Minkowski vacua can be metastable
MMinkowski vacua can be metastable
Jaume Garriga , Benjamin Shlaer , Alexander Vilenkin . Departament de F´ısica Fonamental iInstitut de Ci`encies del Cosmos, Universitat de Barcelona,Mart´ı i Franqu`es 1, 08028 Barcelona, Spain Institute of Cosmology, Department of Physics and Astronomy,Tufts University, Medford, MA 02155, USA
Abstract
We investigate the recent suggestion that a Minkowski vacuum is either absolutely stable, or it hasa divergent decay rate and thus fails to have a locally Minkowski description. The divergence comesfrom boost integration over momenta of the vacuum bubbles. We point out that a prototypicalexample of false-vacuum decay is pair production in a uniform electric field, so if the argumentleading to the divergence is correct, it should apply to this case as well. We provide evidence thatno catastrophic vacuum instability occurs in a constant electric field, indicating that the argumentcannot be right. Instead, we argue that the boost integration that leads to the divergence isunnecessary: when all possible fluctuations of the vacuum bubble are included, the quantum stateof the bubble is invariant under Lorentz boosts. a r X i v : . [ h e p - t h ] S e p . INTRODUCTION In an interesting recent paper [1], Dvali suggested a simple argument indicating that aMinkowski vacuum cannot be metastable. The argument can be summarized as follows.Suppose we have a theory admitting a zero-energy metastable (false) Minkowski vacuumwhich can tunnel to a negative-energy, anti-de Sitter (AdS) true vacuum through bubble nu-cleation. The tunneling process conserves energy, so the critical Coleman-De Luccia (CDL)bubble has zero energy, with the negative energy in the bubble interior compensated by thepositive energy of the bubble wall [2]. A bubble whose size is larger than critical and whoseboundary is at rest will have negative total energy. This state of negative energy may thenbe created from the false vacuum together with a compensating blob of ordinary positive-energy particles, perhaps with some separation between the positive and the negative energyblobs.One might think that the negative-energy blob is like a ghost, that is, a negative-energypoint-particle. If the energy-momentum vector of the bubble is p µ , the positive energyblob must have energy-momentum vector − p µ , and the Lorentz invariance of the Minkowskivacuum implies that the rate for this process per unit spacetime volume can depend onlyon p . Given a positive plus negative blob at nucleation, we can obtain other possibleconfigurations by applying arbitrary Lorentz boosts. This will give configurations witharbitrarily high positive and negative energies, adding up to zero. Since p does not changeunder boosts, the nucleation rate for all these configurations should be the same. Then thetotal vacuum decay rate, obtained by integrating over the boosts, is infinite. The conclusionis that the concept of a metastable Minkowski vacuum is inconsistent; such a vacuum simplycannot exist. We shall refer to this as the “boost integration argument.”As noted in [1], a possible caveat is that a metastable Minkowski vacuum can exist only fora finite time and thus cannot be Poincar´e-invariant. This results in a cutoff for the divergentboost integral. (This issue will be discussed in more detail in subsequent publications [3].)However, according to the standard analysis, the bubble nucleation rate can be arbitrarilylow, resulting in an arbitrarily high boost cutoff and an arbitrarily large enhancement of thevacuum decay rate. This would still call for a drastic revision of the standard approach. Inwhat follows we shall use the term “metastable Minkowski vacuum” with the understandingthat this vacuum is only locally Poincar´e-invariant.2t is important to note that for theories without gravity the argument in [1] does not relyin any way on the existence of negative energies. Given a theory with a metastable zero-energy vacuum, decaying to a true negative-energy vacuum, we can always add a constantto the Lagrangian of the theory, so that both vacua become positive-energy. This of coursedoes not change the dynamics, and the argument should still go through, except now theenergy-momentum vector of the bubble p µ should be understood as the difference betweenthe energy momentum vectors of the false vacuum with and without a bubble. If the phasespace integral in the decay rate was divergent in the original theory, it will still be divergent.Hence, if the argument is correct, then all non-gravitational theories with metastable vacuashould be inconsistent.The boost integration argument is reminiscent of the problem encountered in the earlytreatment of vacuum decay [4] by Voloshin, Kobzarev, and Okun (VKO). If we think ofa CDL bubble as materializing at rest in some frame of reference, this nucleation eventis not boost-invariant. VKO argued that the nucleation rate should therefore include anintegration over boosts, resulting in a divergent total rate. (They suggested a cutoff relatedto the radius of the universe.) The key insight by Coleman [5] was that the bubble evolutionafter nucleation is Lorentz invariant. Hence, integration over boosts is unnecessary, since itamounts to multiple counting of the same final states.Coleman’s result is supported by an independent calculation of the vacuum decay ratein terms of the imaginary part of the vacuum energy density [6]. In this approach, the issueof boost integration does not arise. Another piece of evidence comes from the calculationsof pair creation rate in a constant electric field [7] – a process closely analogous to vacuumdecay. Starting with the work of Heisenberg and Euler [8] and Schwinger [9], pair creationhas been studied by a variety of methods. Results obtained using different methods are infull agreement and do not include any divergent boost integrals.Here we are going to argue that Dvali’s conundrum can be resolved in a similar way.An expanding bubble can emit particles, so the asymptotic states at future infinity doinclude positive-energy particles with a compensating negative-energy bubble. In fact, thetotal number of emitted particles and their total energy are going to be infinite in thislimit. Suppose the bubble were formed together with a photon, as in the boost integrationargument. This process cannot be distinguished from bubble formation with a subsequentemission of the photon at about the same time. Now, if we apply a Lorentz boost, the3mission point of the photon will move to the future (or to the past). Large Lorentz boostswill move it to the very distant future. Integration over boosts would thus account for allpossible photon emission points. Now, the probability that the bubble will emit a singlephoton in its entire history is zero. As we said, the total number of emitted photons is infinite,and all final states having nonzero probability correspond to a certain average number ofphotons emitted per unit area of the bubble wall per unit proper time. Such states remainunchanged under Lorentz boosts.Another way of looking at this is to note that the bubble wall has the geometry of a(2+1)-dimensional de Sitter space. Any quantum field interacting with the field(s) of thebubble will be in a de Sitter invariant quantum state around the wall [10]. This quantumstate accounts for all particles that will ever be emitted by the wall. Since the de Sitterspace is characterized by an intrinsic temperature, things more substantial than particleswill sometimes also pop out. There will be blobs of matter, occasional Boltzmann brains,etc. This will be happening all along the domain wall in a de Sitter invariant fashion.Now, Lorentz boosts in the bulk spacetime correspond to de Sitter transformations on theworldvolume, and since the quantum state is de Sitter invariant, it does not change underboosts. Hence, including configurations of negative and positive energies does not break theLorentz invariance of the final state, but is necessarily part of the quantum description of aLorentz invariant bubble.To substantiate this view, we shall investigate how the boost integration argument playsout in the case of pair production in electric field. We shall first focus on the (1+1)-dimensional case, which is a prototypical example of vacuum decay. Starting with a briefoverview in the next section, we shall analyze some simple massless (1+1)D models in SectionIII. Massive particle production in (3+1)D is discussed in Section IV, and our conclusionsare summarized in Section V. II. PAIR PRODUCTION IN AN ELECTRIC FIELD
A constant electric field in (1+1) dimensions, F µν = E(cid:15) µν with E = const, is invariantunder Poincar´e transformations. Its energy-momentum tensor has the vacuum form, T νµ = E δ νµ . If the field is coupled to particles of electric charge e and mass m , particle-antiparticlepairs will be spontaneously produced with a critical separation d c = 2 m/eE , so that the rest4nergy of the particles, 2 m is compensated by the potential energy, − eEd c . (For definitenesswe assume that E and e are both positive.) The electric field in the space between the pairis reduced to the value ( E − e ), and the vacuum energy is reduced accordingly. The createdparticles play the role of bubble walls in this model. The rate of pair creation per unit lengthis (see, e.g., Ref. [11] and references therein) Γ = eE π exp (cid:18) − πm eE (cid:19) . (1)If the particles are coupled to themselves or to another particle species, then pairs sep-arated by a distance d > d c can be produced. Such pairs have negative energy, and theirformation must be accompanied by production of compensating positive-energy particles.It is clear that the boost integration argument is precisely concerned with this situation. Ifthe argument is correct, the pair production rate should diverge when particle interactionsare included. We shall now examine whether or not such a divergence actually occurs. III. MASSLESS PAIRS
Massless QED in (1+1) dimensions, also known as the Schwinger model, is described bythe action S = (cid:90) d x (cid:20) ¯ ψγ µ ( i∂ µ + eA µ ) ψ − F µν F µν (cid:21) . (2)It is equivalent to the bosonic theory [13] S = (cid:90) d x (cid:20)
12 ( ∂φ ) + gA µ (cid:15) µν ∂ ν φ − F µν F µν (cid:21) , (3)where g = e/ √ π .In terms of the boson, the electric current is given by j µ = e ¯ ψγ µ ψ = g(cid:15) µν ∂ µ φ. (4)The number density of charge carriers n is related to the electric current j = − g ˙ φ by j = 2 e n . It follows that the pair production rate is given byΓ = 12 e djdt = − ¨ φ √ π . (5) This expression is exact in external field (corresponding to the limit e (cid:28) E ) . Backreaction can easily betaken into account in the instanton approximation (see, e.g., [12]). In this case, the result is given by (1),but with E replaced by ¯ E = E − e/
2. The instanton approximation is valid in the limit m (cid:29) eE . φ is (cid:3) φ = − gE, (6)where E is the electric field. For a constant external field, φ develops an expectation value (cid:104) φ (cid:105) = φ ( t ) = − gEt , (7)and Eq. (5) gives Γ = e π E. (8)As one might expect, there is no exponential suppression of pair production for masslessfermions.We note that the massless pair production rate (8) is in agreement with the zero-masslimit of Eq. (1). (For a detailed discussion of this limit, see [11].) It was first derived byWitten [14] in the context of superconducting cosmic strings.In this paper we will not be interested in the effect of back-reaction of the created pairson the electric field. However, if needed, this effect can be easily taken into account [11, 15].From Maxwell’s equation ˙ E = − j = g ˙ φ, (9)and using (6) we have E = E + gφ (10)and (cid:3) φ + g φ = − gE , (11)where E = const . The solution of this equation for a spatially homogeneous φ with theinitial condition φ = ˙ φ = 0 at t = 0, is φ ( t ) = − g − E [1 − cos( gt )] , (12)and Eq. (10) gives E = E cos( gt ) . (13)This shows that the screening of the electric field by the produced pairs occurs on a timescale t ∼ g − . At smaller times t (cid:28) g − , Eq. (12) is well approximated by Eq. (7).The above results are valid to all orders in e , and show that catastrophic vacuum decaydoes not occur in the Schwinger model. One may be concerned that this model is somewhat6pecial. Note that there are no propagating photons in (1+1) dimensions. Hence, if weneglect the dynamics of the electric field, the model does not include any particles thatcould compensate for the increased negative energy of the pairs. On the other hand, if wedo include a dynamical electric field, the model is still equivalent to a free bosonic theory,and hence it may be considered non-generic. In order to account for more generic situationswith propagating particles in final states, we now consider more complicated interactingtheories. A. Interactions
The Schwinger model (2) can be modified by adding self-interaction terms for the field ψ . We first consider a simple interaction of the form L I = λ j µ j µ , (14)where λ is a coupling constant. Apart from the Maxwell field, the action (2) with thisinteraction term corresponds to the massless Thirring model. Naively, the four fermioninteraction could lead to pair creation processes which are not boost invariant, so one mightexpect catastrophic vacuum decay once an electric field is applied.This, however, does not happen. In the bosonized form of the model, the interactionterm (14) becomes L I = λg ( ∂φ ) , (15)and amounts to a finite renormalization of the kinetic term for φ . Thus the model isequivalent to the bosonic Schwinger model, with the replacement g → g (cid:48) = (1 + 2 λg ) − / g .If we apply a constant electric field, the electric current will grow at a constant rate, asbefore: djdt = e π eE (1 + 2 λg ) . (16)Clearly, no catastrophic vacuum decay occurs in this model.Turning now to more generic interactions, we introduce an additional scalar field χ , whichis coupled to ψ via L I = H ( χ ) (cid:15) µν ∂ µ j ν , (17)where the function H ( χ ) is assumed to be O ( χ ) at small χ and is otherwise arbitrary. Thisform of interaction is chosen so that the model can be rewritten in an equivalent bosonic7orm. We could also choose a coupling G ( χ ) j µ j µ , but since the current has an expectationvalue linearly growing with time (at least to the lowest order in the interaction), this wouldresult in a time-dependent, growing mass for χ .Using Eq. (4) to express the current in terms of the bosonic field φ , we have (cid:15) µν ∂ µ j ν = 2 g (cid:3) φ, (18)so the bosonic action takes the form S = (cid:90) d x (cid:20)
12 [( ∂φ ) + ( ∂χ ) ] − V ( χ ) + H ( χ ) (cid:3) φ − gEφ (cid:21) , (19)where we have also added a self-interaction potential for χ .To lowest order in the interactions, the expectation values of φ and χ are (cid:104) φ (cid:105) = φ ( t ) and (cid:104) χ (cid:105) = 0, with φ ( t ) from Eq. (7). Introducing ˆ φ ≡ φ − φ , the action transforms into S = (cid:90) d x (cid:20)
12 [( ∂ ˆ φ ) + ( ∂χ ) ] − V ( χ ) + H ( χ )( (cid:3) ˆ φ − gE ) (cid:21) . (20)For a constant electric field, the term proportional to gE amounts to a finite renormalizationof the mass and self-couplings of χ .We see that even though the classical solution φ ( t ) is growing with time, the model(20) describing the quantum theory on this background has a time-independent Lagrangianand shows no signs of instability. Once again, it seems clear that nothing catastrophic canhappen to the vacuum in this model. IV. MASSIVE QED
We finally consider pair production in massive QED, with photons playing the role ofthe compensating positive-energy particles. Since there are no photons in 2D, we considerthe 4-dimensional case. A constant, spatially homogeneous electric field is invariant underlongitudinal boosts, so according to the boost integration argument, the rate for processeslike Vacuum → e + e − γ, Vacuum → e + e − γγ, (21)etc., should include integration over such boosts and should therefore be divergent.The total vacuum decay rate, including all particle production processes, is related to theimaginary part of the vacuum energy density,Γ vac = 2Im ρ vac , (22)8 " FIG. 1: Diagrams contributing to the decay of an electric field. The solid lines represent theelectron propagator in the presence of the electric field, and the wavy lines represent the photonpropagator. and can be found by evaluating the diagrams shown in Fig. 1. Solid lines in these diagramsstand for electron propagators in a constant external field, and wavy lines represent photonexchange. The one-loop diagram in Fig. 1(a), which does not include any virtual photonlines, gives the Heisenberg - Euler - Schwinger (HES) result,Γ (1) = ( eE ) π exp (cid:18) − πm eE (cid:19) . (23)The higher-loop diagrams of Figs. 1(b)-(c) account for the processes (21), where pair pro-duction is accompanied by photons, as well as for the radiative corrections to the basicpair production process, Vacuum → e + e − . The diagram in Fig. 1(d) represents correlatedformation of two pairs and radiative corrections to the photon propagator. If the boostintegration argument is correct, the multi-loop diagrams should diverge.It should be noted that the pair production rate Γ is generally different from the vacuumdecay rate Γ vac defined by Eq. (22). At the one-loop level, the exact result for Γ is given byEq. (23), while Γ vac is [9] Γ vac = ( eE ) π ∞ (cid:88) n =1 n exp (cid:18) − nπm eE (cid:19) . (24)9or an illuminating discussion of this distinction, see, e.g., Ref. [11]. Here, we will beinterested in the weak field limit, E (cid:28) m /e, (25)when Γ ≈ Γ vac to very good accuracy. Hence, we will not distinguish between Γ vac and Γ inwhat follows.Higher-loop corrections to the HES formula (23) have been extensively studied in theliterature. Affleck, Alvarez and Manton (AAM) [16] derived a remarkably simple formula,which includes contributions of all multi-loop diagrams containing a single electron loop,Γ ( all loop ) = ( eE ) π exp (cid:18) − πm eE + e (cid:19) . (26)It was argued by AAM that diagrams with more than one electron loop are subdominant inthe weak field limit (25); then Eq. (26) is the full multi-loop pair creation rate in this limit.Comparing it to (23), we see that the correction term is definitely finite and small, as onemight naively expect.The AAM derivation was not rigorous, as it relied on the steepest descent evaluationof the path integral for the QED effective action and on a somewhat cavalier treatment ofmass renormalization. However, direct calculations [17–19] of the two-loop contribution inFig. 1(b) also yielded a finite result, which is in agreement with (26).It may be instructive to consider how the infinite boost integration plays out in the QEDcontext. In the semiclassical approximation, electrons (and positrons) move along hyperbolictrajectories with a constant proper acceleration, a = eE/m . The characteristic energy of theemitted photons in the rest frame of the electron is ω ∼ a , and in the weak field limit (25)we have ω (cid:28) m . This indicates that photon emission has negligible effect on the motion ofthe electrons, and one can simply consider radiation from a charged particle moving alonga fixed classical trajectory.For a charge in hyperbolic motion, photons will be emitted with the same characteristicenergy in the rest frame of the charge, but will be boosted to arbitrarily high energiesin the observer’s frame. However, the transverse momentum of the photons, in the planeperpendicular to the electric field, will not be boosted. Hence, the appropriate quantity to The distinction is particularly important in the massless (1+1)-dimensional case, when Γ vac becomesformally divergent, while the pair production rate (8) remains finite [11]. p ⊥ .This calculation has been done, in the lowest order of perturbation theory, in Ref. [20] forphotons and in Ref. [21] for the emission of massless spin-zero particles by a scalar source.The resulting probability is proportional to the divergent integral over the particle’s propertime (which can also be thought of as an integral over boosts, since a boost causes a shiftin the proper time of emission). How do we interpret this divergence?Of course, the probability of photon emission cannot be greater than one, so the diver-gence indicates a breakdown of the perturbation theory. The origin of the problem can beunderstood if we consider a situation where the electric field is turned on only for a finiteperiod of time T . We shall assume this period to be short enough, so that the probabilityof emitting more than one photon in time T is small. (Note that we can treat the electriccharge e as a free parameter. For small values of e , photons will be emitted very rarely, sothe time T can be very long.) In this case there will be no divergence. The probability ofphoton emission will be proportional to the proper time τ spent by the charge in the electricfield. This simply reflects the fact that there is a fixed photon emission probability per unitproper time.What happens when we increase the time T ? For very large values of T , the probabilityof emitting a single photon becomes very small. Photon emission events along the particle’strajectory can be regarded as independent, so the probability for emitting n photons will begiven by the Poisson distribution P n = ¯ n n n ! e − ¯ n , (27)where ¯ n is the average number of the emitted photons (which is proportional to τ ). In thelimit T → ∞ , we have ¯ n → ∞ , so the probability of emitting any finite number of photonsgoes to zero. On the other hand, ¯ n is proportional to e , so a formal perturbative expansiongives a divergent result for the emission of a single quantum, P ( pert )1 = ¯ n + O ( e ). The situation here is similar to the divergence encountered in emission of soft photons, known as theinfrared catastrophe in QED; see, e.g., [22]. . CONCLUSIONS In this paper we have analyzed Dvali’s boost integration argument indicating that aMinkowski vacuum can either be absolutely stable or cannot exist at all. Any decay channelallowing negative-energy bubbles would supposedly result in a divergent vacuum decay rate.This would imply good news for inhabitants of a Minkowski vacuum: if your vacuum survivedfor any finite time, it is guaranteed to endure forever. Since our vacuum is close to Minkowski,by continuity we should then expect it to be extremely long lived.Here, we argued that if the boost integration argument is correct, it should also apply topair production of charged particles in a constant electric field. The pair production rateshould then become infinite when processes like (21), where the pairs are produced togetherwith some other particles, are included.We do not know of any calculations that would exhibit such a divergence. On the otherhand, here we have presented evidence that such divergences do not occur in a class ofmassless (1+1)-dimensional theories and in (3+1)-dimensional (massive) QED.The divergence found in the boost integration argument arises due to the integration overthe momenta of the vacuum decay products. A perturbative calculation of the probabilityfor nucleating a negative energy bubble accompanied by a single positive energy blob ofmatter in the final state contains a divergent integral over the momentum of the blob.However, as we discussed in Section IV, this divergence only indicates a breakdown of per-turbation theory. Physically, the reason is the following. Fluctuations such as the emissionof a blob by an expanding bubble occur with a constant probability per unit wall area perunit proper time. The boost integral in the naive perturbative calculation only changes thevalue of proper time on the worldsheet at which a blob is emitted. If the bubble expands fora finite time, i.e., short enough that the emission of one blob is unlikely, then perturbationtheory applies, and the boost integral will correctly account for the fact that the probabilityof emission is linear in the proper time interval available. However, it is clear that this inte-gral is unrelated to the probability of nucleating the bubble itself. As the bubble expands,the probability of having the bubble with the blob increases, and the probability of havingthe bubble without the blob decreases. Once the probability for having more than one blobis significant, perturbation theory has broken down.In a Lorentz invariant situation, where the bubble expands for an infinite amount of12ime, the correct answer for the probability of having just one blob (or any finite number ofthem) in the final state is actually zero. In other words, the final state consists of a Lorentzinvariant distribution of infinitely many blobs, generated as the bubble expands into thefalse vacuum. But of course this does not imply an infinite rate of bubble nucleation.A metastable Minkowski vacuum cannot exist forever and thus cannot be truly Poincar´einvariant. However, such vacua can be formed as bubbles in the inflationary multiverse andwill decay by producing AdS bubbles. Our analysis indicates that the corresponding decayrate can be calculated using the standard Coleman-de Luccia method and does not involveany divergent boost integrations. Acknowledgements
We are grateful for comments by and discussions with Gia Dvali, Gregory Gabadadze,Slava Mukhanov, and Tanmay Vachaspati. This work was supported in part by grantsDURSI 2009 SGR 168, MEC FPA 2007-66665-C02 and CPAN CSD2007-00042 Consolider-Ingenio 2010 (JG) and by the National Science Foundation grant PHY-0855447 (BS andAV). [1] G. Dvali, “Safety of Minkowski Vacuum,” arXiv:1107.0956 [hep-th].[2] S. R. Coleman and F. De Luccia, “Gravitational Effects on and of Vacuum Decay,” Phys.Rev. D , 3305 (1980).[3] G. Gabadadze and A. Vilenkin, work in progress; G. Dvali, C. Gomez and V. Mukhanov, workin progress.[4] M. B. Voloshin, I. Y. Kobzarev, and L. B. Okun “Bubbles in Metastable Vacuum,” Sov. J.Nucl. Phys. , 644 (1975) [Yad. Fiz. , 1229 (1974)]. The bubble nucleation rate can be affected by the initial conditions, representing how the metastablevacuum was set up. However, this is a transient effect, which we expect to be important on a timescaleof the order of the instanton size. Note that in the massless limit of QED, when the size of the instantonis zero, the pair creation rate depends only on the local value of the electric field, regardless of the initialconditions.
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