Bounce Existence, Symmetry, and Multifield Geometry

A least-action bounce exists and may be taken $O(d)$ symmetric only under hypotheses on the Euclidean functional, the false vacuum, and the admissible field space. For several scalar fields, rotational symmetry in spacetime can still follow, but it does not reduce the trajectory in field space to a guessed one-dimensional path. The decisive test is stationarity in every normal field-space direction.

Required background. Bounce solutions and false-vacuum boundary conditions supplies the radial boundary-value problem. Field variations and boundary terms supplies the variational derivation with fixed asymptotic data. Second variation, Hessians, and Jacobi fields supplies the stationarity and spectrum tests used below.

Helpful background. Negative modes and instability indices explains why a candidate with the wrong instability index is not the canonical leading decay saddle.

Hypotheses behind a least-action bounce

Begin with $n$ canonically normalized real scalars $\boldsymbol\phi$ in flat Euclidean dimension $d>2$. Shift the false vacuum to $\boldsymbol\phi_{\mathrm f}=\boldsymbol 0$ and its energy to $U(\boldsymbol 0)=0$. A useful sufficient framework assumes:

• $U$ is continuously differentiable, with $\nabla U(\boldsymbol 0)=0$;
• the Hessian at $\boldsymbol 0$ is positive definite, so the false vacuum is locally stable and has exponential tails;
• $U$ is negative somewhere, so a lower-energy region is accessible;
• the reduced variational problem defined below has an admissible minimizer with finite kinetic and potential integrals.

The last condition is substantive. It can follow, for example, for broad classes of stabilized finite-order polynomial potentials, but it can fail for unbounded runaways or when minimizing sequences escape to arbitrarily large field values. Under the stated existence assumption, the multifield extension of the Coleman–Glaser–Martin construction proves rotational symmetry of the least-action bounce Blum et al. 2017, §§2–4. The original one-field theorem likewise requires a scalar potential and $d>2$ Coleman, Glaser, and Martin 1978, Theorems A and B.

To see why a reduced problem enters, write a configuration’s kinetic and potential contributions as

$$T[\boldsymbol\phi] = \frac12\int \mathrm d^d x\, \partial_\mu\boldsymbol\phi\mathbin{\cdot}\partial_\mu\boldsymbol\phi, \qquad V[\boldsymbol\phi] = \int \mathrm d^d x\,U(\boldsymbol\phi).$$

Under $\boldsymbol\phi(\mathbf x)\mapsto\boldsymbol\phi(\mathbf x/\lambda)$,

$$T\mapsto\lambda^{d-2}T, \qquad V\mapsto\lambda^dV.$$

Stationarity under this dilation gives the virial identity

$$(d-2)T+dV=0.$$

Thus every nontrivial finite-action bounce has $V<0$. One may first minimize $T$ at fixed negative $V$, then choose the dilation that satisfies the virial identity. Existence of that reduced minimizer is the premise—not a consequence—of the symmetry statement.

It is useful to keep four questions separate:

1. Does an admissible finite-action stationary point exist?
2. Is a least-action representative $O(d)$ symmetric in Euclidean spacetime?
3. Has the full multifield equation, rather than only a restricted ansatz, been solved?
4. Does the resulting Hessian have the one relevant negative mode and the expected translation zero modes?

A positive answer to one question does not supply the others.

Multifield radial equations

For a general positive-definite field-space metric $G_{ab}(\boldsymbol\phi)$, the $O(d)$-symmetric action difference is

$$B = \Omega_{d-1}\int_0^\infty \mathrm dr\,r^{d-1} \left[ \frac12G_{ab}(\boldsymbol\phi)\phi^{\prime a}\phi^{\prime b} +U(\boldsymbol\phi)-U(\boldsymbol\phi_{\mathrm f}) \right],$$

where $\Omega_{d-1}=2\pi^{d/2}/\Gamma(d/2)$. Its Euler–Lagrange equation is

$$\frac{D\phi^{\prime a}}{\mathrm dr} +\frac{d-1}{r}\phi^{\prime a} = G^{ab}\partial_bU, \qquad \frac{D\phi^{\prime a}}{\mathrm dr} = \phi^{\prime\prime a} +\Gamma^a{}_{bc}\phi^{\prime b}\phi^{\prime c},$$

with boundary conditions

$$\phi^{\prime a}(0)=0, \qquad \boldsymbol\phi(r)\longrightarrow\boldsymbol\phi_{\mathrm f} \quad(r\to\infty).$$

The familiar mechanical analogy now describes a particle moving on a curved target space in the inverted potential $-U$, with radial friction $(d-1)/r$. In one field, the original false-vacuum construction uses overshoot and undershoot to order initial conditions Coleman 1977, §III, pp. 2930–2932. There is generally no comparable total ordering in a multidimensional initial-value space, so a multidimensional boundary-value or path-deformation method is needed.

The normal-force test for a proposed path

Suppose a calculation restricts the bounce to a curve $\boldsymbol\gamma(s)$ in field space, parametrized by arc length, and writes

$$\boldsymbol\phi(r)=\boldsymbol\gamma(s(r)), \qquad G_{ab}\dot\gamma^a\dot\gamma^b=1.$$

Projecting the full equation normal to the curve gives

$$P_\perp{}^a{}_b \left[ (s')^2\frac{D\dot\gamma^b}{\mathrm ds} -\nabla^bU \right] =0, \qquad P_\perp{}^a{}_b = \delta^a{}_b-\dot\gamma^a\dot\gamma_b.$$

The friction term is tangent and therefore drops out of this equation. A one-dimensional solution along $\gamma$ is a stationary point of the full theory only if the path curvature balances the transverse potential force at every sampled point. For a straight path in flat field space this reduces to

$$P_\perp\nabla U=0 \quad\text{along the entire bounce.}$$

Checking only that the endpoints are extrema is insufficient.

First application: a two-field false vacuum

Consider

$$\begin{aligned} U(x,y) ={}& \frac{\lambda}{4}(x^2-v^2)^2 +\frac12M^2y^2 +\alpha y(x^2-v^2)\\ &+\frac{3\epsilon}{4} \left(\frac{x}{v}-\frac{x^3}{3v^3}\right), \end{aligned}$$

with $\lambda>0$, $M^2>0$, $0<\epsilon<4\lambda v^4/3$, and

$$\alpha^2 < \frac{M^2}{2} \left( \lambda-\frac{3\epsilon}{4v^4} \right).$$

The points $(x,y)=(\pm v,0)$ remain stationary; $(v,0)$ is the false vacuum and $(-v,0)$ is lower by $\epsilon$. The displayed inequality makes the false-vacuum Hessian positive definite:

$$\det H_{\mathrm f} = M^2 \left( 2\lambda v^2-\frac{3\epsilon}{2v^2} \right) -4\alpha^2v^2 >0.$$

It also implies $\alpha^2<\lambda M^2/2$, which makes the large-field quartic direction stable after minimizing over $y$, because

$$\begin{aligned} &\frac{\lambda}{4}(x^2-v^2)^2 +\frac12M^2y^2+\alpha y(x^2-v^2)\\ &\quad= \frac12M^2 \left[y+\frac{\alpha}{M^2}(x^2-v^2)\right]^2 +\left(\frac{\lambda}{4}-\frac{\alpha^2}{2M^2}\right) (x^2-v^2)^2. \end{aligned}$$

The tempting straight-line ansatz $y(r)=0$ fails immediately:

$$\left.\frac{\partial U}{\partial y}\right|_{y=0} = \alpha(x^2-v^2),$$

which is nonzero through the wall whenever $\alpha\ne0$. The curve that minimizes the potential at fixed $x$,

$$y_\star(x)=-\frac{\alpha}{M^2}(x^2-v^2),$$

lowers the barrier, but substituting it into the potential alone is not enough: its curvature also changes the kinetic term. A controlled reduction must use the induced line element

$$\mathrm ds^2 = \left[1+\left(\frac{\mathrm dy_\star}{\mathrm dx}\right)^2\right] \mathrm dx^2$$

and must still satisfy the normal-force equation. In practice, solve the coupled radial equations, compare actions of all found saddles, evaluate the full residual in both components, and compute the full fluctuation spectrum. This example is a concrete witness against replacing multifield tunneling by an arbitrary straight line.

What rotational symmetry does not cover

The $O(d)$ conclusion above applies to canonical scalar functionals—or to positive field-space metrics after the relevant variational hypotheses have been re-established—and to the least-action bounce whose reduced minimizer exists. It does not by itself cover:

• higher-derivative or nonlocal effective actions;
• singular or indefinite field-space metrics;
• gauge configurations before constraints, ghosts, and gauge fixing are handled;
• boundaries, inhomogeneous media, finite Euclidean-time circles, or curved spacetime;
• a stationary point that is not the least-action decay saddle;
• a potential for which the reduced minimizer does not exist.

For these cases, symmetry must be demonstrated for the actual functional rather than imported from the canonical scalar theorem.

Shared calculation. The bounce control map shows where existence, full-field-space stationarity, and the spectrum enter before a rate may be quoted.

Shared comparison. The instanton–bounce boundary and mode comparison distinguishes the return-to-false-vacuum boundary data used here from interpolation between vacua.

Common pitfalls

Assuming that $O(d)$ symmetry makes the problem one-dimensional in field space. The symmetry reduces dependence on Euclidean coordinates to $r$. All scalar components can still vary, and their coupled radial equations must be satisfied.

Calling a low-action restricted path a bounce. Minimization inside an ansatz proves only restricted stationarity. Evaluate the residual normal to the ansatz and the Hessian in the full fluctuation space.

Treating an unbounded Euclidean potential as automatically admissible. A local false vacuum and a lower region do not guarantee a minimizing bounce. Check the behavior of minimizing sequences and the boundedness assumptions of the existence result being used.

Exercises

1. Derive the covariant radial equation from the action with field-space metric $G_{ab}$.

Solution

Varying the kinetic term gives

$$\delta B_{\mathrm{kin}} = \Omega_{d-1}\int_0^\infty\mathrm dr\,r^{d-1} \left[ -G_{ab}\left( \frac{D\phi^{\prime b}}{\mathrm dr} +\frac{d-1}{r}\phi^{\prime b} \right) \right]\delta\phi^a,$$

after the endpoint term vanishes by regularity and fixed false-vacuum asymptotics. Adding $\partial_aU\,\delta\phi^a$ and multiplying by $G^{ab}$ gives the stated equation.

2. For a straight path $\boldsymbol\phi=\boldsymbol\phi_0+\mathbf n\,s(r)$ in flat field space, show that solving the projected scalar equation is insufficient unless $(\mathbf 1-\mathbf n\mathbf n^{\mathsf T})\nabla U=0$ along the profile.

Solution

The full equation is

$$\mathbf n \left[s''+\frac{d-1}{r}s'\right] = \nabla U.$$

Its component parallel to $\mathbf n$ is the projected scalar equation. Applying the normal projector $\mathbf 1-\mathbf n\mathbf n^{\mathsf T}$ annihilates the left-hand side and leaves the additional condition

$$(\mathbf 1-\mathbf n\mathbf n^{\mathsf T})\nabla U=0.$$

3. Minimize the two-field potential above with respect to $y$ at fixed $x$, and identify the stability condition on the effective quartic coefficient.

Solution

The equation $\partial_yU=M^2y+\alpha(x^2-v^2)=0$ gives

$$y_\star(x)=-\frac{\alpha}{M^2}(x^2-v^2).$$

Substitution yields the quartic coefficient

$$\frac{\lambda}{4}-\frac{\alpha^2}{2M^2}.$$

It is positive precisely when $\alpha^2<\lambda M^2/2$. This condition stabilizes the large-$x$ direction but does not prove that the valley curve is an exact bounce path.

References

• Blum, K., Honda, M., Sato, R., Takimoto, M., and Tobioka, K. (2017). “O($N$) Invariance of the Multi-Field Bounce.” Journal of High Energy Physics 2017(5), 109; Erratum 2017(6), 60. doi:10.1007/JHEP05(2017)109. Erratum.
• Coleman, S. (1977). “The Fate of the False Vacuum. I. Semiclassical Theory.” Physical Review D 15, 2929–2936. doi:10.1103/PhysRevD.15.2929.
• Coleman, S., Glaser, V., and Martin, A. (1978). “Action Minima among Solutions to a Class of Euclidean Scalar Field Equations.” Communications in Mathematical Physics 58, 211–221. doi:10.1007/BF01609421.