Tunneling decay of false kinks
Éric Dupuis, Yan Gobeil, Richard MacKenzie, Luc Marleau, M. B. Paranjape, Y. Ung
UUdeM-GPP-TH-14-238
Tunneling decay of false kinks ´Eric Dupuis a , ∗ Yan Gobeil a , † Richard MacKenzie a , ‡ Luc Marleau b , § M. B. Paranjape a , ¶ and Yvan Ung a ∗∗ a Groupe de physique des particules, Universit´e de Montr´eal,C. P. 6128, Succursale Centre-ville,Montr´eal, QC, Canada, H3C 3J7 and b D´epartement de physique, de g´enie physique et d’optique,Universit´e Laval, Qu´ebec, QC, Canada G1K 7P4
Abstract
We consider the decay of “false kinks,” that is, kinks formed in a scalar field theory with a pairof degenerate symmetry-breaking false vacua in 1+1 dimensions. The true vacuum is symmetric.A second scalar field and a peculiar potential are added in order for the kink to be classicallystable. We find an expression for the decay rate of a false kink. As with any tunneling event, therate is proportional to exp( − S E ) where S E is the Euclidean action of the bounce describing thetunneling event. This factor varies wildly depending on the parameters of the model. Of interestis the fact that for certain parameters S E can get arbitrarily small, implying that the kink is onlybarely stable. Thus, while the false vacuum itself may be very long-lived, the presence of kinks cangive rise to rapid vacuum decay. PACS numbers: 11.27.+d, 98.80.Cq, 11.15.Ex, 11.15.Kc ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] a r X i v : . [ h e p - t h ] J un . INTRODUCTION Kinks are topological solitons in 1+1-dimensional field theories with a real scalar field φ and spontaneously-broken discrete symmetry φ → − φ . The potential has degenerate vacuaat field values φ = ± v ; the kink interpolates between the two. The same model in higherdimensions gives rise to extended objects: linelike defects in 2+1 dimensions, domain wallsin 3+1 dimensions, and so on.We are interested in models which “demote” the vacua at ± v to false vacuum status,there being a lower-energy true vacuum at φ = 0. A specific example is L φ = 12 ( ∂ µ φ ) − V ( φ ) (1)where (see Fig. 1) V ( φ ) = λ ( φ − δv )( φ − v ) (2)with 0 < δ <
1. In the true vacuum the symmetry is of course restored. The motivation forstudying such a model is that the presence of topological defects can have a dramatic effecton the quantum mechanical stability of the false vacuum. Using cosmological language forconvenience, if the universe is in a false vacuum (without kinks) throughout space, it willdecay through quantum tunneling [1], yet the decay rate per unit volume can be exceedinglysmall, to the point where the observable universe could be trapped in a false vacuum fortimes exceeding the age of the universe. Such a scenario is invoked in certain models offundamental physics; see for instance [2]. If so, the presence of topological defects can havea dramatic effect on the decay rate. (It should be noted that if they exist, topologicaldefects will necessarily be formed during a phase transition, so their presence is not merelya possibility; it is a certainty [3].) This situation has been examined previously for magneticmonopoles [4], vortices in 2+1 dimensions [5], and cosmic strings [6].Since there is a unique true vacuum in the class of model we are considering, thereare no classically stable nontrivial static solutions. For instance, a kinklike configurationinterpolating between the two false vacua would bifurcate into two halves, one interpolatingbetween φ = − v and φ = 0 and the other between φ = 0 and φ = v . The potential energydensity being lower between the two halves of the kink than in the exterior region, the kinkswould essentially repel each other and fly off to spatial infinity at speeds approaching thatof light, leaving an ever-expanding region of true vacuum.2 φ V ( φ ) FIG. 1. Example potential (2) with symmetric true vacuum and symmetry-breaking false vacua.Here λ = v = 1 , δ = 0 . Thus we must consider a more complicated model if we wish to study the decay ofclassically stable kinks formed in a symmetry-breaking false vacua. The paper is outlined asfollows. In Section 2, we introduce a generalized model and argue that it does have classicallystable kinks. In Section 3, we find numerical solutions for a variety of parameters. Thesesolutions, although classically stable, will tunnel to the true vacuum, a process analyzedin Section 4. The comparison between kink-mediated vacuum decay and ordinary vacuumdecay is analyzed in Section 5. We then present conclusions and suggestions for furtherwork.
II. A MODEL WITH CLASSICALLY STABLE KINKS
As argued above, classically stable kinks do not exist in the simplest model in which thevacuum structure is as outlined above (two symmetry-breaking false vacua, symmetric truevacuum). One way to obtain stable kinks is to add a second scalar field to the model with anunusual potential. Consider the following model, which is to be viewed more as an examplehaving the desired vacuum structure and stable kinks rather than as a realistic model for aparticular physical system. L = 12 (cid:0) ( ∂ µ φ ) + ( ∂ µ χ ) (cid:1) − V ( φ, χ )3here V ( φ, χ ) = λ ( φ − δ v )( φ − v ) + λ φ + γv (cid:2) ( χ − v ) − ( δ / χ − v )( χ + v ) (cid:3) The model is symmetric under φ → − φ . The potential has seven parameters, two of whichcan be eliminated by rescaling the fields and x to dimensionless variables. Doing so, wecan rewrite the (dimensionless) Lagrangian as follows, having chosen the constants in sucha way as to simplify the equations of motion: L = 12 ( ∂ µ φ ) + α β ( ∂ µ χ ) − V ( φ, χ ) (3)where V ( φ, χ ) = ( φ − δ )( φ − + αφ + γ (cid:20) ( χ − − δ χ − χ + 1) (cid:21) . (4)There are now five parameters, α, β, δ , δ , γ , which we take to be positive; furthermore, wesuppose 0 < δ < < δ < / α = β = 1 forsimplicity.The potential is the sum of two terms. The first term depends on φ only and is, apartfrom rescaling, the potential (2) of the original model (see Fig. 1). The second term is aproduct of two factors. The second of these (in square parentheses, written V ( χ ) below; seeFig. 2) depends on χ only. With δ in the above-mentioned range, V has two minima: aglobal minimum, of zero energy density, at χ = − δ , at χ = +1. The first factor can be viewed as a modulating function which varies the“strength” of the second factor depending on the value of φ . In particular, if φ = 0 (its truevacuum), the modulating factor is maximal, so that if χ passes from one minimum to theother, the cost in potential energy where χ (cid:39) φ, χ ) = (0 , − V (0 , −
1) = − δ . There arefalse vacua at ( φ, χ ) = ( ± , − V ( ± , −
1) = 0. There is also a maximumin the vicinity of ( φ, χ ) = (0 ,
0) which can be quite pronounced if γ (cid:28)
1. Finally, there arepossible extrema at χ = 1 which are less important but not entirely irrelevant to us, so it isworth examining briefly the potential there: V ( φ,
1) = ( φ − δ )( φ − + δ φ + γ . (5)4 χ V ( χ ) FIG. 2. Last factor of the potential (4). Here δ = 0 . This is the sum of the potential depicted in Fig. 1 and a Lorentzian function. It is ofinterest to ask if V ( φ,
1) can be negative, since as we will see it gives rise to an instabilityof the soliton (though not the type of instability which is of primary interest to us). If theLorentzian is broad and sufficiently large in amplitude, it will raise the potential near φ = 0so that it is nowhere negative (for χ = 1). However if the Lorentzian is narrow, it can raisethe potential at φ = 0 to a positive value while leaving negative regions on either side. Thusthere are two cases. In the first case V ( φ,
1) is minimized at φ = 0 so that ( φ, χ ) = (0 ,
1) isa local minimum of the potential. This occurs if γ > δ δ . (6)In this case the minimum of V ( φ,
1) is negative if δ < δ γ. (7)If (6) is not satisfied, V ( φ,
1) is minimized at a pair of nonzero values of φ straddling φ = 0. These points are local minima (false vacua) of the potential while ( φ, χ ) = (0 ,
1) is asaddle point. Now the minimum of V ( φ,
1) is negative if the following surprisingly unwieldycondition is satisfied: (cid:18) − δ − δ
64 + 9 δ − δ (cid:19) + γ (cid:18) − δ
32 + 13 δ − δ (cid:19) − γ (cid:18)
764 + 13 δ
64 + δ (cid:19) − γ (cid:18)
964 + 9 δ (cid:19) − γ
512 + (cid:26)(cid:18) −
164 + 3 δ − δ
256 + 9 δ (cid:19) − γ (cid:18) δ − δ (cid:19) − γ (cid:18) δ (cid:19) − γ (cid:27) (cid:112) (2 − γ − δ )) + 16(2 γδ + γ − δ ) + δ < . (8)5 χ V ( φ , χ )(a) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 0 0.5 1 -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 φχ (b) FIG. 3. (a) 3-dimensional plot and (b) contour plot of V ( φ, χ ) for ( δ , δ , γ ) = (0 . , . , . φ, χ ) = (0 , − φ, χ ) = ( ± , − χ = 1, the nature of which depend onthe values of the parameters, as described in the text. (In the figure, they are local minima – falsevacua – at ( φ, χ ) = (0 ,
1) and very near ( ± , φ, χ ) = (0 , We can understand qualitatively why there could be stable solitons interpolating betweenthe two false vacua ( φ, χ ) = ( ± , − φ has half-kinks at ± l φ , one interpolating between φ = − v and φ = 0 and theother between φ = 0 and φ = v , these being “enveloped” by a kink-antikink of χ at ± l χ ,where l χ > l φ . (To simplify the discussion, we will call all of these objects kinks.) There arefive regions where the fields are approximately constant, two pairs of which are related bysymmetry; these regions are denoted (i) (between the two φ kinks), (ii) (the regions betweenthe φ and χ kinks), and (iii) (exterior to the χ kinks). In these regions the energy densitycomes entirely from the potential energy and is easily evaluated: V ( i ) = − δ + 1 γ δ , V ( ii ) = 11 + γ δ , V ( iii ) = 0 . It is easy to see that the configuration depicted in Fig. 4 cannot be a solution. Considera family of such configurations parameterized by l φ , l χ , where the positions but not the6 -l χ -l φ l φ l χ x ( ii )( ii ) ( iii )( iii ) φχ ( i ) FIG. 4. Configuration to illustrate intuitively the existence of stable solitons. shapes of the transitions vary. For the configuration to be a solution, its energy must bestationary as a function of l φ , l χ . For small displacements ∆ l φ , ∆ l χ , the variation in energyis ∆ E = 2( V ( i ) − V ( ii ) )∆ l φ + 2( V ( ii ) − V ( iii ) )∆ l χ . Since V ( iii ) < V ( ii ) , the energy is not stationary in l χ ; indeed, if left to evolve dynamically,the two χ kinks would move towards the origin to reduce the static energy. Similarly, since V ( iii ) < V ( i ) it is energetically advantageous for region (i) to collapse to zero, converting falsevacuum to true.This variational argument is not powerful enough in itself to determine if a configurationsuch as that depicted in Fig. 4 would evolve into a stable one, because as soon as the kinksoverlap they interact and the energy is no longer a straightforward function of l φ , l χ . If theconfiguration can be deformed to an unstable one without encountering an energy barrieralong the way, it will not evolve into a stable one.One obvious way to go from Fig. 4 to an unstable configuration would be to deform χ to aconstant, χ ( x ) = − φ kinks will fly apart, as described above). This couldbe done by moving the χ kinks towards one another so that they annihilate, or alternativelyby deforming the value of χ between the kinks from +1 to −
1. In either case, the fields passthrough the large potential energy barrier at ( φ, χ ) (cid:39) (0 , III. NUMERICAL SOLUTIONS
The static equations of motion that follow from (3) (with α = β = 1) are: φ (cid:48)(cid:48) − φ ( φ − φ − − δ ) + 2 φ ( φ + γ ) (cid:20) ( χ − − δ χ − χ + 1) (cid:21) = 0 (9) χ (cid:48)(cid:48) − φ + γ ) ( χ − χ − δ /
4) = 0 (10)Given a static solution, the energy is E [ φ, χ ] = (cid:90) dx (cid:18) (cid:0) φ (cid:48) + χ (cid:48) (cid:1) + V ( φ, χ ) (cid:19) . (11)We look for kinklike solutions interpolating between the false vacua ( φ, χ ) = ( ± , − φ to be odd and χ even under space reflection, so we can solve the equations on thehalf-line x ≥ φ (0) = 0 , χ (cid:48) (0) = 0 , (12) φ ( x ) → , χ ( x ) → − x → ∞ . (13)The numerical approach used is an adaptation of the relaxation algorithm explained beau-tifully in [7]. Of course, numerically we do not integrate to infinity so (13) must be handleddifferently. We integrate to some suitably large x max . The most obvious boundary condi-tion would be to impose ( φ, χ ) | x = x max = (1 , − x max . Instead, we linearize the equations (9,10) about( φ, χ ) = (1 , −
1) and insist that the fluctuations tend to zero exponentially as x → ∞ .This gives linear relations between ( φ, φ (cid:48) ) x = x max and between ( χ, χ (cid:48) ) x = x max which we adoptas boundary conditions at x max . With these, the algorithm produces a solution which wouldextrapolate to ( φ, χ ) = (1 , −
1) as x → ∞ .Figure 5 illustrates a typical solution, displaying the functions φ ( x ) , χ ( x ) in (a) and asa parametric plot in the φχ plane superimposed on a contour plot of the potential in (b).The latter is interesting because the equations of motion (9,10) have a mechanical analogy:if x is interpreted as a time coordinate and ( φ, χ ) as Cartesian spatial coordinates, they arethe equations of motion of a particle of unit mass moving in a potential − V ( φ, χ ). Thus, forinstance, the fact that φ goes beyond 1 at around x = 3 (which at first sight might appear8uspect) is actually perfectly reasonable, since the gradient of V ( φ, χ ) along the positive φ -axis points towards the origin beyond x = 1. In other words, the particle of the mechanicalanalogy feels a force towards the origin, giving rise to the gently curved trajectory as theparticle’s position crosses χ = 0 (see Fig. 5(b)). Also of interest, as we shall see, is the factthat in the centre of the soliton χ reaches +1 and is essentially constant while φ sweeps fromnear − -1-0.5 0 0.5 1 -8 -6 -4 -2 0 2 4 6 8 x Kink ( δ =0.01, δ =0.01, γ =0.01)(a) φχ -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 φχ (b) kink solution-1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 φχ (b) kink solution FIG. 5. (a) Graph and (b) parametric plot of solution for parameters ( δ , δ , γ ) = (0 . , . , . A search for stable kinks was undertaken over a wide range of values of δ , δ for sixvalues of γ . Where found, as a rule they look much like the one displayed in Fig. 5. Thekink energy is displayed in Fig. 6. Although of course the details vary from one graph tothe next, they share several striking features:1. The energy is almost independent of δ , and increases as a function of δ .2. For δ large (approaching its maximum value of unity) and δ (cid:28)
1, there is no solution;this region increases with γ . (The nature of this stable/unstable transition will beexplained below.)3. For δ somewhere in the neighbourhood of unity (the value depending on γ but verynearly independent of δ ), there is no solution.9 table kink energy ( γ =0.001) 0.001 0.01 0.1 δ δ E Stable kink energy ( γ =0.003) 0.001 0.01 0.1 δ δ E Stable kink energy ( γ =0.010) 0.001 0.01 0.1 δ δ E Stable kink energy ( γ =0.032) 0.001 0.01 0.1 δ δ E Stable kink energy ( γ =0.100) 0.001 0.01 0.1 δ δ E Stable kink energy ( γ =0.316) 0.001 0.01 0.1 δ δ E FIG. 6. Energy as a function of δ , δ for six values of γ . Where no solution was found (for example,the near corner of each plot), the energy was set to zero. The latter two points are best illustrated in a plot of the stability regions in the δ δ plane,for the various values of γ , shown in Fig. 7. The six lower curves indicate an instability of thekink of the type discussed earlier (see the discussion around Eqs. (6-8)). As one approachesthe stability boundary (imagine decreasing δ from above), there is an easily-overlookedchange of behaviour of φ at the centre of the soliton. Two possibilities are observed (seeFig. 8(a,b)), depending on whether or not (6) is satisfied. If it is, the slope of φ decreases to10 δ δ γ =0.001 γ =0.003 γ =0.010 γ =0.032 γ =0.100 γ =0.316 FIG. 7. (color online) Stability region (between the curves) in the δ δ plane for each value of γ considered. zero at the origin, while if it is not, φ develops nonzero flat regions just off-centre (negativeto the left, positive to the right). In both cases, φ is settling in towards the minimum (or thepair of minima) of V ( φ, false vacuum (or pair of false vacua separated by a kink with χ = 1 and φ interpolating between the false vacua) in its wake. In fact, each of the lowerlines in Fig. 7 is somewhat blurry because it is not one line but two that virtually coincide.One of these lines is the analytically-calculated stability line mentioned earlier (that is tosay, (6) satisfied and (7) saturated, or alternatively (6) not satisfied and (8) saturated). Theother line is the stability line found by the numerical scan of parameter space as describedabove. Obviously the excellent agreement between the two is a pleasing confirmation thatour numerical work is behaving as expected.The six upper curves indicate an instability of a very different type. As δ increases, thepotential V ( χ ) (see Fig. 2) becomes more and more asymmetric and the barrier between χ = +1 and χ = − χ at the centre of the kink no longer reaches+1 (Fig. 8(c)). The instability, then, is towards a configuration where χ no longer encirclesthe less-imposing potential barrier near φ = χ = 0, after which it is energetically preferablefor χ to reduce to − x , leaving the true vacuum in the centre of the soliton, which11 x Kink near boundary, ( δ , δ , γ )=(.0014,.00015,.100)(a) φχ -1-0.5 0 0.5 1-10 -5 0 5 10 x Kink near boundary, ( δ , δ , γ )=(.800,.118,.100)(b) φχ -1-0.5 0 0.5 1 -3 -2 -1 0 1 2 3 x Kink near boundary, ( δ , δ , γ )=(.800,.814,.100)(c) φχ FIG. 8. Kinks near the stability boundary. (a) Lower boundary with inequality (6) satisfied. (b)Lower boundary with inequality (6) not satisfied. (c) Upper boundary. expands rapidly, converting false vacuum to true.
IV. KINK DECAY VIA TUNNELING
As we have seen, kinks are found over a wide range of parameters. They are classicallystable, but since they are built out of the false vacuum, they (like the false vacuum itself)will decay via quantum tunneling. We will explore the decay of an isolated false kink inthis section. In principle, we should solve the Euclidean field equations for the instanton(bounce) solution; the decay rate is expressible in terms of the action of the instanton, aswe will see below.However, this is a formidable task; even for ordinary vacuum decay the action has onlybeen evaluated in a certain limit, where the false vacuum energy density is only slightlyhigher than that of the true vacuum. Then the so-called “thin-wall approximation” is valid,and the field theory problem is reduced to a single degree of freedom: the wall radius.Here we will make a similar approximation, reducing the fields to a single degree offreedom which, at least near the stability boundary, should be a reasonable approximation12o the field theory instanton. The latter has the lowest possible action, giving rise to thefastest possible decay rate; by making an approximation we will calculate an upper boundto the true instanton action which results in a lower bound to the decay rate of the kink.Since we are interested in decay to the true vacuum, it is the second type of instabilitydiscussed in the previous section which is relevant, where the amplitude of the deviation of χ from its true vacuum value χ = − φ k ( x ) , χ k ( x )), where the deformation parameter, written h ( t ), modulates the amplitude of χ k while not affecting φ k : χ k → χ h ( x, t ) ≡ h ( t )( χ k ( x ) + 1) − .χ h interpolates between the true vacuum at h = 0 and the kink at h = 1. The energy ofa static deformed configuration is U ( h ) = E [ φ, χ h ]. Direct substitution and straightforwardalgebra yields U ( h ) = Xh − Y h + Zh + (cid:90) dx (cid:18) φ (cid:48) k + V ( φ k ) (cid:19) (14)where X ≡ (cid:90) dx ( χ k + 1) φ k + γ , Y ≡ (cid:18) δ (cid:19) (cid:90) dx ( χ k + 1) φ k + γ , Z ≡ Y − X. (15)We note that X , Y and Z (which depend both explicitly and implicitly on δ , δ and γ ) arepositive; thus, the potential has a minimum at h = 1, as indeed it must.Fig. 9 shows U ( h ), for γ = 0 .
01, displayed in two ways, highlighting the effect of δ forthree values of δ in (a) and vice versa in (b). The relative unimportance of δ , already notedin Fig. 6, is readily seen in both of these. Changing γ produces only quantitative changes;thus for definiteness in the remainder of this section we will (unless stated otherwise) consider δ = γ = 0 . h = 1, from aconfiguration of the same energy with h near zero. This energy barrier must be overcome ina tunneling event. Clearly from Fig. 9(b), the energy barrier decreases as δ increases; thisis as expected since we are approaching the second type of kink instability alluded to earlier.As δ decreases, the energy barrier increases, which might be surprising given that here toowe are approaching an instability. However, this instability is created by tunneling not tothe true vacuum but rather to one of the false vacua at χ = +1 which arise for sufficiently13 U ( h ) Kink energy ( γ =0.010)(a) δ =0.48 δ =0.11 δ =0.01 δ hU ( h ) U ( h ) Kink energy ( γ =0.010)(b) δ =0.01 δ =0.001 δ =0.0001 δ hU ( h ) FIG. 9. (color online) Kink energy as a function of the deformation parameter h . small δ , in which case there is no reason to think that it will be particularly easy (in termsof the size of the energy barrier) to deform χ to the true vacuum.The recipe for calculating the amplitude for quantum tunneling, which is related to thedecay rate of the unstable state, is well-known [1]. For kink tunneling, having reducedthe two fields to a single degree of freedom h ( t ), the task is particularly straightforward.From the field theory Lagrangian density, we derive an effective Lagrangian for h by thesubstitution φ → φ k , χ → χ h ( t ) = h ( t )( χ k + 1) − . This yields L = 12 M ˙ h − U ( h )where M = (cid:90) dx ( χ k + 1) (16)and U ( h ) is as above. Thus, the parameter h can be thought of as the coordinate of aparticle of mass M in a potential U ( h ). The tunneling amplitude is related to the Euclideanaction of a particle moving in the potential U E ( h ) = − ( U ( h ) − U (1)). The relevant solution,the bounce, written h B , starts at h = 1 at Euclidean time τ = −∞ , rolls down towards theturnaround point h + (see Fig. 10) at τ = 0, and returns to h = 1 as τ → ∞ . Using standardmethods, the action of the kink bounce is S B , k = √ M (cid:90) h + dh (cid:112) U ( h ) − U (1) . Deformed kink energy ( δ = δ = γ =0.01) hh + U ( h ) U (1) FIG. 10. Energy as function of deformation parameter h . Now, U ( h ) − U (1) is a quartic polynomial, and as can be seen from Fig. 10 it has fourreal roots: the double root h = 1, h + , and another (negative) root which we will call h − .Thus we can write U ( h ) − U (1) = X ( h − ( h − h + )( h − h − );in terms of X and Y defined earlier, h ± = Y − X ± (cid:112) Y ( Y − X )2 X . (Note that it is easy to show that
Y > X .) So the kink bounce action is S B , k = √ M X (cid:90) h + dh (1 − h ) (cid:112) ( h − h + )( h − h − ) . (17)The integral can be evaluated analytically, although the result is not terribly transparent;it is: (cid:112) (1 − h + )(1 − h − )24 (cid:40) h +2 − h + h − + 3 h − − h + + h − ) + 4 (cid:41) + ( h + − h − ) ( h + + h − − (cid:16) (cid:16)(cid:112) − h + + (cid:112) − h − (cid:17) − log( h + − h − ) (cid:17) , (18)which can also be written in terms of ˆ Y ≡ Y /X :124 (cid:114) −
32 ˆ Y (cid:16) Y −
12 ˆ Y + 16 (cid:17) + ˆ Y ( ˆ Y − Y − − ˆ Y + 2 (cid:113) − ˆ Y (cid:113) ˆ Y ( ˆ Y − (19)15his expression depends on a very complicated way on the parameters. It is displayed inFig. 11 as a function of δ for three values of δ and for γ = 0 .
01. The action goes to zero as δ approaches the upper limit of the stability zone, as expected. Also as expected (see forexample Fig. 9), the action is virtually independent of δ . (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) (cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43)(cid:43) ^ ^ ^ ^ ^ ^ ^ ^ ^ ^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ∆ S B , k ^ ∆ (cid:61) (cid:43) ∆ (cid:61) (cid:42) ∆ (cid:61) FIG. 11. Kink bounce action as a function of δ for three values of δ ; in all cases γ = 0 . The kink decay rate is given by Γ k = A k e − S B , k , (20)where the prefactor A k is fairly difficult to calculate and is beyond the scope of this paper.Nonetheless we will discuss it briefly in the next section. V. KINK-MEDIATED VACUUM DECAY?
In order to determine if and under what circumstances kinks have a significant effect onfalse vacuum decay, we must take into consideration three factors: the kink decay rate (20),the kink density, and the ordinary vacuum decay rate. Let us examine the latter factor. Wesuppose that prior to tunneling the universe is in one of the false vacua ( φ, χ ) = ( ± , − φ = +1. The tunneling will reach the true vacuum ( φ, χ ) = (0 , − φ the potential V ( φ, χ ) is minimized at χ = − χ . Thus the bounce is determined by a φ -dependent Lagrangianobtained by substituting χ = − L φ = 12 ( ∂ µ φ ) − ( φ − δ )( φ − . Interestingly, the Lagrangian (and therefore any physical implications) depends only on δ .As discussed in [1], the bounce is the minimum-action solution of the Euclidean equationof motion (writing τ for the Euclidean time)( ∂ τ + ∂ x ) φ = ddφ V ( φ )with φ → r = √ x + τ , the bounce solution based on the false vacuum φ B , v ( r ) is thesolution of the following, where the prime denotes differentiation with respect to r : φ (cid:48)(cid:48) + 1 r φ (cid:48) = ddφ V ( φ ) , φ (cid:48) (0) = 0 , lim r →∞ φ ( r ) = 1 . If we interpret φ as the position of a particle and r as the time, this is the equation of motionof a particle moving in a potential − V ( φ ) (see Fig. 12) with a time-dependent dissipativeviscosity term; the particle starts at rest at a position to be determined and reaches φ = 1at time infinity. The initial point φ + must have positive potential energy to compensate φ + φ -V ( φ ) FIG. 12. Potential for a mechanical analogy helpful to understand the bounce ( δ = 0 . for the dissipation; a continuity argument [1] indicates that there is always a solution. Thesolution appears as a “thin wall” if δ (cid:28)
1, in which case the solution and its action can bedetermined analytically; the latter is S B , v = π δ . δ ; the analytical and nu-merical actions are displayed in Fig. 13. We see that the two agree for δ (cid:28)
1, as theymust. ∆ S B , v numericalthin (cid:45) wall FIG. 13. Vacuum bounce action as a function of δ , calculated analytically using the thin-wallapproximation (expected to be valid for δ (cid:28)
1) and numerically. The two agree in the domain ofvalidity of the thin-wall approximation.
The vacuum decay rate per unit length is given byΓ v /L = A v e − S B , v , (21)where again the prefactor is difficult to evaluate; we will discuss it briefly below.Given the decay rate of a single kink (20) and the decay rate per unit length of the vacuum,we can now make a formal statement regarding the importance of kinks for vacuum stability.Imagine that the kink density (that is, the average number of kinks per unit length) is ρ .Then in a universe of length L there are ρL kinks, and the decay rate (that is, the rate for one of the kinks to decay) is ρL Γ k = ρLA k e − S B , k . We must compare this rate with that fora bubble to nucleate in the vacuum (that is, far from any kink). This is, assuming the totallength occupied by kinks is much smaller than the total length, (Γ v /L ) L = A v e − S B , v L . Wesee there is a critical density of kinks where these two rates are equal: ρ c = A v A k e − ( S B , v − S B , k ) . This expression may look questionable because the density must have dimension L − ,whereas the right hand side appears dimensionless. However, the factors A k , A v are pro-portional to ratios of square roots of determinants, and as we will now argue, they have18ifferent dimensions. In the case of A v , it is [9] A v = (cid:18) S B , v π (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) det (cid:48) ( − ∂ τ − ∂ x + V (cid:48)(cid:48) ( φ B , v ))det( − ∂ τ − ∂ x + V (cid:48)(cid:48) ( φ ) | φ =1 ) (cid:12)(cid:12)(cid:12)(cid:12) − / (22)Let us focus on the determinant factors. These are determinants of the second variationof the Euclidean action about the solution (bounce in the numerator, false vacuum in thedenominator). The denominator is straightforward since it is essentially the product ofeigenvalues of a Klein-Gordon operator, whose eigenfunctions are plane waves. The numer-ator is more complicated since φ B , v depends on x, τ . Thus the eigenfunctions are no longerplane waves. Furthermore, since the location of the bounce can be anywhere in the xτ plane,the operator has two zero modes: ∂ x φ B , v and ∂ τ φ B , v . The full determinant of this operatoris therefore zero; however the prime in (22) serves to remove these two zero eigenvalues fromthe determinant. Dimensionally, therefore, A v has dimension (mass) .In the case of A k , we have formally a similar expression: A k = (cid:18) S B , k π (cid:19) / (cid:12)(cid:12)(cid:12)(cid:12) det (cid:48) ( − ∂ τ + U (cid:48)(cid:48) ( h B ))det( − ∂ τ + U (cid:48)(cid:48) ( h ) | h =1 ) (cid:12)(cid:12)(cid:12)(cid:12) − / (23)where h B is the bounce configuration of the deformation variable h as described in theprevious section. One important difference is that there is now only one zero mode to removein the determinant in the numerator (corresponding to time translation of the bounce); thisis directly related to the change in power of the prefactor [9]. It also tells us that A k hasdimension (mass).There is no reason for the ratios of determinants to do anything pathological, so we canimagine that they are on the order of a fundamental mass scale of the problem (that of thescalar fields; call it m ) to the appropriate power. Thus ρ c ∼ m S B , v (cid:112) πS B , k e S B , k − S B , v . It is clear that as the parameters – and therefore the actions – vary, the dominant effect on ρ c is due to the exponential factor (indeed, this is why virtually all discussions of tunnelingignore the prefactor). Fig. 11 tells us that S B , k is essentially independent of δ over a widerange, while it depends substantially on δ . In contrast, S B , v (Fig. 13) depends only on δ .Examining these figures in detail, we see that if δ is small (much less than 1, say) while δ is of order 1, the exponential factor will be miniscule and even a very dilute presence ofkinks will have a dramatic effect on vacuum decay.19 I. CONCLUSIONS
We have examined the effect of topological solitons on vacuum decay in a toy model in1+1 dimensions with symmetry-breaking false vacua and a symmetry-restoring true vacuum.It is far from automatic that such a model would have solitons; for instance, in the simplestsuch model any configuration interpolating between the false vacua would be unstable, withtwo “half-solitons” repelling each other and leaving true vacuum in their wake. The modelwe study is not particularly realistic but it does show the existence of models with classicallystable solitons.We find that solitons can indeed have an important effect on vacuum stability. Thisis essentially because the energy barrier which makes the soliton classically stable, on theone hand, and that which makes the vacuum itself classically stable, on the other, areindependent. In particular, while maintaining a large barrier between true and false vacua(resulting in a very long-lived vacuum), the barrier stabilizing the kink can be made small.In this case, the presence of even a very dilute gas of kinks would cause the vacuum to decayrapidly via nucleation of a bubble of true vacuum in the vicinity of one of the kinks.Of course, any model with solitons in 1+1 dimensions will have domain walls in 3+1dimensions. The decay through tunneling of a domain wall in 3+1 dimensions would bevery different than that of the corresponding kink in 1+1 dimensions: the wall would de-velop a bulge somewhere and the bulge would then expand, analogous to the correspondingphenomenon with stringlike solitons [6]. This situation is currently under investigation.
ACKNOWLEDGEMENTS
This work was financially supported in part by the Natural Science and EngineeringResearch Council of Canada. [1] Sidney Coleman. Fate of the false vacuum: Semiclassical theory.
Phys. Rev. D , 15:2929–2936,May 1977.[2] Shamit Kachru, Renata Kallosh, Andrei Linde, and Sandip P. Trivedi. de Sitter vacua in stringtheory.
Phys. Rev. D , 68:046005, Aug 2003.
3] T.W.B. Kibble. Topology of cosmic domains and strings.
J. Phys. A , 9:1387, 1976.[4] Brijesh Kumar, M. B. Paranjape, and U. A. Yajnik. Fate of the false monopoles: Inducedvacuum decay.
Phys. Rev. D , 82:025022, Jul 2010.[5] Bum-Hoon Lee, Wonwoo Lee, Richard MacKenzie, M. B. Paranjape, U. A. Yajnik, and Dong-han Yeom. Tunneling decay of false vortices.
Phys. Rev. D , 88:085031, Oct 2013.[6] Bum-Hoon Lee, Wonwoo Lee, Richard MacKenzie, M. B. Paranjape, U. A. Yajnik, and Dong-han Yeom. Battle of the bulge: Decay of the thin, false cosmic string.
Phys. Rev. D , 88:105008,Nov 2013.[7] William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling.
Numericalrecipes in fortran: the art of scientific computing, second edition . Cambridge University Press,1992.[8] S. Coleman, V. Glaser, and A. Martin. Action minima among solutions to a class of euclideanscalar field equations.
Comm. Math. Phys. , 58(2):211–221, 1978.[9] Curtis G. Callan and Sidney Coleman. Fate of the false vacuum. ii. first quantum corrections.
Phys. Rev. D , 16:1762–1768, Sep 1977., 16:1762–1768, Sep 1977.