PHYSICAL REVIEW D 100, 104007 (2019) Fresh look at the calculation of tunneling actions including gravitational effects J. R. Espinosa Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology (BIST), Campus UAB, 08193 Bellaterra, Barcelona, Spain, ICREA, Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain and Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid, 28049 Madrid, Spain (Received 6 August 2019; published 5 November 2019) Recently, the calculation of tunneling actions that control the exponential suppression of the decay of metastable vacua has been reformulated as an elementary variational problem in field space. This paper extends this formalism to include the effect of gravity. Considering tunneling potentials V t ðϕÞ that go from the false vacuum ϕþ to some ϕ0 on the stable basin of the scalar potential VðϕÞ, the tunneling action is the R minimum of the functional SE ½V t  ¼ 6π 2 m4P ϕϕþ0 ðD þ V 0t Þ2 =ðV 2t DÞdϕ, where D ≡ ½ðV 0t Þ2 þ 6ðV − V t ÞV t = m2P 1=2 , V 0t ¼ dV t =dϕ, and mP is the reduced Planck mass. This one-line simple result applies equally to anti–de Sitter, Minkowski, or de Sitter vacua decays and reproduces the Hawking-Moss action in the appropriate cases. This formalism provides new handles for the theoretical understanding of different features of vacuum decay in the presence of gravity. DOI: 10.1103/PhysRevD.100.104007 I. INTRODUCTION ðV − V t Þ2 sðV t Þ ≡ 54π 2 ≥ 0; ð2Þ The calculation of the tunneling action that controls the ð−V 0t Þ3 exponential suppression of the decay of metastable states has been reformulated recently in [1]. The new method find the V t ðϕÞ that minimizes S½V t . The minimum action offers an alternative to the standard solution by Coleman thus found is the tunneling action corresponding to the [2], which is based on the calculation of a tunneling bounce decay of the false vacuum at ϕþ . For details about the by solving a differential equation in Euclidean space. derivation of this simple result, see [1]. In a nutshell, the new approach works as follows. The new method of calculation has a number of attractive Consider a potential VðϕÞ with a false vacuum at ϕþ features: it can be considered as a generalization of the and a true vacuum at ϕ− > ϕþ , as in the examples shown in thin-wall case for arbitrary potentials; it allows a fast and Fig. 1. Take a tunneling potential V t ðϕÞ ≤ VðϕÞ that flexible numerical estimate of S, including the case of connects the false vacuum with some point ϕ0 on the multifield potentials [3]; it can be used to generate slope beyond the barrier, in the basin of the true vacuum, potentials that admit analytic solutions to the tunneling with V t ðϕþ Þ ¼ Vðϕþ Þ ≡ V þ and V t ðϕ0 Þ ¼ Vðϕ0 Þ ≡ V 0 . problem; it can be readily extended to the case of decays by To any such function V t ðϕÞ, associate the action thermal fluctuations etc.; see [1]. Moreover, it is useful to have alternative formulations of important problems as Z ϕ0 ðV − V t Þ2 different approaches can offer a better handle in dealing S½V t  ≡ 54π 2 dϕ; ð1Þ ϕþ ð−V 0t Þ3 with different issues. Last but not least, the new formulation is extremely direct and simple to state. The purpose of this paper is to extend the work in [1] where V 0t ¼ dV t =dϕ. Then, under the condition that the including the effect of gravity, which can be quite relevant action density should satisfy for vacuum decay in cosmological settings or in discussions of the population of vacua in the string landscape etc. The solution to this problem in the Euclidean bounce Published by the American Physical Society under the terms of formulation dates back to the work of Coleman and De the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Luccia in [4]. the author(s) and the published article’s title, journal citation, In the new approach, the inclusion of gravitational and DOI. Funded by SCOAP3. effects simply modifies the action density to 2470-0010=2019=100(10)=104007(11) 104007-1 Published by the American Physical Society J. R. ESPINOSA PHYS. REV. D 100, 104007 (2019) The rest of the paper is organized as follows. In Sec. II, the standard Euclidean approach of Coleman and De Luccia to the calculation of the semiclassical tunneling exponent for vacuum decay in the presence of gravity is reviewed. In Sec. III, the main result (3) is obtained, and the corresponding Euler-Lagrange equation for the instanton contribution to the tunneling exponent is derived. After some comments in Sec. IV on the new action, the new method is applied in “reverse gear” to obtain analytically a potential VðϕÞ from a simple V t ðϕÞ in Sec. V. After drawing some conclusions, several appendixes are devoted to more technical (but important) discussions. Appendix A contains the proof that the new formulation agrees with the standard formulation. Appendix B derives the thin-wall limit for the tunneling action using the new formulation. II. EUCLIDEAN ACTION VIA THE TUNNELING BOUNCE Take a single scalar field ϕ in four dimensions, with a potential VðϕÞ that has a metastable false minimum at ϕþ and a deeper minimum at ϕ− ; see Fig. 1. For simplicity ϕþ ¼ 0 is chosen in many of the plots, without loss of generality. The decay of this false vacuum proceeds by fluctuations FIG. 1. Tunneling potential V t ðϕÞ (red) for decay out of an AdS that nucleate bubbles of the energetically preferred phase, vacuum (upper plot) or a dS vacuum (lower plot) in some which grow and eat out the metastable phase. For suffi- example potentials VðϕÞ (blue). ciently long-lived states, the vacuum decay rate (per unit volume) Γ=V is exponentially suppressed. For the decay 6π 2 ðD þ V 0t Þ2 by quantum fluctuations Γ=V ¼ Ae−ΔSE =ℏ , where ΔSE is sðV t Þ ≡ ; ð3Þ the difference between two Euclidean actions, one for the κ2 V 2t D instanton mediating the decay (a bounce/instanton con- figuration that connects the two phases) and the other for where κ ≡ 1=m2P , with mP ¼ 2.435 × 1018 GeV being the the false vacuum background. reduced Planck mass; D is a generalization of the field The bounce configuration results from solving a non- derivative of V t that includes gravitational corrections linear differential equation, a Euler-Lagrange Euclidean qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi equation of motion with appropriate boundary conditions. D ¼ DðϕÞ ≡ ðV 0t Þ2 þ 6κðV − V t ÞV t : ð4Þ In the presence of gravity one also has to deal with the cross-talk between the Euclidean space-time metric and the instanton configuration. Assuming that the instanton sol- The problem to solve is as before: find the V t that R ϕ0 ution that dominates the decay has Oð4Þ symmetry, the minimizes the action ϕþ sðV t Þdϕ. Now the action density most general Oð4Þ-symmetric metric can be written as (3) is explicitly positive definite, but it should be real, too, of course, so V t is constrained to give real D. ds2 ¼ gμν dxμ dxν ¼ dξ2 þ ρðξÞ2 dΩ23 ; ð5Þ This remarkably simple formulation applies to the decay of any type of vacua: anti–de Sitter (AdS) (V þ < 0), where ξ is a radial coordinate that measures the radial Minkowski (V þ ¼ 0) or de Sitter (dS) (V þ > 0) and, in distance along lines normal to three spheres of radius of the latter case, it reproduces the Hawking-Moss exponent curvature ρðξÞ while dΩ23 is the line element on a unit three- [5] in the appropriate cases. Examples of V t functions that sphere. The Ricci curvature scalar for this metric is minimize the tunneling action are shown by the red curves in Fig. 1 for an AdS vacuum (upper plot) and a dS one 6 R¼ ð1 − ρ̈ρ − ρ_2 Þ; ð6Þ (lower plot). The qualitatively different behavior of the two ρ2 cases is apparent (the Minkowski case is similar to the AdS one). where the dots stand for derivatives with respect to ξ. 104007-2 FRESH LOOK AT THE CALCULATION OF … PHYS. REV. D 100, 104007 (2019) The instanton action for the decay of the metastable ϕþ with ρ ¼ 0 at both ends of the interval. For more details on (false) vacuum is obtained [2] by finding an Oð4Þ- the standard picture, see [4,6]. symmetric bounce ϕb ðrÞ (or Euclidean bubble) that inter- The Euclidean action, from which the previous equations polates between the false vacuum and (the basin of) the true for ϕ and ρ follow, is vacuum at ϕ− . This bounce solves the Euclidean equation Z   4 pffiffiffi 1 μν R of motion SE ½ϕ ¼ d x g g ∂ μ ϕ∂ ν ϕ þ VðϕÞ − þ SGHY 2 2κ ̈ þ 3ρ_ ϕ_ ¼ V 0 ; ϕ ð7Þ ð12Þ ρ where SGHY is the (Euclidean) Gibbons-Hawking-York where a dot (prime) represents a derivative with respect to ξ (GHY) boundary term [7]. Applying this general expres- (ϕ). On the other hand, from Einstein’s equations it follows sion to the Euclidean bounce, one gets that ρðξÞ satisfies the equation Z ξ    3 1 _2 max 2   SE ½ϕb  ¼ 2π ρ ϕ þ VðϕÞ κρ2 1 _ 2 0 2 _2 ρ ¼1þ ϕ −V ; ð8Þ  3 2 3ρ 2 þ ðρ̈ρ þ ρ_ − 1Þ dξ þ SGHY ; ð13Þ κ where κ ≡ 1=m2P . The system of coupled differential equations (7) and (8) where it is understood that ξmax ¼ ∞ for noncompact should be solved with the boundary conditions bounces, and ρ2 ρ_ ξ¼ξmax ϕ_ b ð0Þ ¼ 0; ρð0Þ ¼ 0 ð9Þ SGHY ¼ −6π 2 : ð14Þ κ ξ¼0 at the origin ξ ¼ 0, with ϕ0 ¼ ϕb ð0Þ being an unknown. Integration by parts gets rid of the ρ̈ term and cancels out For the decay of a dS vacuum, ρðξÞ returns to 0 at the GHY term. One gets some point Z ξ     3 1 _2 3ρ 2 max 2 ρðξmax Þ ¼ 0; ð10Þ SE ½ϕb  ¼ 2π ρ ϕ þ V − ðρ_ þ 1Þ dξ: 0 2 κ and the bounce is compact. Instead, for AdS and ð15Þ Minkowski vacuum decay, ρðξÞ keeps growing indefinitely Furthermore, both expressions (13) and (15) for the bounce and the bounce is noncompact. In these cases we take action can be simplified by using the equation of motion (7) ξmax ¼ ∞. The boundary condition on the bounce at that and the constraint (8) to get the two expressions point is Z ξ max 2 SE;1 ½ϕb  ¼ −2π ρ3 Vdξ; ð16Þ ϕ_ b ðξmax Þ ¼ 0: ð11Þ 0 Z ξ   max 3 In the presence of gravity it is not guaranteed that SE;2 ½ϕb  ¼ 4π 2 ρ3 V − ρ dξ: ð17Þ 0 κ ϕb ðξmax Þ ¼ ϕþ : for dS vacua ϕb ðξmax Þ ≡ ϕ0þ can be quite different from ϕþ (although it is always on the basin of the The tunneling exponent that suppresses vacuum decay is false vacuum). the difference between this action evaluated for the bounce Identifying ξ with time, Eq. (7) corresponds to the and the action associated to the background (false vacuum) classical motion of a particle in the inverted potential field configuration. −VðϕÞ with a velocity and time dependent friction force. The background action can be obtained by substituting The solution can be found by undershooting and over- in the actions above the potential V by its value at the false shooting, changing the value of the field at the center of the vacuum, V ¼ V þ , and the radius ρðξÞ by the corresponding Euclidean bubble, ϕb ðξ ¼ 0Þ ≡ ϕ0 , till the boundary con- solutions of (8) in such a background, dition at ξ → ∞ is satisfied. At the same time, Eq. (8) has to be solved, with different asymptotic behaviors, depending ρdS ðξÞ ¼ ρ̄ sinðξ=ρ̄Þ; on the type of vacua Vðϕþ Þ is. For AdS and Minkowski ρM ðξÞ ¼ ξ; vacua, the bounce is noncompact, with ρ extending over the infinite range ξ ¼ ð0; ∞Þ. At ξ → ∞, for Minkowski ρAdS ðξÞ ¼ ρ̄ sinhðξ=ρ̄Þ; ð18Þ vacua ρðξÞ ∼ ξ while for AdS, ρðξÞ ∼ expðξ=ρ̄Þ, with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=ρ̄2 ¼ −V þ κ=3. For dS vacua, instead, the bounce is with ρ̄ ≡ 3=ðκjV þ jÞ for decays from dS, Minkowski, or compact and ρ extends over a finite range ξ ¼ ð0; ξmax Þ, AdS vacua, as indicated. One can obtain the background 104007-3 J. R. ESPINOSA PHYS. REV. D 100, 104007 (2019) actions SEþ analytically. In the case of Minkowski or AdS thin-wall expression can be used. The standard derivation is decays, which occur through a noncompact bounce (with not discussed here but Appendix B contains a simple and ξmax ¼ ∞), the background action SEþ diverges, but the direct derivation of this formula using the alternative divergence cancels against a similar divergence coming approach presented in this paper. from the bounce action SE ½ϕb  and one ends up with a finite tunneling exponent, III. EUCLIDEAN ACTION VIA A TUNNELING POTENTIAL ΔSE ≡ SE ½ϕb  − SEþ : ð19Þ Let us follow the approach of [1] and introduce an auxiliary function, V t ðϕÞ, the tunneling potential. Its It is customary to regulate such actions using a cutoff in the connection with the standard bounce method is ξ integrals matching both solutions to enforce the cancel- lation, but it is more convenient to leave the SEþ integral 1 unevaluated and rewrite it in terms of an integral over the V t ðϕÞ ≡ VðϕÞ − ϕ_ 2b ; ð25Þ 2 bounce ξ-coordinate, as shown below. Consider the AdS vacuum decay first. Using form (16), where it is understood that ϕ_ b is considered as a function of the background action for the AdS false vacuum (V þ < 0) the field ϕ. can be written as The properties of V t ðϕÞ are nearly the same as in [1]: Z ∞ (1) obviously V t ðϕÞ ≤ VðϕÞ, with V t ðϕ0 Þ ¼ Vðϕ0 Þ and SEþ ¼ −2π 2 ρ3AdS V þ dξAdS : ð20Þ V t ðϕ0þ Þ ¼ Vðϕ0þ Þ, as ϕ_ b ð0; ξmax Þ ¼ 0 at these end points 0 of the bounce; (2) V t ðϕÞ is a monotonic function for the case of noncompact bounces but it is not monotonic for One can change the integration variable from ξAdS to the compact bounces. This difference in behavior follows from bounce ξ-coordinate by identifying ρAdS ðξAdS Þ ¼ ρðξÞ, the fact that V t ðϕÞ is minus the Euclidean energy, which from which one gets varies with ξ as   dξAdS ρ_ d 1 _2 3 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð21Þ ϕb − Vðϕb Þ ¼ − ρ_ϕ_ 2b : ð26Þ dξ 1 − κV þ ρ2 =3 dξ 2 ρ Using this to rewrite SEþ one ends up with the expression For noncompact bounces ρ_ > 0 and the Euclidean energy is dissipated by the friction term in (7). However, for compact Z ∞   V þ ρ_ bounces, ρ_ changes sign at some intermediate value ξ ¼ ξt ΔSE;1 ¼ −2π 2 ρ3 V − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dξ: ð22Þ in the interval (0, ξmax ). For ξ > ξt, ρ_ < 0 and there is 0 1 − κV þ ρ2 =3 antifriction. Noting that the bounce is also a monotonic If form (17) of the action is used instead, following the function of ξ (intuitively clear from the “motion in an same procedure one arrives at inverted potential” picture1), there is a one-to-one corre- spondence between the slope of ρ and that of V t ðϕÞ. So, for Z   noncompact bounces (for decay from AdS or Minkowski ∞ 3ρ ðρ3 V þ − 3ρ=κÞρ_ ΔSE;2 ¼ 4π 2 3 ρ V − − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dξ: ð23Þ vacua) V t is monotonically decreasing (V 0t < 0) as in the 0 κ 1 − κV þ ρ2 =3 case without gravity. For compact cases, corresponding to decays from dS space, the slope of V t is negative on the side Of course one should get ΔSE;1 ¼ ΔSE;2 . Both tunneling closer to the true vacuum but turns positive on the side actions (22) and (23) also reproduce the Minkowski result closer to the false vacuum (corresponding to ξ > ξt ). simply setting V þ ¼ 0, as the procedure followed goes Examples of these different behaviors of V t ðϕÞ are shown through with dξM =dξ ¼ ρ_ [which is the V þ ¼ 0 limit in Fig. 1. of (21)]. Proceeding as in [1] one can remove any reference to the For the dS case this rewriting of SEþ is not needed and bounce (and the four-dimensional Euclidean space in one simply has, using (16) which it lives) in favor of V t ðϕÞ. From (25) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24π 2 ϕ_ b ¼ − 2½VðϕÞ − V t ðϕÞ; ð27Þ ΔSE ¼ SE ½ϕb  þ : ð24Þ κ2 V þ where the minus sign, chosen due to ϕþ < ϕ− , would be a As in the case without gravity, analytical results for these plus if one takes ϕþ > ϕ− . Equation (27) can then be used tunneling exponents are generically not possible and one resorts to numerical solutions of the differential equations. 1 Nonmonotonic oscillating Coleman-De Luccia bounces exist One exception is the case when the false vacuum is nearly [8] but for the purposes of this paper only their monotonic part degenerate with the true one, a case in which an analytical starting at ξ ¼ 0 is relevant. 104007-4 FRESH LOOK AT THE CALCULATION OF … PHYS. REV. D 100, 104007 (2019) to remove any ξ-derivative of ϕb in terms of V t (and V). However, one can reverse engineer the problem and find The Euclidean radius ρ can be obtained from (7) and (8) as what action density reproduces Eq. (31) under variations pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with respect to V t . Following that route, one can determine 3 2ðV − V t Þ the tunneling action density up to an arbitrary term that only ρ¼ ; ð28Þ depends on V t . Comparing with the bounce action of the D standard approach that constant is fixed and one arrives at where the combination Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6π 2 ϕ0 ðD þ V 0t Þ2 S½V t  ¼ 2 dϕ : ð35Þ D ¼ DðϕÞ ≡ ðV 0t Þ2 þ 6κðV − V t ÞV t ð29Þ κ ϕþ V 2t D first appears. Further derivatives of ρ can be eliminated in The detailed proof of the exact correspondence between this the same way, e.g., action and the standard one is presented in Appendix A. In that appendix it is shown that, for the decay of Minkowski or V 0t κ AdS vacua, one has ρ_ ¼ − ; ρ̈ ¼ − ρð3V − 2V t Þ: ð30Þ D 3 S½V t  ¼ ΔSold E ≡ SE ½ϕb  − SEþ ; ð36Þ _ t Þ for compact bounces is related to a Notice that the 0 in ρðξ 0 in V 0t (at some ϕt away from the end points), in agreement so that the new action encapsulates in a single expression the with the previous discussion. Notice that, to keep D real, difference of the bounce and background actions of the this requires V t ðϕt Þ > 0. By continuity, V 0t > 0 [in the standard formulation. interval (ϕþ , ϕt )] also requires V t > 0. The dS case requires a more detailed discussion as in Taking a derivative of (28) with respect to ξ one gets the general ϕb ðξmax Þ ≡ ϕ0þ ≠ ϕþ , so that the field range differential equation for V t, covered by the compact bounce, (ϕ0þ , ϕ0 ) does not correspond to the range (ϕþ , ϕ0 ) in the integral of ð4V 0t − 3V 0 ÞV 0t ¼ 6ðV t − VÞ½V 00t þ κð3V − 2V t Þ ; ð31Þ Eq. (35). Remarkably, an equality like (36) also holds in the dS case if one extends the definition of V t outside the which takes the place of (7) in the new formulation of the instanton field range, taking V t ¼ V in the interval (ϕþ , tunneling problem (at least for Minkowski and AdS vacua): ϕ0þ ). Such extended V t is shown in Fig. 1, lower plot. With find a ϕ0 and a V t ðϕÞ that solve (31) with the boundary that extension, one has conditions, Z Z 24π 2 V 0t 6π 2 ϕ0 ðD þ V 0t Þ2 ϕ0þ V t ðϕþ Þ ¼ Vðϕþ Þ; V t ðϕ0 Þ ¼ Vðϕ0 Þ: ð32Þ S½V t  ¼ dϕ þ dϕ κ2 2 ϕþ V t κ 2 ϕ0þ V 2t D   Z Equation (31) also leads [assuming V 0 ðϕþ Þ ¼ 0] to 24π 2 1 1 ϕ0 ¼ 2 − þ sðϕÞdϕ: ð37Þ κ V þ V 0þ ϕ0þ V 0t ðϕþ Þ ¼ 0; V 0t ðϕ0 Þ ¼ 3V 0 ðϕ0 Þ=4: ð33Þ In Appendix A it is shown that, again, this exactly The case of dS vacua, for which the instanton does not reproduces the standard result, so that (36) also holds for dS. reach all the way to ϕþ , is discussed below. Note that V t ¼ V is not a solution of the instanton Note that the limit κ → 0 of Eq. (31) reduces to the equation of motion (31). Nevertheless, an important prop- differential equation for the case without gravity discussed in erty of the action (35) is, by construction, that its functional [1], as it should. Notice also that with the new formulation variation with respect to V t ðϕÞ returns the equation of there is no need to keep track of the two functions ρðξÞ and motion (31). More explicitly, one gets ϕðξÞ but just one: V t ðϕÞ. For later use, notice that Eq. (31) can be rewritten in δS ðV − V t Þ ¼ −108π 2 EoM; ð38Þ terms of D as δV t D5 d 3V 0 − 4V 0t where EoM≡ð4V 0t −3V 0 ÞV 0t þ6ðV −V t Þ½V 00t þκð3V −2V t Þ. log D ¼ : ð34Þ dϕ 6ðV − V t Þ From the additional (V − V t ) factor one sees that the extension V ¼ V t away from the instanton range also Once V t has been found, one still needs to calculate the extremizes the action. action associated to it. In the case without gravity Derrick’s Another interesting consequence of Eq. (37) is that it theorem [9] was used to select one particular form of the returns the Hawking-Moss action when the Coleman-De bounce action to be transformed to the new language [1], but Luccia bounce disappears. In that case, ϕ0þ and ϕ0 there is no recourse to a Derrick’s theorem with gravity. converge to ϕT , the field value corresponding to the top 104007-5 J. R. ESPINOSA PHYS. REV. D 100, 104007 (2019) of the barrier. In that case the instanton part of the action For a thermal description of this two-step dS tunneling (37) vanishes and one gets, with V T ≡ VðϕT Þ, see [10].   24π 2 1 1 S½V t  ¼ 2 − ≡ SHM ; ð39Þ IV. COMMENTS ON S κ Vþ VT Some additional comments on the action (35) are the precisely the Hawking-Moss result [5]. following. To see this in more quantitative terms, assume that (the (a) The action density is explicitly positive definite, instanton part of) V t is very flat, say V t ≃ V i, where V i is a positive constant (with V þ ≤ V i ≤ V T ). Then one can 6π 2 ðD þ V 0t Þ2 neglect V 0t and pull V t out of the integral for S½V t . s≡ ≥ 0; ð44Þ Using (37) κ2 V 2t D   pffiffiffi 24π 2 1 1 6π 2 3σðV i Þ and should be integrated over a finite field interval. As a S≃ 2 − þ ; ð40Þ trivial consequence, this expression is better suited for κ Vþ Vi ðκV i Þ3=2 numerical evaluation than the standard ones, which can with take both negative and positive values. For illustration, Z Fig. 2 shows the new action density (red curve, upper plot) ϕi− pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and the two versions of the standard action densities σðV i Þ ≡ 2ðV − V i Þdϕ; ð41Þ ϕiþ corresponding to (22) and (23), rewritten as functions of ϕ (lower plot) for a particular potential (described in the where ϕiþ;i− are the two solutions of VðϕÞ ¼ V i . For a next section). Although the final action is SE ≃ 870, the more realistic trial solution, σðV i Þ would be replaced by a standard action densities take very large peak values, which more complicated integral, but the rough approximation can lead to lower precision of numerical evaluations (that with flat V i can be used to estimate parametrically the behavior of the tunneling solution. The noninstanton piece of (40) starts at 0 for V i ¼ V þ and grows to the Hawking-Moss action for V i ¼ V T , the top of the barrier. The instanton piece instead starts at some positive value for V i ¼ V þ [with σðV i Þ corresponding then to the usual wall tension if the thin-wall limit is applicable], and goes to 0 when V i ¼ V T . For the restricted type of approximate V t configurations considered, but also in general, the action is minimized by the interplay of these two opposite tendencies. If κV þ grows [keeping the shape of VðϕÞ unchanged], the noninstanton piece of (40) decreases as 1=ðκV þ Þ2 while the instanton piece decreases at the slower rate 1=ðκV þ Þ3=2 . Therefore, for sufficiently large κV þ the noninstanton part wins and the action is minimized by the Hawking-Moss configuration. In the opposite limit of κV þ → 0 the noninstanton part diverges and the instanton part dominates the tunneling (with V i → V þ and ϕ0þ → ϕþ ). When the minimum of the action occurs at some intermediate value of V i , that value could be calculated by solving dS ¼ 0; ð42Þ dV i which leads to FIG. 2. Comparison of the new action density of Eq. (44) rffiffiffiffiffi Z ϕ Vi 3 i− V i dϕ (upper plot) with standard tunneling action densities sE;1;2 of 4 ¼ σðV i Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð43Þ Eqs. (22) and (23) (lower plot), for the potential of Sec. V, with 3κ 2 ϕiþ 2ðV − V i Þ C ¼ 100, κ 0 ¼ 1, V þ =μ4 ¼ −0.5. 104007-6 FRESH LOOK AT THE CALCULATION OF … PHYS. REV. D 100, 104007 (2019) require a cancellation between contributions of opposite should be satisfied. An explicit case of this behavior is sign much larger than the end result). shown in the discussion of Appendix A. (b) When the gravitational effects are small one can (d) Already from the pioneering work of Coleman and expand the action density in powers of κ. For the decay De Luccia [4] it is known that gravitational effects could from Minkowski or AdS, V 0t ≤ 0, and the expansion is quench vacuum decay forbidding decays that would be allowed without gravity (see Appendix B for a rederivation   ðV − V t Þ2 2 6κðV − V t ÞV t of this famous result in the thin-wall limit). In the new s ¼ 54π 1− þ Oðκ2 Þ: ð45Þ formalism, gravitational quenching of vacuum decay can ð−V 0t Þ3 ðV 0t Þ2 be understood quite simply as the result of the impossibility of finding tunneling paths with real D for some potentials. The κ 0 term reproduces the action presented in [1]. In the Minkowski or AdS cases, the monotonicity of V t sets Concerning the OðκÞ terms, noting that V t ≤ 0 in such a limit on the average values of V 0t and V t that might not decays, one sees that the small gravitational effects always be sufficiently large to compensate the negative value make the vacua more stable (higher tunneling action). of 6κðV − V t ÞV t . In integral form, V t should satisfy the In assessing this effect there is no need to worry about the inequality OðκÞ effect of gravity on the V t calculation itself as the zero-gravity action is stationary and such corrections affect Z ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 the action only at Oðκ2 Þ. ΔV þ0 ≡ V þ − V 0 ≥ 6κðV − V t Þð−V t Þdϕ; ð49Þ ϕþ For the decay of dS vacua, there is always a region of field space for which V 0t ≥ 0, V t ≥ 0. In that region one gets and it is easy to imagine cases in which this is not possible. a different expansion, For instance, large enough κðV − V þ Þ would lead to gravitational quenching. V 0t (e) As in the case without gravity, one expects that it s ¼ 24π 2 κ V 2t 2 should be simple to estimate V t ðϕÞ for a given potential   ðV − V t Þ2 6κðV − V t ÞV t making educated guesses (as in [1]) to get an accurate þ 54π 2 1− þ Oðκ2 Þ: ð46Þ approximation to the tunneling action using (35), although ðV 0t Þ3 ðV 0t Þ2 no dedicated study of this application is performed in this paper. The success of such a numerical approach rests on As explained before, the 1=κ2 term plays an important the fact that the action (1) is not only an extremal for the V t role in connecting the tunneling action (35) to the that solves the corresponding Euler-Lagrange equation but Hawking-Moss action. When the effects of gravity are is in fact an absolute minimum. In the case with gravity, the small, however, ϕ0þ is exponentially close to ϕþ and action (35) is, by construction, an extremal for the right V t the divergent contribution from the 1=κ 2 term goes to 0, that solves (31) (supplemented by an interval with V t ¼ V as we have discussed before. in the case of dS vacua). Although we expect that also with (c) The action density s would blow up if either V t or D gravity present the action (35) is an absolute minimum at goes to 0. When V t crosses 0 (as it must happen for a decay the right V t , we have not been able to prove it in full from dS to AdS), an expansion in powers of V t gives generality and leave such a proof for future work. qffiffiffiffiffiffiffiffiffiffiffi 12π 2 h i V. POTENTIALS WITH EXACT s¼ ðV 0t Þ2 þ V 0t þ OðV 0t Þ: ð47Þ κ 2 V 2t TUNNELING SOLUTIONS Having potentials that allow one to solve the tunneling The divergence is absent if V 0t < 0, which is satisfied for dS problem analytically is quite useful and there are a number decays at the point where V t crosses 0 (remember that V 0t of papers in the literature that provide such potentials using starts positive from ϕþ , reaches 0 at some intermediate different methods of attack. In the presence of gravity, a value where V t > 0, and only later becomes negative). possibly incomplete list of such previous work is [11]. Concerning D, which is positive definite, the equation of The new approach to tunneling action calculations based motion for V t, written in the form (34), shows that D on the tunneling potential is also useful for the purpose of approaches 0 only exponentially, because its derivative is finding such analytical potentials, as was demonstrated in proportional to itself. So, one does not expect to find D ¼ 0 [1]. Here this application is extended to the case with gravity. at some intermediate value of ϕ but D → 0 happens at ϕþ Instead of starting from V and solving for V t, one for AdS or Minkowski decays. In order to get a finite value postulates a given V t and integrates (31) to obtain the of S the condition corresponding V. While in the case without gravity a closed form solution for V could be given, this does not seem V 0t lim ðϕ − ϕþ Þ ¼ 0 ð48Þ possible with gravity. Nevertheless, one can still solve for V ϕ→ϕþ D for particular choices of V t . 104007-7 J. R. ESPINOSA PHYS. REV. D 100, 104007 (2019) lives. Moreover, it allows one to formulate the problem of obtaining such actions in rather elementary terms, as a quite simple variational problem: find a function V t that mini- mizes the actional functional (1) connecting the metastable and stable phases. An obvious extension of the work in [1] was to include the effects of gravity, which can be very relevant, and it was not clear whether the simple alternative prescription would also hold in this rather more complicated situation. In this paper it is shown that this is indeed the case, with the problem being reduced to a similar variational problem, with just a slightly more complicated action density, as given in (3). It is remarkable that such a simple prescription encompasses FIG. 3. Potential VðϕÞ of Eq. (51), with C ¼ 100, κ0 ¼ 1, decays from all types of vacua (dS, Minkowski, and AdS) in V þ =μ4 ¼ −0.5 (blue) and its corresponding V t ðϕÞ, from a single universal formula and also reproduces the Hawking- Eq. (50), (red). Moss exponent in the appropriate limit. Sidney Coleman would have been pleased with it. In spite of its very appealing features in terms of simplicity Take, for instance, and ease of implementation, this new method cannot be taken as an alternative to completely replace the bounce V t ¼ V þ − μ4 sin2 φ; ð50Þ approach, which is certainly more fundamental. To begin with, the alternative method relies on the Oð4Þ symmetry of where V þ ≤ 0 (only Minkowski or AdS vacua) and the bounce that dominates the decay (an assumption that φ ≡ ϕ=M, with μ, M being some mass scales that control remains unproven in the case with gravity) and cannot be the depth and slope of V t ðϕÞ. It is possible to integrate (31) applied if that symmetry is not present. Moreover, the obtaining quantum corrections [12] to the semiclassical decay rate   have to be computed by considering fluctuations over the 0 −ð1þαÞκ 0 μ8 s22φ μ4 cακ φ sφ bounce and these are explicitly not Oð4Þ symmetric. V ¼ Vt − 1þ ; ð51Þ 6κ 0 V t 2½AðφÞ þ CV t Rather than as a replacement for the bounce method, the new approach is complementary to it and can be most where sφ ≡ sin φ, cφ ≡ cos φ, κ 0 ≡ κM 2 , α ≡ V þ =μ4 − 1. useful not only in numerical applications but also in gaining The function AðφÞ is given in terms of the Appell theoretical insight in particular problems in which such hypergeometric function of two variables, F1 ða; b1 ; insight is more difficult to gather from the standard b2 ; c; x; yÞ, as approach. Hopefully it will be fruitful also in this respect. c2þακ φ 0 ACKNOWLEDGMENTS AðφÞ ≡ F1 ða; b1 ; b2 ; c; c2φ ; −c2φ =αÞ; ð52Þ α2 ð2 þ ακ 0 Þ I thank Pepe Barbón and Jean-François Fortin for interest- ing discussions and comments. This work has been supported with a ¼ 1 þ ακ 0 =2, b1 ¼ ð1 þ αÞκ 0 =2, b2 ¼ 2, and by the European Research Council (ERC) Grant No. 669668 c ¼ 2 þ ακ 0 =2. Finally, C is an integration constant that —NEO-NAT—ERC-AdG-2014, the Spanish Ministry can be expressed in terms of ϕ0 solving Vðϕ0 Þ ¼ V t ðϕ0 Þ. Ministerio de Economía y Competitividad (MINECO) under An example of this potential is given in the upper plot of Grants No. 2016-78022-P and No. FPA2014-55613-P, the Fig. 1, corresponding to the parameter choice C ¼ 4, Severo Ochoa excellence program of MINECO (Grants κ0 ¼ 0.8, and V þ =μ4 ¼ −0.25. The potential (51) also No. SEV-2016-0588 and No. SEV-2016-0597) and by the has a thin-wall limit for κ 0 ≪ 1. As an example, for Generalitat de Catalunya Grant No. 2014-SGR-1450. C ¼ 100, κ 0 ¼ 0.01, V þ =μ4 ¼ 0.01 both VðϕÞ and V t ðϕÞ are plotted in Fig. 3. Rather than a thorough search for the simplest analytical APPENDIX A: PROOF THAT S = ΔSE cases, here we just give this example. No doubt other That the new tunneling action, given in (35), agrees with interesting potentials can be found using this method. the conventional one can be proven by showing that the two action densities differ by an exact differential that vanishes VI. CONCLUSIONS at the boundaries. Let us write the action functional (35) as The new formulation of [1] shows how to calculate Z ϕ 0 tunneling actions purely in field space, without reference to S¼ s dϕ; ðA1Þ an auxiliary Euclidean space in which the standard bounce ϕþ 104007-8 FRESH LOOK AT THE CALCULATION OF … PHYS. REV. D 100, 104007 (2019) with the (positive definite) action density Close to ϕþ , let us approximate the potential by keeping up to its second derivative 6π 2 ðD þ V 0t Þ2 s¼ : ðA2Þ 1 κ2 V 2t D VðϕÞ ¼ V þ þ m2 ðϕ − ϕþ Þ2 þ … ðA10Þ 2 Consider first the decay from Minkowski or AdS vacua and take as the conventional action Eq. (22). Transforming the Solving the equation of motion (31) for the above potential integral to a field-space integral using the procedure leads to the following expansion for the tunneling potential2 explained in the main text one gets 1 Z V t ðϕÞ ¼ V þ þ Bðϕ − ϕþ Þ2 þ Bα ðϕ − ϕþ Þ2þα þ … ϕ0 2 ΔSE ¼ sE dϕ; ðA3Þ ϕþ ðA11Þ with with   sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ðV − V t Þ V 0t V þ 3κV þ 4m2 sE ¼ −108π 2 V þ ; ðA4Þ B¼ 1þ 1− < 0; ðA12Þ D3 Dþ 2 3κV þ where and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dþ ≡ D2 − 6κðV − V t ÞV þ : ðA5Þ 2κV þ α¼ > 0: ðA13Þ B Then it is straightforward to check that, for V t satisfying its equation of motion, On the other hand, Bα is a free constant that cannot be determined by solving (31) around ϕþ but is fixed instead dF by the boundary condition at ϕ0 . Such behavior is expected, s − sE ¼ ; ðA6Þ as V t ðϕÞ must depend on the shape of the potential far from dϕ ϕþ . From this result it follows that with V 0t ; Dþ ∼ ðϕ − ϕþ Þ; D ∼ ðϕ − ϕþ Þ1þα=2 ; ðA14Þ  12π 2 V 3V 0 ðV 0 Þ3 FðϕÞ ¼ − 2 1− t þ t − t3 which shows how the leading terms of ðV 0t Þ2 and 6κðV − κ Vt V þ 2D 2D   V t ÞV t cancel out in D, which is then controlled by the VD D2 þ t þ 3 − þ2 : ðA7Þ subleading term. 2V þ D D The expansion of Fðϕþ Þ around ϕþ gives terms that are clearly 0 except for a term proportional to To find FðϕÞ it was useful to follow the homotopy operator method, as explained, e.g., in [13]. In the Minkowski case ðV 0t Þ3 (for V þ → 0), this function has the finite limit ðϕ − ϕþ Þ2 ∼ ðϕ − ϕþ Þ2−3α=2 : ðA15Þ D3    6π 2 V 0t 2 V 0t From Eqs. (A12) and (A13), it follows that α < 4=3 and F0 ðϕÞ ¼ − 2 1þ 2− : ðA8Þ so the quantity (A15) goes to 0 for ϕ → ϕþ ensuring that κ Vt D D Fðϕþ Þ ¼ 0. Integrating (A6) in ϕ one gets An alternative way of obtaining the same result is to transform FðϕÞ back to the bounce language and use the S − ΔSE ¼ Fðϕ0 Þ − Fðϕþ Þ: ðA9Þ asymptotic behavior of ρðξÞ and ϕðξÞ for ξ → ∞ (see, e.g., Appendix A of [14]). Noting that Dðϕ0 Þ ¼ Dþ ðϕ0 Þ ¼ −V 0t ðϕ0 Þ, it follows that In the Minkowski case, remember that −V 0t =D ¼ ρ_ Fðϕ0 Þ ¼ 0. To evaluate Fðϕþ Þ, notice that one has asymptotes to ρ_ ¼ 1 at ξ → ∞, so that limϕ→ϕþ V 0t = Dðϕþ Þ ¼ Dþ ðϕþ Þ ¼ 0 [assuming that V 0 ðϕþ Þ ¼ 0 at the D ¼ −1, while limϕ→ϕþ V t ðϕÞ ¼ 0. One has therefore that false vacuum]. To calculate the ratios of these quantities, FM ðϕþ Þ ¼ 0. In conclusion, for Minkowski and AdS consider first the AdS case. Both V 0t =D and Dþ =D diverge at ϕ → ϕþ and one needs to know in more detail how 2 This is a Frobenius type of expansion, as expected given the V 0t ; D; Dþ approach 0. fact that ϕþ is a regular singular point of (31). 104007-9 J. R. ESPINOSA PHYS. REV. D 100, 104007 (2019) decays the vanishing of F and F0 at the boundaries ϕ0 and which is readily integrated to get ϕþ proves that S ¼ ΔSE , as promised. Consider next the dS case, for which D2 ¼ ðV 0t Þ2 þ 6κðV − V t ÞV t ≃ CðV − V t Þ; ðB2Þ Z ϕ0 24π 2 where C is an integration constant that can be expressed in ΔSE ¼ sE dϕ þ ; ðA16Þ ϕ0þ κ2 V þ terms of the wall tension as follows. Rewrite the previous equation as with pffiffiffiffiffiffiffiffiffiffiffiffiffiffi −V 0t ðV − V t ÞV V − V t ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: ðB3Þ sE ¼ −108π 2 ; ðA17Þ C − 6κV t D3 From this, one gets which is obtained translating (16) to field space. One also has Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Z ϕ0 ϕ0 V 0t dϕ Z ϕ σ≡ 2ðV − V t Þdϕ ¼ − 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0 ϕþ ϕþ C − 6κV t S¼ s dϕ pffiffiffi ϕþ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z   ≃ ð C − 6κV − − C − 6κV þ Þ; ðB4Þ ϕ0 24π 2 1 1 3κ ¼ s dϕ þ − ; ðA18Þ ϕ0þ κ2 V þ V 0þ where Vðϕ0 Þ ≃ V − , valid in the thin-wall case, has been where s is as given in (A2) and V 0þ ≡ Vðϕ0þ Þ. used. From this, C is extracted as For this dS case one can check that 1 C ¼ 6κV þ þ ð4ΔV − 3κσ 2 Þ2 ; ðB5Þ dF0 8σ 2 s − sE ¼ ; ðA19Þ dϕ where ΔV ≡ V þ − V − . with F0 ðϕÞ given in (A8). Integrating this in the interval Using the results above, the tunneling action (A2) can be (ϕ0þ , ϕ0 ) one has rewritten as Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24π 2 6π 2 ϕ− V − V pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S − ΔSE ¼ F0 ðϕ0 Þ − F0 ðϕ0þ Þ − 2 : ðA20Þ Stw ¼ 2 dϕ 2 pffiffiffiffi t ð C − C − 6κV t Þ2 : κ V 0þ κ ϕþ Vt C ðB6Þ Now, at ϕ0 one still has D ¼ −V 0t , so that F0 ðϕ0 Þ ¼ 0, while at ϕ0þ one has V 0t > 0 so that D ¼ V 0t , and This integral can be performed if one gets rid of the only V F0 ðϕ0þ Þ ¼ 24π 2 =ðκ 2 V 0þ Þ. Plugging this in (A20) leads appearance in the integrand using (B3) to get to the claimed equality S ¼ ΔSE. 12π 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼V APPENDIX B: THIN-WALL CASE Stw ¼ 2 ð1 − 1 − 6κV t =CÞjV tt ¼V þ− : ðB7Þ κ Vt When the potential difference between false and true vacua is very small, one expects to be in the thin-wall limit This agrees with the known result in the literature; see, e.g., (with the bounce having a sharp transition between ϕ0 ≃ ϕ− [6,15,16]. In the case of small gravitational effects an and ϕþ at some ξ). In such cases an analytical expression expansion in κ gives for the tunneling action can be obtained, in terms of the wall   tension σ [4,6]. ðκ¼0Þ 3κhViσ 2 2 Stw ¼ Stw 1− þ Oðκ Þ ; ðB8Þ The derivation of the analytic thin-wall expression for ΔV 2 the tunneling action using the tunneling potential approach proceeds as follows. Consider first the Minkowski and where AdS cases. When the barrier separating the false and true minima in V is high compared to V t one has V 0t ≪ 27π 2 σ 4 ðκ¼0Þ ðV − V t Þ0 . Using this, one can approximate the equation Stw ¼ ; ðB9Þ 2ΔV 3 of motion for V t, written in the form (34) as and hVi ≡ ðV þ þ V − Þ=2. d ðV − V t Þ0 In the Minkowski limit (V þ → 0) one recovers from the log D ≃ ; ðB1Þ dϕ 2ðV − V t Þ general expression (B7) the Coleman result [4] 104007-10 FRESH LOOK AT THE CALCULATION OF … PHYS. REV. D 100, 104007 (2019) ðκ¼0Þ ΔStw Generalizing this discussion to include also AdS decays, ΔStw ¼ ; ðB10Þ ð1 − κ=κ c Þ2 one gets pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi with 4ð −V þ − −V − Þ2 κc ≡ ; ðB13Þ 3σ 2 4ΔV κc ≡ ; ðB11Þ with decays forbidden if κ=κ c > 1. 3σ 2 In the dS case the near degeneracy of false and true vacua displaying the gravitational quenching of the decay for does not necessarily imply the thin wall case. In terms of κ=κc → 1. For κ=κc > 1 the decay is forbidden. 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