Tachyons in Throat Cosmology
aa r X i v : . [ h e p - t h ] J a n HD-THEP-09-26 10 December 2009MPP-2009-200
Tachyons in Throat Cosmology
S. Halter a , B. v. Harling b , A. Hebecker c a Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut), F¨ohringer Ring 6,D-80805 M¨unchen, Germany b School of Physics, University of Melbourne, Victoria 3010, Australia c Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16 und 19,D-69120 Heidelberg, Germany ( [email protected] , [email protected] and [email protected]) Abstract
Tachyonic 5d scalars are generically present in Randall-Sundrum-like models. In par-ticular, they are known to be part of the 5d effective description of the Klebanov-Strasslerthroat. When moving from the IR to the UV region, the 5d bulk profile of Kaluza-Kleinexcitations of tachyons decays more slowly than that of massless scalars or the graviton.As a result, tachyons in many cases dominate the coupling between IR- and UV-localizedsectors, leading to a very significant enhancement of energy-transfer or decay rates fromthe IR to the UV. This can dramatically affect the reheating of the Standard Model afterbrane inflation and the decay of throat dark matter.
Introduction
Strongly warped regions or ‘throats’ are a generic feature of type IIB flux compactifi-cations [1, 2]. It can even be argued that, in certain regions of the ‘landscape’, they arestatistically unavoidable [3]. The prime example is the Klebanov-Strassler (KS) throat (orwarped deformed conifold) [4]. It can be viewed as a stringy version of a Goldberger-Wise-stabilized Randall-Sundrum (RS) model [5] and is thus phenomenologically interestingfor all the well-known reasons making the RS model so attractive. Moreover, throats playa key role in mechanisms realizing de Sitter vacua [6] and in string-theoretic models ofinflation, such as brane-antibrane inflation [7].In warped compactifications, couplings between fields localized in the UV and IRregions (regions of weak and strong warping, respectively) are generically suppressed bypowers of the warp factor [8]. In cosmology, two implications of this suppression are ofparticular interest: One is the long lifetime of certain IR-localized modes, which canhence play the role of ‘throat dark matter’ [9–14]. The other is the small energy transferrate from a plasma of such modes (created at the end of warped brane inflation) to anyUV-localized sector [15–19]. This can cause problems for the reheating of the Standardmodel after inflation in a throat.The main point of the present paper is to demonstrate that tachyonic 5d scalars canplay a central role in coupling IR- and UV-localized sectors and to discuss some of theimmediate cosmological implications.The importance of tachyons in this context is easily understood: Recall first that thepresence of tachyons is AdS is natural from the perspective of 5d supersymmetry [21,22]and observed in concrete examples, such as the compactification of type IIB supergravityto AdS on T , [23]. It is well-known that tachyons do not destabilize infinite AdS aslong as their negative mass-squared respects the Breitenlohner-Freedman bound [24].On a slice of AdS , tachyons generically introduce an instability. In supersymmetricmodels on a slice of AdS , stability is maintained by a positive mass-squared operatorlocalized at the UV brane [22]. Such a brane-mass term can also be used to ensure thestability of non-supersymmetric Randall-Sundrum models with tachyons [25–27]. Now,for any massive 5d scalar, its efficiency in mediating interactions between IR and UVbrane is determined by the 5d bulk profile of its lowest Kaluza-Klein modes. Generically,the value of IR-localized modes is exponentially suppressed near the UV brane. Thisexponential suppression is governed by the 5d mass-squared. Thus, it is natural to expectthat tachyonic scalars will provide the strongest coupling between the UV and IR brane.In the following, we study this general idea at the quantitative level.Our main point can be made in a simple 5d toy model: a single tachyonic scalar ona slice of AdS . Any of its low-lying IR-localized KK modes can be taken to represent a In non-cosmological applications of warped geometries, such as supersymmetry breaking and flavormodel building, related effects are known as ‘sequestering’ (see e.g. [20]). The technical questions arisingin this context are somewhat different and we will not address them in the following. Alternatively, one can impose the Dirichlet boundary condition at the UV brane [28]. This corre-sponds to the limit of sending the boundary mass to infinity. F term of this theory through its value at the UV-brane. We calculate the correspondingdecay rate and find it to be of the same order of magnitude as that resulting from themediation of the graviton. This result arises from two compensating effects: On the onehand, the modes of a tachyon decay more slowly than those of a graviton when movingfrom the IR to the UV. On the other hand, they are subject to an extra suppression dueto the UV-localized mass term (which is necessary for stability).From the above, it is clear that in many relevant situations the tachyon is, in fact,bound to dominate: Namely, massless 5d scalars are expected to obtain a large UVmass term in generic, fully stabilized models. The 5d graviton, which has the profile of amassless scalar and no UV mass term, can not mediate the decay of spin-zero states. Thus,the tachyon dominates the decay of low-lying IR-localized spin-zero states or ‘glueballs’.We illustrate this in a simplified KKLMMT-type setup for warped reheating. It turns outthat tachyons dominate the energy transfer to the Standard Model. They can thus helpto avoid the complete dissipation of inflationary energy density into gravitons, therebysolving a generic problem of (warped) brane inflation.Our paper is organized as follows: Section 2 provides a pedagogical introductionto tachyons in a slice of AdS . Building on this discussion, Sect. 3 gives a simple andintuitive derivation of the decay rate of the IR-localized modes of a 5d tachyon to aUV-localized sector. It is followed by a discussion of situations in which the tachyondominates the decay of IR-localized states in Sect. 4. In Section 5, we explain our resultsfrom the perspective of the dual CFT, finding further support for our intuitive pictureand our calculations. We then discuss the applicability of our 5d analysis to presumablymore realistic and general throat geometries in Sect. 6. Finally, in Sect. 7, we turn tocosmological applications, in particular to effects on reheating and dark matter decay.Our Conclusions are followed by an Appendix in which we provide a Bessel functionanalysis supporting our order-of-magnitude calculation in the main text.Cosmological implications of tachyonic 5d scalars in a warped throat were discussedin [10] and, more recently, in [14, 29]. The analysis of [14] includes processes which, inour language, can be interpreted as tachyon-mediated decays from the IR- to the UV-localized sector. We discuss the relation to our work in Sect. 6. In [29], the characteristic5d profile of tachyons is used in the construction of the waterfall-sector of a model ofhybrid inflation. with metric ds = e − ky ( η µν dx µ dx ν ) + dy , (1) The bulk profile of modes of the 5d graviton is the same as that of a massless 5d scalar. k = 1 /R is the inverse curvature radius. The slice is bounded by two 3-branes at y UV = 0 and y IR = ℓ . In the following, we will focus on the dynamics of a scalar Φ withaction S d = Z d xdy √− G
12 Φ (cid:2) ∇ − M (cid:3) Φ , (2)where ∇ = ( √− G ) − ∂ M ( √− GG MN ∂ N ) and G MN is the AdS metric. Capital indicesrun over { , . . . , } and y . We assume that the model is already stabilized, e.g. via theGoldberger-Wise mechanism [31], and that Φ is an additional scalar field, not related tothe stabilization.It will be convenient to use the coordinate z = k − e ky and the corresponding confor-mally flat metric ds = 1( kz ) ( η µν dx µ dx ν + dz ) . (3)Expanding Φ in KK modes, Φ( x, z ) = X n ϕ n ( z ) χ n ( x ) , (4)we have a set of canonically normalized 4d scalars χ n with masses m n , the wave functionsof which satisfy the equations (cid:18) z ∂ z − z∂ z + m n z − M k (cid:19) ϕ n ( z ) = 0 . (5)To specify boundary conditions, it is convenient to view the extra dimension as an S / Z orbifold. Thus, our ϕ n are defined on the double cover of the interval [ z UV , z IR ],where z UV = k − and z IR = k − e kℓ . We assume that Φ is even under the Z symmetrytransformation, implying that ϕ n is symmetric under ( z − z UV ) → − ( z − z UV ) and periodicwith period 2( z IR − z UV ).In terms of the rescaled field ψ n ( z ) ≡ ( zk ) − / ϕ n ( z ) , (6)Equation (5) takes the form of a 1-dimensional Schr¨odinger equation [30] (cid:0) − ∂ z + V ( z ) (cid:1) ψ n ( z ) = m n ψ n ( z ) (7)with ‘energy’ m n and potential V ( z ) = α − z − z UV δ ( z − z UV ) + 3 z IR δ ( z − z IR ) . (8)Here α ≡ p M /k , and the δ -function contributions come from the rescaling in thepresence of boundary conditions. The solutions of Eq. (7) are Bessel functions of order α (cf. Appendix 8) and all that follows could be derived from a careful analysis of thebehaviour of these functions. However, we find it more illuminating to make our mainqualitative points using a parametric analysis of approximate but explicit solutions.4ccording to Eqs. (7) and (8), we are basically looking for the solutions of a quantummechanical problem on an interval. The coefficient of the 1 /z term in the potential issubject to the Breitenlohner-Freedman bound [24] M ≥ − k (corresponding to α ≥ vacuum. Figures 1 and 2 show the potential forthe particularly interesting cases M = 0 and M = − k . Standard KK-mode intuitiontells us that all m n are positive for M >
0, while m = 0 for M = 0. We then clearlyexpect the presence of negative ‘energy eigenvalues’ m n for ‘tachyonic’ M <
0. Thisis not tolerable from the perspective of the 4d effective field theory. Indeed, while theBreitenlohner-Freedman bound is sufficient to ensure the stability of infinite AdS space,it does not guarantee the stability of a slice of AdS.
V z z IR z UV Figure 1: Potential in the effective Schr¨odinger equation for a 5d mass M = 0. V z z IR z UV Figure 2: Potential in the effective Schr¨odinger equation for a 5d mass M = − k . Thedotted δ -peak appears when a large mass term on the UV brane is added. More precisely, we are looking for the even solutions on the S covering space of that interval. S UV = − Z d xdy √− G λk Φ ( x, y ) δ ( y − y UV ) , (9)where we have written the dimensionful coefficient as a product of k and a dimensionlessparameter λ ≥
0. This mass term modifies the potential, V ( z ) = α − z + 2 λ − z UV δ ( z − z UV ) + 3 z IR δ ( z − z IR ) , (10)such that, for λ > , the δ -peak at the UV brane becomes repulsive (cf. Fig. 2). It isintuitively clear that such a modification of the potential lifts the energy eigenvalues andcan therefore remove tachyonic modes.Indeed, it will become apparent in the next section that, in the limit z IR → ∞ (theRS II limit [33]), a zero mode appears if λ = λ ≡ − α . This is the minimal value of λ required for stability – for λ > λ all modes are massive. In the RS I case (for finite z IR )the minimal value of λ may differ from λ , depending on the specific boundary conditionimposed at the IR brane. However, this difference goes to zero together with z UV /z IR andits precise value will not be important. Moreover, for the specific case of supersymmetricboundary conditions (when the IR brane carries as mass term of opposite value w.r.t.the UV-brane mass term [22]), the minimal value continues to be λ . Thus, it will beconvenient for us to assume such supersymmetric IR boundary conditions (modifyingEq. (10) appropriately) and to think of λ = 2 − α as of the minimal value of λ , also atfinite z IR .To conclude our preliminary discussion of tachyons in a slice of AdS , we brieflydescribe the KK mass spectrum. As long as λ > λ , the spectrum does not differ quali-tatively from the familiar massless or positive-mass-squared case. As we already empha-sized, we are dealing with a quantum-mechanical problem on a compact space of size ∼ ( z IR − z UV ) ≃ z IR . We thus naively expect m n to be quantized in units of z − IR , towhich we will from now on refer as our IR scale m IR ≡ z IR − = k e − kℓ . Indeed, the Besselfunction analysis of App. 8 shows that m n ≃ ( n + α − ) π k e − kℓ for 1 ≪ n ≪ e kℓ . Thisapproximate formula remains rather accurate all the way down to relatively small n .We finally note that a technically related analysis of the wave functions of a tachyonicscalar has recently appeared in [27] in a rather different physical context. We refer tothat paper for more details on the quantum mechanical analogue and correspondingreferences. This is naive in the sense that the potential introduces the further dimensionful quantity z UV intothe calculation. However, in the limit z UV /z IR → /z term together with the UV-brane δ functionbasically just serve to provide a certain boundary condition at one side of the interval. Hence the scale z UV does not affect the (low-lying part of) the spectrum. This will become apparent after the analysisof the next section. Decays mediated by AdS fields As one can see from Fig. 1, a massless scalar has to tunnel through an effective potentialbarrier before it can reach the UV brane. By contrast, a tachyonic scalar with M = − k has no such barrier to surmount (cf. Fig. 2). Thus, one expects the tachyon tocouple more strongly to the UV brane. There is, however, a compensating effect: TheUV-brane mass term of the tachyon suppresses its value at the brane. To determinethe relative importance of these two effects, we now estimate the decay rate of the KKmodes of a bulk scalar to a gauge theory living on the UV brane. Our interest is in thedependence on the parameters α and λ . We assume that Φ interacts with a gauge theory localized at the UV brane via the action Z d x dz √− G k − Φ( x, z ) tr ( F µν F µν ) δ ( z − z UV ) . (11)We have arbitrarily set the coefficient of this interaction term to one in units of k sincewe will anyway perform only an order-of-magnitude calculation. Using Eq. (4), we seethat the coupling of the n -th KK mode to the gauge theory is given by Z d x k − ϕ n ( z UV ) tr ( F µν F µν ) χ n ( x ) . (12)The coupling constant between the n -th KK mode and two gauge bosons, g n ∼ k − ϕ n ( z UV ) = k − ψ n ( z UV ) , (13)has mass dimension [ g n ] = − n ∼ g n m n . (14)In order to calculate g n , we need a detailed understanding of the shape of the KKmodes. To achieve this, we introduce the coordinate b z = m n z (for each m n = 0) andtranslate the δ functions back to boundary conditions at b z UV and b z IR . We then have tosolve (cid:20) ∂ b z + 1 − (cid:18) α − b z (cid:19)(cid:21) ψ n ( b z ) = 0 , (15)subject to (cid:0)b z ∂ b z ψ n (cid:1) | b z = b z UV = (cid:18) λ − (cid:19) ψ n | b z = b z UV and (cid:0)b z ∂ b z ψ n (cid:1) | b z = b z IR = (cid:18) λ − (cid:19) ψ n | b z = b z IR , (16) This is the same situation as for the graviton (cf. [33]). As we will discuss in Sect. 6 in more detail, this term can be viewed as an effective 5d descriptionof the dilaton coupling to the gauge fields on a D-brane stack. n b z b z IR b z UV Figure 3: Sketch of the wavefunction ψ n as a function of b z where we have also modified the IR boundary condition as mentioned earlier.It is straightforward to solve Eq. (15) separately in the regions b z ≪ b z ≫ We findthat for small b z ψ n, UV ( b z ) ≃ N α (cid:16)b z + α + B α b z − α (cid:17) for α > N (cid:18)b z + B b z ln 1 b z (cid:19) for α = 0 , (17)while for large b z ψ n, IR ( b z ) ≃ A α cos( b z + C α ) , (18)where N α , B α , A α and C α are constants of integration. The qualitative behaviour of theresulting wave function is sketched in Fig. 3.The canonical normalization of χ n implies, together with Eq. (6), the normalizationcondition 1 m n Z b z IR b z UV d b z ψ n ( b z ) = 1 (19)for our wave function. Let us assume that b z IR ≫ b z UV and b z ∼ λ isgeneric). It is then apparent that the overall normalization is dominated by the cosinesolution in the IR region 1 ≤ b z ≤ b z IR , implying A α ∼ pb z IR /m n = m − / IR . (20) We assume α − ∼ O (1) for simplicity, excluding the special region α ≃ . Note that, with the redefinitions N ≡ N α B α and B ≡ α (1 − B α )1 + B α , the second line of Eq. (17) can be recovered as the α →
8f course, we are actually interested in the lowest-lying modes, so that z IR ∼ O (1)rather than z IR ≫
1. Nevertheless, the approximate cosine region (which in this casecontains only a few oscillations) continues to contribute an O (1) fraction to the totalnormalization. Thus, Eq. (20) continues to be correct up to O (1) factors.To summarize, we now know that the UV solution has to be matched to the IRsolution with approximate value A − α ∼ m / IR at b z ∼
1. For the decay rate, we need theUV-brane value of this UV solution, which we will now derive.The first boundary condition of Eq. (16) together with Eq. (17) gives B α = − (cid:18) − αλ − α (cid:19) b z α UV , (21) B = − (cid:18) − λ −
2) ln(1 / b z UV ) + 1 (cid:19) / b z UV ) . (22)Now, for λ = λ + O (1) and α = O (1), we see that both the b z + α solution and the b z − α solution, as well as the full solution ψ n, UV are of comparable size at b z UV . Thus,only the more strongly growing solution is important for the matching at b z ∼
1. Thevalue of ψ n at the UV brane is suppressed according to the behaviour of this solution.By contrast, for α = 0 the b z solution and the b z ln(1 / b z ) solution cancel almost exactlyat the UV brane, leading to an extra suppression factor 1 / ln(1 / b z ) in the brane value of ψ n . In summary, we have ψ n ( b z UV ) ∼ b z + α UV m for α > b z ln(1 / b z UV ) m for α = 0 . (23)Note that the α = 0 result could also have been derived by using the expression for B α in Eq. (21) and carefully taking the limit α →
0. It is valid for α < ∼ ln(1 / b z UV ) − .Using Eqs. (13) and (14) and recalling that b z UV = m n /k , one finds the decay ratesΓ n ∼ (cid:16) m n k (cid:17) α m IR for α > (cid:16) m n k (cid:17) m IR ln ( k/m n ) for α = 0 . (24)These rates represent one of our main results and will be used in the cosmological appli-cations later on. Equation (21) suggests that the point λ = λ = 2 − α may require special attention:Near this point, the UV-brane value of the b z − α mode (the falling or weakly growing9ode) is significantly enhanced compared to the value of the b z + α mode (the stronglygrowing mode). We already know that, for λ = λ , the lowest mode is massless since, forthe specific choice of IR boundary conditions discussed earlier, λ is the minimal allowedvalue. This mode can not decay unless the tuning is imperfect. We will not discuss thepossible decay rate of such an ‘almost massless’ mode.Furthermore, in the tuned case the UV-brane δ -function is least-repulsive (or mostattractive). Thus, we expect the decay rates of higher modes to be larger for λ = λ thanfor the generic case of λ > λ . These enhanced decay rates are our main interest in thissubsection.Let us first consider the case 0 < α < λ = λ . Taking Eq. (21) at face value, B α is infinite, which simply means that in the solution of Eq. (17) only the b z − α termis present. When moving from b z = O (1) to b z = b z UV , the wave function then changes bya factor b z − α UV . Repeating the arguments which lead us to Eq. (24), this gives the decayrate Γ n ∼ (cid:16) m n k (cid:17) − α m IR for 0 < α < λ = λ . (25)In the case α = 0, we have B = 0 at λ = λ according to Eq. (22). Hence the wavefunction falls by a factor b z . This impliesΓ n ∼ (cid:16) m n k (cid:17) m IR for α = 0 and λ = λ . (26)As expected, a tuned value of λ enhances the rates in the whole range 0 ≤ α <
1, withthe enhancement being more pronounced at larger α .Apparently, nothing in the above calculation depends on the restriction to α < α >
1. Indeed, if only the b z − α mode is relevant for all b z ≪
1, the ψ n normalization integral is dominated by the regionnear b z UV . Thus, the relevant mode is UV localized. Obviously, this can not be the casefor all modes with m n ≪ k (to which our argument formally applies).At the technical level, the above problem comes from the insufficient accuracy of theapproximation made in Eq. (17). This accuracy can be improved by taking into accounthigher-order terms in the small-argument expansions of the Bessel functions. Includingonly the first-order correction, we find ψ n, UV ( b z ) ≃ N α (cid:20)b z + α (cid:18) − b z α ) (cid:19) + B α b z − α (cid:18) − b z − α ) (cid:19)(cid:21) for α = 11 N (cid:20)b z (cid:18) − b z (cid:19) + B b z − (cid:18) b z ln(1 / b z )2 (cid:19)(cid:21) for α = 1 . (27)In fact, the b z correction to the b z + α solution will not play any role and we drop it in thefollowing. With this simplification, Eq. (27) together with the UV boundary conditions Alternatively, they can be determined by solving Eq. (15) iteratively, treating the ‘1’ as a pertur-bation. B α = 4 α (1 − α ) b z α − UV , (28) B = 42 ln(1 / b z UV ) + 1 . (29)This result can be intuitively understood as follows: It is clear that, for generic λ , theboundary condition Eq. (16) can only be fulfilled if both solutions to the wave equation, b z + α + · · · and B α b z − α + · · · , are of comparable size at the UV brane. This immediatelygives B α ∼ b z α UV as in Eq. (21). Now, for tuned λ = 2 − α , the function B α b z − α fulfills theboundary condition already by itself, as can be easily checked. Therefore, the boundarycondition now forces the leading correction to that function, B α b z − α , to have the samesize as the other solution, b z + α , at the UV brane. This gives B α ∼ b z α − UV in agreementwith Eq. (28).We now observe that, as long as α <
1, the b z − α solution is dominant everywherebetween b z = b z UV and b z = O (1). This justifies our previously derived rate of Eq. (25) andshows that the presently discussed b z corrections are not important for 0 ≤ α < α >
1. In this case, we see that the b z − α solutiondominates for b z UV < b z < b z − /α UV . By contrast, the b z + α solution is larger for b z − /α UV < b z < O (1). Although Fig. 3 does not describe this regime correctly, one can check thatthe normalization is nevertheless dominated by the IR region. It is then easy to see that ψ n ( b z UV ) ∼ z α − m for α > λ = λ , (30)giving the decay rateΓ n ∼ (cid:16) m n k (cid:17) α m IR for α > λ = λ . (31)It can be checked that, in spite of the extra logarithms appearing at intermediate steps,the special case α = 1 is correctly reproduced by simply taking the appropriate limit ofeither Eq. (25) or Eq. (31).The 5d graviton with indices µ, ν has the same equation of motion as a massless 5dscalar without boundary mass (i.e. for λ = λ = 0). We assume M ∼ k in the following.The coupling of the 5d graviton to the energy-momentum tensor on the UV brane isthen suppressed by a factor k − / as in Eq. (11). Hence Eq. (31) with α = 2 also appliesto the graviton and we haveΓ n ∼ (cid:16) m n k (cid:17) m IR for graviton KK modes , (32)in agreement with the literature on graviton tunneling between two throats (see e.g.[8, 34, 35]). These results are relevant since the throat to which the KK modes decayis dual to a UV-brane-localized gauge theory [35]. Of course, Eq. (32) corresponds tothe case where this gauge theory has O (1) degrees of freedom. We finally note that thecoupling strength of graviton KK modes to the UV brane has been given, e.g., in App. Aof [84]. For M ∼ k , it reads g n ∼ √ m n m IR /k . The decay rate then follows from Eq. (14)and also reproduces Eq. (32). 11 Situations dominated by the tachyon
A detailed discussion of the application of our 5d analysis to the Klebanov-Strasslerthroat is the subject of Sect. 6. However, to explain the relevance of tachyon decay rateswe have to jump somewhat ahead and mention certain facts concerning warped fluxcompactifications already in this section: In flux compactifications, scalars obtain a largemass in the unwarped part of the compact space. In our 5d model, this corresponds to alarge and generically detuned mass term on the UV brane. It is then evident from Eq. (24)that a maximally tachyonic scalar ( α = 0) has a considerably larger decay rate than amassless scalar ( α = 2). This can be easily understood from the quantum mechanicalanalog: The suppression effect of the UV mass term is present for both fields, but onlythe massless scalar has to surmount an additional potential barrier. Thus, assuming thata certain KK mode decays only via scalars and that all relevant scalars have a large UV-brane mass term, the tachyon governs the decay. Clearly, the assumption that decaysproceed only through scalars is non-trivial: The graviton decay rate, Eq. (32) is largerthan the tachyon rate, Eq. (24) (albeit only by a factor (ln( k/m n )) − ∼ ( kℓ ) − ). However,this is only relevant for spin-two KK modes which can mix with the graviton.Next, we observe that there are specific situations where, in contrast to what wassaid above, the tachyon decay rate is larger than that of the graviton. This is possi-ble because the tachyon wavefunction rises while the graviton wavefunction falls whenmoving away from the UV brane. Thus, if there is a probe brane in the throat which islocalized somewhere between the UV and IR brane, the decay to this probe brane canbe dominated by the tachyon (even if a corresponding graviton decay is allowed). Weuse the simple ansatz Eq. (11) for the coupling to gauge fields on the probe brane at b z δ (where b z UV < b z δ < b z IR ). We now demonstrate the enhancement of the decay rates quantitatively. Let theprobe brane be localized at y = δ for some δ ≪ ℓ . For the light modes, b z δ = b z UV e kδ ∼ ne − k ( ℓ − δ ) is small and we can use the approximate wavefunctions of Eq. (17). We see that,as b z δ increases, the tachyon wave function ( α = 0) grows like b z (we neglect the additionallogarithmic dependence). By contrast, the KK graviton wave function ( α = 2) falls like b z − α . This is because, as explained before Eq. (30), the falling solution dominates nearthe UV brane. Thus, the ratio of the two wave functions at b z δ is enhanced, relative toits UV brane value, by a factor ( b z δ / b z UV ) . Given that the tachyon decay rate to the UVbrane is suppressed relative to the corresponding graviton rate by the logarithmic factormentioned above, we conclude thatΓ tachyon Γ graviton ∼ ( b z δ / b z UV ) ln ( b z IR / b z UV ) (33) We assume that the K¨ahler moduli are stabilized as well (which requires effects other than flux). As a string-theoretic realization of this situation, consider a D7 brane wrapping a 3-cycle of the T , in a Klebanov-Strassler throat and extending from the compact Calabi-Yau to a certain lowest point inthe throat. For a discussion of such embeddings and applications see e.g. [36–39]. Even though, in 5dlanguage, the D7-brane gauge theory lives everywhere between b z UV and b z δ , we model this situation bya gauge theory localized at b z δ . This is a reasonable approximation since the coupling to throat fields isdominated by interactions at the largest relevant values of b z . b z δ / b z UV ) − becomes smaller than 1 / p ln( b z IR / b z UV ). This isobviously a very weak requirement.Finally, we mention a third situation in which the tachyon dominates the decay ratesto the UV sector. We can not argue from the 5d perspective how natural or unnaturalit is to have a tuned UV mass term, λ = λ . (We will return to this question in thestring-theoretic context in Sect. 6.) However, a tuned value of λ is clearly a legitimatepossibility. Combining Eqs. (25) and (31), the corresponding decay rates can be writtenas Γ n ∼ (cid:16) m n k (cid:17) | α − | m IR for λ = λ . (34)For 0 < α <
2, this obviously dominates the decay rates corresponding to masslessscalars and the graviton. The maximal enhancement is realized for α = 1, i.e., for atachyon the negative mass-squared of which is half that of the Breitenlohner-Freedmanbound. We now present physical arguments for the decay rates of Sect. 3 using the ‘CFT-dual’ description of a Randall-Sundrum model as a strongly coupled 4d field the-ory. In other words, we are going to appeal to a simplified, purely field-theoreticversion of the AdS/CFT correspondence [40]. In this description, a scalar bulk fieldwith mass squared M corresponds to a CFT operator O ∆ with conformal dimension∆ = 2 + p M /k = 2 + α . Without a UV brane, the generating functional of CFTcorrelators can be obtained from the gravity description via the relation [42, 43] Z D φ CFT e − S CFT [ φ CFT ] − R d x φ O ∆ = Z φ D Φ e − S d [Φ] . (35)Here φ ( x ) specifies the behaviour of the bulk field Φ( x, z ) near the boundary of AdS (taking z →
0) to be Φ( x, z ) ≃ φ ( x ) z − ∆ k / . It thus acts as a source for the corre-sponding operator in the generating functional. The value of φ is kept fixed in thefunctional integral on the right-hand side and φ CFT denotes the dynamical CFT fields.In the Randall-Sundrum model, there is a brane at z = k − , corresponding to a UVcutoff Λ UV = k in the CFT. The source φ ( x ) = k / − ∆ Φ( x, k − ) now becomes a physicaldegree of freedom and has to be included in the functional integral [45]: Z D φ e − S UV [ φ ] Z Λ UV D φ CFT e − S CFT [ φ CFT ] − R d x φ O ∆ = Z D φ e − S UV [ φ ] Z φ D Φ e − S d [Φ] . (36) As we will discuss below, for α <
1, also the relation ∆ = 2 − α can be realized [41]. More precisely, Φ( x, z ) ≃ φ ( x ) z − ∆ k / + A ( x ) z ∆ k / , where A ( x ) can be interpreted as theexpectation value of the dual operator O ∆ and φ ( x ) as the source. The factors k / have been introducedfor dimensional reasons.
13e may think of the l.h. side of Eq. (36) as being defined by the r.h. side. In our case,the UV-brane action of the bulk scalar field, S UV [ φ ], consists of the UV-brane mass termand the interaction term with the brane-localized gauge theory. (We have suppressed thegauge fields in the functional integral in Eq. (36) for notational simplicity.) Thus, at theUV scale Λ UV , the complete Lagrangian on the CFT side of the duality reads L = L CFT [ φ CFT ] + M φ φ + 1Λ ∆ − UV φ O ∆ + 1Λ UV φ tr ( F µν F µν ) , (37)where we have redefined φ by an appropriate power of Λ UV to give it mass dimensionone.When running the action corresponding to Eq. (37) down to scales below Λ UV , theCFT induces a kinetic term ∼ ( ∂φ ) for the scalar φ , which now manifestly becomesa propagating field [44]. Similarly, the tachyonic instability which arises for M < φ with M φ = 0 at the UV scaleto a negative mass squared at the IR scale if ∆ < λ = λ of the UV mass described earlier can be interpreted in the CFT languageas a tuning of M φ which ensures that φ becomes massless at the IR scale.We first focus on the case of a generic, large UV mass term, λ = λ + O (1). In thiscase, φ has a mass of the order Λ UV even after running down to the IR scale m IR . Since φ remains heavy all the way from Λ UV down to m IR , its effect on the CFT dynamicsand, in particular, on the dimension of O ∆ is negligible. Thus, the IR scale Lagrangianis still that of Eq. (37), modified only by a kinetic term and a mass correction for φ .We can now integrate out φ entirely, finding the coupling ∼ ∆ − UV M φ O ∆ tr ( F µν F µν ) (38)between the brane-localized gauge theory and the operator O ∆ . At low scales, the confor-mal symmetry of L CFT is broken by the IR brane. In the case of the Klebanov-Strasslerthroat, this breaking of (approximate) conformal invariance is a dynamical effect withinthe strongly-coupled gauge theory. The low-lying glueball states of this gauge theory cor-respond to (linear combinations [46] of) the KK modes of the bulk fields. Assuming thatthese glueballs have non-vanishing overlap with O ∆ , they decay to the brane-localizedgauge theory via the operator in Eq. (38). Since we assumed that M φ ∼ Λ UV even atthe IR scale, the decay rate Γ ∼ (cid:18) m IR Λ UV (cid:19) m IR (39)follows by dimensional analysis. Recalling that Λ UV = k and ∆ = 2 + α , we see immedi-ately that this agrees with the first line of Eq. (24).We now turn our attention to the tuned case, λ = λ , making use of the holographicpicture developed in [26, 45]. As before, we run the Lagrangian of Eq. (37) down to thescale m IR . In contrast to our previous discussion, we can not appeal to the large mass14igure 4: Decay of a glueball G via the light field φ into two gauge fields A µ .of φ to ignore its influence on the CFT. Indeed, as explained earlier, M φ approacheszero as the energy scale approaches m IR . However, provided that α > > φ is negligible because of its small mixing with CFTstates. Indeed, for ∆ >
3, the operator φ O ∆ is irrelevant. Thus, we assume that theIR scale effective action contains light glueballs and a light field φ , with a mixing termsuppressed by 1 / Λ ∆ − UV .The decay of glueballs to fields of the UV-brane gauge theory now proceeds via themassless or light ‘source field’ φ , cf. Fig. 4. The rate follows by dimensional analysis:Γ ∼ (cid:18) m IR Λ UV (cid:19) − m IR . (40)This result can also be immediately obtained from Eq. (39): That equation secretlycontains a suppression factor 1 / (Λ UV ) coming from the massive φ propagator. Replacingthis mass by m IR , i.e. multiplying Eq. (39) by (Λ UV /m IR ) , we obtain Eq. (40). This rateagrees with the result of the gravity calculation for α > α < <
3) the above line of argument breaks down since the mixing between φ and the CFT states can not any more be considered a small perturbation. This canalso be understood from our gravity-side discussion in Sect. 3. Indeed, we are dealingwith a situation where the UV-brane mass term is tuned to allow for an exact zero mode.The bulk profile of this zero-mode is ψ ∼ z − α . Thus, its normalization is UV-localizedfor α > α < α >
1, thissupports our previous statement that φ does not affect the CFT dynamics significantly.By contrast, in the α < α < λ = λ from the CFT perspective,we will first describe the analysis of this section from an equivalent but technically slightlydifferent point of view. When running the Lagrangian of Eq. (37) down to smaller scales,we are consecutively integrating out φ modes with lower and lower 4-momenta k . This15nduces corrections which are schematically of the form ∼ − UV O ∆ ( k ) M φ ( k ) + c ( k ) k . (41)Here the function c is zero at the high scale, c (Λ UV ) = 0, and grows at lower scales to theextent that a kinetic term for φ is induced by CFT loops. If M φ ( k ) becomes small andthe induced operator ∼ O ∆2 is not suppressed by Λ UV (i.e. for ∆ < λ < λ .Alternatively, we may immediately integrate out φ , before considering any RG run-ning or loop effects. This induces a coupling of the type given in Eq. (38) (but with M φ = M φ (Λ UV )). It also induces the term ∼ O (Λ UV ) − M φ (42)which, in contrast to the non-local corrections given schematically in Eq. (41), is a localoperator. In other words, the presence of a UV brane with a brane-localized mass termcan be viewed, on the CFT side, as a correction by a ‘double trace operator’ as discussedin [48].It is now immediately clear that, as M φ is large in the detuned case, this O cor-rection is too small to affect the CFT dynamics at low energy scales (and in particularthe dimension of O ∆ ) significantly. Hence, the coupling O ∆ F can be directly used toestimate the decay rate of any light glueball with non-vanishing overlap with O ∆ . Thisreproduces Eq. (39).It is also clear that, from this perspective, the value ∆ = 2 (corresponding to α = 0)deserves special attention. Indeed, in this case the O operator is marginal rather thanirrelevant. As discussed in Sect. 4 of [48], this causes multiplicative renormalization ofthe operator O ∆ with logarithmic running: O ∆ ( m IR ) ∼ ln(Λ UV /m IR ) O ∆ (Λ UV ) . (43)Upon rescaling the operator in Eq. (38), we obtain an additional factor of ln(Λ UV /m IR ) − in the decay rate Eq. (39), in agreement with the factor ln( k/m IR ) − in the second lineof Eq. (24).We now consider the case α < λ = λ , corresponding tothe largest consistent coefficient of the O correction. From the discussion in Sect. 3 (seein particular Eqs. (17) and (28)) we see that, in this case, the UV behaviour of the AdSscalar changes from Φ ∝ z α to Φ ∝ z − α . According to the general discussion of [41],the dual description of an AdS field theory with such boundary conditions and with α < O ∆ ′ with dimension ∆ ′ = 2 − α (insteadof ∆ = 2 + α ). When we integrate φ out at the UV scale, we obtain the coupling ∼ ∆ ′ UV O ∆ ′ tr ( F µν F µν ) (44)16s well as a term of the type given in Eq. (42). One may in fact say that it is this lattercorrection with an appropriately tuned coefficient which forces the dimension of O tochange to ∆ ′ = 4 − ∆ in the infrared [27] (see also [48, 49]). From Eq. (44), we canestimate the decay rate of glueballs with non-vanishing overlap with O ∆ ′ asΓ ∼ (cid:18) m IR Λ UV (cid:19) ′ m IR , (45)in agreement with Eq. (25).While a more systematic and quantitative study of the CFT description of the decayrates under consideration may be worthwhile, we believe that we have now suppliedenough additional physical intuition for the purposes of the present paper. In particular,we have fully confirmed the results of our gravity-side calculation. We now discuss the applicability of our results to throat geometries in type IIB stringtheory. For definiteness, we focus on the Klebanov-Strassler (KS) throat [4, 47, 50, 51]in the following. We expect, however, that our results can similarly be applied to othertypes of throats.The curvature scale R = k − of the KS throat varies logarithmically along the radialdirection and the geometry smoothly terminates in the IR. We neglect this logarithmicvariation in the following. Sufficiently far away from the IR tip, the KS throat can thenbe approximated by the space AdS × T , . In the UV, the throat is smoothly gluedinto a compact manifold [1]. We focus on compactifications in which the size L of thismanifold is not hierarchically larger than the AdS scale R of the throat. We can thenneglect the ‘thickness’ of the compact manifold and approximate it by the UV brane ofan RS model. Similarly, we model the IR end of the throat by an IR brane (see Ref. [5]for more details on the KS throat as a RS model).The KK reduction of type IIB supergravity on T , in an AdS × T , background hasbeen carried out in [23]. The resulting spectrum of KK modes contains scalars with var-ious tachyonic masses, some of which saturate the Breitenlohner-Freedman (BF) bound.Note that the presence of scalars saturating this bound is expected on general grounds:The theory in AdS , which results from the KK reduction, is supersymmetric. Further-more, the isometries of T , ensure the presence of massless vector multiplets in thespectrum. Such multiplets contain a scalar with maximally tachyonic mass [21, 22, 32].We will now discuss tachyons from such multiplets in more detail. The KK reductionof type IIB supergravity on AdS × T , contains eight massless 5d vectors. One ofthese vectors comes from the compactification of the 4-form potential on the 3-cycle in The throat can only be glued smoothly into the compact manifold if L & R . The distribution of fluxvacua strongly favors vacua where the volume L is small in string units, cf. [52] and references therein.Thus, the generic compactification has no large hierarchy between R and L . , ∼ S × S . The corresponding abelian symmetry of the solution is called U (1) B .The seven remaining vectors are associated with the SO (4) × U (1) R isometry of T , .Since the U (1) R vector is part of the massless graviton multiplet [23] which contains noscalars (and in particular no tachyon), we end up with seven BF tachyons in total. Inthe classification of [23], these scalars belong to shortened versions of vector multiplet I.In a KS throat, the symmetries are reduced with respect to AdS × T , . In theregion between the IR and UV end, which is the most symmetric part of the throat, thesymmetry is reduced to SO (4) × U (1) B × Z M ⊂ SO (4) × U (1) B × U (1) R . The U (1) R isbroken to Z M by the M units of 3-form-flux on the S , which are also responsible forthe logarithmic variation of the AdS scale [53]. Note, however, that this breaking doesnot affect our previous counting of BF tachyons since, as mentioned before, the U (1) R vector has no scalar partner.The symmetry is further reduced at the IR and UV end of the KS throat: In the IR,the SO (4) stays intact, but the U (1) B is completely broken, and the Z M is broken to Z [4,54–56]. In the UV, on the other hand, the SO (4) × U (1) R isometry of T , is brokensince a compact Calabi-Yau has no continuous isometries. Concerning the U (1) R factor,we are anyway only interested in its Z subgroup which survives in the IR region. Thisdiscrete Z symmetry (or a larger discrete subgroup of the T , isometry) may or maynot be respected by the compact Calabi-Yau. The latter is certainly the generic case.In addition, the KS throat has another Z symmetry (called I -symmetry in [56]) whichinterchanges the two 2-sphere in the T , ( T , is an S bundle over S × S ) and reversesthe sign of the 2-form potentials of type IIB supergravity [47, 57]. This symmetry isunbroken in the IR and may or may not be broken in the UV. Furthermore, the U (1) B can be broken or remain unbroken in the UV. This is not essential for us since thecorresponding BF scalar, being in the adjoint, is uncharged. Thus, the couplings of thisscalar to various UV localized fields, which are our main interest, are not forbidden by thissymmetry. We finally note that the symmetry breaking in the UV is mediated to the IR byirrelevant operators and is thus suppressed by powers of the warp factor [10, 11, 60, 61]. We now discuss specifically the effects of the tachyon in the U (1) B vector multiplet(also known as Betti multiplet). This tachyon is an SO (4) singlet, but odd under the I -symmetry Z [55]. In the IR region of the KS throat, where the approximation as As long as the throat is infinite, we can move along a one-parameter family of solutions, the so-called baryonic branch of the dual gauge theory, and the KS solution is a special point which respectsthe I -symmetry [55,56,58,59]. However, the metric deformation which corresponds to moving along thebaryonic branch does not respect the conformal-Calabi-Yau condition [58]. Hence, it is not clear how toglue such a deformed throat into a compact UV space in the framework of warped flux compactifications[1]. For this reason, we do not consider such deformations. It is, of course, nevertheless possible that the I -symmetry is broken by the Calabi-Yau (instead of the throat), corresponding to a Z -breaking on theUV brane. As discussed in [73], relevant perturbations (which grow towards the IR) are also possible as longas they start with a sufficiently small amplitude in the UV. We note that, in the infinite throat limit, this 5d scalar has a massless 4d mode [55, 62]. Whenthe throat is glued into a compact manifold, this mode obtains a mass which is parametrically largecompared to the IR scale m IR if the volume of the compact manifold is small. Thus, this mode is notimportant in our context. × T , becomes unreliable, this tachyon mixes with a scalar of mass M = 5 k [62, 63]. In the UV, on the other hand, the two scalar fluctuations decouple, as we willnow demonstrate. The equations of motion of this system are given in Eqs. (47) and (48)in [62]: ˜ z ′′ − τ ˜ z + ˜ m I ( τ ) K ( τ ) ˜ z = ˜ m · / K ( τ ) ˜ w (46)˜ w ′′ − cosh τ + 1sinh τ ˜ w + ˜ m I ( τ ) K ( τ ) ˜ w = 169 K ( τ )˜ z . (47)Here, primes denote derivatives with respect to the radial coordinate τ of the KS throat(using the metric convention of [4]) and ˜ m is related to the 4d mass of the state. Thefunctions I ( τ ) and K ( τ ) are e.g. given in Appendix B in [62]. In the UV, τ ≫
1, thesefunctions can be approximated by I ( τ ) ∼ τ e − τ/ and K ( τ ) ∼ e − τ/ . The equations ofmotion then indeed decouple and simplify to˜ z ′′ = 0 (48)˜ w ′′ − ˜ w = 0 . (49)This is solved by ˜ z = τ + const. and ˜ w = e τ + const. e − τ (neglecting the overall factors).For τ ≫
1, the coordinate τ is related to our coordinate z by z ∝ e − τ/ and the wavefunc-tions read ˜ z = ln z + const. and ˜ w = z − + const. z . Up to an overall factor of z (whichis related to a field redefinition ) these are indeed the wavefunctions of a tachyon whichsaturates the BF bound and a scalar with mass M = 5 k . In particular, we see that thewavefunctions from the KK decomposition on AdS × T , are a good approximation inthe UV of the KS throat.The mass spectrum of 4d KK modes from this system of two scalars was determinedin [62] and found to contain the lightest state which is known so far for the KS throat(see e.g. [64–69] for other parts of the mass spectrum). Since heavier KK modes decayvery quickly to lighter states via various processes (which we will discuss in Sect. 7), thislightest KK mode generically contains an O (1)-fraction of the energy density of a heatedKS throat. Due to the effect of the U (1) B tachyon, these particles decay to the UVsector with the rate determined in Sect. 3.The approximate SO (4) symmetry of the IR region suppresses decays violating thetotal SO (4) charge. Therefore, a sizeable fraction of the energy density in a KS throat isgenerically in the form of charged KK modes. We will now argue that tachyons are alsorelevant for the decay of these states. Namely, the spectrum in [23] contains tachyonswith various charges, though not all of them have the maximally allowed negative mass-squared. For example, the tachyon in the SO (4) vector multiplet is in the adjoint of In particular, the field ˜ z is related to the fluctuation ψ = δg = δg of the 5-dimensional compactmanifold in the throat by the field redefinition ˜ z = z − ψ . In the UV of the throat, where the approx-imation as AdS × T , is applicable, the field ψ is a 5d scalar with a standard kinetic term (comingfrom the 10d Einstein-Hilbert term). More precisely, this state is part of a massive vector multiplet of 4d N = 1 supersymmetry [63]. Asimilar fraction of the energy density is therefore stored in the superpartners of this scalar. O (4). We expect that these tachyons mix with other scalars with the same charge inthe IR region of the KS throat. Various IR-localized states can therefore decay via agiven tachyon to the UV sector, and the decay rates from Sect. 3 apply. As anotherexample, we can consider KK modes which are dual to glueballs created by the operatorof lowest dimension, ∆ = , in the KS theory. These states are scalars in the ( , ) of SU (2) × SU (2) ∼ SO (4). In [63], the lightest KK mode in this tower was proposed asa candidate for the lightest state in the KS spectrum. A scalar operator of dimension∆ = is dual to a tachyon with α = 2 − ∆ = (cf. Sect. 5). The decay rate of thecorresponding KK modes then follows from the formulas in Sect. 3.We now discuss the UV-brane mass term of the various tachyonic scalars. As empha-sized before, we focus on compactifications where all moduli are stabilized by fluxes andnon-perturbative effects along the lines of [1, 6]. Let us first assume for simplicity thatboth complex structure and K¨ahler moduli are stabilized at the UV scale. In 5d languagethis means that the UV brane theory has only one energy scale: k ∼ R − ∼ L − ∼ M string .(For simplicity, we ignore the hierarchy between L − and M string , assuming it to be small.)Since the full construction is stable, we know that all tachyonic scalars will receive a UV-brane mass term with λ ≥ λ in this context. Given that there is no small energy scalearound (which could correspond to λ − λ ≪
1) and no obvious reason for the tuning λ = λ , we make the assumption that λ = λ + O (1). As we will see in Sect. 7, evenwith this conservative assumption the cosmological effects of tachyonic scalars can bedramatic.There are, however, at least two possible loopholes in this conclusion. The first loop-hole is related to the question where the massless mode of a 5d tachyon with tunedUV-brane mass is localized. To see this, let us first strengthen our previous argumentfor detuning in the following way: Recall that the wavefunction of the massless mode forsupersymmetric boundary conditions, ψ ∝ z − α , is UV-localized for α > α >
1, there is always anexponentially light UV-localized mode in the spectrum. However, with all Calabi-Yaumoduli stabilized at the high scale, there should, in 5d language, be no light fields ator near the UV brane. Thus, the UV-localized mode can not be light, implying that theUV mass term has to be detuned. For α <
1, however, this argument does not apply asthe wavefunction is IR-localized in this case. The zero mode of such a tachyon can belight without violating the requirement that there are no light fields in the UV sector.Furthermore, for generic IR-boundary condition, this mode picks up a mass of the orderthe IR scale. Thus, for α <
1, the UV mass term can in principle be tuned. Of course,we still have no reason for the required tuning, but we can also not dismiss the tunedsituation on general grounds.The second loophole lies in our assumption that there are no light fields in the UV sec-tor. There is obviously the possibility that the K¨ahler stabilization scale is much smallerthan the flux stabilization scale, which may be the natural choice for low-scale SUSY.In this case, the UV-brane theory has a second, smaller energy scale with correspondinglight fields. In the 5d description, these fields could be the lowest KK modes of tachyons.20ore precisely, for tuned λ = λ and α >
1, such a mode is massless and localized inthe UV, corresponding to an unstabilized modulus. Stabilization of the K¨ahler moduli,in the 5d description, is then due to a detuning of the mass on the UV brane. If theK¨ahler stabilization scale is lower than the flux stabilization scale (which we assume tobe of the order the AdS scale), this detuning is small, λ − λ ≪
1, giving the modu-lus a small mass compared to the AdS scale. Of course, as far as some generic K¨ahlermodulus is concerned, such a connection is far from obvious because the throat has no2- or 4-cycles. However, such an approximate tuning can potentially be related to theuniversal K¨ahler modulus. If the detuning is sufficiently small, the higher KK modesof the corresponding tachyon would decay with the enhanced rates derived in Sect. 3.2.In summary, we have very good reasons to consider decays mediated by tachyons inthroat cosmology. While the situation with detuned UV mass term appears to be generic,we can not exclude the possibility that certain tachyonic fields have a UV mass tuned tothe minimal allowed value. Establishing this, which would imply even larger decay ratesthan we discuss in the following, would be an interesting subject for future research.Finally, we note a recent paper [14] which presented a survey of decay channels ofKK modes in a KS throat. In particular, the decay rate of KK modes to moduli andtheir axionic partners was determined. As the moduli can be viewed (at least partially)as fields living on the UV brane, it is interesting to compare this decay rate with ourresults. More precisely, Eq. (5.5) in [14] gives the decay rate of a KK mode into axionicpartners of the moduli. For M p ∼ M s ∼ k and using our notation w = m IR /k and ν ⋆ = α , their Eq. (5.5) reads Γ ∼ (cid:16) m IR k (cid:17) α m IR . (50)This agrees with our Eq. (24) (apart from the logarithmic suppression in the case α = 0).However, we believe that this rate also applies to the tunneling of KK modes to otherthroats. As discussed in [35] in some detail, via the AdS/CFT correspondence, otherthroats can be described by large- N gauge theories which live on the UV brane. Thetunneling rate of KK modes to these sectors is then given by Eq. (50) times the numberof degrees of freedom, ∼ N , of the dual gauge theories. Note that T , is isomorphic to S × S and thus has a nontrivial 2-cycle [47]. However, this S shrinks to zero at the IR tip of the throat and thus is not a cycle of the Calabi-Yau. It is clear that the universal K¨ahler modulus belongs to the UV sector if we can work in a K¨ahler-Weyl frame (the Brans-Dicke frame). In this frame, the metric in the IR region is not affected by a (small)shift of the universal K¨ahler modulus. Namely, such a shift corresponds to changing the prefactor ofthe 4d Einstein-Hilbert term, which is dominated by the compact Calabi-Yau and the UV end of thethroat. However, it has recently been argued that this is not always the appropriate frame [70, 71]. Inother frames, the universal K¨ahler modulus may be viewed (at least partially) as a throat field. Note that the decay rate in their Eq. (5.4) is additionally suppressed as it involves a transitionbetween states with different charges under the approximately conserved symmetries of the throat. Cosmology
To illustrate the relevance of tachyons, we will now discuss the reheating of the StandardModel (SM) after warped brane inflation. For definiteness, we focus on a specific real-ization of warped brane inflation, the KKLMMT scenario [7]. In this scenario, inflationis driven by a D3-brane which slowly rolls towards an anti-D3-brane located at the tipof a KS throat. As discussed in Appendix C of [7], for given parameters (string coupling,string scale, etc.), the hierarchy of this throat can be fixed by observational data. Inthe following, we will use m IR ∼ GeV and k ∼ GeV for numerical estimates.We assume that the SM is realized on D-branes which are localized in the CY outsidethe throat. In the effective 5d description, the SM then lives on the UV brane of thecorresponding RS model. Furthermore, we assume that the inflationary throat is theonly strongly warped region in the compact space.Inflation ends with the annihilation of the D3-brane and the anti-D3-brane. Theresulting energy is deposited in KK modes localized at the tip of the throat [15–18, 72].Temperature and energy density of this gas, which is initially marginally non-relativistic,are set by the IR scale: T ∼ m IR and ρ ∼ m IR . These KK modes decay to the SM atlater times. The resulting reheating temperature of the SM can be estimated as usual:Most of the energy is transferred when the Hubble rate is comparable to the decayrate. The Hubble rate is related to the total energy density by H = ρ tot / M . Forradiation at temperature T and with g SM degrees of freedom, the energy density is givenby ρ tot = π g SM T . The reheating temperature can then be estimated as T RH ∼ (cid:18) g SM (cid:19) p M Γ . (51)It is probable that the energy density in the throat is in the form of different speciesof KK modes with different lifetimes. After one species of KK modes has decayed tothe SM, KK modes with longer lifetimes can easily come to dominate the total energydensity as their energy density scales like matter (and not like radiation). Their decayto the SM then leads to a new phase of reheating and the final reheating temperaturewill be determined by the most stable KK modes. Let us now assume that the throathas no tachyonic scalars. There certainly are scalar KK modes in the spectrum of the KSthroat (e.g. from the dilaton). Without tachyons, their largest possible decay rate wouldbe that for a massless scalar, corresponding to α = 2. As discussed in Sect. 6, we expectthat scalars generically obtain a mass on the UV brane (in the 5d description) and thatthis mass is not tuned to the special value λ . Using the corresponding decay rate for a We use the term ‘Standard Model’ to refer to a sector which contains the SM, e.g. the MSSM.For definiteness, we use g SM = O (100) for the number of effective relativistic degrees of freedom at thereheating temperature T RH . For example, the SM itself has g SM ≈
107 for temperatures T &
300 GeVwhile the MSSM has g SM ≈
229 for T & m / (with the familiar prefactor 7 / Some fine-tuning is generically required to actually achieve slow roll inflation. A systematic studywas carried out in [73] (see also [74, 75] for earlier work). Some of the mechanisms used in these papersrely on D7-branes wrapping 3-cycles of the throat (see e.g. [76] for an explicit realization). For ourpurposes, we assume a proper fine-tuning such that a KKLMMT-type scenario is realized. T RH . . (52)This is dangerously low. In particular, it is difficult to obtain sufficient baryogenesis atsuch low temperatures. Moreover, reheating temperatures below 1 MeV are excluded bynucleosynthesis [78]. As tachyons lead to higher decay rates, the reheating temperaturecan also be higher. We will now discuss this in more detail.We first consider processes in the throat sector which happen on timescales shorterthan those relevant for decays to the SM. Warped KK modes are described by an effec-tive field theory with a low cutoff, ∼ m IR . We do not assume a large hierarchy between k and M (or, from the 10d perspective, k , M s and M ) which would suppress quan-tum corrections. All possible n-point-interactions, which are allowed by the symmetries,will therefore be induced by quantum effects (if they are not present at tree-level). Inparticular, the effective action includes couplings of the type HHGG , (53)which involves two different species of KK modes H and G and allows for the processes2 · H ↔ · G . Here, we have suppressed all symmetry-indices (which are appropriatelycontracted). Let G denote the lightest KK mode in the spectrum, whereas H is anyheavier KK mode. As we will see in a moment, the KK modes are in thermal equilibriuminitially. For temperatures below the mass of the heavier KK mode, the process 2 · H → ·G then occurs with a much higher rate than the inverse process. The heavier KK modesthus begin to annihilate to the lighter KK modes, leading to an exponential decrease ofthe relative number density of the two states, n H n G = e − ( m H − m G ) /T . (54)Here, m H and m G are the masses of the two states and T is the temperature of the gasof KK modes. This exponential decrease continues until the heavier KK modes are sodilute that they decouple. This happens, when n H · h σv i ∼ H , (55)where h σv i is the thermally averaged product of cross section and relative velocity forthe scattering process and H is the Hubble rate. The latter is dominated by the energydensity ρ of the gas of KK modes, H ∼ √ ρ/M . Due to the exponential dependence onthe temperature in Eq. (54), the heavy KK modes decouple when the temperature is stillof the order m IR . By dimensional analysis, it then follows that h σv i ∼ m − IR as well as ρ ∼ m IR and n G ∼ m IR . Using Eq. (55), we find n H n G ∼ m IR M ∼ − . (56) More precisely, the condition on the decoupling temperature T dec follows from Eqs. (54) and(55). Defining m ≡ m H − m G (and assuming that m ∼ m IR ), the condition can be written as e − m/T dec ∼ ( m/M ), where we have neglected powers of m/T dec multiplying the right-hand side. For m IR /M ∼ − , we then find that T dec ∼ m IR / − times less abundant.There are other decay processes in the throat sector. In particular, the effectiveaction will also contain various trilinear couplings. If kinematically allowed, KK modescan therefore decay to two lighter KK modes. If these processes are not suppressed dueto the approximate symmetry SO (4) × Z × Z of the effective action (cf. Sect. 6), thecorresponding decay rate is ∼ m IR (as follows by dimensional analysis). Moreover, weexpect various couplings of KK modes to 4d gravitons in the effective action. While suchcouplings are forbidden by KK-mode orthogonality at the level of the two-derivative,quadratic action [9, 11], they are certainly present in the strongly-coupled and highlynon-linear effective field theory relevant for the dynamics of the low-lying KK modes.Indeed, integrating out the heavier states, all types of multi-particle and higher-derivative vertices are generated in the effective action for the lowest-lying KK modes.Since any derivatives appearing in this action are covariant, various quantities derivedfrom the Riemann tensor naturally arise. As we assume no significant hierarchy betweenthe AdS scale and the Planck scale, the only scale relevant for this argument is the IRscale m IR . Thus, for example, we naturally expect a term of the type ∼ m IR R G (57)to arise. Here, R is some scalar quadratic in the 4d Riemann tensor and the scalar KKmode G is a singlet with respect to the symmetries of the effective action (otherwise,the term would be forbidden or suppressed). We therefore believe that such terms aregenerically present in the effective action and that they are not suppressed by powersof the warp factor, a possibility raised in [14]. As discussed in [14, 77], the couplingEq. (57) allows for the decay of G to two 4d gravitons. By canonically normalizing thegravitons, the relevant vertex is seen to be suppressed by 1 /M , leading to the decay rate ∼ m IR /M . Similar arguments can be made for couplings involving two KK modes anda graviton. Heavier KK modes can therefore decay to lighter KK modes via the emissionof a graviton. If these processes are not suppressed due to the approximate symmetriesof the effective action, the corresponding decay rate is ∼ m IR /M .Let us summarize what we have found: At late times, the energy density is dominantlyin the form of the lightest KK mode. As we have discussed in Sect. 6, the lightest knownKK mode in the KS throat mixes with the tachyon from the Betti multiplet. We willassume that this KK mode is indeed the lightest state. It thus decays to the SM withthe rate for a maximally tachyonic scalar:Γ ∼ (cid:16) m IR k (cid:17) m IR ln ( k/m IR ) . (58)As the lightest known KK mode is odd with respect to the I -symmetry (cf. Sect. 6),the decay to two gravitons is forbidden or strongly suppressed (compared to the rate ∼ m IR /k for k ∼ M ). Thus, the decay to the SM is the dominant decay channel forthis state. 24he heavier KK modes are 10 − times less abundant than the lightest KK mode.In addition, several of these states decay to lighter KK modes via trilinear couplingsand graviton emission. The timescales for these decays are ∼ m − IR and ∼ M /m IR ,respectively. At later times, only the lightest KK modes with given charges under SO (4) × Z × Z may survive. Moreover, as this symmetry is only approximate, eventu-ally all KK modes can decay to the lightest KK mode. Such charge violating decays are,however, suppressed by additional powers of the warp factor [10, 11, 60, 61]. If the decayrate is nevertheless higher than Eq. (58), all KK modes will first decay to the lightest KKmode. The latter subsequently decay to the SM. Using Eq. (51), the resulting reheatingtemperature of the SM is T RH ∼ GeV . (59)This is considerably higher than 1 GeV, due to the enhanced decay rate mediated by atachyon. It is also possible, however, that the total decay rate of the heavier KK modes(to the SM and to the lightest KK mode) is smaller than Eq. (58). These states will thenstill be stable when the lightest KK mode has already decayed to the SM. As their energydensity scales like matter whereas the energy density of the SM scales like radiation, theheavier KK modes might come to dominate the total energy density. Their decay to theSM then leads to a new phase of reheating with a lower reheating temperature. Althoughwe have not analysed this possibility in more detail, it is clear that tachyons may alsoenhance the relevant decay rates (and thus the reheating temperature) in this case.Finally, we briefly consider the decay of fermionic KK modes. If supersymmetry isbroken outside the throat, we expect that the fermionic decay rates are of the sameorder of magnitude as those of the scalars in the same multiplet. However, the decays offermions are somewhat model-dependent as the supersymmetry breaking scale (or thegravitino mass m / ) determines the allowed decay channels. If supersymmetry is brokenat low scales, m / < m IR , the fermionic KK modes can decay to lighter KK modesunder the emission of a gravitino. Another possible channel is the decay to a gravitonand a gravitino. These processes are the analogue of the decay of bosonic KK modesto gravitons. The resulting abundance of gravitinos leads to the well-known gravitinoproblem. If the supersymmetry breaking scale is high, m / > m IR , decays to gravitinosand decays to superpartners of standard model particles are kinematically forbidden. Aswe have assumed that the inflationary throat is the only strongly warped region (and thefermionic KK modes can thus not decay to such sectors), the fermionic KK modes maythen be absolutely stable. In order to avoid the resulting overclosure of the universe, onecan invoke the neutrino portal which allows their decay to a neutrino and a Higgs [12,82]. In this paper, we have analysed the effect of tachyonic scalars on couplings betweenIR- and UV-localized sectors in warped compactifications. We gave an introduction to In some scenarios of brane inflation the gravitino mass provides an upper bound on the inflationaryHubble scale, H I . m / [79, 80]. Since H I ∼ m IR /M , the case m / < m IR is possible also in thesescenarios. in Sect. 2. In particular, we explained the origin ofan instability and its removal via a UV-brane-localized mass term for the field, bothof which can be easily understood in a quantum mechanical analogue. In Sect. 3 weconsidered decays of IR-localized KK-modes to a UV-localized gauge theory in a 5dRandall-Sundrum toy model and derived the dependence of the decay rates on the warpfactor. As expected, this dependence is governed by the 5d mass of the scalar. There isalso a further suppression due to the UV-localized mass term.We have developed the dual CFT description of our results in Sect. 5. In this ap-proach, the dependence on the 5d mass M is encoded in the dimension of the dualoperator, given by ∆ = 2 + p M /k . As long as the UV-brane mass term takesgeneric values above the boundary for stability, its effect on decay rates can be under-stood as a simple propagator suppression on the CFT side. If this boundary mass termis tuned to its minimal allowed value, decay rates are correspondingly enhanced. Whilethis is straightforward to see for ∆ >
3, the analysis of such a tuned situation in theregime with ∆ < ′ = 4 − ∆.In Sect. 6, we have worked out the applicability of our 5d analysis to throat geometriesin string compactifications. As a specific example, we have considered the Klebanov-Strassler throat with its approximate AdS × T , geometry in the UV region. Due to the(gauge) symmetries of the solution, tachyonic 5d scalars saturating the Breitenlohner-Freedman bound are present. We have argued that, generically, the UV-brane mass ofthese tachyons is detuned from its minimal value. The spectrum of light KK modesin the Klebanov-Strassler throat has been studied (cf. e.g. [62–69]) and the lighteststate found so far is a singlet under the continuous symmetries of the throat. This stateresults from the mixture of two 5d scalars, one of which has maximally tachyonic 5dmass. Using the equations of motion of the two coupled scalars, we have checked thatthe two fluctuations decouple in the UV and that one of the resulting wavefunctionscorrectly describes a maximally tachyonic scalar. The lightest (known) KK mode of theKlebanov-Strassler throat thus decays to the UV brane with the decay rate derived inSect. 3.In Sect. 7, we have used a specific example, the reheating of the standard model afterwarped brane inflation, to demonstrate the relevance of tachyons in throat cosmology.Since tachyons affect the decay rate to the UV brane, their role is decisive in settingswhere the SM is localized outside the inflationary throat. We have focused on such sce-narios assuming, for definiteness, that the SM is realized in the bulk of the Calabi-Yau. An important consequence of tachyons is that they lead to a larger reheating tempera-ture of the standard model. More precisely, without tachyons, one finds an upper boundon the reheating temperature T RH . We have, however, also pointed out potential loopholes in this argument. Thus, the case of a tunedUV mass term can not be completely dismissed. Such a tuning would allow for decay rates which areeven higher than those which we found in the generic situation. We believe, however, that our results also apply to situations where the SM lives in another throat.In this case, our decay rates have to be multiplied by the number of degrees of freedom, ∼ N , of thelarge- N gauge theory which is dual to the additional throat. RH ∼ GeV. Charged KK modes may have smaller decay rates and correspondinglylarger lifetimes. Their decay may lead to another phase of reheating, resulting in a lowerreheating temperature. Although we have not determined the decay rate of a genericcharged KK mode, it is likely that their decay rates are also enhanced by tachyons.To discuss this in more detail, we recall that the UV region of the Klebanov-Strasslerthroat can be approximated as AdS × T , . In this region, the KK reduction on T , ,which was performed in [23], is applicable. Close to the IR end of the throat, the approx-imation as AdS × T , becomes unreliable. We expect that a given AdS field (comingfrom the KK reduction on T , ) mixes with all other fields with the same quantumnumbers in this region. This is likely already the case at tree-level as follows e.g. fromthe analysis in [62, 63, 68]. Moreover, there generically is no large hierarchy between the10-dimensional Planck scale and the AdS curvature scale which could strongly suppressquantum corrections. Thus, we expect all bilinear mixing terms, which are allowed bythe symmetries, to appear in the effective 5d action. A 4d KK mode with given quantumnumbers will therefore couple to the UV-brane via all AdS fields with the same quantumnumbers. It is then clear that the decay rate of a scalar KK mode will be determined bythe AdS field with the smallest (or possibly most tachyonic) mass-squared for the givencharges. As the spectrum of these fields is known [23], it should be possible to determinethe decay rate of a generic scalar KK mode from its charges under the (approximate)symmetries of the throat. This would be an interesting project for future research.Another important cosmological application of our results is to throat dark mat-ter [9–14, 16]. Let us first consider scenarios in which the standard model is localizedoutside the throat. Measurements of the cosmic diffuse γ -ray background give a lowerbound of ∼ s for the lifetime of dark matter with decay channels to photons (as-suming an O (1) branching ratio for decays via hadrons) [81]. As we have discussed inSect. 6, the lightest known KK mode of the Klebanov-Strassler throat couples to theUV-brane via a maximally tachyonic scalar. Generically, a considerable fraction of thedark matter will be in the form of this lightest state. In order to fulfill the above bound,the dark matter throat then has to have an IR scale which is smaller than ∼ GeV(assuming k ∼ GeV). This effectively excludes KK modes in an inflationary throat`a la KKLMMT [7] as a viable dark matter candidate. Dark matter in sufficiently longthroats, which may be produced by tunneling from the inflationary throat [9, 16] or ther-mally from the standard model [12], can however still be viable. It may, of course, turnout that the lightest KK mode of the Klebanov-Strassler throat is not the one consideredin Sect. 6. However, the mass of a glueball can roughly be expected to decrease with thedimension of the operator which creates the state [62, 63]. Or, correspondingly, the massof KK modes generically decreases with the mass of the corresponding AdS field [10].It is therefore likely that any lighter KK mode is also associated with a 5d tachyonwhich determines its decay rate. Finally, dark matter can be formed by fermionic KKmodes which can decay to the standard model via the neutrino portal [12,82]. The decayrate is likely to be determined by the superpartner of a tachyonic 5d scalar and can besuppressed by (approximate) R-parity.Alternatively, in scenarios in which the standard model is located in the dark matterthroat [9–11,14,16], our results are relevant for the decay of dark matter to other throats27which are likely to be present in a given compactification [3]). If the dark matter consistsof the lightest (known) KK mode , the requirement that the dark matter lives longerthan the current age of the universe gives an upper bound of ∼ GeV on the IR scaleof the throat (again assuming k ∼ GeV). The preferred scale ∼ TeV thus still allowsfor a viable dark matter candidate (provided it is sufficiently stable against decay to thestandard model). As we have discussed above, even if the dark matter instead consistsof other charged KK modes, tachyons are again likely to determine their decay rate tothe UV sector.
Acknowledgements
We would like to thank Johanna Erdmenger, Tony Gherghetta, Dam T. Son and Gian-massimo Tasinato for helpful comments and discussions. A.H. is grateful to the BerkeleyCenter for Theoretical Physics for hospitality. This work was supported by the GermanResearch Foundation (DFG) within the Transregional Collaborative Research CentreTR33 “The Dark Universe”.
Appendix: Decay Rates with Bessel Functions
The purpose of this appendix is to compute the decay rates estimated in Sect. 3 explicitlyin terms of Bessel functions. Similar calculations to those below can be found e.g. in[8, 22, 83, 84].The analytic solution to Eq. (15) in terms of Bessel functions is given by ψ n = 1 a α,n b z / [ J α ( b z ) + b α,n Y α ( b z )] , (60)where a α,n and b α,n are constants determined from the normalization of ψ n , Eq. (19),and the boundary conditions, Eq. (16).In order to relate the analytic solution to the previous estimates, we consider theasymptotic expansions of the Bessel functions in the regions of small and large argument[85]. For 0 < z ≪ √ α + 1: J α ( z ) ≃ α + 1) (cid:16) z (cid:17) α (61)and Y α ( z ) ≃ π ( γ − ln 2 + ln z ) for α = 0 − Γ( α ) π (cid:18) z (cid:19) α − cos( πα )Γ( − α ) π (cid:16) z (cid:17) α for α > . (62) Note that this state is odd under a Z -symmetry and may therefore be stable against decay to thestandard model. γ ≃ . α ) is the gamma function. The second termin the expansion for Y α is important in the limit α →
0. Furthermore, for z ≫ (cid:12)(cid:12) α − (cid:12)(cid:12) : J α ( z ) ≃ r πz cos (cid:16) z − απ − π (cid:17) (63)and Y α ( z ) ≃ r πz sin (cid:16) z − απ − π (cid:17) . (64)From these expansions it is clear that the Bessel functions explicitly realize the matchingof UV and IR solutions discussed in Sect. 3, cf. Eqs. (17) and (18).Moreover, the Bessel functions satisfy certain identities which are useful in the cal-culation of the decay rates below. For any linear combination C α ( z ) of Bessel functions J α ( z ) and Y α ( z ) in which the coefficients are independent of α and z , the followingidentities are valid [85]: 2 αz C α ( z ) = C α +1 ( z ) + C α − ( z ) , (65)2 ddz C α ( z ) = C α − ( z ) − C α +1 ( z ) (66)and Z ba dz z C α = z (cid:2) C α ( z ) − C α +1 ( z ) C α − ( z ) (cid:3) | ba . (67)In addition, for integer α , C α satisfies: C − α ( z ) = ( − α C α ( z ) . (68)Furthermore, the following relation holds for all α and z : J α +1 ( z ) Y α ( z ) − J α ( z ) Y α +1 ( z ) = 2 πz . (69)We will use these identities frequently throughout the computations to simplify theresults.By inserting the solution Eq. (60) into the boundary conditions Eq. (16), we find: − b α,n = b z IR J α − ( b z IR ) + (2 − α − λ ) J α ( b z IR ) b z IR Y α − ( b z IR ) + (2 − α − λ ) Y α ( b z IR ) = b z UV J α − ( b z UV ) + (2 − α − λ ) J α ( b z UV ) b z UV Y α − ( b z UV ) + (2 − α − λ ) Y α ( b z UV ) . (70)From Eq. (70) we see that the masses m n are determined by[ b z IR J α − ( b z IR ) + (2 − α − λ ) J α ( b z IR )] + b α,n [ b z IR Y α − ( b z IR ) + (2 − α − λ ) Y α ( b z IR )] = 0 . (71)Using the asymptotic expansions for small arguments, Eqs. (61) and (62), as well asEq. (70), we find that (up to O (1) prefactors) b α,n ∼ b z α UV for generic λα (1 − α ) b z α − UV for λ = λ . (72)29or ke − kℓ ≪ m n ≪ k and generic α and λ , one then finds that the masses m n aredetermined by the zeros of J α − ( b z IR ). This gives (see e.g. [22]): m n ≃ (cid:18) n + α − (cid:19) πke − kℓ . (73)As can be seen from the second line of Eq. (72), the masses are also determined by thezeros of J α − ( b z IR ) for a tuned UV mass term, as long as α >
1. For α <
1, the massspectrum is instead given by the zeros of Y α − ( b z IR ). However, this only leads to a shiftin the spectrum.Next, we want to verify the approximate results for the decay rates, Eqs. (24), (25)and (31). Defining Z α ( z ) ≡ J α ( z ) + b α,n Y α ( z ), with b α,n given by Eq. (70), the boundaryconditions can be rewritten as Z α − ( b z IR ) = − − α − λ b z IR Z α ( b z IR ) and Z α − ( b z UV ) = − − α − λ b z UV Z α ( b z UV ) . (74)Furthermore, we have to determine the normalization constant a α,n from Eq. (19). Usingthe above identities for Bessel functions and the boundary conditions Eq. (74), we find a α,n = 1 m n p ( b z IR + c ) Z α ( b z IR ) − ( b z UV + c ) Z α ( b z UV ) , (75)where c = 2 α (2 − α − λ ) + (2 − α − λ ) ∼ O (1) for generic λ .From the definition of the coupling constant, Eq. (13), and our result for the nor-malization constant, Eq. (75), we see that we have to compute the ratio of Z α evaluatedat the IR brane to Z α evaluated at the UV brane. By repeatedly using the boundaryconditions Eq. (74), as well as Eq. (70) and the identities given above, this ratio can besimplified to Z α ( b z IR ) Z α ( b z UV ) = b z UV Y α − ( b z UV ) + (2 − α − λ ) Y α ( b z UV ) b z IR Y α − ( b z IR ) + (2 − α − λ ) Y α ( b z IR ) . (76)Since we assume that b z UV ≪
1, we can use the small argument expansion, Eq. (62),to simplify the nominator. Furthermore, assuming that b z IR ≫
1, we obtain the follow-ing approximation (again up to O (1) prefactors) for the denominator using the massquantization condition Eq. (71): b z IR Y α − ( b z IR ) + (2 − α − λ ) Y α ( b z IR ) ∼ pb z IR for generic α and λ pb z IR for α > λ = λ pb z IR b z − α UV for α < λ = λ . (77)Combining the resulting simplified expression for Eq. (76) with Eqs. (13) and (75), we30an estimate the decay rates on dimensional grounds:Γ n ∼ (cid:16) m n k (cid:17) α m IR for generic α > λ (cid:16) m n k (cid:17) m IR ln ( k/m n ) for α = 0 and generic λ (cid:16) m n k (cid:17) α m IR for α > λ = λ (cid:16) m n k (cid:17) − α m IR for α < λ = λ . (78)Here, we have replaced b z UV by m n /k and b z IR by m n /m IR . The logarithmic factor for α = 0 and generic λ arises from the logarithm in the expansion of Y for small argument,Eq. (62), inserted into Eq. (76). The decay rates in Eq. (78) agree with the results of oursimplified calculation, Eqs. (24), (25) and (31). References [1] S. B. Giddings, S. 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