123520 (2006).[22] R.-G. Cai, B. Hu, and S. Koh, Phys. Lett. B , 181 (2009).[23] L. Susskind, hep-th/0302219; R. Bousso and J. Polchinski, J. High Energy Phys. 06 (2000)006; S. Kachru, R. Kallosh, A. Linde, and S. P. Trivedi, Phys. Rev. D , 046005 (2003); B.Freivogel and L. Susskind, Phys. Rev. D , 126007 (2004); D. I. Podolsky, J. Majumder,and N. Jokela, J. Cosmol. Astropart. Phys. 05 (2008) 024; Q.-G. Huang and S.-H. HenryTye, Int. J. Mod. Phys. A , 1925 (2009); D. Podolsky and K. Enquvist, J. Cosmol.Astropart. Phys. 02 (2009) 007.[24] A. Vilenkin, Phys. Rev. D , 2848 (1983); A. D. Linde, Phys. Lett. B , 395 (1986); A.H. Guth, Phys. Rep. , 555 (2000); A. Vilenkin, gr-qc/0409055; D. I. Podolsky, Grav.Cosmol. , 69 (2009).[25] C. G. Callan, Jr. and S. Coleman, Phys. Rev. D , 1762 (1977).[26] T. Tanaka and M. Sasaki, Prog. of Theor. Phys. , 503 (1992); T. Tanaka, Nucl. Phys. B556 , 373 (1999); A. Khvedelidze, G. Lavrelashvili, and T. Tanaka, Phys. Rev. D ,083501 (2000); G. Lavrelashvili, Nucl. Phys. Proc. Suppl. , 75 (2000); G. Lavrelashvili,Phys. Rev. D , 083513 (2006).[27] C. H. Lee and W. Lee, J. Korean Phys. Soc. , S1 (2004).[28] W. Israel, Nuovo Cimento B , 1, (1966); ibid. B , 463(E) (1967).[29] J. Hansen, D.-i. Hwang, and D.-h. Yeom, J. High Energy Phys. 11 (2009) 016; B.-H. Lee,W. Lee, and M. Minamitsuji, Phys. Lett. B , 160 (2009); E. I. Guendelman and N.Sakai, Phys. Rev. D , 125002 (2008); B.-H. Lee, C. H. Lee, W. Lee, S. Nam, and C.Park, Phys. Rev. D , 063502 (2008); B.-H. Lee, W. Lee, S. Nam, and C. Park, Phys.Rev. D , 103506 (2007).[30] J. W. York, Phys. Rev. Lett. , 1082 (1972); G. W. Gibbons and S. W. Hawking, Phys.Rev. D , 2752 (1977).[31] E. J. Weinberg, Phys. Rev. D , 4614 (1993); G. V. Dunne and H. Min, Phys. Rev. D ,125004 (2005); D. Metaxas, Phys. Rev. D , 065023 (2007); ibid. D , 063533 (2008).1532] D. A. Samuel and W. A. Hiscock, Phys. Lett. B , 251 (1991); Phys. Rev. D , 3052(1991).[33] U. H. Gerlach, Phys. Rev. D , 761 (1983).[34] A. Vilenkin, Phys. Lett. , 177 (1983).[35] J. Ipser and P. Sikivie, Phys. Rev. D30