The WKB approximation and tunneling in theories with non-canonical kinetic terms
Mariana Carrillo Gonzalez, Ali Masoumi, Adam R. Solomon, Mark Trodden
TThe WKB approximation and tunneling in theories withnon-canonical kinetic terms
Mariana Carrillo Gonz´alez, ∗ Ali Masoumi, † Adam R. Solomon, ‡ and Mark Trodden § Center for Particle Cosmology, Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA Institute of Cosmology, Department of Physics and Astronomy,Tufts University, Medford, MA 02155, USA (Dated: March 6, 2017)
Abstract
Tunneling is a fascinating aspect of quantum mechanics that renders the local minima of apotential meta-stable, with important consequences for particle physics, for the early hot stageof the universe, and more speculatively, for the behavior of the putative multiverse. While thisphenomenon has been studied extensively for systems which have canonical kinetic terms, manytheories of fundamental physics contain fields with non-canonical kinetic structures. It is thereforedesirable to have a detailed framework for calculating tunneling rates and initial states aftertunneling for these theories. In this work we present such a rigorous formulation and illustrate itsuse by applying it to a number of examples. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - t h ] M a r ONTENTS
I. Introduction 2II. WKB for arbitrary Hamiltonians 3III. Computing the decay rate 6IV. WKB in a general scalar quantum field theory 8V. Application: Decay rates in general scalar-field theories 9VI. Discussion 17Acknowledgments 17A. Galileons 17References 20
I. INTRODUCTION
The process of quantum tunneling allows transitions out of local minima of an energyfunctional to vacua of lower (or higher) energies. This occurs through a first-order phasetransition mediated by the nucleation of bubbles of the new vacuum inside the old. Thisprocess can have important consequences, not only for particle physics, but also for cosmologysince, starting from its hot initial state, the universe may have gone through several of thesephase transitions before it settled into its current vacuum. It is quite possible that the initialstate for inflation may have been set by the state after quantum tunneling. It is even possiblethat our own vacuum may be susceptible to such transitions. Indeed, the Higgs potentialwith the currently accepted values of top quark and Higgs masses is metastable and, in theabsence of new physics, can decay, albeit after a rather long time (see, for example, Ref. [1]).Decay rates are (almost) always calculated in a semiclassical regime using the WKBapproximation. The generalization of the WKB approximation to cases with more than onedegree of freedom was first presented in Refs. [2, 3]. This was extended to field theories inseveral important works [4–6] and later to cases which include gravitational back-reaction ontunneling [7, 8]. Some analytic approximations for tunneling rates in thin-wall regime weredevised in Refs. [5, 9].The possibility of the existence of the string landscape, and the attendant possibilityof many phase transitions in such a complex potential has attracted further interest invacuum decay processes. However, despite progress in understanding vacuum tunneling,our only analytic insight, through the thin-wall approximation, is solely applicable to caseswhere the tunneling action is large, and as such is only relevant to a very specific class ofreal-world processes. Accounting for gravity and spacetime curvature brings about a newset of problems. There are many conceptual complications in the presence of gravity, suchas the measure problem (see Ref. [10] for a review) or the interpretation of Hawking-Mosschannels of tunneling [11]. A further computational issue is that we do not know whether thetunneling rate in the presence of gravity is dominated by solutions which are O (4) symmetric2n Euclidean space, despite some effort in this direction [12, 13]. Furthermore, the stringlandscape and most other putative landscapes usually have a large number of fields. Theprocess of tunneling here is plagued with many computational difficulties, although thesewere recently circumvented in an efficient numerical package [14].Our goal in this paper is to provide a careful analysis of another important issue in anumber of models relevant to cosmology, that of the problem of tunneling in theories withnon-canonical kinetic terms (for a review of such models, see Refs. [15, 16]). These theoriesappear in many cases in modern cosmological models, and, as we shall see, the decay ratecan be highly non-intuitive (see also Ref. [17]).This paper is organized as follows. In section II we study a general formalism for theWKB approximation for arbitrary Hamiltonians in quantum mechanics, and in section IIIwe calculate decay rates. We generalize these results to quantum field theory in section IV,and provide several applications of our results in section V before concluding in section VI. II. WKB FOR ARBITRARY HAMILTONIANS
In a system described by a Hamiltonian H ( q , p ), we can find the classical motion bysolving the Hamilton-Jacobi equation, H ( q , ∇ S ) + ∂S∂t = 0 , (II.1)where q = ( q , · · · , q n ) are the coordinates, p = ( p , · · · , p n ) are the canonical momenta, and S is the Hamilton principal function given by S ( q , α ; t ) = (cid:90) q p ( q (cid:48) , α ) · d q (cid:48) − (cid:90) H d t , (II.2)satisfying ∇ S = p . The corresponding quantum system is described by the Hamiltonianoperator ˆ H , related to the classical one byˆ H = 1(2 π (cid:126) ) n (cid:90) d p d q d u d v F ( u · v / (cid:126) ) H ( q , p , t ) e ( i/ (cid:126) )[( q − ˆ Q ) · u +( p − ˆ P ) · v ] , (II.3)where ˆ Q , ˆ P are the coordinate and momentum operators respectively, and F ( u · v / (cid:126) ) is thetransformation function [18] that defines the operator ordering and must be real in order toensure that ˆ H is Hermitian. This Hamiltonian appears in the Schr¨odinger equation thatdescribes the quantum system, i (cid:126) ∂ψ ( q , t ) ∂t = ˆ H (cid:18) q , − i (cid:126) ∂∂ q ; t (cid:19) ψ ( q , t ) . (II.4)The semi-classical solution, often referred to as the WKB approximation, for this equationup to O ( (cid:126) ) is given by [20, 21] ψ ( q , t ) = N (cid:115) det (cid:18) ∂ S∂ q ∂ α (cid:19) e i/ (cid:126) S ( q , α ; t ) , (II.5) If the Hamiltonian is not Hermitian, an extra exponential term appears in the WKB wave function [19]. As long as F (0) = 1 and F (cid:48) (0) = 0, which is satisfied for the most common transformation functions [20]. FV q TV qV ( q ) qq q FV TVTP
FIG. 1. An example of a potential where tunneling can happen from the false vacuum q FV to thetrue vacuum q TV . where N is a normalization constant, α i with i = 1 , · · · , n are integration constants that are determined by the initial conditions. We may fix the first constant as α = E ,while the remaining α i ’s are chosen depending on the system at hand. For example, if H = H ( x )+ H ( y )+ H ( z ), we can pick α = E tot = H , α = E x = H ( x ), and α = E y = H ( y ),whereas if we have spherical symmetry, then some of the α i ’s will correspond to angularmomenta. We can see that the time-independent wave function is approximated in thesemi-classical limit as ψ ( q ) = N (cid:115) det (cid:18) ∂ S∂ q ∂ α (cid:19) e i/ (cid:126) (cid:82) q p ( q (cid:48) , α ) · d q (cid:48) . (II.6)To O ( (cid:126) ), we may neglect the pre-factor in eq. (II.6), keeping only the leading-order exponen-tial behavior. This can be understood more easily by recalling that WKB is a semi-classicalapproximation in (cid:126) ; that is, ψ WKB = e i ( σ + (cid:126) σ ) / (cid:126) . The order (cid:126) factor is σ ≡ i (cid:90) q p ( q (cid:48) , α ) · d q (cid:48) , (II.7)while the order (cid:126) contribution, σ , is logarithmic and gives rise to the aforementionedpre-factor. The WKB approximation is widely used to solve tunneling problems. Theone-dimensional case is straightforward, since there is only one tunneling path to follow. Themulti-dimensional case becomes more complicated due to the different paths through whichtunneling is possible. Banks, Bender, and Wu [2] solved this problem by considering themost probable escape paths (MPEPs), which are expected to dominate the amplitude. Fromeq. (II.6) we can see that the largest contribution to the amplitude comes from paths whichminimize the WKB exponent, i.e., the MPEPs are the paths that satisfy δ (cid:90) q TP q FV p · d q = 0 , (II.8)where q FV and q TP are the locations of the false vacuum and the turning point, defined by V ( q FV ) = V ( q TP ) = E ; a typical setup (compressed to one dimension) is illustrated in fig. 1.In the classically forbidden region, through which tunneling occurs, p is imaginary and thus In the Hamilton-Jacobi formalism these are the new momenta; the fact that they are constant in timefollows from the requirement that the transformed Hamiltonian be identically zero. Q ( λ )parametrized by λ and notice that, in the classically-forbidden region, we have ∇ σ · ∇ σ = | p | , (II.9)where the gradient is taken with respect to q . We can expand the gradient in terms of thetangent vector to the curve Q , v (cid:107) = ∂ Q /∂λ , and the vectors orthogonal to Q , v i ⊥ , as ∇| q = Q = v (cid:107) | v (cid:107) | (cid:0) v (cid:107) · ∇ (cid:1)(cid:12)(cid:12) q = Q + (cid:88) i v i ⊥ | v i ⊥ | (cid:0) v i ⊥ · ∇ (cid:1)(cid:12)(cid:12) q = Q . (II.10)This decomposition is useful here because MPEPs are defined as the paths that satisfy v i ⊥ · ∇ σ (cid:12)(cid:12) q = Q = 0 ∀ i . (II.11)To take advantage of this, let us reparametrize the curve as Q ( λ ( s )), with s the properdistance along Q , d s = | d Q | = (cid:114) d Q d λ · d Q d λ d λ = | v (cid:107) | d λ , (II.12)so that we have ∇ σ | q = Q = v (cid:107) | v (cid:107) | (cid:0) v (cid:107) · ∇ σ (cid:1)(cid:12)(cid:12) q = Q = v (cid:107) | v (cid:107) | d σ d s . (II.13)Using this in eq. (II.9), we can finally rewrite eq. (II.8) as δ (cid:90) s ( q TP ) s ( q FV ) | p ( Q ( s ) , E ) | d s = 0 , (II.14)where | p | is found by solving H ( q , p ) = E . The variation in eq. (II.14) keeps the startingpoint fixed but not the end point, with energy conserved along the path. The fact that theendpoint is not fixed gives rise to the boundary conditiond q d λ (cid:12)(cid:12)(cid:12)(cid:12) q = q TP = 0 . (II.15)Now let us choose the parameter λ such thatd s d λ = (cid:12)(cid:12)(cid:12)(cid:12) ∂H∂ p (cid:12)(cid:12)(cid:12)(cid:12) , (II.16)in which case eq. (II.14) translates todd λ | p | (cid:12)(cid:12)(cid:12) ∂H∂ p (cid:12)(cid:12)(cid:12) d Q d λ − (cid:12)(cid:12)(cid:12)(cid:12) ∂H∂ p (cid:12)(cid:12)(cid:12)(cid:12) ∇| p | = 0 . (II.17) This is because in the multidimensional case there is generally not a single point q TP but rather a surfaceof points satisfying the condition V ( q TP ) = E .
5n the following, we will assume that there is a well-defined Legendre transformation thatallows us to switch between the Hamiltonian and Lagrangian formulations. A careful analysis,taking into account that we are in the classically forbidden region, shows that eq. (II.17) canbe written as dd λ (cid:32) ∂L E ∂ d Q d λ (cid:33) − ∂L E ∂ Q = 0 , (II.18)where we have used Hamilton’s equations and L E is the Euclidean Lagrangian. This showsthat the MPEP can be found by solving the Euclidean equations of motion. Note that,since analytic continuation can lead to multi-valued functions, the MPEP Q ( λ ) could bemulti-valued.The fact that the MPEP can be found by solving the Euclidean equations of motion haspreviously been shown for canonical kinetic terms and here we have extended the proof forgeneric kinetic terms of the form T ( q , ˙ q ). That this result applies for generic kinetic terms T ( q , ˙ q ) is one of the main results of this paper. Later, we will show that this result alsoholds for scalar fields with second-order equations of motion.
III. COMPUTING THE DECAY RATE
Once we have an approximation for the wave function, we can use it to calculate thedecay rate in a potential with two non-degenerate minima as in fig. 1. The decay rate of asystem is defined as Γ = − P FV dd t P FV , (III.1)where P FV is the probability of being in the false vacuum. As discussed in Refs. [24, 25],this definition is only meaningful for times t slosh (cid:28) t (cid:28) t non-lin , where t slosh = ω − with ω FV the frequency of oscillation in the false vacuum and t non-lin the scale at which non-linearitiesbecome important. During t < t slosh , high energy modes in the initial wave function willdecay, and it is not until these modes decay that we truly observe the decay rate of the falsevacuum. We may write the decay rate asΓ = 1 m (cid:82) | ψ E ( q TP ) | p TP · d q TP (cid:82) FV d q | ψ E ( q ) | , (III.2)where TP is the turning point, with the integration over all possible turning points, FVstands for the false vacuum, and ψ E is an energy eigenstate. Using the WKB approximationup to O ( (cid:126) ), this translates toΓ = det (cid:16) ∂ S∂ α ∂ q (cid:17)(cid:12)(cid:12)(cid:12) q = q TP | p TP | m (cid:82) q FV det (cid:16) ∂ S∂ α ∂ q (cid:17) d q e − B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = Q , B ≡ i (cid:126) (cid:90) q TP q FV p · d q , (III.3)where Q is the MPEP and B is the WKB exponent. For a canonical kinetic term in onedimension we have det (cid:18) ∂ S∂α∂q (cid:19) = 2 m | p | , (III.4)6 FV x TV x t xV x FV x TV x t x - VEuclidean picturequantum tunneling Classicalroll down quantum tunnelingLorentzian picture
FIG. 2. Lorentzian and Euclidean pictures of the false vacuum tunneling; in the Euclidean picturethe potential is inverted. which leads to the well-known resultΓ = | p FV | m | q FV | e − B (cid:12)(cid:12)(cid:12)(cid:12) q = Q . (III.5)In the case of a canonical kinetic term, the pre-factor has a clear physical interpretation:writing it as v FV / | q FV | , it can be understood as the rate at which the wave function hits thebarrier. However, for the case of non-canonical kinetic terms, it is not simple to find a similarinterpretation, and the rest of this paper will be concerned solely with the exponent B .We now review the calculation of the WKB exponent to leading order for the tunnelingof the false vacuum, illustrated in fig. 2. The tunneling between an unstable vacuum and alower energy (local or global) vacuum, is commonly called the bounce , which is just a specifickind of instanton . The WKB exponent is calculated by solving the Euclidean equations ofmotion, i.e., with the potentials inverted. In the bounce potential, the particle rolls up tothe turning point and then falls back down to the false vacuum (this is, of course, the originof the term “bounce” for this process).To relate the WKB exponent with the Euclidean action S E = iS , we begin by usingeq. (II.2), which tells us that iS ( q ) = i (cid:90) q p · d q (cid:48) + iS ( q FV ) , (III.6)where we have used the fact that the kinetic energy at q FV vanishes, so that we can set (cid:90) H d t = (cid:90) V ( q FV )d t = − S ( q FV ) , (III.7)where H = E is conserved. Given this and being careful with the integration limits, we canwrite the exponent B for the tunneling of the false vacuum as B bounce = S E ( q ) − S E ( q FV ) . (III.8) In general, an instanton is a configuration with a finite, non-zero action that solves the classical equationsof motion.
7t is important to note that this relation only holds at stationary points of B (and S ), i.e.,when the equations of motion are satisfied. One should realize that the path in Euclideanspace goes from q FV at τ = −∞ to q TP at a finite τ (which can generally be taken to be τ = 0) and back to q FV at τ = ∞ ; this path gives the correct factor in eq. (III.8). Giventhis, the unstable vacuum decay rate to O ( (cid:126) ) is written asΓ = e − B (cid:12)(cid:12) q = Q = e − (cid:126) ( S E ( q ) − S E ( q FV )) (cid:12)(cid:12)(cid:12) q = Q , (III.9)which is a well-known result. IV. WKB IN A GENERAL SCALAR QUANTUM FIELD THEORY
In this section we generalize the results obtained in sections II and III for multi-dimensionalquantum mechanics to quantum field theory with a scalar field, again closely followingRefs. [22, 23]. Crucially, we will allow for a general enough kinetic structure for our formalismto cover all Lorentz-invariant scalar-field theories with equations of motion that are secondorder, and therefore avoid the Ostrogradski ghost instability. As discussed in appendix A, theLagrangians for these theories take the form (up to boundary terms) L = L ( φ, ˙ φ, ∇ φ, ∇ φ ),where ∇ φ = ∂ i ∂ j φ is a matrix (rather than the scalar Laplacian). Defining the canonicalmomentum as usual, Π = d L/ d ˙ φ , we can therefore write the Hamiltonian in the form H = (cid:90) d q (cid:2) T ( φ, Π , ∇ φ, ∇ φ ) + G ( ∇ φ, ∇ φ ) + V ( φ ) (cid:3) . (IV.1)Consider a wave functional ψ [ φ ], a functional of φ ( q ) whose squared norm is the probabilitydensity for a configuration φ ( q ). This will obey the generalized Schr¨odinger equation (cid:20)(cid:90) d q T (cid:18) φ, − i (cid:126) δδφ ( q ) , ∇ φ, ∇ φ (cid:19) + U [ φ ] (cid:21) ψ [ φ ] = Eψ [ φ ] , (IV.2)where the functional U [ φ ] is the potential energy that determines the possibility of tunneling,defined by U [ φ ] = (cid:90) d q (cid:0) G ( ∇ φ, ∇ φ ) + V ( φ ) (cid:1) . (IV.3)The classically-forbidden region is given by E < U [ φ ]. The configuration space is the spaceof real-valued functions on R (or the relevant space depending on the problem at hand)satisfying the appropriate boundary conditions.We proceed to make a semi-classical approximation as in the quantum mechanics case; todo so, we expand the wave function as ψ [ φ ] = e i (cid:126) σ [ φ ] = e i (cid:126) ( σ [ φ ]+ (cid:126) σ [ φ ]+ ··· ) . (IV.4)In the following, we solve for the wave function to O ( (cid:126) ). Substituting the semi-classicalexpansion in eq. (IV.2) gives, at leading order, (cid:90) d q T (cid:18) φ, δσ [ φ ] δφ ( q ) , ∇ φ, ∇ φ (cid:19) + U [ φ ] = E . (IV.5)8xpressing the canonical momentum as a function of φ and its gradients, by making use ofthe conservation of energy equation, we find that the leading-order contribution is σ [ φ ] = (cid:90) φ d φ (cid:48) Π( φ (cid:48) , ∇ φ (cid:48) , ∇ φ (cid:48) ) . (IV.6)The next step is to find the MPEP, i.e., the curve in the space of real-valued functions (orfield configurations) that minimizes σ [ φ ]. We will call this curve Φ( λ, q ), parametrized by λ ,denote the vector parallel to this curve by v (cid:107) ( λ, q ) = ∂ Φ /∂λ , and label the continuous set ofperpendicular vectors v ⊥ ( λ, q ; q ). In this case, the condition defining the MPEP is (cid:90) d q v ⊥ ( λ, q ; q ) δσ [ φ ] δφ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) φ =Φ = 0 ∀ q . (IV.7)We reparametrize the curve as Φ( λ ( s ) , q ), with s the proper distance along the curve, givenby d s = | dΦ | = (cid:115)(cid:90) d q (cid:18) dΦd λ (cid:19) d λ . (IV.8)Using this parametrization, we find that the MPEP satisfies δ (cid:90) Π(Φ , ∇ Φ , ∇ Φ)d s = 0 , (IV.9)and since d s d λ = (cid:12)(cid:12)(cid:12)(cid:12) ∂H∂ Π (cid:12)(cid:12)(cid:12)(cid:12) , (IV.10)we then find that eq. (IV.9) translates intodd λ (cid:32) Π (cid:12)(cid:12) ∂H∂ Π (cid:12)(cid:12) dΦd λ (cid:33) + ∇ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ∂H∂ Π (cid:12)(cid:12)(cid:12)(cid:12) ∂ Π ∂ ∇ Φ (cid:19) − ∇ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ∂H∂ Π (cid:12)(cid:12)(cid:12)(cid:12) ∂ Π ∂ ∇ Φ (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) ∂H∂ Π (cid:12)(cid:12)(cid:12)(cid:12) ∂ Π ∂ Φ = 0 . (IV.11)This is again equivalent to finding the Euclidean equations of motions,dd λ (cid:32) ∂L E ∂ dΦd λ (cid:33) + ∇ (cid:18) ∂L E ∂ ∇ Φ (cid:19) − ∇ (cid:18) ∂L E ∂ ∇ Φ (cid:19) − ∂L E ∂ Φ = 0 , (IV.12)that is, the MPEP is a stationary solution of the Euclidean action. The generalization ofthis calculation to include higher-order gradients in T and G is straightforward, although weremind the reader that for the most general scalar field theories with second-order equationsof motion, these terms only depend on spatial gradients up to ∇ φ . We conclude that evenin the presence of non-canonical kinetic terms, the dominant contribution to the tunnelingrate comes from paths which extremize the Euclidean action. V. APPLICATION: DECAY RATES IN GENERAL SCALAR-FIELD THEORIES
To this point we have established a rigorous formalism for computing the decay rates fortunneling processes in scalar field theories with kinetic terms of the form T ( φ, ˙ φ, ∇ φ, ∇ φ, · · · ).9n this section we explicitly compute decay rates for general theories of a single scalar fieldwith second-order equations of motion, known broadly as galileons. We find a simple andfamiliar expression for the decay rate, and discuss how decay of the false vacuum could occurconsiderably more quickly than in theories with just a canonical kinetic term.Consider a scalar field φ defined on flat space and endowed with a potential V ( φ ) withtwo minima, one at slightly higher potential than the other, as shown in fig. 1. A statelocalized in the false vacuum, denoted by V FV , can decay to the true vacuum at V TV . We willdenote the value of φ at these minima by φ + and φ − , respectively. We have shown abovethat, regardless of the choice of kinetic term, the decay rate per unit volume for this processis given by Γ V ∼ e − B , (V.1)where B ≡ ∆ S E is the difference between the Euclidean action for two different solutions: a“bounce,” in which the scalar field rolls from the true vacuum to the false vacuum, and asolution in which the field lives at the false vacuum for all time. The analysis in this sectionlargely follows the classic work of Coleman [5].Before diving into general cases, with all their attendant abstraction, let us start byconsidering a particularly simple example of a non-canonical kinetic term: P ( X ) theories,with an action of the form S = (cid:90) d x [ P ( X ) + V ( φ )] , (V.2)where X ≡ − ( ∂φ ) and we assume (without loss of generality) that P (0) = 0. The conditionsfor the bounce are consistent with an O (4)-symmetric solution for φ [5], so the bouncesolution is generally taken to have this symmetry. For a solution with this symmetry, theEuclidean action is given by S E = 2 π (cid:90) ρ ( P + V ) d ρ , (V.3)with X = ˙ φ , where ρ is the Euclidean O (4) radial coordinate.Following Ref. [17] (in which tunneling was studied in a particular P ( X ) theory) we willmake a slightly non-standard definition of L as the Lagrangian with the spherical measurefactor divided out, S E ≡ π (cid:90) ρ L d ρ , (V.4)and define a similarly non-standard canonical momentum as π φ ≡ ∂L∂ ˙ φ . (V.5) These theories are introduced in appendix A; in particular, the galileons Lagrangians are given by eq. (A.2).We emphasize that these Lagrangians completely cover theories of a single scalar field on a flat backgroundwith second-order equations of motion. The field rolls from true vacuum to false because motion in Euclidean time can be thought of as motion inthe inverted potential. In fact, we can consider a function P ( φ, X ) without affecting our results; however, for clarity we will startoff by cleanly separating the kinetic and potential terms. The more general case is discussed later in thissection. For a canonical kinetic term it can be proven that e − B is extremized for an O (4)-symmetric solution [26],though no such proof currently exists for non-canonical terms. L = P + V , the canonical momentum is π φ = ∂L∂ ˙ φ = 2 P X ˙ φ = 2 P X √ X , (V.6)so that the Hamiltonian defined with respect to this L is H = π φ ˙ φ − L = 2 P X X − P − V . (V.7)This Hamiltonian is not conserved, since the spherical measure induces a friction term in theequation of motion. The “true” conserved Hamiltonian is ρ H , whose associated canonicalmomentum is ∂ ( ρ L ) /∂ ˙ φ = ρ π φ . Hamilton’s equations then imply˙ π φ = − ∂H∂φ − ρ π φ . (V.8)Now let us consider the bounce and false-vacuum solutions for φ in the thin-wall approxi-mation in which (cid:15) ≡ V FV − V TV (V.9)is small. In this approximation the thickness of the wall is very small compared to the radiusof the wall, ¯ ρ , which we can define as the point at which φ ( ¯ ρ ) = ( φ + + φ − ). Moreover, inthis limit our nonstandard H is approximately conserved: since the field should be stationaryin the two vacua, the difference in H from one side of the wall to the other should just beproportional to the difference in the potentials, and therefore to (cid:15) . Accordingly we can write H + O ( (cid:15) ) = E for a conserved E , implying2 P X X − P = E + V + O ( (cid:15) ) . (V.10)We may obtain the energy E by evaluating this for ρ > ¯ ρ , where both the bounce solutionand the always-false-vacuum solution are in the false vacuum, φ = φ + . Since ˙ φ has to vanishat this point in both solutions, and P (0) = 0 by construction, the left-hand side vanishes,so we have E = − V FV . We can simplify this further by defining a new function, V ( φ ), as adeformation of the potential which vanishes, along with its first derivative, at the two vacua,i.e., V ( φ ) ≡ V ( φ ) − V FV + O ( (cid:15) ) , V ( φ ± ) = V (cid:48) ( φ ± ) = 0 . (V.11)Up to O ( (cid:15) ) we may simply replace the right-hand side of eq. (V.10) with V ( φ ),2 P X X − P = V + O ( (cid:15) ) . (V.12)To calculate the bounce factor B , B = S E ( φ ) − S E ( φ + ) , (V.13)with φ the bounce solution, we split the computation up into three different regions: in thetrue vacuum, in the false vacuum, and on the wall, i.e., B = B FV + B wall + B TV . (V.14)Equivalently, this can be thought of as splitting the integrals into pieces from 0 to ¯ ρ (thetrue vacuum), near ¯ ρ (the wall), and from ¯ ρ to ∞ (the false vacuum).11n the false vacuum we simply have B FV = [ S E ( φ + ) − S E ( φ + )] | ∞ ¯ ρ = 0, where in each S E we are only integrating from ρ = ¯ ρ to ρ = ∞ . In the true vacuum, B TV = 2 π (cid:90) ¯ ρ ρ ( V FV − V TV )d ρ = − π ¯ ρ (cid:15) . (V.15)Finally, on the wall we have ρ ≈ ¯ ρ , so that in the thin-wall approximation, (cid:82) wall ρ d ρf ( φ, ˙ φ ) =¯ ρ (cid:82) φ + φ − d φf / ˙ φ , for some generic function f ( φ, ˙ φ ). We can then calculate the portion of B onthe wall as B wall = 2 π ¯ ρ (cid:90) φ + φ − P + V ˙ φ d φ − π ¯ ρ (cid:90) φ + φ − V FV ˙ φ d φ = 2 π ¯ ρ S , (V.16)to leading order in (cid:15) , where S ≡ (cid:90) φ + φ − π φ d φ (V.17)is the tension of the bubble wall. Putting all these together we find the well-known result [5], B = 2 π ¯ ρ S − π ¯ ρ (cid:15) . (V.18)We can determine ¯ ρ by demanding that it extremize B ; i.e. that ∂B/∂ ¯ ρ = 0, yielding¯ ρ = 3 S (cid:15) . (V.19)This gives us the usual result, B = 27 π S (cid:15) . (V.20)Our main result for decay rates in P ( X ) theories, summarized in eqs. (V.17) and (V.20),reduces to the classic result when we choose a canonical kinetic term [5], and also includesthe results of Ref. [17], which studied the case of a Dirac-Born-Infeld (DBI) kinetic term,which is a P ( X ) theory with P ( X ) ∼ f − ( √ f X − S (cf. eq. (V.17)). This shows up in the tunneling rate as e − ( ··· ) S , so minor alterations tothe kinetic structure of a theory can affect its tunneling rate by several orders of magnitude.We can see this explicitly by solving for π φ using the conservation equation for theHamiltonian, H = V + O ( (cid:15) ) , (V.21)in order to determine S in terms of P ( X ) and V ( φ ). For example, taking a canonical kineticterm, P ( X ) = X/
2, we have π φ = ˙ φ = √ V , leading to S = (cid:90) φ + φ − (cid:112) V d φ , (V.22)which appears in the standard result for the tunneling rate [5]. The analogous result for ageneral P ( X ) is obtained by solving eq. (V.12) for π φ = 2 P X √ X . This can lead to important12hanges in π φ and therefore, through S , in the decay rate Γ. We emphasize that by phrasingour result in terms of the non-standard canonical momentum π φ , we can write the decay ratefor P ( X ) theories in a simple form that incorporates both the classic result for a canonicalkinetic term [5] as well as more recent extensions [17].Now let us add one layer of abstraction by considering a general Lagrangian dependingon φ and ˙ φ ; in practice this amounts to a P ( X ) theory with φ dependence, but it will provea useful arena for building a more abstract calculation of the decay rate which we can thenapply to the general second-order scalars.Energy conservation gives H = π φ ˙ φ − L = E + O ( (cid:15) ) , (V.23)and by evaluating this expression at φ = φ + we find E = − L ( φ + , φ = 0 at thispoint, E is a constant and can be thought of as analogous to − V FV . Calculating B in threeparts as above, we find B FV = 0, B TV = 2 π (cid:90) ¯ ρ ρ [ L ( φ − , − L ( φ + , ρ = − π ¯ ρ (cid:15) , (V.24)where we have defined (cid:15) ≡ L ( φ + , − L ( φ − , B wall = 2 π ¯ ρ (cid:90) φ + φ − π φ d φ ≡ π ¯ ρ S , (V.25)with the rest of the calculation of Γ /V following as above. We conclude that for a generalLagrangian depending on φ and ˙ φ , the tunneling rate is given by a simple generalization ofthe classic result, Γ V ∼ e − B , (V.26)where B = 27 π S (cid:15) , (V.27) S = (cid:90) φ + φ − π φ d φ . (V.28)Finally, let us extend our calculation of the decay rate to the full set of scalar field theorieswith second-order equations of motion, the well-known galileons and their generalizations.While these Lagrangians can depend on second derivatives of φ in specific, antisymmetriccombinations (cf. eq. (A.2)), integrations by parts can eliminate the dependence of L on ¨ φ at the expense of introducing explicit ρ dependence, as shown explicitly in appendix A. Thisis a consequence of the galileon structure, which ensures that the equations of motion aresecond-order, and would not remain true for Lagrangians with general functions of ∂ φ .We can therefore consider the full slate of healthy theories of a single scalar field bygeneralizing the above analysis to L = L ( φ, ˙ φ, ρ ). We will assume that L loses its ρ dependence when ˙ φ = 0, i.e., ∂L ( φ, , ρ ) ∂ρ = 0 , (V.29)13s this holds for the galileons and rather simplifies the analysis. Note that this impliesthat, away from the wall, L is constant.Most of the features of the above calculation proceed practically unchanged by theadditional ρ dependence in L , with the final result taking the form B = 2 π ¯ ρ S − π ¯ ρ (cid:15) , (V.30) S = (cid:90) φ + φ − π φ ( φ, ˙ φ, ρ )d φ . (V.31)On the face of it, the entire structure of the decay rate up to this point is unaffected by the ρ dependence. However, the crucial difference is that S now depends on ¯ ρ , so that whenwe calculate ¯ ρ by minimizing B , as above, we will find that the structure of S can play anadditional role, since ∂B/∂ ¯ ρ = 0 now yields3 S − (cid:15) ¯ ρ + ¯ ρ ∂S ∂ ¯ ρ = 0 . (V.32)As a concrete example, consider the cubic galileon with a canonical kinetic term, S E = (cid:90) d x (cid:20)
12 ( ∂φ ) + 1Λ ( ∂φ ) (cid:3) φ + V ( φ ) (cid:21) , (V.33)corresponding to L = 12 ˙ φ + 2Λ ˙ φ ρ + V . (V.34)The canonical momentum is π φ = ˙ φ + 6Λ ˙ φ ρ , (V.35)so that the surface tension of the bubble wall is S ( ¯ ρ ) = (cid:90) φ + φ − (cid:32) ˙ φ + 6Λ ˙ φ ¯ ρ (cid:33) d φ ≡ S can1 + 1¯ ρ S gal1 , (V.36)where S can1 and S gal1 are defined so as not to depend on ¯ ρ . Plugging this into eq. (V.30), wecan minimize B to find ¯ ρ as usual,¯ ρ = 3 S can1 (cid:15) (cid:34) (cid:114) λ (cid:35) , (V.37)where we have defined λ ≡ S gal1 (cid:15) ( S can1 ) . (V.38) When ˙ φ = 0, we will write quantities with two arguments rather than three, e.g., writing L ( φ, , ρ ) as L ( φ, ρ . B = 27 π ( S can1 ) (cid:15) ∆ (cid:18) λ ∆ (cid:19) , (V.39)with ∆ ≡ (cid:32) (cid:114) λ (cid:33) . (V.40)In these expressions for ¯ ρ and B we have not yet taken a thin-wall limit, and it is nothard to see why: the correct limit to take depends on whether the canonical term or thegalileon dominates S , i.e., whether S gal1 (cid:15) ( S can1 ) (cid:29) , or S gal1 (cid:15) ( S can1 ) (cid:28) . (V.41)This depends on the free parameters of the theory: (cid:15) , which controls the difference betweenthe potentials of the two vacua; ∆ φ ≡ φ + − φ − , the difference between the field values at thetwo vacua; and Λ, which controls the size of the galileon term. Given these parameters, wecan estimate the dominant contribution to S as follows. Let us approximate the field profileas φ (cid:39) ∆ φ tanh ( ∆ φ ( ρ − ¯ ρ )); while this simple ansatz will not exactly solve the equations ofmotion (although it does in the absence of the galileon and in the limit (cid:15) → λ to the theory parameters. Evaluating this field profile on S can1 and S gal1 , we find λ ≡ S gal1 ( S can1 ) (cid:15) = 6 (cid:15) ∆ φ Λ . (V.42)We see that the canonical kinetic term dominates the decay rate if (cid:15) ∆ φ (cid:28) Λ , and the galileondominates the rate if (cid:15) ∆ φ (cid:29) Λ . Note that (cid:15) ∆ φ = ∆ V ∆ φ is the overall slope of the potentialbetween the two vacua.We are now in a position to take the thin-wall limit and evaluate the decay rate in thepresence of a cubic galileon. In the limit where the canonical kinetic term dominates we havethe usual decay rate, B can = 27 π ( S can1 ) (cid:15) , (V.43)while when the galileon dominates, we find B gal = 2 π ( S gal1 ) (cid:15) . (V.44)In fig. 3, we can observe the change of the WKB exponent in both limits, when the canonicalterm dominates and when the galileon term dominates. We see that the change in the decayrate will be drastic when the galileon term dominates. We conclude that the galileon canlower the decay rate, potentially by a rather large amount, compared to a canonical scalar.In fact, we can apply this reasoning to the full range of galileons (and therefore of healthyscalar theories). It is not too difficult to show that a general galileon Lagrangian, allowingfor all the galileon terms with any functions of φ and X in front, leads to a Euclidean actionof the form S E = 2 π (cid:90) ρ L d ρ , (V.45)15 c = × B gal = × B c = × B gal = × FIG. 3. In this figure, we observe the behavior of L ( φ bounce ) for two different limits. On the leftside we see the case where the canonical term dominates and on the right side the case where thegalileon term dominates. B can and B gal are the WKB exponents for a canonical scalar field andfor a canonical + cubic galileon scalar field respectively. It is clear that, when the galileon termdominates we see a drastic change in the decay rate. with L = (cid:88) n =0 f n ( φ, ˙ φ ) ρ n . (V.46)Note that f receives contributions from P ( X ) terms and the cubic galileon, f from thecubic and quartic terms, f from the quartic and quintic terms, and f from the quintic term.The bubble tension is S = (cid:88) n =0 g n ( ˙ φ )¯ ρ n , (V.47)where we have defined g n = (cid:90) φ + φ − ∂f n ∂ ˙ φ d φ . (V.48)Solving for ¯ ρ by minimizing B we obtain (cid:15) ¯ ρ + (cid:88) n =0 ( n − g n ¯ ρ n = 0 . (V.49)Note that the n = 3 piece does not contribute, so (after multiplying by ¯ ρ ) this is a cubicequation for ¯ ρ , (cid:15) ¯ ρ − g ¯ ρ − g ¯ ρ − g = 0 . (V.50)We have already addressed above the special case where g = 0 and this equation is quadratic,i.e., when only the cubic galileon and a P ( X ) term are present. If this equation is cubic, wecan perform a similar analysis; solving for B we find B = 27 π g (cid:15) (cid:20) g g (cid:15) + 4 (3 g g + 2 g )27 g (cid:15) + · · · (cid:21) , (V.51)and while neglecting higher-order terms in (cid:15) is tempting, the same lesson we learned aboveholds: if the g n terms, with n ≥
1, are larger than g , one should keep a different set of terms16n eq. (V.51). In this case, we have three expansion parameters given by g n g n +10 (cid:15) n , n = 1 , , , (V.52)for which an analysis similar to the cubic galileon one can be performed, given a specificaction. VI. DISCUSSION
Scalar field theories with non-canonical kinetic structures play an important role inbuilding phenomenologically interesting models of both the early and late universe. Someclasses of such theories arise naturally in supergravity and string theory, and others ariseas limits of massive gravity and brane-world constructions. In each case, it is interesting towonder whether the nonperturbative physics of these theories might provide a novel wayto constrain and test them, and whether they can yield results significantly different fromcanonical fields.In this paper we have examined tunneling in general scalar field theories, allowing for theexistence of non-canonical kinetic structures, while demanding the the resulting equationsof motion be second order, and hence ghost-free. We have shown how to construct thegeneral tunneling formalism for such theories and applied it to several well-known examples,in the thin-wall limit. While the formal structure of the expressions for the decay rates arethe same for both these theories and for canonical ones, the resulting tunneling rates canbe dramatically altered by the presence of non-canonical terms, giving rise to significantdifferences in the decay rates.
ACKNOWLEDGMENTS
We are thankful to Garrett Goon, Kurt Hinterbichler, Austin Joyce, Matteo Vicino, AlexVilenkin and Erick Weinberg for useful comments and discussion. Calculations have madeuse of xAct. The work of M.C. and M.T was supported in part by NASA ATP grantNNX11AI95G. The work of A.M. was supported by NSF grant: PHY-1518742. The workof A.R.S. was supported by funds provided to the Center for Particle Cosmology by theUniversity of Pennsylvania. M.T. was also supported in part by US Department of Energy(HEP) Award DE-SC0013528.
Appendix A: Galileons
In this appendix we briefly discuss the galileons [27] and their generalizations, which arethe most general scalar field theories with second-order equations of motion [28, 29], andshow how they fit into the formalisms discussed in sections IV and V. we are led to the Lagrangian [28, 29] L = D (cid:88) n =2 G n ( φ, X ) L n (A.1)in D spacetime dimensions, where G n ( φ, X ) are arbitrary functions of φ and X = − ( ∂φ ) .In n = 4 the individual galileon Lagrangians are L = ( ∂φ ) , L = ( ∂φ ) (cid:3) φ, L = ( ∂φ ) (cid:2) ( (cid:3) φ ) − φ µν (cid:3) , L = ( ∂φ ) (cid:2) ( (cid:3) φ ) − (cid:3) φφ µν + 2 φ µν (cid:3) , (A.2)where we have defined φ µ ≡ ∂ µ ∂ ν φ . We will frequently refer to L , L , and L as the cubic,quartic, and quintic galileons, respectively.First we will justify the form (IV.1) of the Hamiltonian we considered, in which dependenceon higher spatial gradients but not on higher time derivatives is permitted. A priori it isnot obvious that the Hamiltonians for the galileons (above L ) fall into that class, since theLagrangians themselves, in their covariant form, contain second derivatives of φ . However,the fact that the resulting equations of motion are second order ensures that we are able toeliminate higher time derivatives up to boundary terms.As an illustration, consider the cubic galileon, with the action S = (cid:90) d x (cid:18) −
12 ( ∂φ ) + 1Λ ( ∂φ ) (cid:3) φ − V ( φ ) (cid:19) , (A.3)where Λ is a constant with units of mass. Performing a 3 + 1 spacetime decomposition wehave S = (cid:90) d t d x (cid:20) (cid:16) ˙ φ − ( ∇ φ ) (cid:17) + 1Λ (cid:16) − ˙ φ + ( ∇ φ ) (cid:17) (cid:16) − ¨ φ + ∇ φ (cid:17) − V (cid:21) ≡ (cid:90) d t d xL . (A.4)We may then eliminate the ¨ φ dependence by integrating by parts. Consider the term (cid:16) − ˙ φ + ( ∇ φ ) (cid:17) (cid:16) − ¨ φ + ∇ φ (cid:17) = ˙ φ ¨ φ − ¨ φ ( ∇ φ ) − ˙ φ ∇ φ + ( ∇ φ ) ∇ φ . (A.5)The first piece is a total derivative in time, ˙ φ ¨ φ = ( ˙ φ )˙. The next term can be eliminatedby a pair of total derivatives,dd t (cid:104) ˙ φ ( ∇ φ ) (cid:105) − ∂ i (cid:104) ˙ φ ∂ i φ (cid:105) = ¨ φ ( ∇ φ ) − ˙ φ ∇ φ , (A.6) This requirement is necessary to avoid the Ostrogradsky instability [30, 31]. This may be loosenedsomewhat when multiple fields are present [32], as in the so-called “beyond-Horndeski” theories [33, 34]and their generalizations [35, 36], but for a single scalar field this loophole is not available. In the above notation, this corresponds to G constant. L = 12 (cid:16) ˙ φ − ( ∇ φ ) (cid:17) + 1Λ (cid:16) − φ + ( ∇ φ ) (cid:17) ∇ φ − V , (A.7)up to boundary terms. We can therefore obtain the canonical momentum,Π = d L d ˙ φ = ˙ φ (cid:18) − ∇ φ (cid:19) , (A.8)and solve for H ( φ, Π , ∂ i φ, ∂ i ∂ j φ ). The cubic galileon therefore fits into the form used insection IV.This property has also been shown to apply to the quartic and quintic galileons [37]. Toround out the list of second-order field theories, we only need to generalize this to include φ and X -dependence in the coefficients G n . For simplicity, let us look at the cubic galileonwith some general φ - and X -dependent coefficient, L = e α ( φ,X ) (cid:3) φ . (A.9)We will find it convenient to explicitly consider how α separately depends on ˙ φ and ∂ i φ , α ( φ, X ) → α ( φ, ρ, ∂ i φ ) , (A.10)where for further convenience we have defined ρ = log( ˙ φ/ Λ ), with Λ a constant withdimensions of mass. Using X = − ˙ φ + ∂ i φ∂ i φ , (A.11)we see that, of course, derivatives of α with respect to ρ and ∂ i φ are related to each other, α ρ = − α X ˙ φ , (A.12) α i = 2 α X ∂ i φ , (A.13)where we have defined α ρ ≡ ∂α∂ρ , α i ≡ ∂α∂∂ i φ , α X ≡ ∂α∂X . (A.14)We now write the Lagrangian explicitly in terms of time and space derivatives, L = − e α ¨ φ + e α ∇ φ . (A.15)The second term is already of the form we want: it depends only on φ , ˙ φ , and spatialderivatives of φ (but not of ˙ φ ). We now work on the first term. Integrating by parts on thetime derivative, and rearranging, we have − e α ¨ φ ∼ e α α ρ (cid:16) α φ ˙ φ + α i ˙ φ∂ i ˙ φ (cid:17) , (A.16)where ∼ denotes equivalence up to boundary terms.This explains our choice to use α and ρ rather than G and ˙ φ . The first term in this expression is of the form we want, but we needto remove the spatial derivative from ˙ φ in the second term. Taking this term separately,integrating by parts on the spatial derivative, and using (cf. eq. (A.13)), that α ρi = ∂α ρ ∂∂ i φ = 2 α ρX ∂ i φ = α ρX α X ∂ i φ , (A.17)19e obtain e α α ρ α i ˙ φ∂ i ˙ φ ∼ − e α α ρ + α ρ − α ρρ + α ρX α X (1 + α ρ ) × ˙ φ (cid:20)(cid:18) α φi + α φ α i − α φρ α i α ρ (cid:19) ∂ i φ + (cid:18) α ij + α i α j − α ρj α i α ρ (cid:19) ∂ i ∂ j φ (cid:21) . (A.18)Similar proofs apply to the quartic and quintic galileons multiplied by general functions.A similar (and more straightforward) calculation justifies the formalism used in section V tocompute Euclidean bounce solutions; in particular, the Euclidean action for O (4)-symmetricsolutions can be written in the form S E = 2 π (cid:90) ρ L ( φ, ˙ φ, ρ )d ρ , (A.19)where the ρ dependence in L ( φ, ˙ φ, ρ ) comes only from the cubic, quartic, and quintic galileonsafter integrating by parts, and ∂L ( φ, , ρ ) ∂ρ = 0 . (A.20) [1] M. Sher, Phys. Rept. , 273 (1989).[2] T. Banks, C. M. Bender, and T. T. Wu, Phys. Rev. D , 3346 (1973).[3] T. Banks and C. M. Bender, Phys. Rev. D8 , 3366 (1973).[4] I. Yu. Kobzarev, L. B. Okun, and M. B. Voloshin, Sov. J. Nucl. Phys. , 644 (1975), [Yad.Fiz.20,1229(1974)].[5] S. R. Coleman, Phys. Rev. D15 , 2929 (1977), [Erratum: Phys. Rev.D16,1248(1977)].[6] C. G. Callan, Jr. and S. R. Coleman, Phys. Rev.
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