PHYSICAL REVIEW D 100, 016011 (2019)

Vacuum decay in real and imaginary time formalisms

† Mark P. Hertzberg1,2,* and Masaki Yamada1, 1Institute of , Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA 2Department of Physics, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8551, Japan

(Received 24 April 2019; published 18 July 2019)

We analyze vacuum tunneling in in a general formalism by using the Wigner representation. In the standard instanton formalism, one usually approximates the initial false vacuum state by an eigenstate of the field operator, imposes Dirichlet boundary conditions on the initial field value, and evolves in imaginary time. This approach does not have an obvious physical interpretation. However, an alternative approach does have a physical interpretation: in quantum field theory, tunneling can happen via classical dynamics, seeded by initial quantum fluctuations in both the field and its momentum conjugate, which was recently implemented in Braden et al. [arXiv:1806.06069]. We show that the Wigner representation is a useful framework to calculate and understand the relationship between these two approaches. We find there are two, related, saddle point approximations for the path integral of the tunneling process: one corresponds to the instanton solution in imaginary time and the other one corresponds to classical dynamics from initial quantum fluctuations in real time. The classical approximation for the dynamics of the latter process is justified only in a system with many degrees of freedom, as can appear in field theory due to high occupancy of nucleated bubbles, while it is not justified in single particle , as we explain. We mention possible applications of the real time formalism, including tunneling when the instanton vanishes, or when the imaginary time contour deformation is not possible, which may occur in cosmological settings.

DOI: 10.1103/PhysRevD.100.016011

I. INTRODUCTION Schrödinger equation. However, our interest here is that of quantum field theory. In this case, a direct solution The subject of quantum mechanical tunneling is an of the Schrödinger equation is notoriously difficult, and essential topic in modern physics, with a range of appli- so approximation schemes are needed. The most famous cations, including nuclear fusion [1], diodes [2], atomic approximation method, which is analogous to the Wentzel- physics [3], quantum field theory [4], and cosmological Kramers-Brillouin approximation in nonrelativistic quan- inflation [5]. In the context of a possible landscape of tum mechanics, involves the computation of the Euclidean classically stable vacua in field theory, motivated by considerations in string theory [6], it is essential to instanton solution from one vacuum to another [10,11]. determine the quantum tunneling rate from one vacuum This leads to the well-known estimate for the decay rate per Γ ∝ e−SE S to the next. This has ramifications for the stability of our unit volume , where E is the bounce action of a current electroweak vacuum [7], as well as for the viability solution of the classical equations of motion in imaginary of inflationary models [8], and may have ramifications for time. This method is generally thought to be accurate when S the cosmological constant problem [9]. the bounce action E is large, which evidently corresponds In ordinary nonrelativistic quantum mechanics of a to exponentially suppressed decay rates. single particle, quantum tunneling can be calculated in Since the above method involves nonintuitive features, principle by a direct solution of the time dependent namely a restriction to Dirichlet boundary conditions on the field and dynamics in imaginary time, it begs the question whether there may be other formulations of the *[email protected] tunneling process. Furthermore, if one moves to more † [email protected] general settings, such as in cosmology, there may not always be the usual instanton solution, so one wonders Published by the American Physical Society under the terms of whether other formulations can be employed instead. In the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to this paper we will investigate under what circumstances an the author(s) and the published article’s title, journal citation, alternative approach to tunneling, from classical evolution and DOI. Funded by SCOAP3. of fields whose initial conditions are drawn from some

2470-0010=2019=100(1)=016011(10) 016011-1 Published by the American Physical Society MARK P. HERTZBERG and MASAKI YAMADA PHYS. REV. D 100, 016011 (2019) approximation to the initial wave function, can provide an ΓI ¼ −2ImE0; ð1Þ alternative formulation for decay.1 This work was motivated by the very interesting work of where Ref. [14]. In that work they numerically obtained a tunneling rate from a false vacuum in (1 þ 1)-dimensional ln Z E0 ¼ −limT →∞ ; ð2Þ by solving for the classical dynamics of a scalar T field starting from initial conditions generated by a Gaussian distribution. The method was to consider many and Z is defined by realizations of initial conditions and then to calculate the −HT ensemble-averaged tunneling rate. For their choice of Z ≡ hϕije jϕii: ð3Þ parameters they found that the tunneling rate was similar Here we denote the (approximate) energy eigenstate around to the one calculated by the instanton method. 2 This leads to several natural questions: (i) What is the a false vacuum as jϕii One may neglect the quantum relationship between these two approaches? One is in an fluctuation around the false vacuum and approximate the ˆ imaginary time formalism, and the other is in a real time energy eigenstate by the eigenstate for the operator ϕ.In formalism; so how, if at all, are they related? (ii) Under this case, Z can be written as what circumstances are the rates comparable to each other? Z ϕðT Þ¼ϕi (iii) Under what circumstances are these approaches valid? Z ≈ Dϕe−SE½ϕ: ð4Þ It is known that the instanton method requires the bounce ϕð0Þ¼ϕi action to be large to justify a semiclassical approximation, but what is the corresponding statement for the other real The path integral can be approximated by the contribu- time method? tion from a saddle point, which is known as the instanton In this paper, we address these questions. We will argue solution. One can also calculate the Gaussian integral for that this real time analysis from classical dynamics is not the perturbation around the instanton solution. The result is identical to, but is very closely related to, the instanton given by the well-known formula tunneling process. We will show that for simple choices of −SE½ϕ parameters, the two rates are parametrically similar. ΓI ∼ ImKe bounce ; ð5Þ However, things are more complicated for potentials with ϕ unusual features, which we will discuss, and there can be where bounce is the so-called bounce solution in an advantages to the real time formulation in special circum- “upside-down” potential, with V → −V. Therefore, the stances. We will make use of the Wigner representation as it decay rate can be calculated from the path integral with will provide a general formalism to cleanly identify these imaginary time T . two complementary approaches. We will discuss under Strictly speaking, however, Eq. (5) is not a tunneling rate what circumstances the classical dynamics is justified, from the false vacuum energy eigenstate because the explain why this would fail in single particle quantum boundary condition for the path integral Eq. (4) implies ˆ mechanics, and discuss some cosmological applications. the transition between eigenstates for the operator ϕ. The Our paper is organized as follows: In Sec. II we recap the difference between the energy eigenstate and the eigenstate standard instanton contribution to the decay. In Sec. III we for the operator ϕˆ is negligible only if the zero-point present a more general formalism, using the Wigner fluctuation around the local minimum is much smaller than representation, which allows us to describe these two the typical scale of the potential. approaches within a single framework. In Sec. IV we In quantum field theory, the number of effective degrees discuss the conditions under which the classical dynamics of freedom (d.o.f.) can be large, and hence the quantum method is applicable. In Sec. V we estimate and compare the fluctuations can accidentally overcome the potential barrier. tunneling rates. Finally, in Sec. VI we discuss our findings. This accidental arrangement and subsequent barrier pen- etration was seen in the simulations of Ref. [14]. Hence, to II. STANDARD EUCLIDEAN FORMALISM focus only on initial conditions that are eigenstates of the field operator, as the usual instanton approach does, is not Let us begin by recapping the standard approach to vacuum decay, which occurs within the confines of a 2 One may think that the imaginary part of the vacuum energy Euclidean, or imaginary time, formalism. In this approach is absent because Z defined in Eq. (3) is real. This issue has been the decay rate can be calculated from the imaginary part of investigated in detail in Refs. [15,16]. They discussed that the the vacuum energy E0 as contour of the path integral should be deformed along the steepest descent contour passing through the false vacuum. Fortunately, the resulting tunneling rate can still be calculated by Eq. (5), 1See Refs. [12,13] for a different approach using complex which is the standard formula to calculate the tunneling rate by classical trajectories. the instanton calculation.

016011-2 VACUUM DECAY IN REAL TIME AND IMAGINARY TIME … PHYS. REV. D 100, 016011 (2019) guaranteed to be the most natural choice of boundary shortly, while ϕq and πq are effectively quantum fluctua- conditions. Therefore, we would like to utilize a formalism tions. The path integral can then be written as that can accommodate general initial conditions on the fluctuations for a more complete analysis of tunneling. In ˆ hOðt1Þi the next section, we analyze tunneling within the Wigner ZZ representation as it will allow us to systematically study ¼ dϕc;0dπc;0W0ðϕc;0; πc;0Þ these different possibilities. Z

× DϕcðtÞDπcðtÞDϕqðtÞDπqðtÞOWðϕcðt1Þ; πcðt1Þ;t1Þ III. MORE GENERAL FORMALISM Z t Since vacuum decay is a time-dependent process, it is 1 _ × exp i dt½2ϕqπ_ c − 2πqϕc natural to calculate it by using a real time formalism (or 0 Schwinger-Keldysh formalism), which can describe the time evolution of observables. As we will see, this will þ HWðπc þ πq; ϕc þ ϕq;tÞ − HWðπc − πq; ϕc − ϕq;tÞ ; allow us to more systematically identify initial conditions for the decay, rather than restricting to only those that are ð11Þ useful in the standard imaginary time analysis. The time evolution of the expectation value of an where the Wigner function W0 is defined as the Weyl ˆ observable O can in principle be calculated from transform of the density matrix ρˆ in the field representation. R R It is given by Oˆ t ρˆT ei Hdt0 Oeˆ −i Hdt0 ; h ð Þi ¼ Tr½ time ð6Þ Z W ϕ ; π dϕ ρ ϕ − ϕ ; ϕ ϕ e2iπcϕq : ρˆ T 0ð c;0 c;0Þ¼ q ð c;0 q c;0 þ qÞ where is an initial density operator and time is ’ Oˆ Schwinger s time-ordered operator. The operator may ð12Þ be taken to be an order parameter of the phase transition. Instead, one may use an operator that gives zero around the Momentarily we focus on only one particular mode for false vacuum and nonzero around the true vacuum. This notational simplicity, but will generalize shortly. The equation implies that the contour for the time integral is functions HW and OW are the Hamiltonian H and observ- given by the one shown in Fig. 1. We can define two kinds able O in the Wigner representation, respectively. In of fields: forward (ϕf) and backward (ϕb) fields, depending particular, on the direction of time evolution. It is convenient to then define Z O ϕ ; π dϕ dπ O ϕ − ϕ ; π π e−2iϕqπq ; 1 Wð c cÞ¼ q q ð c q c þ qÞ ϕc ¼ ðϕf þ ϕbÞ; ð7Þ 2 ð13Þ 1 π π π ; ˆ c ¼ 2 ð f þ bÞ ð8Þ where Oðϕ; πÞ is the function obtained from the operator O by direct substitution ϕˆ → ϕ and πˆ → π. For related work, 1 ϕ ϕ − ϕ ; see Refs. [17,18]. q ¼ 2 ð f bÞ ð9Þ 1 A. Classical approximation πq ¼ ðπf − πbÞ; ð10Þ 2 Now we shall rewrite the above path integral by using some approximations. We will discuss the meaning and where π is the canonical conjugates of the field ϕ. Here ϕc π justification of these assumptions in the next section. and c are effectively classical fields, as we will explain If the quantum fluctuations are much smaller than the classical quantities, we can approximate HW as

Forward HWðπc þ πq; ϕc þ ϕq;tÞ − HWðπc − πq; ϕc − ϕq;tÞ

∂HWðπc; ϕc;tÞ ∂HWðπc; ϕc;tÞ ≃ 2πq þ 2ϕq : ð14Þ Backward ∂πc ∂ϕc

FIG. 1. Integration contour for the time variable in the real time Then we can perform the integrations over ϕq and πq, formalism. which give delta functions of the form

016011-3 MARK P. HERTZBERG and MASAKI YAMADA PHYS. REV. D 100, 016011 (2019) dϕ ∂H dπ ∂H O θ ϕ−ϕ c W c W This can be realized by, e.g., taking ¼ ð Þ δ − δ þ : ð15Þ ϕ dt ∂πc dt ∂ϕc with being the field value at the other side of the potential barrier. Then we can count the number of nucleated bubbles This means that indeed fϕc; πcg obey the classical equation per unit space as an exponential function of time. The of motion. The integrals over the fields are determined by tunneling rate is the coefficient of the time variable at the the delta functions, and the result is given by exponent [14]. Z Oˆ ≃ dϕ dπ W ϕ ; π O ϕ ; π ;t : h i c;0 c;0 0ð c;0 c;0Þ Wð c c Þjclassical B. Relation to the instanton calculation The same result can be obtained by the saddle point ð16Þ approximation for Eq. (11). In the classical limit, the path integral can be approximated by saddle points of the If the initial quantum fluctuations are sufficiently small exponent of the integrand. Varying it with respect to ϕc, at the false vacuum, we can approximate the potential by a ϕq, πc, and πq, and eliminating πc and πq, we obtain quadratic form; we will return to discuss under what conditions this approximation is valid. We denote the mass ̈ 0 0 2ϕc ¼ −V ðϕc þ ϕqÞ − V ðϕc − ϕqÞ; ð20Þ parameter as m at the false vacuum, i.e., V00ð0Þ¼m2. The ground state wave function can then be approximated as ̈ 0 0 2ϕq ¼ −V ðϕc þ ϕqÞþV ðϕc − ϕqÞ: ð21Þ

1 2 ψ ≃ e−ωkjϕc;0j =2; 0 1=4 ð17Þ One of the solutions to this equation is ϕq ¼ 0 with ϕc ðπ=ωkÞ being the solution to the usual classical equation of motion, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 2 with initial conditions drawn from the initial Wigner where ωk ¼ m þ k . The initial Wigner distribution W0 distribution. This saddle point corresponds to Eq. (16). can then be estimated by the one for a free field: We refer to this as the real time formalism from classical Z dynamics, seeded by nontrivial initial conditions that 2iπc;0ϕq;0 W0 ¼ dϕq;0ψ ðϕc;0 þ ϕq;0Þψðϕc;0 − ϕq;0Þe ultimately arise from a choice for the initial wave function. Indeed we note that this process is absent if we were simply 1 2 2 to assume trivial initial conditions W0 δ ϕc;0 δ πc;0 , ∝ exp − ωkjϕc;0ðkÞj þ jπc;0ðkÞj : ð18Þ ¼ ð Þ ð Þ ωk which would be the “purely” classical behavior. Now we show that there is another, related, contribution This can be regarded as a probability distribution for the to Eq. (11) that is nonzero even if we were to set ϕ π initial field values, with and treated as independent ϕc;0 ¼ πc;0 ¼ 0. Assuming W0 ¼ δðϕc;0Þδðπc;0Þ,we d 1 random variables. For quantum field theory in þ rewrite Eq. (11) as spacetime dimension, the full result for the initial Z Wigner distribution is approximated as Oˆ Dϕ Dϕ O ϕ ϕ =2;t Z h i¼ f b Wðð f þ bÞ 1Þ ddk 1 W ∝ − ω ϕ k 2 π k 2 ; 0 exp d kj c;0ð Þj þ j c;0ð Þj × exp ½iSf − iSb; ð22Þ ð2πÞ ωk

ð19Þ where we assume that OW is independent of πc and Z where we integrate over all k-modes. t1 Sf;b ¼ i dtLf;b ð23Þ We can calculate the tunneling rate by evolving the 0 classical dynamics with an initial condition generated from Eq. (19). Since the initial condition is generated randomly are the actions for the forward and backward fields, by the Wigner distribution, we should perform a large respectively. This can be rewritten as number of simulations to obtain a statistically reasonable Z result. The tunneling rate is therefore given by the statistical ˆ hOi¼ DϕOWðϕðt1Þ;t1Þ exp ½iS; ð24Þ average of many realizations as done in Ref. [14]. We note that the system has a translational symmetry and there is no 0 strong correlation between two distant points. This implies by defining t ∈ ð−t1;t1Þ and ϕðtÞ¼ϕfðtÞ for t ∈ ð0;t1Þ that we can replace the ensemble average of many and ϕðtÞ¼ϕbð−tÞ for t ∈ ð−t1; 0Þ. This path integral can simulations by the spatial average of a single simulation be calculated by the standard instanton method by with a large simulation box. deforming to imaginary time. Since we assume ϕc;0 ¼ ˆ The operator O should be taken such that it is nonzero πc;0 ¼ 0 in this calculation, this saddle point corresponds to around the true vacuum and is zero around the false vacuum. the transition from vanishingly initial classical fields. It also

016011-4 VACUUM DECAY IN REAL TIME AND IMAGINARY TIME … PHYS. REV. D 100, 016011 (2019) corresponds to the vanishing initial quantum field for and classical mechanics comes from the commutation ϕq ¼ 0, though it leaves the initial condition for the relation for quantum operators. In particular, the commu- quantum field π unspecified. This is identical to the one tation relation between creation and annihilation operators calculated by the instanton method discussed earlier in is given by Sec. II. It is therefore associated with going from a field † † eigenstate with Dirichlet boundary conditions and again aˆ iaˆ j − aˆ i aˆ j ¼ δij: ð25Þ returning, in imaginary time, to a field eigenstate with Dirichlet boundary conditions. It is the so-called bounce However, the effect of the right-hand side is negligible † solution in imaginary time. Importantly, the difference from when the occupation numbers haˆ i aˆ ii are large. We then the saddle point solution corresponding to Eq. (16) is the expect that the high occupancy limit corresponds to the initial condition (or the boundary condition at t ¼ 0). classical limit of quantum systems. This implies that the Let us comment on how to rotate the time variable in the approximation Eq. (14) is justified when the number of imaginary space. If we naively take ϕq ¼ 0 in Eq. (11), the particles in the system is extremely large. exponent vanishes. This is not consistent with Eq. (24), In Ref. [19], we have shown that the expectation values where the action does not vanish and gives the Euclidean of quantum operators are approximated by a corresponding action in the imaginary time. This inconsistency comes classical ensemble average over many classical microstates, from the naive analytic continuation of the time variable. with initial conditions drawn from the initial quantum We can use the epsilon prescription to specify a possible wave function. Equation (16) is a mathematical expression way to change the integration contour in the imaginary of this statement. It can be understood as an extension of space. The Hamiltonian should include an imaginary mass this discussion to the quantum regime, where the initial term that specifies the way to change the integration state is not a high occupancy state, but a (quasi-)vacuum contour. Therefore the time variable for the Hamiltonian state with zero point fluctuations. Due to the possible for ϕf should be rotated in the opposite way to the one for production of bubbles, which arises due to rare accidental ϕb. This is the reason that we obtain a nonzero exponent arrangements from the nontrivial initial conditions, the even if we take ϕq ¼ 0 in Eq. (11). Later we will comment occupation number can be large enough to use the classical on more general situations, which may occur in cosmology, description. ϕ where this rotation to imaginary time may be more In this case, the approximation q ≪ ϕc is satisfied, problematic. except for the initial condition, and we can evolve ϕc by the classical equation of motion. In the regime before the ϕ ≪ ϕ C. Comparison tunneling, q c may not be satisfied. However, we can still use Eq. (16) if the amplitude of fluctuations is small In summary, there are two basic sets of initial conditions enough to neglect terms in the potential that are higher one may utilize to implement the saddle point for the path order than quadratic. This is because the Wigner approxi- integral Eq. (11). The first one is given by Eq. (16), where mation is exact for the free-field theory. We will discuss the initial condition is given by some approximation to the situations in which the neglecting of these higher order initial wave function and the time evolution is purely given terms may not be valid. by the classical equation of motion. The second one is given by the saddle point of Eq. (24), where the initial A. Tension and pressure condition is ϕ ¼ 0 and the time evolution is deformed into the complex plane to the imaginary time axis. Equation (16) can describe the classical dynamics of the At first sight it may seem surprising that the first should field after the bubble nucleation. This is different from the be associated with tunneling. But indeed tunneling can instanton method, where we need to connect the Lorentzian occur because of the nontrivial initial conditions it can and Euclidean regimes to describe the dynamics of the make for rare events to take place even within the bubble after nucleation. The tunneling process calculated framework of classical dynamics. This is the tunneling by Eq. (16) can therefore describe the tunneling process process that was calculated in Ref. [14]. In this sense, this itself as well as the dynamics of nucleated bubble after the contribution is complementary to the instanton contribu- nucleation. tion, though in appropriate regimes that we will discuss, Since the nucleated bubble obeys the classical equations they can approximate each other quite well. of motion, its behavior can be understood easily, particu- larly for the thin-wall case. The bubble wall tends to shrink to a point due to its tension while it tends to expand due to IV. CONDITIONS FOR THE CLASSICAL the pressure of the vacuum energy. As we evolve the field APPROXIMATION classically with an initial condition, a lot of small bubbles In this section we discuss conditions to calculate a are nucleated, but most of them do not have enough tunneling rate by Eq. (16) in the context of quantum field pressure to overcome the tension of the wall. In order theory. We first note that the distinction between quantum for the bubble to expand after the nucleation, the pressure

016011-5 MARK P. HERTZBERG and MASAKI YAMADA PHYS. REV. D 100, 016011 (2019) Z of the vacuum energy should overcome the tension of the pffiffiffiffiffiffi pffiffiffiffiffiffi σ ¼ dϕ 2V ∼ v Vh: ð29Þ bubble. For a thin-wall bubble, this requires

d−1 d Ad−1R σ ≲ VdR ϵ; ð26Þ Using Eq. (26), we obtain a typical radius of the nucleated bubble as where R is the radius of the bubble, σ is the tension of the σ ϵ R ≃ d : wall, and is the difference of the vacuum energy. Here we b ϵ ð30Þ define area and volume factors in the unit d-dimensional sphere: Let us first report on the Euclidean instanton action associated with the bubble, as it is the standard quantity to d=2 2π compute tunneling in the literature, as Ad−1 ≡ ; ð27Þ Γðd=2Þ 2Rdσ 2dd σdþ1 d=2 SE ≃ ≃ ð31Þ π d þ 1 d þ 1 ϵd Vd ≡ : ð28Þ Γðd=2 þ 1Þ d 2ðdþ1Þ=ðd−1Þ ðd−1Þ=2 Vh v ∼ : ð32Þ A similar type of inequality is expected to be satisfied for a ϵ Vh thick-wall bubble. However, in order to justify the alternative real time B. Occupation number tunneling from classical dynamics, we need to compute the bubble’s occupancy number. We define it by the gradient Now we examine under what conditions the occupation energy of the bubble in the unit of m, number of the quanta describing nucleated bubbles is much Z larger than unity. In this case the nucleated bubble is 1 1 N ≡ ddx ∇ϕ 2 essentially coherent and can be treated within the frame- m 2 ð Þ ð33Þ work of classical field theory. bubble We estimate the occupation number of nucleated bubble A Rd−1σ in two simple cases. First we consider the case where the ≃ d−1 m ð34Þ scalar potential is described by typical values of the curvature scale around vacua m, field value v, height of ℏ V (if we were to reinstate factors of , the actual occupancy the potential barrier h, and the difference of the vacuum number would be this divided by ℏ). It is roughly given by energy ϵ (see Fig. 2). We note that Vh must be smaller than 2ðdþ1Þ=ðd−1Þ of order v for d>1 because of the unitarity V d−1 v2ðdþ1Þ=ðd−1Þ d=2−1 v 2=ðd−1Þ N ∼ h : bound (e.g., in 3 þ 1 dimensions this is related to the d−1 =2 ϵ Vh mð Þ familiar idea that the quartic coupling λϕ4 obeys λ ≲ 1 to be in a weakly coupled regime). ð35Þ

1. Thin wall Note that every factor on the most right-hand side is larger than unity for weakly coupled field theories, so the occupa- ϵ ≪ V We assume h and use the thin-wall approximation tion number can in fact be quite large, N ≫ 1. [Note that for now. In this case, the wall tension is given by 2 2 2 1=2 for d ¼ 1, it simplifies to N ∼ v ðVh=ðm v ÞÞ .]

2. Thick wall The above suggests that the occupation number can be as small as of order unity when the vacuum energy is not degenerate and the difference of the vacuum expectation value is as small as m. This implies that the bubble is a thick-wall type and so we should reexamine the above analysis. Here the scalar self-coupling could be as large as Oð1Þ. We check that the occupation number is larger than unity in this extreme case, too. We consider the following potential: 2 3 1 ϕ λ3 ϕ FIG. 2. Schematic picture of a typical potential and parameters VðϕÞ¼U0 þ ; ð36Þ describing its shape. 2 Λ 3! Λ

016011-6 VACUUM DECAY IN REAL TIME AND IMAGINARY TIME … PHYS. REV. D 100, 016011 (2019)

2ðdþ1Þ=ðd−1Þ where U0 and Λ (U0 ≲ Λ ) are dimension-full development of bubbles over time as the system explores parameters and λ3 [≲Oð1Þ] is a dimensionless constant. phase space. Since ϕ ¼ 0 is a false vacuum, it can tunnel to the other side of the potential hill. The tunneling action and the occupa- V. TUNNELING RATE tion number are given by Now we shall compare the tunneling rate (per unit volume) via the standard Euclidean instanton to the Λ2ðdþ1Þ=ðd−1Þ ðd−1Þ=2 −2 tunneling rate via classical dynamics with initial conditions SE ≃ cSλ3 Ad ; ð37Þ U0 drawn some approximation to the wave function. The latter one can be estimated in the following way. 2ðdþ1Þ=ðd−1Þ ðd−1Þ=2 −2 Λ Ideally the relevant initial fluctuations that ultimately N ≃ cN λ3 Ad−1 ; ð38Þ U0 lead to the formation of the bubble are sufficiently small that the potential can be approximated to be a quadratic where the numerical constants are given by cS ≃ 4.1 × 10 form around a false vacuum; we will revisit this shortly. 2 and cN ≃ 1.0 × 10 for d ¼ 3. Even in this extreme case, Then the initial distribution of fluctuations is given by the the tunneling action and the occupation number are larger Wigner distribution of a free massive scalar field with mass 2 m around the false vacuum Eq. (19). This distribution does than Oð10=λ3Þ. This justifies that the nucleated bubble can be described classically (or a scalar condensate). not change much even if we allow the field to classically evolve in time. The tunneling rate can therefore be estimated by the probability that ϕcðkÞ and πcðkÞ are large C. Single particle quantum mechanics enough to nucleate a classical bubble. A classical bubble One may wonder if these arguments can extend to the that expands after the nucleation must satisfy the condition problem of single particle tunneling in ordinary nonrela- Eq. (26), and hence its radius must be larger than Rb ∼ σ=ϵ. tivistic quantum mechanics. In this case there is obviously Now in order to completely determine the probability for no such thing as a “bubble” that can be formed. So there is tunneling, one should perform a simulation of this non- no obvious sense in which there is any object at high linear system of classical equations of motion, with the occupancy. appropriate initial conditions specified above. However, we Nevertheless, we can formally view this problem as a can give an estimate of the probability of bubble formation quantum field theory in 0 þ 1 dimensions. So, for the sake by utilizing the initial wave function’s statistical distribu- of completeness, let us formally take the result in Eq. (35) tion as a guide; we will return to this shortly. In order to and take d → 0. Then we formally obtain form a bubble there are two conditions that need to be satisfied: (a) the field needs to be on the far side of the ϵ barrier, and (b) the bubble needs to have sufficient energy to N ∼ : ð39Þ m avoid collapse. Let us estimate these probabilities in turn. First, in order for the bubble to be on the other side of the Now we should note that in this case m, which is the barrier, we need that the field value in position space obeys (square root) of curvature of the potential in quantum field ϕc ≳ v. In the k-space representation this condition means theory at the false vacuum, is just the characteristic ðthÞ that we need ϕcðkÞ > ϕc ðkÞ where frequency of oscillation of the particle ω0 around the Z metastable minimum in quantum mechanics. ðthÞ d ikx d However, what is important is that in quantum mechan- ϕc ðkÞ¼ d xe ϕðxÞ ∼ Rbv; ð40Þ ics, energy conservation tells us that the tunneling process bubble requires that the particle tunnel to a point at the same for a bubble of radius Rn to be nucleated. The probability potential energy as its starting values. Hence, ϵ here should Pa that ϕcðkÞ exceeds this threshold can be estimated from ϵ 0 be the potential height difference, so it is in fact just ¼ . the Wigner distribution as This implies N ¼ 0. So there is no sense in which one is at Z high occupancy. This means that this procedure of sam- ddk − P ∼ ω ϕðthÞ k 2 ∼ ω v2Rd; pling from some Gaussian approximation to the wave ln a d kj c ð Þj b b ð41Þ ð2πÞ function and using this to determine tunneling will typi- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cally fail in ordinary quantum mechanics. Conversely, it ω m2 k2 ∼ m2 R−2 can be applicable in field theory in higher dimensions with where b ¼ þ b þ b is some character- bubbles of high occupancy, as we discussed above. istic frequency, associated with the bubble associated with a Furthermore, in single particle mechanics, with a con- characteristic wave number kb ∼ 1=Rb. fining potential, the system does not exhibit ergodicity. However, this is not a sufficient condition for nucleation, On the other hand, some ergodicity is exhibited in the because if the bubble appears on the other side of the barrier field theory, providing the adequate time evolution for the with arbitrarily low energy, then it can collapse. Suppose

016011-7 MARK P. HERTZBERG and MASAKI YAMADA PHYS. REV. D 100, 016011 (2019)

R that a small bubble with radius ini and kinetic energy This allows us to make the estimate d 2 R πc forms due to fluctuations. The kinetic energy must pffiffiffiffiffiffi ini γ ∼ v V Rd; be larger than the energy of the bubble with radius Rb so I h b ð49Þ that the bubble can expand after the nucleation. This means ðthÞ for the instanton tunneling exponent. In this final expres- we need πc > πc where d sion we have still kept a factor of Rb for convenience, since this is a common factor that appears in Eq. (45) also. Rd πðthÞ 2 ∼ Rdϵ: inið c Þ b ð42Þ

ðthÞ B. Examples Note that this πc is the conjugate momentum in π k We now use the above results to compare the tunneling position space. It can be written in termsR of ð Þ in the ðthÞ d d −ikx rates that we have estimated in these different formalisms. momentum space as πc ¼ 1=ð2πÞ d ke πcðkÞ ∼ ðthÞ −d πc k ∼ 1=R R . The probability Pb to have sufficient ð iniÞ ini 1. Weakly broken Z2 Symmetry energy can then again be estimated from the Wigner Let us consider a potential of the form distribution as Z V ϕ λ ϕ2 − v2 2 δV ϕ ; 50 ddk 1 ϵ ð Þ¼ ð Þ þ ð Þ ð Þ − P ∼ πðthÞ k 2 ∼ Rd: ln b d j c ð Þj b ð43Þ ð2πÞ ωk ωb where δVðϕÞ is a term that weakly breaks the Z2 symmetry. This potential is similar to the kind of potential shown 4 in Fig. 2 with Vh ∼ m =λ. In this case the bubble thickness A. Tunneling rate is approximately set by the Compton wavelength as λC ∼ 1=m. However, the bubble radius is at least this large, According to the Wigner representation, the variables ϕc i.e., mRb ≳ 1. This ensures the frequency ωb can be and πc are taken as independent random variables. This ω ≃ m ϵ says that the probability that both (a) and (b) occur is the approximated by the mass b . By noting that is bounded to be of the order of or much smaller than product PaPb. This allows us to estimate the tunneling rate V ∼ v2m2 P ≲ P (per unit volume) within this real time formalism as , we can conclude that the probability a b. Hence, the rate γR is approximated as

−γR ΓR ∼ cRe ; ð44Þ 2 d γR ∼ mv Rb: ð51Þ with 2 Then from Eq. (49), with Vh=m ∼ mv , we have γI ∼ γR.

2 d ϵ d γR ¼ aωbv Rb þ b Rb; ð45Þ ωb 2. Standard Model Higgs As another example, let us consider the Higgs potential a b O 1 where and are ð Þ prefactors. in the minimal standard model. Upon renormalization We can compare this to the usual result for tunneling group running of the Higgs self-coupling λ, the top mass, using the Euclidean imaginary time formalism given earlier and other couplings, one finds that the Higgs potential turns in Eq. (5) as over and then goes negative. This happens at around v ∼ 1011 GeV, or so. In this example, the potential is Γ ∼ c e−γI ; I I ð46Þ dominated by the quartic term near the tunneling point. In this case, we can use the above formula by taking with 4 ϵ → Vh ∼ λv . The bubble radius is now of order or larger dþ1 than γI ¼ SE ∼ ϵRb : ð47Þ 1 −10 −1 Rb ∼ pffiffiffi ∼ 10 GeV ð52Þ The instanton rate is calculated for the thin-wall case, but λv the result is not qualitatively different for the thick-wall case once we identify ϵ as the difference of the vacuum (using λ ∼ 0.01 in this regime). This radius is much smaller energy between the false vacuum and the tunneling point. than the Compton wavelength of the Higgs which is −1 −2 −1 Since this involves an extra factor of Rb compared to the m ∼ 10 GeV . Hence now we are in a regime in γ scaling in R, we need an estimate for the bubble radius, which ωb ∼ 1=Rb. In this regime, both Pa and Pb are which is roughly comparable, and they both give ffiffiffiffiffiffi vpV 1 R ∼ h : γ ∼ b ϵ ð48Þ R λ ð53Þ

016011-8 VACUUM DECAY IN REAL TIME AND IMAGINARY TIME … PHYS. REV. D 100, 016011 (2019)

(we naturally focus here on the physical case of 3 þ 1 classical field theory dynamics seeded by quantum zero dimensions). This is comparable to the instanton rate point fluctuations. We note that this ensemble average can be γI ∼ 1=λ, so we again have γI ∼ γR. practically realized by a spatial average of a single simu- We note that in this case with Rb ≪ m, giving ωb ≫ m, lation by appealing to a form of ergodic theorem. Since and probing deep into the quartic term in the potential, it we use the Wigner approximation for the initial wave was not guaranteed that the Gaussian approximation based function in the real-time approach, we do not expect that on the free theory would suffice. However, parametrically it the resulting rate is exactly the same as the one calculated in is of the right order. the imaginary-time approach. However, we have checked that the exponent in the tunneling rate is parametrically the 3. Flat hilltop same in both approaches in several examples. Suppose the hilltop is very flat, more so than it appears in A. Classicality Fig. 2. To be clear, let us imagine that it is so flat that 2 2 Vh ≪ m v , which would be the naive value based on In order to justify the classicality of the field in this latter dimensional analysis. Such a potential is perhaps unusual approach, the quantum fluctuations have to organize into a from the microscopic point of view, but it is allowed in bubble and the occupation number has to be much larger principle. In this case the instanton gives an exponent than unity. This can be realized only if the d.o.f. in the (normalized to bubble volume) that is linear in the barrier system are large enough, as is possible in quantum field d theory, as it is for the nucleated bubbles. We note, however, width γI=Rb ∝ v. On the other hand, if we turn to the real time formalism, we obtain different estimates. From Eq. (43) that much of the would remain at low occupancy, the contribution from the kinetic energy effect gives so it is not entirely guaranteed that the classical dynamics is d 0 extremely accurate, but perhaps only roughly accurate. γR=Rb ∝ v , which is too small. On the other hand, from Eq. (41) the contribution from the need to be on the other Furthermore, this approach is ordinarily not valid in single d 2 particle quantum mechanics as the notion of high occu- side of the barrier gives γR=R ∝ v , which is too large. b pancy there does not seem to be valid. In this case, the Gaussian approximation for the initial One may wonder if the tunneling rate depends sensi- wave function is not accurate, since it assumes that the tively on the initial fluctuations. This is actually the case fields’ mass is m, but for such a potential, the effective mass when the number of d.o.f. in the system is not much larger in the barrier is smaller. Instead we need to alter our simple than of order unity. However, we are interested in the estimates. We need to essentially replace the frequency of tunneling process in quantum field theory, where the the bubble by some appropriate effective mass, fromffiffiffiffiffiffi the m ∼ pV =v number of relevant d.o.f. can be quite large. In this case, effective curvature of the potential, namely eff h . all relevant modes may interact with each other somewhat This leads to chaotically and the distribution will be randomized after an pffiffiffiffiffiffi ergodic time. So one expects dynamical evolution to wash − P → m v2Rd ∼ v V Rd; ln a eff b h b ð54Þ away some features of the initial condition. (We may have to wait for a time scale longer than the time scale of which is indeed of the order of the instanton rate. oscillation around a false vacuum so that some features of On the other hand, if we persist with the original Wigner the initial condition are washed away. This condition is distribution, we believe that it is plausible that a simulation T=T → ∞ “ ” similar to slosh for the direct method that was can arrive at roughly the correct tunneling rate anyhow. introduced in Refs. [15,16].) This is because even though the initial distribution is not an However, our simple estimates for the tunneling rates did accurate representation of the false vacuum eigenstate, involve a dependence on the mass of the field defined these initial conditions may be partially washed away in around the initial false vacuum, as it affects the initial the simulation, leading to the appropriate rate. Gaussian approximation to the wave function. So these simple estimates involve some sensitivity to initial con- VI. DISCUSSION ditions, especially in the case of potentials with extreme We have used a general Wigner representation to establish features. But in the case in which the bubble has character- k k ≲ m two formulations of tunneling with slightly different boun- istic wave number values ( ), we are not sensitive to dary conditions and dramatically different dynamics: in the UV behavior of the initial conditions. Furthermore, addition to the usual formulation of the imaginary time more general estimates could be made in more extreme saddle point contribution to the decay amplitude with situations also. Dirichlet boundary conditions on the field, there is another real time formulation based on classical dynamics with B. Applications initial conditions set by some estimate for the initial wave As an application of these results, suppose there is an anti function. While the former one is the familiar one from the de Sitter vacuum between two dS vacua. The tunneling rate instanton action, the latter one is an ensemble average of from a dS vacuum to the other dS vacuum cannot be

016011-9 MARK P. HERTZBERG and MASAKI YAMADA PHYS. REV. D 100, 016011 (2019) calculated by using the standard instanton method because the contour to the imaginary time axis. If there is (quasi-) there is no instanton solution. However, the transition rate periodic behavior in the time domain, it will reorganize into must be nonzero because anything can happen in quantum growing exponential behavior in imaginary time, which theory according to the path integral expression [20].In may be an obstruction to an efficient implementation of the fact, the “classical tunneling” discussed in this paper is Euclidean instanton analysis. Furthermore, if there is some expected to give a nonzero transition rate. This is the only form of nonanalytic structure to the time dependence, such practical way we are aware of to calculate the transition rate as from a steplike time behavior, then this may be an in such a case. This transition process is complementary to obstruction to deforming the contour. In these cases it may the standard instanton tunneling process. In this sense, the be more intuitive and more practical to perform a real time result gives a lower bound on the tunneling rate. analysis. As another application, consider a dynamical setting, such as during preheating after inflation. In this case a field ACKNOWLEDGMENTS may exhibit a strongly time dependent effective potential from its interactions with the inflaton or the metric, etc. If We thank Jonathan Braden, Alexander Vilenkin, and such a field is also trapped in a type of false vacuum, then it Mohammad Hossein Namjoo for useful discussions. may be highly nontrivial to implement the standard M. P. H. is supported by National Science Foundation instanton tunneling procedure, as this requires deforming Grant No. PHY-1720332 and a JSPS fellowship.

[1] A. B. Balantekin and N. Takigawa, Rev. Mod. Phys. 70,77 [12] N. Turok, New J. Phys. 16, 063006 (2014). (1998). [13] S. F. Bramberger, G. Lavrelashvili, and J.-L. Lehners, Phys. [2] L. Esaki, Phys. Rev. 109, 603 (1958). Rev. D 94, 064032 (2016). [3] G. Binning, H. Rohrer, C. Gerber, and E. Weibel, Phys. Rev. [14] J. Braden, M. C. Johnson, H. V. Peiris, A. Pontzen, and S. Lett. 49, 57 (1982). Weinfurtner, arXiv:1806.06069. [4] P. H. Frampton, Phys. Rev. D 15, 2922 (1977). [15] A. Andreassen, D. Farhi, W. Frost, and M. D. Schwartz, [5] A. H. Guth, Phys. Rev. D 23, 347 (1981); Adv. Ser. Phys. Rev. Lett. 117, 231601 (2016). Astrophys. Cosmol. 3, 139 (1987). [16] A. Andreassen, D. Farhi, W. Frost, and M. D. Schwartz, [6] M. R. Douglas, J. High Energy Phys. 05 (2003) 046. Phys. Rev. D 95, 085011 (2017). [7] M. Sher, Phys. Lett. B 317, 159 (1993); 331, 448(A) (1994). [17] A. Kamenev and A. Levchenko, Adv. Phys. 58, 197 [8] A. H. Guth and Y. Nomura, Phys. Rev. D 86, 023534 (2009). (2012). [18] A. Polkovnikov, Ann. Phys. (Amsterdam) 325, 1790 [9] L. F. Abbott, Phys. Lett. 150B, 427 (1985). (2010). [10] S. R. Coleman, Phys. Rev. D 15, 2929 (1977); 16, 1248(E) [19] M. P. Hertzberg, J. Cosmol. Astropart. Phys. 11 (2016) (1977). 037. [11] C. G. Callan, Jr. and S. R. Coleman, Phys. Rev. D 16, 1762 [20] A. R. Brown and A. Dahlen, Phys. Rev. Lett. 107, 171301 (1977). (2011).

016011-10