Vacuum Decay and Euclidean Lattice Monte Carlo Jiayu Shenabc 1 Patrick Draperabc and Aida X. El-Khadraabc aIllinois Quantum Information Science and Technology Center Urbana Illinois 61801

2025-04-26 0 0 993.82KB 34 页 10玖币

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Vacuum Decay and Euclidean Lattice Monte Carlo

Jiayu Shena,b,c,1, Patrick Drapera,b,c, and Aida X. El-Khadraa,b,c

aIllinois Quantum Information Science and Technology Center, Urbana, Illinois 61801

bIllinois Center for Advanced Studies of the Universe, Urbana, Illinois 61801

cDepartment of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois

61801

Abstract

The decay rate of a metastable vacuum is usually calculated using a semiclassical

approximation to the Euclidean path integral. The extension to a complete Euclidean

lattice Monte Carlo computation, however, is hampered by analytic continuations that

are ill-suited to numerical treatment, and the nonequilibrium nature of a metastable

state. In this paper we develop a new methodology to compute vacuum decay rates

from Monte Carlo simulations of Euclidean lattice theories. To test the new method,

we consider simple quantum mechanical systems systems with metastable vacua. This

work can be extended to Euclidean ﬁeld theories, which we discuss in the Conclusions.

1jiayus3@illinois.edu

arXiv:2210.05925v2 [hep-lat] 26 May 2023

1 Introduction

The decay of a metastable vacuum state is an old and well-studied problem in quantum

mechanics (QM) and quantum ﬁeld theory (QFT). It is well-known how to compute the

tunneling rate in QM using semiclassical methods, and these techniques can be extended in

a natural way to QFT [1, 2]. In recent years the theory of tunneling has received renewed

attention [3–11].

Since the standard semiclassical analysis is performed using the Euclidean path integral,

it is natural to ask whether Euclidean lattice theory can also be used to study vacuum

decay. In addition to ordinary barrier penetration problems, lattice methods could be useful

for quantitative studies of vacuum decay in situations where the semiclassical methods are

inadequate, such as the decay of vacua that emerge from strong dynamics (see e.g. Ref. [12]).

Formulating and reﬁning a lattice approach to these problems might also yield methods of

more general interest and applicability.

However, Euclidean Monte Carlo (MC) simulations of false vacua are not without sub-

tleties. A conﬁguration which begins in a metastable state, or in a false vacuum (FV), will

evolve in Monte Carlo time to eventually thermally ﬂuctuate over the barrier. In the semi-

classical limit, the barrier “peak” is a saddle point of the classical action, a solution known as

the bounce [1], and the Monte Carlo time evolution can be thought of schematically as “false

vacuum →bounce →true vacuum.” If the true vacuum (TV) is deep, as a practical matter,

the system will never return to the false vacuum after thermalization, so all conﬁgurations in

the thermalized ensemble describe the true vacuum. They are exponentially more important

than the bounce and they are only rendered innocuous after a ﬁnal analytic continuation

back to real time, a point emphasized in the study of Ref. [3] which sought to place the

problem of vacuum decay on more rigorous footing. This analytic continuation is more or

less straightforward in semiclassical analyses, but it is impractical in an MC approach.

In this paper, we develop a new framework to compute approximate but accurate decay

rates from Euclidean lattice simulations. To test the approach, we consider QM tunneling

problems as illustrated in Fig. 1. Our primary results are the deﬁnition of a new observable

that approximates the decay rate of a quantum mechanical metastable vacuum, a prescription

for its computation in Euclidean Monte Carlo simulations, and numerical simulations testing

the accuracy of the method.

The remainder of this paper is organized as follows. In Sec. 2 we develop the necessary

theoretical tools, deﬁne our computational approach, and describe the systematic uncer-

tainties introduced by the associated approximations. In Sec. 3 we apply the method to

a representative family of potentials. An advantage of studying QM tunneling problems

is the ability to compute the decay rate by solving the time-dependent Schr¨odinger equa-

tion (TDSE). We perform three-way comparisons between results obtained from solving the

TDSE (“exact”), from Euclidean lattice Monte Carlo computations (“lattice”), and from

semiclassical analyses. We ﬁnd good agreement between the results over several decades in

the decay rate, thus establishing the accuracy of our lattice method. In Sec. 4 we turn our

attention to very long lifetimes, where computing the rate from ensembles of practical sizes

requires a diﬀerent approach. We propose the “constrained ensemble reweighting” method

and illustrate it with an example. Our conclusions are presented in Sec. 5, where we further

outline how our framework can be extended to Euclidean quantum ﬁeld theories.

2 Vacuum Decay in Euclidean Lattice Theory

2.1 Preliminaries

xFV b

xTV

Figure 1: Example potential V(x). xFV is the local potential minimum corresponding to the false

vacuum. xTV is the starting position of a global-minimum plateau region of the potential. bis the

classical turning point that satisﬁes V(b) = V(xFV). R={x|V(x)< VFV}={x|x>b}is the

classically allowed region.

We consider single-particle quantum mechanics with a tunneling potential. An example

potential is shown in Fig. 1. The continuum Euclidean action is

SE=Zdt 1

2dx

dt 2

+V(x)!.(1)

In this normalization xis treated as a 0 + 1D ﬁeld: the kinetic term has a dimensionless

coeﬃcient 1/2, so that the dimension of xis [x] = [E−1/2]. This deﬁnition of xis used

throughout this paper. With the false vacuum positioned at xFV = 0, we parametrize the

leading term in the expansion of the potential around xFV as V(x) = 1

2m2x2+. . .. Since

this term has the same form as the mass term in scalar ﬁeld theories, we can consider the

dimensionful parameter mas the mass of the particle. A more detailed description of the

potential is given in Sec. 3.1.

The continuum Euclidean path integral facilitates a convenient semiclassical treatment

of false vacuum decay. One ﬁrst constructs the bounce, a solution xb(t) to the Euclidean

equations of motion that asymptotes to the classical false vacuum at early and late times.

The leading order (LO) decay rate is governed by the bounce action, Γ ∼e−SE[xb]. The next-

to-leading-order (NLO) correction is given by the quadratic ﬂuctuation integrals around

the bounce. In these integrals the low lying modes of the ﬂuctuation operator must be

treated separately. Zero modes associated with symmetries can be treated with a collective

coordinate method. More importantly, the bounce is always associated with a single mode

of negative eigenvalue. The integral over the amplitude of this mode is divergent and is

generally deﬁned by analytic continuation.

On the lattice, a simple choice for the discretized action is

Slat =a

i=1 −1

2xi

xi+1 −2xi+xi−1

a2+V(xi),(2)

where ais the lattice spacing and NT= 2T/a is the total number of sites (2Tis the total

time). The diﬀerence between the lattice action and the continuum action is O(a2) due to

the discrete second-order derivative.

In order to study vacuum decay in Euclidean lattice Monte Carlo simulations, we must ﬁrst

identify an observable that can be related to the desired decay rate and computed with Monte

Carlo methods. We show that the probability density to ﬁnd the particle at the classical

turning point has the desired properties and describe its computation with Euclidean path

integrals and its relation to the decay rate in Sec. 2.2.

Any continuum calculation in Euclidean time must be analytically continued to real time.

However, such continuations are impractical in lattice Monte Carlo computations because

they require exponential sensitivity. We elaborate on the problem in Sec. 2.3 and deﬁne a

procedure that avoids the need for analytic continuation, removing the exponential sensitivity

requirement, at the cost of introducing a systematic error.

2.2 Probability Densities from Euclidean Path Integrals

The probability density for the system to be in the state |x⟩at time t, given that we started

from a normalized state ψat t= 0, is

ρ(x, t) = |⟨x, t|ψ, 0⟩|2.(3)

When |ψ⟩=|FV⟩, a metastable state localized near the classical false vacuum, the decay

rate is deﬁned as

Γ = −lim

T→∞

P(FV, T )

dP (FV, T )

dT ,

P(FV, T )≡ZFV

dx ρ(x, T ) = ZFV

dx |⟨x, T |FV,0⟩|2,

P(R, T )≡ZR

dx ρ(x, T ) = 1 −P(FV, T ).(4)

The result for Γ should not be sensitive to the exact deﬁnition of the FV region, as long

as it reasonably contains the point xFV and does not extend beyond b. The long Tlimit

of Eq. (4) is satisﬁed when Tis large compared to the “escape attempt time” ∼1/m in

the false vacuum, 1/m ≪T. If we consider times within the long Tlimit that are short

compared to 1/Γ, then the probability P(FV, T )≈1, and the decay rate can be estimated

Γ≈ − ˙

P(FV, T ) = ˙

P(R, T ),1/m ≪T≪1/Γ (5)

in this regime.

Now let us relate ˙

P(R, T ) to ρ. We have

P(R, T ) = ZR

dx ˙ρ(x, T ) = j(b, T ).(6)

Here j(b, T ) is a probability current ﬂowing through x=b, and we have used the continuity

equation ˙ρ(x, T ) = −∂xj(x, T ). We can also deﬁne a probability ﬂow velocity uthrough

j(x, T )≡u(x, T )ρ(x, T ).(7)

Semiclassically, the probability ﬂow velocity can be estimated from the classical deﬁnition

of the kinetic energy EFV −V(x) = (1/2)ucl(x, T )2, where EFV ≈(1/2)mis the quantum

vacuum energy of the approximate quadratic potential centered at xFV. For 1/m ≪T≪1/Γ

and x=b, we have u(b, T )≈√m.

The relationship u(b, T )≈√mis easily validated for speciﬁc examples by the numerical

solution of the time-dependent Schr¨odinger equation (TDSE). In Fig. 2, we compare u=j/ρ

from the full quantum mechanics and the approximation ucl =p2(EFV −V(x)), exhibiting

good agreement when x≳b.ucl is not expected to match j/ρ in the classically forbidden

region, i.e., when xis substantially smaller than b.

Therefore, if ρ(b, T ) can be computed by other means, then the decay rate can be estimated

Γ≈˙

P(R, T ) = j(b, T )≈√mρ(b, T ).(8)

The advantage of this formulation is that ρ(b, T ) can be evaluated with a Euclidean path

integral and is approximately independent of Tin the time range of interest 1/m ≪T≪1/Γ

described above. We deﬁne a Euclidean transition amplitude,

A(ψ, b;T) = ⟨b|e−HT |ψ⟩=Zdy ψ(y)K(y, b;T),(9)

where the Euclidean propagator over time Tbetween some yand zis

K(y, z;T) = Zx(τ=T)=z

x(τ=0)=yDx e−RT

0dτLE[x].(10)

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VacuumDecayandEuclideanLatticeMonteCarloJiayuShena,b,c,1,PatrickDrapera,b,c,andAidaX.El-Khadraa,b,caIllinoisQuantumInformationScienceandTechnologyCenter,Urbana,Illinois61801bIllinoisCenterforAdvancedStudiesoftheUniverse,Urbana,Illinois61801cDepartmentofPhysics,UniversityofIllinoisatUrbana-Champaign,...

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Vacuum Decay and Euclidean Lattice Monte Carlo Jiayu Shenabc 1 Patrick Draperabc and Aida X. El-Khadraabc aIllinois Quantum Information Science and Technology Center Urbana Illinois 61801

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