Experimental Probes of the Randall-Sundrum Infinite Extra Dimension

The phenomenological possibilities of the Randall-Sundrum non-compact extra dimension scenario with the AdS horizon increased to approximately a millimeter length, corresponding to an effective brane tension of (TeV)4, are investigated. The corrections to the Newtonian potential are found to be the only observationally accessible probe of this scenario, as previously suggested in the literature. In particular, the presence of the continuum of KK modes does not lead to any observable collider signatures. The extent to which experimental tests of Newtonian gravity can distinguish this scenario from the scenario of Arkani-Hamed, Dimopoulos, and Dvali with one and two millimeter size extra dimensions is explicitly demonstrated. ∗Electronic mail: [email protected] †Electronic mail: [email protected] ‡Electronic mail: [email protected] 1 Much of the recent excitement and theoretical speculation regarding large/warped compact extra dimensions found for example in Refs. [1–4] has been fueled by the amelioration/novel rephrasing of the various hierarchy problems. In addition, the models of Refs. [1,4] have raised some hopes that upcoming experiments may reveal signatures of the existence of higher dimensions [5–7]. On the other hand, the novelty of having phenomenologically consistent four-dimensional gravity in the presence of non-compact higher dimensions has been the main motivation for the investigations of “warped bulk” models based on the idea of Randall and Sundrum [8,9] without much motivation from upcoming experimental signature possibilities. In this paper, we investigate the possible experimental signatures of the model of Randall and Sundrum of Ref. [8] in the case that the warping scale is sufficiently lowered. Since most of the qualitative aspects of this scenario have already been discussed in Ref. [10] and Ref. [11], this paper will focus on the details of the experimental signatures. Unlike the non-warped scenarios and some of the variations of the warped model for which the gravity is quasilocalized (see also [9,12,13]), the original warped bulk model of Randall and Sundrum [8] (henceforth referred to as RS) did not seem to have any low energy phenomenological implications (perhaps with the exception of black hole physics [14–16,11]), mainly because the cosmological constant contribution coming from the constant energy density on the Planck brane had always been identified with the string scale for naive naturalness reasons. (We will refer to this constant energy density as the RS brane tension.) However, as explicitly noted for example by Kraus [17], if one attempts to identify the RS brane tension with that of a collection of D3 branes, one finds a discrepancy: the D3 brane tension is 2/3 of the brane tension needed for the RS brane tension. More recently, a different class of SUGRA solutions [18] (albeit singular) have been discovered which share the warped bulk spacetime of the RS scenario (although the noncompact limit has not been explored). Even more recently, there has also been progress in embedding the SUGRA containing the RS solution within a particular compactification of Type IIB string theory [19]. One emerging picture is that the RS brane does not correspond to any particular D-brane but is an “effective geometry” arising from a combination of a 2 stack of negative tension branes stuck at an orbifold fixed point. Importantly, the positive quantity that was previously naively identified with the brane tension is not really the brane tension, but is actually a term arising from the combination of the positive curvature from the orbifold fixed point singularity and the negative tension of the branes confined there. In this picture, the RS brane tension is of the order of Vbrane ∼ M plk, (1) where 1/k is the AdS horizon of the RS brane embedding spacetime (the warping scale). It is given by [17,19] 1 k = (4πgN) Mst , (2) whereMst is the string scale, g is the string coupling, and 2N ≫ 1 is the number of D3-branes stacked to form the RS brane. Hence, if N is taken large enough, the warping scale can be lowered to k−1 ∼ 0.1 mm such that the RS brane tension will be at O(1 TeV), which may be a useful scale for model building (such as for particle/sparticle mass splitting). Additionally, given that the cosmological constant scale is O(10/mm), it may not be unreasonable to expect the 10/mm scale to enter the effective theory. In any case, the RS model with small warping will predict signatures for the upcoming experiments testing submillimeter behavior of Newton’s law. We will refer to this model as the RS model. As is well known, the leading order multiplicative correction to Newton’s law has a functional behavior of 1 + 1/(kr) for the RS model at large distances. Hence, one would naively expect this scenario to behave just like that of Ref. [1] (ADD model) with two extra dimensions (n=2), which would imply that there would be collider signatures for the RS model. On the other hand, as pointed out by Ref. [10], this 1+1/(kr) correction becomes 1/(kr) at distances much shorter than the AdS horizon length of 1/k. In other words, the 1 Although we restrict our investigations in this paper to the RS model, our conclusions should easily extend to its generalizations such as that of Ref. [20]. 3 RS model for √ s ≫ k is effectively ADD (n=1), since at large √s the curvature can be ignored. Hence, at collider distances (1/TeV), for k ∼ (0.1mm)−1 the corrections from the continuum of KK states do not enhance the gravitational processes sufficiently to cause them to be observable. However, it is not clear that there will be absolutely no effects in this case because the brane tension given by Eq. (1) is at the collider scale. It can be argued that since the RS brane is confined to an orbifold fixed point, the brane should not behave as a thick or “fat” brane even when the collider energy is above the scale of the brane tension. On the other hand, it is not obvious whether the stacking of such a large number of D-branes is consistent with the orbifold fixed point idealization. If the brane can fluctuate at the collider scale, the brane tension may serve as a cutoff for the standard model theory confined to the brane and may help solve the hierarchy problem (in analogy with the lowering of the fundamental scale in the ADD scenarios). Even independently of the string theory picture (note that the string theory realization of the RS model is not known to be unique) and assuming somehow that a field theory can be well defined in the presence of orbifold fixed point singularities, the RS brane tension may not always reflect the scale of the tension of the object sitting at the orbifold fixed point. For example, suppose an extended object of tension Tb is sitting at the orbifold fixed point but is not confined to the fixed point. Given that gravity itself is a derivative expansion of a more fundamental theory and that the curvature at the orbifold fixed point is singular, one would expect there to be large corrections to Einstein’s equation arising from higher derivative curvature terms. Hence, one would find that the effective RS brane tension TRS is a sum of an infinite number of higher curvature terms and Tb: e.g. schematically, TRS = Tb + α1R 2 + α2R (k/M) + . . . , (3) where the R’s correspond to curvature quantities, the αi correspond to coefficients of the 2For aspects of thick brane physics, see for example [21,6,22,23]. 4 derivative expansion, and M is the five-dimensional Planck scale. Therefore, even if TRS = O(TeV), one may in principle have a Tb that is much larger, depending upon the precise nature of αi and R. In that case, even without the “confining” effect of the orbifold fixed point, the brane would not “fatten” at a TeV scale. On the other hand, the brane may fatten such that there may be associated collider signatures and an amelioration of the hierarchy problem. In this work, we assume that the RS brane can be treated as an idealized thin domain wall and neglect any possibility of brane fattening. As a partial follow-up to the work of Ref. [10], we investigate both the gravitational and collider experimental prospects for this scenario. We find that although there is no collider phenomenology, the corrections to Newton’s law distinguish the RS scenario from the ADD scenario. We find that not only the functional behavior of the gravitational correction is different but the magnitudes are sufficiently different to distinguish the two scenarios. Indeed, identifying the scale R of the extra dimension in ADD with the AdS horizon length 1/k of RS, we show that the RS gravitational correction is much larger at the reach of the upcoming experiments. Owing to the larger number of moduli fields present in the the ADD scenario than in the RS scenario, other signatures may distinguish the two scenarios. However, these more model dependent questions will not be addressed in this paper. We first calculate the Newtonian potential generated by a point source of mass msource localized on the Planck brane. Explicitly, we consider the effects of the KK tower on the effective four-dimensional gravitational potential following the procedure of Garriga and Tanaka [15]. The details of this calculation can be found in Ref. [15], and several of the relevant results (including a derivation of the effective action to fix our notation and conventions) are listed in the Appendix. To determine for completeness the correction associated 3An investigation of domain wall solutions in the presence of higher derivative curvature terms can be found in Ref. [24].