Revisiting Coleman-de Luccia transitions in the AdS regime using holography
Jewel K. Ghosh, Elias Kiritsis, Francesco Nitti, Lukas T. Witkowski
PPreprint typeset in JHEP style - HYPER VERSION
CCTP-2021-01ITCP-IPP-2021/1
Revisiting Coleman-de Luccia transitions in the AdSregime using holography
Jewel K. Ghosh a,b , Elias Kiritsis c,d , Francesco Nitti c , Lukas T. Witkowski e a Independent University Bangladesh (IUB),Plot 16, Block B, Aftabuddin Ahmed Road, Bashundhara R/A, Dhaka, Bangladesh b International Centre for Theoretical Sciences, Tata Institute of Fundamental Research,Shivakote, Bengaluru 560089, India c APC, AstroParticule et Cosmologie, Universit´e de Paris, CNRS/IN2P3, CEA/IRFU,Observatoire de Paris,10, rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France d Crete Center for Theoretical Physics, Institute for Theoretical and ComputationalPhysics, Department of Physics, Voutes University Campus,University of Crete, 70013, Heraklion, Greece e Institut d’Astrophysique de Paris, GReCO, UMR 7095 du CNRS et de SorbonneUniversit´e, 98bis boulevard Arago, 75014 Paris, France
Abstract:
Coleman-de Luccia processes for AdS to AdS decays in Einstein-scalar theoriesare studied. Such tunnelling processes are interpreted as vev-driven holographic RG flowsof a quantum field theory on de Sitter space-time. These flows do not exist for genericscalar potentials, which is the holographic formulation of the fact that gravity can actto stabilise false AdS vacua. The existence of Coleman-de Luccia tunnelling solutions ina potential with a false AdS vacuum is found to be tied to the existence of exotic RGflows in the same potential. Such flows are solutions where the flow skips possible fixedpoints or reverses direction in the coupling. This connection is employed to constructexplicit potentials that admit Coleman-de Luccia instantons in AdS and to study theassociated tunnelling solutions. Thin-walled instantons are observed to correspond to dualfield theories with a parametrically large value of the dimension ∆ for the operator dualto the scalar field, casting doubt on the attainability of this regime in holography. Fromthe boundary perspective, maximally symmetric instantons describe the probability ofsymmetry breaking of the dual QFT in de Sitter. It is argued that, even when suchinstantons exist, they do not imply an instability of the same theory on flat space or on R × S . a r X i v : . [ h e p - t h ] F e b ontents
1. Introduction and summary of results 2
2. Coleman-de Luccia processes and holographic RG flows 14 O ( D )-instantons and flat domain-walls 142.2 Holographic RG flow solutions in Einstein-scalar theories 172.3 O ( D )-instantons as holographic RG flows 212.4 O ( D )-instanton action 232.5 No-go result for tunnelling to AdS minima or maxima 252.6 Tunnelling from AdS minima is non-generic 282.7 O ( D )-instantons and exotic holographic RG flows 33
3. Flat AdS domain-walls and a thin-wall limit 36 O ( D )-instantons 40
4. O(4)-instantons in a sextic potential 41 O ( D )-instantons 424.2 Numerical results: Tunnelling from AdS minima 444.3 Thin-walled O (4)-instantons 494.4 Numerical results: Tunnelling from AdS maxima 52
5. The Lorentzian continuation 54
6. From the sphere to the cylinder 68
Acknowledgements 78Appendices 79A. Expansions of A ( u ) and ϕ ( u ) near UV fixed points 79B. Sufficient condition for tunnelling solutions from AdS minima 81C. Flat domain-walls with a thin-wall limit: an alternative approach 83 – 1 – . Comparison with thin-wall approximation of Coleman-de Luccia 84
1. Introduction and summary of results
Systems with multiple ground states are common in many areas of physics. The transitionsfrom a higher energy ground state (false vacuum) to a lower energy one (true vacuum)can proceed via tunnelling, with the formation of a bubble which subsequently expands.As was understood by Coleman, this process is described by instanton solutions in theEuclidean theory [1]. The Euclidean instanton provides initial conditions for the subsequentLorentzian evolution. In the absence of gravity, such solutions always start in the falsevacuum in the far past and end in the true vacuum in the far future.The story becomes richer and more complex in the presence of gravity. In [2], thegeometry mediating vacuum decay in the context of General Relativity was first studied.The corresponding solution is generally referred to as the Coleman-de Luccia (CdL) in-stanton. When coupling a field theory to general relativity, the values of the ground stateenergies (and not only the energy difference) become important. In particular, it makes abig difference if the end point of the vacuum decay process has a negative vacuum energy,compared to the case of a positive or zero vacuum energy. In the latter case, the truevacuum space-time has the geometry of de Sitter or Minkowski, whereas in the former caseit has anti-de Sitter (AdS) geometry.As already noticed in [2], decays to de Sitter or Minkowski space are not qualitativelyvery different from the corresponding processes in the absence of gravity. This is notthe case for decays to AdS for at least two reasons: 1) Tunnelling is not always allowed,i.e. gravity can stabilize the false vacuum, and, 2) when tunnelling is allowed, the genericendpoint of the process (when continued to Lorentzian signature) is not the true vacuumAdS space-time, but it is rather an open Friedmann-Robertson-Walker (FRW) universewhich initially expands but is expected to undergo a big crunch singularity.The problem of vacuum decay from/to AdS space-time has received a renewed attentionin recent years, for two main reasons. On the one hand, the string theory landscape of vacuadisplays a (very) large number of AdS solutions with different values of the cosmologicalconstant. If one thinks of each of these vacua as a local extremum of a (loosely defined)potential energy landscape, the question whether these vacua can decay to lower andlower ones by bubble nucleation, and the corresponding decay rates, is very important forphenomenology.On the other hand, the AdS/CFT correspondence relates space-times with asymptoti-cally AdS geometry to lower-dimensional asymptotically conformal quantum field theories(QFTs). One can then try to use the duality to understand vacuum decay from the per-spective of the dual QFT. Furthermore, attempts were made to give a QFT description ofthe singularities in the Lorentzian “crunching AdS” space-time, with the hope of giving anon-perturbative resolution to cosmological singularities (see e.g. [3, 4, 5, 6]).– 2 –t the same time, the AdS/CFT interpretation poses a few puzzles to the problem ofvacuum decay. As emphasized in [7, 8], the gravitational path integral must be performedwith fixed boundary conditions at the AdS boundary. If these correspond to the falsevacuum AdS, then a complete vacuum decay can never occur, since it would have to bemediated by a solution which in the future approaches asymptotically to the true vacuumAdS. This indicates that CdL-like solutions in AdS cannot always be given an interpretationin terms of vacuum decay, but rather have to be interpreted as holographic RG flows definedon curved space-times, a point of view which is also put forward in [9].From the technical standpoint, computing the decay rate of the false vacuum is oftena difficult task. Given the Euclidean field equations for the metric g ab and the otherdynamical fields ϕ i , one first has to find a solution { ¯ g ab , ¯ ϕ i } which interpolates between thefalse vacuum and the true vacuum. The tunnelling rate is then given semiclassically byΓ = Ae − S E [ { ¯ g ab , ¯ ϕ i } ] (1.1)where the exponent is given by the Euclidean action evaluated on the solution, and A is apre-factor (which is a one-loop determinant).In order to perform this task, two hypotheses were made in [2], and are often used inthe literature since then. The first one is that the tunnelling is described by a solutionwith maximal symmetry. In the Euclidean case, for a D -dimensional space-time, this is an O ( D )-invariant instanton of the form ds = dξ + a ( ξ ) d Ω d , ϕ i = ϕ i ( ξ ) , ξ ∈ [0 , + ∞ ] , (1.2)where d = D − d Ω d is the metric on the d -dimensional unit sphere. The solutionmust be regular at the center ξ = 0, where a ( ξ → → ξ , and has to approach the falsevacuum metric as ξ → + ∞ . This solution describes a bubble of the (near-) true vacuum(interior region) inside a space-time which asymptotically approaches the false vacuum.The second assumption which is widely used is the so-called thin-wall approximation:this means that the region where the solution departs significantly from either vacuum isa thin shell around some value ξ , whose thickness is small compared to its radial size, a ( ξ ). Under this assumption, it is often possible to obtain an analytic expression for thedecay rate, [2]. Moreover, in this approximation, it is common practice to replace theoriginal problem (with dynamical variables including various fields beyond gravity) by amuch simpler problem in which one freezes the fields to their vacuum values on either sideof a codimension-one hypersurface (the bubble wall). The problem is now that of finding apure gravity solution on each side of the wall, gluing them using Israel’s junction condition,then extremizing the action with respect to the remaining variable, i.e. the position of thewall. In certain examples, in top-down AdS/CFT models related to N = 4 super Yang-Mills theory, the thin wall was identified with a D -brane separating AdS solutions withdifferent units of RR flux [10, 11]. Or rather, a field value close to the true vacuum. This semiclassical expression is valid approximately when S E [ { ¯ g ab , ¯ ϕ i } ] (cid:29) – 3 –hile the thin-wall approximation is very useful in practice, it is not always evidentunder which conditions it can be applied. In the absence of gravity, the thin-wall approx-imation holds when the potential energy difference between the true and false vacua issmall compared to the height of the barrier separating them. However in the presence ofgravity, and particularly when tunnelling to AdS, things are not so clear, and this is oneof the issues that we shall analyze in this work.In this paper, we define the gravitational theory using the holographic correspondence.We shall therefore consider transitions in the AdS regime. We perform a detailed studyof CdL solutions without relying on the thin-wall limit, in an AdS/CFT setup in D = d + 1 dimensions, which consists of Einstein gravity minimally coupled to a single scalarfield ϕ . The scalar field potential V ( ϕ ) has several extrema where the potential itselftakes negative values, each corresponding to an AdS solution. This gravitational theory,we assume to be holographically dual to a QFT. In this theory we focus on Euclidean O ( D )-invariant solutions of the form (1.2), which describe tunnelling towards the lowestenergy AdS minimum, starting either from another (higher) AdS minimum or from an AdSmaximum.Starting from an AdS false vacuum has the advantage that one can in principle give aholographic interpretation to both the starting point and the end point of the tunnellingprocess. In particular, this makes the boundary problem at ξ = ∞ in the metric (1.2) welldefined.In the dual, d -dimensional field theory, each extremum corresponds to a different con-formal field theory. The usual holographic interpretation of a gravity solution interpolatingbetween two AdS extrema along the radial direction is that of a field theory RG flow be-tween conformal fixed points. Near each fixed point, the geometry is well approximatedby the AdS boundary region (UV fixed point) or the AdS interior geometry (IR fixedpoint). The scalar field corresponds to an operator responsible for breaking conformal in-variance, either because its (relevant) coupling is non-zero, or because it acquires a vacuumexpectation value.The above interpretation usually refers to RG flows of field theories defined in flat(Euclidean or Lorentzian) space-time. Here however the situation is different. Since theconstant- ξ slices are spheres all the way to the boundary of AdS space-time at ξ = + ∞ , ac-cording to the holographic dictionary the dual UV field theory is defined on a d -dimensionalsphere ( S d ) in the Euclidean case, and on d -dimensional de Sitter space (dS d ) in theLorentzian case. The solutions (1.2) can therefore be interpreted as holographic RG flowsof QFTs defined on a d -dimensional space-time of constant positive curvature, with theUV fixed point corresponding to the false vacuum theory. In this case, as we shall reviewbelow, the flow cannot reach the true vacuum (i.e. the lower AdS minimum) but at theendpoint ξ →
0, it stops at a field value ϕ where V (cid:48) ( ϕ ) (cid:54) = 0, [13]. In the subsequentevolution in real time, the scalar field continues to roll towards the true minimum but itis not guaranteed (and in fact, as we shall see, generically it does not occur) that it settlesin the true minimum. This is for simplicity. Although the multiscalar case can be involved, its structure of solutions has beenstudied in detail, [12], and our qualitative conclusions are expected to hold more generally. – 4 –n order to describe vacuum tunnelling, the false vacuum theory must correspond toa conformal fixed point, without any relevant couplings turned on. This is automaticallythe case when we consider tunnelling from an AdS minimum: in such cases the scalarcorresponds to an irrelevant operator in the dual field theory, whose QFT coupling mustnecessarily be set to zero in the UV. If one starts from an AdS maximum instead, thescalar field is dual to a relevant operator and in general one has the choice of turning onthe corresponding coupling. If the coupling is non-zero, we are in the presence of a source-driven RG flow, in which conformal invariance is explicitly broken in the UV. If instead thecoupling is set to zero, one can still have a non-trivial flow (which we call a vev-driven flow),where the breaking of conformal invariance is spontaneous, and the boundary conditionsare the same as in the fixed-point theory. It is only the latter case which corresponds tovacuum tunnelling, since these solutions are in the same class (satisfy the same boundaryconditions ) as the pure AdS solution which sits at the fixed point.Based on the discussion above, studying AdS to AdS vacuum decay via O ( D ) CdL in-stantons is equivalent to constructing vev-driven holographic RG flows on constant positivecurvature space-times. Such solutions were recently studied in detail in [13].Vacuum decay of/to AdS, in connection to the AdS/CFT correspondence, has beenwidely considered in the literature [2, 14, 3, 4, 10, 11, 15, 9, 5]. In this paper, we make useof the results of [13], to draw general results about AdS to AdS vacuum decay in Einstein-scalar theories, without relying on the thin-wall approximation. In this context, we shalladdress the question of the existence (or not) of CdL bubbles in theories with generic scalarpotentials, the existence of a thin-wall limit, and the fate of the real-time solution insidethe bubble. We shall analyse these questions in general, and with the help of numericalanalyses in a few explicit (but generic) examples.In the rest of this section, we briefly summarize our results. When is AdS vacuum decay possible?
A question we address is under what circum-stances O ( D )-instanton solutions mediating AdS decay exist in a potential.Although it is hard to single out exactly which local features of a potential allow forinstanton solutions (i.e. height and curvature of the extrema), we are able to connect theexistence of CdL instantons to another feature of the scalar potential: the existence of exotic holographic RG flows, [16]. These are RG flows which are characterized by a non-monotonic behavior of the coupling, also referred to as a ‘bounce’ , and/or by ‘skipping’of intermediate fixed points. Such a behavior is possible in holographic theories, but it isnot allowed in perturbative QFTs. These solutions where found and analyzed in detailedin [16] in the case of a flat-sliced bulk theory (corresponding to the dual QFT defined inMinkowski space), and in [13] and [17, 18] in the case of theories on constant curvaturespaces and at finite temperature, respectively. Exotic flows exist for special classes of It should be clear that when we talk about AdS boundary conditions, we mean the value of the sourceterm in the asymptotic expansion near the AdS boundary. All the results listed below apply to the case we analyse here, i.e. tunnelling via maximally symmetricinstantons. It is possible that considering instantons with a lower degree of symmetry will lead to differentconclusions. We leave this as an open question. This is not related to Coleman’s “bounce”. – 5 – ϕ uv ϕ f ϕ uv ϕ ϕ t V ( ϕ ) (a) ϕ f ϕ ϕ t V ( ϕ ) (b) Figure 1: (a)
Potential V ( ϕ ) with two minima at ϕ f (false vacuum) and ϕ t (true vacuum)and two maxima at ϕ uv and ϕ uv . The potential admits an O ( D )-instanton describingtunnelling from ϕ f to ϕ (black arrow). This potential will then also admit holographicRG flow solutions from the UV fixed point at ϕ uv to an end point in the red regionthat skips past the other maximum at ϕ uv . This is a so-called skipping flow of [16, 13](red arrow). In addition, the potential will allow for flows leaving ϕ uv to the left beforechanging direction and flowing to an end point in the green region. This is a so-calledbouncing flow of [16, 13] (green arrow). The holographic RG flows are for a QFT definedon S d . (b) Potential V ( ϕ ) with a maximum at ϕ f and a minimum at ϕ t permitting an O ( D )-instanton describing tunnelling from ϕ f to ϕ t (black arrow). This potential willalso exhibit bouncing holographic RG flows (green arrow) from ϕ f to the green region, inaddition to standard source-driven holographic RG flows (orange arrow). Again, the RGflows are for theories defined on S d .potentials. Their existence is a global feature of the potential and it is likely not possibleto relate it to local features of the extrema. Here, we show that O ( D )-instantons exist if and only if the theory admits such exoticflows. More precisely: • When the false vacuum is a minimum of the potential, a CdL instanton exists if andonly if the potential also permits exotic RG flows for theories on S d of both theskipping and bouncing type, see fig. 1a. Flat-sliced RG flows also play an importantrole: If the potential admits a flat-sliced RG flow of skipping type, this is a sufficientcondition for the potential to also permit an O ( D )-instanton. Our observation is that potentials with an AdS false vacuum in general do not admit exotic RG flowsand hence tunnelling from AdS is non-generic. In contrast, in the limit where gravity decouples from thescalar dynamics (while keeping the scalar potential fixed) O ( D )-symmetric tunnelling solutions always existas follows from the overshoot/undershoot argument in [1]. This implies that as the gravitational couplingconstant is ‘decreased’, the space of potentials that admit O ( D )-instantons is expected to increase. Wethank Daniel Harlow for pointing this out to us. However, note that in this limit the AdS radius becomesparametrically large and a holographic interpretation of tunnelling along the lines of AdS/CFT becomesquestionable. – 6 – When the false vacuum is a maximum of the potential, a CdL instanton exists ifand only if the theory admits exotic RG flows for the theory on S d which display abounce, see fig. 1b. We also find that the condition for the minima to be nearly degenerate is not sufficientfor an instanton solution to exist. The requirement of near-degenerate minima was intro-duced in [2] as a condition for the applicability of the thin-wall approximation. However,the near-degeneracy of minima does not guarantee that such a solution exists, which weobserve here for AdS decays. This confirms and generalizes previous observations using thethin-wall approximation. Such observations presuppose that minima are nearly degenerateand then claim that gravity tends to stabilize a metastable vacuum with negative curvature[2, 11, 15].
Tunnelling rate.
When a solution of the form (1.2) exists, we compute the correspond-ing semiclassical tunnelling rate by evaluating the bulk action on the solution, see (1.1). Weshow on general grounds that, for solutions corresponding to normalizable boundary asymp-totics (i.e. zero source in the dual field theory), the tunnelling rate is finite. In contrast,the decay rate vanishes for non-normalizable scalar field asymptotics due to uncancellednear-boundary divergences. This confirms the expectation that source-driven solutions donot describe vacuum decay, but only have an interpretation in terms of holographic RGflows. This expectation is completely natural from the dual field theory point of view:to consider spontaneous decay of the fixed point CFT, all conformal symmetry-breakingcouplings must to be set to zero.
How to construct bulk potentials which support CdL instantons.
Having foundthat AdS vacuum decay is non-generic, we describe a method which allows to designbottom-up bulk potentials which permit instanton solutions. We start with a fine-tunedpotential, which allows for a flat domain wall-solution interpolating between two extrema,in which the metric is sliced by flat radial hypersurfaces, ds = dξ + a ( ξ ) η µν dx µ dx ν , ξ ∈ ( −∞ , + ∞ ) . (1.3)The bulk scalar obeys special asymptotics approaching the false vacuum in the UV: ϕ ( ξ ) (cid:39) ϕ f + ϕ + e ∆ ξ , ξ → −∞ , (1.4)where ϕ f is the scalar field value corresponding to the false vacuum, and ∆ > The potential depicted in fig. 1a also permits a CdL instanton describing tunnelling from ϕ uv in virtueof the existence of the bouncing flows. We refrained from depicting this, for simplicity. Here near-degenerate means that the energy separation between the two vacua is small compared tothe height of the potential barrier between them. – 7 –ows. This makes it generically impossible to impose the regularity conditions wherethe scale factor vanishes, i.e. at the locus a ( ξ ) →
0. However, it is possible to fine-tunethe bulk potential in such a way that solutions like (1.3-1.4), that are also regular in theinterior, exist.Starting from such a tuned potential, we show that, by deforming it in an appropriateway (therefore relaxing the fine-tuning), the resulting potential admits spherical O ( D )-instantons. In this way, one can obtain continuous families of potentials in which one ofthe AdS extrema is unstable to O ( D )-invariant bubble nucleation, provided we have thecurved boundary conditions for the asymptotic metric. The thin-wall approximation.
In this work we also assess the validity of the the thin-wall approximation.In the limit of an infinitely thin wall with non-zero tension, the approximation consistsof taking the scalar field to be constant and equal to the value extremizing the potential oneach side of the wall, and the metric to be exactly AdS, with the curvature scale determinedby the value of the potential on each side. The two AdS space-times are then connectedusing Israel’s junction conditions at the wall location, see e.g. [11, 15, 19].One of the main points of our analysis is that, independently of the specific form ofthe bulk potential, there is no limit of the Einstein-scalar setup in which the above thin-wall setup is strictly valid, i.e. where the CdL bubble reduces to an infinitely thin-wallwith finite tension separating two AdS vacua, such that 1) the two vacua have both finite(negative) potential energy and 2) they have a finite separation in field space. The reasonis that in Einstein’s equations, it is impossible to match the Dirac distributions that arisein this limit. As a consequence, although in specific examples the wall can be parametrically thin andthe transition between two AdS space-times may be quite fast, the scalar field dynamicsmay never be completely neglected and can be important for obtaining correct results.This is a serious caveat that has to be kept in mind when one considers models whichuse this simplified picture, which are very common in the literature in various contexts[20, 11, 15, 19, 21].This analysis does not apply to cases when the bubble has a different origin than a bulkscalar field domain wall, e.g. when the object mediating vacuum decay is genuinely lower-dimensional. Examples include nucleation of probe D3-branes in type IIB string theory [11],which do not have a simple interpretation in terms of bulk scalars interpolating betweentwo minima of a potential. More generally, this applies to cases where vacuum decay ismediated by excitations in the open string sector.It is nevertheless interesting to analyze in more detail, in which cases the bubble wallis parametrically thin (although infinite thinness cannot be achieved in any consistent For source-driven flows the scalar field approaches the false vacuum as exp( d − ∆) ξ , with ∆ < d . Theyexist only if the false vacuum is a maximum of the potential. This is unlike what happens in the absence of gravity, where the thin-wall approximation can beunderstood as a consistent limiting procedure. In this case only the jump equation in the scalar field mustbe satisfied. – 8 –imit). To this end, we consider a class of analytically-engineered potentials constructed asexplained in the previous paragraph, and which depend on a few controllable parameters:the values of the potential at the minima and their curvatures there, which in turn controlthe dimension ∆ of the operator dual to the bulk scalar in the two CFTs living at theextrema. In this parametrization, we study examples based on a bulk potential given by asextic polynomial. Numerical analyses of the solutions lead to the following observations: • Most importantly, for a solution to describe a thin-walled bubble, the dimension ∆of the dual CFT operator both in the false and true minimum CFTs must becomeparametrically large. This is needed for the interpolation between false and (near-)true minimum to be sufficiently ‘rapid’ as characteristic for a thin wall. Hence weexpect this finding to hold beyond the family of potentials studied. • If a thin-walled CdL instanton solution exists, we find that the corresponding poten-tial has a large barrier between the two minima compared to the energy differencebetween the minima. This is in agreement with the heuristic observations of Colemanand de Luccia in their original work [2]. • Computing the solution and the on-shell action numerically, we observe that thedecay rate is faster for thick walls than for thin walls, as suggested in [2]. • CdL instantons describing tunnelling from an AdS maximum can also be found. How-ever, these are observed to always be thick-walled solutions. This is consistent withour previous finding as the dimension ∆ of the dual CFT operator describing defor-mations from a maximum is bounded as ∆ < d and cannot become parametricallylarge.These results show that Coleman’s thin-wall approximation (for which there is always asolution based on Israel’s junction condition, if the domain wall tension is taken to besufficiently small) may sometimes underestimate the decay rate. Also, the fact that thethin-walled solutions require the dimension ∆ of the corresponding CFT operator in thefalse and true minimum to be very large is quite problematic: it is believed that there areupper bounds on the lowest dimension irrelevant operator in a CFT (as suggested fromresults from the conformal bootstrap program, see e.g. [22]). If, on the other hand, the vac-uum CFT contains an operator of lower dimension, then this corresponds to an additionalbulk scalar of lower mass, whose dynamics must be included in the analysis, invalidatingthe results obtained in the single-scalar model we are using. We can, however, give aqualitative argument that extrapolates our results to the multiscalar case. At a genericsaddle point, one has both positive mass scalar directions (dual to irrelevant operators) andnegative mass ones (dual to relevant directions). For an instanton to exist, regular curvedvev-flow solutions must exist. Generically, several instanton solutions may be also possi-ble. The larger the slope in the direction of the associated flow, the thinner the domainwall, and the larger the instanton action will be. However, we expect that in the presenceof several low-dimension operators, the associated vevs and flow will be mostly in theirdirection and the domain wall will be thick. A detailed analysis is, however, necessary inorder to establish whether this is a generic phenomenon, or if it always happens.– 9 – ubble interior and crunch singularity.
We analyze the fate of the solution in realtime, after the bubble has nucleated. As mentioned earlier, it has been observed (startingwith [2] and later with [23]) that, in the thin-wall limit, the space-time inside the bubbleis a cosmological space-time which grows at first, but then crunches into a singularity.This is based on the fact that, in the thin-wall limit, the space-time in the interior is AdS,which has a big-crunch coordinate singularity in the bubble interior. This is expected tobecome a physical singularity when deviations from AdS due to the scalar field dynamicsare included, [2].Here, we show that the big crunch is generically unavoidable. This can be tracedto the fact that, inside the bubble, the scalar field does not reach the true vacuum, butin real time it starts forced oscillations which eventually destabilize the system towards asingularity. The latter however is hidden behind a horizon (the cone swept by the real-timeevolution of the center of the bubble, ξ = 0) which cloaks it from the AdS boundary. Boththe horizon and the singularity reach the boundary in the infinite future as measured bythe boundary QFT living on de Sitter space-time. Therefore, from the boundary QFTpoint of view, the solution looks like an RG flow on d -dimensional de Sitter space-time atall times, and the interior of the bubble is akin to the interior of an eternally expandingblack hole, which however cannot send signals to the boundary.Although this qualitative picture is well established, here we show that this is thecase generically and independently of the details of the potential and of the thin-wallapproximation. The interpretation of the solution from the dual CFT point of view.
In theLorentzian patch of the geometry, the near-boundary solution is that of the (false vacuum)CFT d defined on dS d space. This is clear from the structure of the sources. The structureof the Lorentzian solutions outside the bubble does not correspond to the ground state ofthe CFT d on dS d , in which the bulk scalar is constant. When the scalar is constant, thestate is dual to the AdS d +1 solution in the bulk, and has the full O (2 , d ) symmetry ofAdS d +1 . The CdL solution corresponds instead to a state with a scalar vev, dual to a non-trivial (non-AdS) geometry in the bulk. The running scalar is sourced by a non-zero vevof an irrelevant operator and breaks (spontaneously) the full symmetry O (2 , d ) to O (1 , d ),the isometry group of dS d .The instanton instability in this context is interpreted as follows. There is a specialstate in the CFT d on dS d that is essentially a Hartle-Hawking state. It is defined byinitial conditions on S d − given by doing the CFT path-integral on a half S d with S d − Although it may be in principle possible to construct a fine-tuned potential with regular interior ge-ometries. This is left as an open question. Based on this picture, it was suggested in [9] that one consider the bubble wall as an IR (broken) CFTwhich encodes the interior degrees of freedom. Despite the conformal anomaly, a CFT on dS d has the same amount of symmetry than a CFT in flat d -dimensional Minkowski space, since dS d has the same number of conformal vector fields as flat space(see e.g. [24]). This is clear from the bulk perspective, as the flat and dS slicings of AdS d +1 result by adifferent choice of coordinates of the same d + 1-dimensional manifold in the embedding space R ,d , whichhas symmetry group O (2 , d ). – 10 –s a boundary. Such a state, for weakly-coupled theories corresponds to the Bunch-Davisvacuum.On the gravity side, this state is dual to the analogous bulk Hartle-Hawking state,[25, 26], whose wave-function is defined by the path-integral over Euclidean geometrieswith EAdS boundary conditions [27].Moreover, there are two semiclassical states for the CFT d on dS d . The first is themaximal symmetry state in which all scalar vevs vanish. The other is the reduced symmetrystate in which the vev of the operator dual to the bulk scalar is non-trivial and the symmetryis broken as O (2 , d ) → O (1 , d ). The latter has a higher Euclidean free energy than theuniform state.Since the Euclidean path integral is dominated by the two semiclassical solutions above,the Hartle-Hawking state has a non-zero overlap with both semiclassical states. It is mostlythe highest symmetry, zero vev state, and it has a small admixture of the non-zero vev, lowersymmetry state. The amplitude for this admixture is given by the instanton amplitude.Therefore, for such a “false vacuum” CFT d , there is a small probability for the scalarobtaining a vev and for the breaking of the O (2 , d ) symmetry. Vacuum decay of different boundary QFT space-time geometries.
As we havestressed earlier, maximally symmetric instantons of the form (1.2) can mediate AdS vacuumdecay only when the asymptotic boundary has the geometry of S d . The dual boundaryQFT is therefore defined on a constant positive curvature space-time and from the boundaryperspective, the decay takes an infinite time, as we discussed in the previous paragraph.The natural question is whether the existence of the maximally symmetric instantonallows to draw conclusions about the same QFT on flat space, or on the cylinder R t × S d − ,which in the gravity dual would correspond to an asymptotic boundary of Poincar´e-AdS d +1 and Global AdS d +1 , respectively.It is important to stress here that, in AdS/CFT (or more generally when dealing withsemiclassical gravity in asymptotically AdS space-times) the boundary conditions on themetric must be included in the definition of the (gravitational) path integral: only solutionssatisfying the same set of asymptotic boundary conditions contribute to the same quantumpath integral. Changing the boundary conditions amounts to turning on fluctuations whichare non-normalizable near the AdS boundary, and correspond to a change in the externalsources (e.g. the background metric) of the dual field theory. If we were to investigatevacuum decay in the flat space QFT, the O ( D ) instanton would not be the appropriatesolution, because it does not obey the appropriate boundary conditions.One can nevertheless hope to say something about the theory on other boundarygeometries, because sometimes one can change the boundary conditions by a (large) dif-feomorphism, and this changes the dual theory by changing the sources, while still havinga solution to the bulk Einstein equation. This procedure therefore provides a semiclassicalsaddle point to a different path integral.Now, it is possible to find such a large coordinate transformation that maps the CdLinstanton (with near boundary leading behavior of the metric corresponding to dS d ) to ageometry whose metric has leading behavior corresponding to R t × S d − . This can be seen– 11 –s a map between the QFT on de Sitter and the QFT on the cylinder. Through thismap, the infinite de Sitter future is mapped to finite global-AdS time. In particular, whenmapped to the cylinder, the crunch singularity of the CdL solution reaches the boundary infinite global time, at which point it is not possible to extend the solution further. One canthink of this as being caused by an instantaneous source at global boundary time t = π/ R t × S d − . Geometriescorresponding to states in these theory are such that the scalar field asymptotes to thefalse-vacuum value (with zero source term) at all times −∞ < t < + ∞ .In contrast, in the solution resulting from mapping the CdL instanton to cylinderslicing, the scalar field asymptotes to the false-vacuum value on the boundary (with zerosource term) only for a finite time interval but then its value jumps by a finite amount in-stantaneously at global time t = π/
2. One can imagine modifying the boundary conditionson the gravity side in order to accommodate the cylinder-CdL geometry as an allowed so-lution. Provided this can be done consistently, this would define a different field theory onthe cylinder, which does not admit the global AdS solution as one its states. This theorycan only be extended up to a finite global time, at which point the time-evolution reachesa singularity. This point of view is the one taken in [11], where the CdL geometry wasinterpreted in terms of an unstable dual field theory with an unbounded Hamiltonian.In conclusion, the existence of the maximally symmetric instanton solution does notimply an instability of the boundary CFT on R t × S d − . This does not necessarily implythat the theory on the whole cylinder is absolutely stable. Decay may occur via othersolutions with less symmetry. Our results show that tunnelling by bubble nucleation between AdS vacua, mediated bymaximally symmetric instantons is non-generic. When it does occur, the thin-walled bubblelimit seems to be very problematic: it requires a very large leading irrelevant operatordimension in the endpoint CFT, and additionally, a very small separation in field space inour examples. Therefore, decay by thin-walled bubbles is the exception rather than thenorm, when dealing with AdS to AdS vacuum tunnelling, which as we just stated, is rathernon-generic in and of itself.Furthermore, although one can define this approximation parametrically, the thin-wallapproximation is never a mathematically consistent limiting procedure in a model with We carry out the detailed analysis for the boundary geometry R × S d − but similar considerations alsohold for a flat Minkowski boundary. – 12 –ravity and scalars. This leads one to question the very widespread use of the exact thin-wall limit to treat in a simplified way systems with gravity and scalars [20, 11, 15, 19, 21].This leaves the exact thin-wall limit to the nucleation of lower codimension objects suchas D-branes in string theory.From the dual QFT perspective, the maximally symmetric instantons we have consid-ered here necessarily require the QFT to be defined on de Sitter space (or on a Euclideansphere). Therefore, vacuum decay in other geometries cannot be mediated by these pro-cesses. This is the case for example of flat space (the boundary of the AdS Poincar´e patch)or the Einstein Static Universe (the boundary of global AdS). Based on the results for O ( D )-symmetric spherical bubbles, one can infer nothing about vacuum stability or decayrate in these other space-times. The existence of lower-symmetry instantons which mediatevacuum decay on these geometry remains an open question. What we can say is that, ifthey exist, these are not coordinate transformations of spherical instantons.As is well known, the interior of the bubble, in the real time evolution after its nucle-ation, is an (open) FRW universe. Here we have shown that, generically and unavoidably,it ends in a big-crunch singularity. One point to be stressed is that the singularity (and infact the whole FRW geometry) are hidden by a bulk horizon, which reaches the boundaryonly in the infinite future. Therefore it is very difficult to investigate the singularity fromthe QFT perspective. This is similar to the fact that it is hard to obtain informationon the interior singularity of a black hole, although this topic is evolving. In particular,certain quantum effects may be able to generate such transmissions, [28, 29, 30], and it isan interesting problem to study this in the present context.It is appropriate here to comment on more general gravitational solutions describingRG flows dual to holographic QFTs defined on a sphere. Such solutions generalize thesolutions studied in this paper, and have been investigated, mostly in the Euclidean regimein [13, 31]. There, their Lorentzian continuation has been studied and it was shown thatthe associated geometry involves dS slices that end in a bulk horizon. This horizon issimilar to the one described here, with the only difference that it extends to the infinitepast. There is, however, the region behind the horizon, similar to the inside of the bubblein figure 12a. We portray the Penrose diagram in this case in figure 12b. To find the fieldsof the orange region of 12b, we must again solve the bulk equations with V → − V andinitial conditions similar to the ones used here. Like it happened here, we expect for thesame reasons that there will be singularities in the two orange quadrants the past and thefuture one. It is an interesting problem to study the maximal extension of such geometries.This work leaves some important questions open for future investigation. As thepresent analysis is limited to maximally symmetric solutions, one is naturally led to askwhether, in the absence of such solutions, vacuum decay may be mediated by lower-symmetry instantons. This is particularly relevant for the stability of global AdS (dual tofield theories on R × S d − ), where one would look naturally for instantons with O ( D − R × S d − .Finally, in this work we have found that, in the explicit example we considered, thebubbles are generically not thin-walled bubbles, unless one looks in very unnatural (fromthe CFT point of view) regions of parameter space. While our example is not in any wayspecial, it would be interesting to investigate whether these facts apply to generic theories.This work is organized as follows. In Section 2 we present our setup, the generalfeatures of O ( D ) instanton solutions, their connection with holographic RG flows of fieldtheories on spheres, and the computation of the decay rate. We explain the generic (non)-existence of O ( D ) instantons and connect them to exotic holographic RG flows. In Section 3we show some results about flat domain walls, how they can be constructed in holographictheories, and what the thin-wall limit entails in this case. In Section 4 we return tospherical instantons and we present a systematic procedure for obtaining these solutionsfrom theories which admit flat domain walls. We then turn to the numerical analysis ofa class of explicit examples, where in particular we discuss the validity of the thin-wallapproximation. In Section 5 we turn to the Lorentzian solution after bubble nucleation,and we show that this generically result in a crunching FRW space-time in the interior. InSection 6 we describe how to relate the spherical instanton to a solution with cylindricalslicing, and what this implies for the dual field theory. Several technical details are left tothe Appendix.
2. Coleman-de Luccia processes and holographic RG flows O ( D ) -instantons and flat domain-walls Tunnelling processes of Coleman-De Lucia type involve spherical instantons of the gravityplus scalar field equations. Such Euclidean solutions are relevant in a theory of Einsteingravity coupled to a scalar field theory with at least two vacua. In this work such vacuawill be assumed to have both negative cosmological constant. The associated Lorentziansignature action in D = d + 1 space-time dimensions is: S = M d − P (cid:90) d d +1 x (cid:112) | g | (cid:18) R ( g ) − g µν ∂ µ ϕ∂ ν ϕ − V ( ϕ ) (cid:19) + S ghy , (2.1)where we use the ‘mostly plus’ convention and R ( g ) is the scalar curvature associated withthe metric g µν . As the solutions considered will exhibit boundaries, we also included theGibbon-Hawking-York term S ghy to have a well-defined variational problem.The corresponding Euclidean action is given by S E = − S . The two AdS vacua inquestion correspond to two minima of V ( ϕ ) which we choose to be located at ϕ = ϕ f (the‘false’ vacuum) and ϕ = ϕ t (the ‘true’ vacuum), respectively, with V ( ϕ f ) ≥ V ( ϕ t ) , and V ( ϕ f ) , V ( ϕ t ) < . (2.2) In a few cases that will be considered separately, ϕ f corresponds to an AdS maximum. – 14 – ( ϕ ) V ( ϕ ) ϕ f ϕ t ϕ ϕ r in ¯ r r out r in ¯ r r out rr ∞ ∞ Figure 2:
LHS:
Potentials with minima at ϕ f (false vacuum) and ϕ t (true vacuum) per-mitting Coleman-de Luccia tunnelling solutions from ϕ f to a generic point ϕ . RHS:
Cartoon of the corresponding O ( D )-symmetric configuration in space-time. The coloringindicates the map of values of ϕ in space-time. Here r in designates the inner limit of thewall, r out the outer limit of the wall and ¯ r is the center of the wall where the field ϕ hasinterpolated half-way between ϕ f and ϕ . Top:
Example with a thick wall, i.e. a gradualtransition from ϕ f to ϕ along the radial direction r . Bottom:
Example with a thin wall,i.e. a sudden transition from ϕ f to ϕ along the radial direction r .See the LHS of figure 2 for two example potentials. The solutions considered in this workshall be of two kinds: • Coleman and de Luccia conjectured and have given sufficient arguments that thedominant instanton solutions must have O ( D ) symmetry [2]. Our primary focustherefore will be on O ( D )-symmetric solutions, which can be identified with CdLinstantons describing tunnelling out of a false vacuum. These solutions also havean interpretation as holographic RG flows for a Euclidean field theory defined on a d -dimensional sphere. • In addition, we also consider flat AdS domain-wall solutions that interpolate betweenthe two AdS minima. In holography these correspond to RG flow solutions dual toa QFT defined on d -dimensional flat space. While these solutions do not describedecay processes, we still consider them here, as they turn out to be useful for un-derstanding certain properties of the O ( D )-symmetric solutions. The reason is thatthe flat domain walls can be understood as a particular limit of the O ( D )-symmetriccase, that at the same time allow for constructing explicit analytic expressions moreeasily.In both cases, the metric and scalar field profile can be written as ϕ = ϕ ( ξ ) , ds = dξ + ρ ( ξ ) ζ µν dx µ dx ν , (2.3)– 15 –ith µ, ν running over the remaining d coordinates. For O ( D )-symmetric solutions ζ µν isthe metric of a d -sphere with radius α and corresponding scalar curvature R ( ζ ) = d ( d − α . (2.4)The coordinate ξ describes a radial direction with range ξ ∈ [0 , ∞ ). For flat domain wallsolutions ζ µν is a metric of the Euclidean space R d and ξ ∈ ( −∞ , ∞ ) is unconstrained.In this work, we primarily use a different set of space-time coordinates, that is morefamiliar from the study of holographic RG flows. These coordinates will be convenient inorder to use insights from RG flows in the tunnelling context. In particular, we introduce u ≡ u − ξ , e A ( u ) ≡ ρ ( ξ ) , (2.5)for some finite value of u . That is, the radial coordinate is now u and for O ( D )-symmetricsolutions covers the range u ∈ ( −∞ , u ]. Using these coordinates the ansatz in (2.3)becomes ϕ = ϕ ( u ) , ds = du + e A ( u ) ζ µν dx µ dx ν . (2.6)We can obtain the (gravitational) equations of motion, by varying the action (2.1) withrespect to the metric and scalar field, finding2( d −
1) ¨ A + ˙ ϕ + 2 d e − A R ( ζ ) = 0 , (2.7) d ( d −
1) ˙ A −
12 ˙ ϕ + V − e − A R ( ζ ) = 0 , (2.8)¨ ϕ + d ˙ A ˙ ϕ − V (cid:48) = 0 , (2.9)with ˙ x ≡ dx/du , X (cid:48) ≡ ∂X/∂ϕ . (2.10)For the case of flat domain walls, we simply set R ( ζ ) = 0 in (2.7),(2.8). Note that (2.9)is identical to the (one-dimensional) equation of motion of a particle of unit mass andcoordinate ϕ , performing damped motion in the inverted potential − V , with u playing therole of time. We frequently employ this mechanical analogue for intuition in the behaviourof the system at hand.In this work we shall be interested in O ( D )-instantons corresponding to tunnellingfrom the false vacuum at ϕ = ϕ f (as defined above) to some point with scalar field value ϕ = ϕ . These will correspond to solutions to (2.7)–(2.9) with the following asymptoticbehavior: O ( D )-instantons: ϕ → ϕ f , ˙ ϕ → , ρ = e A → ∞ , for u → −∞ , ξ → + ∞ , (2.11) ϕ → ϕ , ˙ ϕ → , ρ = e A → , for u → u , ξ → . The flat domain-wall solutions studied here will interpolate between the extrema at ϕ f and ϕ t . Therefore, the relevant asymptotic behavior is:Flat domain-walls: ϕ → ϕ f , ˙ ϕ → , ρ = e A → ∞ , for u → −∞ , ξ → + ∞ , (2.12) ϕ → ϕ t , ˙ ϕ → , ρ = e A → , for u → + ∞ , ξ → −∞ . – 16 –n the following, we also examine under which circumstances O ( D )-instantons are ofthe ‘thin-wall’ type. By construction, O ( D )-solutions are configurations corresponding toconcentric circles of physical radius r = αρ = αe A , with ϕ constant on each sphericalshell, but interpolating between its boundary values (here ϕ f and ϕ ) along the radialdirection. The wall is identified as the interval in the radial direction where the bulk ofthis interpolation happens. We refer to the wall as thin if the physical width of the wallis small compared to the radius at the center of the wall. See figure 2 for two cartoons of O ( D )-solutions with a thick and a thin wall, respectively. We shall find it useful to makethis quantitative by introducing a parameter η that measures the thickness (or rather the‘thin-ness’) of the wall.First, we define the radius at the center of the wall ¯ r as the radius at the locus where ϕ has interpolated half way between its boundary values ϕ f and ϕ , i.e.¯ r ≡ αe A (¯ u ) , with ϕ (¯ u ) = ϕ f + ϕ . (2.13)The wall itself denotes the region where the interpolation between ϕ f and ϕ effectivelyoccurs. There is no universal definition for this and here we make a choice. We define thewall as the interval [ u in , u out ] with ϕ ( u in ) = ϕ f + ϕ − γ ϕ f − ϕ , ϕ ( u out ) = ϕ f + ϕ γ ϕ f − ϕ , (2.14)with a parameter γ < ϕ ( u in ) is to ϕ and ϕ ( u out ) to ϕ f . Inall practical examples we choose γ = 0 . We can then define r out = αe A ( u out ) and r in = αe A ( u in ) as the radii corresponding to the outer and inner edge of the wall. Puttingeverything together, we define the ‘thin-ness’ parameter η as η ≡ r out − r out ¯ r = e A ( u out ) − e A ( u in ) e A (¯ u ) . (2.15)We refer to a wall as ‘thin’, if η (cid:28) η → η (cid:38) O ( D )-instantons and flat domain walls permit an interpre-tation in terms of holographic RG flows. Hence, in the next section we briefly review therelevant holographic RG flow solutions. According to the holographic principle, a strongly-coupled large- N quantum field theory(QFT) is dual to a weakly-coupled gravitational theory [32, 33]. Physical quantities on theQFT side can be obtained by computing analogous quantities on the gravity side [34, 35].In particular, the RG flow on the QFT side is geometrized on the gravity side as the conceptknow as ‘holographic RG flow’.A canonical setting (but not the most general one) for the study of holographic RGflows is Einstein-scalar theory with a non-trivial scalar potential [36, 37, 38, 39, 40, 41], This implies that ϕ ( u in ) = ϕ + 0 .
12 ( ϕ f − ϕ ) and ϕ ( u out ) = ϕ f + 0 .
12 ( ϕ − ϕ f ). – 17 –.e. the theoretical framework introduced in (2.1) and (2.2), together with the ansatz (2.6).We begin by reviewing some general aspects of holographic RG flows in this framework. • AdS solutions on the gravitational side are dual to conformal field theories (CFTs).In the theory (2.1), such solutions correspond to ϕ =constant and coinciding withan extremum ϕ ext of V with V ( ϕ ext ) <
0. Here, the potential exhibits at least twosuch extrema at ϕ = ϕ f and ϕ = ϕ t , which hence represent two different CFTsrespectively. The corresponding gravitational duals are two AdS d +1 space-times,characterised by their AdS lengths (cid:96) f and (cid:96) t , respectively, given by (cid:96) f ≡ − d ( d − V ( ϕ f ) , (cid:96) t ≡ − d ( d − V ( ϕ t ) . (2.16) • The scalar field ϕ is dual to a scalar operator O perturbing the CFT associated witha given extremum and inducing a RG flow. The scaling dimension of this operatorfor a given extremum of V is given by∆( ϕ ext ) ≡ d (cid:32) (cid:115) d − d V (cid:48)(cid:48) ( ϕ ext ) | V ( ϕ ext ) | (cid:33) , (2.17)which crucially depends on the curvature V (cid:48)(cid:48) of the potential at that locus. In par-ticular, for a maximum ( V (cid:48)(cid:48) <
0) the scaling dimension is bounded as ∆ < d andthe operator O corresponds to a relevant deformation of the CFT. In contrast, fora minimum ( V (cid:48)(cid:48) >
0) the scaling dimension takes values ∆ > d and the perturbingoperator O is irrelevant . In the following, unless specified differently, ∆ will referto the scaling dimension of the operator associated with the extremum at ϕ = ϕ f ,i.e. the false vacuum. • The coordinate u (or, equivalently, ξ ) can be used as a parameter along the flowinduced by O . Here we define flows such that the direction of increasing u (i.e. de-creasing ξ ) corresponds to the direction of flow towards the infrared. In particular,the locus u → −∞ (i.e. ξ → + ∞ ) will denote the deep UV, that corresponds to theUV fixed point from which the flow originates. In this work we shall be interested insolutions with boundary conditions (2.11) (i.e. ‘tunnelling from ϕ f ’). In the languageof holography, these solutions correspond to RG flows away from a UV fixed pointgoverned by the CFT associated with ϕ f . In all but a few special cases that will beconsidered separately, ϕ f will correspond to a minimum of V . Therefore, the flowsconsidered here will be flows induced by the vev of an irrelevant operator. The bulkgeometry exhibits a boundary at u → −∞ which can be identified as the boundaryof a AdS d +1 space-time with AdS length (cid:96) f . If the maximum separating the two minima at ϕ = ϕ f and ϕ = ϕ t is an AdS maximum, one can alsoassociate another CFT with this extremum. – 18 – The metric ζ µν in (2.6) describes the background on which the dual d -dimensional fieldtheory is defined. Here, we consider ζ µν describing a d -sphere or flat d -dimensionalspace, corresponding to RG flows for field theories on the respective backgrounds.In the context of holographic RG flows, it is often convenient not to work with thefunctions A ( u ) and ϕ ( u ), but with a different set of dynamical quantities W ( ϕ ), S ( ϕ ) and T ( ϕ ) obeying first-order differential equations. These are defined as W ( ϕ ) ≡ − d −
1) ˙
A , S ( ϕ ) ≡ ˙ ϕ , T ( ϕ ) ≡ e − A R ( ζ ) . (2.18)This can be done piecewise along a flow ϕ ( u ), for intervals separated by loci where ˙ ϕ = 0.In terms of these functions, the equations of motion (2.7)–(2.9) become S − SW (cid:48) + 2 d T = 0 , (2.19) d d − W − S − T + 2 V = 0 , (2.20) SS (cid:48) − d d − SW − V (cid:48) = 0 , (2.21)which are coordinate-independent, first-order non-linear differential equations. In the fol-lowing sections, when appropriate, we shall refer to RG flow solutions in terms of theirsolutions for W , S and T .The dual QFT is further specified by the UV value j for the source of the operator O and its vev (cid:104)O(cid:105) . For the ansatz (2.6), the QFT is additionally characterised by the valueof the scalar curvature R ( ζ ) of the QFT background space-time. The parameters j and R ( ζ ) act as boundary conditions in the UV when solving for A ( u ) and ϕ ( u ) and hence willbe referred to as boundary data. The vev (cid:104)O(cid:105) is then determined by the solution for agiven set of boundary data.For a given holographic RG flow, the QFT data j , R ( ζ ) and (cid:104)O(cid:105) can be read off fromthe asymptotic behavior of the corresponding solutions, in the vicinity of a UV fixed point.This can be done using the expressions for A ( u ) and ϕ ( u ), but here we focus on W ( ϕ ).The UV fixed point is identified with an extremum of the potential, so we need to considerthe solutions for W in the vicinity of an extremum. Here we just present the results. Moredetails and the derivation can be found in [16, 13].In particular, one has to distinguish between maxima and minima of V . Near a maxi-mum of V , there exist two branches of solutions for W , which we denote by ( − ) and (+).In contrast, near a minimum of V only the (+)-branch solution exists. Expanding about More precisely, the metric of the background for the dual QFT is in the same conformal class as ζ µν .By choosing an integration constant appropriately when solving for A ( u ), one can always ensure that thebackground for the field theory is described by ζ µν exactly. Strictly speaking, one needs to specify the UV value of the scalar curvature of the QFT backgroundspace-time R uv = lim u →−∞ e − u/(cid:96) f R (ind) , where R (ind) = e − A ( u ) R ( ζ ) is the induced curvature on a fixed- u -slice. As we describe in [13], we can always set R uv = R ( ζ ) without loss of generality, so that a choice of R ( ζ ) is equivalent to fixing R uv . – 19 –n extremum at ϕ = ϕ f = 0 the solutions on the two branches are given by [13]: W − ( ϕ ) = W reg − ( ϕ ) + R d(cid:96) f | ϕ | / ( d − ∆) (cid:0) . . . (cid:1) + C(cid:96) f | ϕ | d/ ( d − ∆) (cid:0) . . . (cid:1) , (2.22) W + ( ϕ ) = W reg+ ( ϕ ) + R d(cid:96) f | ϕ | / (∆) (cid:0) . . . (cid:1) , (2.23)where R , C and R are dimensionless parameters and the ellipses denote subleading terms.The functions W reg ± ( ϕ ) have a regular expansion in powers of ϕ and are given by: W reg − ( ϕ ) = 2( d − (cid:96) f + d − ∆2 (cid:96) f ϕ + O ( ϕ ) , (2.24) W reg+ ( ϕ ) = 2( d − (cid:96) f + ∆2 (cid:96) f ϕ + O ( ϕ ) . (2.25)The superpotentials (2.22) and (2.23 give rise to solutions with scalar field near-boundaryasymptotics: ϕ ( ξ ) (cid:39) ϕ f + ϕ − (cid:96) d − ∆ f e ( d − ∆) u/(cid:96) f + ϕ + (cid:96) ∆ f e ∆ u/(cid:96) f + . . . u → −∞ (2.26)where ϕ − = j is the source of the dual operator O , and ϕ + is related to its vev by: (cid:104)O(cid:105) = (2∆ − d )( M (cid:96) f ) d − ϕ + . (2.27)In the case of a W + -type solution, the source ϕ − = 0 and the scalar field has only subleadingboundary asymptotics. The solution is controlled by a single parameter, ϕ + .The numerical parameters R , C and R encode the QFT data. In particular, they aredimensionless combinations of the QFT data j , R ( ζ ) and (cid:104)O(cid:105) [42, 13]: R = R ( ζ ) | j | − / ( d − ∆) , (2.28) C = d − ∆ d (cid:104)O(cid:105) ( M (cid:96) f ) d − | j | − ∆ / ( d − ∆) , (2.29) R = (2∆ − d ) / ∆ ( M (cid:96) f ) d − / ∆ R ( ζ ) (cid:104)O(cid:105) − / ∆ . (2.30)The above expressions also hold for RG flows for theories on flat space, in which case onejust sets R ( ζ ) = 0 and hence R = 0 = R in the above.The fact that the solution on the (+)-branch only contains the parameter R but not R or C can then be understood as follows. The (+)-branch describes a RG flow drivenpurely by a non-zero vev (cid:104)O(cid:105) , i.e. the source vanishes, j = 0. The solution W + hence onlycontains R as this is the only dimensionless combination of QFT data that does not contain j . For RG flows from a maximum of V , as originally observed in [43], it can be shownthat the (+)-branch solution can be understood as the limit j → (cid:104)O(cid:105) = const. and R ( ζ ) = const. of a ( − )-branch solutions. Therefore, for a maximum as a UV fixed point, the(+)-branch solution can be seen as a special case from the family of ( − )-branch solutions. In [13] the parameters R and R appearing in (2.22) and (2.23) were denoted by the same symbol R .In the more detailed analysis of [13] this could be done without creating confusion, but here we choosedifferent symbols for clarity. – 20 –ne can also show, see e.g. [13], that if a potential admits RG flow solutions from the( − )-branch, these solutions come as a continuous one-parameter family. The reason is asfollows. As reviewed above, ( − )-branch solutions depend on the two numerical parameters R and C . The constraint that the holographic RG flow is regular at the IR end point,denoted by ϕ , fixes one combination of R and C . This leaves one free parameter, whichcan be shown to be equivalent to a choice of the value of the IR end point ϕ , [13]. Incontrast, a (+)-branch solution only contains one free parameter R . As a result, there isnot enough freedom to ensure regularity of a (+)-type flow at any potential IR end point ϕ and hence (+)-branch solutions do not come in continuous families. Instead, (+)-typesolutions, if they exist at all in a given potential, exist as isolated solutions for selectedvalues of the IR end point ϕ . Each such solution comes with a fixed value of the parameter R . Having discussed holographic RG flow solutions in some detail, we now turn our at-tention to their role in describing Coleman-de Luccia tunnelling processes. An importantquestion at this stage is whether all holographic RG solutions satisfying the metric ansatz(2.6) with R ( ζ ) (cid:54) = 0 automatically also possess an interpretation as an O ( D )-instanton.The answer to this is ‘no’, as we explain in the following section. In particular, we shallargue that only (+)-branch solutions admit an interpretation as O ( D )-instantons while( − )-branch solutions cannot be understood as mediating tunnelling. O ( D ) -instantons as holographic RG flows O ( D )-instantons are solutions to (2.7)–(2.9) with R ( ζ ) (cid:54) = 0. From the discussion in theprevious section it follows that such solutions, when describing tunnelling from AdS vacua,also describe holographic RG flows with the following properties: • The holographic RG flow corresponding to an O ( D )-instanton is a flow for a fieldtheory defined on a d -sphere, since in this case the metric ζ µν in (2.6) is a metric ona d -sphere. This in turn implies that holographic RG flows for field theories on flatspace-time can never have an interpretation in terms of an O ( D )-instanton. • An O ( D )-instanton describing tunnelling away from an AdS extremum ϕ f will cor-respond to an RG flow away from a UV fixed point at ϕ f . The dimension ∆ of theoperator O perturbing the UV CFT is related to the potential and its curvature at ϕ f via (2.17).The fact that the O ( D )-instantons permit an interpretation in terms of holographicRG flows implies that there is a way of describing tunnelling in AdS from the dual QFTpoint of view. In particular, an important set of quantities that characterise a holographicRG flow are the boundary QFT data. These are given by j , the UV value of the source ofthe scalar operator O dual to ϕ , and, for a field theory on a d -sphere, the scalar curvature R ( ζ ) of the d -sphere. To describe an O ( D )-instanton from the QFT perspective thereforecorresponds to specifying the relevant QFT data that will reproduce the holographic RGflow corresponding to the desired O ( D )-instanton.– 21 –n the following we argue that, from the QFT point of view, a given O ( D )-instantonsolution will correspond to not just one QFT, but a family of QFTs with boundary data j = 0, R ( ζ ) (cid:54) = 0 with R ( ζ ) otherwise arbitrary. The fact that O ( D )-instantons correspond toQFTs with j = 0 can be understood as following from the requirement that the tunnellingrate is well-defined. According to Coleman, the tunnelling probability per unit time andper unit volume will be denoted by Γ /V and can be computed in the semiclassical limit as[1, 44, 2]: Γ V = A e − B (cid:126) [1 + O ( (cid:126) )] , (2.31)where (cid:126) is shown explicitly here, but will be suppressed in the following. The prefactor A depends on the one-loop determinant of the theory under consideration and we leave itundetermined here. The exponent B is given by [1, 44, 2]: B = S E, inter − S E, false (2.32)where S E, inter is the Euclidean on-shell action for the interpolating O ( D )-tunnelling solu-tion and S E, false is the background Euclidean action, i.e. the action for the solution withthe false vacuum throughout space-time. Here, for tunnelling from an AdS vacuum, both S E, inter and S E, false exhibit divergences due to the infinite volume of the AdS boundary inthe UV, with the divergent terms depending explicitly on the UV data. These divergencesonly cancel (and therefore B is well-defined) if the boundary data of the interpolating solu-tion and the background solution are the same. This is shown in detail for D = d + 1 = 4in appendix A. The background solution corresponds to the UV CFT associated withthe false vacuum and therefore has j = 0 by definition. This in turn implies that the O ( D )-tunnelling solution must also have j = 0. In the terminology introduced in the pre-vious section, this implies that O ( D )-instantons correspond to holographic RG flows onthe (+)-branch, as it is these solutions that have j = 0, while ( − )-branch solutions have j (cid:54) = 0.We turn to the remaining QFT data. As stated before, the O ( D ) symmetry of thetunnelling solution implies that the corresponding field theory is defined on a d -sphere,and hence we require R ( ζ ) (cid:54) = 0. But what about the value of R ( ζ ) ? How is this related toproperties of the O ( D )-tunnelling solution?To answer this question, we return to our result that O ( D )-instantons correspond to(+)-branch type RG flows. In (2.23) we recorded the near-boundary expansion of W ona (+)-branch solution. Note that this depends on R ( ζ ) only through the dimensionlesscombination R , which has the interpretation of the UV curvature R ( ζ ) in units of the vev (cid:104)O(cid:105) . That is, a given O ( D )-instanton corresponds to a holographic RG flow with a fixedvalue of R , not R ( ζ ) . This implies that we can pick the value of the boundary curvature R ( ζ ) freely, and the vev (cid:104)O(cid:105) will adjust accordingly in the solution so that the dimensionless The definition of the on-shell action typically includes a set of counterterms that regulate the UVdivergences and render the action finite. These counterterms depend explicitly on the boundary dataand also contain free parameters whose choice corresponds to picking a renormalization scheme. For theColeman instanton action B in (2.32) to be scheme-independent the QFT data of the two theories comparedneeds to coincide, which again enforces that the interpolating solution must have a vanishing source, j = 0. – 22 –ombination R remains fixed. The only constraint on R ( ζ ) is that we have to choose thesame value for the O ( D )-tunnelling solution and for the background solution, so that theinstanton action B in (2.32) is again well-defined.To conclude: Holographic RG flow solutions that correspond to O ( D )-instantons de-scribe QFTs with vanishing UV source, j = 0, defined on a d -sphere with curvature R ( ζ ) (cid:54) = 0, but otherwise unconstrained. O ( D ) -instanton action For a tunnelling event mediated by an O ( D )-instanton, the tunnelling probability perunit time and per unit volume was recorded above in (2.31), and depends on the one-loop determinant A of the theory under consideration and the so-called instanton action denoted by B [1, 44, 2]. In the following, we describe how the instanton action B iscomputed explicitly for the tunnelling processes considered in this work.Following [1, 44, 2], the instanton action given in (2.32) is defined as the differencebetween the on-shell action for the tunnelling solution S E, inter and the on-shell action forthe background solution S E, false with the false vacuum filling all of space-time. For AdSfalse vacua, both of these diverge due to the infinite volume of the AdS boundary and(2.32) should be modified as B = lim u uv →−∞ S E, inter − S E, false , (2.33)where u uv is a UV-cutoff that regulates the divergences.Then, both S E, inter and S E, false can be computed by evaluating the action in (2.1) onthe corresponding solutions. Both the instanton and the background solutions obey theansatz (2.6), in which case the action in (2.1) can be written as: S E, O ( D ) = 2( d − M d − V d (cid:2) e dA ˙ A (cid:3) u uv − d M d − R ( ζ ) V d (cid:90) u u uv du e ( d − A , (2.34)with V d ≡ (cid:90) d d x (cid:112) | ζ | = Vol( S d ) = ˜Ω d (cid:0) R ( ζ ) (cid:1) − d/ . Then S E, inter can be computed by evaluating S E, O ( D ) in (2.34) on a solution A ( u ) for theinterpolating case. Similarly, S E, false is given by (2.34) evaluated on the solution A ( u )corresponding to the false vacuum. For the latter, we can give an explicit expression for A ( u ). The false vacuum corresponds to a solution to (2.7)–(2.9) with ϕ = ϕ f and ˙ ϕ = 0.One can check that for R ( ζ ) > A ( u ) is given by A ( u ) = ln (cid:20) − (cid:96) f α sinh (cid:18) u − u f, (cid:96) f (cid:19)(cid:21) , (2.35) There is a caveat though, if A turns out to be imaginary (this happens if the determinant has an oddnumber of negative modes) then Γ is imaginary and the decay is unsuppressed, [44]. For the ansatz (2.6) one can re-express R ( g ) as a function of A , ˙ A and ¨ A . Using the equations of motion(2.7)–(2.9) it is then possible to eliminate all explicit appearances of ¨ A , ˙ ϕ and V to arrive at the expressionin (2.34) (see e.g. [13, 31] for more details). – 23 –here (cid:96) f was defined in (2.16) and α in (2.4). The parameter u f, is an integration constantand we can choose it freely. Here we set it to u f, = ( (cid:96) f /
2) log(4 α /(cid:96) f ).We can then calculate B in (2.33) by inserting the regulated expressions for S E, inter and S E, false obtained from (2.34) and removing the regulator by setting u uv → −∞ .The calculation parallels the one presented in [31] for source-driven RG flows. Weobserved that: • The first term in equation (2.34) cancels exactly when taking the difference (2.33).The reason is that the finite contribution to this term comes purely from the termproportional to C in the expansion of W , equation (2.22-2.23), and this term is absentfor W + -type solution. This is unlike the case analyzed in [31] where the C terms givesthe leading contribution to the free energy. • The second term in (2.34) gives a finite result (after subtracting the AdS contribution)and can be written as: S E, inter − S E, false → − ( M (cid:96) f ) d − B (cid:90) d d x (cid:112) | ζ | (cid:16) R ( ζ ) (cid:17) d/ , for u uv → −∞ , (2.36)where we introduced a dimensionless constant B which does not depend on curva-ture, which is uniquely determined for a given interpolating solution, like the valueof the parameter R appearing in equations (2.23-2.30).The integral in equation (2.36) gives a curvature-independent result, since the volume of S d scales as α d where α is the sphere radius given by equation (2.4). This leads to the finalresult: B = − ( M (cid:96) f ) d − ˜Ω d B , (2.37)where ˜Ω d is a geometric factor related to the volume of the unit d -dimensional sphere,defined in (2.34).Although B can only be computed numerically, we can determine its sign withoutdoing any extra computation. The sign of B is very important, since by equation (2.31)it governs the decay rate of the false vacuum: a negative B would imply that the decay isunsuppressed and the false vacuum is actually unstable, rather than metastable.To determined the sign of equation (2.37), we can go back to the expression (2.34),and rewrite (recall the first term gives no contribution for W + -flows): B = (cid:18) − d M d − R ( ζ ) V d (cid:19) lim u uv →−∞ (cid:20)(cid:90) u u uv du e ( d − A inter ( u ) − (cid:90) u f, u uv du e ( d − A false ( u ) (cid:21) (2.38)where A inter ( u ) and A false ( u ) are the warp factors of the CdL instanton and of the AdSfalse-vacuum solution, respectively. It is easy to see that the quantity in the square bracketsis negative, because the scale factor e A decreases faster in the CdL solution than in AdS.One can see this using the definition (2.18) and the relation (cfr. eq. (2.23), (2.25)): W + > W f = 2( d − (cid:96) f . (2.39) To make contact with [31], B is related to the vev-parameter appearing in the the U -function. – 24 –he above argument shows that, regardless of its numerical value, B > B computes the difference betweenthe free energies on S d of the two semiclassical states corresponding to the CFT stuck atthe UV fixed point ϕ f and the RG-flow driven away from the fixed point by the non-zerovev ϕ + : B = F RG-flow − F fixed-point > . (2.40)The positivity of B implies the semiclassical state of lower free energy is the pure AdSsolution with no flow. In this section we show that a process described by an O ( D )-instanton can never result in ϕ arriving at exactly an AdS extremum of the potential. One consequence of this is thattunnelling from a false vacuum to exactly the true vacuum is forbidden. This result is notnew and has already been observed in the foundational work by Coleman and de Luccia[2]. However, in the approximation considered in [2] the end point of tunnelling can bemade arbitrarily close to the true vacuum, so that this observation was not emphasisedthere. This may leave casual readers of the topic with the false impression that tunnellingexactly to a true vacuum is possible, while in fact is is not. The purpose of this section isto emphasize this fact about tunnelling. No tunnelling to exactly an AdS extremum
We demonstrate this by assuming that a solution exists that arrives in the vicinity of anAdS extremum of V and then show that there is no solution of O ( D ) type that can reachthe extremum exactly. To start, note that in the vicinity of an AdS extremum at ϕ = ϕ ext ,the potential can always be written as V ( ϕ ) = − d ( d − (cid:96) − ∆( d − ∆)2 (cid:96) ( ϕ − ϕ ext ) + O (cid:0) ( ϕ − ϕ ext ) (cid:1) , (2.41)where we have used (2.17) to re-express V (cid:48)(cid:48) ( ϕ ext ) in terms of the dual scaling dimension∆. We now construct the solution for an O ( D )-instanton in the vicinity of the extremum byperturbing about the solution at the extremum. At the extremum, i.e. for ϕ = ϕ ext = const,the solution for the space-time is just D -dimensional AdS space with AdS length (cid:96) . Weencountered the corresponding solution for A ( u ) before in (2.35) for the case f ext = ϕ f .Denoting this solution by A ( u ), we have A ( u ) = ln (cid:20) − (cid:96)α sinh (cid:18) u − u (cid:96) (cid:19)(cid:21) , (2.42)with α defined in (2.4). To find solutions in the vicinity of ϕ ext , we then expand about thesolution at ϕ = ϕ ext . We choose the ansatz A ( u ) = A ( u ) + (cid:15) A ( u ) + O ( (cid:15) ) ,ϕ ( u ) = ϕ ext + (cid:15)ϕ ( u ) + O ( (cid:15) ) , (2.43)– 25 –here (cid:15) ∼ | ϕ − ϕ ext | is a small parameter measuring the departure from the extremum.This will allow for the construction of solutions for ϕ ( u ) and A ( u ) iteratively, by solvingorder by order in (cid:15) upon inserting this ansatz into the equations of motion (2.7)–(2.9).Here, we are mainly interested in whether interpolating solutions exist that smoothlyarrive at ϕ = ϕ ext . Therefore, we focus on the correction ϕ ( u ) to the constant solution ϕ = ϕ ext . The corresponding equation of motion is given by:¨ ϕ ( u ) + d(cid:96) coth (cid:18) u − u (cid:96) (cid:19) ˙ ϕ + ∆( d − ∆) (cid:96) ϕ ( u ) = 0 . (2.44)We need to complement this with the appropriate boundary conditions. We are interestedin O ( D )-instantons that correspond to tunnelling to ϕ = ϕ ext . The end point of a tunnellingprocess by an O ( D )-instanton is the centre of the solution, where the radius e A → u → u .Therefore, a solution corresponding to tunnelling to ϕ ext would have to satisfy the boundaryconditions, see (2.11): ϕ ( u ) = 0 , ˙ ϕ ( u ) = 0 . (2.45)The task now simply is to solve (2.44) and implement the boundary conditions. While thiscan be done in all generality (see e.g. [31]), a simplified analysis will be sufficient to provethe point without sacrificing rigour. In particular, by restricting attention to the leadingterms for u → u we can expand the coth-term in (2.44) for u → u and solve (2.44) as ϕ ( u ) = u → u c (cid:20)(cid:16) u − u (cid:96) (cid:17) − ( d − + O (cid:18)(cid:16) u − u (cid:96) (cid:17) − ( d − (cid:19)(cid:21) (2.46)+ c (cid:20) − ∆( d − ∆)2( d + 1) (cid:16) u − u (cid:96) (cid:17) + O (cid:18)(cid:16) u − u (cid:96) (cid:17) (cid:19)(cid:21) , with c and c integration constants. It is now easy to verify that O ( D )-solutions arrivingat the extremum are not allowed. The boundary condition ˙ ϕ ( u ) = 0 can only be satisfiedby setting c = 0 and similarly the boundary condition ϕ ( u ) = 0 requires c = 0. As aresult the only possible solution is that ϕ vanishes identically. For the ansatz in (2.43), if ϕ vanishes then all higher corrections, i.e. A etc., will also vanish, as one can verify fromthe equations of motion. Therefore, we arrive at the result that a solution that arrives at ϕ ext does not exist. This confirms what we have set out to prove: there is no solution of O ( D ) type can describe tunnelling to exactly an extremum of the potential .Instead, tunnelling processes via O ( D )-instantons deposit ϕ at generic points of thepotential V that are not extrema. For completeness, we record the expressions for ϕ ( u )and A ( u ) in the vicinity of such a generic point, denoted by ϕ = ϕ . More details regardingthe derivation can be found in [13]. In particular, near ϕ = ϕ we can expand ϕ ( u ) and For d = 3, which is the case of most relevance for our 4-dimensional universe, the first subleadingcorrection in the term with integration constant c should be O (cid:0) log( u − u (cid:96) ) (cid:1) . However, this does not affectanything that follows. – 26 – ( u ) for u → u as A ( u ) = ln (cid:16) − u − u α (cid:17) + O (cid:0) ( u − u ) (cid:1) , (2.47) ϕ ( u ) = ϕ + V (cid:48) ( ϕ )2( d + 1) ( u − u ) + O (cid:0) ( u − u ) (cid:1) , (2.48)with α defined as in (2.4). Note that an end point ϕ with V (cid:48) ( ϕ ) < can only be approachedfrom the left, while an end point with V (cid:48) ( ϕ ) > can only be arrived at coming from theright. A quick look at flat domain walls
In contrast to O ( D )-instantons, flat domain walls interpolate between two extrema of thepotential, as we briefly review here. Consider again an extremum at ϕ = ϕ ext with V inits vicinity given by (2.41). Approaching an extremum, the bulk space-time asymptotes toAdS d +1 , which for R ( ζ ) = 0 implies that A ( u ) = − u(cid:96) . (2.49)Inserting this into (2.9) one finds¨ ϕ ( u ) + d(cid:96) ˙ ϕ + ∆( d − ∆) (cid:96) ϕ ( u ) = 0 , (2.50)which can be solved as ϕ ( u ) = ϕ ext + ϕ − (cid:96) ( d − ∆) e ( d − ∆) u/(cid:96) + ϕ + (cid:96) ∆ e ∆ u/(cid:96) , (2.51)with ϕ − and ϕ + integration constants. The UV and IR ends of the domain wall are givenby the loci e A → + ∞ and e A →
0, respectively, corresponding to u → −∞ and u → + ∞ .At both the UV and IR end points we need to implement the boundary conditions ϕ ( u → ±∞ ) = ϕ ext , ˙ ϕ ( u → ±∞ ) = 0 . (2.52)We can then make the following observations: • Both maxima and minima of V can be UV ends of flat domain walls. In the case ofa minimum of V , ∆ > d , the boundary conditions require setting ϕ − = 0, but ϕ + can be non-zero. • Maxima of V , ∆ < d , cannot be IR end points of flat domain walls. The boundaryconditions can only be satisfied by setting both ϕ − = 0 = ϕ + and hence the wallcannot arrive there. • Instead, flat domain walls have their IR end points at minima of V (∆ > d ). Theboundary conditions require setting ϕ + = 0, but ϕ − can be non-zero.– 27 – .6 Tunnelling from AdS minima is non-generic Without gravity, a physical system with two non-degenerate ground states will genericallypermit tunnelling from the false to the true vacuum [1]. However, in the presence ofgravity this is no longer the case. The existence of two non-degenerate ground states doesnot automatically imply the existence of an O ( D )-instanton solution. This has alreadybeen observed in the foundational work by Coleman and de Luccia [2] in the context ofthin-wall solutions. Here, we present a general argument for this observation valid beyondthe thin-wall approximation, and link it to known features of holographic RG flows. Inaddition, we derive a sufficient condition on a potential for possessing a O ( D )-instantondescribing tunnelling out of an AdS minimum. A mechanical analogue –
A useful tool for constructing the argument of this sectionwill be a mechanical analogue to the problem at hand, originally used by Coleman in [1].The observation is that the equation of motion (2.9) is identical to that for the Lorentzianproblem of a particle of unit mass performing one-dimensional motion in the invertedpotential − V . The field ϕ plays the role of the coordinate of the particle and the term ∼ ˙ A ˙ ϕ implements friction. For positive friction the time direction for the motion of theparticle needs to be in the negative u -direction. This follows from the equation of motion(2.8) together with the boundary conditions (2.11) which imply that ˙ A is strictly negative,˙ A <
0. Hence, for positive friction the time variable t for the particle has be taken as t = − u . As a result (2.9) can be written as d ϕdt + b ( t ) dϕdt − V (cid:48) = 0 , with b ( t ) ≡ − d ˙ A ( u ) > , (2.53)which can be identified as the equation of motion for damped motion in the invertedpotential − V .Due to the identification t = − u , the initial conditions for the particle are set bythe boundary conditions at the end point of the tunnelling process. An O ( D )-solutiondescribing tunnelling from ϕ f to some locus ϕ exhibits the behaviour ϕ → ϕ f , ˙ ϕ → , for u → −∞ , ϕ → ϕ , ˙ ϕ → , for u → u , This implies that in the mechanical analogue the particle is initially released from ϕ atrest ( dϕdt = 0). To describe tunnelling from an extremum at ϕ = ϕ f , the particle then hasto come to rest again at ϕ f . As a result, an O ( D )-solution describing tunnelling from anAdS minimum (maximum) at ϕ f to some generic point ϕ has the following mechanicalanalogue: It corresponds to a solution, where the particle starts rolling from rest from ϕ and after performing damped motion comes to rest again at the maximum (minimum) ϕ f of the inverted potential.We can also consider the mechanical analogue of a flat domain-walls. For flat domain-walls interpolating between ϕ f and ϕ t one has ϕ → ϕ f , ˙ ϕ → , for u → −∞ , ϕ → ϕ t , ˙ ϕ → , for u → + ∞ . See (2.48) for the precise behaviour near u = u . – 28 – V ( ϕ )0 V ( ϕ ) ϕ f ϕ (cid:63) ϕ t ϕ f ϕ (cid:63) ϕ t Figure 3: Potential V ( ϕ ) with two AdS minima at ϕ = ϕ f (‘false vacuum’) and ϕ = ϕ t (‘true vacuum’) with V ( ϕ f ) > V ( ϕ t ), separated by a barrier. The inverted potential − V ( ϕ )exhibits two maxima at ϕ f and ϕ t separated by a valley. The locus ϕ (cid:63) is defined as thepoint with V ( ϕ (cid:63) ) = V ( ϕ f ), but on the other side of the barrier/valley from ϕ f . A particlereleased from rest and performing damped motion in the inverted potential − V cannotreach ϕ f if it is initially released outside the shaded region.The mechanical analogue now corresponds to a particle being released at rest from ϕ t andcoming to rest again at ϕ f .In the following, we test whether a potential admits an O ( D )-instanton solution orflat domain-wall solution by using the mechanical framework and releasing a test particlefrom various points ϕ start with the appropriate initial conditions, and follow its subsequenttrajectory. In fact, we can discuss both O ( D )-instanton solutions and flat domain-wallsolutions in a unified way as in both cases the particle has to be released from rest. Tunnelling from AdS minima is non-generic –
Here we present the main argument ofthis section, starting with the mechanical picture, before turning to a holographic point ofview. As explained above, solutions describing tunnelling from a minimum of V correspondto mechanical solutions describing a particle coming to rest exactly at a maximum of − V .In the mechanical framework it is then easy to see that minima of − V are attractorswhile maxima are not. That is, if the particle is released from some point in the invertedpotential, the particle will roll down the inverted potential and, as a result of friction,generically come to rest again in a minimum of − V . In contrast, to come to rest at amaximum of − V the trajectory of the particle has to be tuned carefully to avoid any over-or undershooting.The observation then is that in some potentials this can be done by adjusting thestarting point ϕ start for the particle, but in other potentials no such starting point can befound. This is best illustrated by an example. Consider a potential with exactly two AdSminima at ϕ = ϕ f and ϕ = ϕ t separated by a barrier by default, with V ( ϕ f ) > V ( ϕ t ) asshown in fig. 3. As described before, solutions describing tunnelling from ϕ f will correspondto mechanical solutions in the inverted potential − V where the particle comes to rest at ϕ f . The inverted potential exhibits a local maximum at ϕ f and a global maximum ϕ t separated by an intermediate minimum, and is also shown in fig. 3 Due to the existence– 29 –f friction, a necessary condition for the particle to reach ϕ f is that its initial potentialenergy is at least as large as the potential energy at ϕ f , i.e. − V ( ϕ start ) > − V ( ϕ f ). Thisimplies that the particle has to start in the interval ϕ start ∈ [ ϕ (cid:63) , ϕ t ] (the shaded region infig. 3) where at ϕ (cid:63) one has V ( ϕ (cid:63) ) = V ( ϕ f ). However, this is not a sufficient condition fora solution to exist. In fact, depending on the precise shape of the potential, there may notexist a choice of starting point so that the particle successfully reaches ϕ f . In particular,friction may be sufficiently strong that for any starting point ϕ start ∈ [ ϕ (cid:63) , ϕ t ] the particlewill not reach ϕ f , but undershoot and subsequently roll down to the bottom of the valley.In the Euclidean picture this implies that no solutions describing tunnelling from ϕ f existin this potential. This shows that trajectories that will allow the particle to come to rest ata maximum of − V only exist for certain potentials, but not generically. In the Euclideanpicture the corresponding finding is that solutions describing tunnelling from a minimumof V are non-generic.This result can also be deduced from a holographic point of view, as we proceed toshow in the following. To begin, we use the formulation in terms of holographic RG flows tore-derive the statement from above that maxima of V (i.e. minima of − V ) are attractorswhile minima of V (i.e. maxima of − V ) are not. To this end we recall the asymptoticbehaviour of holographic RG flow solutions near an extremum. For example, considerthe solutions for W ( ϕ ) near an extremum of V that represents the UV fixed point for aholographic RG flow. We encountered these before in (2.22) for an expansion around amaximum of V , and in (2.23) for an expansion around a minimum of V . The observationwas that in the vicinity of a maximum, W depends on two free parameters R and C , whilein the vicinity of a minimum there is only one free parameter R . That is, solving near amaximum of V , solutions for W come as a two-parameter family, while near a minimumof V , solutions for W represent a one-parameter family:Expansion near max. of V : W ( ϕ ) = W R ,C ( ϕ ) , (2.54)Expansion near min. of V : W ( ϕ ) = W R ( ϕ ) . (2.55)In a complete RG flow, these parameters are fixed by the boundary/ regularity conditionson the other end of the flow. In particular, setting the value of ϕ where the flow ends on theother side fixes one combination of the parameters. This does however not yet guaranteethat the solution is regular, i.e. regularity is an additional requirement to be satisfied.The observation is that in the case of a RG flow from a UV fixed point at a maximumof V , we can generically ensure regularity by adjusting the remaining free parameter. Asa result, in a generic potential V with a maximum, there generically exist holographicRG flows that are regular away from this maximum. In other words, maxima of V areattractors for holographic RG flows.For RG flow emanating from a UV fixed point at a minimum of V , there is no furtherfree parameter that can be adjusted to ensure regularity and typically no regular solutioncan be found. Therefore, we confirm once more what we endeavoured to show in thissection: In a generic potential, regular holographic RG flows emanating from an AdSminimum of V generically do not exist. Using the identification between O ( D )-instantons– 30 – V ( ϕ )0 V ( ϕ ) ϕ f ϕ (cid:63) ϕ ϕ t Figure 4: Potential V ( ϕ ) with two minima at ϕ = ϕ f (‘false vacuum’) and ϕ = ϕ t (‘truevacuum’) with V ( ϕ f ) > V ( ϕ t ), separated by a barrier. The inverted potential − V ( ϕ )exhibits two maxima at ϕ f and ϕ t separated by a valley. For the depicted potential, usingthe mechanical analogue of tunnelling solutions, a particle released at rest from ϕ t towardsthe left will overshoot ϕ f , as depicted the uppermost red trajectory. A particle released atrest from ϕ (cid:63) with V ( ϕ (cid:63) ) = V ( ϕ f ) will undershoot ϕ f and eventually settle at the bottomat the valley as depicted by the lower red trajectory. By continuity, there exists a locus ϕ ,so that if the particle is released at rest from there, it will neither over- nor undershoot,but come to rest again at exactly ϕ f . This trajectory is shown in purple.and holographic RG flows, this confirms that tunnelling from AdS minima is non-generic, aswe have concluded previously using the mechanical picture. The exceptions are potentialswhich are to some extent tuned, so that the boundary and regularity condition can besatisfied simultaneously by adjusting just one parameter.An important question then is: how can one identify potentials that admit O ( D )-instantons for tunnelling from an AdS minimum? In the following, we derive a sufficientcondition on V for such solutions to exist. Tunnelling from AdS minima: a sufficient condition on V – Here we again considera potential V with two AdS minima at ϕ = ϕ f and ϕ = ϕ t with V ( ϕ f ) > V ( ϕ t ), separatedby a barrier. To employ the mechanical analogue of O ( D )-instantons, we now turn to theinverted potential − V . This has two maxima ϕ f and ϕ t with − V ( ϕ f ) < − V ( ϕ t ), with avalley between them. Both the potential and inverted potential are shown in blue in fig. 4.To determine whether the potential V permits a O ( D )-instanton solution we shall studywhether the corresponding mechanical solution in − V exists.In the following, we employ a form of the overshoot/undershoot argument originallyintroduced in [1]. Consider starting at rest at ϕ t and letting the particle roll towards ϕ f . There are three possibilities for the subsequent behaviour of the particle: In practice, when solving numerically, for the particle to roll away in finite time, one has to start somefinite distance away from ϕ t . – 31 –. Overshooting case:
As the initial potential energy at ϕ t is higher than the potentialenergy at ϕ f , the particle can pass the trough and overshoot the maximum of − V at ϕ f .2. Undershooting case:
If the friction is sufficiently large, enough kinetic energy canbe dissipated so that the particle undershoots the maximum of − V at ϕ f . Instead,the particle eventually settles at the bottom of the valley, after performing dampedoscillations about this minimum of − V .3. Interpolating case:
The particle can cross the valley and come to rest exactly at ϕ f . In the original Euclidean interpretation, this corresponds to a flat domain-wallsolution. This is not expected to occur in a generic potential, which will either leadto the particle overshooting or undershooting.To continue, we first examine the overshooting case from above, i.e. we assume thatthe potential is such that the particle overshoots ϕ f when released from rest at ϕ t . This isshown by the upper red trajectory in fig. 4. Now consider releasing the particle not fromthe top of the hill of − V at ϕ t , but further down the slope. In particular, as a startingpoint consider the locus ϕ = ϕ (cid:63) where the particle has the same potential energy as at ϕ = ϕ f , i.e. − V ( ϕ (cid:63) ) = − V ( ϕ f ), but on the other side of the trough from ϕ f , as shown infig. 4. Even though the particle initially has just sufficient potential energy to exactly reach ϕ f on the other side of the valley, it will undershoot and become trapped in the valley as aconsequence of friction (see the lower red trajectory in fig. 4). This also applies to all otherstarting points that lie further down the slope than ϕ (cid:63) . To summarise, here, when releasedfrom the top of the hill at ϕ t , the particle will overshoot ϕ f , but when released from ϕ (cid:63) orlower down the slope it will undershoot ϕ f . By continuity it then follows that there existssome locus ϕ = ϕ ∈ [ ϕ (cid:63) , ϕ t ] for which the particle will neither over- nor undershoot, butexactly come to rest at ϕ f . This is the mechanical version of the solution we seek: theEuclidean analogue will be an O ( D )-solution describing tunnelling from ϕ f . In fig. 4 thisis indicated by the purple trajectory.We now turn to the undershooting and interpolating cases from above, i.e. we assumethat the inverted potential − V is such that the particle either undershoots or arrives atrest at ϕ f when starting from ϕ t . Now consider releasing the particle further down thehill. In this case the initial potential energy is lower, so it will be more difficult for theparticle to reach ϕ f on the other side of the valley. At the same time, the velocity of theparticle, when passing the valley will be lower, which lowers friction, so that less kineticenergy is dissipated. While at this stage we cannot exclude that these two effects maycancel one another, so that one may find a solution that comes to rest at ϕ f exactly, this isnot expected to happen generically. As before, once the starting point is lowered beyond ϕ (cid:63) , i.e. − V ( ϕ start ) < − V ( ϕ (cid:63) ) = − V ( ϕ f ), the particle again must undershoot. All thingsconsidered, we do not expect potentials from the undershooting and interpolating casesabove to permit O ( D )-instanton solutions. Correspondingly, we did not find any examples for this in any of the numerical examples studied. – 32 –ased on these observations, we can identify a sufficient condition on the potential V to permit a solution describing tunnelling from ϕ f : A potential V with two AdS minima at ϕ = ϕ f and ϕ = ϕ t with V ( ϕ f ) > V ( ϕ t ) willpermit an O ( D ) -instanton describing tunnelling from ϕ f , if in the mechanical formulation,the particle overshoots ϕ f when released from rest at ϕ t . Given a potential V with two non-degenerate minima ϕ f and ϕ t , the above conditionallows for an easy test to determine whether the false minimum ϕ f is prone to decayvia O ( D )-instantons. One simply has to solve (2.7)–(2.9) with the boundary conditionscorresponding to the particle released from rest at ϕ t . In the Euclidean picture, these arethe boundary conditions for a flat domain-wall solution ending at ϕ t i.e. ϕ ( u → + ∞ ) → ϕ t , e A ( u → + ∞ ) → . (2.56)In section 4 we shall use this criterion to design potentials that admit O ( D )-tunnellingsolutions. O ( D ) -instantons and exotic holographic RG flows We now show that the existence of an O ( D )-instanton describing tunnelling from an AdSminimum is tied to the existence of a holographic RG flow solution that exhibits ‘exotic’phenomena, such as reversals of the flow in ϕ -space and the skipping of fixed points. Suchexotic RG flows have been studied in detail in [16] for field theories on flat space-timeand in [13] for field theories on constant curvature backgrounds. Here, we argue that apotential that admits an O ( D )-instanton, describing tunnelling from an AdS minimum,will also admit holographic RG flows with these exotic features. Conversely, the existenceof exotic RG flows will imply the existence of O ( D )-instanton solutions.To be specific, consider once more a potential with two minima at ϕ = ϕ f and ϕ = ϕ t with V ( ϕ f ) > V ( ϕ t ). Now we also assume that the potential admits an O ( D )-instantonsolution describing tunnelling from ϕ f to some locus ϕ = ϕ that is not an extremum( V (cid:48) ( ϕ ) (cid:54) = 0). We shall argue that this potential will also admit holographic RG flows withthe exotic features mentioned above.We turn once more to the mechanical picture introduced in the previous section. There,we found that the O ( D )-instanton can be identified as the solution at the interface ofsolutions where the particle overshoots or undershoots ϕ f . That is, if the particle is releasednot at ϕ , but slightly further up the slope of − V , it will overshoot. If instead it is releasedjust below ϕ it will undershoot. In the following, we focus on the solutions in the over-and undershooting case. These are solutions to the equations of motion (2.7)–(2.9) andhence also possess an interpretation in terms of holographic RG flows. We find that thesesolutions can be matched with the so-called exotic holographic RG flows. Undershooting solutions as bouncing flows –
Consider releasing the particle not from ϕ but further down the slope of − V from ϕ so that the particle undershoots ϕ f . Once In practice, the boundary conditions are implemented at some large but finite value of u . The appro-priate boundary conditions can be read-off from the near-extremum behaviour of A ( u ) and ϕ ( u ) shown in(2.49) and (2.51). – 33 – V ( ϕ )0 V ( ϕ ) ϕ f ϕ b ϕ ϕ a ϕ t ϕ uv ϕ uv Figure 5: Potential V ( ϕ ) with two AdS minima at ϕ f and ϕ t with V ( ϕ f ) > V ( ϕ t ), sep-arated by a maximum at ϕ uv (blue region). The potential outside the region [ ϕ f , ϕ t ] isshown in green. Beyond the minimum at ϕ f the potential exhibits another maximum at ϕ uv . The inverted potential exhibits two maxima at ϕ f , ϕ t and two minima at ϕ uv , ϕ uv .The potential admits an O ( D )-instanton describing tunnelling from the false minimum ϕ f to the locus ϕ . The mechanical analogue of this is shown in purple denoting the trajectoryof a particle released at rest from ϕ and rolling to ϕ f . If the particle is released from ϕ b somewhat below ϕ it undershoots ϕ f , reverses direction and eventually settles at ϕ uv (lower red trajectory). In the Euclidean framework this corresponds to a holographic RGflow from a UV fixed point at ϕ uv to an IR end point at ϕ b that ‘bounces’, i.e. reversesdirection in ϕ (upper orange flow). If the particle is released from ϕ a somewhat above ϕ it overshoots ϕ f and comes to rest in ϕ uv (upper red trajectory). In the Euclideanframework this corresponds to a holographic RG flow from a UV fixed point at ϕ uv to ϕ a (lower orange trajectory). This flow ‘skips’ over the other UV fixed point at ϕ uv .it undershoots, the particle is trapped in the valley of − V between ϕ f and ϕ t and willeventually settle at the bottom. In the Euclidean framework, solutions that end at thebottom of the valley of − V between ϕ f and ϕ t correspond to holographic RG flows froma UV fixed point at the maximum that is located at the top of the barrier separating ϕ f and ϕ t . This UV fixed point is denoted as ϕ uv in fig. 5.The observation is that if the particle is released sufficiently close to ϕ , it will notjust roll to the bottom of the valley directly, but it will oscillate about the bottom of thevalley (at least once). The reason is that if the particle is released sufficiently close to ϕ , it should just fail to reach ϕ f on the other side of the valley and hence passes thebottom of the valley at least once before eventually settling there. This is shown in fig. 5as the red trajectory starting at ϕ b . In the Euclidean picture this will correspond to an RGflow that leaves the UV fixed point at ϕ uv towards ϕ f , reverses direction and passes the– 34 –aximum at least once, before finally ending at ϕ b . This is shown as the orange flow from ϕ uv in fig. 5. The reversal of flow direction in ϕ -space was termed a ‘bounce’ in [16, 13].The observation is that a potential that admits O ( D )-tunnelling from an AdS minimumwill typically also permit bouncing flows from the UV fixed point ϕ uv , i.e. the top of thebarrier separating the false and true minimum. Overshooting solutions as skipping flows –
Now consider releasing the particle notfrom ϕ but further up the slope of − V from ϕ so that the particle overshoots ϕ f . Todiscuss the subsequent evolution of the particle, we need to extend the trajectory of theparticle beyond ϕ f . As ϕ f is a minimum, the potential beyond it can either rise mono-tonically and eventually diverge, or it could reach another maximum. For the divergentcase, no regular solution for the subsequent evolution of the particle can be found and weignore this possibility.We therefore consider the case that V possesses another maximum beyond ϕ f , whichin fig. 5 we denoted by ϕ uv . In the inverted potential − V the maximum at ϕ uv translatesinto a minimum. As seen before, minima of − V are attractors for the particle. Therefore,once the particle overshoots ϕ f and enters the valley of ϕ uv it will typically settle at thebottom of this minimum (or another minimum of − V if the particle overshoots the valleyof ϕ uv as well). This is shown as the red trajectory originating from ϕ a in fig. 5In the Euclidean picture this will correspond to a RG flow from a UV fixed-point at ϕ uv to ϕ a . Most importantly, this RG flow skips the other potential UV fixed point at ϕ uv . Flows that bypass viable fixed points at extrema of the potential were denoted as‘skipping’ flows in [16, 13]. From the above reasoning we expect that a potential thatadmits O ( D )-tunnelling from an AdS minimum will also permit skipping flows. For theskipping flow the relevant UV fixed point will be another maximum outside the range[ ϕ f , ϕ t ], here denoted by ϕ uv .The same argument can also be run in reverse: the existence of curved skipping RGflows which overshoot ϕ f and of curved bouncing RG flows which undershoot it imply theexistence of a single flow as the limiting case which separates the two classes of solutions,and which connects ϕ f with point ϕ on the slope of the potential. This flow is the O ( D )-instanton which describes decay of the false AdS D vacuum.In the discussion above, the RG flows were for QFTs defined on S d , but interestingstatements can also be made for flat-sliced flows that connect extrema of the potential.Consider for example a potential that admits a flat-sliced skipping flow from ϕ uv to ϕ t .This is the RG flow equivalent of the overshooting mechanical solution for a particle startingat rest from ϕ t , which we identified as a sufficient condition for a potential to admit an O ( D )-instanton.An explicit example for a potential that exhibits the features discussed here can befound in [16, 13]. In particular in [13] the various RG flow solutions for all possible flowend points in this potential were derived and their holographic interpretation was discussedsystematically. We refer readers to this work for more details. We ignore the possibility that V remains constant beyond ϕ f as this seems highly artificial and we haveno such examples in string theory. – 35 – . Flat AdS domain-walls and a thin-wall limit In this section we shall construct explicit solutions for flat AdS domain-walls and thecorresponding potential permitting these solutions. While these solutions do not describetunnelling processes, we find it nevertheless convenient to discuss them for the followingreasons. The potentials that we derive here, will be good starting points for constructingpotentials that will admit O ( D )-instanton solutions, our primary target. This is valuable, asgeneric potentials typically do not permit these solutions as discussed above. Furthermore,flat domain-wall solutions can be understood as a limit of O ( D )-instanton solutions wherethe curvature of the bubble wall is taken to zero. As a result, the analytic expressions weshall compute for flat domain-walls, will be helpful for making analytic statements about O ( D )-instanton solutions in a certain limit, at least approximately. Here we construct families of flat AdS domain-walls that will explicitly admit a limit inwhich the domain wall becomes infinitely thin. It behoves to point out that this ‘thin-walllimit’ for flat domain walls is a priori completely unrelated to the more familiar thin-walllimit of O ( D )-instantons. For O ( D )-symmetric solutions, the bubble wall is deemed thin, ifthe interval in the physical radius r over which the interpolation occurs, is small comparedto the bubble radius ¯ r , as described in section 2.1. For flat domain walls there is nonotion of a radial coordinate and a different definition is needed. In this case we definethe wall as the interval in the coordinate u over which the interpolation between the twoAdS backgrounds occurs. To decide whether this wall is thin, we need to compare this toanother length scale. For AdS domain-walls, there exist two further length scales in theform of the AdS lengths of the two AdS backgrounds connected by the wall. Therefore, wedefine a flat AdS domain-wall to be thin if the length of the interval in u where interpolationoccurs is small compared to either AdS length scale . This gives an invariant definition inthe gravitational theory, but has no clear meaning in the dual QFT.We now proceed to constructing such a family of solutions explicitly. The relevantansatz is given by (2.6) with ζ µν a metric on flat Euclidean space. The correspondingequations of motion are given in (2.7)-(2.9) with R ( ζ ) = 0. We seek solutions that: • interpolate between two AdS space-times with length scales (cid:96) f and (cid:96) t , respectively,with (cid:96) t < (cid:96) f ; • interpolate at the same time between the values ϕ = ϕ f and ϕ = ϕ t .AdS solutions for the ansatz (2.6) with ζ µν describing Euclidean space correspond to ascale factor A ( u ) = − u/(cid:96) , ˙ A ( u ) = − /(cid:96) , with (cid:96) the relevant AdS length. The two demandsin the list above are hence equivalent to the following boundary conditions:˙ A = − (cid:96) f , u → −∞ , − (cid:96) t , u → ∞ . , ϕ = ϕ f , u → −∞ ,ϕ t , u → ∞ . (3.1)– 36 –e now turn to another demand on our solution, the existence of a thin-wall limitin the sense described above. The limit of a vanishingly thin wall corresponds to thecase where the interpolation happens instantaneously, i.e. ˙ A and ϕ jump discontinuouslybetween the two boundary values at some locus u = ¯ u . To implement this, we try theansatz ˙ A ∼ Θ( u − ¯ u ), ϕ ∼ Θ( u − ¯ u ), with Θ referring to the Heaviside theta function,which in turn implies ¨ A ∼ δ ( u − ¯ u ) and ˙ ϕ ∼ δ ( u − ¯ u ). However, one can easily confirmthat in this case, none of the equations of motion (2.7)–(2.9) can be solved, i.e. the ansatzis invalid. This does not imply that an extreme thin-wall limit is impossible, it just showsthat not both of ¨ A and ˙ ϕ can simultaneously be a δ -function. One way of overcomingthe problem is to demand that only one of ¨ A or ˙ ϕ becomes a δ -function in the extremethin-wall limit. This is what we do here, choosing that ˙ ϕ ∼ δ ( u − ¯ u ) in the extreme limit.The alternative analysis for ¨ A ∼ δ ( u − ¯ u ) can also be done and is shown in appendix C.One way of ensuring that ˙ ϕ permits an extreme thin-wall limit with ˙ ϕ ∼ δ ( u − ¯ u )is to propose a solution for ˙ ϕ in terms of a resolved δ -function. There are many ways ofresolving δ -functions , but inspired by Coleman’s analysis in [1, 44] we write˙ ϕ ( u ) = ∆4 (cid:96) f (cid:37) cosh (cid:0) ∆2 u − ¯ u(cid:96) f (cid:1) , (3.2)with ∆ a numerical constant and (cid:37) a free parameter to be ultimately fixed by the boundaryconditions. We also included (cid:96) f for dimensional reasons. The extreme thin-wall limitcorresponds to ∆ → + ∞ , for which ˙ ϕ → (cid:37) δ ( u − ¯ u ) as required. The choice of the symbol∆ here is no accident. So far we have used ∆ to refer to the scaling dimension of the operatorperturbing the CFT associated with an AdS extremum of V . As will be confirmed later,for ϕ f a minimum of V , the parameter ∆ introduced above will indeed correspond to thescaling dimension, consistent with our previous use of ∆. If ϕ f is a maximum of V , thisidentification does not always hold, as we shall explain later. This will not be a problem,as in this paper ϕ f always describes a minimum of V unless explicitly stated otherwise.Using (3.2), the equations of motion (2.7)–(2.9) and the boundary conditions (3.1) wecan construct the desired solutions for ϕ ( u ), A ( u ) and even the potential V ( ϕ ). Integrating(3.2) and implementing the boundary conditions for ϕ we find ϕ ( u ) = ¯ ϕ + (cid:37) (cid:18) ∆2 u − ¯ u(cid:96) f (cid:19) , with ¯ ϕ ≡ ϕ f + ϕ t , (cid:37) ≡ ϕ t − ϕ f , (3.3)i.e. the boundary conditions for ϕ fix the value of (cid:37) . Inserting this into (2.7) we canintegrate to obtain the expression for ˙ A . After implementing the boundary conditions in(3.1) one finds˙ A ( u ) = − (cid:96) f − ∆ (cid:37) d − (cid:96) f (cid:18) ∆2 u − ¯ u(cid:96) f (cid:19) + tanh (cid:16) ∆2 u − ¯ u(cid:96) f (cid:17) cosh (cid:16) ∆2 u − ¯ u(cid:96) f (cid:17) , (3.4)together with the relation 1 (cid:96) t = 1 (cid:96) f + ∆ (cid:37) d − (cid:96) f , (3.5) We can use any smooth interpolating function for this argument. – 37 –.e. out of the four parameters (cid:96) f , (cid:96) t , ∆ and (cid:37) only three can be chosen freely. Integratingonce more we finally obtain A ( u ) = A − u(cid:96) f − (cid:37) d − ∆ u(cid:96) f + 2 log cosh (cid:18) ∆2 u − ¯ u(cid:96) f (cid:19) −
12 cosh (cid:16) ∆2 u − ¯ u(cid:96) f (cid:17) , (3.6)with A an integration constant. We can evaluate the Kretschmann invariant which forthe ansatz (2.6) gives R µν ; ρσ R µν ; ρσ = 4 d ( ¨ A + ˙ A ) + 2 d ( d −
1) ˙ A = d ( d − (cid:16) d −
1) ˙ A − ˙ ϕ (cid:17) + 2 d ( d −
1) ˙ A , (3.7)where in the second step we have used (2.7). One can easily check that for the solutionsconsidered here, this is regular everywhere. Therefore, we have succeeded in finding analyticsolutions for ϕ ( u ) and A ( u ) for flat AdS domain-walls, that admit a thin-wall limit in thesense described above, which is reached for ∆ → + ∞ .Finally, using (2.8) (with R ( ζ ) set to zero) and inserting our solutions for ˙ ϕ and ˙ A from (3.2), (3.4) we can arrive at an expression for a potential V that admits this flatdomain-wall solution. To write the potential as a function of ϕ we invert (3.3) to eliminate u in favour of ϕ . To remove clutter, here we set ϕ f = 0 such that ¯ ϕ = ϕ t / (cid:37) = ϕ t .After some algebra one finds: V ( ϕ ) = − d ( d − (cid:96) f − ∆( d − ∆)2 (cid:96) f ϕ + ∆( d − (cid:96) f ϕ t ϕ − ∆ (cid:0) − d + dϕ t (cid:1) d − (cid:96) f ϕ t ϕ + d ∆ d − (cid:96) f ϕ t ϕ − d ∆ d − (cid:96) f ϕ t ϕ . (3.8)We also compute the corresponding solution W ( ϕ ) = − d −
1) ˙ A . This can obtained fromour expression for ˙ A in (3.4) and eliminating u in favour of ϕ by inverting (3.3). Settingagain ϕ f = 0 we find: W flat ( ϕ ) = 1 (cid:96) f (cid:20) d −
1) + ∆2 ϕ − ∆3 ϕ t ϕ (cid:21) , (3.9)where we added the subscript ‘flat’ to indicate that this describes a flat AdS domain-wall.The potential (3.8) has the following features: • It has two AdS extrema at ϕ f = 0 and ϕ t with V ( ϕ f ) > V ( ϕ t ). The extremum at ϕ t is always a minimum. The extremum at ϕ f = 0 is a maximum for ∆ < d and aminimum for ∆ > d . In the latter case there is an intermediate maximum at some ϕ ∈ [ ϕ f , ϕ t ]. • The potential has three parameters that we are free to control: (cid:96) f , ∆ and ϕ t . Theparameter (cid:96) f adjusts the value of the potential at ϕ f = 0. By dialing ∆ we can adjustthe height of the barrier between ϕ f and ϕ t . By choosing ϕ t we can set the separationin ϕ between the two extrema at ϕ f = 0 and ϕ t . Alternatively, an adjustment in ϕ t can also be used to set the value of the potential at that minimum through therelation (3.5). – 38 –he above solution also has an interpretation in terms of a holographic RG flow for afield theory on flat space(-time), as we briefly describe in the following. • The above solutions correspond to RG flows from a UV fixed point at the extremumof V at ϕ f = 0 to an IR fixed point at the extremum ϕ t . The identification of ϕ f with the UV fixed point follows from the fact that when ϕ f is reached as u → −∞ ,simultaneously one finds e A → ∞ , as expected for a UV fixed point. Similarly, when ϕ t is reached as u → + ∞ , one finds e A →
0, as required for an IR fixed point.One can also check that for u → −∞ the solution exhibits the expected asymptoticbehaviour near a UV fixed point, as detailed in e.g. [16]. • For ∆ > d/ V (cid:48)(cid:48) ( ϕ f ) via (2.17), as canbe checked explicitly. This implies that in this case, ∆ corresponds to the scalingdimension of the operator perturbing the UV CFT associated with the AdS extremumat ϕ f , consistent with our use of ∆ throughout this work. Note that this includesthe case where ϕ f is a minimum of V , i.e. ∆ > d , which is the situation of primaryinterest in this work. Comparing (3.9) with (2.24, 2.25) we also conclude that for∆ > d/ • For 0 < ∆ < d/ V and ∆ is not (2.17), but instead (note theminus sign) ∆ ≡ d (cid:32) − (cid:115) d − d V (cid:48)(cid:48) ( ϕ ext ) | V ( ϕ ext ) | (cid:33) . (3.10)Therefore in this case the solution corresponds to the ( − )-branch, i.e. the UV CFTis perturbed by the presence of a source for a relevant operator. The thin-wall limit of flat domain-wall solutions:
The solutions derived above havebeen designed to admit a thin-wall limit in the sense described at the beginning of thissection, which is reached for ∆ → + ∞ . In the following, we discuss which potentials V correspond to thin-walled flat domain walls. We can distinguish two cases:1. Consider taking ∆ → + ∞ with the parameters (cid:96) f and (cid:37) (or ϕ t for ϕ f = 0) fixed.The effect on V in (3.8) is that the height of the barrier separating the two minimadiverges, which is best checked numerically. However, at the same time the potentialdifference between the two minima V ( ϕ f ) − V ( ϕ t ) also diverges. This follows from(3.5), which implies that (cid:96) t → → + ∞ with (cid:96) f and (cid:37) fixed. Therefore, for finiteseparation ϕ t − ϕ f , the extreme thin-wall limit of a flat AdS domain-wall requiresa potential with diverging barrier and a diverging potential difference between theminima.2. Alternatively, we can also consider the extreme thin-wall limit with the potentialdifference between the two minima fixed. That is, while increasing ∆ we keep both (cid:96) f and (cid:96) t constant. The potential barrier between the minima again diverges for– 39 – → + ∞ . This can e.g. be seen analytically by evaluating V ( ϕ t / → + ∞ can be expanded as V ( ϕ t /
2) = 3( d − (cid:96) f − (cid:96) t (cid:96) f (cid:96) t ∆ + O (∆ ) . (3.11)From (3.5) it further follows that for fixed (cid:96) f and (cid:96) i , the separation (cid:37) between theminima decreases as ∆ is increased. That is, we can reach the extreme thin-wall limitwith a finite potential difference between the minima at the expense of the minimagetting infinitesimally close to one another in ϕ -space.Although we obtained the result using a specific interpolation, any other interpolationproduces qualitatively similar results. The thin-wall limit from a holographic viewpoint –
So far in this section ∆ hasbeen mainly treated as a free parameter. However, in the holographic interpretation of flatdomain-wall solutions as holographic RG flows, ∆ is the scaling dimension of the scalaroperator in the UV CFT. Therefore, there may be bounds on ∆ coming from consistencyconditions on the field theory side. The thin-wall limit looks particularly challenging fromthe dual field theory point of view. This corresponds to a diverging scaling dimension,∆ → ∞ , implying that the UV CFT is deformed by an operator that is de-facto infinitelyirrelevant, but even a finite but large value of ∆ may be problematic from the field-theorypoint of view. For example, results for 3d CFTs from the application of conformal bootstrapideas have shown that their lowest lying operator must have scaling dimensions ∆ (cid:46) (cid:46)
12 (for parity odd operators) [22]. Therefore, field theorieswith excessively large values for ∆ may not be physical and thus, the thin-wall limit maynever arise in practice. One way of avoiding this would be to construct solutions thatpermit a thin-wall limit that does not rely on ∆ → ∞ . We leave this as an interestingquestion for future work. O ( D ) -instantons In this section we argue that the analytic results for flat domain-walls can be used toderive approximate expressions for certain O ( D )-instantons. We begin by comparing O ( D )-symmetric solutions and flat domain-walls, which are both captured by the ansatz (2.6).The difference between O ( D )-symmetric solutions and flat domain-walls is in the geometryof the slices at fixed coordinate u : In the former case a fixed- u slice describes a d -spherewhile in the latter case it is d -dimensional Euclidean space. That is, for O ( D )-instantonsfixed- u slices are curved while for domain-walls they are flat.The physical radius of the d -sphere slices in the O ( D )-symmetric case is given by r = αe A ( u ) . Near the centre of the O ( D ) bubble, i.e. for r →
0, the corresponding d -sphere is highly curved and we hence expect the behaviour of a O ( D )-instanton to departsignificantly from that of a flat domain-wall solution. This is indeed the case: at some finitevalue u , one reaches the centre of the O ( D ) bubble, while a flat domain-wall continuesuntil u → + ∞ . In contrast, for large radii, r → ∞ , the curvature of the d -sphere slicesbecomes vanishingly small and consequently only affects the solution at subleading order.– 40 –t large radius the behaviour of a O ( D )-instanton is therefore well-approximated by acorresponding flat domain-wall. The picture that emerges is, that at sufficiently largeradius, any O ( D )-instanton behaves like a flat domain-wall, but departs from it more andmore as the radius is reduced.An interesting situation occurs if the wall of an O ( D )-instanton, i.e. the radial intervalin which the interpolation effectively happens, is located entirely in the region where the O ( D )-solution can still be well-approximated by a flat domain-wall. This is not just atheoretical exercise, as in section 4 we shall find numerical examples that exhibit this.Consider for example an O ( D )-instanton that is well-approximated deep into the interiorof the bubble by a flat domain wall solution of the type derived in section 3. In this case wecan use the analytic expressions from section 3 to describe the O ( D )-instanton includingthe wall. For such solutions we can then, for example, give an approximate expression forthe ‘thin-ness’ parameter η defined in (2.15), i.e. η flat = e A flat ( u out ) − e A flat ( u in ) e A flat (¯ u ) , (3.12)where the subscript ‘flat’ indicates that this is an approximate expression calculated usingthe exact solution for a flat domain-wall.We can give an explicit expression for η flat for the flat domain-wall solution derived inthe previous section. For u out and u in we use the implicit definition that we introduced in(2.14) for the O ( D )-symmetric case, but with ϕ replaced by ϕ t . Then, by employing theanalytic solution for ϕ ( u ) in (3.3) we can solve for u out and u in explicitly. Using this andinserting for A flat ( u ) with (3.6), after some algebra we arrive at η flat = 2 e − (cid:96)f /(cid:96)t − [ γ − − γ ) ] sinh (cid:18) (cid:96) f /(cid:96) t ∆ tanh − γ (cid:19) , (3.13)where we also used (3.5) to eliminate ϕ t . Recall that γ is the parameter introduced in (2.14)specifying the extent of the wall in ϕ -space. In all numerical examples we set γ = 0 . O ( D )-instantonsolutions in section 4.3. The reason that (3.13) will be useful there is that the potentialsused in this section, will correspond to deformations of the potential (3.8), for which theflat domain-wall solution was derived.
4. O(4)-instantons in a sextic potential
In this section we present a strategy for designing potentials that admit O ( D )-instantonsolutions.We shall then obtain numerical solutions for O ( D )-instantons in a family of potentialsconstructed using this method. We aim at answering the following questions: • How to construct potentials that admit O ( D )-instantons? • What kind of potentials admit thin-walled O ( D )-instantons? • What is the instanton action and hence the decay rate for the various solutions?– 41 – − V V − Vϕ f ϕ t ϕ f ϕ t ϕ Figure 6:
Left:
Potential V that admits a flat domain-wall solution interpolating betweenthe two minima at ϕ f and ϕ t , indicated by the orange arrow. In the mechanical picture inthe inverted potential − V this corresponds to a trajectory for a particle that is releasedfrom rest at ϕ t and coming to rest again at ϕ f (red trajectory). Right:
Potential V thatdiffers from V only by exhibiting a lower barrier separating the two minima at ϕ f and ϕ t .In turn the inverted potential − V exhibits a shallower valley separating ϕ f and ϕ t than − V . A particle released from ϕ t in − V will overshoot ϕ f as the shallower valley leadsto a lower velocity and hence less friction. By reducing the initial potential energy andreleasing the particle lower down the slope at some ϕ , a trajectory can be found so thatthe particle neither over- nor undershoots, but comes to rest exactly at ϕ f (red trajectory).In the Euclidean picture, this corresponds to an O ( D )-instanton describing tunnelling from ϕ f to ϕ (orange arrow).A challenge for constructing O ( D )-instanton solutions is that they do not exist for genericpotentials as shown in section 2.6. Therefore, to find O ( D )-instanton solutions one firstneeds to identify potentials that do admit them. Interestingly, this can be done by startingwith a potential admitting a flat domain wall solution, and then deforming it, as we arguenext. O ( D ) -instantons Here we present a strategy for constructing potentials that admit O ( D )-instanton solutionsfor tunnelling from an AdS minimum. An important instrument in the construction of theargument will be the mechanical analogue to Euclidean tunnelling/domain-wall solutionsalready introduced in section 2.6. This is the observation that the equation of motion(2.9) is identical to that for a particle of unit mass performing damped motion in theinverted potential. In section 2.6 we also identified a sufficient condition for a potential withtwo non-degenerate minima ϕ f and ϕ t to admit O ( D )-tunnelling solutions. In particular,an O ( D )-instanton describing tunnelling from the false vacuum ϕ f will exist, if in the– 42 –echanical picture, the particle overshoots ϕ f if released from rest at ϕ t in the invertedpotential. In the following, we describe how a potential satisfying this condition can beconstructed. An equivalent statement is that the potential admits flat skipping flows.We begin with a potential V ( ϕ ) with a false vacuum at ϕ f , a true vacuum at ϕ t , and admitting a flat domain-wall solution interpolating between the two . This is a convenientstarting point, as such a potential V can be easily reverse-engineered from a given domain-wall solution. The idea is to propose some interpolating solution for ϕ ( u ), solve for theassociated geometry and then use the equations of motion to deduce the correspondingpotential. This is for example what we have done in section 3.1, arriving at a sexticpotential admitting flat domain-walls.The inverted potential − V has two maxima at ϕ f and ϕ t separated by a trough.Both V and − V are depicted on the left of fig. 6. The flat domain-wall solution has thefollowing interpretation on the mechanical side. It implies that if the particle is releasedfrom rest at ϕ t , it will roll through the valley and come to rest again exactly at ϕ f . Thisis indicated by the red trajectory on the left in fig. 6. The shape of the potential V is justright so that there is just enough friction to make the particle come to rest at ϕ f , avoidingan overshoot or an undershoot.Now consider a potential V , that possesses the same minima as V , but exhibits asmaller barrier separating ϕ f and ϕ t . That is, to obtain V we deform V by just loweringthe barrier, as indicated by the purple arrow on the bottom right in fig. 6. In the invertedpotential − V , shown on the top right in fig. 6, the lowered barrier of V manifests itselfas a shallower trough between the two maxima. Consider once more releasing the particlefrom rest at ϕ t , but now in the inverted potential − V . As the trough separating the twomaxima is now shallower, the particle will reach a lower terminal velocity at the bottomcompared to its trajectory in − V . As the friction term in the equation of motion for theparticle, (2.53), is proportional to the velocity ˙ ϕ , we expect that there will not be sufficientfriction to stop the particle before arriving at ϕ f . As a result, we expect that the particlewill overshoot ϕ f , therefore realising the condition for the existence of an O ( D )-instantonsolution.Then, following the arguments laid out in section 2.6, the particle can be made tocome to rest at ϕ f , if it is released from some value ϕ = ϕ further down the the invertedpotential, as shown by the red trajectory on the right of fig. 6. On the Euclidean side thiswill correspond to an O ( D )-instanton solution describing tunnelling from ϕ f to ϕ .What we observe is that for all example potentials V constructed according to theinstruction above, we always succeeded in finding an O ( D )-instanton solution. We take thisas evidence that the strategy presented above is indeed a practical method for constructingpotentials V that will admit O ( D )-instantons describing tunnelling from ϕ f .Interestingly, the above arguments can also be used to construct potentials with afalse minimum at ϕ f , that we expect to be stable against tunnelling via O ( D )-instantons.Consider again a potential V admitting a flat domain-wall, but now we deform it byincreasing the barrier between ϕ f and ϕ t , leaving the minima untouched. The invertedpotential now has a deeper trough between the maxima, leading to a higher terminalvelocity at the bottom and hence increased friction. This will cause all possible trajectories– 43 – .2 0.4 0.6 0.8 1.0 (cid:45) (cid:45) (cid:45) ϕϕ f ϕ t (cid:96) f V ( ϕ )0 (cid:96) f V ( ϕ ) v = 1 v = 2 v = 5 v = 10 v = 20Figure 7: Potential V ( ϕ ) in (4.2) in units of (cid:96) − f with ∆ = 30 . ϕ t = 1 for differentvalues of v . Note that the barrier separating the two AdS minima exhibits a region with V > v = 20.permitted by (2.9) in the inverted potential to undershoot, and no starting point can befound so that the particle comes to rest at ϕ f . In the Euclidean setting, this implies thatthere should not exist any O ( D )-symmetric interpolating solutions starting at ϕ f .However, we cannot exclude that ϕ f may still be unstable with respect to instantonswith less symmetry, which exhibit more complicated equations of motions, and where oursimple mechanical analogue need not apply. In this section we set D = d + 1 = 4 throughout and present numerical O (4)-instantonsolutions for tunnelling from an AdS minimum. We begin by specifying the types ofpotentials V that will be employed here and which are constructed following the strategyin sec. 4.1. In the following, we again set ϕ f = 0.The potentials V , are obtained as deformations of a potential V exhibiting a flatdomain-wall solution. To be explicit, for V we use the sextic potential derived in sec. 3.1,admitting a flat domain-wall with tanh-profile for ϕ ( u ), and which we reproduce here for d = 3 for convenience: V ( ϕ ) = − (cid:96) f − ∆(3 − ∆)2 (cid:96) f ϕ + ∆(1 − ∆) (cid:96) f ϕ t ϕ + ∆ (cid:0) − ϕ t (cid:1) (cid:96) f ϕ t ϕ + ∆ (cid:96) f ϕ t ϕ − ∆ (cid:96) f ϕ t ϕ (4.1)This has two AdS minima at ϕ f = 0 and ϕ t separated by a barrier. According to thestrategy in sec. 4.1, we can arrive at a potential V admitting O (4)-instanton solutions bydefining V ( ϕ ) = V ( ϕ ) + v ( ϕ ) , (4.2)– 44 – = 30 . , ϕ t = 1 , η flat = 0 . v ϕ R η B/ ( M (cid:96) f ) O (4)-instanton solutions in a family of potentials (4.2) with∆ = 30 . ϕ f = 0, ϕ t = 1. Data for five potentials is shown that differ in their value of v .For the O (4)-instantons the tunnelling end point ϕ , the dimensionless boundary curvature R from (2.30), the thin-ness parameter η and the instanton action B are recorded.with v ( ϕ ) a contribution that will lower the barrier while leaving the minima unaffected.To be specific, here we choose v ( x ) = − v (cid:96) f ϕ ( ϕ t − ϕ ) ϕ t , (4.3)with v constant. Having a holographic interpretation of our solutions in mind, animportant quantity is the curvature of the potential at the minima, setting the dimensionsof the operators deforming the associated CFTs. By having chosen v ( x ) to only give cubicand higher contributions when expanded about ϕ f = 0 and ϕ t , the operator dimensionsare determined by V alone. Using (2.17) it is easy to verify that the dimension of theoperator perturbing the CFT at ϕ f = 0, i.e. the UV CFT, is given by ∆.For illustration, in fig. 7 we plot a family of potentials given by (4.2) obtained from apotential V with ∆ = 30 . ϕ t = 1 for different choices of v . Note, that the barrierseparating the AdS minima, typically exhibits a region with V > In the following, we derive numerically O (4)-instanton solutions for various potentials V ( ϕ ) given by (4.2). In practice this is done by solving the relevant equations from a tenta-tive tunnelling end point ϕ , shooting towards ϕ f . If the trial solution over- or undershoots ϕ f , the point ϕ is adjusted accordingly. This is to be iterated until a solution is found thatapproximates the desired solution coming to rest at ϕ f up to the desired accuracy. For agiven example solution, we then record the value of ϕ , extract the dimensionless curvature R given in (2.30), compute the thin-ness parameter η defined in (2.15) and calculate theinstanton action B given in (2.33). The factor 64 ϕ − t is chosen for convenience such that v ( ϕ t /
2) = − v /(cid:96) f , which is the approximatevalue by which the barrier is lowered. In this work we also discuss the interpretation of O ( D )-instantons as holographic RG flows. It maytherefore seem worrying at first, that the potential exhibits a region with V >
0, raising questions aboutthe applicability of holography ´a la AdS/CFT in this context. This worry is unfounded as the holographicRG flow solutions obtained here will always interpolate between regions of the potential with
V < – 45 – (cid:45) (cid:45) (cid:45) ϕ ( u ) u/(cid:96) f (a) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) A ( u ) u/(cid:96) f (b) Figure 8: ϕ ( u ) (a) and A ( u ) (b) for an O (4)-instanton solution ( red ) and a flat domainwall solution ( blue, dashed ). The O (4)-instanton is obtained for the potential V ( ϕ ) givenin (4.2) with ∆ = 30 . ϕ t = 1 and v = 1 and was found to have dimensionless curvature R = 0 .
36. The flat domain wall solution arises from the potential V ( ϕ ) given in (4.1)with the same values of ∆ and ϕ t . The integration constants in A ( u ) were adjusted suchthat for u → −∞ , the functions A ( u ) asymptote to − u/(cid:96) f , which is denoted by the brownline. The vertical dot-dashed lines demarcate the wall, defined as the interval [ u out , u in ]with ϕ ( u out ) = 0 . ϕ and ϕ ( u in ) = 0 . ϕ , consistent with γ = 0 .
76 in (2.14). The dottedvertical line indicates ¯ u , defined as the locus ϕ (¯ u ) = 0 . ϕ . Note that the O (4)-instantonsolution for A ( u ) is effectively indistinguishable from the solution A flat ( u ) at the locus ofthe wall.For the one-parameter family of potentials with ∆ = 30 . ϕ t = 1 plotted in fig. 7the numerical results are shown in table 1. We make the following observations: • The lower the barrier between the minima (i.e. the larger v ), the further the point ϕ is removed from the true vacuum at ϕ t = 1. This can be understood usingthe mechanical analogue introduced in section 2.6: the lower the barrier, the weakerfriction becomes and hence ϕ needs to be further away from ϕ t to avoid overshootingbeyond ϕ f . • We further observe that the closer ϕ to ϕ t , the smaller R and η and the larger B/ ( M (cid:96) f ) , i.e. schematically: ϕ → ϕ t : R ↓ , B/ ( M (cid:96) f ) ↑ . The trends for R and B can be understood intuitively. The flat domain-wall solutioncan be interpreted as the R → O (4)-instanton with B → ∞ as conse-quence of its stability. The observations above are then consistent with the fact that O (4)-instantons asymptote towards a flat domain-wall for ϕ → ϕ t . The quantities η and η flat depend on the parameter γ trough the definition of u in and u out in (2.14).Here and throughout this work we choose γ = 0 . This is a relative observation. Note that in absolute terms ϕ is very close to the true vacuum ϕ t = 1in all cases displayed in table 1. – 46 – (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) ϕ ( u ) u/(cid:96) f (a) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) A ( u ) u/(cid:96) f (b) Figure 9: ϕ ( u ) (a) and A ( u ) (b) for an O (4)-instanton solution ( red ) and a flat domainwall solution ( blue, dashed ). The O (4)-instanton is obtained for the potential V ( ϕ ) givenin (4.2) with ∆ = 30 . ϕ t = 1 and v = 20 and was found to have dimensionless curvature R = 8 .
3. The flat domain wall solution arises from the potential V ( ϕ ) given in (4.1) withthe same values of ∆ and ϕ t . The integration constants in A ( u ) were adjusted such thatfor u → −∞ the functions A ( u ) asymptote to − u/(cid:96) f , which is denoted by the brown line.The vertical dot-dashed lines demarcate the wall, defined as the interval [ u out , u in ] with ϕ ( u out ) = 0 . ϕ and ϕ ( u in ) = 0 . ϕ , consistent with γ = 0 .
76 in (2.14). The dottedvertical line indicates ¯ u , defined as the locus ϕ (¯ u ) = 0 . ϕ . Note that the O (4)-instantonsolution for A ( u ) differs significantly from the solution A flat ( u ) at the locus of the wall. • Another observation is that O (4)-instantons with thicker walls have a lower instantonaction. Therefore, a false vacuum permitting an O (4)-instanton with a thicker wallis more prone to decay than a false vacuum permitting an O (4)-instanton with athinner wall.We find it instructive to also display the solutions for ϕ ( u ) and A ( u ) for a few examplecases. Hence, in fig. 8 we display ϕ ( u ) and A ( u ) for the O (4)-instanton solution from table1 with v = 1. These are given by the red curves in fig. 8 while in blue we overlay ϕ ( u ) and A ( u ) for the flat domain-wall solution solution in the potential V with the same values of ∆and ϕ t . The vertical dot-dashed lines define the inner and outer limits of the wall, definedvia (2.14) with γ = 0 .
76. The vertical dotted line indicates the centre of the wall where ϕ has interpolated half way between ϕ f and ϕ . The brown line in fig. 8b corresponds to − u/(cid:96) f , to which A ( u ) asymptotes to for u → −∞ .The main observation here is that the flat domain-wall solution effectively coincideswith the O (4)-symmetric solution well into the interior of the O (4)-bubble: Firstly, the twosolutions for ϕ ( u ) are near-indistinguishable throughout; secondly, the solutions for A ( u )only start departing from one another significantly for u (cid:38) − .
6. This example thereforeexplicitly realises the scenario conjectured in section 3.2, i.e. the case of a O (4)-instantonthat can be well-approximated by a corresponding flat domain-wall solution.In fig. 9, the red curves show ϕ ( u ) and A ( u ) for the O (4)-instanton solution from table 1– 47 – = 50 . , ϕ t = 1 , η flat = 0 . v ϕ R η B/ ( M (cid:96) f ) . , ϕ t = 0 . , η flat = 0 . v ϕ R η B/ ( M (cid:96) f ) O (4)-instanton solutions in two families of potentials (4.2)with ∆ = 50 . ϕ f = 0, ϕ t = 1 and ∆ = 30 . ϕ f = 0, ϕ t = 0 .
25, respectively. Data forpotentials differing in their value for v is shown. For the O (4)-instantons the tunnellingend point ϕ , the dimensionless boundary curvature R from (2.30), the thin-ness parameter η and the instanton action B are recorded.with v = 20. The blue curves again denote ϕ ( u ) and A ( u ) for the flat domain-wall solutionsolution in the potential V . Here, the flat domain-wall solution already departs from the O (4)-instanton solution well outside the wall and is hence not a good approximation forthe O (4)-instanton. This is not surprising, as for v = 20 the potential V is significantlyaltered compared to V as can be seen in fig. 7.We now return to the results in table 1, focusing on how the thin-ness parameter η varies across this family of solutions. Note that η decreases as v is lowered. More precisely,as v is decreased the thin-ness parameter η asymptotes towards the value η flat , whichappears to set the lower bound for η in this family of potentials. This can be understoodwith the help of the observations made from the plots in figs. 8 and 9. There we observedthat if v is decreased, the O ( D )-instanton solutions is increasingly well-approximated bythe flat domain-wall solution that exists in the potential V . As a result, the thin-nessparameter η approaches the value η flat obtained for the flat domain-wall solution.In table 2 we collect data for two additional one-parameter families of potentials cor-responding to the parameter choices (∆ = 50 . , ϕ t = 1) and (∆ = 30 . , ϕ t = 0 . v is increased are also reproduced bythese additional examples. – 48 – ��� (cid:1) �� η flat1∆ (cid:96) t /(cid:96) f Figure 10: Contours of constant η flat as given in (3.13) on the ( (cid:96) t /(cid:96) f )-1 / ∆-plane for γ =0 .
76. The locus η flat → → ∞ with ∆ (cid:96) t /(cid:96) f → ∞ . O (4) -instantons Thin-walled O (4)-instanton solutions have played an important role in the study of tun-nelling processes. The reason is the existence of the ‘thin-wall approximation’ of Colemanand de Luccia [2], which allows for an analytic computation of relevant quantities such asthe decay rate. In this section we explore thin-walled O (4)-instanton solutions without re-lying on the approximation scheme of Coleman and de Luccia. Instead, here we use resultsfrom the previous section to develop an analytic understanding of the thin-wall limit. Inparticular, we shall be interested in the requirements on potentials to admit thin-walled O (4)-instantons. We then compare our results with the findings of Coleman and de Luccia.The results here-obtained will be derived for potentials V ( ϕ ) that fall into the categorydefined in (4.2), i.e. potentials based on the sextic potential V ( ϕ ) in (4.1) with the barrierseparating the two minima lowered by an amount − v /(cid:96) f . At the end of this section wethen argue how we expect our findings to generalise beyond this class of potentials. In fact,we explain why we expect our main results to be robust and reveal universal properties ofthin-walled instanton solutions.How can we then construct a thin-walled O (4)-instanton in the family of potentials(4.2)? First, consider a subfamily of potentials of type (4.2) with a fixed choice for V ( ϕ ) butdifferent values of v . As argued before, every such potential will admit an O ( D )-instantonand we can calculate the corresponding ‘thin-ness’ parameter η defined in (2.15). From theresults in section 4.2, recall that for all O ( D )-instantons in this subfamily of potentials,the parameter η flat given in (3.13) represents the lower bound for η , i.e. η > η flat . Moreprecisely, by letting v → + we found that η → η +flat . That is, we can construct an O ( D )-instanton with a value of η arbitrarily close to η flat by choosing v sufficiently small.It hence follows that to construct an O ( D )-instanton solution with a small value of η we must consider subfamilies with sufficiently small η flat in the first place. That is, η canbe made small to the extent that we can make η flat small. Recall that the value of η flat is unique for a given choice of V ( ϕ ). – 49 –ere we shall be interested in the extreme thin-wall limit η →
0. From the above weconclude that such solutions can only occur in families of potentials that exhibit η flat → η flat as a function of the two dimensionless parameters∆ and (cid:96) f /(cid:96) t . Note that (cid:96) f /(cid:96) t > V ( ϕ f ) > V ( ϕ t ). Themain observation then is that η flat → only be obtained for ∆ → ∞ , that is η flat → η flat → (cid:96) f /(cid:96) t , i.e. ∆ (cid:96) t (cid:96) f → ∞ . These results are summarised infig. 10 where we plot contours of constant η flat on the ( (cid:96) t /(cid:96) f )-1 / ∆-plane.We are now in a position to describe the corresponding potentials V that permit O ( D )-instantons in the extreme thin-wall limit η →
0. We can proceed without specifying v in (4.3) as we find that v will only enter the subsequent expressions at subleading orderfor ∆ → ∞ . Here we can work for general D = d + 1 so instead of using the expression(4.1) for V ( ϕ ) valid for d = 3 we revert to the general expression in (3.8). We make thefollowing observations: • In the limit η flat → (cid:37) → ϕ t → ϕ f . This follows from (3.5) and the requirement that for η flat → → ∞ and ∆ (cid:96) t (cid:96) f → ∞ . Contrast this with flat domain walls, where therewere two ways of obtaining a thin flat domain wall, summarised in the enumeratedlist close to the end of section 3.1. In case (1) (cid:37) remains finite while in case (2) (cid:37) → O ( D )-instantons there is only one way, but it coincides with case (2) of the flatdomain wall analysis. • The barrier separating the two minima becomes infinitely tall compared to the poten-tial separation V ( ϕ f ) − V ( ϕ t ) between false and true minimum. We can e.g. compute(again using (3.5) to substitute for ϕ t ): V (cid:0) ϕ f + ϕ t (cid:1) − V ( ϕ f ) V ( ϕ f ) − V ( ϕ t ) = ∆ →∞ , ∆ (cid:96)t(cid:96)f →∞ (cid:96) t ∆8 d ( (cid:96) f + (cid:96) t ) + O (∆ ) , (4.4)which diverges in the relevant limits ∆ → ∞ and ∆ (cid:96) t (cid:96) f → ∞ as postulated. • Another observation is that in the thin-wall limit the dimensions ∆( ϕ f ) and ∆( ϕ t )of the operators perturbing the false vacuum and true vacuum CFTs diverge. Bydefinition, what we refer to as ∆ is the operator dimension for the false vacuum CFT.For the true vacuum CFT the operator dimension can be calculated from (2.17) sothat we find. ∆( ϕ f ) = ∆ , ∆( ϕ t ) = ∆ (cid:96) t (cid:96) f (cid:16) − (cid:96) t (cid:96) f (cid:17) . (4.5)These indeed diverge in the thin-wall limit ∆ → ∞ and ∆ (cid:96) t (cid:96) f → ∞ . The expression (3.13) also contains the parameter γ , whose value determines which part of the inter-polating solution we define as the wall. Its choice, while to some extent a matter of personal taste, is madeonce and for all and hence γ is not an adjustable parameter. – 50 – omparison with Coleman and de Luccia: Our results from the family of sexticpotentials (4.2) imply that thin-walled O ( D )-instantons exist in potentials that exhibit alarge barrier separating false and true minima compared to the potential difference betweenthe minima. How does this compare to criteria for applicability of the thin-wall approxi-mation as laid out by Coleman and de Luccia? In [2] Coleman and de Luccia write thatthe thin-wall approximation applies “in the limit of a small energy difference between thetwo vacuums.” As the energy difference is dimensionful, it can only be small compared toanother quantity with the same dimensions. The only other appropriate quantity in [2] isthe barrier height and hence our conditions agree. We also provide a more quantitativecomparison in appendix D. However, note that our analysis implies that the criterion of[2] is not sufficient for an instanton solution to exist. As we have argued in section 2.6, O ( D )-instantons do not exist in generic potentials and this also holds for thin-walled ones.What we find is that if a potential admits a thin-walled O ( D )-instanton, the potentialswill be of the type required by [2], but the converse is not generically true. Thin-walled O ( D ) -instantons in more general potentials: In the class of potentialsin (4.2), instanton solutions are thin-walled to the extent that ∆ is chosen large. Here wedescribe to what extent we expect this finding to be valid beyond this class of potentials.An instanton solution is thin-walled if the interpolation in ϕ effectively happens over a smallradial interval r out − r in when compared to the radius of the bubble ¯ r . We observe that inthis case the interpolation is also ‘rapid’ in terms of the coordinate u , i.e. the interpolationbetween ϕ f and ϕ occurs in a smaller u -interval the thinner-walled the solution, comparee.g. the solutions in figs. 8a (thinner wall) and 9a (thicker wall).In the following, it will be useful to invoke the mechanical picture of tunnelling interms of a particle crossing a well, introduced at the beginning in section 2.6, with t = − u the time variable. In this formulation, a thin-walled solution corresponds to the particlestaying at rest at ϕ for an extended period of time, before experiencing a short period ofstrong acceleration followed by short period of strong deceleration, before coming to restagain at ϕ f . An important observation is that the initial acceleration from rest and finaldeceleration to come to rest again are effectively controlled by V (cid:48) , which follows from theequation of motion (2.9) or, equivalently, (2.53). For the particle to stay effectively at restat ϕ for an extended period of time, one thus requires V (cid:48) ( ϕ ) ≈
0, which implies that ϕ needs to be close to the extremum at ϕ t . That is, thinner-walled instantons deposit ϕ closer to the true minimum, which is indeed what we observe for the examples in tables 1and 2. Now, to achieve strong initial acceleration and final deceleration, we require that | V (cid:48) | becomes large immediately once the particle departs from ϕ or approaches ϕ f . Bydefinition, a change in V (cid:48) due to a step in ϕ is quantified by the curvature V (cid:48)(cid:48) of thepotential. Thus, a strong initial acceleration and strong final deceleration is achieved if | V (cid:48)(cid:48) ( ϕ ) | ≈ | V (cid:48)(cid:48) ( ϕ t ) | and | V (cid:48)(cid:48) ( ϕ f ) | are ‘large’ (in appropriate units). Using (2.17), this canbe translated into the requirement ∆( ϕ f ) (cid:29) ϕ t ) (cid:29)
1, i.e. the dimensions of theoperators perturbing both the false vacuum and true vacuum CFTs need to be large. This Other possible dimensionful quantities for comparison are the absolute values of the potential at theminima. However in [2] either V ( ϕ f ) = 0 or V ( ϕ t ) = 0 so that these values are either zero or given by thepotential difference itself. – 51 –s indeed what we observed when analysing the thin-wall limit of instantons in the familyof sextic potentials (4.2).Identifying thin-walled instantons as solutions with a period with strong accelerationfollows by a period strong deceleration, we can define a general condition on potentials toadmit such solutions. As stated above, we require | V (cid:48) | to grow quickly once the particledeparts from ϕ or approaches ϕ f . The most generic way of achieving this is to have | V (cid:48)(cid:48) ( ϕ f ) | and | V (cid:48) ( ϕ t ) (cid:48) | large or, equivalently, ∆( ϕ f ) (cid:29) ϕ t ) (cid:29)
1. One could inprinciple also imagine the case where | V (cid:48)(cid:48) ( ϕ f ) | and | V (cid:48) ( ϕ t ) (cid:48) | are small but some higherderivative is extremely large at and close to ϕ f and ϕ t . This would eventually also leadto a strong acceleration/deceleration, but this is an even more artificial and tuned choicethan just having ∆( ϕ f ) (cid:29) ϕ t ) (cid:29) The thin-wall limit from a holographic point of view –
As already discussed at theend of section 3.1 one may ask to what extent ∆ can be taken large. The reason is thatfor the AdS-to-AdS-tunnelling considered here ∆ also has an interpretation as the scalingdimension deforming the UV CFT associated with the false vacuum. Results from theconformal bootstrap program applied to 3d CFTs show that the scaling dimension of thelowest-lying scalar operator is bounded in consistent theories [22]. This in turn would implythat consistency of the dual field theory sets a limit on how thin-walled O ( D )-instantonscan be. One way of avoiding this would be to construct solutions that permit a thin-walllimit that does not rely on ∆ → ∞ . We leave this as an interesting question for futurework. Here we consider O ( D )-instanton solutions describing tunnelling from ϕ f , where ϕ f isnow an AdS maximum of V . There is one important difference when ϕ f is a maximumcompared to a minimum: Following the discussion in sec. 2.2, for ϕ f describing a maximumthere exist two types of O ( D )-symmetric solutions with ϕ ( u → −∞ ) = ϕ f , which are the( − )-branch solutions and the (+)-branch solutions, while for ϕ f describing a minimum theonly possible solutions are (+)-branch solutions. However, as discussed before in sec. 2.4only the (+)-branch solutions permit an interpretation as instantons.In sec. 2.6 we employed the fact that (+)-branch solutions only have one free parameter R near ϕ f to argue that (+)-branch solutions do not exist in generic potentials. Whilethere it was assumed that ϕ f is a minimum of V , nothing in the argument based on thenumber of free parameters hinged on this fact, and hence the argument also holds for ϕ f describing a maximum of V . Hence, for a generic potential V with an AdS maximum at ϕ f , one does not expect O ( D )-instantons describing tunnelling from ϕ f to generically exist.Therefore, as for the case of tunnelling from an AdS minimum, tunnelling from anAdS maximum only occurs in potentials that are sufficiently ‘tuned’. Here we could oncemore consider the sextic potential in (4.2), which allows for O ( D )-instantons describingtunnelling from the UV fixed point at the maximum between ϕ f and ϕ t . However, to alsoshow examples that do not rely on the potential (4.2), in the following we will considera family of potentials hat can be written as a degree-12 polynomial, and that has been– 52 – ϕ ϕ ϕ V ( ϕ ) ϕ ϕ Figure 11: Cartoon of a degree-12 polynomial potential V ( ϕ ) used in sec. 4.4. The explicitexpression for V ( ϕ ) is given in [16]. The potential admits an O ( D )-instanton solutionsdescribing tunnelling from the AdS maximum at ϕ (black arrow), corresponding to aholographic RG flow of (+)-type. At the same time it admits non-bouncing (orange arrow)and bouncing (green arrow) RG flow solutions of ( − ) type from ϕ . The end point ϕ sits at the interface of possible end points for non-bouncing (orange region) and bouncingsolutions (green region). ϕ ϕ ϕ ϕ ∆( ϕ ) ϕ R η B/ ( M (cid:96) f ) ϕ , ϕ , ϕ , ϕ , ∆( ϕ )) used to construct a degree-12 polynomial (explicit expression in [16]) admitting O (4)-instanton solutions describingtunnelling from a maximum at ϕ . The values of the tunnelling end point ϕ , the dimen-sionless boundary curvature R from (2.30), the thin-ness parameter η from (2.15) and theinstanton action B for the respective O (4)-instanton solutions are also recorded.previously analysed in [16] in the context of exotic RG flows. O ( D )-symmetric solutions of(+)-type were observed to exist in this family in [13], including such flows emanating froma maximum. For the explicit expression for V we refer readers to [16] and instead showa cartoon in fig. 11. The potential can be defined by specifying values for ( ϕ , ϕ , ϕ , ϕ ),which describe the loci of four extrema of V . Without loss of generality one furtherextremum is fixed at ϕ = 0. In addition, there is the freedom to pick the value of ∆ These are the flows ending at ϕ ∗ in Figure 30 of [13]. – 53 –t one extremum of choice. We once more set D = d + 1 = 4.In table 3 we record four choices of values for ( ϕ , ϕ , ϕ , ϕ , ∆( ϕ )) for which thepotential permits O (4)-solutions of (+)-type from a maximum at ϕ . That is, in thisexample ϕ plays the role of ϕ f . Here ∆( ϕ ) corresponds to the value of ∆ computedfrom (2.17) at the maximum ϕ . In table 3 we further record the value of R for the O ( D )-solution, the thin-ness parameter η as well as the value of the instanton action B .Note that for all examples in table 3 we have η >
1. These are thick-walled solutionswith the interpolation between ϕ f and ϕ happening gradually. This is consistent with ourfindings of sec. 4.3: There, we observed that in this class of potentials, thin-walled solutionsrequire ∆ → ∞ . However, for ϕ f to be a maximum of V , the value of ∆ is bounded as∆ < d . Therefore, we expect that tunnelling solutions from an AdS maximum can neverbe of the thin-walled type in this class of potentials. This is supported by the examples intable 3.Interestingly, we observe that potentials that permit tunnelling solutions from an AdSmaximum also permit so called ‘bouncing’ flows from the same AdS maximum. For ex-ample, this is the case for the degree-12 polynomial potentials considered here and studiedin [13]. This can be understood as follows. The tunnelling solutions corresponds to aholographic RG flow on the (+)-branch, i.e. a flow with vanishing value j = 0 for the UVsource. Since here the UV fixed point is an AdS maximum, in addition to the (+)-branchsolution it will also permit a family ( − )-branch solutions for any value j (cid:54) = 0 of the UVsource. Now consider such a ( − )-branch solution with j = (cid:15) with (cid:15) arbitrarily small butnon-zero. By continuity, the corresponding ( − )-branch solution will only depart slightlyfrom the (+)-branch solution with j = 0. In particular, by choosing (cid:15) sufficiently smallthe end point of the ( − )-branch flow can be made arbitrarily close to ϕ , the end point ofthe (+)-branch solution. Alternatively, we can consider a solution with j = − (cid:15) . Again, bychoosing (cid:15) sufficiently small we can ensure that the end point of this ( − )-branch solutionis sufficiently close to ϕ . Now note that for the ( − )-branch solution with j = (cid:15) the flowdeparts the UV fixed point towards the right (larger values of ϕ ), while for j = (cid:15) the flowdeparts the UV fixed point towards the left (smaller values of ϕ ). For these two flows toboth end in the vicinity of ϕ at least one of the flows needs to reverse direction (and alsopass the UV fixed point one more time), therefore constituting an exotic flow of bouncingtype. Note that in general, this will be a bouncing flow for a QFT defined on S d . Thuswe conclude that solutions describing tunnelling generically correspond to the interface insolution space between bouncing and non-bouncing holographic RG flows, see again fig. 11for illustration. Another way of phrasing our finding is that the existence of bouncing flowsfor theories on S d is in one-to-one correspondence with the existence of an O ( D )-instantondescribing tunnelling from a maximum.
5. The Lorentzian continuation
In this section we will consider the Lorentzian geometry, as a solution to the bulk equationsof motion, subject to the initial conditions specified by the instanton solution.– 54 – .1 Lorentzian solutions inside and outside of the bubble
To describe the Lorentzian space-time after a tunnelling event mediated via an O ( D )-instanton, we have to analytically continue the Euclidean O ( D )-symmetric tunnelling solu-tion to Lorentzian signature. How to do this has already been explained in the foundationalwork [2]. To briefly review this, here we find the conventions in [45] useful.For an O ( D )-instanton the metric can be written as in (2.3) with ζ µν a metric on thesphere S d . In the following it will be useful to specify coordinates on S d to write (2.3) as: ds = dξ + ρ ( ξ ) α (cid:0) dθ + sin ( θ ) d Ω d − (cid:1) = dξ + r ( ξ ) (cid:0) dθ + sin ( θ ) d Ω d − (cid:1) , (5.1)where in the second step we introduced r ( ξ ) = αρ ( ξ ) corresponding to the physical radiusof a shell at coordinate locus ξ .To describe the complete Lorentzian space-time after the tunnelling event two analyticcontinuations will be required. The reason is the following. Consider the center of the CdLbubble at r (0) = 0 as the origin of the Lorentzian coordinate system. As we later show, thefirst analytic continuation will allow us to only access points that are space-like separatedfrom the origin. This is also referred to as the region ‘outside the bubble’. To access thetime-like separated region we shall require a different analytic continuation. To find thespace-time in this region, denoted as ‘inside the bubble’, we need to solve the correspondingequations of motion. The Euclidean solution only provides the initial conditions in thiscase.To describe the region ‘outside the bubble’ the angle θ on S d is continued as [45]: θ = π iχ . (5.2)Substituting this into (5.1) the metric becomes ds = dξ + r ( ξ ) (cid:2) − dχ + cosh ( χ ) d Ω d − (cid:3) . (5.3)The expression multiplying r ( ξ ) can be identified as a metric of d -dimensional de Sitterspace with unit radius. One can check that for this continuation, the equations of motionfor the metric and scalar field are unchanged compared to the Euclidean case, i.e. they arestill given by (2.7)–(2.9). As the equations of motion are unchanged, this implies that theEuclidean solution itself can be continued into this region. That is, the expression r ( ξ ) in(5.3) and also ϕ ( ξ ) are just given by the solutions obtained for the Euclidean instanton.As one can easily confirm, continuing the Euclidean solution results in the entire bubbleincluding wall and bubble exterior being mapped into the region described by (5.3).To access the region ‘inside the bubble’ we instead continue as [45]: ξ = iτ, θ = iη , (5.4)and define a ( τ ) = − ir ( iτ ) (5.5)– 55 –nserting this into (5.1) we find: ds = − dτ + a ( τ ) (cid:0) dη + sinh η d Ω d − (cid:1) , (5.6)which describes a FRW universe with hyperbolic slicing with scale factor a ( τ ). In Euclideansignature, the dilaton ϕ = ϕ ( ξ ) was just a function of ξ . In this Lorentzian continuationwe instead have ϕ = ϕ ( τ ). The equations of motion are now given by2( d −
1) ¨ˆ A + ˙ ϕ + 2 d R ( ζ ) e − A = 0 , (5.7) − d ( d −
1) ˙ˆ A + 12 ˙ ϕ + V ( ϕ ) + R ( ζ ) e − A = 0 , (5.8)¨ ϕ + d ˙ˆ A ˙ ϕ + V (cid:48) ( ϕ ) = 0 . (5.9)where we introduced the scale factor ˆ A ( τ ) via a ( τ ) = αe ˆ A ( τ ) and a dot ˙ now denotes aderivative with respect to τ . The difference with the equations of motion (2.7)-(2.9) in theEuclidean case is that the sign in front of every appearance of V and V (cid:48) is flipped comparedto (2.7)-(2.9).As we show shortly, the metric (5.3) describes the space-time for the space-like-separated region from the center of the bubble at r (0) = 0 whereas the metric (5.6) isvalid in the time-like separated region. For a continuous space-time and solution for thescalar, the two metrics and the scalar field have to be matched at the center of the bubble.The relevant matching conditions are ϕ ( ξ = 0) = ϕ ( τ = 0) , ddξ ϕ ( ξ ) | ξ =0 = ddτ ϕ ( τ ) | τ =0 ,r ( ξ = 0) = a ( τ = 0) , ddξ r ( ξ ) | ξ =0 = ddτ a ( τ ) | τ =0 . (5.10)One can show that the curvature invariants (both ‘outside’ and ‘inside the bubble’) can bewritten as functions of ˙ ϕ [13]. Therefore, the conditions (5.10) imply that all the geometricquantities are continuous across r (0) = 0.In the region ‘inside the bubble’ the equations of motion differ from those in the Eu-clidean setting and hence the Euclidean solution itself cannot be continued into this region.Instead, we have to solve afresh for a ( τ ) and ϕ ( τ ). However, the Euclidean instanton willprovide the initial conditions for this analysis via the matching conditions (5.10). Causality structure of the Lorentzian continuation –
To close this section, we con-struct the Penrose diagram for the continued Lorentzian space-time. To this end, it ishelpful to bring the Lorentzian space-time into the form ds = f ( ξ ) (cid:0) − dt + dx + dy + dz (cid:1) , (5.11)describing a manifestly conformally flat space-time. We begin by showing how the metric– 56 –5.3) ‘outside the bubble’ can be brought into the form (5.11). To this end we define t = σ ( ξ ) sinh χ , (5.12) x = σ ( ξ ) cosh χ cos ψ (5.13) y = σ ( ξ ) cosh χ sin ψ cos φ , (5.14) z = σ ( ξ ) cosh χ sin ψ sin φ . (5.15)Inserting this into (5.11) one finds: ds = f ( ξ ) (cid:2) σ (cid:48) ( ξ ) dξ + σ ( ξ ) (cid:0) − dχ + cosh ( χ ) d Ω (cid:1)(cid:3) , (5.16)which we can identify as (5.3) as long as f ( ξ ) σ (cid:48) ( ξ ) = 1 , f ( ξ ) σ ( ξ ) = r ( ξ ) . (5.17)These conditions are satisfied for σ ( ξ ) = σ e (cid:82) ξξ dξ (cid:48) r ( ξ (cid:48) ) , f ( ξ ) = r ( ξ ) σ ( ξ ) , (5.18)where ξ is a fiducial initial point and σ can be chosen at will.We can also confirm that the point r ( ξ = 0) = 0 in the coordinates (5.3) coincides withthe origin t = x = y = z = 0 in (5.11). To this end, note that for ξ → r ( ξ ) → ξ ,which follows from e.g. (2.47). Using (5.18) this implies that σ ( ξ ) (cid:39) Cξ ξ → , (5.19)where C is a constant which can be set to unity by an appropriate choice of σ in equation(5.18). From (5.12-5.15) we see that t = x = y = z → ξ →
0. To conclude, themetric (5.3) ‘outside the bubble’ can be brought successfully into the form (5.11) via thecoordinate transformations (5.12)–(5.15).What can we learn from this? Note that from the definitions (5.12)–(5.15) it followsthat x + y + z − t = σ ( ξ ) > . (5.20)i.e. the region that can be accessed via the coordinates t, x, y, z as defined in (5.12)–(5.15)are the points that are space-like separated from the origin / center of the bubble. Notethat t, x, y, z cover the space-time given in (5.3) termed the region ‘outside the bubble’.Here, we find that this region fills the part of space-time that is space-like separated fromthe origin / center of the bubble. Recall that the Euclidean solution itself is continued intothis region.We now turn to the metric (5.6) ‘inside the bubble’ and show how it can be broughtinto conformally flat form. However, as the coordinate ξ has been eliminated from (5.6) infavour of τ , instead of (5.11) for the conformally flat metric we write ds = g ( τ ) (cid:0) − dt + dx + dy + dz (cid:1) , (5.21)– 57 –o bring this into the form (5.6) we now define t = ω ( τ ) cosh η , (5.22) x = ω ( τ ) sinh η cos ψ , (5.23) y = ω ( τ ) sinh η sin ψ cos φ , (5.24) z = ω ( τ ) sinh η sin ψ sin φ , (5.25)which inserted into (5.16) gives ds = g ( τ ) (cid:2) − ω (cid:48) ( τ ) dτ + ω ( τ ) (cid:0) dη + sinh ( η ) d Ω (cid:1)(cid:3) . (5.26)This reduces to (5.6) for g ( τ ) ω (cid:48) ( τ ) = 1 , g ( τ ) ω ( τ ) = a ( τ ) . (5.27)which can be solved as ω ( τ ) = ω e (cid:82) ττ dτ (cid:48) a ( τ (cid:48) ) , g ( τ ) = a ( τ ) ω ( τ ) , (5.28)with τ is a fiducial point where ω takes the value ω .For t, x, y, z as defined in (5.22)–(5.25) we now have x + y + z − t = − ω ( τ ) < . (5.29)i.e. the coordinates t, x, y, z cover only the region that is time-like separated from the origin.This implies that the space-time described by (5.6) describes the region that is time-likeseparated from the origin / center of the bubble.We now have all ingredients to draw the Penrose diagram for the CdL geometry in-cluding the Lorentzian continuation. This is given by the plot of the space-time as the | (cid:126)x | - t -plane and is shown in fig. 12a. On the lower half of the plot, we show the Euclideansolution, while the upper half shows the Lorentzian continuation with t >
0. The centerof the CdL bubble is located at t = 0 = | (cid:126)x | . In the region that is space-like separatedfrom the origin, (shown in green), the space-time is described by the metric (5.3). As re-marked before, the entire CdL bubble solution is mapped into this region. The Euclideanbubble can be described as a series of concentric shells labelled by the coordinate ξ , with ϕ ( ξ ) and r ( ξ ) varying across shells. In the Lorentzian continuation, surfaces of constant ξ (cid:63) correspond to the hyperbolic curves | (cid:126)x | − t = σ ( ξ (cid:63) ) , (5.30)as follows from (5.12)–(5.15). In fig. 12a we plot one such surface corresponding to theboundary of the space-time. As the boundary is only reached for ξ → ∞ , for illustrationpurposes we plot the boundary contour for some ξ that is finite but very large. Thecorresponding surface is shown as a black dashed line in fig. 12a. The region that is time-like separated from the origin (shown in orange) is described by the metric (5.6). Note thatany event that occurs in this region, cannot reach the green space-like separated region and– 58 –uclidean geometry H o r i z o n H o r i z o n | (cid:126)x | t Time-like regionSpace-like region Space-like region S i n g u l a r i t y Boundary (a) H o r i z o n H o r i z o n | (cid:126)x | t Time-like regionSpace-like region Space-like region S i n g u l a r i t y S i n g u l a r i t y B o und a r y B o und a r y H o r i z o n H o r i z o n Time-like region (b)
Figure 12: (a):
Space-time diagram of the CdL geometry including the Lorentzian con-tinuation. The purple region is the Euclidean geometry. Green and orange are space-likeand time-like regions from the origin respectively. The boundary is denoted as the dashedblack line whereas the blue dashed line denotes the horizon. Inside the time-like region,there is a singularity which is denoted as the red dashed line. This singularity is hiddenbehind the horizon. (b) : Space-time diagram for a holographic RG flow for a QFT ondS d , shown for comparison with the CdL case. The causal structure is identical to twocopies of the Lorentzian continuation of the CdL geometry, connected at the surface t = 0.The solution in the green regions has been previously studied in [13]. The solution in theorange regions can be computed using the same method as in the CdL case. By analogy,we once more expect singularities in the orange regions.the surface t = | (cid:126)x | acts as a horizon for the interior region. In the time-like separated region,we also display one further feature that we have not discussed yet, but will be the focus ofthe next section. In this region, at some time τ (cid:63) , the space-time will generically exhibit abig crunch, as is known since the foundation paper [2]. This big crunch singularity at τ (cid:63) corresponds to t − | (cid:126)x | = ω ( τ (cid:63) ) > , (5.31)which is a hyperbola in the | (cid:126)x | - t -plane and shown as the dotted red line in fig. 12a. In the foundational work of Coleman and de Luccia [2] it was observed that when tunnellingto an AdS region of the potential the subsequent evolution leads to a big crunch of thespace-time in the region ‘inside of the bubble’, i.e. the space-time described by (5.6). While The crunching is generically observed whenever the O ( D )-instanton deposits the field in a region of thepotential with V < – 59 – .2 0.4 0.6 0.8 1.0 1.2 1.42000400060008000100001200014000 τ a ( τ ) α = e ˆ A ( τ ) (a) ϕϕ ϕ = ϕ t ˆ W ( ϕ ) ˆ W goes to infinity (b) Figure 13: Plots of a ( τ ) /α ( (a) ) and ˆ W ( ϕ ) ( (b) ) for the Lorentzian continuation of an O ( D )-instanton in the space-time region described by (5.6). Here the O ( D )-instantoninterpolates between ϕ f = 0 to ϕ = 0 . . ϕ t and v = 10. Red arrows show the direction of time flow. The scale factor a ( τ ) increasesuntil it reaches a maximum before decreasing and reaching zero again. The plot for ˆ W ( ϕ )shows that the dilaton ϕ oscillates about the true minimum of the potential at ϕ t = 1, firstwith a decreasing amplitude, then with increasing amplitude, until it diverges as | ϕ | → ∞ when a ( τ ) drops to zero, where W ( ϕ ) → ∞ also diverges.in [2] this was examined in the context of the thin-wall approximation, the existence ofthe big crunch does not rely on the applicability of this approximation, but is a genericoutcome of tunnelling to a region with V <
0, as is also emphasised by Tom Banks ine.g. [7].Here we briefly present arguments for the generic existence of a big crunch singularity,after tunnelling to an AdS region, and comment on the significance of the singularity forthe holographic interpretation of the tunnelling solution as an RG flow.To understand the appearance of the big crunch, it will be useful to consider an explicitexample. To this end, we revisit one example of an O ( D )-instanton from section 4.2 andcompute the continuation in the region described by (5.6).To find the solution in (5.6) we need to solve (5.7)–(5.9) with the initial conditions givenby (5.10). That is, the initial conditions for a ( τ ) and ϕ ( τ ) in the Lorentzian continuationare set by the behaviour of r ( ξ ) and ϕ ( ξ ) or equivalently A ( u ) and ϕ ( u ) at the center ofthe Euclidean bubble geometry. The behaviour of A ( u ) and ϕ ( u ) near the center has beenrecorded in (2.47) and (2.48) which via (5.10) imply that: a ( τ ) = τ → τ + O ( τ ) , ϕ ( τ ) = τ → ϕ − V (cid:48) ( ϕ )2( d + 1) τ + O ( τ ) . (5.32)Here, in addition to solving for a ( τ ) and ϕ ( τ ), we introduce a new set of dynamical quan-tities that will satisfy first order differential equations. To this end, parallel to what wedid in the Euclidean case in (2.18), we introduce the quantitiesˆ W ( ϕ ) = − d −
1) ˙ˆ
A , ˆ S ( ϕ ) = ˙ ϕ , ˆ T ( ϕ ) = R ( ζ ) e − A . (5.33)– 60 – ˆ S ( ϕ ) ϕ ˆ S goes to infinityFigure 14: Plot of ˆ S ( ϕ ) for the Lorentzian continuation of an O ( D )-instanton in the space-time region described by (5.6). Initially, after starting from rest at ϕ the scalar field ϕ oscillates about the minimum at ϕ t leading to the cyclic trajectory in ˆ S . Eventually, theamplitude of this oscillation increases and ϕ is ejected from the potential well with | ˆ S | diverging.In terms of these, the equations of motion (5.7)–(5.9) just become (2.19)–(2.21), but withthe sign in front of every instance of V and V (cid:48) flipped. Using (2.19) and substituting into(2.20,2.21) we can eliminate ˆ T to arrive at the following two equations: d d −
1) ˆ W + ( d −
1) ˆ S − d ˆ S ˆ W (cid:48) − V = 0 , (5.34)ˆ S ˆ S (cid:48) − d d −
1) ˆ W ˆ S + V (cid:48) = 0 . (5.35)One can further show that the initial conditions (5.32) imply the following boundary con-ditions for ˆ W and ˆ S :ˆ W ( ϕ ) = ϕ → ϕ ˆ W (cid:112) | ϕ − ϕ | + O (cid:0) | ϕ − ϕ | (cid:1) , ˆ S ( ϕ ) = ϕ → ϕ ˆ S (cid:112) | ϕ − ϕ | + O (cid:0) | ϕ − ϕ | (cid:1) , (5.36)with ˆ S = (cid:114) | V (cid:48) ( ϕ ) | d + 1 , ˆ W = − ( d −
1) ˆ S . (5.37) A numerical example –
We now solve for a ( τ ), ˆ S ( ϕ ) and ˆ W ( ϕ ) for the Lorentziancontinuation in the region (5.6), in the aftermath of the O ( D )-instanton admitted by thepotential given in (4.2) with ∆ = 30 . ϕ t and v = 10. This was shown in section 4.2 todescribe tunnelling from ϕ f = 0 to ϕ = 0 . a ( τ ) (fig. 13a) and ˆ W ( ϕ ) (fig. 13b). In fig. 14 the result for ˆ S is shown.We observe that a ( τ ) first grows, before reaching a maximum and then decreasing againuntil it vanishes again at τ (cid:63) which we identify as a big crunch singularity. The fact that a ( τ ) initially increases follows from the initial conditions (5.32) which initially set a = 0,– 61 – a >
0. The fact that the curve for a ( τ ) eventually turns around is a direct consequenceof the equation of motion (5.7). Rearranging, the equation implies that ¨ a ≤
0, so that˙ a can only decrease. This eventually turns ˙ a negative, ˙ a <
0, at which point the crunchbecomes unavoidable. The solution for ˆ W ( ϕ ) is a multivalued function. This results fromthe fact that the solution for ϕ ( τ ) oscillates about the minimum at ϕ t = 1. Intuitioninto the behaviour of ϕ ( τ ) can be obtained from (5.9), which can be identified as theNewtonian problem of a particle performing motion in the potential V (not the invertedpotential in this case), with ˙ a > a > a > a ( τ ) reaches the maximum, i.e. ˙ a = 0 we haveˆ W = 0 following from (5.33). After that, the amplitude of the oscillation increases after ˙ a changes sign to act as anti-friction. Eventually the anti-friction is so strong that the field ϕ is ejected towards infinity for τ → τ (cid:63) , with ˆ W → + ∞ also diverging. This interpretationis also supported by the behavior of ˆ S as shown in fig. 14. By definition, ˆ S ∼ ˙ ϕ vanisheswhenever ϕ comes to rest. The oscillation of ϕ about ϕ t = 1 therefore manifests itself asthe cyclic trajectory in fig. 14. Ultimately, ˆ S diverges when anti-friction leads to ϕ beingejected out of the potential well.To demonstrate that the locus τ (cid:63) where a ( τ ) crunches is indeed a singularity, wecompute the curvature scalar R AB R AB . In particular, in the region ‘inside the bubble’ thisis given by [13]: R AB R AB = d (cid:32) − ˆ S d + Vd ( d − (cid:33) + dV ( d − , (5.38)which we have written as a function of ˆ S and V . This diverges as R AB R AB → ∞ at thelocus a ( τ → τ (cid:63) ) = 0 in virtue of | ˆ S | → ∞ . Therefore, the fate of the space-time ‘inside thebubble’ is indeed a big crunch singularity.Are there examples where the crunch singularity can be avoided in principle? One canimagine that at τ = τ (cid:63) with a ( τ (cid:63) ) = 0 the field ϕ is not ejected to infinity, but comes torest at some locus ϕ (cid:63) = ϕ ( τ (cid:63) ). The geometry could be regular there as it is at τ = 0, wherethe scale factor also vanishes, a ( τ = 0) = 0, and one has ϕ ( τ = 0) = ϕ . For a genericpoint ϕ (cid:63) , i.e for V (cid:48) ( ϕ (cid:63) ) (cid:54) = 0, this would correspond to a ( τ ) = τ → τ (cid:63) − ( τ − τ (cid:63) ) + O (cid:0) ( τ − τ (cid:63) ) (cid:1) , ϕ ( τ ) = τ → τ (cid:63) ϕ (cid:63) − V (cid:48) ( ϕ (cid:63) )2( d + 1) ( τ − τ (cid:63) ) + O (cid:0) ( τ − τ (cid:63) ) (cid:1) , (5.39)which one can confirm to be the regular solution to (5.7)–(5.9) that satisfies the boundaryconditions ϕ ( τ (cid:63) ) = ϕ (cid:63) , ˙ ϕ (cid:63) = 0 and a ( τ (cid:63) ) = 0. One observation is that for a given value ϕ (cid:63) the solution (5.39), just like the solution for τ → ϕ (cid:63) , a tentative solution (cid:0) a ( τ ) , ϕ ( τ ) (cid:1) has to approach the value ϕ (cid:63) in a unique way determined by (5.39) to end there in a regularway. However, in the interior region of the bubble the solution is determined by the initialconditions at τ = 0. Thus there is no further freedom to ensure that the solution canapproach any value ϕ (cid:63) in ‘the right way’ to end there in a regular fashion. This wouldrequire a tuned potential and thus does not happen generically.– 62 –hat is not possible either is that the field settles at the minimum, i.e. ϕ ( τ → τ (cid:63) ) → ϕ t while at the same time a ( τ → τ (cid:63) ) →
0. The argument is the same as that presented insection 2.5, where we show that the Euclidean tunnelling solution cannot deposit the field ϕ exactly at the AdS minimum at ϕ t . Note that this argument, both for the Lorentzianand the Euclidean case, only holds for R ( ζ ) (cid:54) = 0, which is the case for O ( D )-instantons andtheir Lorentzian continuation.Alternatively, one could try to avoid the crunch singularity by ensuring that the space-time expands forever. For this to happen, it must be avoided that ˙ a turns negative, asthen the crunch becomes unavoidable due to ¨ A ≤
0. For this to be realised, one requiresthat ¨ˆ A → a is still positive or approaches zero. Note that from (5.7) it follows thatfor ¨ˆ A → ϕ → A → ∞ . Again, while we cannot excludethat such a solution may exist, it does not seem generic that all these conditions can besatisfied simultaneously in a generic potential. Indeed, in the example considered above,this did not occur. Hence, we conclude that while big crunch singularities may be avoidedin principle, in practice they are unavoidable once a CdL process deposits the field ϕ in anAdS region. On this topic, also see the discussion in [7], where it is also argued that thebig crunch singularity cannot be avoided in general. In this section we would like to ask again the question: what is the significance of the bigcrunch singularity, and more generally of the CdL O ( D )-symmetric instanton, from thepoint of view of the (dual) boundary field theory ?One possibility is that the singularity signals that the process is unphysical, i.e. thatdecays into spaces with negative cosmological constant do not occur in quantum gravity, aviewpoint advocated in e.g. [7, 46]. Alternatively, if the O ( D )-symmetric solution interpo-lates between a dS and an AdS region, it was argued in [46, 47] that this solution shouldnot be interpreted as a tunnelling process but instead as a recurrence to a low-entropystate represented by the crunching region (see also [8] for a brief summary of this idea).The holographic duality however casts a different light on the big crunch singularityand on the tunneling process as a whole. If one takes the point of view of the boundary fieldtheory, the AdS-to-AdS tunnelling by the O ( d +1)-instanton has a consistent interpretationin terms of a holographic RG flow for a Euclidean field theory defined on the sphere S d or (in the Lorentzian continuation) on d -dimensional de Sitter space dS d . This RG floworiginates from a CFT, dual to the false-vacuum AdS extremum of the bulk potential at ϕ = ϕ f . When defined on dS d , this CFT admits a non-trivial RG flow with subleadingscalar field asymptotics of ϕ + type, as in equation (2.26). The flow is driven by the vev (cid:104)O(cid:105) of the operator dual to ϕ , given by equation (2.27), with the source j set to zero. Thisoperator is relevant if the false vacuum is a maximum and irrelevant if the false vacuum isa minimum. We remind the reader that d denotes the dimension of the boundary and D = d + 1 is the dimensionof the bulk. – 63 –y equation (2.30), the value of the vev (cid:104)O(cid:105) is fixed by the value of the curvature R ( ζ ) of the de Sitter space-time on which the boundary theory is defined, because for a W + -typeflow the parameter R is uniquely fixed to a specific value as explained at the end of section2.2. If the big crunch introduced a pathology into the system, one would expect this tomanifest itself on the boundary. Hence, here we examine to what extent the boundary isaffected by the existence of the crunch. To this end, we return to the space-time diagramin fig. 12a. The O ( D )-symmetric solution is identified with the holographic RG flow of theEuclidean theory. This is shown in fig. 12a as the Euclidean solution in the lower half. Itis also mapped by analytical continuation into the green region that is space-like separatedfrom the origin. This region is dual to the RG flow on dS d of the boundary theory. Thedashed black line indicates the UV boundary of the space-time, which is formally locatedat ξ → ∞ , but is shown here for illustration purposes at some large, but finite ξ .From the Penrose diagram in figure 12a we can make two observations:1. The singularity and the horizon both reach the boundary only in the asymptotic deSitter future χ → + ∞ .2. No information from the orange region in the future lightcone of the origin canreach the green region, nor the boundary. This then also applies to the big crunchsingularity (shown as the red dotted line) as this is confined to the future lightconeof the origin.Therefore, from the boundary point of view, the big crunch singularity is cloaked by ahorizon at t = | (cid:126)x | and hence the boundary is unaffected by the crunch. The considerations above may be taken as a hint that, despite the existence of thebig crunch singularity, a holographic interpretation of the O ( D )-instanton geometry is stillperfectly possible, i.e. it successfully describes a holographic RG flow of a consistent theoryon de Sitter space-time. This point of view is also the one advocated in [9]. If the existenceof the big crunch does introduce a pathology on the field theory side, it must do so in amore subtle way, which leaves an interesting direction for future work.We therefore continue under the assumption that the existence of the CdL solutiondoes not constitute a pathology of the holographic theory, and we turn to discussing whatits existence implies for the boundary theory.For one thing, the CdL instanton certainly does not describe a vacuum transition ordecay process on the field theory side. As we discussed above, the boundary theory sees onlythe exterior geometry, whose boundary is de Sitter space-time with a constant curvatureand a non-zero but constant vev for an irrelevant operator (in the case the false vacuumis a minimum) which induces an RG flow. The boundary field theory is the false-vacuumQFT at all times, and no vacuum decay occurs.What makes the situation interesting is that we have found two distinct regularLorentzian geometries with the same boundary asymptotics: AdS-to-AdS CdL processes have also been considered in [11] from a holographic point of view. There,the CdL process in the bulk was argued to lead to a dual decay instability of the boundary theory, dual toglobal AdS. We will come back to this point in Section 6. – 64 –. The solution with constant scalar field set to the false vacuum value, ϕ = ϕ s andmetric given by AdS d +1 foliated by dS d ;2. The vev-driven RG-flow solution with non-trivial ϕ = ϕ ( ξ ), and metric which deviatesfrom AdS d +1 as we move towards the interior.In both solutions the source term in the near-boundary expansion of the scalar field isvanishing, and the metric source is the same. Therefore (1) and (2) represent two different semiclassical states of the same dual QFT on de Sitter, which we will denote by | C (cid:105) and | C (cid:105) . They have the property: (cid:104) C |O| C (cid:105) = 0 , (cid:104) C |O| C (cid:105) = ( M (cid:96) f ) ( d − (2∆ − d ) ϕ + , (5.40)where O is the operator dual to the bulk scalar field.The two semiclassical states have different symmetry properties: | C (cid:105) enjoys the fullAdS d +1 invariance SO (2 , d ), whereas | C (cid:105) is only invariant under the SO (1 , d ) subgroup,i.e. the group of dS d isometry of the constant- ξ slices.An interesting point is that both solutions have the same holographic stress tensor.Indeed, in pure vev solutions the backreaction of the running scalar on the metric appearat orders which are subleading with respect to the terms giving the boundary stress tensor[13]. However the difference may be seen at the level of higher-point correlation functionswhich will bear the signs of the spontaneously broken conformal invariance. So far we have described the field theory interpretation of the two different Lorentziansolutions. What is the interpretation of the CdL instanton, and what is the meaning onthe field theory side of the tunneling rate e − B , with B given in (2.33)?The gravitational path integral with Euclidean AdS d +1 boundary conditions corre-sponds to the Hartle-Hawking state [25]. Specifying these boundary conditions in Eu-clidean time on the gravity side corresponds to putting the field theory in a specific state,which we call | HH (cid:105) . Although the boundary field theory is not coupled to a dynamicalmetric, one can say that this state obeys the “no-boundary” condition of [25], since thegeometry on which the field theory lives is S d in the Euclidean past. The | HH (cid:105) state is not a semiclassical state, i.e. it does not correspond holographicallyto a specific classical geometry. In fact, we have just seen that there are at least twoclassical solutions of the bulk theory which obey the Hartle-Hawking condition in theEuclidean regime: these are the ones obtained by gluing solutions (1) and (2) above to thecorresponding Euclidean solutions (the one with trivial scalar and the one with runningscalar) on a space-like hypersurface at χ = 0. In each of these two solutions, we denote by This phenomenon for a CFT defined on flat space was discussed in [48, 49]. The original no-boundary prescription of Hartle and Hawking is adapted here to AdS space-times, andbecomes the condition that in the Euclidean path integral the geometry asymptotes to Euclidean AdS withno sources turned on [27]. Hartle and Hawking argued that this state may represent the vacuum state of the gravity theory. Ifone accepts this interpretation, then one may suppose that one can take this state as the vacuum of theQFT on de Sitter. This is the case for example in a (near) free QFT on de Sitter, where the Bunch-Daviesvacuum does correspond to the Hartle-Hawking state (see e.g. [50]) – 65 – ξχ = 0AdS solution RG flow solution ϕ f ϕ Figure 15: No-boundary Euclidean solutions with two different spatial metric and dilatonat the fixed time-slice θ = π/ χ = 0). The left figure represents the false-vacuum solution, in which the scalar field is constant, ϕ = ϕ f , and the metric is AdS D slicedby S d . On the right, the solution corresponding to the CdL instanton, a.k.a. holographicRG flow on S d , in which the scalar field flows from ϕ f (for ξ → + ∞ ) to ϕ at the center,and the metric deviates from AdS in the interior. γ ab ( ξ ) and ϕ ( ξ ) the metric and scalar field profile induced on the χ = 0 surface. The twocorresponding Euclidean geometries are represented in figure 15: on the left we have (thelower half of) the Euclidean AdS false-vacuum solution with constant scalar field; on theright (the lower half of) the CdL instanton with scalar field interpolating from ϕ f to ϕ .Both solutions have the same sources at the (conformal) boundary.According to the standard interpretation of the Hartle-Hawking prescription, the quan-tity e − S E [ γ ,ϕ ] represents (up to an overall normalization) the semiclassical approximationto the quantum amplitude for obtaining the ( γ ( ξ ) , ϕ ( ξ )) geometry in the HH state. Inother words, the Hartle-Hawking condition provides a state, i.e. a wave-functional of clas-sical spatial geometries, and the instanton action is the probability of finding a givensemiclassical configuration, in that state. Equivalently, e − S E [ γ ,ϕ ] gives, in the semiclassical approximation, the wave-function Ψof the HH state (which is a functional of a fixed-time geometry), evaluated at the classical“point”( γ ab , ϕ ), which can be taken to be solution (1) or (2) respectively,Ψ[ γ , ϕ ] = (cid:104) HH | C (cid:105) ∝ e − S E [ γ ,ϕ ] , Ψ[ γ , ϕ ] = (cid:104) HH | C (cid:105) ∝ e − S E [ γ ,ϕ ] (5.41)According to this interpretation, the bounce action gives the relative probability of“measuring” one of the two classical configurations when the system is in the HH state: |(cid:104) HH | C (cid:105)| |(cid:104) HH | C (cid:105)| = e − ( S − S ) , (5.42) The same as in quantum mechanics | ψ ( x ) | = |(cid:104) ψ | x (cid:105)| is the probability (density) of finding the particleat x in the state corresponding to the wave-function ψ . In our case, the classical position variable x isreplaced by the classical d -dimensional geometry and scalar field on a space-like hypersurface. – 66 – O(cid:105)V
ΨFigure 16: Quantum mechanical analogy of the space of classical no-boundary solutions interms of a double-well potential V (black curve). The horizontal axis represents the spaceof solutions (characterized in this case by the vev of the operator O dual to ϕ ). The twominima of the potential are the semiclassical solutions: The minimum on the left representsthe AdS solution, the one on the right the RG flow solution. The red curve is a sketch ofthe would-be ground-state wave-function of this quantum mechanical system. It should beremarked that the potential here is not the bulk scalar potential.(up to subleading 1 /N corrections which correspond to quantum effects in the bulk), where S is the Euclidean bounce action and S is the Euclidean false vacuum AdS action. Theexponent on the right hand side of equation (5.42) is exactly the negative of the quantity B in (2.33).One way of visualizing the situation qualitatively (although with many caveats) isby making the analogy with a quantum mechanical system with a potential having two(non-degenerate) minima, as in figure 16. Each minimum of the potential (black curve)corresponds to a classical state (the particle on the left or on the right of the barrier), butthe ground state wave-function (red curve) is delocalized. If the energy difference betweenthe two minima is large, the wave-function will be mostly peaked on the lower-energyminimum. A measurement of the position of the particle will have with high probabilitythe classical outcome “the particle is localized on the left”.In our case, the peak in Ψ on the right corresponds to the CdL instanton geometry,which has higher free energy than the pure AdS solution (i.e. B >
0, as we have found insection 2.4). Therefore the HH state has an (exponentially) larger overlap with the AdSsolution than with the RG flow. Nevertheless, the full AdS symmetry is broken in the HHstate, albeit by exponentially small effects.The quantum mechanical analogy cannot be taken too seriously, however. First of all,the free energy of the field theory on the sphere is not the same as the eigenvalue of the– 67 –amiltonian. Furthermore, we are dealing with a QFT in de Sitter, which does not havea global time-like killing vector, so there is not really a Hamiltonian to speak of, and theconcept of a ground state is rather vague. What we can say is that there is a state withhigher symmetry, which is the semiclassical state | C (cid:105) . The HH state on the other handhas lower symmetry since it is a linear combination of (at least) | C (cid:105) and | C (cid:105) . It wouldbe very interesting if one could give an independent characterization of the HH state fromthe field theory side. This is possible for example in perturbative field theories on de Sitter(the Bunch-Davies vacuum).To summarize this section, while the CdL instanton has a dynamical interpretation inthe bulk as computing a decay rate between the false AdS vacuum and (something close to)the true AdS vacuum, the boundary field theory interpretation is completely different, as itdoes not describe a transition: it concerns only the “false vacuum” QFT, and it computesthe relative overlap of two states, which have a semiclassical counterpart each, with a thirdstate (the HH state) which does not. In this picture, there is no obvious trace of the “truevacuum” AdS CFT in the boundary field theory (as it is hidden behind the horizon).
6. From the sphere to the cylinder
In the previous sections we discussed vacuum decay by bubble nucleation described bya maximally symmetric instanton. As we have emphasized, this only applies when theasymptotic boundary of the Euclidean bulk space-time has the geometry of S d . Conse-quently, the dual QFT is defined on a round sphere or, in the Lorentzian solution describingthe expanding bubble, on de Sitter space-time.In this section we ask the question whether these solutions give any information aboutvacuum stability for different asymptotic boundary geometries. For example, can we sayanything about the stability of a dual QFT defined on d -dimensional Minkowski space-timeor on a (generalized) cylinder?The reason this question makes sense is that in AdS/CFT there exist bulk diffeo-morphisms which have various effects on the boundary QFT data. In general, in locallyasymptotically AdS space-times, one can identify three types of bulk diffeomorphisms,depending on their near-boundary asymptotics:1. Diffeomorphisms which change the leading near-boundary behavior of the metricand bulk fields. These transformations change the sources of the UV QFT (i.e. themetric and coupling constants of relevant operators), therefore they are maps betweendifferent boundary theories.2. Diffeomorphisms which preserve the leading boundary asymptotics (sources) butchange the subleading near-boundary behavior of the metric and bulk fields (i.e. thevevs of the dual operators). These are maps which change the state of the boundarytheory, but not the theory itself.3. Diffeomorphisms which vanish fast enough close to the boundary, as to change neitherthe sources nor the vevs (therefore they do not affect the Fefferman-Graham expan-sion to any order). Such transformations are preserving the state of the dual QFT. As– 68 –ll data of the QFT are computed from vevs near the boundary, such transformationsact trivially on all QFT data and in particular on all QFT correlation functions. Theydo not have a corresponding description in the dual quantum field theory but are truegauge transformations (i.e. redundancies) of the bulk gravitational description.In all of these cases, diffeomorphisms may be accompanied (and typically they do) bythe appearance of horizons.As we shall discuss below, one can construct type 1 diffeomorphisms which connectthe QFT on de Sitter space-time to the same field theory on the (generalized) cylinder orflat space.Consider the case of an exact AdS geometry (with no bubble). There, one can make aglobally defined (as opposed to only asymptotically) change of coordinates which transformsthe S d (or dS d ) radial slicing to a slicing by R t × S d − or by flat space-time. Here,the CdL tunnelling geometries we considered are asymptotically AdS in spherical slices,and hence one may define appropriate coordinate transformations which, asymptotically,change the boundary structure from S d to flat space-time or the generalized cylinder. Thecorresponding geometry will still be a (regular) solution of Einstein equation, and one maywonder, whether this solution still describes vacuum decay of the same dual QFT theory,when defined on flat-space or on a cylinder.In the case we analysed so far, the exterior solution, in which the bubble expands,extends all the way to the infinite de Sitter future. As χ → ∞ , with χ the de Sitter time(as defined in equation (5.3)), both the null horizon separating the interior and the exteriorsolution, and the big-crunch singularity in the interior, approach the asymptotic boundaryof the ( d + 1)-dimensional space-time. The fact that this happens in the infinite futureindicates that the dual QFT on the boundary can exist indefinitely .It may appear, however, that this situation changes drastically when we go from thesphere to flat space or the cylinder. In fact, as we will review below, in the exact AdScase, future infinity in the de Sitter slicing of AdS corresponds to a finite time in boththe cylinder (Global AdS) and Minkowski slicing (Poincar´e AdS). Therefore, it may seemthat the solutions we found would, upon an appropriate coordinate transformation, yieldsolutions with different boundary conditions which describe a QFT which ceases to existafter a finite time (when the singularity reaches the boundary). This observation had beenalready made by several authors, who argued that the existence of an O ( D )-symmetricinstanton in the bulk implies a runaway vacuum decay at finite time for the correspondingdual field theory defined on the cylinder [11].Here, we argue that this conclusion is too rash: the O ( D )-symmetric instanton, uponcoordinate transformation to a different slicing, does not in fact describe a spontaneous decay in a finite time of the dual field theory. Rather, it describes a driven decay, notunlike what one would obtain by turning on, in the UV CFT, a time-dependent sourcewhich becomes singular at a finite time.Below, we will carry out the analysis for the cylindrical boundary, but similar conclu-sions can be reached for a flat-space boundary.– 69 – .1 AdS in different radial slicings Before we turn to the full vacuum decay problem, we review the various ways to sliceAdS d +1 corresponding to different boundary geometries.Global AdS d +1 space-time with length (cid:96) is described by the metric:Cylinder slicing: ds = dλ − (cid:96) cosh λ(cid:96) dψ + (cid:96) sinh λ(cid:96) d Ω d − , (6.1)where d Ω d − is the metric on the unit ( d − ψ hasthe domain ( −∞ , + ∞ ). The boundary region is reached as λ → + ∞ , where the metricasymptotes to: ds → dλ + e λ/(cid:96) (cid:96) (cid:2) − dψ + d Ω d − (cid:3) , λ → + ∞ . (6.2)The de Sitter slicing we have been using so far is given in equation (5.3) (we will onlyneed the exterior geometry, i.e. the one which is space-like separated from the center of thebubble), which we record below:de Sitter slicing : ds = dξ + (cid:96) sinh ξ(cid:96) (cid:2) − dχ + cosh χd Ω d − (cid:3) . (6.3)The metric in the square brackets is the de Sitter metric in global coordinates, with time χ running from −∞ to + ∞ . The radial coordinate ξ takes values in (0 , + ∞ ). The asymptoticboundary is reached as ξ → + ∞ , where the metric asymptotes to: ds → dξ + e ξ/(cid:96) (cid:96) (cid:2) − dχ + cosh χd Ω d − (cid:3) , ξ → + ∞ . (6.4)In writing (6.3) we have chosen coordinates so that the asymptotic de Sitter scale is 1 /(cid:96) ,but any other choice would have been possible by a shift in ξ .The two metrics (6.1) and (6.3) are diffeomorphic to each other, and they are connectedby the coordinate transformation: sinh λ(cid:96) = sinh ξ(cid:96) cosh χ , tan ψ = tanh ξ(cid:96) sinh χ , (6.5)and its inverse: sinh ξ(cid:96) = cos ψ (cid:0) sinh λ(cid:96) − tan ψ (cid:1) / , cosh χ = sinh λ(cid:96) cos ψ (cid:0) sinh λ(cid:96) − tan ψ (cid:1) / . (6.6)The corresponding coordinate transformation in the Euclidean signature is obtained byusing the identification (5.2), which connects the R t E × S d − and the S d slicings of EuclideanAdS: sinh λ(cid:96) = sinh ξ(cid:96) sin θ , tanh t E = tanh ξ(cid:96) cos θ , sinh ξ(cid:96) = cosh t E (cid:0) sinh λ(cid:96) + tanh t E (cid:1) / , sin θ = sinh λ(cid:96) cosh t E (cid:0) sinh λ(cid:96) + tanh t E (cid:1) / . (6.7)– 70 – π ξχ λψ π π λ = ¯ λ Figure 17: Global vs. de Sitter slicing of AdS. The Black region is the interior (not coveredby the de Sitter slicing); Black curves are contours of constant ξ (with ξ = 0 correspondingto the null surface which separates the interior from the exterior), red curves are contoursof constant χ (with χ = 0 coinciding with the horizontal axis). The dashed blue line isa fixed- λ surface, which eventually crosses all constant- ξ curves as global time increasestowards π/ S d − , whichare identified in the two metrics) also summarized in figure 17.1. The two radial coordinates ξ and λ (and in fact the full spatial geometries) coincideat ψ = χ = 0. We can therefore identify the initial spatial hypersurfaces at thatpoint, and focus on the future evolution, ψ > χ > ξ = 0 coincides with the center of global AdS at ψ = 0, and subsequently it describes the curve (for ψ > λ/(cid:96) = tan ψ . (6.8)One can check that this is a null hypersurface, and it divides global AdS in tworegions, which we will call interior and exterior. The former is the black region infigure 17, while the latter is the clear region.3. The coordinate system ( ξ, χ ) in (6.5) covers only the exterior region sinh λ/(cid:96) > tan ψ ,and it only extends in the future to times ψ < π/ ψ < π/
2. The exterior region “shrinks” as global time increases towards ψ = π/ χ → + ∞ – 71 –except for the point λ = ψ = 0). The “end” of global time, ψ = π/
2, is of course onlya coordinate singularity, but to continue past this time one has to go to a differentpatch with de Sitter slicing.4. In the Euclidean case on the other hand, the coordinate system ( ξ, θ ) in (6.7) coversthe full Euclidean AdS. Here, ξ = 0 is the single point λ = 0 , t E = 0 and there areno“hidden” regions. Notice that Euclidean time t E is not periodic, so this solutiondescribes a QFT at zero temperature.5. A general ξ = ¯ ξ hypersurface is described by the curve:sinh λ(cid:96) = 1cos ψ (cid:114) sin ψ + sinh ¯ ξ(cid:96) . (6.9)6. The boundary of the de Sitter slicing ξ → + ∞ is reached as λ → + ∞ and ψ fixed.In this limit, the coordinate transformation (6.5-6.6) simplifies to: λ (cid:39) ξ + log (2 cosh χ ) , cos ψ (cid:39) χ . (6.10)7. The boundary of the cylinder slicing, λ → + ∞ , can instead be reached in two differentways: either sending ξ → + ∞ with χ fixed, or by keeping ξ = ¯ ξ fixed and sending χ → ∞ . The latter option corresponds to going towards the boundary by followingone of the curves (6.9) as ψ → π/ λ hypersurface, no matter how far towards the boundary of global AdS,will eventually cross the curve (6.8) and cross over into the interior patch. In otherwords, the surface spanned by the horizon ξ = 0 will intersect any fixed large- λ hypersurface λ = ¯ λ at a time ψ max approximately given by equation (6.8): ψ max (cid:39) π − e − λ/(cid:96) , λ/(cid:96) (cid:29) . (6.11)The last property is the crucial one for our purposes, when we go from an exact AdSsolution to the expanding bubble solution in the next section. We shall now discuss the case where the scalar field undergoes a non-trivial radial flow.We start from the solution in dS slicing, ds = dξ + r ( ξ ) (cid:2) − dχ + cosh χd Ω d − (cid:3) , ϕ = ϕ ( ξ ) , (6.12)where r = e A and we chose the coordinate ξ such that the boundary is at ξ → + ∞ and theflow ends at ξ = 0 in the interior. As we have seen in section 2, the scalar field behavesasymptotically as ϕ ( ξ → + ∞ ) (cid:39) ϕ f + ϕ + (cid:96) ∆ f e − ∆ ξ/(cid:96) f + . . . , and ϕ (0) = ϕ , (6.13) In the notation of section 2, we define ξ = u − u . – 72 –here ϕ f is the UV minimum of the potential (the false vacuum) and ϕ is the valuereached by the scalar field at the center of the bubble.In the UV, the metric has the same asymptotic behavior as in equation (6.4): r ( ξ → + ∞ ) (cid:39) (cid:96) f e ξ/(cid:96) f , (6.14)Close to the endpoint ξ = 0, one can show that the scale factor behaves as: r ( ξ ) = (cid:96) (cid:34) ξ(cid:96) + 16 (cid:18) ξ(cid:96) (cid:19) + O ( ξ ) (cid:35) , (cid:96) = − d ( d − V ( ϕ ) . (6.15)Notice that equation (6.15) is the same as the expansion to cubic order of the scale factorsinh ξ/(cid:96) appearing in the exact AdS solution (6.3) , except that (cid:96) is replaced by (cid:96) . This canbe traced to the fact that close to the IR endpoint, the bulk Ricci tensor is approximately[13]: R AB (cid:39) V ( ϕ ) d − g AB . (6.16)Since the UV and IR asymptotics are AdS-like (albeit with two different scales), theremust exist a coordinate transformation of the form ξ = F ( λ, ψ ) , χ = G ( λ, ψ ) , (6.17)which, in both asymptotic regions, implements the change from de Sitter to cylinder slicing,i.e. it approaches the transformation (6.6) both near the boundary ξ → ∞ and close to theendpoint ξ →
0. Even without having an explicit form for this transformation, from theprevious considerations we can conclude that it will have the following properties:1. The Lorentzian solution in the λ, ψ coordinates will have the general form ds = dλ − f ( λ, ψ ) dψ + g ( λ, ψ ) d Ω d − , ϕ = ϕ ( λ, ψ ) , (6.18)such that: f ( λ, ψ ) (cid:39) g ( λ, ψ ) (cid:39) (cid:96) f e λ/(cid:96) f , λ → ∞ , ψ fixed . (6.19)We took the above limit with ψ fixed, because this corresponds, in the pure AdS case,to going to the boundary of the spherical slicing by sending ξ → + ∞ as in (6.14). Itis in this limit, that the coordinate transformation (6.17) has to approach (6.6)The metric (6.18) describes a complicated time-dependent solution of the equationsof motion, evolving out of an initial state which corresponds to the nucleation ofthe bubble at ψ = 0. The time-evolution is with respect to a time-coordinate whichmatches onto the global time-coordinate ψ when we approach the boundary. This can be done by integrating the first order equations (2.18) near the flow endpoint, with theexpansion of the scalar functions W , S and T which can be found in Appendix F of [13]. We can choose the functions F and G in (6.17) such that the off-diagonal components vanish and setthe λλ -component to one. – 73 –. Close to the center of the bubble, ξ = 0, the coordinate transformation can be madeto match the one we found in pure AdS, i.e. the ξ → (cid:96) replacedby (cid:96) . With this choice, the ξ = 0 hypersurface which separates the exterior fromthe interior geometry will again be described in ( λ, ψ ) coordinates by the null curvesinh λ/(cid:96) = tan ψ , which will again approach the boundary as ψ → π/ ξ → + ∞ , global time ψ will again come to an end at ψ = π/
2. However this time, unlike in the previoussubsection, the metric will have a true singularity in the interior patch, which willalso approach the curve ξ = 0 from the interior. This means that this time, ψ = π/ ξ . Itsnear-boundary limit as ξ → + ∞ is given in equation (6.13) and we can use theasymptotic form of the coordinate transformation for large ξ , equation (6.10), tofind: ϕ ( λ, ψ ) (cid:39) ϕ f + ϕ + (cos ψ ) ∆ e − ∆ λ/(cid:96) f + . . . , λ → + ∞ . (6.20)From the point of view of the dual QFT which lives on R ψ × S d − , these asymptoticsrepresent a theory with the relevant coupling set to zero , but with a time-dependentvev of the dual operator, (cid:104)O(cid:105) ∝ ϕ + (cos ψ ) ∆ . (6.21)This agrees with what was already noticed in [11].From equations (6.20-6.21) it may seem that the (cid:104)O(cid:105) becomes infinite at ψ = π/ before we reach the final time. The reason is that for ψ arbitrarilyclose to π/
2, the scalar field (6.20) is not a small perturbation around the fixed point value ϕ f , and the metric is not close to the UV AdS metric. The usual expansion in leading(source) term and subleading (vev) term breaks down.To make the situation clearer, it is convenient to introduce a cut-off at a finite radius.The cut-off surface should have the same geometry as the one seen by the dual QFT.Therefore, if we want information about the fate of the theory on the cylinder, we have tointroduce a cut-off at fixed λ = ¯ λ . For convenience, we also define an energy cut-off:Λ = 1 (cid:96) f e ¯ λ/(cid:96) f . (6.22)The cut-off geometry is R × S d − , where R is parametrized by global time ψ . On thishypersurface, the scalar field takes values: ϕ cut-off ( ψ ) = ϕ (¯ λ, ψ ) . (6.23)– 74 – f ϕ Figure 18: This figure is a representation (not obtained from the actual solution) of theembedding of the constant- λ hypersurfaces (thick red lines) in the de-Sitter sliced RGflow geometry. The red arrows indicate the direction of increasing λ . The color-shadingindicates the value of the bulk scalar field, which interpolates between ϕ f ( ξ → + ∞ ) and ϕ ( ξ = 0).As ψ increases, the cut-off surface at fixed λ will approach the null trajectory spannedby the center of the bubble, which is approximately described by equation (6.9) (with (cid:96) replacing (cid:96) ). Along the way, the scalar field value will smoothly interpolate between ϕ f and ϕ , and only for small ψ (i.e. as long as ϕ stays close to ϕ f ) will expression (6.20) hold.In a time approximately given by π/ − / (Λ (cid:96) ), the scalar field on the cut-off surface willhave moved close to the value at the center of the bubble, ϕ .The situation can be described in an equivalent way, if we trace the cut-off surface λ = ¯ λ in the de Sitter-sliced geometry, as we show in figures
18 and 19. As one can see infigure 18, the cylinder cut-off surface has a trajectory which approaches the center of thebubble. As time moves forwards, on any cut-off surface the value of the scalar field willinevitably depart from the asymptotic value ϕ f and will approach the value at the centerof the bubble, ϕ , as one can see in figure 19.In the limit Λ → ∞ we can think of the situation as if the boundary value of the scalarfield receives an instantaneous “kick” at the global time ψ = π/
2, which displaces the filedby a finite amount from its fixed point value ϕ f . This is particularly clear in the bottomfigure 19. We can think of this as a Dirac δ -like source concentrated at ψ = π/
2. Assuch, the cylinder version of the CdL solution does not describe a solution with the same These figures are qualitative and not obtained from the full solution since they would require knowingthe full coordinate transformation (6.17). In practice they are obtained by neglecting the backreaction ofthe scalar field on the metric and using the transformation (6.5). – 75 – = Λ = Λ = Λ = ϕ f ϕ ϕ (¯ λ, χ ) χ Λ = Λ = Λ = Λ = ϕ f ϕ ϕ (¯ λ, ψ ) ψ Figure 19: The different curves illustrate the time-dependent behavior of the scalar field ϕ evaluated on different constant- λ cutoff hypersurfaces, as a function of de Sitter time χ (top figure) or the global time ψ (bottom figure). The cutoff in the figure is definedas (cid:96) f Λ ≡ sinh ¯ λ/(cid:96) f . At early times the scalar field displays the near-boundary behavior ϕ (cid:39) ϕ f + (cid:15) ∆ (cosh χ ) ∆ , where (cid:15) ≡ Λ − .boundary conditions obeyed by the global AdS solution: the latter satisfies source-less AdSboundary conditions on the cylinder for all times −∞ < ψ < + ∞ .From the point of view of the dual field theory on the cylinder, there are two possibleways to interpret the CdL solution, depending on two possible definitions of the gravitypath integral:1. We can define the dual field theory by imposing on the gravity side the usual boundaryconditions which accommodate the global AdS solution, i.e. that the asymptoticboundary is the full cylinder R × S d − with zero source term for the scalar field. Inthis case, the CdL solution does not correspond to a state of the theory, and it does– 76 –ot indicate a pathology of the CFT on the cylinder.2. We can instead define the boundary conditions on the gravity side in order to ac-commodate the CdL geometry as an allowed solution. Provided one can do thisconsistently , this defines a different field theory on the cylinder, which does notadmit the global AdS solution as one its states. This theory can only be extendedup to a finite global time, at which point the time-evolution reaches a singularity asa consequence of an infinite instantaneous “kick” at ψ = π/
2. This point of view iscloser to the one taken in [11].Notice that, although at the initial time (say ψ = 0, if we prepare the states usinganalytic continuation from the Euclidean solution) the CdL and global AdS states lookexactly the same from the point of view of the boundary sources, this does not contradictthe fact that they may have a different future evolution, because in AdS the initial conditionalone does not uniquely specify the state at subsequent times: in order to do that, one hasto specify boundary conditions for all future times. In this language, the two theories abovehave different time evolution because in theory 2 there is an instantaneous perturbationwhich occurs at a finite time. Therefore, the picture above is consistent and describesthe evolution of a stable CFT on the cylinder (case 1) versus an unstable one (case 2).In the second case, the work of [11] constituted a proposal for identifying this instabilityas originating from an unbounded CFT Hamiltonian, and curing it by a stabilizing extraterm.We have argued that the O ( d +1)-instanton solution we discussed in this paper mediatesvacuum decay of the field theory on dS d . This cannot be mapped to a solution describinga similar spontaneous vacuum decay process of the same theory on the full R × S d − . Onemay wonder if such a process can be mediated by different solutions with O ( d )-symmetry.If such solutions exist, they are certainly not found by looking at holographic RG flows on R × S d − in which the bulk field evolves only in the radial direction, of the form: ds = dλ − f ( λ ) dψ + g ( λ ) d Ω d − , ϕ = ϕ ( λ ) . (6.24)Solutions of this form are expected to exist, which have an IR endpoint λ such that either f ( λ ) = 0, with g ( λ ) finite, or g ( λ ) = 0 with f ( λ ) finite. In the first case, (6.24) describesa black hole. However, if we insist that the theory starts in its zero-temperature groundstate, then we must take Euclidean time to be non-compact, and regularity requires f ( λ )to be nowhere vanishing. The second case corresponds to non-trivial RG flows in R × S d − in which the S d − shrinks to zero size at the IR endpoint. Although these solutions areexpected to exist, they cannot mediate vacuum decay: since they are time-independentand the time-direction is infinite, the Euclidean solution cannot have a finite action. It is not entirely clear to us how to consistently define the boundary condition which allow the CdLstate in terms of the usual leading/subleading classification of the boundary behaviour of the scalar field,because on any cut-off surface the asymptotic expansion in terms breaks down after a finite time. Theremay be a distributional limit for the source term which exactly matches the behavior of the CdL solutionas Λ → ∞ , but this is not evident. Similar solutions where discussed for general S d × S d foliations in pure gravity [51], and recently for S × S foliations in Einstein-dilaton theory [52]. – 77 –y the above argument, any finite-action instanton preserving O ( d ) symmetry mustbe time-dependent: it has to reduce to the vacuum for the Euclidean time τ → −∞ , for allvalues of the radial coordinate. These instantons must therefore have the more general form(6.18). We emphasize however, that they are not obtained as coordinate transformationsof O ( d + 1)-symmetric instantons. Acknowledgements
We would like to thank Jos´e Barbon, Francesco Bigazzi, Aldo Cotrone, Roberto Emparan,Daniel Harlow, Javier Mas, Mukund Rangamani, Eliezer Rabinovici, Nick Tetradis forfruitful discussions. We also thank Jos´e Barbon, Daniel Harlow and Eliezer Rabinovici fora critical reading of the manuscript.This work was supported in part by the European Union via the ERC Advanced GrantSM-GRAV, No 669288. LW also acknowledges support from the European Union via theERC Starting Grant GEODESI, No 758792. JKG acknowledges the postdoctoral programat ICTS for funding support through the Department of Atomic Energy, Government ofIndia, under project no. RTI4001 and partial support through the Simons FoundationTargeted Grant to ICTS-TIFR “Science without Boundaries”.– 78 –
PPENDIXA. Expansions of A ( u ) and ϕ ( u ) near UV fixed points Here we record the asymptotic form for the scale factor ϕ ( u ) and A ( u ) near a UV fixedpoint that is reached for u → −∞ . As pointed out in section 2.2, the near-UV expansionsencode the QFT data of the UV fixed point in a precise way. There we focused on thenear-UV expansions for the function W ( ϕ ) finding two branches of solutions denoted by( − ) and (+) respectively. Here, for completeness, we record the corresponding expressionsfor ϕ ( u ) and A ( u ).Given the near-UV expressions for W ( ϕ ) and S ( ϕ ) one can derive the correspondingexpressions for ϕ ( u ) and A ( u ) in virtue of (2.18). The relevant expansions are given in[16, 13] and we refer the reader to these works for details. Without loss of generality weconsider a UV fixed point at an extremum of V at ϕ = 0 with the potential in the vicinitygiven by V ( ϕ ) = − d ( d − (cid:96) f − ∆( d − ∆)2 (cid:96) f ϕ + O ( ϕ ) (A.1)with ∆ = d (cid:32) (cid:115) d − d V (cid:48)(cid:48) (0) | V (0) | (cid:33) . (A.2)For the ( − )-branch of solutions one then finds: ϕ ( u ) = ϕ − (cid:96) ( d − ∆) f e ( d − ∆) u/(cid:96) f (cid:104) O (cid:16) R ( ζ ) (cid:96) f e u/(cid:96) (cid:17) + . . . (cid:105) (A.3)+ Cd | ϕ − | ∆ / ( d − ∆) ( d − ∆)(2∆ − d ) (cid:96) ∆ f e ∆ u(cid:96) f (cid:104) O (cid:16) R ( ζ ) (cid:96) f e u/(cid:96) f (cid:17) + . . . (cid:105) + . . . ,A ( u ) = − u(cid:96) f − ϕ − (cid:96) d − ∆) f d − e d − ∆) u/(cid:96) f − R ( ζ ) (cid:96) f d ( d − e u/(cid:96) f (A.4) − ∆ C | ϕ − | d/ ( d − ∆) (cid:96) df d ( d − − d ) e du/(cid:96) f + . . . . where ϕ − is an integration constants and the ellipses denote subleading terms for u → −∞ On the (+)-branch we instead obtain: ϕ ( u ) = ϕ + (cid:96) ∆ f e ∆ u/(cid:96) f (cid:104) O (cid:16) R ( ζ ) (cid:96) f e u/(cid:96) f (cid:17) + . . . (cid:105) + . . . , (A.5) A ( u ) = − u(cid:96) f − ϕ (cid:96) f d − e u/(cid:96) f − R ( ζ ) (cid:96) f d ( d − e u/(cid:96) f + . . . , (A.6)where ϕ + is an integration constants.The QFT data can be read from this as follows. First note the appearance of theboundary curvature R ( ζ ) which is the curvature of the manifold on which the QFT isdefined. On the ( − )-branch of solutions we further identify ϕ − with the source for theoperator O deforming the UV CFT. The parameter C has the interpretation of the vev of– 79 – in units of the source. In particular: (cid:104)O(cid:105) − = ( M (cid:96) f ) ( d − Cdd − ∆ | ϕ − | ∆ / ( d − ∆) , (A.7)where the subscript indicates that this only holds for ( − )-branch solutions. On the (+)-branch the source vanishes and the corresponding flows are purely driven by the vev of O .This is related to the parameter ϕ + as (cid:104)O(cid:105) + = ( M (cid:96) f ) ( d − (2∆ − d ) ϕ + . (A.8) The instanton action and the cancellation of divergences:
The instanton action asgiven in (2.33) is defined as the difference between the on-shell action for the interpolating(tunnelling) solution S E, inter and the on-shell action evaluated on the false vacuum solution S E, false . If the false vacuum at ϕ f is an AdS extremum, both individual expressions S E, inter and S E, false are formally divergent, with the divergence arising from the infinite volume ofthe AdS geometry associated with ϕ f . For the interpolating solution to describe tunnellingwith a non-zero decay rate the divergences have to cancel between S E, inter and S E, false sothat the instanton action B gives a finite value. Here we show that this is the case if theinterpolating solution is a (+)-branch solution but not if it is a ( − )-branch solution. To doso, we compute the divergent contributions to S E, inter and S E, false explicitly. In the lattercase this can be done as the full expression for A ( u ) is known, while in the former case thenear-boundary expressions recorded above are sufficient to obtain the relevant results.To be specific, in the following we restrict attention to D = d + 1 = 3 + 1. To regulatethe divergences we introduce the quantityΛ ≡ e A ( u uv ) (cid:96) f , (A.9)which diverges as Λ → ∞ for u uv → −∞ . This is convenient for making contact with pre-vious literature on holographic RG flows (see e.g. [31, 53]), where Λ has the interpretationas a UV cutoff for the dual field theory.We begin with S E, false . To this end we insert A ( u ) as given in (2.35) into (2.34).Eliminating u uv in favour of Λ using (A.9), after some algebra one finds: S E, false ( M (cid:96) f ) = 8 π (cid:26) − (cid:16) R ( ζ ) + 1 (cid:17) / (cid:27) (A.10)= − √ π (cid:26) ( R ( ζ ) ) / + Λ( R ( ζ ) ) / − √ (cid:27) + (vanishing for Λ → ∞ ) , where on the second line we isolated the two manifestly divergent terms for Λ → ∞ . Wenow turn to S ( − ) E, inter for a solution on the ( − )-branch, focusing on the divergent terms for u uv → −∞ . To this end we insert the near-boundary expansion (A.4) into (2.34). Again,– 80 –liminating u uv in favour of Λ using (A.9) we find the following: S ( − ) E, inter ( M (cid:96) f ) = Λ →∞ − √ π (cid:26) ( R ( ζ ) ) / (cid:16) O (cid:0) | ϕ − | Λ − − ∆) (cid:1)(cid:17) (A.11)+ Λ( R ( ζ ) ) / (cid:16) O (cid:0) | ϕ − | Λ − − ∆) (cid:1)(cid:17)(cid:27) + (manifestly finite for Λ → ∞ ) . We can repeat the analysis for S (+) E, inter , i.e. the action evaluated on a solution on the (+)-branch. Inserting (A.4) into (2.34) and re-expressing in terms of Λ one now finds: S (+) E, inter ( M (cid:96) f ) = Λ →∞ − √ π (cid:26) ( R ( ζ ) ) / (cid:16) O (cid:0) | ϕ + | Λ − (cid:1)(cid:17) (A.12)+ Λ( R ( ζ ) ) / (cid:16) O (cid:0) | ϕ + | Λ − (cid:1)(cid:17)(cid:27) + (manifestly finite for Λ → ∞ ) . We are now in a position to check to what extent the divergent terms cancel between S ( ± ) E, inter and S E, false . We begin with S ( − ) E, inter , i.e. the action evaluated on a solution onthe ( − )-branch. These solutions only exist if the UV fixed point reached for u → −∞ is a maximum of V . From (A.2) this in turn implies that these solutions only exist for d/ < ∆ < d , i.e. 3 / < ∆ < d = 3 as we consider here. It is then easy to seethat S ( − ) E, inter contains divergences which do not appear in S E, false and hence do not cancelbetween them. In particular, the subleading term on the first line of expression (A.11)scales as ∼ ( R ( ζ ) ) − / | ϕ − | Λ − , (A.13)which is always divergent for 3 / < ∆ <
3. No such divergence exists in S E, false andsubtracting S E, false from S ( − ) E, inter does hence not lead to a finite expression for the instantonaction B . As a result, ( − )-branch solutions cannot be understood as describing tunnelling.In contrast to ( − )-branch solutions, (+)-branch solutions can depart from both maximaand minima of V . Therefore, from (A.2), it follows that (+)-branch solutions can existfor any value ∆ > d/
2, i.e. ∆ > / d = 3. By inspecting the expressions (A.12) and(A.10) it is then easy to see that S (+) E, inter and S E, false share the same divergent terms. As aresult, the instanton action B obtained by subtracting S E, false from S (+) E, inter will be finite,as expected if the (+)-branch solution describes tunnelling. B. Sufficient condition for tunnelling solutions from AdS minima
In section 2.6 we identified a sufficient condition for a potential to admit O ( D )-instantonsdescribing tunnelling from a false AdS vacuum. In the mechanical picture introduced there,this condition is that a test particle in the inverted potential − V , when released from restat ϕ t , will overshoot ϕ f when released in that direction. An alternative formulation of that– 81 – ϕϕ f ϕ t B ( ϕ ) W flat ( ϕ )Figure 20: If the W flat ( ϕ ) solution from the true vacuum ϕ t does not touch B ( ϕ ) at ϕ f ,then there exists a curved W + solution.condition is that the potential admits a flat domain-wall solution/ flat-sliced holographicRG flow that has its IR end point at ϕ t .Here we record a quantitative version of this condition. This will be phrased in termsof solutions for the function W ( ϕ ) introduced in (2.18). One observation is that (for R ( ζ ) ≥
0) the equation of motion (2.20) implies that W ( ϕ ) ≥ B ( ϕ ) where B ( ϕ ) is definedas B ( ϕ ) = (cid:113) − d − d V ( ϕ ). Furthermore, W ( ϕ ) = B ( ϕ ) only occurs at critical points ofthe flow, i.e. at fixed points or where the flow in ϕ reverses direction.Now consider W flat ( ϕ ) by which we denote the tentative flat domain-wall solutionending at ϕ t whose existence would suffice to prove the existence of CdL instantons in thesame potential. In terms of W , the statement that the particle overshoots ϕ f is equivalentto the statement that ϕ f is not a critical point and hence W flat ( ϕ f ) > B ( ϕ f ). This isillustrated in fig. 20. The conditions for the existence of a CdL instanton are thus W flat ( ϕ t ) = B ( ϕ t ) , W flat ( ϕ f ) > B ( ϕ f ) . (B.1)In the following, we will exploit that W flat satisfies (2.19)–(2.21) with T = 0, which alsoimplies that S flat = W (cid:48) flat . Then we can write: W flat ( ϕ t ) − W flat ( ϕ f ) < B ( ϕ t ) − B ( ϕ f ) , (B.2) ⇒ (cid:90) ϕ t ϕ f dϕW (cid:48) flat ( ϕ ) < d − (cid:18) (cid:96) t − (cid:96) f (cid:19) , (B.3) ⇒ (cid:90) ϕ t ϕ f dϕ (cid:115) d d − W flat + 2 V < d − (cid:18) (cid:96) t − (cid:96) f (cid:19) . (B.4)where we have used B ( ϕ t ) = d − (cid:96) t , B ( ϕ f ) = d − (cid:96) f and (2.20).This condition can be used as follows. We can compute W flat ( ϕ ) by integrating (2.20)starting from ϕ t , where we implement the boundary condition W flat ( ϕ t ) = d − (cid:96) t , all the– 82 –ay to ϕ f . If inserted into (B.4), this condition is satisfied, the potential will admit a CdLinstanton describing tunnelling from ϕ f C. Flat domain-walls with a thin-wall limit: an alternative approach
In section 3.1, we constructed analytic flat domain-wall solutions that admitted a thin-walllimit in the sense described in 3.1. In particular, in the limit of an infinitely thin wall˙ ϕ ∼ δ , that is ˙ ϕ takes the form of a δ -function. Here, we consider the an alternative case:In particular, here we demand that in the limit of an infinitely thin wall ¨ A ∼ δ .The relevant equations of motion are again given in (2.7)–(2.9) with R ( ζ ) = 0 for a flatdomain-wall. As in section 3.1 we seek solutions that: • interpolate between two AdS space-times with length scales (cid:96) f and (cid:96) t , i.e.˙ A ( u → −∞ ) = − (cid:96) f , ˙ A ( u → + ∞ ) = − (cid:96) t , (C.1)with (cid:96) t < (cid:96) f : • interpolate at the same time between the values ϕ = ϕ f and ϕ = ϕ t .As before, in the extreme thin-wall-limit we require:˙ A = − (cid:96) f , u < u , − (cid:96) t , u > u . , ϕ = ϕ f , u < u ,ϕ t , u > u . (C.2)As stated at the beginning, here we satisfy the above by requiring that in the extremethin-wall limit we observe ¨ A ( u ) ∼ δ ( u − u ), while in the main text we instead required˙ ϕ ( u ) ∼ δ ( u − u ). We therefore propose the following ansatz for ¨ A ( u ):¨ A ( u ) = a (cid:15)L u − u (cid:15)L , (C.3)with a a parameter to be fixed later by boundary conditions, L is a fiducial length scaleand (cid:15) another numerical parameter that can be used to realise the thin-wall limit by letting (cid:15) → + . One can confirm that in this limit lim (cid:15) → ¨ A = a L δ ( u − u ) as demanded.Integrating once we obtain: ˙ A = a L + a L tanh u − u (cid:15)L , (C.4)where a is another integration constant. Implementing the boundary conditions (C.1) onethen finds: a = L(cid:96) f − L(cid:96) t , a = − L(cid:96) f − L(cid:96) t . (C.5) Recall that simultaneously requiring ¨ A ( u ) ∼ δ ( u − u ) and ˙ ϕ ( u ) ∼ δ ( u − u ) is not permitted by theequations of motion. – 83 –ntegrating ˙ A once more we obtain A ( u ) = ¯ A + a uL + a (cid:15) (cid:16) cosh u − u (cid:15)L (cid:17) , (C.6)with ¯ A an integration constant.We now turn to ϕ ( u ). Using the equation of motion (2.7) we can then obtain anexpression for ˙ ϕ , given by ˙ ϕ = 1 L (cid:114) − ( d − a (cid:15) u − u (cid:15)L . (C.7)Integrating, this gives ϕ ( u ) = ϕ + (cid:112) − ( d − a (cid:15) arctan (cid:16) sinh u − u (cid:15)L (cid:17) . (C.8)Finally, from (2.8) we obtain an expression for V . We can write this as a function of ϕ by inverting (C.8). The result is V = − d ( d − a L − d ( d − a a L sin χ − d ( d − a L sin χ − ( d − a (cid:15)L cos χ , (C.9)where we defined χ ≡ ϕ − ϕ f (cid:112) − ( d − a (cid:15) − π . (C.10)This is a potential with two minima at χ = ± π corresponding to ϕ = ϕ f and ϕ = ϕ t , respectively, separated by a barrier in-between. In the limit (cid:15) → ϕ i and ϕ f approach one another. D. Comparison with thin-wall approximation of Coleman-de Luccia
In [2] Coleman and de Luccia consider a potential with one true and one false minimum(which for consistency we label again as ϕ f and ϕ t ), which can be written as U ( ϕ ) = U ( ϕ ) + (cid:15)u ( ϕ ) . (D.1)Here U is a potential with two degenerate minima at ϕ f and ϕ t . The separation in energyinto false and true minimum is then achieved by adding (cid:15)u ( ϕ ) where (cid:15) is defined as thepotential difference between true and false minimum: (cid:15) ≡ U ( ϕ f ) − U ( ϕ t ) . (D.2)According to Coleman and de Luccia the thin-wall approximation applies “in the limit ofa small energy difference between the two vacuums,” that is as long as (cid:15) is small. Thisstatement is problematic, as (cid:15) is a dimensionful parameter and can hence only be smallcompared to another quantity sharing its dimensions. Here we assume that (cid:15) is small– 84 –ompared to the barrier separating the minima at ϕ f and ϕ t . To summarise, according toColeman and de Luccia the thin-wall approximation applies if the potential can be splitinto a part with degenerate minima and a correction that splits the degeneracy, but issuppressed compared to the degenerate part. The parameter that establishes the hierarchybetween the two parts is the ratio between the energy difference of the two minima andthe barrier height.In this work we studied CdL tunnelling in the sextic potential V defined in (4.2).In section 4.3 we then showed that potentials of this type admit thin-walled CdL bubblesolutions in the limit ∆ → ∞ , ∆ (cid:96) t (cid:96) f → ∞ . In the following, we show that in this thin-walllimit the potential V from (4.2) can indeed be brought into the form (D.1). To split thepotential into a part with degenerate minima and a correction, we separate it its symmetricand anti-symmetric parts about ϕ = ϕ f + ϕ t , respectively. In particular, we define U ( ϕ ) = 12 (cid:18) V ( ϕ ) + V ( ϕ f + ϕ t − ϕ ) (cid:19) + 12 (cid:18) V ( ϕ f ) − V ( ϕ t ) (cid:19) , (D.3) (cid:15)u ( ϕ ) = 12 (cid:18) V ( ϕ ) − V ( ϕ f + ϕ t − ϕ ) (cid:19) − (cid:18) V ( ϕ f ) − V ( ϕ t ) (cid:19) . (D.4)The constant shifts are included to ensure that U ( ϕ f ) = U ( ϕ t ) = V ( ϕ f ). Using theexplicit expression for V ( ϕ ) in (4.2) with the help of (4.1,4.3) we can then compute U ( ϕ )and (cid:15)u ( ϕ ) in the limit ∆ → ∞ , ∆ (cid:96) t (cid:96) f → ∞ . Using (3.5) to eliminate ϕ t we find, settingagain ϕ f = 0 for convenience: U ( ϕ ) = 1 (cid:96) f (cid:20) − d ( d −
1) + a ∆ (cid:16) O (cid:0) (cid:1)(cid:17) ϕ + a ∆ / (cid:16) O (cid:0) (cid:1)(cid:17) ϕ (D.5)+ a ∆ (cid:16) O (cid:0) (cid:1)(cid:17) ϕ + a ∆ / (cid:16) O (cid:0) (cid:1)(cid:17) ϕ + a ∆ (cid:16) O (cid:0) (cid:1)(cid:17) ϕ (cid:21) ,(cid:15)u ( ϕ ) = 1 (cid:96) f (cid:20) b ∆ (cid:16) O (cid:0) (cid:1)(cid:17) ϕ + b ∆ / (cid:16) O (cid:0) (cid:1)(cid:17) ϕ (cid:21) , (D.6)where a , , , , and b , are coefficients depending on (cid:96) t /(cid:96) f and v . The observation nowis that the potential (cid:15)u is suppressed by one power of ∆ compared to U . That is, themonomial ϕ in (cid:15)u comes with a power of ∆, while in U the same monomial is premultipliedby ∆ . Similarly, the monomial ϕ in (cid:15)u is proportional to ∆ / , while in U the cubicterm has a prefactor ∆ / . Recall from section 4.3 that in the thin-wall approximation,the ratio of the energy difference between the minima and the barrier height scales as ∼ / ∆, see (4.4). Therefore, 1 / ∆ behaves exactly as the small parameter in the thin-wallapproximation of Coleman and de Luccia. We consequently conclude, that in the thin-walllimit (∆ → ∞ ), the sextic potential V in (4.2) can indeed be brought into the schematicform (D.1) proposed by Coleman and de Luccia. References [1] S.R. Coleman,
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