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 let $t>0$ be real Lorentzian time. The false-vacuum survival 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; it is the chapter overview’s zero-temperature quantity $\gamma_0$. The semiclassical calculation instead uses a Euclidean extent $\mathcal T_E$: one analytically continues the false-vacuum amplitude and deforms its integration cycle onto the metastable contour. Writing

$$\mathcal V_E=\mathcal T_E V_{d-1}, \qquad Z_f(\mathcal T_E)\sim \exp[-\mathcal E_f\mathcal V_E],$$

defines the analytically continued false-vacuum energy density $\mathcal E_f$. Returning to Lorentzian time and taking the modulus squared gives

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

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

$$\frac{Z_1}{Z_0} =\frac{i}{2}\,\mathcal V_E\,A\,e^{-B}.$$

The factor $i/2$ is fixed by choosing the contour for the decaying metastable state through the single negative direction; the conjugate, growing solution uses the opposite contour. Widely separated identical bounces give $1/n!$, so

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

Comparing this exponential with $Z_f\sim\exp[-\mathcal E_f\mathcal V_E]$ gives

$$\delta\mathcal E_f =-\frac{i}{2}A e^{-B}, \qquad \gamma=-2\operatorname{Im}\mathcal E_f=Ae^{-B},$$

and therefore

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

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, §6.5, p. 337.

For the canonical least-action one-field bounce, minimization at fixed negative potential contribution gives at most one negative direction; the scale deformation gives at least one, because the second scale variation is negative for $d>2$. Together these statements give exactly one Coleman 1985, ch. 7, §6.5, pp. 337–338. Coleman later explained the one-and-only-one result in terms of the most probable escape path for broader classes of particle, scalar-field, and gauge-field systems under mild technical restrictions Coleman 1988, pp. 178–186.

This is not a theorem about every Euclidean stationary point that resembles a bounce: other solutions can have several negative modes and then do not represent the canonical tunneling saddle. Excited or oscillating solutions, gravity, constraints, restricted ansätze, and multifield or gauge problems outside the cited hypotheses require a fresh count in the full physical fluctuation space.

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

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

They are self-adjoint with radial inner product

$$\langle f,g\rangle = \int_0^\infty \mathrm dr\,r^{d-1}f(r)g(r).$$

For an isolated, nondegenerate canonical bounce, the 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. Exact scale invariance, internal symmetries, or accidental moduli add zero or soft modes that must be treated separately.

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 negative eigenvalue—or an additional one that persists under box and mesh refinement—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 $\mathcal T_E V_{d-1}=\mathcal V_E$; 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. “Quantum Tunneling and Negative Eigenvalues.” Nuclear Physics B 298 (1988): 178–186. 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.