Metastability and Vacuum Decay

Metastable vacuum decay is not determined by a barrier height alone. A rate claim requires a false-vacuum persistence observable, a finite-action bounce with return-to-false-vacuum boundary conditions, the correct instability and zero modes, a renormalized prefactor, and quantitative control of approximations and environmental effects. This chapter develops that chain in flat spacetime at zero temperature and gives exact transfers when multifield geometry, thermal physics, or gravity changes the problem.

Helpful background. What an interacting Lagrangian does and does not specify supplies the state, stability, regulator, and observable data that a scalar potential does not fix by itself. Cross sections and decay rates supplies the distinction between a total decay probability, a rate, and a rate density.

Enter this chapter

The core flat-space result in $d$ spacetime dimensions has the form

$$\gamma_0 := \frac{\Gamma}{V_{d-1}} = A\,e^{-B}, \qquad [\gamma_0]=\text{mass}^d.$$

Here $B$ is the false-vacuum-subtracted Euclidean action of the relevant bounce, and $A$ contains the renormalized nonzero-mode determinant, collective-coordinate Jacobians, and normalization factors. This compact formula is the end of the calculation, not its starting assumption.

The pages use

$$\Omega_{d-1} = \frac{2\pi^{d/2}}{\Gamma(d/2)}$$

for the area of the unit $(d-1)$-sphere. At zero temperature in noncompact flat spacetime, an $O(d)$ bounce has $d$ translation zero modes. The standard leading decay saddle has exactly one relevant negative mode under the canonical least-action hypotheses.

Choose the calculation you need

Your question	Start here	Result and next check
What boundary-value problem describes escape from a false vacuum?	Bounce solutions and false-vacuum boundary conditions	Derive the $O(d)$ radial equation, bracket the solution, and distinguish a decay bounce from an interpolating instanton.
How does a Euclidean saddle become a rate density?	Decay rates, the negative mode, and prefactors	Track the negative direction, translation Jacobians, determinant, spacetime volume, and mass dimension.
Is spacetime rotational symmetry justified, and may several fields be replaced by one path?	Bounce existence, symmetry, and multifield geometry	State the existence theorem’s hypotheses and test the full normal field-space residual.
When are radius and surface-tension formulas accurate?	Thin-wall control and corrections	Derive $R_{\mathrm{tw}}$ and $B_{\mathrm{tw}}$, then compare them with a converged full bounce at the logarithmic accuracy required.
Why do the effective potential and profile depend on gauge or scale while the rate does not?	Gauge and renormalization-scale dependence	Use Nielsen and RG identities at one consistent order across the profile, derivative terms, determinant, and prefactor.
Can the flat zero-temperature result be quoted in the declared environment?	Validity limits and thermal and gravitational handoffs	Return a pass, qualified pass, or fail decision and transfer to the finite-temperature or gravitational treatment when required.

If you already have a numerical profile, do not begin with its action. Start with the boundary-condition page to verify that it solves the correct false-vacuum problem, then use the existence and spectrum pages before forming a rate.

The logic of a rate claim

The necessary sequence is:

1. Define the observable. Specify the prepared false-vacuum state and its persistence probability per unit spatial volume.
2. Find a stationary saddle. Solve the regular Euclidean boundary-value problem and subtract the false-vacuum action.
3. Justify reductions. Establish any $O(d)$ symmetry and test every discarded multifield or constrained equation.
4. Classify fluctuations. Verify one relevant negative mode, all collective-coordinate zero modes, and a controlled nonzero spectrum.
5. Renormalize the prefactor. Use one regulator, subtraction prescription, and parameter convention for the saddle and determinant.
6. Test approximations. Compare thin-wall, derivative, loop, gauge, scale, finite-volume, and numerical errors with the target error in $\ln\gamma_0$.
7. Test the environment. Replace the calculation when the Euclidean time circle, plasma dynamics, curvature, or gravitational backreaction is resolved.

The chapter’s bounce control map gives the same sequence visually, including the conditions that stop or redirect the flat-space calculation.

A threaded tilted-quartic example

The pages share the one-field potential

$$U(\phi) = \frac{\lambda}{4}(\phi^2-v^2)^2 +\frac{3\epsilon}{4} \left( \frac{\phi}{v}-\frac{\phi^3}{3v^3} \right),$$

with false vacuum $\phi_{\mathrm f}=+v$, true vacuum $\phi_{\mathrm t}=-v$, and energy-density difference $\epsilon$. The sequence is:

• The bounce page derives the radial equation and false-vacuum tail and uses overshoot/undershoot to bracket the center value.

• The rate page identifies the dilation instability, $d$ translations, determinant ratio, and the units of $A$.

• The existence page shows what must change if the scalar is coupled to another field and supplies a normal-force test that can reject a straight-line ansatz.

• The thin-wall page derives

  $$R_{\mathrm{tw}} = \frac{(d-1)\sigma_0}{\epsilon}, \qquad B_{\mathrm{tw}} = \frac{\Omega_{d-1}}{d}(d-1)^{d-1} \frac{\sigma_0^d}{\epsilon^{d-1}},$$

  and compares the result with converged full bounces.

• The gauge-and-scale page explains which additional terms enter if this barrier belongs to a gauge theory or is generated radiatively.

• The final page decides whether the result remains flat and zero-temperature or belongs to a thermal or gravitational calculation.

This threaded example also shows why successive checks cannot be collapsed into one number. A very accurate bounce action says nothing by itself about the number of negative modes, determinant normalization, or environmental validity.

Boundaries of this chapter

This chapter owns the flat-space, zero-temperature semiclassical construction and the decision to transfer it. It does not develop:

• finite-temperature statistical and real-time nucleation prefactors, which belong to thermal bounces and nucleation rates;
• coupled scalar–gravity saddles and cosmological boundary conditions, which belong to the curved-vacuum-decay regime contract and Coleman–De Luccia bounces;
• model-specific, date-sensitive vacuum-lifetime assessments;
• rigorous resonance and exceptional-case existence theory beyond the stated variational hypotheses.

The instanton–bounce boundary and mode comparison is the quickest check when the physical observable may have been confused with tunneling level splitting.

Review the chapter

1. Why is a solution that approaches the true vacuum at one Euclidean-time end and the false vacuum at the other not the standard false-vacuum decay bounce?

Answer

The decay bounce is a localized fluctuation inside the false-vacuum persistence amplitude. It must return to the false vacuum at large Euclidean distance. Heteroclinic endpoint data instead describe interpolation between distinct asymptotic sectors and lead to a different observable and mode structure.

2. A two-field calculation solves the radial equation along a straight line joining the vacua. What additional equation must be checked?

Answer

The gradient of the potential normal to the line must vanish everywhere sampled by the profile:

$$P_\perp\nabla U=0.$$

More generally, path curvature must balance the normal potential force. Endpoint stationarity alone does not establish full stationarity.

3. In four dimensions, derive the mass dimension of the determinant contribution after the four translation zero modes are removed.

Answer

Removing four eigenvalues, each of dimension mass squared, makes the primed determinant ratio have dimension $\text{mass}^{-8}$. Raising its absolute value to $-1/2$ gives $\text{mass}^4$. The translation Jacobian is dimensionless in the conventional normalization, so the complete $A$ has the required dimension $\text{mass}^4$.

4. Why can a $0.5\%$ error in a bounce action be unacceptable?

Answer

The logarithmic rate contains $-B$. If $B=10^5$, a $0.5\%$ error is an absolute error of about $500$ in $\ln\gamma_0$, corresponding to an exponentially large multiplicative uncertainty. Control targets must therefore be placed on the absolute error in the logarithmic rate.

5. Name two changes that occur when a calculation moves from zero temperature to the high-temperature static regime.

Answer

Euclidean time becomes a circle of circumference $\beta=1/T$, and the dominant static saddle is generally $O(3)$ rather than $O(4)$, with exponent $S_3/T$. In addition, the complete nucleation prefactor includes thermal statistical and real-time dynamical information; it is not just the zero-temperature determinant with $B$ replaced.