Mass Renormalization in Lattice Simulations of False Vacuum Decay

False vacuum decay, a quantum mechanical first-order phase transition in scalar field theories, is an important phenomenon in early universe cosmology. Recently, real-time semi-classical techniques based on ensembles of lattice simulations were applied to the problem of false vacuum decay. In this context, or any other lattice simulation, the effective potential experienced by long-wavelength modes is not the same as the bare potential. To make quantitative predictions using the real-time semi-classical techniques, it is therefore necessary to understand the redefinition of model parameters and the corresponding deformation of the vacuum state, as well as stochastic contributions that require modeling of unresolved subgrid modes. In this work, we focus on the former corrections and compute the expected modification of the true and false vacuum effective mass, which manifests as a modified dispersion relationship for linear fluctuations about the vacuum. We compare these theoretical predictions to numerical simulations and find excellent agreement. Motivated by this, we use the effective masses to fix the shape of a parameterized effective potential, and explore the modeling uncertainty associated with non-linear corrections. We compute the decay rates in both the Euclidean and real-time formalisms, finding qualitative agreement in the dependence on the UV cutoff. These calculations further demonstrate that a quantitative understanding of the rates requires additional corrections.


I. INTRODUCTION
False vacuum decay is a fascinating phenomenon in quantum field theory involving many of its most challenging aspects: it is dynamical, nonlinear, and nonperturbative. It has applications to a variety of fundamental scenarios in cosmology, and is also relevant to a broad range of condensed matter systems. While direct experimentation with cosmological systems is beyond current technologies, recent proposals to create analog relativistic false vacuum systems in the lab [1][2][3][4][5][6][7][8][9][10] provide the exciting possibility of experimentally probing some aspects of false vacuum decay. To make maximal use of such experiments and compare to theory, a thorough understanding of the theoretical modeling of false vacuum decay is required. The calculations in this work demonstrate the importance of understanding renormalization effects on the properties of the false vacuum.
Relativistic vacuum decay is the process by which a large region of space trapped in a local potential minimum (i.e., a false vacuum) transitions to a new state that is no longer localized in the potential minimum. The process is expected to occur via the formation of bubbles within the ambient false vacuum. Inside these "true vacuum" bubbles, the field is on the opposite side of the barrier and no longer trapped in the false vacuum. The bubbles then expand, with the speed of the walls rapidly approaching the speed of light. In many contexts the bubbles eventually coalesce, resulting in a phase of bubble collisions and the conversion of the initial false vacuum into an (excited) state localized around the true vacuum. In other cases, such as when the space in the false vacuum regions is undergoing rapid accelerated expansion, the bubbles fail to coalesce, leading to an incomplete phase transition and a state consisting of the ambient false vacuum with bubbles embedded within.
The standard approach to vacuum decay, referred to as the bounce formalism [11,12], works in Euclidean time, thus obscuring the fundamentally time-dependent nature of the problem. In this formalism, a high degree of symmetry is imposed on the nucleating bubbles, with contributions from inhomogeneities assumed perturbative in ℏ and relegated to (frequently neglected) correction terms. Furthermore, bubble nucleations are interpreted as quantum tunneling events. Therefore, there is no analog of classical time evolution to describe the formation of a bubble. The basic input in this approach is to specify a potential for the mean field, which we will call the tree-level Euclidean potential and denote V T . From this we can compute expected rates of bubble nucleation and the typical profile of a nucleated bubble.
Reference [13] introduced an alternative approach to the problem of vacuum decay, which instead works in real time by evolving realizations of the false vacuum forward in time (see also [14][15][16][17][18][19][20][21] for subsequent related work). Analogous techniques are used in a variety of other contexts, such as preheating at the end of inflation [22][23][24][25], evolution of dilute gas cold atom Bose-Einstein condensates [26,27], relativistic heavy ion collisions [28,29], and nonequilibrium and thermalization properties of relativistic fields [30][31][32]. In this approach, the problem is treated with classical statistical field theory by sampling and evolving initial configurations of the false vacuum. In this framework, the purely classical time evolution leads to the dynamical formation of "true vacuum" bubbles from the (Gaussian) realizations of the false vacuum. In analogy to the Euclidean view, we will refer to the emergence of a bubble in these simulations as the decay of the false vacuum. These bubbles then expand and eventually coalesce, causing the system to transition from the false vacuum to a state localized around the true vacuum. The distribution of bubble nucleation times is well modeled by an exponential decay law, allowing for a straightforward determination of the decay rate. Thus, these simulations give an explicitly time-dependent description of the formation of bubbles, providing a classical time-dependent connection between the initial Gaussian vacuum state and the subsequent phase of an expanding bubble. In particular, we do not assume an initial bubble shape (unlike previous investigations [33]), nor do we assume that a "tunneling" event describes the sudden appearance of a bubble. Further, since we have access to much more fine-grained information, many additional questions that are inaccessible in the Euclidean framework can be explored, such as the correlation between bubble nucleation events [21]. This approach fully captures nonlinearity (i.e., backreaction and rescattering effects) and places no symmetry assumptions on the resulting solutions, but does not incorporate interference effects between different initial realizations. In addition to a potential, which we will call the bare lattice potential and denote V bare , the real-time approach requires an assumption about the statistics of the initial vacuum field fluctuations.
Summarizing, in the Euclidean approach the bubbles are assumed to be highly symmetric and in the standard interpretation appear via a quantum tunneling process with no classical real-time description. Meanwhile, the semiclassical real-time approach makes no symmetry assumptions about the bubbles and provides a classical description of the moment of nucleation. However, the latter approach is unable to capture interference between different histories. Given the drastically different interpretations and approximations used in the two approaches, we would like to understand if they are different descriptions of the same decay process or if they describe different mechanisms by which bubbles nucleate in the false vacuum. A first step is to compare the decay rates between the two approaches. If we identify the tree-level Euclidean potential with the bare lattice potential, we find that the decay rate extracted from the real-time approach is much larger than that of the Euclidean approach. Specifically, the log of the decay rate decreases more slowly as the amplitude of the quantum fluctuations is decreased (relative to the width of the barrier) for the real-time simulations than the Euclidean prediction. This raises an interesting puzzle about the relationship between the two approaches, pointed out in Refs. [13,15].
However, as noted in Ref. [13], identifying the tree-level Euclidean potential with the bare lattice potential is not correct because it ignores important renormalization effects associated with the lattice simulations. To make a proper comparison of the decay rates, we must determine which tree-level Euclidean potential corresponds to the bare lattice potential employed by the real-time formalism. The main focus of this paper is to make progress toward this goal.
In more detail, while the bare lattice potential correctly encodes the dynamics for the full inhomogeneous field realizations, to obtain the dynamics of the mean field alone, we must integrate out the inhomogeneities. This results in a modification of the mean field dynamics, similar to the effects of renormalization in quantum field theory. Since the Euclidean potential encodes the dynamics of the mean field, it is reasonable to conjecture that the corrections to the bare lattice potential should be included in the mapping between the lattice and Euclidean calculations. In what follows, we will refer to this as the effective lattice potential and denote it V eff .
We determine the effective force felt by the spatially averaged mean field assuming Gaussianity of the fluctuations, an improvement on a one-loop approximation. From this, we compute the effective mass squared (i.e., effective potential curvature) for mean field values located at symmetric minima of the bare potential. Unfortunately, the situation is significantly more complicated when the potential is no longer symmetric about the minimum, or if we displace the mean field from the potential minimum. To explore the potential impacts of these deformations on the Euclidean decay rate, we write down some simple parametrizations of the full effective potential, with tunable parameters that can be calibrated using either our analytic calculations of the effective mass, or measurements made in simulations. An illustration of this approach can be found in Fig. 1. While there is some uncertainty in our theoretical modeling, we place reasonable bounds on the sensitivity of the Euclidean decay rate to the details of the shortwavelength fluctuations.
Although we focus on false vacuum decay, the formalism outlined here is universally applicable to tackling renormalization issues in real-time semiclassical stochastic lattice simulations. Therefore, the methods developed here may also be of use in other settings, such as generating initial conditions for preheating simulations.
The remainder of this paper is organized as follows. In Sec. II we review the real-time lattice framework and the particular potential we consider in this paper. In Sec. III we compute the expected renormalization of the field mass and compare it to results from lattice simulations, finding excellent agreement across a range of parameters. We then explore the possible impacts of these deformations on the predicted Euclidean decay rate in Sec. IV. Finally, we describe how some simple modifications of the initial fluctuations modify the real-time decay rates extracted from lattice simulations. We then conclude in Sec. VI. Details of our effective potential modeling to compute the Euclidean decay rates are provided in the Appendix. Unless otherwise stated, we work in units with ℏ ¼ c ¼ 1 throughout.

II. STOCHASTIC LATTICE SIMULATIONS
We work within the semiclassical stochastic lattice framework, which we now briefly review (see Ref. [13] for further details). The key assumption is to model quantum effects by sampling field and momentum configurations of the quantum vacuum. Generating samples of the fully self-consistent interacting vacuum is a highly nontrivial undertaking. Therefore, we approximate the vacuum state as Gaussian and initialize whereα n andβ n are independent complex Gaussian random deviates satisfying hjα n j 2 i E ¼ 1 ¼ hjβ n j 2 i E , α −n ¼ α Ã n , β −n ¼ β Ã n , and h·i E denotes an ensemble average. V is the simulation volume, and ω n is the oscillation 1. An illustration of how we determine the effective potential V eff seen by long-wavelength fluctuations. We develop analytic and empirical techniques to measure the corrections to the curvature about the False (m 2 FV ) and True (m 2 TV ) vacua defined by the bare potential V bare (light solid blue line). Modeling V eff by an expansion that respects the periodic symmetry of the bare potential, we use the measured m 2 FV and m 2 TV to fix two free parameters. We can then explore the modeling uncertainty arising from additional terms in the expansion (relevant in the shaded region of the potential), which affect the properties of the critical bubble (instanton solution) that determines the Euclidean rate obtained from V eff . frequency of the nth mode. Provided the fluctuations primarily probe the quadratic region of the local potential minimum, we expect this to be a good approximation to the true vacuum state. Since we work on a discrete lattice of finite size, we have truncated the spectrum above some wave number k cut ≡ 2π L n cut below the Nyquist wave number k Nyq ¼ π dx where dx is the grid spacing. For the results presented here, we used convergence testing to verify that k cut was sufficiently large to capture all relevant dynamical evolution of the fluctuations. In particular, for our choice of lattice and model parameters this demonstrates that there is minimal transfer of power between the "short-wavelength" modes with k > k cut and "long-wavelength" modes with k < k cut . Note that since our lattice has a finite volume, initializing modes directly in Fourier space does not exactly reproduce the real space two-point correlation function. Therefore, we also confirmed the finite box size has a negligible effect on our results, which is expected since the mass of the field cuts off the infrared fluctuation power per ln k bin. 1 We then propagate these initial conditions forward in time using the classical equations of motion As in our previous work [13], we use a 10th order Gauss-Legendre integrator for time evolution, and a Fourier collocation discretization of space (see the appendices of Ref. [34] for more details). Expectation values of quantum operators are approximated by a classical ensemble average over realizations of the initial fluctuations. The philosophy of this approach is to model quantum effects through the randomly drawn initial conditions δφ and δΠ, but not directly in the time evolution. Heuristically, we are enforcing the uncertainty principle in the initial conditions by drawing individual field and momentum Fourier modes as Gaussian random deviates with the appropriate width. Meanwhile, we are neglecting interference effects in the path integral between different classically-evolved trajectories, which are not captured by our classical time evolution. This approach is expected to work well when the "quantum" fluctuations are initially linear, and at the level of reproducing operator expectation values can be shown to incorporate the first two orders of a systematic expansion of the full quantum dynamics in ℏ (see for example the Supplemental Material of Ref. [13]). In the above, we have restricted ourselves to spatially homogeneous background solutions that are at rest. Via appropriate generalizations of the mode functions, this can be extended to the case with nonzero initial field momentum, or spatial inhomogeneity. Although we will not pursue it here, modifying the covariance structure of thê α k 's andβ k 's allows us to explore nonvacuum states, such as squeezed states or thermal states.
The stochasticity, therefore, refers to the particular realization of the initial conditions, not additional noise terms appearing during the time evolution. This approach is related, at least in spirit, to alternative stochastic approaches such as stochastic tunneling (see e.g., Ref. [35]), stochastic inflation (see e.g., Refs. [36][37][38]) or the Langevin approach to thermal field theory on the lattice (see e.g., [39][40][41]), where the short-wavelength modes are modeled as additional noise contributions during the time evolution. The primary difference is that rather than externally imposing a model for these additional noise terms, we are self-consistently solving for their dynamics as the system evolves. For example, suppose we restrict ourselves to consideration of the spatial average of the field, which we will refer to as the zero mode. From the viewpoint of the effective dynamics of the zero mode, all of the realized inhomogeneous modes effectively act as stochastic contributions to the dynamics. However, rather than having to explicitly model the dynamics of these bath modes, we are selfconsistently evaluating the statistical properties of the noise by evolving the realized modes.
In this paper we focus on the analog false vacuum decay potential with corresponding potential force For λ 2 > 1, this potential possesses an infinite sequence of false vacuum minima at ϕ FV ¼ 2πnϕ 0 and a corresponding sequence of true vacuum minima at ϕ TV ¼ ð2n þ 1Þπϕ 0 with n ∈ Z. Since we have in mind the application of this potential to false vacuum decay, in this paper we will primarily concentrate on states initially localized around the false vacuum ϕ ¼ 0. To make the most direct connection with existing work [13,15,21], we will also limit ourselves to one spatial dimension. We will provide a more detailed exploration in higher dimensions in a future publication.

III. RENORMALIZATION OF THE FALSE VACUUM
As outlined above, we are interested in developing an effective potential description of our lattice simulations, with the goal of connecting to the potential appearing in Euclidean calculations of the false vacuum decay rate. The appropriate dynamical variable is therefore the mean field, ϕ ≡ hϕi. For lattice simulations, rather than work directly with the potential, it is more convenient to study the additional force experienced byφ. Roughly, this corresponds to studying the derivative of the effective potential.
To obtain the effective dynamics of the mean field, we average the equation of motionφ where we have assumed that the mean field is spatially homogeneous so we can discard the Laplacian term. For now we allow h·i to represent either an ensemble h·i E or a volume h·i V average. In a given realization (i.e., simulation) these two quantities deviate from each other, but if we average the volume average over an ensemble of (finitevolume) numerical simulations, we expect it to converge to the full ensemble average. For notational convenience, we define the (negative of the) effective force acting on the mean field If F eff is a function of the mean field alone we can identify V 0 eff ¼ F eff and integrate to obtain an effective potential, although this need not hold in general situations.
We now define ϕðx; tÞ ¼φðtÞ þ δϕðx; tÞ and assume that we can Taylor expand the potential around the instantaneous mean field valueφ, transforming hV 0 bare i into a weighted sum over the moments of δϕ. One can resum the Gaussian contributions, and for the analog BEC potential (3) we obtain where σ 2 ϕ ≡ hδϕ 2 i and F NG encodes all non-Gaussian contributions to the one-point distribution of δϕ. In the following theoretical development, we will neglect the effects of these non-Gaussian corrections. Under the assumptions of Gaussianity and a Taylor-expandable bare potential, we have therefore reduced the expression for the effective force acting on the mean field to the theory of a single function of the mean field, σ 2 ϕ ðφÞ.

A. Vacuum fluctuations
To make further progress we must provide a model for σ 2 ϕ . Since we are interested in relativistic vacuum fluctuations, we assume We work in the discrete limit where L is the side length of our simulation volume and Δk ¼ 2π L . In linear perturbation theory we would identify m 2 PS ¼ V 00 bare . However, the presence of the realized fluctuations leads to a deformation of the vacuum state. A key result of this work is to calculate an improved value of m 2 PS describing the corrected vacuum. More generally, we can allow m 2 PS to be a free parameter describing the initial state of our system, and then explore the subsequent response of the system to this class of (generally nonvacuum) initial conditions. In the remainder of the paper, we will refer to this class of initial conditions as pseudovacuum initial conditions.
It is straightforward to obtain the field variance in these pseudovacuum states, which we denote σ 2 PS . Restricting to one spatial dimension, we have where we have assumed UV (k UV ) and IR (k IR ) cutoffs on the spectrum. In the context of lattice simulations, the UV cutoff arises naturally from the spectral truncation (1) required to initialize our fluctuations. Meanwhile, the IR cutoff can arise either from the existence of the fundamental mode in our simulations or the dynamical emergence of localized spatial structures. Assuming the fundamental mode is the relevant IR scale, and accounting for the implicit centered binning of our Fourier modes, we have which will be confirmed in Sec. III C below. We are typically interested in the limits k 2 UV ≫ jm 2 PS j and Within this framework, we have swapped the unknown variance σ 2 PS for the unknown effective mass m 2 PS .

B. Renormalized false vacuum mass
Now that we have an understanding of how the fluctuations modify the dynamics of the mean field, we want to determine how the effective mass of the field is deformed from its bare value. There are two masses appearing in our formulation: the mass m PS that determines the variance of the fluctuations, and a dynamical mass associated with the curvature of the effective potential, which we will denote m eff . Assuming the system is in a pseudovacuum state, we have where σ 2 PS is given by (9). The emergence of a nonlinear algebraic equation (12) for m 2 eff is a consequence of the evenness of the potential about the minimum, which leads to two simplifications: (i) the location of the false vacuum does not shift in response to the presence of the fluctuations, and (ii) derivatives of σ 2 PS with respect toφ do not appear.
In the self-consistent vacuum state, the potential curvature (m 2 eff ) must match the squared effective mass in the initial conditions (m 2 PS ). Denoting this special value by m 2 G , we must solve In the remainder of the paper, we will call this the nonlinear Gaussian approximation.
In the absence of fluctuations, we have We obtain a simple analytic approximation by substituting σ 2 PS ðm 2 B Þ ≡ σ 2 B into the rhs of (13) m 2 G;ð1Þ We refer to this as the Gaussian bare mass approximation. We can also obtain a perturbative approximation in the fluctuation amplitudes by first expanding (12) in σ 2 PS . To leading nontrivial order we have which we refer to as the one-loop approximation. Equating m 2 eff and m 2 PS as before, we can then solve the resulting nonlinear equation. We will call this the nonlinear one-loop approximation and denote m 2 1-loop;NL . Alternatively, we can further approximate σ 2 PS ≈ σ 2 B to obtain which we will refer to as the one-loop bare mass approximation. These predictions are illustrated in Fig. 2 as ϕ −2 0 is varied, while holding fixed the bare potential parameter λ 2 , as well as the UV and IR spectral cutoffs. Since σ 2 ϕ =ϕ 2 0 controls the amount of nonlinear potential structure probed by the fluctuations, as ϕ 0 → ∞ we return to the limit of the bare lattice potential. For ðσ 2 ϕ =ϕ 2 0 Þ ≪ 1 all of the approximation schemes closely match each other. However, as ϕ 0 is decreased and the fluctuations probe more of the nonlinear potential structure, the curves begin to deviate significantly.
For the potential considered here, we find that the oneloop approximation always overestimates the magnitude of the mass correction. Further, we find the one-loop bare mass approximation (where the field variance is estimated as that in the bare vacuum) is more accurate than a full nonlinear solution to the one-loop equation (15). Presumably, this is because the nonlinear one-loop solution is implicitly including higher orders in σ 2 ϕ in the solution, while not self-consistently including these higher orders in the defining equation. Comparing to the nonlinear Gaussian approximation, we see the Gaussian bare mass approximation (14) provides a significant improvement over either of the one-loop approximations.

C. Lattice measurements of the renormalized mass
We now test the preceding analytic predictions directly in lattice simulations by studying the time evolution of field realizations from pseudovacuum initial conditions. To avoid causal interaction of a point with itself, we integrate for a single light-crossing time (i.e., set our integration time to the box length L). Since we want to study properties of the false vacuum, in any given realization we must restrict ourselves to times before a bubble nucleates. Therefore, in this section we take λ 2 ≫ 1 to ensure that most of our simulations do not decay (i.e., nucleate a bubble) within a light-crossing time. The rare simulations that decayed were excluded from the ensemble averages. We chose lattice cutoffs k cut sufficiently large to ensure that the flow of fluctuation power into UV modes beyond k cut was strongly suppressed for the duration of the simulations. As a complementary check, we confirmed that our results were insensitive to changes in the grid spacing at fixed side length L and spectral cutoff k cut .
Given the preceding predictions for the shift in effective mass of the field, we expect a corresponding shift in the effective oscillation frequency of small amplitude modes. Since the oscillation frequency sets the fluctuation spectrum for linear vacuum fluctuations, a self-consistent determination of the vacuum fluctuations must account for this effect. In Fig. 3 we show a coarse-grained demonstration of this by studying the variance of the field fluctuations in the simulations as we allow m 2 PS in our pseudovacuum initial conditions to vary. If the fluctuations begin in the vacuum, they should have time-stationary statistics. 2 In particular, σ 2 ϕ should be time independent, at least on timescales much shorter than the decay time of the metastable state. The left panel of Fig. 3 shows the time evolution of σ 2 ϕ for a few select values of m 2 PS . We see that in all cases considered here, σ 2 ϕ initially experiences a rapid transient before undergoing damped oscillations around a shifted time-averaged mean σ 2 ∞ . This suggests that the initial excited state has settled into some sort of statistically stationary state that differs from the initial pseudovacuum. Crucially, when we choose m 2 PS to be consistent with our prediction for the modified vacuum mass, the field variance is found to be in a stationary state as expected, and does not undergo damped oscillations. This is further illustrated in the right panel of the figure, where we show the initial field variance σ 2 0 and late-time field variance σ 2 ∞ as a function of the initial m 2 PS . For reference, we also include the analytic result for the initial variance σ 2 PS in (9) with cutoffs k IR and k UV given by (10). The excellent agreement with the empirically measured curve confirms the relationship between the lattice parameters and cutoffs in the continuum calculations. A necessary condition for the initial state to be a vacuum is that it be time stationary. Therefore, the initial and asymptotic variances must be equal for the correct m 2 PS . In accordance with our prediction above, we see that the two variances coincide at the special choice m 2 PS ¼ m 2 G , indicating that a self-consistent study of the vacuum should include a modification of the initial conditions. This point also appears to coincide with the minimum of the asymptotic variance. A possible explanation for this is that rescattering effects remain subdominant so that each Fourier mode effectively acts as an independent harmonic oscillator with frequency set by the renormalized effective mass. In this case, for each Fourier mode the pseudovacuum initial conditions with smallest time average field variance is the one where m 2 PS þ k 2 matches the oscillation frequency.
The coarse-grained view provided by the field variance σ 2 ϕ indicates that the bare lattice vacuum lacks timetranslation invariance when accounting for the effects of interactions between realized fluctuations. We now take a more fine-grained look at the field fluctuations and study the evolution of the individual Fourier modes. In Sec. III we showed that the potential force acting on the homogeneous mode of the field receives a correction that can be interpreted as a shift in the effective mass. Although our preceding result is not strictly applicable, we expect that inhomogeneous modes will experience a corresponding shift in their effective mass. In fact, for symmetric minima we have shown that truncating the dynamical equation for δϕ at cubic order in the amplitude (i.e., the leading nonlinear correction), produces a shift in the frequency of the modes whose σ 2 ϕ dependence matches that of the oneloop approximation (16) above.
We illustrate the mass shift in Fig. 4, where we show the power spectral density with δφ ∝ R dtdxe ikx−iωt ϕðk; tÞ. In these spectra, the presence of small amplitude oscillatory modes with a welldefined dispersion relationship ωðkÞ manifests as a sharp feature that traces out the line ðk; ωðkÞÞ. For concreteness, we show the results from simulations using the corrected m 2 PS ¼ m 2 eff determined as described above. 3 Figure 4 clearly demonstrates the presence of a relativistic dispersion relationship ω 2 ¼ k 2 þ m 2 eff for the inhomogeneous field fluctuations. Further, the value of m 2 eff agrees with our predictions in the preceding section, indicating a distortion of the effective mass of the field due to the presence of the fluctuations. We also see that our nonlinear Gaussian resummation (13) continues to be valid even in regimes where the one-loop approximation has failed.

IV. RENORMALIZATION AND THE EUCLIDEAN DECAY RATE
In the preceding sections we showed that the effective dynamics for the lattice mean field is modified from that encoded in the bare lattice potential. This raises interesting questions about the relationship between the bare potential, V bare , appearing in our lattice simulations and the tree-level potential, V T , appearing in the bounce calculation. Both the real-time and Euclidean approaches approximate the full quantum dynamics using an expansion in ℏ [11][12][13][14]. Therefore, to compare the two formalisms, we should truncate them at the same order in ℏ. With this in mind, let us briefly review the nature of both expansions.
Recall that in the Euclidean formalism (see, for example, Refs. [11,12,42]), the decay rate per unit length takes the form We see the Gaussian prediction for the vacuum corresponds to the special point σ 2 0 ¼ σ 2 ∞ , which also coincides with the minimum of the asymptotic variance. Our analytic result for the pseudovacuum variance (9) is shown as a light red line. We used numerical ensembles of 1000 lattice simulations in a box of side length λmL ¼ 128, N ¼ 2048 lattice sites, and spectral cutoff n cut ¼ 1023. We verified the results are insenstive to the choice of grid spacing at fixed L and n cut , and that the dependence on the box size was that expected from adjusting k IR . We empirically estimated the sampling errors and found they were comparable to the width of the lines, consistent with the agreement between the σ 2 0 (empirical) and σ 2 PS (theory) lines. 3 We also ran simulations for initial conditions specified by the bare vacuum mass, m 2 PS ¼ V 00 bare ðϕ FV Þ ¼ λ 2 − 1, and we found similar results for the model and lattice parameters used here. This correspondence breaks down if we take ϕ 0 sufficiently small. A possible explanation is that σ 2 ∞ is relatively flat around m 2 eff , so that for initial bare masses near the corrected mass the effective mass of the field is insensitive to the initial value of the field fluctuations.
where we have temporarily restored ℏ. Here S E is the Euclidean bounce action, obtained by solving the bounce equation, andD is a (dimensionless) prefactor encoding information about fluctuations about the bounce. We will refer toD as the fluctuation determinant below. μ is some inverse length scale characterizing the bounce that has been factored to makeD dimensionless. When we set ℏ ¼ 1, μ becomes a mass scale and will be referred to as such below. The leading order contribution in ℏ appears nonperturbatively in the e − S E ℏ factor. Meanwhile, the first quantum correction is encoded in the fluctuation determinant, which is expected to have the formD ∼ 1 þ OðℏÞ. As usual in quantum field theory, this determinant is naively divergent and must be renormalized. Now, compare this to the real-time semiclassical lattice calculations. The classical evolution captures contributions of classical trajectories to the path integral, which appear as e i ℏ S cl . Viewed as an ℏ expansion, this appears at the same order as the leading order contribution in the Euclidean formalism. Further, in our lattice simulations the initial fluctuations have variance OðℏÞ, and thus contribute at the same order as the corrections due to the fluctuation determinant. Combined with the fact thatD encodes information about fluctuations around a highly symmetric saddle point (the bounce), this suggests that the lattice effective potential force is capturing the renormalization effects present in the fluctuation determinant.
This comparison of expansion orders suggests that Euclidean calculations must include renormalization effects in order to make a fair comparison with the real-time results. 4 However, the standard Euclidean expansion (18) only includes one-loop quantum corrections, which as demonstrated above fail to properly describe the effective lattice potential. A reasonable conjecture is that we should identify the effective lattice potential with an "effective tree-level Euclidean potential" V E;eff , and perform our bounce calculations in V E;eff . Since the lattice results automatically include a UV cutoff, this substitution may capture the effects of the divergent part of the fluctuation determinant in the usual Euclidean formalism. We leave a detailed study of the applicability of these conjectures to future work, and here we simply explore a few potential consequences if they are true.
Examining the Euclidean decay rate (18), we see that fluctuation corrections can impact the predicted Euclidean rate in three ways: (1) modifying the Euclidean action S E , (2) modifying the characteristic mass scale μ, or (3) modifying the OðℏÞ correction. Roughly, we expect that μ 2 ∼ jV 00 eff;max j is set by the curvature of the potential at the local maximum, which receives corrections just as the vacuum masses do. However, for technical reasons, this shift is much more difficult to compute than masses at the symmetric local minima. Similarly, understanding the modification to the OðℏÞ piece requires a detailed study of the fluctuation determinant. Since our purpose is to get a flavor of the types of corrections we can expect, we will focus only on the changes to S E . A reasonable guess is that μ 2 's behavior may be similar to m 2 FV and m 2 TV . However, for this model even simple combinations such as m 2 TV − m 2 FV and m 2 TV þ m 2 FV have different ϕ 0 dependence, so that more detailed investigation is needed. With this in mind, we now explore how the Euclidean bounce action changes if we replace the bare lattice potential with an (approximate) effective lattice potential. In Sec. III we showed how the curvature of the effective lattice potential is modified at a symmetric local minimum of the potential. A first principles derivation of the effective eff using the Gaussian approximation, while the dashed lines are using the one-loop bare mass approximation. We show results for ϕ 2 0 ¼ 9 (blue) and ϕ 2 0 ¼ 2.5 (red). In the former case, the oneloop and Gaussian resummed approximations match, and both accurately describe the results of the simulation. However, for the latter case the one-loop approximation deviates noticeably from the simulation results, while the Gaussian resummation continues to accurately capture the shifted frequency. For reference, the bare effective mass prediction is shown as a red dashed line. All simulations used a lattice size mL ¼ 64, N ¼ 1024 lattice sites, spectral cutoff n cut ¼ 511, and time step dt ¼ dx 8 . The potential parameter was λ ¼ 2. The averaging was performed over ensembles of 100 simulations. We remove the (rare) simulations that nucleate bubbles through a cut on hcos ϕi. To reduce sampling artifacts, we interpolated the data points onto curves of constant y ≡ ω − ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2 G þ k 2 p and then applied a Gaussian filter along the constant y lines. 4 More precisely, the shifts in the bare parameters observed in the lattice simulations must be accounted for in the Euclidean calculations. potential requires extending these techniques to locations on the bare potential that are neither minima nor possess even symmetry. This significantly complicates the required analysis, and we leave a more detailed development of the theory to future work. Here, we instead take an effective theory (EFT) approach and expand the effective potential in a set of basis functions that respect the symmetries of the bare lattice potential. The unknown coefficients in this expansion must then be set by measurements. In keeping with our EFT inspired approach, we compute m 2 TV and m 2 FV by solving for the Gaussian resummed masses at ϕ ¼ ϕ 0 π and ϕ ¼ 0, respectively. These numerical "measurements" are then used to fix two of the coefficients in our effective potential expansions. Our potential parametrizations are specified by the number of basis functions we retain. Within a fixed parametrization we can explore how modifying the properties of the fluctuations translates into the Euclidean action. Alternatively, we can test the sensitivity of our results to modeling choices by allowing some of the unfixed parameters to float, providing us with a "theory error bar" of sorts. The interested reader can find more details of the potential modeling in the Appendix. These parametrizations are not meant to be fully realistic, but rather to give a flavor of the impact that using the effective lattice potential in tree-level bounce calculations can have on the inferred decay rate.
In the first two rows of Fig. 5 we show the response of the Euclidean action to changes in the fluctuation statistics. For illustration we use the two term truncation of the cosine series, allowing us to uniquely fix the potential via determination of m 2 FV and m 2 TV . In the absence of corrections and in one spatial dimension, the Euclidean action has the form where C depends only on λ, and not ϕ 0 or m. In the top row of the figure we fix the dynamic range of fluctuations, encoded in k IR and k UV , while allowing ϕ 0 to vary. This adjusts the field variance relative to the curvature scale of the potential minimum. Despite generating relatively large corrections to the effective potential, we see that (in the range of ϕ 2 0 values shown in the plot) the main effect of the corrections is to shift S E by a ϕ 2 0 independent constant. In other words, in the leading approximation to ln Γ as a function of ϕ 2 0 , the intercept with respect to ϕ 2 0 shifts while holding the slope fixed. Of course, since the two curves must converge as ϕ 2 0 → ∞, this cannot be a complete description, and we confirmed that for ϕ 2 0 → ∞, the two curves converge.
In the middle row, we additionally allow the UV cutoff on the fluctuations to vary. The corresponding changes to V eff at a fixed value of ϕ 2 0 while k 2 UV is varied (in logarithmically spaced intervals) are illustrated in the left panel. Somewhat surprisingly, we find the behavior observed in the first row persists throughout the range of k UV values considered here, with the intercept of −S E with respect to ϕ 2 0 increasing with k UV . A priori, it is rather remarkable that the seemingly complicated changes to the effective potential manifest in such a simple way in the Euclidean action. However, we can provide a partial understanding applicable in the regime where the mass deformations are sufficiently small. From our previous analysis, we know that Δm 2 ≈ −αϕ −2 0 for some constant α and sufficiently large ϕ 0 . For dimensional reasons, we expect ΔS ∼ Δm 2 ϕ 2 0 ∼ −α, resulting in a constant positive shift in ΔS. The detailed calculations provided in Fig. 5 seem to indicate that similar scalings apply even when the individual mass corrections no longer scale as ϕ −2 0 , as shown in Fig. 2. In the investigations described above, we truncated our cosine expansions at second order, leaving only two unknown coefficients. Specifying two independent mass measurements uniquely determined the resulting effective potential, so that any uncertainties in this specific truncation arose solely from corrections to our computations of the effective masses. However, since our effective potentials are not derived from first principles, there is significant modeling uncertainty. Some obvious examples of this are our choice of truncation order and our choice to directly expand the effective potential (instead of another quantity such as σ 2 ϕ ) in a cosine series. To illustrate one possible effect of this modeling uncertainty, in the bottom row of Fig. 5 we instead truncate the expansion at third order, introducing a single tunable parameter. As outlined in the Appendix, it is convenient to take this free parameter to be the vacuum energy splitting, as it is both physically transparent and straightforward to directly relate to the expansion coefficients. Since the energy splitting can be expressed as it encodes (integrated) information about the shape of the potential force between the two minima. In the bottom row of Fig. 5, we allow the dimensionless energy splitting Δρ m 2 ϕ 2 0 to vary, resulting in the potential deformations shown in the left panel. From the right panel, we see that unlike the previous cases, the primary effect of these variations is to change the slope of S E with respect to ϕ 2 0 . Of course, such modification can only apply in a limited window of ϕ 0 , since as ϕ 0 → ∞ at fixed k UV we must recover the bare potential, for which the vacuum energy splitting Δρ ¼ 2m 2 ϕ 2 0 is known.

V. FLUCTUATION DEPENDENCE OF REAL-TIME DECAY RATES
As shown above, the effective frequency of small amplitude oscillations around local potential minima is FIG. 5. An illustration of the impact of potential deformations on the Euclidean action obtained in the bounce formalism. In all cases, we use a bare lattice potential with λ ¼ 2 and IR spectral cutoff k IR ¼ π 64 . Unless otherwise specified, the fluctuations have a UV spectral cutoff k UV ¼ 2π 64 × 511.5. In the left column, we illustrate the assumed potential deformations. Meanwhile, the right column shows the corresponding response of the Euclidean action. The first two rows show the impact of changing the fluctuation properties, assuming the two-term cosine expansion is valid, while the final row demonstrates some of the modeling uncertainty by considering a three-term cosine expansion. The Euclidean action is only directly influenced by the parts of the potential probed by the bounce solution. We have indicated these regions with dark shading in the left plots. The extension of the potentials beyond the region probed by the bounce is indicated with partially transparent shading. Top row: we vary ϕ 0 , thus adjusting the fluctuation amplitude relative to the bare curvature scale of the false vacuum. The effective potentials in the left panel are color coded by the value of ϕ 0 , which match the corresponding Euclidean actions in the right panel. For reference, the bare lattice potential and corresponding Euclidean action are shown as a black dashed line. Middle row: in the middle row we show the impact of adjusting the spectral cutoff of the fluctuations. The left panel shows how the effective potential changes at a fixed value ϕ 2 0 ¼ 5, again assuming the two-term cosine expansion. The color coding increases in darkness as k UV is increased. The response in the Euclidean for the same values of k UV , but scanning across ϕ 0 values, is shown in the right panel. Bottom row: finally, the bottom row demonstrates one possible impact of potential modeling uncertainty by extending to the three-term cosine expansion. For concreteness, we fix the new free parameter by tuning the dimensionless energy splitting Δρ The effective mass squared is itself determined by the properties of the initial fluctuations. In a similar fashion, we expect that the effective potential for the long-wavelength field modes will be modified, with a resulting change to the measured real-time decay rates. To investigate this, we now show how the real-time decay rates respond to two natural modifications of the initial fluctuation state. Analysis for a broader range of parameters will be presented in future work. Before presenting the results, we briefly summarize our procedure to extract the decay rate. We first determine the volume average hcosðϕ=ϕ 0 Þi V ≡ c for each simulation. We then define the decay time t decay of a single simulation to be the first time that c crosses the threshold value c cut ¼ 0.7. From the collection of decay times t ðiÞ decay for a given ensemble, we compute the empirical survival probability P survive as a function of time. Finally, we carry out a linear fit ln P survive ¼ −Γt þ A to extract Γ. 5 Our choice of spectral cutoffs was guided by the requirement that the nucleation times in each individual simulation were insensitive to the spatial resolution of the grid for a fixed choice of initial condition.
Since the vacuum state of a free harmonic oscillator is dictated by its oscillation frequency, an accurate model of the Gaussian false vacuum should use pseudovacuum initial conditions with the self-consistently determined m 2 PS ¼ m 2 eff . However, existing analyses initialize the fluctuations based on linear perturbation theory with m 2 PS ¼ V 00 bare ðϕ FV Þ, thus effectively starting the simulations from an excited state. Further, we demonstrated above that the field variance, and thus the effective mean field force, undergoes nontrivial dynamical evolution if the fluctuations are initialized away from the self-consistently determined vacuum. It is therefore of interest to understand if the FIG. 6. Decay rates (per unit volume) extracted from real-time lattice simulations for two deformations of the initial conditions. For illustration, and to compare with previous work, we have used a bare lattice potential with λ ¼ 1.2. In all cases, we restricted our fits to times less than the light-crossing time, although we verified that our results were insensitive to using decays within three light-crossing times. Left: we vary the pseudovacuum initial conditions by scanning the mass parameter m 2 PS =m 2 B . Since the corrections to m 2 B are negative for our fiducial potential, we vary m 2 PS =m 2 B in the interval [0. . Each data point was extracted from an ensemble of 1000 simulations. Right: the measured decay rates as we vary the spectral cutoff k cut , holding all other model and lattice parameters fixed. For these simulations, we used a box size mL ¼ 32 on pseudovacuum initial conditions with bare vacuum mass m 2 PS ¼ m 2 ðλ 2 − 1Þ. Based on the left panel, we do not expect the results to depend on the choice of m 2 PS . The change in the decay rate is approximated by a vertical shift in lnðΓ=m 2 LÞ while holding the slope with respect to ϕ 2 0 fixed. This agrees with the qualitative behavior of the Euclidean rates in Fig. 5. The decay rates were extracted using ensembles of 3000 simulations. The light shaded regions are an estimate of the statistical errors associated with the finite size of our ensembles. 5 For some choices of k cut and ϕ 2 0 , the survival probabilities displayed two different exponentially decaying regimes. In these cases, we have treated the earlier exponential as a transient and fit the later time exponential. Specifically, if the survival fraction after one-light-crossing time was less that 0.1, we fit to the region 0.1 ≤ P survive ≤ 0.6. Otherwise, we use all of the decays with P survive < 0.99. We verified that the extracted decay rate curves were insensitive to these particular choices of probability cutoffs (with the caveat that we fit only a single exponential region), as well as to the particular choice of threshold c cut . decay rate is sensitive to the choice of m 2 PS . The left panel of Fig. 6 explores this for a range of pseudovacua initial conditions with m 2 PS ∈ ½0.1m 2 B ; m 2 B and the particular choice λ ¼ 1.2. Remarkably, we see that the measured decay rate is independent of the initial m 2 PS , at least over the range of values considered here, which indicates that the measured decay rates are robust to deformations of the vacuum state arising from modifications of the effective potential. This is despite the fact that we expect the fluctuations to dynamically rearrange themselves, and we have effectively started the system in an excited state. A possible explanation is that the asymptotic field variance is relatively flat over the range of mass deformations we simulated, as seen (for a different choice of λ) in Fig. 3. As a result, the system may be rapidly equilibrating itself to a near-vacuum state that is not so different from the Gaussian vacuum. A more definitive understanding requires further investigation.
Finally, we examine the dependence of the real-time decay rates on the spectral cutoff k cut used in the lattice simulations in the right panel of Fig. 6. By increasing the cutoff, we are explicitly modeling more and more of the short-wavelength modes, rather than assuming they can be captured by simple modifications to the IR dynamics. This type of modification of the initial conditions is of great interest, as it connects directly to the standard view of renormalization. Analogous to the variation in the pseudovacuum initial conditions, we considered a range of values for k cut in the model with λ ¼ 1.2. Within the range of ϕ 0 values considered here, we see that impact of changing k cut is well modeled as a constant shift in the numerical value of ln Γ at fixed ϕ 2 0 , while maintaining its slope as a function of ϕ 2 0 . This qualitative behavior matches what we found for the decay rates predicted by the Euclidean formalism as the UV cutoff k UV was increased. We leave a more detailed comparison between the k cut dependence in the real-time simulations and the k UV dependence of the Euclidean rates to future work. For now we simply note that this can provide either a nontrivial way to differentiate the predictions of the two approaches, or an additional parameter to constrain our effective potential construction.

VI. CONCLUSIONS
We studied renormalization effects that result from realized vacuum fluctuations in nonlinear lattice simulations. Our focus is on relativistic scalar fields sitting at local potential minima. First, we derived a quantitative prediction for the shift in the effective mass of the field and compared this prediction to measurements of oscillation frequencies in lattice simulations, finding excellent agreement. Further, we demonstrated the variance of the field fluctuations was constant in time for this special choice of mass-a necessary condition for a vacuum state. We then used our calculations of the mass modifications to partially fix the shape of the modified effective potential and studied the resulting changes to the false vacuum decay rates Γ predicted by the Euclidean formalism, showing reasonable deformations can both shift the overall scale and slope of ln Γ with respect to ϕ 2 0 . Finally, we considered the sensitivity of decay rates measured in real-time simulations to variations of the effective mass (m 2 PS ) and spectral cutoff (k cut ) defining the initial fluctuation spectra. The measured decay rates were insensitive to the specific choice of m 2 PS , but had a similar dependence on k cut as the Euclidean rates. Fully understanding the impact of these fluctuations on predictions of measurable quantities (such as decay rates) is important for comparing the results of real-time lattice simulations both with other theoretical approaches and with experiments. In light of upcoming analog relativistic false vacuum decay experiments [1][2][3][4][5][6][7][8][9][10], these studies are particularly important to make optimal use of the experimental results.
Under the assumption of Gaussian fluctuations, we obtained the form of the effective potential force in terms of the field fluctuation variance. From this, we derived the expected change in the effective mass of the field at the local minima of the potential. We compared these analytic predictions to full nonlinear lattice simulations, and found excellent agreement across the range of parameters explored here. In particular, the regime of validity of our Gaussian resummation approach was significantly larger than the one-loop approximation. If we allow for increasingly large fluctuation amplitudes, we expect non-Gaussian corrections to become important, but we leave an exploration of this regime to future work.
Motivated by these observed modifications to the effective mass, we studied possible impacts on the decay rate predicted by the Euclidean formalism. The periodicity and symmetry properties of the bare potential motivated an expansion of the effective potential in a cosine basis. Truncating this expansion at second order allowed us to fix the potential coefficients using the true and false vacuum masses predicted by our formalism. Under these assumptions, we found that the primary effect of the renormalization corrections was to shift the Euclidean action by a ϕ 0 independent, but k cut dependent constant. Finally, we explored sensitivity to our specific modeling choice by including an additional term in the expansion and fixing it by allowing the vacuum energy splitting to vary. We found that this modification adjusted the slope of the Euclidean action with respect to ϕ 2 0 . These findings motivate a more detailed study of the impacts renormalization corrections can have on the Euclidean action, which we leave to future work.
Finally, we explored the impact of modifying the fluctuation content on the decay rates extracted from lattice simulations. In particular, we considered the sensitivity of the decay rate to two deformations of the initial conditions: (i) a variation in the initial pseudovacuum initial mass, and (ii) changes to the spectral cutoff k cut . We found the decay rate was insensitive to the initial mass used in the spectrum. Meanwhile, increasing the spectral cutoff k cut led to an approximately ϕ 0 independent shift in the logarithmic decay rate. This latter observation matches the qualitative behavior observed in the Euclidean decay rates under variations of k cut .
This work suggests a number of future directions. While we presented some simplified models for the renormalization effects in the effective potential, it would be interesting to perform a more systematic construction of the potential away from the minimum, including the effects of the non-Gaussian contributions neglected here. This includes pushing the fluctuations into a regime where the Gaussian approximation begins to fail. It is also of interest to understand how the fluctuations restructure themselves when starting from an excited pseudovacuum state. Further, the potential considered here was special in the sense that the local minima were symmetric. Extending the analysis to asymmetric minima introduces new effects, which will be presented in future work. Finally, it is of interest to more rigorously explore the connection the effective lattice potential studied here, and the effective potential identified in Euclidean path integral calculations.

Calibration to effective mass measurements
A full first principles calculation of the c n 's is beyond the scope of this paper. However, we have an empirically verified theoretical understanding of the effective masses around symmetric potential minima. Therefore, we take a more pedestrian approach and assume the squared effective masses at the false (m 2 FV ) and true (m 2 TV ) vacua are known, and use these to fix the unknown coefficients in our potential expansions.
For the case of the cosine expansion, the expansion coefficients c n must be calibrated so that V 00 eff ð0Þ ¼ m 2 FV and V 00 eff ðπÞ ¼ m 2 TV . At this level of approximation, we can then uniquely fix the potential by truncating after the n ¼ 2 term where the integration constant V FV is fixed to the value of the potential at the false vacuum. Alternatively, we could continue to incorporate additional information or uncertainties by including additional terms in the expansion. For example, truncating at third order and assuming we have also measured the energy splitting between the false and true vacuum, we have V eff m 2 ϕ 2