Flyover Transitions in Vacuum Decay

• Flyover transitions are semiclassical false-vacuum decay processes where a classical scalar field fluctuation overcomes the potential barrier without quantum tunneling.
• The mechanism relies on initial kinetic energy to trigger bubble nucleation, differentiating it from standard instanton tunneling in both thin-wall and thick-wall regimes.
• In multi-vacuum landscapes, flyover transitions can produce nested bubble configurations with implications for gravitational waves and baryogenesis.

Searching arXiv for papers on “flyover transitions” and related vacuum-decay work. Searching arXiv for “flyover vacuum decay” and “flyover transitions”. Flyover transitions are a semiclassical, classically allowed form of false-vacuum decay in which an initially false-vacuum field configuration evolves into the true vacuum without tunneling through the barrier quantum mechanically. In this framework, decay is triggered by initial kinetic energy in the scalar field: a localized fluctuation in the time derivative of the field is large enough, and/or spread over a large enough region, that classical evolution carries part of the configuration over the barrier and nucleates an expanding true-vacuum bubble. In the literature on vacuum decay, the term therefore denotes a hybrid quantum-classical mechanism: the initial fluctuation probability is computed quantum mechanically, while the subsequent nonlinear evolution is treated classically (Wang, 2019).

1. Conceptual status within vacuum decay theory

Flyover transitions were introduced as an alternative to Euclidean tunneling. In the conventional instanton picture, the field penetrates the barrier through a bounce solution in Euclidean time; in the flyover picture, the field instead acquires a sufficiently large fluctuation in $\dot\phi$ and then crosses the barrier by real-time classical evolution. The scalar field starts in a local vacuum, a rare quantum fluctuation of the field velocity occurs in a finite region, and if the fluctuation is large enough the field is pushed over the barrier rather than tunneling through it. After that, the subsequent evolution is treated classically and can form an expanding bubble of the new vacuum (Blanco-Pillado et al., 2019).

This distinction is not merely terminological. The rate estimates in flat space and for downward de Sitter transitions have the same parametric form as instanton tunneling rates, differing only by an $\mathcal{O}(1)$ numerical factor in the exponent, but the dynamical history of bubble formation can be qualitatively different. An important exception arises for upward transitions from a lower-energy de Sitter vacuum to a higher-energy de Sitter vacuum state, where the flyover rate is parametrically different and can be many orders of magnitude higher than tunneling. The literature therefore treats flyover decay not as a rephrasing of Coleman-style nucleation, but as a distinct semiclassical channel with overlapping but not identical phenomenology (Blanco-Pillado et al., 2019).

A recurrent misconception is that flyover decay requires a violent local overshoot at the initial time. The later development of the subject shows that the original local-threshold criterion is sufficient but not necessary. This broadening is central to the modern usage of the term.

2. Classical initial-value problem and barrier-crossing data

A standard realization uses a single scalar field $\phi(t,r)$ in flat space with spherical symmetry, starting from a homogeneous false vacuum,

$\phi(t=0,r)=0.$

The field evolves according to

$$\ddot{\phi}=\phi''+\frac{2}{r}\phi'-\frac{dV}{d\phi},$$

with regularity condition

$\phi'(t,r=0)=0.$

In the model used to study the occurrence of semiclassical vacuum decay, the potential is

$$V(\phi)=\phi^2-2(1+2\omega)\phi^3+(1+3\omega)\phi^4.$$

It has a false vacuum at $\phi_+=0$ with $V_+=0$, a true vacuum at $\phi_-=1$ with $\mathcal{O}(1)$0, vacuum energy difference $\mathcal{O}(1)$1, and a barrier top at $\mathcal{O}(1)$2 (Wang, 2019).

The initial fluctuation is inserted into the time derivative rather than into the field value: $\mathcal{O}(1)$3 Thus the field starts at the false vacuum, but with a Gaussian kick in velocity. A convenient normalized form is written using

$\mathcal{O}(1)$4

so that $\mathcal{O}(1)$5 is special: at $\mathcal{O}(1)$6 the field velocity equals the critical velocity

$\mathcal{O}(1)$7

A critical radius $\mathcal{O}(1)$8 is defined as the largest radius where the local velocity exceeds the barrier-crossing threshold, namely where $\mathcal{O}(1)$9. If $\phi(t,r)$0, then $\phi(t,r)$1 does not exist: the velocity is nowhere locally large enough to cross the barrier (Wang, 2019).

The earlier stochastic treatment uses the same basic logic in a more generic notation,

$\phi(t,r)$2

and gives the essential barrier-crossing condition as

$\phi(t,r)$3

Here the factor $\phi(t,r)$4 reflects the requirement that the Gaussian fluctuation be sufficiently large over a region of size $\phi(t,r)$5 rather than only at a single point (Blanco-Pillado et al., 2019).

3. Original flyover criterion and its geometric interpretation

The original flyover picture imposed two simultaneous requirements. First, there had to be a region where the local velocity was supercritical,

$\phi(t,r)$6

Second, that supercritical region had to be at least as large as the minimal expanding bubble size,

$\phi(t,r)$7

where

$\phi(t,r)$8

is the wall tension (Wang, 2019).

In this formulation, flyover decay is not produced by an arbitrarily narrow overshoot. A localized excursion can cross the barrier and still fail to seed a growing bubble if the excited region is too small. The relevant object is therefore a finite patch of initial data that is both supercritical in velocity and wide enough to nucleate an expanding configuration. This geometric requirement aligns with the earlier analytic estimates for thin-wall bubbles, where the minimal bubble size is taken as

$\phi(t,r)$9

and the rate estimate becomes

$\phi(t=0,r)=0.$0

to be compared with Coleman’s

$\phi(t=0,r)=0.$1

The parametric dependence is the same, but the exponent differs by an $\phi(t=0,r)=0.$2 numerical factor (Blanco-Pillado et al., 2019).

This criterion gave the earliest operational meaning of “flyover”: there must exist a locally supercritical patch, and that patch must be wide enough to behave as a critical bubble rather than a transient local excursion. Subsequent work retained this regime as a valid decay channel, but no longer treated it as exhaustive.

4. Broad subcritical profiles and the “pop-up” generalization

A central result in the later development is that local supercriticality is not necessary. Vacuum decay can still happen even when the initial velocity is nowhere locally large enough to classically cross the barrier, provided the velocity profile extends over a sufficiently large region. This is the “pop-up vacuum decay” mechanism: the field initially lacks any point with $\phi(t=0,r)=0.$3, but the total kinetic energy is large enough, nonlinear classical evolution redistributes energy, gradient energy builds up, a bubble wall emerges later, and the bubble then expands (Wang, 2019).

The analysis introduces integrated energies over the simulation region,

$\phi(t=0,r)=0.$4

with total energy

$\phi(t=0,r)=0.$5

For the Gaussian initial profile,

$\phi(t=0,r)=0.$6

The key claim is that there exists a critical energy threshold $\phi(t=0,r)=0.$7 such that decay occurs when

$\phi(t=0,r)=0.$8

Numerically, the onset of bubble formation is tied to a characteristic gradient-energy scale,

$\phi(t=0,r)=0.$9

For the thick-wall case, where $$\ddot{\phi}=\phi''+\frac{2}{r}\phi'-\frac{dV}{d\phi},$$0, this becomes

$$\ddot{\phi}=\phi''+\frac{2}{r}\phi'-\frac{dV}{d\phi},$$1

Using $$\ddot{\phi}=\phi''+\frac{2}{r}\phi'-\frac{dV}{d\phi},$$2, the approximate condition becomes

$$\ddot{\phi}=\phi''+\frac{2}{r}\phi'-\frac{dV}{d\phi},$$3

and, in the thick-wall estimate,

$$\ddot{\phi}=\phi''+\frac{2}{r}\phi'-\frac{dV}{d\phi},$$4

The control parameter is therefore roughly $$\ddot{\phi}=\phi''+\frac{2}{r}\phi'-\frac{dV}{d\phi},$$5, that is, the total kinetic energy rather than the existence of a locally supercritical point (Wang, 2019).

This broadens the concept of flyover transitions. Originally, flyover meant locally supercritical velocity over a wide enough region. In the generalized picture, global energy content and spatial extent matter even if no point is individually supercritical. A plausible implication is that the physically relevant distinction is not between “over the barrier” and “under the barrier” at a single point, but between initial data that can or cannot nonlinearly seed an expanding bubble.

5. Bubble formation dynamics, rates, and gravitational settings

The dynamical history of flyover nucleation differs from the standard instanton picture. In thin-wall numerical simulations in flat space, the field at the center crosses the barrier first, but the region is initially too small to make a stable expanding bubble. The center oscillates between false and true vacuum; only later does a shell form at radius $$\ddot{\phi}=\phi''+\frac{2}{r}\phi'-\frac{dV}{d\phi},$$6; the shell separates an oscillating interior from the false-vacuum exterior; the shell has an inner and outer boundary; and both boundaries propagate at nearly the speed of light. In the thick-wall regime, by contrast, the fluctuation region is large enough that almost the whole region goes over the barrier together, no prominent shell structure appears, and the bubble resembles the usual expanding true-vacuum bubble more closely, although the interior still oscillates around the true vacuum (Blanco-Pillado et al., 2019).

The probability for the initial fluctuation is Gaussian,

$$\ddot{\phi}=\phi''+\frac{2}{r}\phi'-\frac{dV}{d\phi},$$7

For flat space, the rms smeared velocity fluctuation is

$$\ddot{\phi}=\phi''+\frac{2}{r}\phi'-\frac{dV}{d\phi},$$8

and

$$\ddot{\phi}=\phi''+\frac{2}{r}\phi'-\frac{dV}{d\phi},$$9

These estimates underwrite the rate comparisons between flyover decay and instanton tunneling in thin-wall and thick-wall limits (Blanco-Pillado et al., 2019).

De Sitter space introduces a sharp division between downward and upward transitions. For downward de Sitter transitions, the flyover rate remains parametrically the same as in flat space. For upward transitions from a lower-energy de Sitter vacuum to a higher-energy de Sitter vacuum, however, the instanton result obeys detailed balance,

$\phi'(t,r=0)=0.$0

whereas the flyover estimate gives

$\phi'(t,r=0)=0.$1

with $\phi'(t,r=0)=0.$2 for $\phi'(t,r=0)=0.$3. The upward flyover rate is therefore parametrically much larger than the instanton prediction. In strong-gravity upward cases, numerical solutions show that a false-vacuum bubble can inflate behind a wormhole and eventually appear to the parent observer as a baby universe inside a black hole (Blanco-Pillado et al., 2019).

These results establish two separate points. First, flyover decay can reproduce the semiclassical parametric scaling familiar from tunneling while preserving a distinct real-time dynamical narrative. Second, gravitational settings can expose genuinely new behavior rather than merely altered prefactors.

6. Multiple vacua, nested bubbles, and scope of the concept

In a scalar field theory with multiple metastable vacua, flyover transitions can produce configurations unavailable to ordinary tunneling dynamics. A three-vacuum toy potential is used to study a landscape with minima $\phi'(t,r=0)=0.$4, two barriers of heights $\phi'(t,r=0)=0.$5 and $\phi'(t,r=0)=0.$6, and energy gaps $\phi'(t,r=0)=0.$7 and $\phi'(t,r=0)=0.$8. The field begins in the highest metastable vacuum $\phi'(t,r=0)=0.$9 and receives the same type of Gaussian velocity fluctuation,

$$V(\phi)=\phi^2-2(1+2\omega)\phi^3+(1+3\omega)\phi^4.$$0

with amplitude $$V(\phi)=\phi^2-2(1+2\omega)\phi^3+(1+3\omega)\phi^4.$$1 and width $$V(\phi)=\phi^2-2(1+2\omega)\phi^3+(1+3\omega)\phi^4.$$2 representing the fluctuation energy and coherence length. To seed a growing bubble, the width must exceed a critical scale,

$$V(\phi)=\phi^2-2(1+2\omega)\phi^3+(1+3\omega)\phi^4.$$3

and the amplitude must satisfy $$V(\phi)=\phi^2-2(1+2\omega)\phi^3+(1+3\omega)\phi^4.$$4 (Wei et al., 31 Aug 2025).

The principal new object is the double-layered vacuum bubble. When $$V(\phi)=\phi^2-2(1+2\omega)\phi^3+(1+3\omega)\phi^4.$$5 and $$V(\phi)=\phi^2-2(1+2\omega)\phi^3+(1+3\omega)\phi^4.$$6 are sufficiently large, the central region of the perturbation has enough kinetic energy to fly over both barriers and reach $$V(\phi)=\phi^2-2(1+2\omega)\phi^3+(1+3\omega)\phi^4.$$7, while the outer region only crosses the first barrier and settles into $$V(\phi)=\phi^2-2(1+2\omega)\phi^3+(1+3\omega)\phi^4.$$8. The result is an inner $$V(\phi)=\phi^2-2(1+2\omega)\phi^3+(1+3\omega)\phi^4.$$9 bubble wrapped inside an outer $\phi_+=0$0 shell. By contrast, although a formal $\phi_+=0$1 bounce exists in the tunneling picture, the direct critical profile does not reliably create a stable double-layered bubble in this model: one segment can grow, but the other collapses. The paper therefore concludes that, in this setting, quantum tunneling behaves dynamically like a $\phi_+=0$2 transition only, whereas flyover decay can stabilize the nested structure (Wei et al., 31 Aug 2025).

The subsequent wall dynamics are controlled by pressure differences across the relevant interfaces. If the inner wall moves faster than the outer wall, it can catch up and collide with it, causing the two layers to merge and destroying the double-layered configuration; if the outer wall moves faster, the double structure can persist. Collision studies with two aligned double-layered bubbles show outer-wall collisions, local conversion from $\phi_+=0$3 toward $\phi_+=0$4, the generation of trapped regions, and eventual relaxation to $\phi_+=0$5. The authors identify gravitational wave production and baryogenesis as promising consequences, while explicitly noting that these observables are not computed in that work (Wei et al., 31 Aug 2025).

Taken together, these developments define the modern meaning of flyover transitions in high-energy theory and cosmology. They are vacuum-decay events driven by classical evolution from false-vacuum fluctuations rather than by Euclidean tunneling; they may be triggered either by a locally supercritical fluctuation or by a sufficiently broad subcritical one whose total kinetic energy exceeds a critical threshold; and, in multi-vacuum landscapes, they can generate bubble configurations that tunneling alone typically cannot maintain.