PHYSICAL REVIEW D 97, 105002 (2018) Thin-wall approximation in vacuum decay: A lemma Adam R. Brown* Physics Department, Stanford University, Stanford, California 94305, USA (Received 16 February 2018; published 1 May 2018) 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. DOI: 10.1103/PhysRevD.97.105002 Z ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Metastable states may decay by quantum tunneling. In σ ≡ f dϕ 2ðV½ϕ − V½ϕ Þ; the semiclassical (small ℏ) limit, the most important min f Zϕà contribution to the decay rate ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ ≡ f dϕ 2ðV½ϕ − V½ϕ Þ: ð Þ max t 3 ϕ rate ∼ A exp½−B=ℏ; ð1Þ t σ Coleman used only min and left Eq. (2) as an uncontrolled is the tunneling exponent B. In a famous paper Coleman approximation, in the sense that we are provided neither explained how to calculate B for a scalar field [1].He with an estimate of its accuracy nor with a bound on its showed that B is given by the Euclidean action of an error. We can do better. The main result of this paper is that, instanton that interpolates from the metastable (or “false”) for any potential VðϕÞ, vacuum toward the target (or “true”) vacuum. For a given field potential VðϕÞ, the instanton can be derived by 1 ð Þ∶ B¯ ½σ ≤ B ≤ B¯ ½σ : ð Þ Lemma no gravity tw min tw max 4 numerically integrating the second-order Euclidean equa- tions of motion. The first inequality is proved in Appendix A1; the second For those who lack either the computing power or the V − V → 0 σ − in Appendix A2.As false true so too max patience to solve the equations of motion, or who seek an σ → 0 min and the two thin-wall approximations converge. intuitive understanding of the parametric dependence of the GV ≤ 0 GV > 0 decay rate, Coleman also showed that we can often use the false false “thin-wall” approximation. While the instanton is a bubble ¯ G always B ≤ B ½σmax? that smoothly interpolates from the false vacuum to near tw B ≥ B¯ G ½σ ? the true vacuum, the thin-wall approximation treats this always tw max transition as abrupt [1]; the decay exponent is then We can partially generalize this to include gravity. As approximated by first calculated by Coleman and de Luccia [2], gravitational backreaction changes B: the decay exponent now depends 27π2 σ4 − B ∼ B¯ ≡ : ð Þ not just on the difference Vfalse Vtrue, but also on Vfalse tw 2 ðV − V Þ3 2 false true and Vtrue separately, since zero-point energy curves space- time. The gravitational generalization of the thin-wall B¯ G ½σ Consider these two expressions for the tension of the approximation to the decay exponent, tw , is given by bubble wall σ, Eq. (B5), and the partial generalization of Lemma 1 is that Lemma2 *[email protected] ðwhenGV ≤ 0Þ∶ B ≤ B¯ G ½σ : ð5Þ Published by the American Physical Society under the terms of false tw max the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to This inequality is proved in Appendix B. The addition of B ≥ B¯ G ½σ the author(s) and the published article’s title, journal citation, gravity means it is no longer always true that tw min : and DOI. Funded by SCOAP3. GV > 0 a de Sitter false vacuum ( false ) with a short but very 2470-0010=2018=97(10)=105002(7) 105002-1 Published by the American Physical Society ADAM R. BROWN PHYS. REV. D 97, 105002 (2018) FIG. 1. The field starts in the false vacuum ϕf and tunnels towards the true vacuum ϕt. The definition of ϕà is such that V½ϕüV½ϕf . Equation (3) gives two different definitions of the tension, with σmax > σmin. σ broad barrier has arbitrarily large min but still decays APPENDIX A: PROVING relatively promptly via a Hawking-Moss instanton [3],so NONGRAVITATIONAL RESULTS B ≪ B¯ G ½σ < B¯ G ½σ that tw min tw max ; and a Minkowski false GV ¼ 0 B<B¯ G ½σ This appendix proves Lemma 1. The two inequalities vacuum ( false ) that has tw min was exhibited B demand two very different proof strategies. numerically in Sec. Vof [4]. (It is still open whether may Consider the decay of the false vacuum of Fig. 1. We will B¯ G ½σ GV > 0 be greater than tw max for false ; I suspect it assume that there are no intervening minima between the may not.) false and true vacua, but otherwise leave the potential One application of these lemmas is diagnosing instability. completely general. I will now review how to calculate the GV ≤ 0 Gravity can stabilize superficially metastable false vacuum decay rate; this is all explained with great clarity in vacua [2], and there is great interest in determining which [1].In[1] it is shown that the tunneling exponent is given vacua decay and which endure (see e.g. [5]). Lemma 2 by the Euclidean action of an instanton. The instanton provides a sufficient condition for instability, which is that ϕ¯ ðτ; x⃗Þ R4 B¯ G ½σ < ∞ lives on and extremizes the Euclidean action tw max , or equivalently Z 1 sufficient condition for instability∶ S ¼ d4x ð∂ ϕÞ2 þ VðϕÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 2 μ −V½ϕ − −V½ϕ Z σ < truepffiffiffiffiffiffiffiffiffi false : ð6Þ 1 max 6πG ¼ 2π2 dρρ3 ϕ_ 2 þ VðϕÞ ; ð Þ 2 A1 GV > 0 No such condition is required for false , since all de 2 2 2 2 2 2 Sitter false vacua are unstable [2]. where we have written ds ¼ dτ þ dx⃗ ¼ dρ þ ρ dΩ3 ϕ We can generalize Lemma 2 by replacing t with any and used that the instanton can be shown [6] to have O(4) ϕ ≤ ϕ < ϕ ¯ ¯ value in the range t mid Ã, as shown in Fig. 2. Thus spherical symmetry ϕ ¼ ϕðρÞ. One boundary condition is the false vacuum is unstable if there exists any ϕ such ϕ¯ → ϕ ρ → ∞ mid that the field returns to the false vacuum f as ; that the other boundary condition is that the field immediately Z ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi after nucleation, given by σ ≡ f dϕ 2ðV½ϕ − V½ϕ Þ mid mid ϕ ∶ϕðt ¼ 0; x⃗Þ¼ϕ¯ ðτ ¼ 0; x⃗Þ; ð Þ midpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi field after nucleation A2 −V½ϕ − −V½ϕ < midpffiffiffiffiffiffiffiffiffi false : ð7Þ has the same energy as the false vacuum (ΔE ¼ 0)and 6πG classically evolves toward the true vacuum. (In fact, the symmetry tells us that it will give rise to a bubble of When there are multiple fields, there are many possible approximately true vacuum that expands out at approaching routes over the barrier, and this condition applies to all of the speed of light.) The instanton gives a path through the them: if we can find any route for the σ integral such that mid space of ϕðx⃗Þs that connects the before-tunneling configu- Eq. (7) holds, the vacuum must be unstable, the field must ϕðτ ¼ −∞; x⃗Þ¼ϕ eventually decay, and spacetime is doomed [2]. ration f to the after-tunneling configura- tion ϕðτ ¼ 0; x⃗Þ. Indeed, the instanton is defined as the solution of minimum Euclidean action that satisfies these ACKNOWLEDGMENTS boundary conditions—in the language of [7],itisthemost 1 Thank you to Sonia Paban and Erick Weinberg probable decay path. for reading a draft of this paper, and to the organizers of TASI ’09 for their hospitality while these lemmas were 1Technically we insist that the decay path ends at its first established. intersection with the ΔE ¼ 0 surface. 105002-2 THIN-WALL APPROXIMATION IN VACUUM DECAY: A LEMMA PHYS. REV. D 97, 105002 (2018) FIG. 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 ϕtrue, and can instead stop anywhere in the range ϕtrue ≤ ϕmid < ϕÃ. Right: when there are multiple fields, there are many escape routes, and the decay path may take any of them. Z ∞ 1 τ ¼ 0 B ¼ S ðϕ¯ Þ−S ðϕ¯ Þ¼2π2 dρρ3 ϕ_ 2 þVðϕÞ−V : (With no constraint on the energy of the configu- E E f false ration, the instanton has exactly one negative mode [8]; this 0 2 negative mode is associated with changing the energy of the ðA6Þ nucleated bubble away from ΔE ¼ 0. In this paper we will freeze this negative mode by only considering paths that Since tunneling conserves energy, immediately after nucle- end with the same energy as the false vacuum ΔE ¼ 0; ation the bubble must have the same energy as the false amongst this set of paths the instanton is a minimum of the vacuum, Euclidean action [9].) Z ∞ 1 To minimize the action, the instanton must satisfy the ΔE ≡ 4π dρρ2 ϕ¯_ 2 þ Vðϕ¯ Þ − V ¼ 0: ð Þ false A7 Euler-Lagrange equation, 0 2 But the energy density is not zero everywhere. Instead, the d 1 _ 2 3 _ 2 ϕ¯ − Vðϕ¯ Þ ¼ − ϕ¯ : ð Þ energy density is positive in the “wall” of the bubble where dρ 2 ρ A3 the field traverses the barrier, and then negative inside the bubble.