False vacuum decay in a two-dimensional black hole spacetime
KKOBE-COSMO-21-02
False vacuum decay in a two-dimensional black hole spacetime
Taiga Miyachi (cid:92) and Jiro Soda (cid:92)(cid:92)
Department of Physics, Kobe University, Kobe 657-8501, Japan (Dated: February 5, 2021)
Abstract
We study a false vacuum decay in a two-dimensional black hole spacetime background. Thedecay rate in the case that nucleation site locates at a black hole center has been calculated in theliterature. We develop a method for calculating the decay rate of the false vacuum for a genericnucleation site. We find that the decay rate becomes larger when the nucleation site is close to theblack hole horizon and coincides with that in Minkowski spacetime when the nucleation site goesto infinity. a r X i v : . [ g r- q c ] F e b . INTRODUCTION The vacuum in the standard model of particle physics (SM) confirmed by the discoveryof Higgs boson [1, 2] might be metastable depending on the top quark mass which has beenyet completely fixed experimentally. The metastable false vacuum state must decay intothe true vacuum [3–5]. Since the life time of the false vacuum in a flat spacetime is muchlonger than the age of the universe [6], it is believed that there is no apparent inconsistencyin the SM. However, there might have been primordial black holes in the early universe.Therefore, it is necessary to know the decay rate of the false vacuum in the presence ofblack holes. Indeed, Hiscock pointed out that the black hole makes the life time of the falsevacuum shorter [7]. Recently, it is shown that the life time of the Higgs vacuum is lessthan the age of the universe in the presence of the small black holes [8–12]. Consideringthe Hawking evaporation of black holes, small black holes would be ubiquitous in the earlyuniverse. Thus, the Higgs vacuum might be unstable. If so, we need to go beyond the SM,namely, the instability suggests a new physics at the energy scale higher than the TeV scale[13–19].
FIG. 1. A bubble formation away from a black hole is depicted. The black region is the black holeand the shaded region is the true vacuum and the other are the false vacuum.
It should be noticed that there are two points to be clarified in previous computationsof the decay rate of the false vacuum with black holes. First, in [7–12], the back-reactionof the false vacuum decay to spacetime geometry is taken into account and the black holespacetime changes after tunneling. The validity of such a quantum gravitational picture isnot apparent at least in the period around the electroweak phase transition. Second, no onehas considered cases where a nucleation site of a true vacuum is away from the black hole(Fig.1). This is because there is a difficulty in investigating the false vacuum decay in thepresence of a black hole. Indeed, the black hole breaks the translation symmetry and hence2he dominant bubble shape away from a black hole is no more spherical. Note that it isworth investigating bubble nucleation away from black holes because the process is relevantto estimation of gravitational waves from bubble collisions [20]. In fact, the presence of blackholes would change the nucleation rate and we need to take into account collisions betweenblack holes and bubbles on top of the bubble collisions.In this paper, as a first step, we consider the false vacuum decay in a two-dimensionalblack hole. To resolve the first issue, we take a semi-classical approach and consider thevacuum decay in a fixed black hole spacetime. As to the second issue, we consider two-dimensions where we can develop a new method by extending the formulation in [21, 22]to calculate the decay rate. Using numerical calculations, we find the decay rate of thefalse vacuum is enhanced in the presence of a black hole in two-dimensions. The decay ratebecomes larger when the nucleation site is close to the black hole horizon and coincides withthat in Minkowski spacetime when the nucleation site goes to infinity.The organization of the paper is as follows. In Sec.II, we briefly review how to calculate adecay rate of a false vacuum in Minkowski spacetime. Then, we introduce a new method forcalculating a nucleation rate of a bubble nucleated at the center of a black hole and comparewith previous researches. In Sec.III, we calculate the decay rate of the false vacuum forwhich nucleation site locates at a generic point. We find the fitting formula for a bubbleformation which yields the decay rate through a bubble nucleation away from the black hole.For comparison, we also study the case of a four-dimensional spacetime by taking annularbubbles and show that the decay rate is also enhanced as in the case of a two-dimensionalspacetime. Of course, the bubble with such a shape is not dominant one in four-dimensionsbut it is useful to see the tendency of the false vacuum decay in the presence of black holes.The final section is devoted to conclusion.
II. NUCLEATION AT THE CENTER OF A BLACK HOLE
In this section, we calculate a decay rate for a false vacuum decay when the nucleationsite locates at the center of a black hole. Recently, this rate has been investigated in [8] byusing Israel junction conditions [23] to treat the dynamics of the bubble. In this paper, weuse the fixed background. Hence, instead of the junction conditions, we extend a method fortreating the dynamics of the bubble in Minkowski spacetime [21, 22] to a two-dimensional3chwarzschild black hole spacetime.
A. False vacuum decay in Minkowski spacetime
Here, we briefly summarize the false vacuum decay in Minkowski spacetime. Let usconsider the following action S = − (cid:90) dx √− g (cid:20) ∂ µ φ∂ µ φ + U ( φ ) (cid:21) , (1)where the potential U ( φ ) has two local minima (FIG.2). In this paper, we take these minimaas follows U ( φ + ) = 0 , U ( φ − ) = − ε < , where φ + < φ − . (2)Here, ε is the difference of the energy density between the true and the false vacuum.We can calculate the decay rate of the false vacuum Γ as follows [3];Γ = Ae − B where B = I decay − I false , (3)where I decay is the classical Euclidean action for the vacuum decay process and I false is thatof the false vacuum. In this setup, I false vanishes and what we have to calculate is I decay .The prefactor A includes quantum corrections [4]. We focus on the leading contribution B in this paper. In a two-dimensional Minkowski spacetime, the exponent B is given by (seeAppendix A), B = πσ ε , (4)where σ is the surface tension of the bubble wall. B. False vacuum decay in black hole spacetime
Now, we move on to the black hole spacetime. We want clarify the effect of a black holeon the false vacuum decay process. The metric of the black hole is given by ds = − f ( r ) dt + dr f ( r ) where f ( r ) = 1 − GMr , (5)where M is the black hole mass and r is the absolute value of a spatial position from thecenter. Here, we assumed the reflection symmetry with respect to r . Now we put the bubbleradius r = R ( t ) and divide the action into three parts S = S + + S − + S wall , (6)4 IG. 2. The potential U ( φ ) with two local minima is plotted. The left zero point is the falsevacuum and the right zero point is denoted as φ . The true vacuum φ − has a negative energy − ε . where S wall , S + and S − are contributions from the wall, outside the bubble and inside thewall, respectively. Assuming that φ is static and homogeneous both inside and outside ofthe bubble, S ± can be easily calculated as S + + S − = 2 ε (cid:90) dt ( R ( t ) − r h ) , (7)where r h ≡ GM is a gravitational radius of the black hole. Since there is no matter insidethe black hole, the volume of the black hole is subtracted. To calculate S wall , we need severalsteps. First, using the spatial reflection symmetry, the action can be written as S = 2 (cid:90) dt (cid:90) ∞ dr (cid:20) f ( r ) − ( ∂ t φ ) − f ( r )( ∂ r φ ) − U ( φ ) (cid:21) . (8)Second, the tangent and normal vector of the wall is given by v µ (cid:107) = (cid:16) , ˙ R (cid:17)(cid:113) f ( R ) − f ( R ) − ˙ R , v µ ⊥ = (cid:16) f ( R ) − ˙ R, f ( R ) (cid:17)(cid:113) f ( R ) − f ( R ) − ˙ R , (9)where a dot denotes a derivative with respect to the time. Defining the following derivativeoperators ∂ (cid:107) ≡ v µ (cid:107) ∂ µ , ∂ ⊥ ≡ v µ ⊥ ∂ µ , (10)we obtain S = − (cid:90) dt (cid:90) ∞ dr (cid:20)
12 ( ∂ ⊥ φ ) −
12 ( ∂ (cid:107) φ ) + U ( φ ) (cid:21) . (11)5e are now in a position to use the thin-wall approximation [3](see Appendix A for moredetails). We envisage the situation that the energy is concentrated in the thin wall of thebubble and assume that ∂ (cid:107) φ << ∂ ⊥ φ . (12)Thus, S wall can be approximated as S wall = − (cid:90) dt (cid:90) wall dr (cid:20)
12 ( ∂ ⊥ φ ) + U ( φ ) (cid:21) . (13)The equation of motion is given by ∂ ⊥ φ = dU ( φ ) dφ . (14)Integrating this equation from the outside of the bubble to the wall, we obtain12 ( ∂ ⊥ φ ) | wall = U ( φ ) | wall . (15)Then S wall can be calculated as S wall = − (cid:90) dt (cid:90) wall dr (cid:20)
12 ( ∂ ⊥ φ ) + U ( φ ) (cid:21) = − (cid:90) dt (cid:90) wall dr ( ∂ ⊥ φ ) = − (cid:90) dt (cid:90) φ − φ + dφ drdφ ( ∂ ⊥ φ ) . (16)In the thin wall approximation, we can deduce the relation dφdr ∼ ∂ ⊥ φ (cid:113) f ( R ) − f ( R ) − ˙ R . (17)Therefore, we obtain S wall = − (cid:90) dt (cid:90) φ − φ + dφ (cid:112) U ( φ ) (cid:113) f ( R ) − f ( R ) − ˙ R ≡ − σ (cid:90) dt (cid:113) f ( R ) − f ( R ) − ˙ R , (18)where σ is the tension of the bubble wall (see (A17)); σ = (cid:90) φ − φ + dφ (cid:112) U ( φ ) . (19)Finally, we obtain the following effective action; S = (cid:90) dt (cid:20) ε ( R ( t ) − r h ) − σ (cid:113) f ( R ) − f ( R ) − ˙ R (cid:21) . (20)6 IG. 3. The effective potentials for
GM/R = 0 .
0, 0.08, 0.15 and 0.25, where R = σ/ε . With the Wick rotation t = − iτ , the effective Euclidean action reads I = − iS = (cid:90) dτ (cid:104) − ε ( R ( τ ) − r h ) + 2 σ (cid:112) f ( R ) + f ( R ) − R (cid:48) (cid:105) , (21)where a prime denotes a derivative with respect to an Euclidean time τ .Let us evaluate the decay rate of the false vacuum. The dynamics of the bubble wallin the Lorentzian spacetime can be deduced from the Hamiltonian derived from the action(20) as H L ≡ ˙ R ∂L∂ ˙ R − L = 2 σf ( R ) (cid:113) f ( R ) − f ( R ) − ˙ R − ε ( R − r h ) . (22)Since the initial energy vanishes, from the energy conservation law, we have H L = 0 . (23)From this, we obtain the following equation˙ R ( t ) = − f ( R ) (cid:18)(cid:16) σε (cid:17) f ( R )( R − r h ) − (cid:19) ≡ − V ( R ) . (24)Performing the Wick rotation t = − iτ , we can also obtain the Euclidean equation R (cid:48) ( τ ) = V ( R ) . (25)In FIG. 3, we plotted the effective potential V ( R ) for various masses of black holes. The leftzero point of V ( R ) corresponds to the location of the black hole horizon; R = r h . The right7ero point of V ( R ) is an initial position of the bubble wall; R = R (0). After the tunneling,the bubble wall appears at R (0) and expands rapidly.First, we calculate the decay rate in the case of a Minkowski spacetime and check the con-sistency with the known result (4). From (25), we can derive an equation for the Minkowskispacetime R (cid:48) ( τ ) = (cid:16) σε (cid:17) R − . (26)With the initial condition ˙ R (0) = R (cid:48) (0) = 0, we obtain R ( τ ) = (cid:113) R − τ , (27)where R ≡ σ/ε . Note that the initial radius R is the same as that derived in Appendix A.Substituting this solution into the Euclidean action (21) and integrating over − R < τ < R ,we obtain the exponent B = (cid:90) R − R dτ (cid:16) − εR + 2 σ √ R (cid:48) (cid:17) = (cid:90) R − R dτ (cid:16) − ε ( R − τ ) + 2 σR ( R − τ ) − (cid:17) = 2 σR (cid:90) π dθ cos θ = πσ ε ≡ B flat . (28)This coincides with the result of (4). FIG. 4. The exponent B for a bubble is plotted. Here, B flat is that of the Minkowski spacetimeand R ≡ σ/ε . B is given by B = (cid:90) β dτ (cid:104) − ε ( R ( τ ) − r h ) + 2 σ (cid:112) f ( R ) + f ( R ) − R (cid:48) (cid:105) , (29)where β is the period of the solution of (25). We numerically evaluated B and plotted it inFIG. 4. It turns out that the black hole makes the decay rate Γ = A exp( − B ) smaller thanthat in the Minkowski spacetime.The present formulation can be extended to the 4-dimensional spacetime as shown inAppendix B. In this case, the decay rate is enhanced for small black holes (FIG. 5). This isconsistent with the result in [8]. Thus, we have shown that the qualitative behavior of thedecay rate depends on the dimensions when the nucleation site is located at the center of ablack hole. FIG. 5. The normalized exponent B for a 4-dimensional bubble is plotted. In this case, B flat isthat of the four-dimensional Minkowski spacetime and we used R ≡ σ/ε . III. NUCLEATION AT A GENERIC LOCATION
In this section, we consider the decay rate of the false vacuum when the nucleation siteis located outside a black hole. There are two bubble walls which are represented by r = P ( t ) , Q ( t ) where r h < P ( t ) < Q ( t ) . (30)In the previous section, we used the reflection symmetry to reduce the dynamics of two wallsin both sides of the black hole to the dynamics of a single wall. In this section, however, we9ave to treat these two walls independently because of the lack of symmetry in the case ofthe bubble nucleation outside a black hole.Let us investigate the dynamics of the bubble walls. Let us consider the following action S = − (cid:90) dx √− g (cid:20) ∂ µ φ∂ µ φ + U ( φ ) (cid:21) . (31)The metric is the same as (5). As in the last section, we can derive the effective action asfollows S = (cid:90) dt (cid:34) ε ( Q ( t ) − P ( t )) − σ (cid:88) R = P,Q (cid:113) f ( R ) − f ( R ) − ˙ R (cid:35) . (32)We have to follow the dynamics of two walls using a Hamilton formulation because it isdifficult to obtain effective potentials from the energy conservation law. In a Lorentzianformalism, the Hamiltonian is given by H L ≡ (cid:88) R = P,Q (cid:113) σ f ( R ) + f ( R ) π LR − ε ( Q ( t ) − P ( t )) , (33)where π LR is the Lorentzian canonical conjugate momentum for P ( t ) or Q ( t ). The Hamiltonequations of motion are as follows;˙ R ( t ) = f ( R ) π LR (cid:112) σ f ( R ) + f ( R ) π LR , (34)˙ π LR ( t ) = ∓ ε − r h R σ + 2 f ( R ) π LR (cid:112) σ f ( R ) + f ( R ) π LR , (35)where R is either P or Q . For the second equation, the minus sign is for P and the other isfor Q . From energy conservation law, we obtain H L = 0 . (36)To make an analytic continuation, we choose the boundary conditions at the turning pointas follows; ˙ P (0) = ˙ Q (0) = 0 . (37)From (34), there are two choices for the wall of P ; π LP (0) = 0 or P (0) = r h and we choosethe former to consider a bubble away from the black hole. Taking into account P ( t ) < Q ( t ),the initial condition for the wall position Q must be π LQ (0) = 0. Then, the two initialconditions read π LP (0) = π LQ (0) = 0 . (38)10rom H L = 0 and π LR (0) = 0, we also obtain (cid:112) f ( P (0)) + (cid:112) f ( Q (0)) − εσ ( Q (0) − P (0)) = 0 . (39)Hence, one constant of integration is left in solutions of Eqs.(34) and (35). The singleparameter determines the location where the bubble appears. Therefore, from (34), (35),(38) and (39), the Lorentzian dynamics of the bubble wall can be solved.In an Euclidean region, the effective action reads I = (cid:90) dτ (cid:34) − ε ( Q ( τ ) − P ( τ )) + σ (cid:88) R = P,Q (cid:112) f ( R ) + f ( R ) − R (cid:48) (cid:35) . (40)The Hamiltonian in this case is given by H E ≡ (cid:88) R = P,Q ( − ) (cid:113) σ f ( R ) − f ( R ) π ER + ε ( Q ( τ ) − P ( τ )) , (41)where π ER is the Euclidean canonical conjugate momentum for P ( τ ) or Q ( τ ). The Hamiltonequations of motion are as follows; R (cid:48) ( τ ) = f ( R ) π ER (cid:112) σ f ( R ) − f ( R ) π ER , (42) π (cid:48) ER ( τ ) = ± ε + r h R σ − f ( R ) π ER (cid:112) σ f ( R ) − f ( R ) π ER . (43)To perform analytic continuation from the Euclidean solution into the Lorentzian solution,we choose the conditions at the transition point; π EP (0) = π EQ (0) = 0 , (44) (cid:112) f ( P (0)) + (cid:112) f ( Q (0)) − εσ ( Q (0) − P (0)) = 0 . (45)Therefore, from (42), (43), (44) and (45), the Euclidean dynamics of the bubble wall can bededuced. A. A consistency check
To check the consistency, we calculate the decay rate of the false vacuum in the Minkowskispacetime (4). The Euclidean Hamilton equations of motion (42) and (43) become R (cid:48) ( τ ) = π ER (cid:112) σ − π ER , (46) π (cid:48) ER ( τ ) = ± ε . (47)11hese equations can be solved easily under the initial conditions π ER (0) = 0; R ( τ ) = ∓ (cid:113) R − τ + C R . (48)where C R is a constant of integration and R ≡ σ/ε . From (45), we see C P = C Q ≡ C = P ( τ ) + Q ( τ )2 . (49)Then, the constant C represents the center of the bubble. Thus, the exponent B is given by B = (cid:90) R − R dτ (cid:32) − ε ( Q ( τ ) − P ( τ )) + σ (cid:88) R = P,Q √ R (cid:48) (cid:33) = (cid:90) R − R dτ (cid:32) − ε (cid:113) τ − τ + 2 σ τ (cid:112) τ − τ (cid:33) = πσ ε = B flat . (50)This is the same result as (4). B. Decay rate for a generic nucleation site
Now, we are in a position to calculate the decay rate for generic cases. In the absence of ablack hole, P ( τ ) and Q ( τ ) have the same period. However, it is not true in the presence of ablack hole (FIG. 6). This does not matter because the two walls contribute to the Euclideanaction (40) independently.First, we have numerically solved the dynamics of bubble walls. In FIG. 6, the trajectoriesof bubble walls for various masses of black holes with a specific initial locations of bubblesare plotted. We analytically continue Lorentzian solutions ( t >
0) and Euclidean solutions( τ <
0) at t = τ = 0. After a bubble nucleation ( t = 0), the bubble wall P falls into blackhole horizon and the other wall Q goes to infinity at the speed of light in Schwarzschildspacetime. In the Minkowski case, the trajectories of P and Q are symmetric. While, whenthe nucleation point of P is close to the horizon, the trajectories become asymmetric in theblack hole cases.For the Schwarzschild black hole spacetime, the exponent B is given by B = (cid:90) β P dτ (cid:16) εP ( τ ) + σ (cid:112) f ( P ) + f ( P ) − P (cid:48) (cid:17) + (cid:90) β Q dτ (cid:16) − εQ ( τ ) + σ (cid:112) f ( Q ) + f ( Q ) − Q (cid:48) (cid:17) , (51)12 IG. 6. The trajectories of bubble walls for real and imaginary time ( t > τ <
0) are plottedfor P (0) /R = 0 .
5. From the top to the bottom, we took the mass
GM/R = 0 . GM/R = 0 . GM/R = 0 .
2, respectively. The vertical dotted line represents a location of the black holehorizon. where β R denotes the period of R ( τ ). We calculate the exponent B numerically and findthat the black hole enhances the decay rate for nucleation process of a bubble (FIG. 7). Thedecay rate is largest at the horizon and asymptotically approaches the result of Minkowskispacetime as the nucleation point goes to infinity. Note that small black holes make the13 IG. 7. The exponent B for generic bubbles. Left: Plots for P (0) /R = 0 . , . , . , . , . P (0). Right: Plots for GM/R = 0 . , . , . , . GM . decay rates larger [8–12].We shall find a fitting formula for the solutions. Notice that, in the absence of a blackhole, the solutions (48) can be written as( R ( τ ) − C R ) + τ = R . (52)In order to find a fitting formula, we take the ansatz in the Euclidean region( R ( τ ) − C ) ( R (0) − C ) + τ τ ∗ R = 1 , (53)where τ ∗ R is the half period of solutions and C = P ( τ ∗ P ) = Q ( τ ∗ Q ). We put these twoparameters as follows τ ∗ R = σε (cid:18) a R r h P (0) (cid:19) , (54) C = P (0) + σε (cid:114) − r h P (0) , (55)where a R are fitting parameters calculated by least squares method. In the absence of ablack hole, we should take the limits r h → P (0) → ∞ . Using these ansatze, the decay14ate (51) can be evaluated as B = (cid:90) τ ∗ P − τ ∗ P dτ (cid:16) εP ( τ ) + σ (cid:112) f ( P ) + f ( P ) − P (cid:48) (cid:17) + (cid:90) τ ∗ Q − τ ∗ Q dτ (cid:16) − εQ ( τ ) + σ (cid:112) f ( Q ) + f ( Q ) − Q (cid:48) (cid:17) = τ ∗ P ε (cid:16)(cid:16) − π (cid:17) C + π P (0) (cid:17) + σ (cid:90) π dθ sin θ (cid:115) f ( P ) + f ( P ) − (cid:18) C − P (0) τ ∗ P (cid:19) cot θ + τ ∗ Q − ε (cid:16)(cid:16) − π (cid:17) C + π Q (0) (cid:17) + σ (cid:90) π dθ sin θ (cid:118)(cid:117)(cid:117)(cid:116) f ( Q ) + f ( Q ) − (cid:32) Q (0) − Cτ ∗ Q (cid:33) cot θ (56)where we performed the transformation of a variable τ = τ ∗ R cos θ in the integral. The decayrate obtained from a fitting formula shows a good agreement with the numerical calculation(FIG. 8). Thus, the fitting formula (53) is a good approximation for the dynamics of thebubble wall in the black hole spacetime. FIG. 8. The exponent B calculated by the fitting formula. Left: Plots for P (0) /R = 1 . a P ∼ . a Q ∼ . P (0) /R = 0 . a P ∼ − . a Q ∼ . After an analytic continuation of the Euclidean solution (53), we obtain a deformedsolution in the Lorentzian region. Since we have fitted the solution in the Euclidean region,it is difficult to capture the feature that the bubble wall P falls into the black hole as is seenin FIG. 6. 15 . Discussion What we wanted to reveal is the effect of a black hole on the decay rate of the falsevacuum in four-dimensions. In this subsection, we discuss a four-dimensional false vacuumdecay.
FIG. 9. An annular bubble is depicted. The black region is a black hole and the shaded region isthe true vacuum and the other are the false vacuum.
As we already mentioned, it is not easy to solve the nucleation process in four-dimensions.As a modest first step, we consider the annular bubble in a four-dimensional black holespacetime (FIG. 9). We extended the formulation in two-dimensions to four dimensions inAppendix B. Apparently, the annular bubble (FIG.9) does not describe a dominant processof the false vacuum decay. However, it is useful to see the tendency of gravitational effectson the decay process in a four-dimensional black hole spacetime.From numerical calculations, we see the black hole also enhances the decay rate of thefalse vacuum even in four dimensions (FIG. 10). Therefore, it is expected that the black holealso enhanced the decay rate of the false vacuum through the dominant bubble nucleationprocess in a four-dimensional black hole spacetime.
IV. CONCLUSION
We studied the false vacuum decay in a two-dimensional black hole spacetime. In par-ticular, we considered the cases the bubble nucleates at a generic point in the black holespacetime. We developed a method for calculating the decay rate in a fixed Schwarzschildblack hole spacetime by extending the formulation in [21, 22]. Using numerical calculations,we showed that the black hole enhances the decay rate when the nucleation occurs outside16
IG. 10. The exponent B for a four-dimensional annular bubble is plotted for P (0) /R = 1 .
0. Wenormalized B by the action B for an annular bubble in the four-dimensional Minkowski spacetime. of the black hole. The decay rate is maximized near the horizon and reduces to the onefor a Minkowski spacetime at infinity. This is natural because away from the horizon thespacetime asymptotically approaches the Minkowski spacetime. Our finding indicates thatit is worth investigating the decay rate of the false vacuum in a four-dimensional black holespacetime in order to discuss gravitational waves from bubbles collisions and black holebubble collisions [20].For future work, it is interesting to apply our method to other O(3) symmetric spacetimessuch as Schwarzschild-deSitter black holes or charged black holes. We expect that thecosmological constant make the decay rate larger and the charge affects oppositely as inthe previous works [8, 10]. It is challenging to study the false vacuum decay in a four-dimensional black hole spacetime because the bubble outside the black hole horizon breaksthe symmetry of the spacetime. It is also intriguing to consider rotating black holes [24]. ACKNOWLEDGMENTS
J. S. was supported by JSPS KAKENHI Grant Numbers JP17H02894 and JP20H01902.17 ppendix A: N dimensional bubble in Minkowski spacetime
In this appendix, we derive the decay rate in a N-dimensional Minkowski spacetime basedon [3].Let us consider the following action S = − (cid:90) dx N (cid:18) ∂ µ φ∂ µ φ + U ( φ ) (cid:19) , (A1)where the potential U ( φ ) has two local minima (FIG.2). For simplicity, we take these minimaas follows U ( φ + ) = 0 , U ( φ − ) = − ε < , where φ + < φ − . (A2)In this setup, I false vanishes and we only have to treat I decay . Performing the Wick rotation t = − iτ , we obtain the following Euclidean action I = − iS = (cid:90) dτ dx N − (cid:18) ∂ µ φ∂ µ φ + U ( φ ) (cid:19) . (A3)From this Euclidean action, the equation of motion for φ is derived as (cid:0) ∂ τ + ∆ N − (cid:1) φ = dU ( φ ) dφ , (A4)where ∆ N − is the Laplacian for a (N-1)-dimensional Euclidean space. To derive a solutionwhich contributes to a vacuum decay, we impose the following boundary conditions; φ | τ = ±∞ = φ + , (A5) φ | |−→ x | =+ ∞ = φ + , (A6) ∂φ∂τ (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = 0 . (A7)Next, we assume O(N) symmetry so that φ depends only on ρ = (cid:112) τ + |−→ x | . Then, (A3),(A4), (A5), (A6) and (A7) become I = T N (cid:90) dρρ N − (cid:32) (cid:18) dφdr (cid:19) + U ( φ ) (cid:33) , (A8) d φdρ + N − ρ dφρ = dU ( φ ) dφ , (A9) φ ( ∞ ) = φ + , (A10) dφ (0) dρ = 0 , (A11)18here T N is the surface area of a unit sphere. The solutions satisfying the boundary condi-tions (A10) and (A11) is called the bounce solution [3].Let us now show the existence of the bounce solution. There is a zero point of U ( φ ) be-tween φ + and φ − and we write it as φ . Since the Euclidean energy monotonously decreases; ddρ (cid:32) (cid:18) dφdρ (cid:19) − U (cid:33) = − N − ρ (cid:18) dφdρ (cid:19) ≤ , (A12) φ will undershoot at φ + when φ (0) ≤ φ . If φ (0) is very close to φ − , the equation of motion(A9) can be linearized as follows; (cid:18) d dρ + N − ρ ddρ − µ (cid:19) ( φ − φ − ) = 0 , (A13)where µ ≡ d U ( φ − ) /dφ . The solution is given by φ ( ρ ) − φ − = 2 N − Γ (cid:18) N (cid:19) ( φ (0) − φ − ) I N − ( µρ )( µρ ) N − , (A14)where Γ( N ) is the Gamma function and I N is the modified Bessel function of the first kind.If we put φ (0) sufficiently close to φ − , φ will stay around φ − for long time ρ . However, thedamping force which is proportional to 1 /ρ can be neglected at large ρ and φ will overshootat φ + . Therefore, there must be intermediate initial position φ (0) for which φ just comes torest at φ + after the infinite time ρ .From the above argument, φ ( ρ ) of the bounce solution stays near φ − for a long time ρ = R and goes quickly to φ + . Let us consider the behavior of φ near ρ = R . For ρ ∼ R ,we can neglect the damping force and the equation of motion becomes d φdχ = dU ( φ ) dφ , (A15)where χ ≡ ρ − R . This equation has the following solution; χ = (cid:90) φ w dφ (cid:112) U ( φ ) . (A16)The action for this solution is given by I wall = T N (cid:90) wall dχ ( χ + R ) N − (cid:32)(cid:18) dφ w dχ (cid:19) + U ( φ w ) (cid:33) = T N R N − (cid:90) φ − φ + dφ (cid:112) U ( φ ) ≡ T N R N − σ , (A17)19here σ is the surface tension of the bubble. Then the bounce solution is as follows; φ ( ρ ) = φ − for ρ << Rφ w ( ρ − R ) for ρ ∼ Rφ + for ρ >> R . (A18)The region where φ = φ − is called bubble . To obtain the decay rate, we assume that R issufficiently larger than the wall width of the bubble. This approximation is called the thin-wall approximation. Then, after substituting (A18) into (A8), we can divide the Euclideanaction into three parts; I = I + + I − + I wall , (A19)where I + ( I − ) denotes outside (inside) of the bubble and I wall is the contribution of thebubble wall. Using (A17), we obtain I = − V N R N ε + T N R N − σ , (A20)where V N is the volume of a unit sphere. Taking the Variation with respect to R δIδR = T N ( − R N − ε + ( N − R N − σ ) = 0 , (A21)we obtain R = ( N − σ/ε ≡ R . Note that the thin-wall approximation can be statedquantitatively as follows; µσε >> . (A22)We finally obtain the exponent BB = I = π N σ Γ( N + 1) (cid:18) ( N − σε (cid:19) N − ≡ B flat , (A23)where we write down V N and T N explicitly. This is an extension of the result in [3] to aN-dimensional Minkowski spacetime. Appendix B: N-dimensional effective action
In this appendix, we extend the effective actions (20) and (32) to a N-dimensional space-time. In both cases, we assume O(N-1) symmetry for the spatial configuration of a scalarfield. Note that an annular bubble is assumed for a generic nucleation. We can take the20ame method as the 2-dimensional cases to derive the effective actions in N-dimensionalspacetime. The action (20) is extended to S = (cid:90) dt (cid:18) εV N − ( R ( t ) N − − r N − h ) − σT N − R ( t ) N − (cid:113) f ( R ) − f ( R ) − ˙ R (cid:19) (B1)and the action (32) is also extended to S = (cid:90) dt (cid:32) εV N − ( Q ( t ) N − − P ( t ) N − ) − σT N − (cid:88) R = P,Q R ( t ) N − (cid:113) f ( R ) − f ( R ) − ˙ R (cid:33) , (B2)where V N is the volume of a N-dimensional unit sphere and T N is the surface area of that.From these actions, the dynamics of the bubbles can be derived and we can calculate thedecay rate by imposing the same boundary conditions as those in 2-dimensional spacetime. [1] G. Aad et al . , “Observation of a new particle in the search for the Standard Model Higgsboson with the ATLAS detector at the LHC,” Phys. Lett. B (2012) 1-29. [arXiv:1207.7214[hep-ex]].[2] S. Chatrchyan et al. , “Observation of a new boson at a mass of 125 GeV with the CMSexperiment at the LHC,” Phys. Lett. B (2012) 30. [arXiv:1207.7235 [hep-ex]].[3] S. R. Coleman, “Fate of the false vacuum. 1. Semiclassical Theory ∗ ,” Phys. Rev. D (1977)2929-2936, Phys. Rev. D (1977) 1248 (erratum).[4] C. G. Callan and S. R. Coleman, “Fate of the false vacuum. 2. First Quantum Corrections ∗ ,”Phys. Rev. D (1977) 1762-1768.[5] S. R. Coleman and F. D. Luccia, “Gravitational Effects on and of Vacuum Decay,” Phys. Rev.D (1980) 3305.[6] S. Chigusa, T. Moroi, Y. Shoji, “State-of-the-Art Calculation of the Decay Rate of ElectroweakVacuum in Standard Model,” Phys. Rev. Lett. ,(1987) 1161.[8] R. Gregory, I. G. Moss and B. Withers, “Black holes as bubble nucleation sites,” JHEP (2014) 081 [arXiv:1401.0017 [hep-th]].[9] P. Burda, R. Gregory and I. Moss, “Gravity and the stability of the Higgs vacuum,” Phys.Rev. Lett. (2015) 071303 [arXiv:1501.04937 [hep-th]].
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