Decay Rates, the Negative Mode, and Prefactors

One relevant negative mode makes the local Gaussian integral about a bounce imaginary on the false-vacuum contour. The $d$ translational zero modes supply the Euclidean spacetime volume and their collective-coordinate Jacobian, while the remaining spectrum gives a renormalized determinant ratio. Under the canonical flat-space, zero-temperature, least-action-bounce hypotheses, these ingredients produce a decay rate per spatial volume with mass dimension $d$.

Required background. Bounce solutions and false-vacuum boundary conditions supplies the $O(d)$ radial saddle and false-vacuum persistence boundary data. Negative modes and instability indices supplies the contour meaning of a negative Hessian eigenvalue and explains why its absolute value alone does not determine a phase.

Helpful background. Fluctuation operators and determinant ratios supplies primed determinants, reference-sector normalization, and finite-box spectral checks.

From persistence amplitude to rate

Let $V_{d-1}$ be a large spatial volume and $T$ a long Euclidean time interval. The false-vacuum persistence probability has the dilute-decay form

$$P_f(T) \simeq \exp\!\left[-\gamma V_{d-1}T\right], \qquad \gamma\equiv\frac{\Gamma}{V_{d-1}},$$

where $\gamma$ is the decay rate per spatial volume. If $\mathcal E_f$ denotes the analytically continued false-vacuum energy density, then

$$\gamma=-2\,\operatorname{Im}\mathcal E_f.$$

This is not an unstable-particle width: no one-particle external state or final-state phase-space integral appears. It is an intensive rate for nucleating a critical configuration somewhere in spacetime.

For one bounce centered at $x_0$, the semiclassical contribution to the Euclidean persistence amplitude takes the form

$$\frac{Z_1}{Z_0} =\frac{i}{2}\,V_d\,A\,e^{-B}, \qquad V_d=T V_{d-1}.$$

The factor $i/2$ is fixed by the metastable contour through the single negative direction. Widely separated identical bounces give $1/n!$, so

$$\frac{Z_f}{Z_0} \simeq \sum_{n=0}^{\infty} \frac1{n!} \left(\frac{i}{2}V_d A e^{-B}\right)^n =\exp\!\left(\frac{i}{2}V_d A e^{-B}\right).$$

Comparing this exponential with $Z_f\sim\exp[-\mathcal E_f V_d]$ yields

$$\boxed{\gamma=Ae^{-B}}$$

at leading semiclassical order. The factor $1/2$ in the amplitude and the factor $-2$ relating $\operatorname{Im}\mathcal E_f$ to the probability rate cancel. Callan and Coleman derive this contour and dilute-bounce normalization in Callan and Coleman 1977, pp. 1762–1765.

Spectrum of the bounce

For one canonical scalar,

$$M_b=-\partial^2+U''(\phi_b), \qquad M_f=-\partial^2+U''(\phi_f).$$

An $O(d)$ bounce has $d$ translational zero modes,

$$u_\mu(x)=\partial_\mu\phi_b(x).$$

Rotational invariance and the bounce scaling identity imply

$$\int \mathrm d^d x\, \partial_\mu\phi_b\,\partial_\nu\phi_b =\delta_{\mu\nu}B.$$

Consequently, replacing the normalized zero-mode coefficients by the center $x_0^\mu$ gives

$$\prod_{\mu=1}^{d}\frac{\mathrm dc_\mu}{\sqrt{2\pi}} \longrightarrow \left(\frac{B}{2\pi}\right)^{d/2}\mathrm d^d x_0.$$

The normalization follows by projecting fluctuations onto normalized translation modes and changing variables from their coefficients to the bounce center Coleman 1985, ch. 7, §6.3, pp. 332–336.

The least-action one-field bounce has one relevant negative eigenvalue. The statement is hypothesis-dependent: it uses the constrained minimum property of the canonical bounce, not the visual fact that its profile crosses a barrier. Coleman, Glaser, and Martin’s constrained variational construction rules out a second independent negative direction for the least-action solution in its class Coleman, Glaser, and Martin 1978, pp. 216–221. Gauge fields, gravity, extra scalars, excited bounces, or a restricted ansatz require a new count.

For a radial numerical check, decompose fluctuations into spherical harmonics. The radial operators are

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

The canonical pattern is one negative eigenvalue in the $\ell=0$ sector, $d$ translation zero modes in the $\ell=1$ sector, and no further nonpositive modes. Regularity requires the radial eigenfunction to behave as $r^\ell$ at the origin. A mode count that changes when the outer boundary is enlarged or the mesh is refined is not yet a continuum result.

In the thin-wall limit, changing the bubble radius supplies an independent check. With

$$B(R) =\Omega_{d-1} \left[ \sigma R^{d-1}-\frac{\epsilon}{d}R^d \right]$$

and $R=(d-1)\sigma/\epsilon$, the curvature of $B(R)$ divided by the norm of the radius deformation gives

$$\lambda_- \simeq-\frac{d-1}{R^2}.$$

The negative sign is therefore the local statement that a critical bubble shrinks if made smaller and expands if made larger.

The renormalized prefactor

Use a prime to omit the $d$ translation zero modes but retain the negative eigenvalue inside the absolute value. In this convention,

$$A =\left(\frac{B}{2\pi}\right)^{d/2} \left| \frac{\det{}'M_b}{\det M_f} \right|_{\mathrm{ren}}^{-1/2} \left[1+O(\hbar)\right].$$

The determinant ratio is not dimensionless after $d$ eigenvalues of mass dimension two have been omitted from its numerator:

$$\left[\frac{\det{}'M_b}{\det M_f}\right] =(\mathrm{mass})^{-2d}, \qquad [A]=(\mathrm{mass})^d.$$

Hence $Ae^{-B}$ has the required units of inverse time per spatial $(d-1)$-volume. Dividing by $V_{d-1}$ a second time, or treating $A$ as dimensionless, is a normalization error.

The subscript “ren” is essential. The bounce and false-vacuum operators must use the same regulator, boundary conditions, and counterterms; local ultraviolet divergences cancel only after the action, determinant, and parameter renormalization are combined. At a fixed perturbative order,

$$\mu\frac{\mathrm d}{\mathrm d\mu} \log\!\left(Ae^{-B}\right)$$

must be of the first omitted order. A small determinant by itself is not evidence of a large rate: it can instead signal an unremoved zero mode, a soft collective coordinate, or loss of semiclassical control.

Practical partial-wave and determinant calculations, including their ultraviolet subtraction, are reviewed in Devoto et al. 2022, §3.5, pp. 40–43.

One-component scalar decay in four dimensions

For the asymmetric quartic model introduced on the bounce page, set $d=4$. A converged radial solution has exponent $B$ and fluctuation operators

$$M_b=-\partial^2+U''(\phi_b), \qquad M_f=-\partial^2+U''(+v).$$

The rate per spatial volume is

$$\frac{\Gamma}{V_3} = \left(\frac{B}{2\pi}\right)^2 \left| \frac{\det{}'M_b}{\det M_f} \right|_{\mathrm{ren}}^{-1/2} e^{-B} \left[1+O(\hbar)\right].$$

Four translation zero modes produce the center integral $\mathrm d^4x_0$ and the factor $(B/2\pi)^2$. In the nearly degenerate regime, the radial mode approaches $\lambda_-\simeq-3/R^2$. These two checks locate common errors before the determinant is evaluated: the wrong number of zero modes gives the wrong mass dimension, while a missing or additional stable negative eigenvalue invalidates the canonical contour interpretation.

Shared calculation. The bounce control map shows where the spectrum, determinant, and volume factors enter relative to existence, thin-wall, gauge, thermal, and gravitational checks.

Shared comparison. The instanton–bounce boundary and mode comparison prevents the level-splitting measure of a tunneling instanton from being substituted for a false-vacuum rate.

Control and failure boundaries

The formula for $\gamma$ is controlled only if:

• $B\gg1$ and the first omitted loop correction is small;
• the selected bounce is the relevant least-action saddle for the false-vacuum contour;
• there is exactly one relevant negative mode and precisely the expected zero modes;
• the determinant ratio is regulator-stable and renormalized with the action;
• the bounce gas is dilute on the scale of the bounce radius;
• the spacetime box is large compared with the bounce and the intensive logarithm is formed before the infinite-volume limit; and
• finite-temperature and gravitational corrections are parametrically negligible.

An additional negative mode is not repaired by taking an absolute determinant. A missing negative mode removes the standard imaginary contribution. An approximate zero mode must be treated as a collective or quasi-collective coordinate rather than hidden in a numerically unstable determinant.

Common pitfalls

Confusing amplitude and probability. The negative-mode contour gives an imaginary contribution to the persistence amplitude. The probability rate follows only after taking the modulus squared, which supplies the factor $-2\,\operatorname{Im}\mathcal E_f$.

Losing the spacetime volume. The center of a $d$-dimensional bounce has $d$ collective coordinates. Their integral gives $T V_{d-1}$; removing that factor defines the intensive rate, while their Jacobians remain in $A$.

Declaring one negative mode by assumption. The count is a spectral result tied to the least-action saddle and the full field space. A restricted path can conceal transverse negative directions.

Exercises

1. Prove the translation-mode normalization

$$\int \mathrm d^d x\, \partial_\mu\phi_b\,\partial_\nu\phi_b =\delta_{\mu\nu}B.$$

Solution

Rotational invariance gives

$$\int \mathrm d^d x\, \partial_\mu\phi_b\,\partial_\nu\phi_b =\frac{\delta_{\mu\nu}}{d} \int \mathrm d^d x\,(\partial\phi_b)^2.$$

Writing $K=\int \mathrm d^d x\,(\partial\phi_b)^2/2$, the scale identity $(d-2)K+dV=0$ implies $B=K+V=2K/d$. Therefore the right-hand side is $\delta_{\mu\nu}(2K/d)=\delta_{\mu\nu}B$.

2. Explain why omitting $d$ zero eigenvalues makes the inverse square root of the determinant ratio have mass dimension $d$.

Solution

Every Hessian eigenvalue has dimension mass squared. Before omissions, the bounce and false-vacuum determinants contain the same regulated number of eigenvalues and their ratio is dimensionless. Removing $d$ numerator eigenvalues multiplies the ratio’s dimension by $(\mathrm{mass}^2)^{-d}$, so its inverse square root has dimension $(\mathrm{mass})^d$.

3. Derive $\lambda_-\simeq-(d-1)/R^2$ for the thin-wall radius mode.

Solution

At the stationary radius, $B''(R)=-\Omega_{d-1}\epsilon R^{d-2}$. The radius deformation is $\partial_R\phi_b\simeq-\phi'_{\mathrm{wall}}$, whose norm is

$$\lVert\partial_R\phi_b\rVert^2 \simeq\Omega_{d-1}R^{d-1} \int\mathrm d\rho\,(\phi'_{\mathrm{wall}})^2 =\Omega_{d-1}R^{d-1}\sigma.$$

Their ratio is $-\epsilon/(\sigma R)=-(d-1)/R^2$ because $\epsilon R=(d-1)\sigma$.

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, 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.
• Devoto, Federica, Simone Devoto, Luca Di Luzio, and Giovanni Ridolfi. “False Vacuum Decay: An Introductory Review.” Journal of Physics G: Nuclear and Particle Physics 49 (2022): 103001. DOI. Open PDF.