SLED Phenomenology: Curvature vs. Volume
PPrepared for submission to JHEP
SLED Phenomenology: Curvature vs. Volume
Florian Niedermann a,c and Robert Schneider a,c a Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität, Theresien-straße 37, 80333 Munich, Germany c Excellence Cluster Universe, Boltzmannstraße 2, 85748 Garching, Germany
E-mail: [email protected] , [email protected] Abstract:
We assess the question whether the SLED (Supersymmetric Large Extra Di-mensions) model admits phenomenologically viable solutions with 4D maximal symmetry.We take into account a finite brane width and a scale invariance (SI) breaking dilaton-branecoupling, both of which should be included in a realistic setup. Provided that the micro-scopic size of the brane is not tuned much smaller than the fundamental bulk Planck length,we find that either the 4D curvature or the size of the extra dimensions is unacceptablylarge. Since this result is independent of the dilaton-brane couplings, it provides the biggestchallenge to the SLED program.In addition, to clarify its potential with respect to the cosmological constant problem,we infer the amount of tuning on model parameters required to obtain a sufficiently small4D curvature. A first answer was recently given in [1], showing that 4D flat solutionsare only ensured in the SI case by imposing a tuning relation, even if a brane-localizedflux is included. In this companion paper, we find that the tuning can in fact be avoidedfor certain SI breaking brane-dilaton couplings, but only at the price of worsening thephenomenological problem.Our results are obtained by solving the full coupled Einstein-dilaton system in a com-pletely consistent way. The brane width is implemented using a well-known ring regular-ization. In passing, we note that for the couplings considered here the results of [1] (whichonly treated infinitely thin branes) are all consistently recovered in the thin brane limit,and how this can be reconciled with the concerns about their correctness, recently broughtup in [2]. a r X i v : . [ h e p - t h ] D ec ontents The SLED model [3] provides a promising candidate for addressing the cosmological con-stant (CC) problem [4]. The main motivation is that for a codimension-two brane, the 4DCC only curves the transverse extra-space into a cone, while the on-brane geometry staysflat. However, it was realized from the very beginning [3] that for compact extra dimensionsthis comes at the price of yet another tuning relation, stemming from the flux quantizationcondition, which in turn is required to stabilize the compact extra space. Alternatively,from a 4D point of view, the problem can be formulated as saying that it is simply theclassical scale invariance (SI) of this theory which leads to a flat brane geometry, in whichcase Weinberg’s general no-go argument [4] applies.To circumvent this problem, a brane-localized flux (BLF) term was later included [5, 6];the idea was that if this term breaks SI, then it is in principle possible that the dilatondynamically adjusts such that flux quantization is fulfilled, thereby avoiding the tuningrelation (or runaway solutions). However, it was recently shown [1] (and also confirmed ina specific UV model [7]) that only SI brane couplings—including the BLF term—ensure a– 1 –at brane geometry. But then, it does not alter the tuning (or runaway) problem either,and we are basically back at square one.However, the mere fact that the 4D curvature is zero in the SI case does not immediatelyrule out the model as a potential solution to the CC problem. It might still be possible toachieve a nonzero but small (compared to standard model loop contributions) curvature ina phenomenologically viable and technically natural way by breaking SI on the brane. Themain purpose of this companion paper to [1] is to investigate this remaining question indetail.The starting point of our analysis is the effective theory that is obtained after solving forthe Maxwell field in a 4D maximally symmetric configuration, and adding a counter-term todispose of divergences which generically arise due to the BLF, as discussed in [1]. The goalhere is to explicitly solve the resulting Einstein-dilaton system for given model parametersand couplings. Explicitly, we will focus on a SI breaking brane tension. Since the standardmodel sector breaks SI on the brane, this term should be included in a realistic setup, andits size will be set by loop contributions of the brane matter fields. Furthermore, we willendow the brane with a finite thickness in order to avoid potential divergences. This shouldnot be viewed as a mere technical regularization, but rather as another physically unavoid-able feature: A realistic brane has to come with some microscopic thickness, which wouldultimately be determined by an underlying UV model. We will find that both sources—thenon SI tension and the brane width—contribute to the 4D curvature independently, anddiscuss them in detail.To endow the brane with a thickness, we choose in Sec. 3.1 a convenient and well-known technique [see e.g. 8, 9] that replaces the infinitely thin brane by a ring of finiteproper circumference (cid:96) . Most importantly, we expect the low energy questions we are goingto ask to be insensitive to this microscopic choice. This setup only admits static solutions ifthere is some additional mechanism that prevents the ring from collapsing. Effectively, thisboils down to adding an angular pressure component p θ , the size of which can be inferredfrom the junction conditions across the brane. This allows us to generalize a previouslyderived formula for the 4D curvature to the regularized setup, thereby enabling us to studythe tuning issue and the phenomenological viability of the model. Prior to that, we checkin Sec. 3.3 whether our result are consistent with the delta-analysis in [1]: We find thatthe delta-results are all recovered in the thin brane limit if and only if p θ → . Since foran infinitely thin object there is no direction this pressure could act in, this is a reasonablephysical assumption. Here, it will also be shown to be true for the case of exponentialdilaton-brane couplings as introduced in Sec. 3.4. These couplings model the SI breakingand are of particular interest with respect to the CC problem as they allow to be close toSI without the need of tuning the coefficients small.A discussion of the model’s phenomenological status is given in Sec. 3.5, leading to anunambiguous conclusion:
Without tuning certain model parameters to be small comparedto the bulk Planck scale, it is not possible to comply with both the observed value of the Nonetheless, it was recently disputed in [2] and used as an argument against the trustworthiness of [1].We comment on this in Appendix A. – 2 – ubble parameter as well as constraints on the size of the extra dimensions.
This nega-tive conclusion applies to both the SI breaking tension and the finite brane width effectsindependently.This—so far analytical—verdict is based on several assumptions that are all confirmedby explicitly solving the brane-bulk system in Sec. 4. To that end, the full set of fieldequations for a 4D maximally symmetric ansatz is integrated numerically, as explainedin Sec. 4.1. Special attention is given to imposing the required regularity conditions at both axes of the compact space, because only then are all integration constants uniquelydetermined. The results and physical implications, both for SI and non SI dilaton-branecouplings, are discussed in Secs. 4.2 and 4.3, respectively. We find that in both cases anacceptably small 4D curvature is typically only achieved by tuning the (dilaton independentpart of the) brane tension, but that this tuning can indeed be alleviated for certain brane-dilaton couplings. However, we also confirm the analytic prediction, so that in either casethe extra dimensions are way too large to be phenomenologically viable.Let us note that the same model was recently analyzed in [10] in a dimensionallyreduced, effective 4D theory. Our present work instead solves the full 6D bulk-brane fieldequations, thus providing an alternative and complementary approach. While confirmingthe result of [10] that a large extra space volume can be achieved for certain parameterswithout the need for putting in large hierarchies by hand, we are also able to go one stepfurther and uncover the tuning that is always needed to get both the 4D curvature and thevolume within their observational bounds. Our conclusions are summarized in Sec. 5.
We first provide a brief review of the thin brane setup. The reader familiar with thecorresponding discussion in our companion paper [1] should feel free to skip this section.The field content of the SLED model comprises the 6D metric g AB , a Maxwell field A B , which stabilizes the compact bulk dimensions, and the dilaton φ , which renders thebulk theory SI. The corresponding action reads [6] S = S bulk + S branes , (2.1)where the bulk part is S bulk = − (cid:90) d X √− g (cid:26) κ (cid:2) R + ( ∂ M φ )( ∂ M φ ) (cid:3) + 14 e − φ F MN F MN + 2 e κ e φ (cid:27) , (2.2)with κ and e the gravitational and U(1) coupling constants, respectively. The 6D Ricciscalar R is built from the 6D metric g AB , and F ≡ d A . The brane contributions are S branes = − (cid:88) b (cid:90) d x √− g (cid:26) T b ( φ ) − A b ( φ ) (cid:15) mn F mn (cid:27) , (2.3) We use the same notation and conventions as in [1]. – 3 –here the index b ∈ { + , −} runs over both branes situated at the north ( + ) and south( − ) pole of the compact space, where the metric function B (see below) vanishes. The 4Dbrane tension is denoted by T b ( φ ) . The second term, controlled by A b ( φ ) , describes thebrane localized flux (BLF). In general, both terms are allowed to have arbitrary dilatondependences; in particular, the SI case corresponds to T b ( φ ) = const and A b ( φ ) ∝ e − φ .In [1] we investigated the theory under the assumption of 4D maximal symmetry andazimuthal symmetry in the bulk. This leads to the following general ansatz, d s = W ( ρ ) ˆ g µν d x µ d x ν + d ρ + B ( ρ )d θ , (2.4a) A = A θ ( ρ )d θ , (2.4b) φ = φ ( ρ ) , (2.4c)where ˆ g µν is 4D maximally symmetric and thus fully characterized by its (constant) 4D Ricciscalar ˆ R . With these symmetries, the Maxwell equations can be integrated analytically,yielding F ρθ = e φ B (cid:34) QW + (cid:88) b δ b πB A b ( φ ) (cid:35) , (2.5)where Q is an integration constant, and δ b is shorthand for the Dirac delta function δ ( ρ − ρ b ) .In the case of a nonvanishing BLF, the second term leads to a divergence ∝ δ (0) in theremaining equations of motion, which can be interpreted as a relict of treating the branesas point-like objects. We proposed a corresponding brane counter term which allowedto consistently dispose of this contribution. After this subtraction, the remaining fieldequations consist of the dilaton equation − κ BW (cid:0) BW φ (cid:48) (cid:1) (cid:48) = e φ (cid:18) Q W − e κ (cid:19) − (cid:88) b δ b πB (cid:26) T (cid:48) b ( φ ) − QW e φ (cid:2) A (cid:48) b ( φ ) + A b ( φ ) (cid:3)(cid:27) , (2.6)and the ( µν ) , ( ρρ ) and ( θθ ) components of Einstein’s field equations, − κ (cid:32) ˆ R W + 3 W (cid:48)(cid:48) W + B (cid:48)(cid:48) B + 3 W (cid:48) W + 3 W (cid:48) B (cid:48) W B + 12 φ (cid:48) (cid:33) = e φ (cid:18) Q W + 4 e κ (cid:19) + (cid:88) b δ b πB T b ( φ ) , (2.7a) κ (cid:32) ˆ R W + 6 W (cid:48) W + 4 W (cid:48) B (cid:48) W B − φ (cid:48) (cid:33) = e φ (cid:18) Q W − e κ (cid:19) , (2.7b) κ (cid:32) ˆ R W + 4 W (cid:48)(cid:48) W + 6 W (cid:48) W + 12 φ (cid:48) (cid:33) = e φ (cid:18) Q W − e κ (cid:19) . (2.7c)Integrating the dilaton equation over an infinitesimally small disc covering one of theaxes yields the boundary condition for φ . For W and B the same is achieved by taking– 4 –ppropriate combinations of the Einstein equations. Explicitly, one finds (cid:2) Bφ (cid:48) (cid:3) ρ = ρ b = κ π C b , (2.8a) (cid:2) B ( W ) (cid:48) (cid:3) ρ = ρ b = 0 , (2.8b) [ B (cid:48) ] ρ = ρ b = 1 − κ π [ T b ( φ )] ρ = ρ b , (2.8c)where we defined C b := (cid:26) T (cid:48) b ( φ ) − QW e φ (cid:2) A (cid:48) b ( φ ) + A b ( φ ) (cid:3)(cid:27) ρ = ρ b , (2.9)which measures the brane coupling’s deviation from SI.Furthermore, integrating a suitable combination of the field equations over the wholecompact extra space yields V ˆ R = 2 κ (cid:88) b W b C b , (2.10)with the 2D volume defined as V := 2 π (cid:90) d ρ BW = (cid:90) d y √ g W . (2.11)Hence, the SI case ( C b = 0 ) implies ˆ R = 0 . Let us now turn to a peculiarity [2] of the delta setup which was not discussed in [1].Multiplying the constraint (2.7b) by B and taking the limit ρ → ρ b yields (assuming that B e φ → ) (cid:26) W (cid:2) B ( W ) (cid:48) (cid:3) + 1 W (cid:2) B ( W ) (cid:48) (cid:3) (cid:2) B (cid:48) (cid:3) − (cid:2) Bφ (cid:48) (cid:3) (cid:27) ρ = ρ b = 0 . (2.12)The terms in square brackets are those which appear in the boundary conditions (2.8), andso we are lead to (assuming that [ T b ( φ )] ρ = ρ b is finite, as it should be for physically relevantsituations) C b = 0 . (2.13)This is in clear contradiction to the SI breaking expectation C b (cid:54) = 0 . In [2], it was arguedthat this uncovers an inconsistency of the delta analysis; we will comment on this in moredetail in Appendix A. Here, let us merely state the other possibility: that (2.13) is in factanother prediction of the delta setup, saying that it is impossible to consistently break SIon a delta-brane, at least on-shell. In this work, we will explicitly verify that this option isindeed realized for a relevant class of couplings. More specifically, starting with exponentialSI breaking couplings of the form C b ∝ e γφ b and a thick brane setup, we will find that φ b → −∞ in the thin brane limit, thereby restoring C b → .At this point, let us also emphasize that the SI case is completely insensitive to thiswhole issue, because then (2.13) is identically fulfilled. Thus, the important achievement– 5 –f [1], namely the first correct identification of those BLF couplings which unambiguouslylead to ˆ R = 0 (and the resulting tuning relation), remains unaffected. However, (2.13) also implies that the actual (nonzero) value of ˆ R for broken SI cannotbe inferred within the pure delta framework (which always predicts ˆ R = 0 ), but requiresstudying a thick brane setup. This also has the advantage that potential singularities areregularized. In order to avoid any singularities and potential ambiguities of the (non SI) delta branesetup, the authors in [7, 11] introduced a specific UV model describing the brane as a vortexof finite width in extra space. We will instead use a different and technically simpler wayof regularizing the system, in which the delta brane is replaced by a ring of circumference (cid:96) [8, 9]. We assume the microscopic details of the regularization to be irrelevant for the lowenergy questions we want to study.Let us note that introducing the regularization scale (cid:96) breaks SI. This, however, doesnot necessarily imply that the underlying UV theory (which would resolve the brane mi-croscopically) breaks SI explicitly. Indeed, a SI mechanism could easily be built, in analogyto the flux stabilization which fixes the large size of the extra dimensions. In that case,the UV model parameters would not determine (cid:96) , but rather the SI combination (cid:96) e φ / . However, this does not change the fact that (cid:96) has to take a specific value in order to complywith observations. For a SI UV model, this would correspond to a spontaneously brokenSI; but the physical conclusions would be the same.For simplicity, the brane at the south pole is chosen to be a pure tension brane withoutdilaton coupling, for which no regularization is required as it only leads to a conical defectof size α − = 1 − κ π T − . (3.1)The northern brane, which breaks SI, is regularized and now sits near the north pole atthe coordinate position ρ + , corresponding to a proper circumference (cid:96) ≡ πB + > . The In fact, the whole analysis of [1] could also be trivially adapted to the point of view of [2] on the SIbreaking case (by simply including an angular pressure p θ ), without changing any of the conclusions. Itwould only add another contribution ∝ p θ to (2.10), which also only vanishes in the SI case. However, weregard an angular pressure for an infinitely thin object as unphysical, cf. Sec. 3.3 and Appendix A. In the proposal of [2] ˆ R (cid:54) = 0 would still be possible for delta branes, but only at the price of allowing p θ (cid:54) = 0 . Note that even though it is not obvious how the BLF term could be consistently adapted to the 5Dbrane in a covariant way at the level of the action, introducing the regularization after the Maxwell fieldhas been solved for is straightforward. (In any case, the BLF term will in the end not be crucial for ourmain conclusions.) This is analogous to the SI GGP solutions [1, 12], where not the extra space volume V is fixed, butonly the combination V e φ . Here and henceforth, evaluation at ρ = ρ , ρ + and ρ − will be denoted by subscripts “ ”, “ + ” and “ − ”,respectively. – 6 – igure 1 . Embedding picture of the numerical solution obtained for the specific parameterchoice (4.2) and V = 256 π (in units of the bulk Planck scale κ ). The regularized northern brane(which breaks SI) is localized along the ring separating the interior (red/dark) from the exterior(green/bright) region. The conical singularity at the south pole is caused by the unregularized (SI)pure tension brane. position of the (regular) axis at the north pole is denoted by ρ ( < ρ + ) . We can perform ashift of the ρ coordinate such that ρ = 0 . Figure 1 depicts the regularized bulk geometryfor the exemplary parameter choice (4.2). The interior of the ring (red/dark) is almost flat,whereas the exterior (green/bright) has the usual rugby ball shape.Since the delta function δ + ≡ δ ( ρ − ρ + ) is now localized at the position of the finitewidth ring, the regularized equations of motion are then formally identical to those presentedin Sec. 2, apart from one crucial further modification: In order to prevent the ring fromcollapsing, it is necessary to introduce an angular pressure component, i.e. to add the term δ + πB p θ (3.2)to the right hand side of the ( θθ ) Einstein equation (2.7c). A possible way of modeling sucha stabilization microscopically was first given in [13] and later also applied to the SLEDmodel [9]: The idea is to introduce a localized scalar field that winds around the compactbrane dimension and is subject to nontrivial matching conditions. As a result, shrinkingthe extra dimensions causes the related field energy to increase, hence implying a stableconfiguration with finite ring size. By integrating out the scalar field, it was explicitlyshown in [9] that it contributes to the ( φ -dependent) tension on the brane and leads toa pressure in angular direction. The tension shift can be taken care of by an appropriate– 7 –enormalization, and the whole stabilizing sector is then solely characterized by an angularpressure component p θ . Thus, without loss of generality, we will work with the renormalizedtheory. As argued in [9], the value of p θ needed to stabilize the ring can be inferred fromthe Einstein equations.The junction conditions across the brane can be readily derived and read [ Bφ (cid:48) ] disc = κ π C + , (3.3a) B (ln W ) (cid:48) ] disc = κ π p θ , (3.3b) [ B (cid:48) ] disc = − κ π (cid:20) T + ( φ ) + 34 p θ (cid:21) ρ = ρ + , (3.3c)where we introduced the notation [ f ] disc := lim (cid:15) → [ f ( ρ + + (cid:15) ) − f ( ρ + − (cid:15) )] , (3.4)for any function f ( ρ ) .Furthermore, we have to impose appropriate boundary conditions at both axes. Sincethe north pole is regularized, the corresponding axis (at coordinate position ρ = 0 ) isrequired to be elementary flat, i.e. φ (cid:48) = 0 , W (cid:48) = 0 , B (cid:48) = 1 , B = 0 . (3.5)In general, the unregularized south pole (at coordinate position ρ = ρ − ) features a conicalsingularity characterized by φ (cid:48)− = 0 , W (cid:48)− = 0 , B (cid:48)− = − α − , B − = 0 . (3.6)Note that only three of the four boundary conditions at each axis are independent, dueto the radial Einstein constraint (2.7b). Let us now count the total number of integrationconstants: There are two second order and one first order equation, leading to a total of fivea priori undetermined integration constants. In addition, there is one integration constantincluded in the metric ansatz (2.4a), namely ˆ R . All of them are fixed by imposing the sixindependent boundary conditions stated above. The closed system for φ , W and B is thusgiven by the off-brane ( ρ (cid:54) = ρ b ) equations (2.6) and (2.7), the junction conditions acrossthe ring (3.3) and the boundary conditions (3.5) and (3.6) at the north and south pole,respectively.After fixing the above boundary conditions, we are left with a one-parameter familyof solutions, parametrized by the Maxwell integration constant Q . However, it cannot bechosen freely, because it contributes to the total flux Φ tot := (cid:82) d ρ d θ F ρθ , which is subjectto the flux quantization condition [5, 14], Φ tot = 2 πQ (cid:90) d ρ e φ BW + (cid:104) A + ( φ )e φ (cid:105) ρ = ρ + ! = 2 πn ˜ e ( n ∈ N ) , (3.7)where in general the U(1) gauge coupling ˜ e can be different from e . For convenience, here and throughout the rest of Sec. 3, we set W + = 1 , which is always possible by a(rigid) rescaling of the 4D coordinates. – 8 – .2 4D Curvature The 4D curvature is crucial in studying the phenomenological viability of the model, solet us again derive its relation to the brane couplings, but now for the regularized model.Repeating the derivation that lead to (2.10) in the thin brane setup, and taking into ac-count (3.2), we now find V ˆ R = κ (2 C + + p θ ) . (3.8)We see that the regularized expression is only modified by the last term proportional to p θ .Next, we will also express p θ in terms of the brane couplings in the thin brane limit. The aim of this section is to explicitly check whether the above relations are compatiblewith the delta results of [1], and to gain further intuition about the regularized system andits stabilization. This will in turn allow us to narrow down physically interesting dilatoncouplings.Whether the brane looks pointlike to a good approximation is determined by the hi-erarchy between brane and bulk size, i.e. by the dimensionless ratio (cid:15) := (cid:96) /V . Thus, thedelta limit corresponds to (cid:15) → , and can be realized by letting (cid:96) → and/or V → ∞ . Inthis work, we will keep (cid:96) fixed at a value not smaller than the bulk Planck length, and let V become large.Let us first check whether the matching conditions (3.3) are compatible with the deltaresults (2.8) in the limit (cid:15) → . Since the geometry is close to flat space in the vicinity ofthe regularized axis, we assume lim ρ (cid:37) ρ + φ (cid:48) = O ( (cid:15) ) , lim ρ (cid:37) ρ + W (cid:48) = O ( (cid:15) ) , lim ρ (cid:37) ρ + B (cid:48) = 1 + O ( (cid:15) ) . (3.9)In that case, Eq. (3.3a) indeed reduces to the dilaton boundary condition (2.8a) as (cid:15) → .On the other hand, Eqs. (3.3b) and (3.3c) show that the boundary conditions for W and B are again modified by a term proportional to p θ . This was also observed in [9].At this point several remarks are in order: • The delta results [1] are recovered if and only if lim (cid:15) → p θ = 0 . • The occurrence of p θ is expected, and a mere consequence of regularizing the setup asa ring. It has the clear physical interpretation as the angular pressure that is neededto stabilize the compact dimension. • From a physical perspective, there is no understanding of an angular pressure for aninfinitely thin object. As a result, we expect the pressure to vanish whenever thereis a large hierarchy between the bulk size V and the regularization scale (cid:96) . Thisexpectation is in accordance with the above observation that for p θ → all results ofthe delta analysis are recovered. Our present analysis allows to go beyond physicalexpectations and to explicitly take the thin brane limit. Specifically, we will set ρ + = √ κ in the numerical examples below, corresponding to (cid:96) ≈ π √ κ . These assumptions were also verified numerically. – 9 –
For the physically relevant class of exponential couplings (which admit a small 4Dcurvature and a large bulk volume), we will confirm the above expectation by showing lim V →∞ p θ = 0 . This result also confirms the correctness of the delta approach in [1]within this class of couplings. While it is possible to construct examples in which p θ (cid:57) , these are typically plagued by some sort of pathology, like a runaway behavioror a diverging brane energy (cf. Sec. 3.4). Again, this is not very surprising, as thereis no meaningful notion of a pointlike angular pressure. • The authors of [2] instead argued that p θ should be nonzero for SI breaking deltabranes. We comment on this in Appendix A.We will now derive an expression for p θ in terms of the dilaton coupling. This inturn enables us to identify and discuss those couplings that are compatible with the deltadescription. As we will see, these are also just the ones that lead to small ˆ R .As pointed out in [9], an expression for p θ can be found by evaluating the radial Einsteinconstraint (2.7b) in the limit ρ (cid:38) ρ + : (cid:0) κ p θ (cid:1) − (cid:0) π − κ T + (cid:1) κ p θ + 4 κ C − (cid:15) V ˆ R + (cid:15) κ V e φ + (cid:18) Q − e κ (cid:19) = O ( (cid:15) ) , (3.10)where we used (3.9) and (3.3) to express the radial derivatives through the brane fields.The terms in the second line are suppressed by (cid:15) and can be neglected in the delta limit.Solving for p θ , we find κ p θ = 43 (cid:40)(cid:0) π − κ T + (cid:1) ± (cid:114) (2 π − κ T + ) − κ C (cid:41) + O ( (cid:15) ) (3.11)where the branch was chosen such that the delta result p θ = 0 is recovered for SI couplingsin the limit (cid:15) → . For vanishing BLF this coincides with the result derived in [9].An important observation from the above equation is that for finite (cid:15) and SI couplingsin general p θ = O ( (cid:15) ) (cid:54) = 0 . The physical reason is that introducing a brane width ingeneral requires a stabilizing angular pressure.The requirement of being close to SI can be made more precise by defining a near SIregime according to κ C + (cid:28) . (3.12)This in turn leads to an approximate expression for the stabilizing pressure, p θ = κ π (cid:18) − κ T + π (cid:19) − C + O ( (cid:15) ) + O ( C ) . (3.13) Note that we only consider subcritical tensions T + < π/κ . There is a special class of SI solutions with W (cid:48) = 0 (no warping), Q = 2 e/κ and ˆ R = 0 for which p θ = 0 as an exact result even for (cid:15) (cid:54) = 0 . Physically, these solutions correspond to the regularized rugby ballsetup. However, with respect to the CC problem this class is of no interest as it requires to unacceptablytune the relative size of both tensions. – 10 –fter inserting this into the formula for ˆ R in (3.8), we arrive at V ˆ R = 2 κ C + + 14 π (cid:18) − κ T + π (cid:19) − κ C + O ( (cid:15) ) + O ( C ) . (3.14)By comparing to its delta counterpart (2.10), we find two small corrections:(i) a term quadratic in C + and hence suppressed (in the near SI regime) relative to theleading linear term;(ii) generic order (cid:15) contributions caused by the finite brane width.Which of the two dominates depends on the details of the dilaton coupling. Later, we willfind that both possibilities can be realized.In summary, we have shown that the delta result for ˆ R receives two corrections whichare small in the near SI regime (which we intend to study) and for a large hierarchy betweenthe brane size and extra space volume. As expected, the near SI regime is of superior phenomenological importance as it leadsto parametrically small values of the 4D curvature due to (3.14). We look for a dilatoncoupling which allows to keep the SI breaking effects small without introducing an a priorihierarchy of the coupling parameters. In principle, this can be realized by using exponentialcouplings [10, 11], i.e. T + ( φ ) = λ + + τ e γφ and A + ( φ ) = Φ + e − φ , (3.15)with φ -independent (and SI) tension λ and constant parameters γ , τ and Φ + . For τ and γ (cid:54) = 0 the tension term breaks SI explicitly. We see that even for (a naturally) large τ , theSI breaking given by T (cid:48) + becomes small when φ + is sufficiently negative. This makes theexponential couplings interesting with respect to the CC problem.By contrast, the BLF term preserves SI. Technically, we could have introduced the SIbreaking also via the BLF term, which would lead to the same outcome. However, itshould be noted that it is physically more imperative to include a SI breaking tension as weexpect loops of localized brane matter, which in general breaks SI, to contribute to τ . Inother words, there is no obvious way of having τ small without imposing a fine-tuning. As aconsequence, when looking for natural solutions, we have to consider a φ -dependent tensionwith generic coefficient τ . On the other hand, in the case of the BLF term, it depends on In fact, we checked this explicitly. The reason is that the terms T (cid:48) + and (e φ A + ) (cid:48) (which lead to SIbreaking if nonvanishing) always occur in the combination (2.9), so technically it makes no difference whichof the two mediates the SI breaking. A SI matter theory would lead to observational problems: As argued in [10], this would imply adirect coupling between brane matter and φ , corresponding to an additional (Brans-Dicke like) force ofgravitational strength. This is clearly ruled out by solar system observations [15] unless a mechanism isincluded to shield the dilaton fluctuations inside the solar system. A complete study of this case is thusbeyond the scope of our present work. – 11 –he details of the matter theory whether we expect loop corrections to Φ + . Following thediscussion in [10], if the matter fields are not coupled directly to the Maxwell sector, theremight be a chance of keeping SI breaking contributions to A + small. In any case, includinga breaking via the BLF term would, due to (3.14), yield an additional contribution to ˆ R and, as we will see, would make it even more difficult to comply with the observationalconstraints.With these couplings we find C + = τ γ e γ φ + , (3.16)leading to an angular pressure p θ = κ πα + (cid:16) τ γ e γφ + (cid:17) + O ( (cid:15) ) + O ( C ) , (3.17)where α + := 1 − κ π λ + . The numerical analysis we conduct in this work (cf. Sec. 4) willshow emphatically that the volume obeys V ∝ e − φ + , (3.18)hence implying p θ ∝ (cid:40) V − γ ( for < γ < / V − ( for γ = 0 or γ > / , (3.19)asymptotically for V /κ (cid:29) . The second line follows from the observation that for γ > / the first expression in (3.17) becomes sub-dominant compared to the O ( (cid:15) ) contribution.The case γ = 0 is special as it corresponds to a SI coupling, where SI is only broken bythe regularization. From (3.17) it is clear that it is not continuously connected to γ (cid:54) = 0 because the first term vanishes identically (irrespective of the value of V ). In both cases, γ = 0 and γ > / , the exponent saturates to the constant value − .The above formula allows us to discuss the consistency of the delta limit. We distinguishtwo cases:1. For γ ≥ , increasing the volume of the compact space leads to a decreasing angularpressure. In other words, when we make the hierarchy between transverse branesize and bulk volume large, the angular pressure tends to zero in accordance withthe physical expectation. Moreover, in this limit the SI case is approached (since C + ∝ γV − γ → ), which renders the above approximations more and more accurate.As an aside, note that this observation, i.e. the concurrency of p θ being small andhaving a small amount of SI breaking, is the loophole to the objections raised in [2].We discuss this more extensively in Appendix A.2. For γ < the situation is different: If τ > , the system eventually hits a point (justbefore it becomes super-critical) where (3.11) yields no real solution for p θ anymore,indicating a runaway behavior. Therefore, a discussion of that case requires the In the special case of a scale invariant coupling ( γ = 0 ) and delta branes, this follows analytically fromthe GGP solutions [12], see [1]. – 12 –nclusion of a general time dependence of the fields which is beyond the scope of thiswork.On the other hand, if τ < , there are static solutions for which p θ grows as V is increased due to (3.11). This is related to the observation that the system getsdriven away from SI ( C + → ∞ ). As a result, the 4D curvature ˆ R cannot be keptunder control for a phenomenologically large V unless the coefficient τ is tuned tobe extremely small. Moreover, the tension tends to −∞ in this case which stronglyquestions the physical consistency of these solutions. So this case is not interesting,neither phenomenologically nor with respect to the tuning issue.In summary, the exponential coupling with γ ≥ is of particular interest, as it allowsto be close to SI, which is important to make the 4D curvature parametrically small. This isachieved by considering a sufficiently large bulk volume. Other types of couplings (includingmonomial and exponential ones with γ < ) either lead to a runaway behavior or areincompatible with being close to SI (if the coefficient is not tuned to be small). The abovediscussion also shows that the physically relevant class of couplings is compatible with thedelta description because p θ (or any hidden metric dependence of the delta function asargued in [2]) vanishes for V → ∞ . We have singled out the exponential tension-dilaton coupling (3.15) as the phenomenologi-cally relevant one, since its contribution to the 4D curvature can be made arbitrarily small.Let us now discuss whether this can lead to phenomenologically viable solutions.At the present stage, there are two main phenomenological inputs the model has tocomply with:(1) In models with large extra dimensions the weakness of 4D gravity is a result of the largeextra dimensions. This is possible because the 4D Planck mass is given, via dimensionalreduction, by [10] M = Vκ . (3.20)Given present tests of the gravitational inverse square-law [16] (see [17] for a review),the upper bound on the size of the extra dimensions is of order of ten microns. Then,(3.20) implies that the bulk gravity scale κ − / is not allowed to be significantly below ∼
10 TeV , which translates into the upper bound Vκ (cid:46) . (3.21)(2) The observed value of the 4D curvature measured in Planck units is notoriously small,viz. [18] ˆ RM ∼ − . (3.22)– 13 –et us now study whether the model is compatible with both requirements. For conve-nience, we will set κ = 1 , i.e. here and henceforth dimensionful quantities are all measuredin units of the bulk gravity scale.We now make use of our central formula (3.14) which permits to express the 4D cur-vature in terms of the extra space volume. Using (3.16), (3.18) as well as (3.20), we thenfind that the leading contribution is ˆ RM = N V − (2+ γ ) + N V − , (3.23)where N i are dimensionless coefficients, with N ∝ γτ and N ∝ (cid:96) . (3.24)The unknown constants of proportionality are due to the unknown coefficients in (3.18)and the O ( (cid:15) ) term in (3.14), respectively. For model parameters which do not contain apriori hierarchies among themselves, we expect them to be roughly ∼ . While at thispoint it is merely a reasonable expectation, it will also be confirmed by the numericalsolutions discussed in Sec. 4, which allow us to explicitly calculate these coefficients. Therelation (3.23) is one of the main results of this work. The two phenomenological boundsabove then require N × − γ ) + N × − (cid:46) − . (3.25)One way how this could in principle be fulfilled is by assuming a cancellation of the twoterms. However, this would only be achieved by tuning the parameters γ and τ veryaccurately. Therefore, we dismiss this possibility and demand both terms to fulfill thebound separately. From (3.24) we know that the first term vanishes identically for a SIcoupling ( γτ = 0 ). If SI is broken, it could only comply with the bound without tuning N (and thus τ ) if γ (cid:38) . . The second term, however, is more problematic: it implies that N (cid:46) − . As expected from (3.24), and explicitly confirmed in Sec. 4, this could onlybe achieved by assuming the brane width (cid:96) to be ∼ orders of magnitude smaller thanthe bulk Planck length. Not only would this again correspond to introducing an a priorihierarchy by hand, but also question the applicability of a classical analysis.As a result, if we do not allow the model parameters to be fine-tuned or to introducelarge hierarchies, the model is ruled out phenomenologically . Either the 4D curvature or thesize of the extra dimensions would be too large to be phenomenologically viable.Before concluding this sections, let us summarize the assumptions that went into thisresult: • The interior profiles are close to their flat space estimates with corrections O ( (cid:15) ) ,cf. Eq. (3.9). • Motivated by the GGP result, the extra space volume is assumed to be proportionalto e − φ + , cf. Eq. (3.18). In Sec. 4, however, we will uncover yet another fine-tuning (imposed by flux quantization) which couldonly be avoided if γ (cid:28) . – 14 – The coefficients in (3.24) are of order unity.They are all quite reasonable, and will indeed all be explicitly confirmed by our nu-merical analysis. Moreover, the numerical treatment will allow us to infer the amount oftuning (due to flux quantization) that is required to get a sufficiently small 4D curvature(albeit corresponding to a too large V ). In this section we present the results of our numerical studies of the regularized modeland discuss their physical implications for the SLED scenario. We will first briefly sketchthe numerical algorithm in Sec. 4.1. Next, in Sec. 4.2, we will discuss the simple caseof SI brane couplings. In this case we know the exact analytic solutions for infinitelythin branes—the GGP solution, reviewed in [1]—and so this provides a useful consistencycheck for our numerical solver. Finally, Sec. 4.3 addresses the actual case of interest: a SIbreaking tension. We derive the solutions of the full brane-bulk system without relying onany approximations, which in turn enables us to explicitly test (and confirm) the analyticalapproximations and results of the last section.
The goal is to determine the ρ -profiles of the dilaton φ and of the metric functions B and W for given model parameters. As explained above, this requires solving the bulkequations (2.6), (2.7), supplemented by the junction conditions (3.3) and the boundaryconditions (3.5), (3.6). We do so by starting at the north pole ( ρ = 0 ) and integratingoutward using the second order equations. Since the constraint (2.7b) is analyticallyconserved, it only needs to be imposed initially at ρ = 0 . For ρ > it can then be used asa consistency check (or error estimator) of the numerical solution. At ρ = ρ + , however, theconstraint must be used once again, because it determines the stabilizing pressure p θ . Inother words, when the integration reaches ρ (cid:37) ρ + , the three junction conditions (3.3) mustbe supplemented by the constraint (evaluated at ρ (cid:38) ρ + ) in order to determine the threeexterior ρ -derivatives and p θ . Afterwards, the integration continues until B → , definingthe south pole ρ = ρ − .Before the equations can actually be integrated in this way, we need to specify thethree a priori unknown integration constants φ , Q and ˆ R . In general, however, all of themare ultimately fixed via (the SI case is exceptional, see Sec. 4.2)(i) flux quantization (3.7),(ii) regularity at the south pole, i.e., φ (cid:48)− = 0 , We used two independent implementations: one in Python, using an explicit adaptive Runge-Kuttamethod, and one in Mathematica, using its “NDSolve” method. The corresponding results were found toagree within the numerical uncertainties. The corresponding regularity condition for W is not independent thanks to the constraint, i.e., W (cid:48)− = 0 automatically whenever φ (cid:48)− = 0 . – 15 –iii) the correct conical defect at the south pole, i.e., B (cid:48)− = − α − .Technically, this can be achieved by a standard shooting method: we choose some initialguesses for φ , Q and ˆ R ; after integrating the ODEs, the violations of (i)–(iii) can becomputed, and finally be brought close to zero via an iterative root-finding algorithm.In this way—and in agreement with the discussion in Sec. 3.1—since there are nointegration constants left (in the non SI case), we also see that the full solution is uniquelydetermined for a given set of model parameters. These consist of the bulk couplings κ = 1 (in our present units), e , the regularization width ρ + , the brane couplings, parametrized by α ± , τ , γ and the BLF parameter Φ + , as well as the gauge coupling ˜ e . Since the latter onlyenters via flux quantization (3.7), it is convenient to introduce the abbreviation N := 2 πn ˜ e , (4.1)so that flux quantization simply reads Φ tot = N .Note that the solution would not be determined uniquely if, for instance, the boundaryconditions ensuring regularity at the south pole were neglected. In this case, it would notbe possible to numerically predict the value of ˆ R , since it could be chosen freely. Thus,in order to compute this quantity numerically, it is crucial to find complete, regular bulksolutions. To our knowledge, this is done here for the first time. The main question is whether it is possible to find solutions for which ˆ R is small enoughand V is large enough to be phenomenological viable without fine-tuning, i.e. for genericvalues of the model parameters. For definiteness, and in order not to introduce any largehierarchies into the model by hand, we will choose the following parameters, e = 1 , ρ + = 1 , Φ + = − . , τ = 0 . × π , α + = 0 . and α − = 0 . . (4.2)(Somewhat different values would not change the main results, though.) The parameter N ,determining the total flux, will be varied, and used as a dial to achieve different values of ˆ R and V .An exemplary numerical solution is shown in Fig. 2, where the three functions B, W, φ ,as well as their ρ -derivatives are plotted, for γ = 0 . and two different choices of N , leadingto two different values of V , as is evident from the profile of B . Since we chose α + (cid:54) = α − ,the solutions are warped—both W and φ have nontrivial profiles. Furthermore, one can already see that the profiles inside the regularized brane ( ρ < ρ + )become more trivial as V increases, as expected. This trend continues, and all functionsand their derivatives at ρ (cid:37) ρ + were always found to approach the corresponding values atthe regular axis ( ρ = 0 ) like V − for V → ∞ , thereby confirming (3.9).All of the ρ -derivatives are discontinuous at the regularized brane ( ρ = ρ + ), as requiredby the junction conditions (3.3). B (cid:48) consistently approaches − α − = − . at the south pole Analytically, the regularity condition also implicitly entered the derivation of (3.14) when integratingover the whole bulk. However, this equation for ˆ R is not yet a prediction solely in terms of model parameters,since it still contains V and φ + , which are a priori unknown. We were only able to infer the explicit valueof ˆ R numerically. Note that here we chose the gauge W = 1 for convenience. – 16 – igure 2 . Complete numerical solutions of the coupled Einstein-dilaton system for the parame-ters (4.2) and γ = 0 . . The axis at the north pole ( ρ = 0 ) is regular ( W (cid:48) = φ (cid:48) = 0 ) and elementaryflat ( B (cid:48) = 1 ), while the axis at the south pole is regular but has a defect angle corresponding to theunregularized pure tension brane ( B (cid:48) = − . ); the regularized brane sits at ρ + = 1 , and producesjumps in the ρ -derivatives. The orange (light) and purple (dark) curves correspond to V = 8 π and V = 16 π , respectively (which were obtained for N = − . and N = − . ). The required 4Dcurvature was ˆ R = 0 . and . , respectively. The constraint violation, i.e. the numericaldeviation of (2.7b) from zero, was always smaller than − in this example, and the numericalerror bars would not exceed the line widths in the plots. and, most importantly, both W (cid:48) and φ (cid:48) vanish there, as required by regularity. By runningthe numerics similarly for different choices of γ and N , we can now systematically learnhow these model parameters determine ˆ R and V . Let us first consider the case τ = 0 corresponding to a SI tension T + = 2 π (1 − α + ) .Incidentally, in this case the dilaton profile is regular, and so the solution can even be– 17 – a) (b) Figure 3 . Numerical results for parameters (4.2) and τ = 0 , corresponding to SI brane couplings.For large volume V , the 4D curvature and the total flux both approach the corresponding GGPvalues which are valid for delta branes. The dashed lines are numerically inferred (and extrapolated)scaling laws. obtained for the idealized, infinitely thin brane, as already discussed in [1]. It is given bythe GGP solution [12], for which ˆ R = 0 . In that case, the integral in the flux quantizationcondition (3.7) can be performed explicitly, yielding πe √ α + α − + Φ + = N . (4.3)The dilaton integration constant φ drops out of all equations due to SI, and thus the abovecounting of constants does not add up, resulting in the tuning relation (4.3) among modelparameters. If we chose parameters which do not fulfill this equation, there would not bea static solution, in accordance with the expected runaway behavior à la Weinberg [4]. Inturn, the extra space volume V , which turns out to be ∝ e − φ [1], can be chosen freely. As aresult, this model could have a phenomenologically viable volume (although a vanishing 4Dcurvature is not compatible with observations), but only at the price of a new fine-tuning.If SI is broken, things will change: on the one hand, φ will be fixed, and thus thetuning relation is expected to disappear. On the other hand, the volume V will also bedetermined, and ˆ R is expected to be nonzero. The question then is if they can satisfy thephenomenological bounds presented in Sec. 3.5, and if so, whether this can be achievedwithout introducing yet another tuning.Let us now present the numerical results for a regularized brane with τ = 0 [all otherparameters as in (4.2)]. In that case SI is already broken by introducing a regularizationscale (cid:96) . Thus, the above discussion applies here as well: φ and V are fixed in terms ofmodel parameters. Moreover, we expect ˆ R (cid:54) = 0 due to O ( (cid:15) ) contributions caused by the– 18 –nite brane width. However, if the thin brane limit is taken by letting V → ∞ (which canbe achieved by adjusting N appropriately), these effects should become suppressed, and weexpect to recover the GGP solution with ˆ R = 0 . This is exactly what happens, as can beseen from Fig. 3a. Specifically, we find that ˆ R ∝ V − as V → ∞ . Furthermore, the angularpressure p θ (not shown) is also nonvanishing, but goes to zero like V − . These findings arein complete agreement with the analytic predictions (3.8), (3.11) (with C + = 0 ).At the same time, the tuning relation (4.3) is also violated, and the static solutionsexist for any choice of parameters. But again this violation, δ Φ := Φ
GGP − N , with Φ GGP := 2 πe √ α + α − + Φ + , (4.4)vanishes (like V − ) as V → ∞ , see Fig. 3b. In summary, we explicitly confirmed that introducing a regularization leads to O ( (cid:15) ) corrections of the GGP predictions ( ˆ R = 0 , Φ GGP = N , p θ = 0 ). In particular, this agreeswith the analytic result of [1] that ˆ R = 0 is only guaranteed in the SI delta model (whichis approached as (cid:15) → ) via a tuning of model parameters ( Φ GGP = N ). Furthermore, thissimple example already shows that a stabilizing pressure p θ is necessary for a thick brane,but also that p θ → as (cid:15) → , allowing for a consistent delta description as in [1].But now we can even make a precise statement about the required tuning beyond theidealized delta brane limit. The phenomenological bound (3.22) together with (3.20) yields(recall that we are working in units in which κ = 1 ) −
120 ! ∼ ˆ RV ∼ δ Φ , (4.5)where the second estimate used (and extrapolated) our numerically inferred scaling relations(neglecting the O (1) coefficients), cf. Fig. 3. Therefore, the parameter N ≡ πn/ ˜ e must betuned close to Φ GGP ≡ πe √ α + α − + Φ + with a precision of ∼ − . This is clearly notbetter than the CC problem we started with. It is crucial to note that this can also directlybe read as a tuning relation for the brane tension λ , since α + = 1 − λ/ π .But—as already anticipated in Sec. 3.5—there is also another problem regarding phe-nomenology, even if we allow for such a tuning: For δ Φ ∼ − , the extra space volumewould be V ∼ , grossly violating the bound (3.21). Thus, by tuning ˆ R small enough,we have at the same time tuned the extra space volume 12 orders of magnitude larger thanallowed. Alternatively, if we require V to satisfy the observational bound (3.21), ˆ R wouldstill be 36 orders of magnitude larger than what is observed. Hence, as it stands, the modelsuffers not only from a tuning problem, but is not even phenomenologically viable.This nicely agrees with the analytic discussion in Sec. 3.5. Explicitly, we confirmedthe relation (3.23) (here for γ = 0 ), finding the coefficient N = 3 . for this specific set ofparameters, i.e. e , ρ + , Φ + and α ± as given in (4.2). Now, since the resulting failure to get This is a qualitative difference to models with two infinite extra dimensions, where a regularized puretension brane still has ˆ R = 0 [19, 20]. Incidentally, it turns out that without warping, i.e. for α + = α − , the scalings are somewhat different: ˆ R ∝ V − , δ Φ ∝ V − and p θ ∝ V − . However, this does not help with the tuning problem discussed below. – 19 –oth ˆ R and V within their phenomenological bounds is the central result of this work, it isworthwhile to discuss its robustness.First, it should be noted that the main reason for this result can be traced back to the O ( (cid:15) ) contributions to the 4D curvature ˆ R , cf. Eq. (3.14), which are caused by endowingthe brane with a finite width. Hence, they are unavoidable in a (realistic) thick branesetup; of course, we did our explicit calculations only in one particular regularization, butthe standard EFT reasoning suggests that the qualitative answer would be the same forany other reasonable regularization. While there are additional contributions to ˆ R if thedilaton couplings break SI, see Eq. (3.23), they can only make things worse (unless therewere a miraculous cancellation—a possibility that we dismiss in the search of a naturalsolution to the CC problem). Again, this will be explicitly confirmed in the followingsection.Next, we checked numerically that the scaling relation, as well as the order of magnitudeof the coefficient N do not change if different tensions (i.e. other generic values for α ± ) arechosen. Furthermore, the parameters Φ + and e have no influence on the result at all; thisis obvious for the BLF Φ + , but also easily seen for the gauge coupling e as follows: For theSI couplings we are considering here, the full (regularized) equations of motion enjoy theexact symmetry e (cid:55)→ ae , Q (cid:55)→ aQ , e φ (cid:55)→ a e φ , (4.6)for any constant a . Hence, after changing e , the new solution is simply obtained from theold one by rescaling Q and e φ appropriately. Since the metric is unaltered, this leaves ˆ R and V unchanged. Hence, the only parameter that could change things is ρ + , determining the regulariza-tion scale (cid:96) ≈ πρ + , in accordance with the discussion below Eq. (3.24). We now turn to the case τ (cid:54) = 0 (and γ > ), where SI is broken explicitly via the tensionterm. The hope is to find values of γ for which no tuning is required in order to achievea large volume and small curvature. As argued above, this suggests focusing on γ > ,because then V → ∞ drives the model towards the SI case which in turn implies ˆ R → .While this case was already discussed in Sec. 3.5 under certain reasonable assumptions, thenumerical analysis independently confirms the previous results and allows to quantify theamount of tuning necessary to get a viable 4D curvature.Figure 4 shows the numerical results for different values of γ > . Again, small ˆ R andlarge V are generically realized for δ Φ → , i.e. if Φ GGP is tuned close to N . Evidently, bothquantities again show a power law dependence on δ Φ , with exponents which now depend One could test this assumption by repeating our analysis e.g. in the UV model proposed in [11]. Note that the (bulk) flux transforms as Φ (cid:55)→ Φ /a , and so N has to be readjusted accordingly. This,however, does not affect the relation between ˆ R and V . The case γ = 0 is still SI and identical to the discussion above after renaming λ + τ → λ . – 20 – a) A small 4D curvature ˆ R is realized for a small vi-olation δ Φ of the GGP tuning relation, thus implyinga highly tuned brane tension. (b) A large extra space volume V is achieved for asmall δ Φ .(c) The angular pressure p θ vanishes in the thin branelimit in accordance with the EFT expectation. (d) The dilaton evaluated at the brane φ + controlsthe extra space volume V via (3.18). Figure 4 . Numerical results for the parameters (4.2) and different values of the SI breakingparameter γ . Each dot corresponds to a separate run; the numerical uncertainties were alwayssmaller than the point sizes. The dashed lines show power law fits with exponents as given in (4.7),(3.19) and (3.18), which are always approached as V → ∞ . Whenever the scaling is γ independent,there are several data points which lie on top of each other. on γ . Empirically, we find the following laws, ˆ R ∝ (cid:40) δ Φ /γ δ Φ , V ∝ (cid:40) δ Φ − /γ ( for < γ < δ Φ − ( for < γ ) , (4.7)– 21 –s δ Φ → . These are plotted in Figs. 4a and 4b as dashed lines, and evidently providevery good fits to the numerical data points. Note that the scalings for γ > are thesame as the ones obtained in the SI case τ = 0 . The transition to this generic scaling lawoccurs because for γ > the finite width effects (which are independent of γ ) dominate,cf. Sec. 3.3. Also note that combining the scaling relations for ˆ R and V exactly reproducesthe analytic prediction (3.23). For completeness, let us mention that the correspondingnumerical coefficients for N in (3.24), i.e. the ratios N / ( γτ ) , were found in the range ∼ to . Likewise, the scaling relations (3.19) for p θ , which are drawn as dashed lines in Fig. 4c,again agree very well with the data. Finally, Fig. 4d shows the relation between the dilatonevaluated at the brane and the volume, confirming (3.18).With these results, we can now turn to the tuning question. For γ > , the discussionis exactly the same as for the SI case ( τ = 0 ) above, because the scaling relations arethe same. But for γ < there is a modification: Using the scaling relations (4.7), thephenomenological bound (4.5) now implies − ∼ δ Φ /γ . (4.8)For γ (cid:46) , δ Φ still has to be tuned tremendously close to zero; but for γ (cid:28) , this is not thecase anymore. Specifically, if we choose γ ≈ / (which is not hierarchically small), thisrelation is already fulfilled if δ Φ ∼ . , i.e. without any fine-tuning of model parameters.So we find the remarkable result that the near-SI tension is capable of producing a small4D curvature and a large volume (as compared to the fundamental bulk scale) withoutfine-tuning, although this was not possible for a SI tension ( τ = 0 ). At first sight, thislooks very promising. However, on closer inspection, there is an even bigger problem withthe volume bound (3.21) than before, since γ ∼ / and δ Φ ∼ . now yields V ∼ ,exceeding the bound by 32 orders of magnitude. In turn, if we chose γ ∼ / , so thatthe volume satisfies the bound for δ Φ ∼ . , then ˆ R ∼ − M , which is orders ofmagnitude larger than its observational bound.In summary, while it is possible to get small ˆ R and large V without tuning Φ GGP extremely close to N , it is not possible for both of them to satisfy their phenomenologicalbounds, in accordance with the general discussion in Sec. 3.5.Let us note that this possibility of getting a large volume without large parameterhierarchies was also recently observed in [10], where the same model was studied in adimensionally reduced, effective 4D theory. However, there it was also assumed that it wouldat the same time be possible to have ˆ R within its bounds (possibly via some independentfine-tuning), so that the model could in this way at least address the electroweak hierarchyproblem (albeit not the CC problem). Here we found that this is not possible, because ˆ R and V are not independent, and so one cannot tune ˆ R without at the same time ruiningthe value of V . The main result of our preceding work [1] was that the SLED model (with delta branes)only guarantees the existence of 4D flat solutions if the brane couplings respect the SI of the– 22 –ulk theory, and that this comes at the price of a fine-tuning (or runaway), as expected [4].Here, we took one step further and asked how large the 4D curvature ˆ R is for SI breakingcouplings and the (more realistic) case of a finite brane width not below the fundamental6D Planck length.Specifically, we worked with a regularization which replaces the delta brane by a ringof stabilized circumference (cid:96) , and considered a SI breaking tension term parametrized as T + = λ + τ e γφ + . This type of dilaton-brane coupling is particularly interesting with respectto the CC problem as it allows to be close to SI without assuming an unnaturally smallcoefficient τ . We then followed two complementary routes:First, we analytically derived a formula for ˆ R . Motivated by the GGP solution, theextra space volume was then assumed to be proportional to e − φ + . This resulted in the rigidrelation (3.23) between ˆ R and the extra space volume V , consisting of two V -dependentcontributions to ˆ R with unknown numerical constants of proportionality N and N . Theyoriginate from the SI breaking dilaton coupling and the finite brane width, respectively.Provided that N / ∼ , we found that either ˆ R or V exceeds its phenomenological bound(by 36 or 12 orders of magnitude, respectively).Second, we solved the full bulk-brane field equations numerically. By enforcing thecorrect boundary conditions at both branes, we were able to calculate all observables, inparticular ˆ R and V , for given model parameters. We thereby confirmed the analyticallyderived scaling relations without relying on any approximations and were able to explicitlycompute the coefficients N / , indeed affirming N / ∼ . The only way to get N (cid:28) would be to either require SI brane couplings—which would ruin solar system tests due toa fifth force [10]—or to fine-tune (either τ or λ ). As for N , the only caveat is provided byallowing the brane width (cid:96) to be much ( ∼ orders of magnitude) smaller than the bulkPlanck scale. This, however, would confront us with the problem how such a hierarchycould arise naturally, and whether one would have to take quantum gravity effects intoaccount.Moreover, the numerical analysis admitted an extensive discussion of the tuning issue.To be precise, we calculated the amount of tuning necessary to realize a large hierarchybetween the bulk scale and V , as is phenomenologically required according to (3.21), withthe following results: • For SI couplings ( τ = 0 ) a sufficiently large V is only achieved by tuning the totalflux (or, equivalently, the brane tension) close to the corresponding GGP value witha precision of ∼ − . • If SI is broken explicitly by a φ -dependent tension, it turns out that the tuningproblem can in fact be avoided for near SI tension couplings γ (cid:28) , in agreementwith [10]. However, the phenomenological problem still persists (and even gets worse).Explicitly, for γ ∼ / , which yields the required volume without tuning, ˆ R wouldbe 63 orders of magnitude above its measured value.In summary, there are no phenomenologically viable solutions in the SLED model ifthe brane width is not smaller than the fundamental bulk Planck length. But even if this– 23 –ere allowed, the required SI breaking dilaton coupling of the brane fields would alwayslead to a way too large 4D curvature or extra space volume, unless some sort of fine-tuningis at work. Acknowledgments
We thank Cliff Burgess, Ross Diener, Stefan Hofmann, Tehseen Rug and Matthew Williamsfor many helpful discussions. The work of FN was supported by TRR 33 “The DarkUniverse”. The work of FN and RS was supported by the DFG cluster of excellence “Originand Structure of the Universe”.
A Validity of Delta-Analysis
The authors of [2] critically assessed our preceding work [1] based on a delta-analysis. Specifically, they argued that the unregularized approach did not take into account a hiddenmetric dependence of the delta-function of the form ∂δ (2) ( y ) ∂g θθ =: C δ (2) ( y ) B , (A.1)which would introduce an additional (localized) term in the ( θθ ) -Einstein equation. In thatcase, the constant C would be constrained by the radial Einstein equation (2.7b) in termsof the brane tension; specifically, we find T + C (cid:39) − κ π T (cid:48) (cid:16) − κ T + π (cid:17) , (A.2)where higher order terms in T (cid:48) + were neglected.The first important observation is that C vanishes for T (cid:48) + = 0 . This shows that theconcerns of [2] do not apply to the SI case. So one of the central results of [1], namely that ˆ R = 0 for SI delta branes (and not for dilaton-independent couplings, as had been claimedpreviously [5, 6]), is insensitive to this issue.But it also looks as if assuming C = 0 , as implicitly done in [1], would be in conflictwith the SI breaking case T (cid:48) + (cid:54) = 0 . This was exactly the argument given in [2]. However,there is a loophole to that reasoning: the right hand side of (A.2) depends on φ evaluatedat the position of the delta brane, so we cannot make any final statement without knowingits value. In particular, φ + could be such that the right hand side vanishes in the case ofan infinitely thin brane.The intuitive explanation for C (cid:54) = 0 in [2] was that a delta function should depend onthe proper distance from the brane and thus implicitly on the off-brane metric. However,this picture is misleading since C is in fact not ∂δ ( y ) /∂g ρρ (which vanishes!), but ∂δ ( y ) /∂g θθ . They only considered the case without BLF, so we will do the same here. This indeed agrees with the finding in [2] up to an irrelevant factor − , which we think got somehowlost in [2]. – 24 –ence, in the parlance of [2] C corresponds to the delta function’s knowledge about theazimuthal distance around a point. Equivalently, and more physically speaking, it is theazimuthal pressure of the point source. This is obvious after noticing that the introductionof C is formally equivalent to introducing p θ as we did in our ring-regularization, uponidentifying lim (cid:15) → p θ ≡ − T + ( φ ) C . Either way, C (cid:54) = 0 seems to be rather unphysical.While the analysis of [1] is in line with the physical (but indeed more qualitative)argument that there is no well-defined notion of an angular pressure for an infinitely thinobject, we think that a rigorous statement requires an explicit calculation of the right sideof (A.2). Since φ can generically diverge at the non SI delta brane, this can only be doneby first introducing a regularization of (dimensionless) width (cid:15) and then letting (cid:15) → . Thiswas (admittedly) not done in [1], but neither in [2, 7, 10, 11]. But it was done in this work,and we were able to give an unambiguous answer: For the relevant case of an exponentialdilaton coupling, p θ → in the delta limit (and thus C = 0 )—in accordance with ourphysical expectation. As a result, the old delta analysis correctly captures the physics ofan exponential dilaton coupling.However, it should be noted that whenever p θ → , also ˆ R → , cf. (3.8) and (3.11).As already mentioned in Sec. 2.2, this was not realized in the delta-analysis [1], where itwould have translated to the impossibility of breaking SI on a delta brane. But this wouldonly have given yet another reason for studying the (more realistic) regularized setup, aswe now did. Nonetheless, it is true that the delta formula for ˆ R gives the correct leadingnonzero contributions that arise for a regularized, near SI brane, as discussed in Sec. 3.3.Now, let us be more specific and explicitly evaluate (A.2). First, for all couplingsstudied, we verified numerically φ + → −∞ ( for V → ∞ ) . (A.3)We start with the physically relevant exponential coupling (3.15) (as already discussed, thisallows to be close to SI without tuning the coefficient). Then, Eq. (A.2) implies a vanishing C in the limit (A.3), hence proving that the loophole is realized.We also considered monomial couplings; physically, they are less interesting as theyeither lead to a diverging negative or super-critical tension in the limit (A.3). Nevertheless,even in these cases, we find C → . For concreteness, consider a linear coupling in φ : Inthat case, it is easy to check that the denominator in (A.2) diverges while the numeratoris a constant, hence implying C → (albeit p θ → const (cid:54) = 0 , which we interpret as beingcaused by the pathological tension).Of course, we could not check the validity of (A.3) for all possible couplings and theremight very well be more complicated ‘designed potentials’ with a different behavior. How-ever, based on our previous findings we conjecture that these potentials either lead to avanishing C or again introduce some sort of pathology.In summary, we agree with the formulas in [2], yet we come to a different conclusionbased on a simple loophole that applies for both exponential and linear couplings (and Note that we checked this not only for the exponential tension coupling as discussed in the main text,but also for the analogous exponential BLF coupling. 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