False-vacuum decay and bounce solutions

Use a canonically normalized real scalar with positive Euclidean kinetic term and subtract the false-vacuum action. The potential difference is measured relative to its false-vacuum value so the subtracted action can be finite. For the usual single-field flat-space zero-temperature problem, seek the dominant O four symmetric bounce as a function of Euclidean radius rho. Its radial equation contains a friction-like term three over rho, regularity imposes zero derivative at the origin, and the field approaches the false vacuum at infinity.

The center value is determined by that boundary problem and need not equal the true-vacuum minimum exactly. Overshoot and undershoot shooting arguments can locate the appropriate trajectory for suitable potentials. This is different from a stable double-well instanton connecting two distinct minima at opposite Euclidean times. Gravity, finite temperature, multiple fields, and noncanonical kinetic terms change the problem, so the formulas below should not be promoted to every vacuum-transition setting.

Let epsilon be the positive false-minus-true vacuum energy density. When the vacuum splitting is small compared with the barrier scale, the bubble radius can be much larger than its wall thickness. Approximate the wall by a planar interface with tension S one computed in the nearly degenerate potential. A Euclidean three-sphere wall costs its area times that tension, while the enclosed four-ball gains epsilon times its volume. These are Euclidean geometric factors, not the three-dimensional static thermal-bubble factors.

Extremizing the radius-dependent action gives three times the wall tension divided by epsilon. Substituting that radius yields the familiar inverse-cube dependence on the energy-density difference. The second derivative at the stationary radius is negative: enlarging or shrinking the bounce size moves along an unstable action direction. This does not invalidate the semiclassical method, but it requires treating the saddle with the correct false-vacuum analytic continuation rather than a convergent real Gaussian in every direction.

The standard dominant bounce has one negative fluctuation direction and four translation zero modes in this flat four-dimensional setting. Translation modes are replaced by collective-coordinate integrations; dividing by total spacetime volume gives a rate density. The negative mode is handled through a specified analytic continuation defining the metastable state. Merely taking an absolute determinant before defining that continuation loses the origin of the imaginary contribution and can introduce a factor-of-two error.

With metastable energy $E_f=E_R-i\hbar\Gamma/2$, the survival amplitude has a factor exponential minus Gamma t over two, and its squared magnitude decays as $e^{-\Gamma t}$. This identifies a rate within the regime where exponential survival applies. The prefactor includes regulated fluctuations, zero-mode Jacobians, and counterterms, so the wall action alone is not a complete numerical lifetime prediction. Competing bounces, breakdown of diluteness, or failure of the thin-wall hierarchy require further analysis. A large exponential sensitivity is a reason to state approximations carefully rather than to attach an unjustified precise lifetime to a schematic potential.