Rest frame of bubble nucleation - arXiv

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Rest frame of bubble nucleation

arXiv:1304.6681v2 [hep-th] 5 Sep 2013

Jaume Garriga,a,c Sugumi Kanno,b Takahiro Tanakac a Departament

de F´ısica Fonamental i Institut de Ci`encies del Cosmos, Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain b Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA c Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

Abstract. Vacuum bubbles nucleate at rest with a certain critical size and subsequently expand. But what selects the rest frame of nucleation? This question has been recently addressed in [1] in the context of Schwinger pair production in 1+1 dimensions, by using a model detector in order to probe the nucleated pairs. The analysis in [1] showed that, for a constant external electric field, the adiabatic “in” vacuum of charged particles is Lorentz invariant, and in this case pairs tend to nucleate preferentially at rest with respect to the detector. Here, we sharpen this picture by showing that the typical relative velocity between −1/3 the frame of nucleation and that of the detector is at most of order ∆v ∼ SE  1. Here, SE  1 is the action of the instanton describing pair creation. The bound ∆v coincides with the minimum uncertainty in the velocity of a non-relativistic charged particle embedded in a constant electric field. A velocity of order ∆v is reached after a time interval of order −1/3 ∆t ∼ SE r0  r0 past the turning point in the semiclassical trajectory, where r0 is the size of the instanton. If the interaction takes place in the vicinity of the turning point, the semiclassical description of collision does not apply. Nonetheless, we find that even in this case there is still a strong asymmetry in the momentum transferred from the nucleated particles to the detector, in the direction of expansion after the turning point. We conclude that the correlation between the rest frame of nucleation and that of the detector is exceedingly sharp.

Contents 1 Introduction

1

2 Timescales

5

3 Pair production

7

4 Charged detector (3-point interaction)

10

5 Neutral detector (4-point interaction) 5.1 Kinematics 5.2 Asymmetry

15 16 17

6 Summary and conclusions

19

A Asymptotic expansion of Dν (Z)

20

B Transition amplitude for the neutral detector B.1 Probing the “in” vacuum state of the φ field B.2 Probing a one particle state for the φ field

23 23 25

1

Introduction

False vacuum decay at zero temperature was first described in the context of field theory in a classic paper by Voloshin, Kobzarev and Okun [2]. These authors proposed that, as a result of quantum tunneling, true vacuum bubbles would nucleate at a certain rate per unit volume. The bubbles would nucleate at rest, with a critical size r0 , and then expand with constant proper acceleration r0−1 , due to the pressure difference between the false and the true vacuum. On the other hand, it was also pointed out in [2] that, due to the Lorentz invariance of the false vacuum, bubbles would not have any preferred reference frame in which to nucleate at rest. At first, this observation seemed to suggest that the total decay rate per unit volume should include an integral over the Lorentz group, in order to account for all possible rest frames of nucleation. Of course, such integral would be divergent, or at least cut-off dependent if a regulator was imposed. However, soon after this idea was proposed, Coleman [3] developed the instanton approach to false vacuum decay, showing that the decay rate is finite – and that integration over the Lorentz group does not play any role in calculating it 1 . Coleman’s argument, however, did not shed much light on a related and somewhat mysterious aspect of bubble nucleation, which requires clarification. If it is true that a long lived metastable false vacuum is approximately Lorentz invariant, what is it that determines the rest frame in which the critical bubble nucleates? 1 The reason is that instanton in this case is O(4) invariant, and its analytic continuation (describing the bubble after nucleation) is invariant with respect to Lorentz boosts. Because of that, the final state in the asymptotic future is independent of the rest frame in which the critical bubble nucleates. Integrating over the Lorentz group would then amount to overcounting the final states. For a recent discussion of related issues, see [4–7].

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BUBBLE   Figure 1. Diagram illustrating the nucleation ofWALL   a bubble of true vacuum. The system is prepared

to be in the metastable false vacuum at some early time t = −t0 . A bubble of size r0 nucleates at rest at time t = 0, and subsequently expands. The semiclassical picture of an expanding bubble is valid for t � τq . r   r  0  

event occurs at the time which we here denote as t = 0, forming a critical bubble, which then expands according to (1.1). Nucleation is a quantum process (indicated in Fig. 1 by a wavy line), and therefore we should not say that it takes place exactly at the turning point hypersurface t = 0. FALSE   The Vsemiclassical picture of an expanding bubble is valid only after a ACUUM   certain time τq has ellapsed, t � τq > 0. (1.2) -­‐  t  0   of the instanton describing vacuum decay is Ini3al   condi3on   ypersurface   As we shall see, in thehregime where the t=   action we have � r0 . In of this sense, it stillvacuum. quite accurate to say that nucleation takes Figure 1. Diagramlarge, illustrating theτqnucleation a bubble of is true The system is prepared placefalse on vacuum a t ≈ const. hypersurface. to be in the metastable at some early time t = −t0 . A bubble of size r0 nucleates at rest Since only the expanding branch of (1.1)ofisan relevant, thebubble actualisprocess at time t = 0, and subsequently expands. The semiclassical picture expanding valid of vacuum decay is not at all Lorentz invariant. In a reference frame S � which moves at high velocity for t & τq . v = tanh φv with respect to the rest frame of nucleation, things look rather different (see Fig. 2). Observers in the new frame will see a piece of the bubble appear at time t� ≈ −r0 sinh φv , moving very fast in the boost direction. The bubble will come to a halt at t� = 0. At It should be that noted that process of bubble nucleation cannotshape: itself be Lorentzofintime thethe bubble wall presents a somewhat awkward it consists one hemisphere variant. To illustrate this point, we may consider the limit when the bubble walls arecannot thin be described attached to a rather fuzzy interface between false and true vacuum, which compared to the bubble size. In this the trajectory of anot vacuum bubble radius semiclassically. The case, nucleation process does conclude untilofthe time rt� as ≈ +r0 sinh φv , a function of timewhen t is given by [2, 3] (see Fig. wall 1) wraps the full sphere. the semiclassical bubble Given its impact on the kinematics of bubble formation, it is in hindsight surprising that 2 r2 − = r02 . has been neglected for over three decades. (1.1) investigation of the frame of tnucleation Recently, however, this issue was addressed in Ref. [1], by considering a model detector which interacts Eq. (1.1) is invariant under Lorentzbubbles. boosts, Several and describes bubble were whichanticipated contracts to from with the nucleaded plausibe ascenarios possibly emerge infinite size to thefrom minimum size r(A) thenbe expands However, only the with the rest 0 , and this study: It could that theagain frametoofinfinity. nucleation simply coincides expanding part offrame this of trajectory is relevant forwords, describing vacum will decay. Indeed,forming we areat rest in her the detectors. In other each detector see bubbles assuming that theown system is prepared in the false vacuum in the far past, remaining in thathistory is not rest frame. (B) It is conceivable that the contracting part of the bubble completely off andAcan be at least partially observed. (C) Awhich third possibility state for a very long period cut of time. tunneling event occurs at the time we here is that the

denote as t = 0, forming a critical bubble, which then expands according to (1.1). Nucleation is a quantum process (indicated in Fig. 1 by a wavy line), and therefore we cannot say that it takes place exactly at the turning point hypersurface t = 0. Rather, the semiclassical picture of an expanding bubble is valid only after a certain time τq has ellapsed, –2–

t & τq > 0.

(1.2)

As we shall see, in the regime where the action of the instanton describing vacuum decay is large, we have τq  r0 . In this sense, it is still quite accurate to say that nucleation takes place on a t ≈ const. hypersurface. Since only the expanding branch of (1.1) is relevant, the actual process of vacuum decay is not at all Lorentz invariant. In a reference frame S 0 which moves at high velocity v = tanh φv with respect to the rest frame of nucleation, things look rather different (see Fig. 2). Observers in the new frame will see a piece of the bubble appear at time t0 ≈ −r0 sinh φv , moving very fast in the boost direction. The bubble then comes to a halt at t0 = 0. At that time the bubble wall presents a somewhat awkward shape: it consists of one hemisphere

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n in a reference frame S � which moves at high velocity v = tanh φv with and the normalization condition, tion. Observers in the boosted frame will see a piece of the bubble appear r’   ving very fast opposite to the boost direction. At � t� = 0 r  0  the bubble wall � ∗ interface between ∗ ape: it consists of one hemisphere attached to a fuzzy i φk (t)∂ − φk (t)∂t φk (t) t φk (t)lasts ch cannot be described semiclassically. The process of formation φv , when the semiclassical bubble wall closes into a full sphere.

= 1.

FALSE  VACUUM   of Eq.(2.9) can be expressed in Linearly independent solutions cylinder functions, eation process does not conclude until the time t� ≈ +r0 sinh φv , ∗ bble wall wraps the full sphere. φ± k (z) ∝ Dν [±(1 − i)z] , 0

2. Bubble nucleation a reference frame S which moves he kinematics ofFigure bubble formation, it isin in hindsight surprising thatat high velocity v = tanh φv with respect to the frame of nucleation. Observers in the boosted frame will see a piece of the bubble appear of where nucleation has been over three decades. Recently, at time t0 ≈neglected −r0 sinh φv , for moving very fast to the boost direction. At t0 = 0 the bubble wall � opposite � has a somewhat awkward shape: it consists of one hemisphere attached to a fuzzy interface between √ detector which dressed in Ref. [1], by considering a model k interacts 1 + iλ false and true vacuum, which cannot be described semiclassically. The process of formation lasts z ≡ eE t + , ν = − , λ es. Several plausibe scenarios were anticipated to possibly emerge untile the time t0 ≈ +r0 sinh φv , when the semiclassical bubble wall closes into a full sphere. eE 2 uld be that the frame of nucleation simply coincides with the rest other words, each detector will see bubbles forming at rest in her The general is interface given by afalselinear superposition φ± conceivable that the solution contracting part of thebetween bubble history not which cannot be of attached to a rather fuzzy and trueisvacuum, described k . We 0 semiclassically. The (C) nucleation process does not is conclude until the time t ≈ +r0 sinh φv , n be at least partially observed. A third possibility that the when the semiclassical bubble wall wraps the full sphere. uenced by how the decaying false vacuum was set up. π ∗ 1 ν Given its impact on the kinematics of bubble formation, it isisomewhat that 4 ∗ [−(1 − i)z] , φ (z) = e Dνsurprising [1] was done byinvestigation using pairof production of a charged scalar field φ k the frame of nucleation has been neglected1/4 for over three decades. Recently, (2eE) d as a model forhowever, bubblethis nucleation in (1+1)in dimensions. To takewhich was adopted there was issue was addressed Ref. [1]. The strategy consider model detector which the interacts with the bubbles, the hypersurfaceto of initiala conditions (where quantum state for so as to reveal their state of motion at the time of interaction. Several plausibe scenarios as taken to the infinite past t → −∞. In this idealized situation, were anticipated to possibly emerge from this study: (A) It could be that the frame of nucleation simply coincides with ” vacuum is Lorentz invariant2 , and therefore cannot determine a the rest frame of the detectors. In other words, each detector will see bubbles forming at rest

her own rest frame. (B) Itfield, is conceivable contracting lly from the boostininvariance of the electric since the that “in” the vacuum must part of the bubble history is not completely cut off and can be at least partially observed. (C) A third possibility is that d in a specific gauge. the frame of nucleation is influenced by how the decaying false vacuum was set up. The investigation in [1] was done by using pair production of a charged scalar field φ in a constant electric field as a model for bubble nucleation in (1+1) dimensions. To take care of option (C), the initial hypersurface where the quantum state for the field φ is prepared was taken to the infinite past t → −∞. In this idealized situation, it was shown that the “in” –3– vacuum is Lorentz invariant2 , and therefore cannot determine a preferred frame of nucleation. To investigate (A) and (B), the model detector was chosen to be a particle of a second charged field ψ (see Fig. 3), interacting with φ through the vertex

–6–

g(φψ ∗ χ + h.c.), 2

(1.3)

This does not follow trivially from the boost invariance of the electric field, since the “in” vacuum must be defined in a given frame and in a specific gauge.

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that,conditions out of the infinite bath of created particles, only those whose momentum relativeattoleast in the limit wh areψ(t, removed far inχ(t, the past. this paper weLagrangian shall consider tree-level laboratory forinteraction studying the bubble nucleation process, charged scalar x) andsufficiently a neutral scalar x), withIn the 3 We shall †c †sense, that the number density of vanishes on†far some initial surface t= const. back this issue inpairs Section 4. (φ the detector is in a certain range have atochance interact it. Ininthis interactions between the pairs and will the detector. The kinematics interactions Lofintthese =with −g ψχ +is ψsuch φχ) ,Inmost Lint −gcons (φ conditions aretoremoved sufficiently the past. this paper we= shall back to this issue in Section 4. † † Despite somewhat unphysical we findof that the that, in outthe of the infinite bath of created particles, those whose momentum to The kinematics interactions between the pairs and therelative detector. these intera Lint = −g their (φ ψχ + ψ only φχ) ,its (1.4)properties, particles bath are invisible, and infinite density is irrelevant. Despite its somewhat unphysical properties, we find that the in-vacuum a u the is in awe certain range athat, chance to with it. In most out of interact the infinite bath of created particles, onlyin those whose momentu laboratory for studying the bubble nucleation process, at least inisby the where gwill isdetector ahave coupling constant. This where model gthis ishas asense, coupling been studied constant. This detail model ha M To detector be specific, model the bystudying introducing two additional scalar fields: agreat laboratory for the bubble nucleation process, at least in the limit where the in where g is a coupling constant. This model has been studied in great detail by Massar and particles in the bath are invisible, and their infinite density is irrelevant. the detector is in a certain range will have a chance to interact with it. In th conditions are removed sufficiently far in past. In shall this paper we sh Parentani [13] et interaction alParentani [14], [13] used and it the to byinvestigate Gabriel et the alconsider [14], Unruh who eff charged scalar a neutral scalar χ(t, x), with the Lagrangian conditions areby far inwho the past. In this we treeParentani [13]ψ(t, andx) by and Gabriel et al whoand used itremoved toGabriel Unruh effect for an To be specific, we model the[14], detector by introducing two the additional scalar fields: a paper particles ininvestigate thesufficiently bath are invisible, and their infinite density is irrelevant. interactions between the pairs and the detector. The kinematics of the accelerated detector. accelerated detector. pairs we andmodel the detector. The kinematics of these accelerated detector. charged scalar ψ(t, x) and a neutralinteractions scalar χ(t, between x), the Lagrangian be the specific, the detector by introducing twointeractions additional issc † Towith † interaction Lthat, = −g (φ ψχ + ψ , (1.4) out of the infinite bath of created particles, only those whose We start with a charged ψ-particle inthat, the initial state. Itφχ) interacts with φ-antiparticles out of the infinite bath of created particles, only those whose momentum relativ int We start with a charged ψ-particle We in the start initial with state. a charged It interacts ψ-particle with in φ-an them charged scalar ψ(t, x) and a neutral scalar χ(t, x), with the interaction Lagran † is †. certain of the pairs via4 ψφ∗ → χ and thus L has lifetime calculate the momentum ∗+ 4have thea=finite detector inτψψaχ range will have a will chance to ∗τinteract with it.calculate Inhas thisa sense, m −g4(φ ψχ φχ) ,shall (1.4) the detector isWe in athus certain range aψ→ chance to interact with int oft  the pairs via ψφ → and has of the a finite pairs lifetime via ψφ . We χ and shall thus finite the m t   †Massar † and distribution of χ-particles in the final state andinuse it bath to deduce the momentum distribution particles the arestudied invisible, and their infinite density isψirrelevant. where g is a coupling constant. This model has been ininvisible, great detail by L = −g (φ ψχ + φχ) , int particles in the bath are and their infinite density is irrelevan of χ-particles in consider theinfinal distribution and of use χ-particles itand to deduce inadditional the final momentum state and d of the created φ-pairs. Todistribution achieve this goal, we have to thestate interaction (1.4) where g[13] is aand coupling constant. model haswill been studied great detail by Unruh Massar To who be specific, we the detector by introducing two an scalar field Parentani by Gabriel etThis al [14], used it tomodel investigate the effect for To be specific, we model the detector by introducing two addi 2 2 of the created φ-pairs. To achieve of this the goal, created we will φ-pairs. have To to consider achieve this the goal, intera with a time-dependent = g exp(−t /T socoupling thata the detector isThis turned Parentani [13] and by coupling, Gabriel etg(t) alcharged [14], who used to investigate thescalar Unruh effect forhas an where g it), isx) a constant. model been studied in great detail b scalar ψ(t, and neutral χ(t, x), on with the interaction Lagrangian accelerated detector. 2of/T 2χ(t, charged scalar ψ(t, and neutral scalar x), the interaction for a finite period of time with ∆t ∼ T We shall also briefly discuss the case when the role accelerated detector. a . time-dependent coupling, witha= agettime-dependent exp(−t ), so coupling, that the g(t) detector = Unruh g exp is Parentani [13] andx) byg(t) Gabriel al [14], who used it towith investigate the † † Wedetector start with a charged ψ-particle in the initial state. It interacts with φ-antiparticles the is played by the neutral χ-particle. All our results point in the direction of L = −g (φ ψχ + ψ φχ) , We start with a charged ψ-particle in the initial state. It∼interacts φ-antiparticles intawith accelerated detector. for a finite period of time ∆t T . for We finite shall also period briefly of time discuss ∆t ∼ the T . case We when shall ∗the † 4 ψφ ∗ → option (A) that frame of thus pair nucleation is the of−g of the via–4via ψφ → χχand hasabubble) afinite finite lifetime τψdetermined . shall We shallbycalculate the Lframe =in (φ ψχ state. + ψ † φχ) , ofpairs the pairs and thus (and has lifetime τψ . with We calculate the momentum int We start a charged ψ-particle themomentum initial It interacts with φ the detector is played by the neutral the χ-particle. detector is played All our by results the neutral point inχ-parti the d where g is a coupling constant. This model has been studied in great detail by Massar the detector. 4 ∗ distribution of χ-particlesin in the the final state and use it to deduce the momentum distribution distribution of χ-particles final state and use it to deduce the momentum distribution of the x   pairs via ψφ → χ and thus has a finitex   lifetime τψ . We shall calculate th The paper is organized as follows. In Section 2, we introduce the in-vacuum state and option (A) – that the frame of pair option (and bubble) (A) – that nucleation the frame is determined of pair (and by bub th Parentani [13] and by Gabriel et al [14], who used it to investigate the Unruh effect fo of created the created φ-pairs. To To achieve achieve this goal, we will have to consider (1.4)use gwe is awill coupling This model hasitbeen studied in great distribution ofhave χ-particles inthe theinteraction final and to deduce the momentum of the φ-pairs. thiswhere goal, toconstant. consider thestate interaction (1.4) review the Schwinger pair coupling, production using the method of 2Bogoliubov coefficients. InisSection 2 /T accelerated detector. the detector. the detector. with a time-dependent g(t) = g exp(−t ), so that the detector turned on 2 2 of the created φ-pairs. To achieve this goal, we will have to consider the int Parentani and by Gabriel ettwo-point al [14], iswho used on it to investigate th with3,awetime-dependent coupling, g(t) invariant. = gstart exp(−t ), soψ-particle that the turned the in-vacuum Lorentz In [13] Section 4, we calculate the with a/T charged indetector the the initial state. interacts with φ-antipart 2It 2 ), so that for a show finitethat period of time ∆tis∼ T . paper WeWe shall also briefly discuss the case when role of introduce with a time-dependent coupling, g(t) = g exp(−t /T the detector The is organized as follows. The In Section paper 2, is organized we as follows. the in-vacuum In Sec accelerated detector. 4 ∗ in this vacuum and∆t discuss the expectation value ofχthe current, pointing out the τψthe for afunction finite period of time ∼ Tof . the We shall also briefly discuss the case when role pairs ψφ → and has a∼finite lifetime . We shallofcalculate thecase momen the detector is played byreview the neutral χ-particle. All our results point the direction forvia a finite period of thus time ∆tin T . Schwinger We shall of also briefly discusscoefficients. the wh E  the density E   Schwinger pair production review using the the method of pair Bogoliubov production using the pathologiesisassociated with the infinite ofWe particles inwith the in-vacuum. We argue that start a charged ψ-particle in the initial state. It interac distribution of χ-particles in the final state and use it to deduce the momentum distribu the detector played by the neutral χ-particle. All our results point in the direction of option (A) – that the frame of pair (and bubble) nucleation is determined by the χ-particle. frame of All our results point in th the detector is played by the neutral 4 To ∗ these can be remedied by 3, considering breaking initial conditions. In show Section 5, will we we that the in-vacuum is Lorentz 3, invariant. that the In Section in-vacuum 4, calculate inva the ofLorentz the φ-pairs. achieve this we have to consider interaction ofcreated the pairs via ψφ → and thus has abubble) finite lifetime τisψisthe .Lorentz We shall cal option – that the frame of show pair (and bubble) nucleation isχ we determined by the frame of we the(A) detector. option (A) – that the frame of goal, pair (and nucleation determined by set up our detector model. function The final with distribution of χ-particles and the observed momentum 2 2 a time-dependent coupling, g(t) = g exp(−t /T ), so that the detector is turne in this vacuum and discuss function the expectation in this vacuum value of and the discuss current, the pointi expe distribution of χ-particles in the final state and use it to deduce the m The paper is organized as follows. In Section 2, we introduce the in-vacuum state and the detector. the detector. distribution of the pairs are calculated in Sections 6period and 7. ofFinally, our∼ results are shall summarized (a)   for aassociated finite time ∆t T . We also briefly discuss the case when the ro (b)   review the Schwinger pair production using the method of Bogoliubov coefficients. In Section pathologies with the infinite pathologies density associated of particles with in the the in-vacuum. infinite density We Thethe paper is introduce organized asthe follows. In 2, we introduce in-vacu of In theSection created φ-pairs. To presented achieve this goal,state we will have totheconsider Thediscussed paper in is Section organized as follows. 2,calculations we in-vacuum and and 8. Some technical details are in theSection the detector isofplayed by the neutral χ-particle. All our results point in thecoefficien directio 2 2 3, we show that the in-vacuum is Lorentz invariant. In Section 4, we calculate the two-point review the Schwinger pair production using the method of Bogoliubov can beusing remedied by considering these Lorentz can be breaking remedied by considering conditions. Lorentz Inthe Sec Appendices. with a that time-dependent coupling, g(t) = g exp(−t /T ), so by that review the Schwinger pair these production the method of Bogoliubov coefficients. Ininitial Section option (A) the frame of pair (andis bubble) nucleation is determined the fram function this“detector” vacuum and discuss expectation value of current, pointing out the (1.3), 3, –wemodel. show that thethe in-vacuum Lorentz invariant. In Section 4, we calculate Figure 3. inThe particle ψ the can anihilate the φThe antiparticle through the vertex proset up our detector final set distribution up our detector of χ-particles model. The and final the observed distributi m for a finite period of time ∆t ∼ T . We shall also briefly discuss the 3, wepathologies show thatassociated the in-vacuum Lorentz invariant. In in Section 4, we calculate the two-point the detector. with the is infinite density of are particles the in-vacuum. We the argue that function inatthis vacuum and discuss expectation valuetoof the current, poi ducing a neutral particle χ. Kinematically, there most two opportunities for the interaction 2 Schwinger pair production distribution of the pairs calculated distribution inthe Sections ofpointing 6density the and pairs 7.ofintroduce Finally, are the calculated our results in Sectio are su the detector is played by neutral χ-particle. Allthe results poi The paper is are organized asoffollows. In Section 2,5, we in-vacuum state function incanthis vacuum and discuss the expectation value the current, out these be remedied by considering Lorentz initial conditions. Section we pathologies associated with infinite inour the in-vacuum. W take place along the hyperbolic trajectory of φ.breaking If the pair nucleates inthe theIn rest frame of theparticles detector review the Schwinger pair production using the method of Bogoliubov coefficients. In Sec option (A) – that the frame of pair (and bubble) nucleation is determ and discussed in Section 8. Some technical and discussed details in of Section the calculations 8. Some technical are presen set consider up ψ, our The final of and theconsidering observed momentum pathologies associated with the infinite ofχ-particles particles in charged the in-vacuum. Wehyperbola, argue that theseplace can be remedied by breaking initial conditions. Ind We ainconstant electric field in distribution (1will +density 1)-dimensions. particles that particle asdetector (a), model. then the collision take inSpin-zero the expanding branch ofLorentz the and 3, we show that the detector in-vacuum isAppendices. Lorentz invariant. In Section we calculate the two-p distribution of thethe pairs calculated Sections 6breaking and 7. Finally, our results are summarized the detector. Appendices. set up our model. The final distribution χ-particles and the observe are being produced byare the Schwinger effect are described ainitial complex scalar field φ(t,In these can bepair remedied by considering Lorentz Section 5,4,we the momentum of decay product χin will be negative p by < 0. On theconditions. other hand, ifx); the pairofof nucleates function in this vacuum and discuss the expectation value the current, pointing out and discussed in Section 8. Some technical details of the calculations are presented in the distribution of the pairs are calculated in Sections 6 and 7.2, Finally, our resultsthe ar The is organized as follows. Inmomentum Section we introduce setinupa 3our detector The distribution ofNonetheless, χ-particles and the observed frame which ismodel. highly with respect to paper the detector, then is ausing good the The number density of pairsboosted is not afinal sharply defined quantity. it can be there defined by the chance pathologies associated with the infinite density of particles inthat thethe in-vacuum. Weare argue Appendices. and discussed in Section 8. Some technical details of calculations prec instantaneous Hamiltonian diagonalization. Such states (with vanishing particle number at t = 0) have been review the Schwinger pair production using the method of Bogoliubov interaction takes place in the contracting branch of the hyperbola. This will lead to a χ particle with distribution of the pairs are2calculated Sections 6production andby 7. considering Finally, our results are summarized theseincan be remedied Lorentz breaking initial conditions. In Section 5 Schwinger pair 2 defined Schwinger Appendices. discussed in Refs. [11] and [15]. The initial number of particles could also be rigorously by considering pair production momentum p > 0, as in (b). Therefore, a strong asymmetry in the momentum distribution of Section 3, we show that the in-vacuum is Lorentz invariant. In we c andpositive discussed in Section 8. Some technical details of calculations are presented inand the set up our detector model. The final distribution of χ-particles the observed4,momen an electric field which vanishes at t → −∞ and then turned on at some time in the past. An abrupt turn-on 2 Schwinger pair production 2 detector encounters the products towards negative momenta can be interpreted as evidence that the at an instant of time was considered in Refs. [15–17], and an adiabatic turn-on with E(t) ∝ 1/ cosh (t/T ) distribution of the pairs are calculated in Sections 6 and 7. Finally, our results are summa function in this vacuum and discuss the expectation value of the Appendices. We consider a constant electricpair fieldWe in consider (1 + 1)-dimensions. a constant electric Spin-zero field charged in (1 +cur par 1) 2 the Schwinger production wasexpanding discussed in Ref. [11]. much more often the branch than contracting branch, consistent with option (A) in and discussed in Section 8. Some technical details of the calculations are presented in 4 consider a constant electric field in pathologies associated with the infinite density ofby particles in the in-va We (1produced + 1)-dimensions. Spin-zero charged particles that are being by the Schwinger are being effect pair are produced described by the a complex Schwinger scalar effect fi Here and below we use asterisk to denotepair antiparticles. the introduction. Appendices. are being pair produced by the Schwingerthese effect are described by a complex scalar x); breaking Wecan consider a constant electric field field in (1φ(t, + 1)-dimensions. Spin-zero charged be remedied by considering Lorentz initial conditi 2 Schwinger pair production 3 3 The number density ofdetector pairs is not a sharply The defined number quantity. density Nonetheless, of described pairs not itaacan sharply beand defined define are being pair produced by the Schwinger effect are by complex scala 3 set up our model. finalby distribution of is χ-particles the The number density of pairs is not a sharply defined quantity. Nonetheless, it canThe be defined using the 2 Schwinger pair production instantaneous Hamiltonian diagonalization. instantaneous Such states Hamiltonian (with vanishing diagonalization. particle number Such at state t= 3 instantaneous Hamiltonian diagonalization. Such states (with vanishing particle number at t = 0) have been of the pairs are calculated Sections 6 and 7. Finally, our The number density of pairs is not charged a sharplyin defined quantity. Nonetheless, it can be defir We consider a constant electric field indistribution (1[11] + 1)-dimensions. Spin-zero particles discussed inThrough Refs. and [15]. The discussed number of in particles Refs. [11] could and [15]. alsothat be The rigorously initial number definedof by p discussed [11] and [15]. The initial number particles could also initial be rigorously bySuch considering where g isin Refs. a coupling constant. this interaction, the ψdefined particle can anihilate the instantaneous Hamiltonian diagonalization. states (with vanishing particle number at t – 6of –discussed and in Section 8. Some technical details of the calculation We consider a constant electric field in (1 + 1)-dimensions. Spin-zero charged particles are being pairfield produced byanthe Schwinger effect are described by a complex scalar field φ(t, x); which vanishes at t → −∞ an and electric then field turned which on at vanishes some time at t → in the −∞ past. and then An ab tu an electric which vanishes atelectric t→ −∞field and then turned on at some time in the past. An abrupt turn-on in particle Refs. [11] and [15]. Thekinematics initial number2of particles could also be rigorously define φ at antiparticle in the pair, producing a discussed neutral χ.Refs. The ofadiabatic this process are being pair produced by the are described by aturn-on complex scalar field∝φ( an instant of time was considered in Refs. [15–17], and an adiabatic turn-on E(t) ∝ 1/ cosh ) considered Appendices. at an instant of time was considered in Schwinger at with an of and time an(t/T was in with Refs. E(t) [15–17], 1 an electric field which vanishes at [15–17], tinstant →effect −∞ and then turned on at some time in the past. An 3 was discussed in Ref.φ[11]. number density of pairs is not a sharply defined quantity. Nonetheless, it can be defined by using the isThe such that the antiparticle has at most two chances, along its hyperbolic trajectory, of at an instant of time was considered in Refs. [15–17], and an adiabatic turn-on with E(t) was discussed was discussed in Ref. [11]. 3 in Ref. [11]. 4 The number density ofvanishing pairs is notparticle a sharply defined quantity. Nonetheless, can be defined by usin Here and below we use asterisk antiparticles. 4 to denote Such instantaneous Hamiltonian diagonalization. states (with number at t of = 0) have beenitto was discussed in Ref.center interacting with the detector. reason is that the of4 Here mass energy the collision HereThe and below we use asterisk to[11]. denote antiparticles. and below we use asterisk denote antipart instantaneous 4Hamiltonian diagonalization. Such states (with vanishing particle number at t = 0) have Here and below we use asterisk to denote antiparticles. 2 Schwinger pair production discussed in Refs. [11] and [15]. The initial number of particles could also be rigorously defined by considering ∗ discussed in Refs. [11] and [15]. The initial number of particles could also be rigorously defined by consid between φ and the detector particle ψ hasturned to be on equal to the rest mass mAn χ of the product. an electric field which vanishes at t → −∞ and then at some in the an electric field which vanishes at t →time −∞ and thenpast. turned onabrupt at some turn-on time in the past. An abrupt tu 2 This selects the magnitude of in theat momentum of was φan tointurn-on ψ, but we have two at an instant of time was considered Refs. [15–17], and adiabatic with E(t) ∝ 1/ cosh (t/Tfor ) with E(t) an We instant of time considered Refs. [15–17], and adiabatic turn-on ∝ 1/ cosh2 consider arelative constant electric field in an (1 + options 1)-dimensions. Spin-zero was discussed in Ref. [11]. sign. in Ref. [11]. wasits discussed being produced by the Schwinger effect are described by a comp – pair 4 are – 4 and6below we use asterisk to denote antiparticles. Here If and below use asterisk to denote antiparticles. the pairwe nucleates in the restHere frame of the detector, as in the left panel –of6 Fig. 3, then – 3 – a6 sharply – the The number density of pairs is not defined quantity. Nonetheless, – it 6 c the collision will take place in the expanding branch of the hyperbola, and momentum instantaneous Hamiltonian diagonalization. Such states (with vanishing particle nu of the decay product χ will be negative p < 0. On the other hand, if the pair nucleates in a discussed in Refs. [11] and [15]. The initial number of particles could also be rigorou frame which is highly boosted with respect to field the which detector, thenatthere is6 –aand good an electric vanishes t→– −∞ thenchance turned that on at some time in the the interaction may take place in the contracting branch of the hyperbola. This at an – instant and lead an adiabatic turn-on w 6 – of time was considered in Refs. [15–17], will was discussed in Ref. [11].right panel in Fig. 3. The results to a χ particle with positive momentum p > 0, as in the 4 Here and below we asterisk to denote of Ref.[1] showed a strong asymmetry in the momentum use distribution of the antiparticles. decay products, towards negative momenta. This was interpreted as evidence that the detector encounters the expanding branch much more often than the contracting branch, consistent with option (A). It was therefore concluded that nucleation takes place preferentially in the rest frame of the detector. –6– The analysis of Ref. [1] was restricted to a range of parameter space where the momentum of the φ antiparticle at the time of collision is highly relativistic. In this sense, the frame of nucleation was probed rather imprecisely, with a tolerance much larger than the size of

–4–

the critical bubble. This still left some room for option (B) to be realized. The purpose of this paper is to sharpen the discussion given in Ref. [1], by tightening the precision to which the rest frame of bubble nucleation can be determined. We start in Section 2 by investigating the timescales which are relevant to bubble nucleation. These include the time τq of quantum fuzziness after nucleation, the size of the critical bubble r0 , the timescale τnuc that it takes for a small quantum fluctuation in the false vacuum to tunnel into a critical bubble, the time t0 ellapsed since the false vacuum was prepared, and the average lifetime τvac of the false vacuum. The hierarchy between these scales as a function of the instanton action SE will be clarified. In Section 3 we briefly review pair production by an electric field in (1+1)-dimensions. In Subsections 4.1 and 4.2 we consider the response of the model detector (1.3), extending the analysis of Ref. [1] to the range of parameter space where the momentum of the φ antiparticle at the moment of collision is only mildly relativistic or non-relativistic. This will allow us to probe the frame of nucleation on scales smaller than r0 and down to the minimum resolution scale ∆t ∼ τq . Beyond that, the semiclassical picture breaks down. In Subsection 4.3 we use a model based on a four-point interaction, where the detector particle is not charged. In this case, we are able to show that even if the interaction takes place within the time interval ∆t ∼ τq in the vicinity of the turning point, there is still a very strong asymmetry in the momentum transferred from the nucleated particles to the detector, in the direction of expansion after the turning point. Our conclusions are summarized in Section 5.

2

Timescales

In the thin wall limit, the action for a vacuum bubble is given by Z Z p S = − M (r) 1 − r˙ 2 dt +  V (r) dt.

(2.1)

Ω rD−1 , D−1

(2.2)

Here, M (r) is the mass of the domain wall of radius r,  is the difference in energy density between the false and the true vacuum, and V (r) is the volume inside the bubble. In D spacetime dimensions, we have M (r) = σ Ω rD−2 ,

V (r) =

where σ is the wall tension, and Ω is the surface area of the unit (D − 2)-sphere. A vacuum bubble has zero energy (relative to the false vacuum configuration without the bubble), and hence it satisfies p p2r + M 2 −  V (r) = 0. (2.3) Here the radial momentum is given by

pr = γM r, ˙

(2.4)

with γ = (1 − r˙ 2 )−1/2 . Up to temporal shifts, the solution of (2.3) is given by Eq. (1.1), with r0 =

(D − 1)σ . 

(2.5)

Eq. (2.5) gives the size of the critical bubble, which is also the acceleration timescale, which is needed for the bubble wall to become relativistic τacc ∼ r0 .

–5–

(2.6)

To relate r0 to other relevant scales, we introduce the dimensionless combination λ≡

σD ∼ SE  1. D−1

(2.7)

Up to a numerical coefficient of order one, this coincides with the instanton action SE , which can be calculated by substituting the Euclidean version of (1.1) into the Euclidean version of (2.1). The semiclassical approach to tunneling is only valid when the last strong inequality is satisfied. This will make λ a useful expansion parameter. Eq. (2.3) can be thought of as the classical limit of a Schr¨odinger equation for the wave function of the bubble. This was used in [2] to estimate the probability that a bubble of vanishing size may tunnel to the critical size. The estimate in [2] is in qualitative agreement with the decay rate per unit volume Γ ∼ e−SE which is obtained by the instanton methods [3]. The lifetime τvac for the false vacuum is therefore exponential in λ, ln τvac ∝ λ.

(2.8)

As mentioned in the introduction, the semiclassical description is not adequate at the classical turning point r = r0 , where the radial momentum pr vanishes. However, it does become very accurate for t  τq , where τq can be determined from the condition [r(τq ) − r0 ] pr (τq ) ∼ 1. This leads to τq ∼ λ−1/3 r0  r0 . (2.9) Hence, the intrinsic uncertainty in the time of nucleation is much smaller than the bubble size (as illustrated in Figs. 1 and 2). The uncertainty τq should not be confused with the timescale τnuc that it takes for the critical bubble to form out of a tiny false vacuum fluctuation. In the semiclassical picture, it is somewhat unclear how to characterize τnuc . Causality suggests that this time should at least be as large as the size of the critical bubble, τnuc & r0 .

(2.10)

The scale τnuc will be characterized more precisely in the next Section, where we consider pair production as a model for vacuum decay in (1+1) dimensions. In summary, we are led to the following hierarchy of scales, τq  r0 . τnuc . t0  τvac .

(2.11)

where the strong inequalities are parametrically enhanced the larger is λ ∼ SE . In (2.11), t0 is the time ellapsed from the hypersurface of initial conditions to the moment when the bubble nucleates. This should be at least marginally larger than τnuc . We impose the last strong inequality on t0 because our primary interest is in bubbles which nucleate in isolation, without interference from collisions with other bubbles. Bubbles that form at t0  τvac are very likely to remain isolated for a period of time which is much larger than all other scales involved in the problem. Finally, it should be noted that τvac grows exponentially with λ, whereas for all other scales the dependence is power law. In this sense, we can practically think of τvac as infinite. In the (1+1)-dimensional example which we consider in the following Section, the electric field producing the pairs is external and the pairs do not interact with each other. In this

–6–

idealized situation, the scale τvac does not play any role, and the relevant parameters in the above discussion are given by σ = m,

 = eE,

λ=

m2 , eE

(2.12)

where m is the mass of the charged particles that nucleate, e is their charge and E is the electric field.

3

Pair production

Here we briefly review Schwinger pair production, which serves as a model for bubble nucleation in (1+1) dimensions. The main advantage is that the nucleated pairs are treated fully quantum mechanically. In the following Sections, we shall consider the detection of the nucleated pairs. Our conventions will follow those of Ref. [1]. Consider a charged scalar field φ, coupled to an external electric field with gauge potential Aµ = δµx A(t). (3.1) By spatial homogeneity, we can separate φ(x) into Fourier modes, which satisfy the equation. φ¨k + wk2 φk = 0,

(3.2)

wk2 = m2 + (k − eA)2 .

(3.3)

where Here, m is the mass of the field φ and e is its charge. For A(t) = −Et, we have a constant electric field E, and (3.2) becomes φ00k + (λ + z 2 )φk = 0, where

√

(3.4)

m2 , (3.5) eE and primes indicate derivatives with respect to z. Eq. (3.4) has normalized positive frequency solutions √ −iπ/4 1 i π4 ν ∗ ∗ [− φk = e D 2e z], (3.6) ν (2eE)1/4 z≡

eE(t + k/eE),

λ≡

where

1 + iλ . (3.7) 2 From the asymptotic expansion of the parabolic cylinder function at large negative z, we have  √ ν ∗ i 2 1 φk ≈ − 2z e2z , for − z  |ν| . (3.8) (2eE)1/4 ν=−

Therefore, leaving aside an irrelevant phase, we have φk ≈ (2wk )−1/2 e−i

Rt

wk dt0

(t → −∞).

(3.9)

The “in” positive frequency mode φk can be expressed as out∗ φk = αk φout , k + β k φk

–7–

(3.10)

5 Conclusions and outlook

1

1

1

Introduction

7

For A(t) = −Et, we have a constant electric field E

Introduction

Introduction

�t

tc −mφ −arg(Z) vs−+ vs+zC2 )φ e−ik φ��k +12(λ

T � τq

=wk0(t

αk �t 1 − + −i wk (t� tc −mφ −arg(Z) tc −mβφk −arg(Z) 2 vs vs C e where As shown originally by Voloshin, Kobzarev and Okun [2],E, a metastable vacuum α As by Voloshin For A(t) = −Et, we have a constant electric field and originally (3.1)false becomes k shown √ in field theory can decay by quantum tunneling. The process occurs locally, by nucleation in field can+decay by quantum k theory z β≡ eE(t k/eE), λ of true vacuum bubbles of a critical size��r0 . In the semiclassical picture, a critical bubble is 2of true As0, shown originally Voloshin, vacuum ofbya critical s z )φacceleration k = initially at rest, and then expands withφconstant proper r0−1 . bubbles Lorentz invariance k + (λ + in field theory can quantum initially at rest, anddecay then expands of the false vacuum, that derivatives bubbles will notwith have any preferred frame and however, primesindicates indicate respect torestby z. Eq. wit (3. of true vacuum bubbles of a critical si thevacuum, total ratehowever, of decay per thethat false indicate where in which to nucleate. This observation seems to suggest [2] solutions 2 √ over the Lorentz unit volume should include an integral group,atto inrest, order to account for all with m 1and initially then expands in which nucleate. This π ∗observatio z ≡ eE(t + k/eE), λ ≡ , i ν possible frames. Such integral would of course be divergent (or= at least cut-off dependent ifindicate a∗ [−(1 of the false vacuum, however, φ e 4 D unit should include an eE νintegr kvolume 1/4 regulator is imposed). Nonetheless, soon after this ideain was proposed, Coleman [3] developed (2eE) which to nucleate. observation possible frames. Such This integral would o and primes indicate derivatives with respect to z. Eq. (3.3) has normalized a instanton method which made it clear that the rate is finite, and that integration overpositive the unit volume should include an integra is imposed). Nonetheless, so 1. Lorentz group does not play any role in calculating it regulator where solutions possible frames. Such integral would o which made it cle Coleman’s arguments, however, do 1not shedi πmuch light on amethod related and somewhat ∗a instanton 1 + iλ ν φnucleation, e 4(3.4).regulator DThe i)z], is−imposed). Nonetheless, soo ν ∗ [−(1 k = ν necessary =not− mysterious of bubble aodinger missing which seems an .role Lorentz group does play Figureaspect 4. The effective potential in the Schr¨ scattering which determines the forany 1/4 Eq. ingredient (2eE) 2 Bogolubov coefficients overbarrier, theitreflection βk isColeman’s exponentially suppressed. method which made it clea adequate description of isthe process.so If is truecoefficient that aa instanton long lived metastable false vacuum arguments, however, Lorentz invariant, what is it that determines the aspect restdoes frame in whichany the role i Lorentz group play where is approximately mysterious ofnot bubble nucleatio From the asymptotic expansion of the parabolic cylinde 1+ iλ Coleman’s critical bubble nucleates? arguments, out (z) = (2eE)−1/4 e−i π4 ν D [(1 + i)z] adequate of the however, process. ν = − .bubbledescription is the positive “out” are mode, which where φ k The question is best illustratedν in the limit when the frequency walls thin compared ∗ � � 2+∞. of bubble nucleatio √ Lorentz ν invariant, the right hand side of (3.9) at late times, t →mysterious is approximately w i 2 1bubble aspect to the behaves bubbleassize (see Fig. 1). The trajectory of a vacuum of radius r as a function z The number density of “out” particles in the “in” vacuum per unit momentum space 2 description of large the φk ≈ adequate 2 |z| at e process. critical bubble nucleates? From of the asymptotic of the parabolic cylinder function z,, we hI time t is given by expansion [2, 3] 1/4 interval is given by (2eE) invariant, w dn The questionLorentz is best illustrated r2 − t21 = r202 . is approximately (1.1) ∗ � � = |β | . (3.11) k √ icritical bubble size nucleates? 1 dk 2π ν to Fig. z 2 thea bubble Eq. (1.1) is invariant under Lorentz boosts, and describes bubble which contracts fromThe tr 2 φ ≈ 2 |z| e , for z �(see − . 1). 2 ),|ν| k Therefore, have Eq. (3.4) can be thought of aswe a1/4 Schr¨ odinger equation with potential Vef f = −(λ + zbest andillustrated i The question is is given by [2, 3] �again infinitethesize (at t → −∞)(2eE) to the minimum size r (atoft time = 0),t and then expands to� t determination of the Bogolubov coefficients αk 0and βk amounts to solving a scattering −1/2 −i w to the bubble size (see Fig. 1). k dtThe tr ≈ (2w (t in that this potential Fig.case 4). is Note that k entersφ Eq. (3.4) only through theecombi1 k its k )continuation Theproblem reason is instanton (see in this O(4) invariant, and Lorentzian (describing of time t is given by [2, 3] Therefore, we have nation k + eEt. This means that the scattering near z = 0 occurs approximately at the time the bubble after nucleation) is invariant with respect to Lorentz boosts. Because of that, the final state in the 1 2

vs−

vs+ C

e−i

�t

wk (t� )dt�

�

1

The reason that instanton in this cas � t ≈ −k/(eE), but it is otherwise independent of k. Eq.t critical (3.11) thennucleates. be is rewritten as −ithe w asymptotic future is independent of the rest frame−1/2 in which bubble Integrating over the k dt can φ ≈ (2w ) e (t → −∞). k k k the For bubble afterdiscussion nucleation) is invariant Lorentz group would then amount to overcounting the final 2states. a recent of related issues, with 1 Γ = dn/dt = eE|β | /(2π), (3.12) k asymptotic future is independent see [4–6]. The reason is that instanton of in the thisrest casef

The “in” positive frequency mode φ can be expressed a

The “in” positive frequency mode φk can be expressed asafterwould out∗ Lorentz group thenout amount to overco the bubble nucleation) is invariant with iπν ∗

φ =α φ

+β φ

where the coefficient βk = e can be easily calculated from a standard linear k k relation k k k see future isand independent between the parabolic cylinder functions appearing in φkasymptotic and[4–6]. φout (see e.g. [1] references of the rest fr k out group would then amount to overcou βk φout∗ therein). This leads to Schwinger φ rate k = αk φk +Lorentz kπ ν , −i −1/4 out see [4–6]. 4 ν k eE1 –−πλ π – Γ= e , (3.13) −i 4 ν −1/4 out φk (z) = (2eE) e Dν [(1 + 2π i)z] is the positive frequency “out” mo

where φ (z) = (2eE) e D [(1 + i)z] is the po where behaves as the right hand side of (3.8) at late times, t inhand (3.13) coincides the Euclidean SE = tπλ, agreement with number the behaves as The theexponent right side ofwith (3.8) at lateaction times, →in +∞. The particles in the in vacuum per unit momentum spacedens in instanton approach, which is valid for large SE . particles in theTheinasymptotic vacuumexpansion momentum space interval is givenforby dn/dk = (3.8) for φk , andof theas analogous one for φout (3.3) canperbeunit thought a Schrodinger equation wit k , are valid |z|  |ν|. Hence the of scattering off the potential inwith Fig. 4, potential where the mixing (3.3) can be thought of timescale as a Schrodinger equation Vefoff = −(λ + the determination of Bogolubov coefficients αk and positive and negative frequency modes occurs, isthe at most of order the determination of the Bogolubov coefficients αk and βk amounts to solving a −1/2 1/2 problem(see in Fig. this potential τmixing??). . ν(eE) ∼ λ(see r0 . Fig. ??). (3.14) problem in this potential only through k only through thekcombination k + the eEt.combination k + eEt. 4 4.1

–8– Detecting the of nucleation 4 frame Detecting the frame of nucleation

3-point interaction

4.1

3-point interaction

On the other hand, this seems to be a rather crude upper bound. Mode mixing will be at its peak when the non-adiabaticity parameter w˙ k fk (t) = 2 (3.15) wk

is at its maximum. The function fk is symmetric around the time t = tk ≡ k/eE (where it vanishes) and has two peaks which are at a distance ∆t ∼ r0 away from tk . Hence, we can estimate τmixing ∼ r0 . (3.16) Identifying this with the semiclassical timescale τnuc for nucleation of a pair out of a vacuum fluctuation suggests that τnuc ∼ r0 . (3.17) This is consistent with the estimate τnuc & r0 which we mentioned in the previous subsection. Stronger evidence for (3.17) can be found by considering the case of an electric field ˆ E(t) which is turned on and off on a timescale t0 : ˆ = E(t)

E . cosh (t/t0 ) 2

(3.18)

The corresponding gauge potential is given by A(t) = −Et0 tanh(t/t0 ), and the mode equation (3.2-3.3), can in this case be solved in terms of hypergeometric functions . The Bogoliubov coefficient is given by (see e.g. [8] and references therein) h q i   cosh2 π (eEt20 )2 − 41 + sinh2 πt20 (w+ − w− ) |βk |2 = , (3.19) sinh(πt0 w+ ) sinh(πt0 w− ) p where w± = m2 + (k ∓ eEt0 )2 . For the mode with conserved momentum k = 0, the physical momentum kphys = k − eA(t) vanishes precisely at the time t = 0, which therefore corresponds to the turning point. For this mode, w+ = w− and (3.19) simplifies to h q i cosh2 π (eEt20 )2 − 41 i. h p |βk=0 |2 = (3.20) sinh2 πt0 m2 + (eEt0 )2

This expression determines, through Eq. (3.12), the rate of pair production at the time when ˆ = 0) = E. Further simplification occurs when the the electric field is at its maximum E(t timescale t0 during which the electric field is switched on is bigger than the scale of quantum fuzziness t0  τq ∼ λ−1/3 r0 . In this case, we have eEt20 > λ1/3  1.

(τq < t0 ).

(3.21)

Now, we can distinguish two cases. For t0  r0 , we have eEt0  m, which leads to |βk=0 |2 ≈ e−2πmt0 ,

(τq < t0  r0 ),

(3.22)

while for t0  r0 , we have eEt0  m, and |βk=0 |2 ≈ e−πλ .

–9–

(t0  r0 ).

(3.23)

In the first case, Eq. (3.22), the pair production rate near t = 0 is independent of the electric field. Clearly, this is unrelated to the Schwinger process, and corresponds to particle creation by the non-adiabaticity of the switching process. On the other hand, we see from Eq. (3.23) that if the electric field changes on timescales which are larger than the radius of the critical bubble, then we obtain the adiabatic Schwinger pair production rate. This strongly suggests that the timescale of nucleation of a pair from a small vacuum fluctuation satisfies τnuc . r0 . In combination with the bound τnuc & r0 which is suggested by causality, we are again lead to the estimate (3.17). In conclusion, out of the scales in (2.11), two of them are of comparable size r0 ∼ τnuc .

4

Charged detector (3-point interaction)

Let us start by reviewing the model detector which was introduced in Ref. [1]. This involves a charged ψ particle interacting with the antiparticle of a nucleated pair of the charged field φ, through the vertex g(φψ ∗ χ + h.c.), (4.1) In the gauge (3.1), the canonical momentum of charged particles is conserved. Denoting by −k the momentum of the φ antiparticle, by q the momentum of the ψ particle, and by p the momentum of the χ particle, we have p = q − k.

(4.2)

The physical momentum of charged particles changes with time, due to the acceleration caused by the electric field, qphys = q + eEt,

kphys = k + eEt.

(4.3)

From (4.2) and (4.3), we also have p = qphys − kphys . Note that the ψ particle detector is accelerated by the electric field. At early times it is highly relativistic with negative velocity, and at late times it is highly relativistic with positive velocity, following a hyperbolic trajectory. As a consequence, the momentum distribution of the resulting χ particles is actually Lorentz invariant [1]: dNχ = C

dp , wχ

where Nχ is the number of χ particles, p is their spatial momentum, q wχ = p2 + m2χ ,

(4.4)

(4.5)

and C is a constant. The form of (4.4) can be understood as follows. The interaction of ψ with nucleated pairs has a constant probability of occurring per unit proper time interval dτ along the trajectory of the ψ particle. This trajectory can also be thought of as an orbit of the Lorentz boosts, with parameter φv = aψ τ , where aψ = eE/mψ is the proper acceleration of the detector particle. If we consider two collisions which look identical in the detector’s rest frame, separated by a time delay dτ , the corresponding momenta p will be related by a boost with parameter dφv = (dp/wχ ) = aψ dτ . Hence, Eq. (4.4) simply expresses the fact that the probability of producing a χ particle per unit proper time is constant. An immediate consequence is that the distribution (4.4) cannot inform us about the frame of nucleation.

– 10 –

This difficulty is avoided by switching on the interaction (1.3) for a short period of time T . This can be implemented by substituting the coupling g in (4.1) by a time dependent coupling with a Gaussian profile [1]: 2 /T 2

g → g(t) = g e−t

.

(4.6)

If we choose q = 0,

(4.7)

then, during the time interval |t| . T , we have |qphys | . eET  mψ < mχ − mφ .

(4.8)

For any time interval T , we may choose the mass mψ of the detector to be heavy enough, so that the above strong inequality holds, and in this case, the detector is essentially at rest within the time interval T before the interaction. The last inequality in (4.8) indicates that we are choosing the mass mχ of the decay product to be larger than mψ + mφ . Otherwise the detection of φ antiparticles would be forbidden by energy conservation (at least in the absence of an electric field and of any time dependence in the coupling g). Within this setup, a significant asymmetry in the momentum distribution for the decay products χ towards negative values of p will indicate that the detector is more likely to be hit from the right, indicating that the interaction is more likely to happen in the “expanding” branch of the hyperbola, when the φ antiparticle is moving to the left. On the other hand, a more symmetric distribution would indicate that there is a significant chance of detecting the anti-particle also in the “contracting” branch (see Fig. 3) . The momentum distribution of χ particles for an interaction of finite duration T was calculated in [1], and it is given by dNχ 1 = |A(p, q = 0, T )|2 , dp 2π where A(p, q = 0, T ) ≡

Z

∞

−∞

dt g(t) ψq=0 (t) φ∗−p (t) χ∗p (t).

(4.9)

(4.10)

Here, the mode function φk is given by (3.6), and similarly for ψq (with the mass mφ replaced with mψ ). The mode function for the neutral particle is simply given by χp = (2wχ )−1/2 e−iwχ t . It follows from Eq. (4.8) that the ψ particle stays non-relativistic throughout the interaction, so we can approximate ψ0 ≈ (2mψ )−1/2 e−1/2 e−imψ t . Here, particle production of the ψ field is also neglected. Within this approximation, the amplitude (4.10) can be calculated explicitly [1], and we have   dNχ CT α(p2 + W 2 ) ≈ exp −2 |Dν (Z)|2 , (4.11) dp wχ (4 + α2 )eE where ν = −(1 + iλ)/2

(4.12)

W ≡ wχ − m ψ .

(4.13)

and

– 11 –

4.1 4.2

3-point interaction 4-point interaction

laboratory for studying the bubble nucleation process, at its least in the limit where the in Despite somewhat unphysical prop

7infinite specific, we model the detector by introducing two additional scalar fields: a that the number density of pairs vanishes that,conditions outToofbethe of created particles, only those whose momentum areψ(t, removed bath sufficiently far in the past. this paper weLagrangian shall consider tree-l laboratory forinteraction studying the bubble nucleatio charged scalar scalar χ(t, x), withIn the 7 x) and a neutral † some that the number density of pairs vanishes on in 3 and Pair back to this issue in Section 4. the detector is in a certain range will have a chance to interact with it. Ininthis interactions between the pairs theproduction detector. The kinematics of these interactions L = −g (φ ψχ +is ψs conditions are removedintsufficiently far the back to issue in ,Section 4.the †this †only branch of the hyperbola, when the φ antiparticle is moving to the left. On other hand, Despite its somewhat unphysical p that, outthe ofthe the infinite bath of created particles, those whose momentum relativ interactions between pairs and the detect L = −g (φ ψχ + ψ φχ) (1.4) particles in bath are invisible, and their infinite density is irrelevant. int 5 Conclusions and outlook 7of detecting 4 Detecting Contents the frame of infinite nucleation Despite its somewhat unphysical properties, wm a more symmetric distribution would indicate that there is a significant chance the detector is in awe certain range athat, chance to with it. In out of interact the bath of created par laboratory for studying the bubble nuclea � r0 where gwill isdetector ahave coupling constant. This where model gthis ishas asense, coup been To be specific, model the bystudying introducing two additional sca |pr |dr the anti-particle also “contracting” branch (see Fig. 3) where .particles The last inequality inconstant. (4.8) laboratory for the bubble nucleation process 4.1and 3-point interaction g isin a(0.1) coupling This model has been studied in great detail by Massar and a the bath are invisible, their infinite density is irrelevant. Γ ∼ine−ther=0 the detector is in a certain range will have conditions are removed sufficiently far in 1 the Introduction 1 Parentani [13] by et interaction alParentani [14], [13] used and it charged scalar scalar χ(t, x), with the Lagrang indicates that we are�choosing mass mχ of the decay product to be than m + mφmodel .a neutral conditions are far inwho the past. In ψ and Parentani [13]ψ(t, andx) by Gabriel et 4.2 al whoand used itremoved toGabriel Unruh effect for an 4-point interaction Tolarger be specific, we the[14], detector by introducing two the additional scalar fields particles ininvestigate thesufficiently bath are invisible, and their interactions between the pairs and the de Otherwise the detection of φ antiparticles energy conservation(at least accelerated detector. accelerated detect Contents D−2would be forbidden byaccelerated interactions between the pairs and the detector. The detector. charged scalar the Lagrangian Γtotal = Γ Ω dφv (sinh φv ) =∞ (0.2)ψ(t, x) and a neutral scalar χ(t, x), be specific, we model the detector 2 Timescales † Towith † interaction in the absence of an electric field and of any time dependence in theWe coupling g). a charged ψ-particle Lthat, = −g (φout +the ψ5It φχ) , created of infinite bath of created start with inthat, the initial state. interacts with φ-antiparticles out of the infinite bath particles, on int start with aψχ charged ψ-particle We in the start initial wit 5 We Conclusions and outlook charged scalar ψ(t, x)ofand a neutral scalar χ 4 ∗ The momentum distribution of χ particles for an interaction of finite duration T was † † of the pairs via ψφ → χ and thus L has lifetime .φχ) We shall calculate the momentum 4(φ ∗+ 4hav thea=finite detector is inτψψaχ certain range will have a will chance −g ψχ , −SEPair production the detector is in a certain range int 3 6 of the pairs via ψφ → and thus has of the a finite pairs lifetim via ψ(t #" Γ it=isAgiven e by (0.3) calculated in [1], and #" distribution of χ-particles in the final state andinuse it bath to deduce the momentum distribution particles the arestudied invisible, and their infinite d where g is a coupling constant. This model has been ininvisible, great detail by L =of −g (φ int particles in the bath aredistribution and thit of χ-particles in theinfinal and use χ-p 1 Introduction 1 Introduction of the created φ-pairs. Todistribution achieve this goal, we have to consider thestate interaction (1.4) where g[13] is (0.4) aand coupling constant. This model haswill been studied great detail by Unruh Massar To who be specific, we model the detector by intro Detecting the frame Parentani of nucleation 6to τq � r0 � t40 → ∞ dN 1 by Gabriel et al [14], used it investigate the χ 2 To be2it),To specific, wedetector model the detec 2 /T = |A(p, q = 0, T )| , with of theetg(t) created φ-pairs. achieve of this the goal, created we φ-p a time-dependent coupling, = g exp(−t isThis turned on Parentani [13] and by(4.9) Gabriel alcharged [14], who used to investigate the Unruh effect for where g isx) asocoupling constant. model scalar ψ(t, and a the neutral scalar χ(t, x),will wit 4.1 interaction 6that dp 3-point 2π detector. � 2of/T SE ≡ πλ �4.2 1 4-point interactionaccelerated (0.5) charged scalar and neutral scala period of time with ∆t ∼ T We shall also briefly discuss the case when the role Tfor �a τfinite accelerated detector. a . time-dependent coupling, witha= aTgettime-depen exp(−t q Parentani [13] andx) byg(t) Gabriel alτq[14], wh 6ψ(t, † Wedetector start with a acharged ψ-particle the initial state.point ItLinteracts φwhere the is with played by the neutral our results in = � ∞ −gdirection (φwith ψχ +of ψ We start charged ψ-particle inin the initial state. with φ-antiparti int accelerated detector. foratcametastable finiteχ-particle. period ofAllvacuum time ∆t It∼interacts T .for We athe finite shall also period br wφ + woriginally (0.6) ψ = wχ ∗the ∗ 4 ψφ ∗ [? As shown Kobzarev and Okun false (A) that frame of thus pair (and nucleation is determined by the frame of of∗−pthe via–4via ψφ → χχ],and hasabubble) afinite finite lifetime τ . We shall calculate the A(p, q 5 = 0,Conclusions T ) ≡ by Voloshin, dt g(t) (t) φ (t)option χpairs (t). pairs (4.10) andψ0outlook 6 L = of → and thus has lifetime τ . We shall calculate the moment ψ pthe int We start with a charged ψ-particle in t ψby Voloshin, Kobzarev and − As shown originally O the detector is played by the neutral the χ-particle. detector is pl A −∞ where g by is athe coupling constant. This model has been the detector. 4 ψφ ∗ the mχ − m mφ by in field theory can quantum tunneling. The occurs locally, nucleation distribution ofprocess χ-particles the final state and usepairs it toit deduce momentum distribu distribution of(0.7) χ-particles inininthe final state and use to deduce the momentum of via → χ and thus has a finit ψ �decay !" field theory can decay byGabriel quantum tunneling. The paper is organized as follows. In Section introduce the in-vacuum state and Here, the mode function φk is given by (3.6), and similarly forof ψthe (with the mass mφ reoption (A) –where that the frame of consider pair (and (A) –used that nu Parentani [13] and by etoption al who ita(p q The created φ-pairs. To achieve this goal, we will have to the interaction g2, iswe awill coupling This mod distribution ofis χ-particles in[14], thebubble) final state 1 Introduction created φ-pairs. To achieve this goal, we have toconstant. consider inter of true vacuum bubbles of aforcritical sizeof rthe In the semiclassical picture, a critical bubble 0 .review T � τq function (0.8) ofdetector. trueg(t) vacuum bubbles aso critical size r0 . InInthe the semi placed with mψ ). The mode the neutral particle is simply given bypair χp production = the Schwinger using the method of of coefficients. Section 2 /T 2Bogoliubov accelerated detector. the the detector. with a time-dependent coupling, = g exp(−t ), that the detector is turned 2 2 −1 of the created φ-pairs. To achieve this goal Parentani and by Gabriel ettwo-point al [14], with a time-dependent coupling, g(t) = gstart exp(−t ), soψ-particle that the Lorentz invariance rest, and then expands constant proper acceleration (2wχinitially )−1/2 e−iwχ tat . Further, it follows from Eq. (4.8)with that the ψ 3, particle stays non-relativistic show in-vacuum isrLorentz In [13] Section 4, we calculate the initially rest, expands with constant We with a/T charged indetector theproper initial 0 T .. at Tthat � τthe forwea−1/2 finite period of time ∆t ∼ Weinvariant. shall also briefly discuss the case the q dN with a4then time-dependent coupling, g(t) = grol The paper isand organized as follows. The Inwhen Section paper 2, ise −1/2 ψ t . Within this accelerated detector. ∗briefly throughout interaction, so wehowever, can approximate ψ0 ≈ (2m )function e e−im in this vacuum and∆t discuss the expectation value ofχthe current, pointing out the for finite period of time ∼ T . We shall also discuss the case when ψa (0.9) of thethe false vacuum, indicates that bubbles will not have any preferred rest frame of the pairs via ψφ → and thus has a finite lifetime of the false vacuum, however, indicates that bubbles wi the detector is played byreview the neutral χ-particle. All our results point in the direction for a finite period of time ∆t ∼ T . We sha dp $" $" t −m the Schwinger pair production review using the the Schwin metho approximation, the amplitudec (4.10) and, up to an irrelevant φ can be calculated explicitly pathologies associated with the infinite density of particles in the in-vacuum. We argue that We start with a charged ψ-particle distribution of χ-particles in the final state and use iti the detector is– played by the neutral Allisobservation our results point in the in to χ-particle. nucleate. This seems suggest initwhich nucleate. As This observation toKobzarev suggest [? ]frame that the total rate ofvacuum decay option (A) thatand theOkun ofwhich (and bubble) nucleation is determined bytothe fram the detector played by the neutral χ-par shown originally byseems Voloshin, [2], apair metastable false phase, is given to by [1] 4 To ∗ these can be remedied by 3, considering breaking initial conditions. In show Section 5, will we we show that the in-vacuum is Lorentz 3, we invariant. that th I eET (0.10) ofLorentz the created φ-pairs. achieve this goal, we of the pairs via ψφ → χ and thus has a option (A) – that the frame of pair (and bubble) nucleation is determined by unit volume should include an integral over the Lorent the detector. �20 �15 �10 �5 5 option (A) –for that the of pair (and 2bufi per unit volume should include integral over thedetector Lorentz group, in order to account in field theory decay tunneling. The process occurs locally, by � canan �by quantum set up our model. The final distribution ofnucleation χ-particles and theframe observed momentum 2 with a time-dependent coupling, g(t) = g exp(−t /T function in this vacuum and discuss function the expectation in this va distribution of isχ-particles in the final stat + δw] of the C vacuumα[p 2 /4detector. Ther(0.11) paper ispairs organized as picture, follows. Section 2, we introduce the in-vacuum state the detector. possible Such integral would of course be diverge p