Bounce Solutions and False-Vacuum Boundary Conditions

An $O(d)$-symmetric bounce describes false-vacuum escape by leaving the false vacuum in the interior of Euclidean spacetime and returning to that same false vacuum at large Euclidean radius. Its regularity condition at the center and false-vacuum asymptotics turn the field equation into a two-point boundary-value problem. These are decay boundary conditions, not the heteroclinic boundary conditions of an instanton that interpolates between distinct vacua.

Required background. Euclidean tunneling saddles and boundary conditions supplies the analytic-continuation and finite-action logic used to define the Euclidean problem. What an interacting Lagrangian does and does not specify supplies the stability, state, regulator, and observable data needed before a local scalar potential can be interpreted as a false-vacuum problem.

Helpful background. Linear ODEs, evolution operators, and Wronskians supplies shooting methods and asymptotic boundary matching for the radial equation.

The radial false-vacuum boundary problem

Consider one canonically normalized real scalar in $d>2$ flat Euclidean dimensions,

$$S_E[\phi] =\int \mathrm d^d x\, \left[\frac12(\partial\phi)^2+U(\phi)\right].$$

For the least-action solution in the canonical one-field class, the reduction to $O(d)$ symmetry follows under the potential and existence hypotheses of the Coleman–Glaser–Martin theorem Coleman, Glaser, and Martin 1978, Theorems A and B. This page solves the resulting radial problem; the multifield page states when that reduction can and cannot be extended.

Let $\phi_f$ be a local minimum of $U$ and let some lower region of the potential be accessible through a barrier. The false-vacuum-subtracted bounce exponent is

$$B=S_E[\phi_b]-S_E[\phi_f].$$

For an $O(d)$-symmetric configuration $\phi_b(x)=\phi_b(r)$, where $r=(x_\mu x_\mu)^{1/2}$, define

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

as the area of the unit $(d-1)$-sphere. Then

$$B =\Omega_{d-1}\int_0^\infty \mathrm dr\,r^{d-1} \left[ \frac12\left(\frac{\mathrm d\phi_b}{\mathrm dr}\right)^2 +U(\phi_b)-U(\phi_f) \right].$$

Varying this functional gives

$$\frac{\mathrm d^2\phi_b}{\mathrm dr^2} +\frac{d-1}{r}\frac{\mathrm d\phi_b}{\mathrm dr} =U'(\phi_b),$$

with boundary conditions

$$\left.\frac{\mathrm d\phi_b}{\mathrm dr}\right|_{r=0}=0, \qquad \lim_{r\to\infty}\phi_b(r)=\phi_f.$$

The first condition removes a conical singularity. If $\phi_0=\phi_b(0)$, the regular initial expansion is

$$\phi_b(r) =\phi_0+\frac{U'(\phi_0)}{2d}r^2+O(r^4).$$

The second condition is stronger than merely reaching the false side of the barrier. With $m_f^2=U''(\phi_f)>0$, the decaying tail behaves as

$$\phi_b(r)-\phi_f \propto r^{-(d-1)/2}e^{-m_f r}$$

up to relative powers of $(m_f r)^{-1}$. A finite numerical interval must reproduce this asymptotic behavior as its outer boundary is moved outward.

The defining boundary condition is return to $\phi_f$ in every Euclidean direction. By contrast, a tunneling instanton used for a level splitting approaches different degenerate vacua at the two ends of Euclidean time. Coleman’s construction makes this distinction before any determinant is evaluated Coleman 1977, pp. 2929–2932.

Overshoot and undershoot

Read $r$ as time and $\phi$ as the position of a particle moving in the inverted potential $-U$. The term $(d-1)\phi_b'/r$ is a positive friction that is strongest near the release point. Start at rest at $\phi_0$ and integrate outward:

• an overshoot crosses $\phi_f$ with nonzero velocity;
• an undershoot turns around or stalls before reaching $\phi_f$;
• the bounce is the boundary between the two behaviors.

For the usual one-field double-well geometry, a release point sufficiently close to the true minimum retains enough inverted-potential energy to overshoot, while a point closer to the barrier undershoots. Continuity then brackets a critical $\phi_0$. The classic argument and its hypotheses are given in Coleman 1985, ch. 7, §6.2, pp. 329–332.

A robust shooting calculation uses the near-origin series rather than starting at $r=0$, records the first crossing or turning event, and bisects only a bracket with opposite classifications. Convergence requires more than a small ODE residual: the action, center value, tail amplitude, and classification must stabilize as the starting radius, outer radius, and integration tolerance are varied.

This reasoning is not a general multifield existence proof. In several fields there is no ordered line on which “between overshoot and undershoot” has an automatic meaning, and a prescribed one-dimensional path can fail the normal component of the field equations. Bounce existence, symmetry, and multifield geometry gives the required replacement.

An asymmetric quartic example

A useful threaded example keeps both vacua at known field values:

$$U(\phi) =\frac{\lambda}{4}(\phi^2-v^2)^2 +\frac{3\epsilon}{4} \left( \frac{\phi}{v}-\frac{\phi^3}{3v^3} \right), \qquad \lambda>0,\quad v>0.$$

The points $\phi_f=+v$ and $\phi_t=-v$ are exact stationary points with

$$U(\phi_f)=+\frac{\epsilon}{2}, \qquad U(\phi_t)=-\frac{\epsilon}{2}.$$

Thus their energy-density difference is $\epsilon$. The false minimum remains locally stable when

$$0<\epsilon<\frac43\lambda v^4,$$

because $U''(+v)=2\lambda v^2-3\epsilon/(2v^2)>0$. The bounce begins at a value $\phi_0$ on the true side of the barrier, not exactly at $\phi_t$, and approaches $+v$ as $r\to\infty$.

When $\epsilon/(\lambda v^4)\ll1$, the wall is locally close to the degenerate quartic profile

$$\phi_{\mathrm{wall}}(\rho) =v\tanh\!\left(\sqrt{\frac{\lambda}{2}}\,v\rho\right),$$

centered near a large radius. The corresponding planar tension is

$$\sigma_0 =\int_{-v}^{v}\sqrt{2U_0(\phi)}\,\mathrm d\phi =\frac{2\sqrt2}{3}\sqrt{\lambda}\,v^3, \qquad U_0(\phi)=\frac{\lambda}{4}(\phi^2-v^2)^2.$$

These formulas give a stringent starting bracket and dimensional check, but they are not the full bounce when the tilt is finite. The exact radial solution must still satisfy both boundary conditions, and thin-wall control and corrections quantifies when the wall approximation is accurate.

From a bounce to a decay rate

A radial solution is only the first gate in a decay calculation. The figure below separates the boundary-value problem from the fluctuation spectrum, renormalized prefactor, optional thin-wall approximation, and environmental handoffs. Inspect especially the stop conditions: a stationary profile with the wrong boundary data, an unjustified symmetry reduction, or the wrong number of relevant negative modes does not define the canonical leading decay rate.

Control map for a flat-space, zero-temperature false-vacuum decay calculation in $d$ Euclidean dimensions. The map is schematic, not to scale: the canonical leading bounce returns to the false vacuum, has exactly one relevant negative mode and $d$ translational zero modes, and yields a rate only after determinant, renormalization, approximation, and thermal or gravitational checks pass.

In words, the sequence is: define the false-vacuum persistence observable; solve the correct boundary problem; justify the symmetry and field-space reduction; verify one negative mode and all translation zero modes; form a renormalized prefactor with units of mass to the power $d$; and only then interpret the result as a rate per spatial volume. A thin-wall formula is an optional approximation inside this sequence, not an independent definition of decay.

Shared comparison. The instanton–bounce boundary and mode comparison contrasts the physical question, endpoint data, zero and negative modes, determinant, and observable for tunneling instantons and decay bounces.

In the dilute regime, well-separated translated copies of the same bounce form approximate multi-bounce configurations. Integrating their centers gives a factor proportional to spacetime volume for each bounce, while division by $n!$ accounts for indistinguishable copies. Summing over $n$ exponentiates the one-bounce contribution; overlap corrections are small only when the expected number of events in a bounce-sized spacetime region is small Callan and Coleman 1977, pp. 1762–1765.

Scope and failure boundaries

The construction above assumes a flat, zero-temperature Euclidean functional integral for a canonically normalized scalar and a well-defined false-vacuum state. It does not establish:

• that the displayed $O(d)$ solution is the least-action saddle in a multifield, gauge, derivative-coupled, or constrained theory;
• that its Hessian has exactly one relevant negative mode;
• that the determinant and local counterterms have been renormalized consistently;
• that a dilute multi-bounce sum is valid;
• that the Euclidean exponent alone supplies a thermal nucleation rate; or
• that metric backreaction is negligible.

Finite temperature makes Euclidean time periodic and can replace an $O(d)$ saddle by a static $O(d-1)$ critical bubble. Curvature makes the metric dynamical and changes both the boundary problem and its negative modes. Those are changes of problem, not small annotations to the flat bounce.

Common pitfalls

Using instanton endpoints for decay. A decay bounce returns to the same false vacuum at large Euclidean distance. A path connecting two distinct vacua computes a different amplitude and does not acquire a decay interpretation merely because its action is exponential.

Starting exactly at the true vacuum. With zero initial derivative, an exact minimum is a constant solution. The bounce starts nearby, at the unique center value selected by the false-vacuum asymptotic condition.

Accepting a visually smooth profile. A smooth curve can be an undershoot, a finite-box artifact, or a solution of a restricted ansatz rather than the full equations. Boundary residuals, action stability, mode counting, and box convergence are independent tests.

Exercises

1. Derive the regular expansion $\phi_b(r)=\phi_0+U'(\phi_0)r^2/(2d)+O(r^4)$.

Solution

Write $\phi_b=\phi_0+a r^2+O(r^4)$. Then $\phi_b'=2ar+O(r^3)$ and $\phi_b''=2a+O(r^2)$. The constant term in the radial equation is

$$2a+2a(d-1)=U'(\phi_0),$$

so $a=U'(\phi_0)/(2d)$.

2. Let $K=\int \mathrm d^d x\,(\partial\phi_b)^2/2$ and $V=\int \mathrm d^d x\,[U(\phi_b)-U(\phi_f)]$. Apply the scale variation $\phi_\alpha(x)=\phi_b(x/\alpha)$ and show that a stationary bounce satisfies $(d-2)K+dV=0$.

Solution

Changing variables to $y=x/\alpha$ gives

$$B(\alpha)=\alpha^{d-2}K+\alpha^dV.$$

Stationarity at $\alpha=1$ implies

$$0=B'(1)=(d-2)K+dV.$$

Hence $B=K+V=2K/d>0$ for $d>2$. This scaling identity is an independent action check for a numerical profile.

3. Explain why $\phi(-\infty)=\phi_t$ and $\phi(+\infty)=\phi_f$ is not a false-vacuum bounce boundary condition.

Solution

Those endpoints define a transition amplitude between distinct field configurations. False-vacuum decay is extracted from a false-vacuum persistence amplitude, so the Euclidean configuration must approach $\phi_f$ at both temporal ends; radial symmetry expresses this as $\phi_b(r)\to\phi_f$ in every direction as $r\to\infty$.

References

• Callan, Curtis G., Jr., and Sidney Coleman. “Fate of the False Vacuum. II. First Quantum Corrections.” Physical Review D 16 (1977): 1762–1768. DOI.
• Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 7, pp. 265–350. DOI.
• Coleman, Sidney. “Fate of the False Vacuum: Semiclassical Theory.” Physical Review D 15 (1977): 2929–2936; erratum, Physical Review D 16 (1977): 1248. DOI.
• Coleman, Sidney, V. Glaser, and André Martin. “Action Minima among Solutions to a Class of Euclidean Scalar Field Equations.” Communications in Mathematical Physics 58 (1978): 211–221. DOI.