The Thin-Wall Approximation in Vacuum Decay: a Lemma

The 'thin-wall approximation' gives a simple estimate of the decay rate of an unstable quantum field. Unfortunately, the approximation is uncontrolled. In this paper I show that there are actually two different thin-wall approximations and that they bracket the true decay rate: I prove that one is an upper bound and the other a lower bound. In the thin-wall limit, the two approximations converge. In the presence of gravity, a generalization of this lemma provides a simple sufficient condition for non-perturbative vacuum instability.

Metastable states may decay by quantum tunneling. In the semiclassical (small ) limit, the most important contribution to the decay rate is the tunneling exponent B. In a famous paper Coleman explained how to calculate B for a scalar field [1]. He showed that B is given by the Euclidean action of an instanton that interpolates from the metastable (or 'false') vacuum towards the target (or 'true') vacuum. For a given field potential V (φ), the instanton can be derived by numerically integrating the second-order Euclidean equations of motion. For those who lack either the computing power or the patience to solve the equations of motion, or who seek an intuitive understanding of the parametric dependence of the decay rate, Coleman also showed that we can often use the 'thin-wall' approximation. While the instanton is a bubble that smoothly interpolates from the false vacuum to near the true vacuum, the thinwall approximation treats this transition as abrupt [1]; the decay exponent is then approximated by Consider these two expressions for the tension of the bubble wall σ, Coleman used only σ min and left Eq. 2 as an uncontrolled approximation. We can do better.
The main result of this paper is that, for any potential V (φ), The first inequality is proved in Appendix A.1; the second in Appendix A.2. As V false −V true → 0 so too σ max − σ min → 0 and the two thin-wall approximations converge. We can partially generalize this to include gravity. As first calculated by Coleman and de Luccia [2], gravitational backreaction changes B: the decay exponent now depends not just on the difference V false − V true , but also on V false and V true separately, since zero-point energy curves spacetime. The gravitational generalization of the thin-wall approximation to the decay exponent,B G tw [σ], is given by Eq. 37, and the partial generalization of Lemma 1 is that Lemma 2 (including gravity, when GV false ≤ 0) : This inequality is proved in Appendix B. The addition of gravity means it is no longer always true that B ≥B G tw [σ min ]: a de Sitter false vacuum (G V false > 0) with a short but very broad barrier has arbitrarily large σ min but still decays relatively promptly via a Hawking-Moss instanton [4]

FALSE FALSE
One application of these lemmas is diagnosing instability. Gravity can stabilize superficially metastable GV false ≤ 0 vacua [2], and there is great interest in determining which vacua decay and which endure (see e.g. [3]). Lemma 2 provides a sufficient condition for instability, which is thatB G tw [σ max ] < ∞, or equivalently sufficient condition for instability : No such condition is required for GV false > 0, since all de Sitter false vacua are unstable [2]. We can generalize Lemma 2 by replacing φ t with any value in the range φ t ≤ φ mid < φ * . Thus the false vacuum is unstable if there exists any φ mid such that When there are multiple fields, there are many possible routes over the barrier, and this condition applies to all of them: if we can find any route for the σ mid integral such that Eq. 7 holds, the vacuum must be unstable, the field must eventually decay, and spacetime is doomed [2]. V V mid mid mid Figure 2: If we can find any path across the barrier that satisfies Eq. 7, the vacuum must be unstable. Left: the σ integral need not be taken all the way to φ true , and can instead stop anywhere in the range φ true ≤ φ mid < φ * . Right: when there are multiple fields, there are many escape routes, and the decay path may take any of them.

A Proving non-gravitational results
This appendix proves Lemma 1. The two inequalities demand two very different proof strategies.
Consider the decay of the false vacuum of Fig. 1. We will assume that there are no intervening minima between the false and true vacua, but otherwise leave the potential completely general. I will now review how to calculate the vacuum decay rate; this is all explained with great clarity in [1]. In [1] it is shown that the tunneling exponent is given by the Euclidean action of an instanton. The instantonφ(τ, x) lives on R 4 and extremizes the Euclidean action where we have written ds 2 = dτ 2 + d x 2 = dρ 2 + ρ 2 dΩ 2 3 and used that the instanton can be shown [6] to have O(4) spherical symmetryφ =φ(ρ). One boundary condition is that the field returns to the false vacuumφ → φ f as ρ → ∞; the other boundary condition is that the field immediately after nucleation, given by field after nucleation : has the same energy as the false vacuum (∆E = 0) and classically evolves towards the true vacuum. (In fact, the symmetry tells us that it will give rise to a bubble of approximately true vacuum that expands out at approaching the speed of light.) The instanton gives a path through the space of φ( x)s that connects the before-tunneling configuration φ(τ =−∞, x) = φ f to the after-tunneling configuration φ(τ = 0, x). Indeed, the instanton is defined as the solution of minimum Euclidean action that satisfies these boundary conditions-in the language of [7], it is the most probable decay path 1 .
(With no constraint on the energy of the τ = 0 configuration, the instanton has exactly one negative mode [8]; this negative mode is associated with changing the energy of the nucleated bubble away from ∆E = 0. In this paper we will freeze this negative mode by only considering paths that end with the same energy as the false vacuum ∆E = 0; amongst this set of paths the instanton is a minimum of the Euclidean action [9].) To minimize the action, the instanton must satisfy the Euler-Lagrange equation, This equation implies that 1 2φ 2 − V (φ) monotonically decreases as ρ increases (in [1] this is 1 Technically we insist that the decay path ends at its first intersection with the ∆E = 0 surface. called 'friction'), which in turn implies that The tunneling exponent is then given by the different in Euclidean action between the instanton and the false vacuum Since tunneling conserves energy, immediately after nucleation the bubble must have the same energy as the false vacuum, But the energy density is not zero everywhere. Instead, energy is spent traversing the barrier in the 'wall' that is then recouped by the negative energy density inside the bubble. Fig. 3 shows Figure 3: A cross-section through a typical bubble at the moment of nucleation. The center of the bubble may have an energy density as low as V true ; but for ρ > ρ * the energy density necessarily exceeds that of the false vacuum 1 The total integrated energy is equal to that in the false vacuum, ∆E = 0. a cross-section of a typical bubble. 'Outside' the bubble (φ > φ * ) the energy density is bigger than V false ; 'inside' the bubble (φ < φ * ) we know only that the energy density is bigger than V true .
For the remainder of Appendix A we will add a constant to the potential to set V false = 0.

A.1 Proving B ≥B tw [σ min ]
Proof strategy: The instanton is a bubble configuration with zero energy, ∆E = 0. In order to have ∆E = 0, any bubble must be big, and if it is big enough it must have B ≥B tw [σ min ].

First let's construct a new potentialV [φ] that decays faster than
The region of V [φ] between φ t and φ * has been excised, with the φ > φ * part glued straight onto the φ < φ t part. This does not affect σ min or V false or V true , but since we have removed part of the barrier the decay rate is faster The shape of the bubble is plotted in Fig. 4. The bubble stays uniformly in the true vacuum until ρ =ρ * , and then proceeds towards the false vacuum with a profile given by Eq. 10. Figure 4:V [φ] is constructed by deleting the part of V (φ) that lies between φ t and φ * . The corresponding bubble instanton has pure true vacuum inside some radius that we will callρ * . The bubble has energy ∆Ê = 0.
Let's calculateB. First notice that by changing variables from ρ to φ, (16) The total energy of the bubble, relative to the false vacuum, is Tunneling conserves energy, ∆Ê = 0, so the bubble must be largê Since ∆Ê = 0 impliesB =B − πρ * ∆Ê 2 , we can use Eqs. 13 & 17 to prove our result Proof strategy: The instanton is the path of minimum action that interpolates from the false vacuum to a ∆E = 0 state on the true vacuum side of the barrier. I will explicitly construct an interpolating path with actionB tw [σ max ].
Consider the one-parameter family of field-profiles φR(ρ) parameterized byR and defined by From the definition it follows that φ t < φR(ρ) ≤ φ f . (This field profile has a discontinuous first derivative at both ρ =R and ρ = 0. This indicates it is not a minimum of the Euclidean action, but the field profile still contributes to the path integral.) The Euclidean action of φR(ρ) is Our family of field-profiles φR(ρ) must contain as 'escape path', in the language of [7]: while small values ofR gives positive energy ∆E > 0, arbitrarily large values ofR give arbitrarily negative energies, so there must be an intervening valueR E=0 such that ∆E[φR(t = 0, x)] = 0. Therefore the most probable escape path has This proof would still have gone through had we replaced V true by V mid ≡ V [φ mid ] in Eq. 26, for any φ mid in the range φ t ≤ φ mid < φ * . Thus a strengthening of the lemma is that for every possible φ mid , and (for multifield potentials) for every possible route over the barrier.

B Proving gravitational result
When gravity is included, the decaying field curves spacetime. Despite this complication, the same general proof strategy will apply as for the non-gravitational case of Sec. A.2.
The formalism that governs the gravitational decay of the false vacuum was described with great clarity in [2]. With gravity included, it has not been proved that the dominant instanton is still O(4)-symmetric; following conventional wisdom, we will assume that it is. The metric may then be written Matter tells space how to curveρ and space tells matter how to move d dξ When the gravitational constraint Eq. 34 is satisfied, the action is given by Eq. 3.9 of [2] as The thin-wall approximation to the tunneling exponent 2 (see also [14]) whereρ is the radius of the bubble wall that maximizes Eq. 37, namely (for GV f ≤ 0) As σ approaches ( for larger values of the tension, the thin-wall approximation predicts that the false vacuum is stable. To prove the gravitational lemma (Eq. 5) we will follow Sec. A.2 in constructing a family of paths and showing that one member of the family is an escape path, and every member of the family has ∆S E ≤B G tw [σ max ]. The family of field-profiles will be parameterized byR, Notice that while this family need not satisfy the equation of motion for the field, Eq. 35, it is required to satisfy the gravitational constraint, Eq. 34, because only such configurations contribute to the gravitational path integral. Notice also thatρ ≥ 1 > 0 so ρ monotonically increases with ξ and the topology is R 4 . The field and metric differ from the false vacuum only inside ρ <R, so the difference in action First consider σ max < ( √ −V t − √ −V f )/ √ 6πG . In this case, both ∆S E [R] and ∆E[R] become unboundedly negative at largeR, so the family contains an escape path; since no member of the family has an action that exceedsB G tw [σ max ], Lemma 2 holds. By contrast, for the case σ max ≥ ( √ −V t − √ −V f )/ √ 6πG the family may not contain an escape path, but sincē B G tw [σ max ] = ∞, Lemma 2 trivially holds. Thus we have proved Lemma 2 for all values of σ max . As in the non-gravitational case, this proof would still have gone through had we replaced φ t with any value in the range φ t ≤ φ mid < φ * , yielding the more powerful condition of Eq. 7.