Negative Modes and Instability Indices

A negative eigenvalue of the Euclidean Hessian means that the real Gaussian contour is not locally convergent. It does not authorize an arbitrary absolute value or phase: the original integration cycle and its analytic continuation fix the steepest direction. For a false-vacuum bounce, one negative mode is the local signature that produces an imaginary false-vacuum energy and hence a decay rate; a stable tunneling instanton has no negative mode after its translation zero mode is removed.

Required background. Fluctuation operators and determinant ratios supplies the Hessian, primed determinant, and common-boundary normalization.

Helpful background. Second variation, Hessians, and Jacobi fields supplies the variational index; Laplace’s method and steepest descent supplies the local contour rotation; Lagrangian submanifolds and generating functions supplies the geometric language for admissible middle-dimensional cycles.

Morse index and the Gaussian contour

For a real Euclidean saddle, let

$$M_\sigma u_n=\lambda_n u_n.$$

After zero modes have been projected out, the Morse index is

$$\mu_\sigma=\#\{n:\lambda_n<0\},$$

counted with multiplicity in the regulated problem. Along one negative-mode coefficient $c$,

$$\exp\!\left[-\frac{\lambda c^2}{2g}\right] =\exp\!\left[+\frac{|\lambda|c^2}{2g}\right]$$

grows on the real axis. Rotating to either steepest direction, $c=\pm is$, gives

$$\int_{\mathcal J_\pm}\frac{\mathrm dc}{\sqrt{2\pi g}}\, e^{+|\lambda|c^2/(2g)} =\pm i\,|\lambda|^{-1/2}.$$

The sign is an orientation of the thimble. In a multidimensional Gaussian the phase is the product of the rotations selected by the integration cycle. Replacing $\det M$ by $|\det M|$ keeps the magnitude but erases precisely this information.

Morse index and physical instability are related but not identical:

• the index is a local property of the second variation with specified boundary conditions;
• contribution to an observable is a global property of the integration cycle;
• interpretation as a decay rate also requires a metastable boundary condition and causal or lateral prescription.

An arbitrary saddle with one negative direction does not by itself prove that a state decays. Conversely, a complex saddle may encode an instability even though a real-valued Morse index is no longer the right invariant.

Stable instanton versus false-vacuum bounce

The quartic-double-well instanton connecting degenerate minima has

$$M_I=-\frac{d^2}{d\tau^2} +4-6\,\operatorname{sech}^2(\tau-\tau_0).$$

Its spectrum contains a translation zero mode, a positive bound state at $\lambda=3$, and continuum beginning at $4$. After removing the zero mode,

$$\mu_I=0.$$

The instanton contributes a real off-diagonal tunneling amplitude and splits parity eigenstates. It is not a false-vacuum decay saddle.

Now let $\phi_b(r)$ be an $O(d)$-symmetric bounce from a false vacuum $\phi_{\rm fv}$ and back,

$$\phi_b''+\frac{d-1}{r}\phi_b'=V'(\phi_b), \qquad \phi_b'(0)=0, \qquad \phi_b(\infty)=\phi_{\rm fv}.$$

Decomposing fluctuations into spherical harmonics gives radial operators

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

The $\ell=1$ sector contains the $d$ translation zero modes $\partial_\mu\phi_b$. For the leading single-field bounce under the usual regularity and asymptotic hypotheses, the $\ell=0$ sector has exactly one negative eigenvalue and higher partial waves are nonnegative. The intuitive variational direction changes the bounce radius: at the stationary critical bubble, increasing or decreasing the radius lowers the action to quadratic order along one direction. Coleman’s overshoot/undershoot construction and Sturm–Liouville nodal argument make the count precise; see Coleman 1977, pp. 2933–2936 and Callan and Coleman 1977, pp. 1763–1768.

The quadratic data are therefore

$$\begin{array}{c|c|c|c} \text{saddle} & \text{translation modes} & \text{negative modes} & \text{leading interpretation}\\ \hline \text{degenerate-vacuum instanton} & 1 & 0 & \text{real level mixing}\\ O(d)\text{ false-vacuum bounce} & d & 1 & \text{imaginary false-vacuum energy} \end{array}$$

The distinction is a useful check on any numerical spectrum. Counting the finite-box lifting of a translation mode as negative, or taking an absolute determinant before counting, can interchange the two physical conclusions.

From the negative mode to a rate

Let

$$B=S_E[\phi_b]-S_E[\phi_{\rm fv}]>0.$$

After analytic continuation from a stable potential or an equivalent false-vacuum prescription, the one-bounce contribution to the Euclidean persistence amplitude has an imaginary part. Schematically, per unit spatial volume,

$$\frac{\Gamma}{V} =\mathcal P\, e^{-B/\hbar}\,[1+O(\hbar)],$$

where $\mathcal P$ contains:

$$\mathcal P \propto \left(\frac{B}{2\pi\hbar}\right)^{d/2} \left| \frac{\det{}'M_b}{\det M_{\rm fv}} \right|^{-1/2}.$$

The prime removes the $d$ translations, while the displayed absolute value reports only the magnitude after the one negative direction has already been treated by its contour. The decay interpretation uses

$$\Gamma=-2\,\operatorname{Im}E_{\rm fv}$$

in quantum mechanics, or the corresponding imaginary effective action per spacetime volume in QFT. The familiar factor of one half in the one-bounce imaginary part follows from the metastable contour prescription, not from a second negative eigenvalue. Callan and Coleman 1977, pp. 1762–1768 derives the determinant and collective-coordinate prefactor.

This conclusion has boundaries. Coupling to gravity changes both the fluctuation problem and the interpretation. Multiple fields can introduce additional negative directions or bifurcating bounces. A configuration with $\mu>1$ usually does not give the leading decay of a metastable state on the conventional contour, although it can contribute to another cycle or describe a higher saddle. These cases require an explicit integration-cycle analysis.

Shared calculation. The saddle-contribution anatomy keeps the contour phase separate from the determinant magnitude. Shared comparison. The canonical saddle comparison contrasts their local fluctuation prescriptions, while the instanton–bounce boundary and mode comparison distinguishes the boundary conditions, zero modes, and negative modes of tunneling and decay saddles.

How to count reliably

A defensible index calculation combines three checks:

1. Variational check. Exhibit trial deformations that lower the action and distinguish them from symmetry variations.
2. Spectral check. Solve each angular-momentum or symmetry sector with its correct measure and boundary conditions; use nodal theorems where applicable.
3. Contour check. Show that the original or analytically continued cycle actually contains the steepest direction and fixes its orientation.

The count must be stable as the box and discretization are refined. Translational modes should approach zero with profiles proportional to $\partial_\mu\phi_b$. A purported additional negative mode that disappears under refinement is a numerical artifact; one that remains is a physical warning that the assumed leading bounce or contour is incomplete.

Common pitfalls

Equating one negative eigenvalue with decay. Decay also requires a metastable state and a contour prescription that produces the relevant imaginary part. The same local Hessian embedded in a different integration problem may not contribute.

Using $|\det M|$ before specifying the contour. The magnitude cannot determine the phase or its sign. Treat negative directions first, then use an absolute value only as notation for the remaining magnitude.

Counting symmetry modes by the sign of a finite-box eigenvalue. Endpoints can lift a zero mode slightly above or below zero. Identify it by its analytic profile and volume scaling before assigning the Morse index.

Exercises

1. Evaluate the two steepest Gaussian contours for a single negative eigenvalue.

Solution

With $\lambda=-|\lambda|$, set $c=+is$ or $c=-is$. Then $\mathrm dc=\pm i\,\mathrm ds$ and

$$e^{-\lambda c^2/(2g)} =e^{-|\lambda|s^2/(2g)}.$$

The normalized real Gaussian in $s$ equals $|\lambda|^{-1/2}$, leaving the oriented phase $\pm i$. Which sign occurs is fixed by how the original contour is deformed.

2. Show that translations of an $O(d)$ bounce lie in the $\ell=1$ sector.

Solution

For a radial profile,

$$\partial_\mu\phi_b(r) =\frac{x_\mu}{r}\phi_b'(r).$$

The functions $x_\mu/r$ are the degree-one spherical harmonics. Differentiating the bounce equation with respect to $x_\mu$ gives $M\,\partial_\mu\phi_b=0$, so these are $d$ zero modes in the $\ell=1$ sector.

3. Why is the radius variation of a thin-wall bubble expected to be negative at the critical radius?

Solution

In $d$ Euclidean dimensions, the thin-wall action has the form

$$B(R)=\Omega_{d-1}\sigma R^{d-1} -\frac{\Omega_{d-1}}{d}\,\Delta V\,R^d.$$

The stationary radius is $R_c=(d-1)\sigma/\Delta V$. Differentiating twice gives

$$B''(R_c) =-\Omega_{d-1}(d-1)\sigma R_c^{d-3}<0.$$

Thus the collective radius direction lowers the action to quadratic order. This thin-wall argument identifies the direction but does not replace the full spectral proof away from the thin-wall regime.

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. “Fate of the False Vacuum: Semiclassical Theory.” Physical Review D 15 (1977): 2929–2936; erratum 16 (1977): 1248. DOI.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, §§1.3–1.7, pp. 10–38. DOI.