On a possible quantum limit for the stabilization of moduli in brane-world scenarios

I consider the implications for brane-world scenarios of the rather robust quantum-gravity expectation that there should be a quantum minimum limit on the uncertainty of all physical length scales. In order to illustrate the possible significance of this issue, I observe that, according to a plausible estimate, the quantum limit on the length scales that characterize the bulk geometry could affect severely the phenomenology of a recently-proposed brane-world scenario. Marie Curie Fellow of the European Union (address from February 2000: Dipartimento di Fisica, Universitá di Roma “La Sapienza”, Piazzale Moro 2, Roma, Italy) An extensive research effort (see, e.g., Refs. [1, 2, 3, 4, 5, 6, 7, 8] and references therein) has been recently devoted to the possibility that the non-gravitational degrees of freedom be confined to one or more p-branes while gravitational degrees of freedom have access to some extra dimensions. With respect to the development of related formalism an important observation is that in certain string theories it is quite natural [2] to obtain this type of different properties for gravitational and nongravitational degrees of freedom. Concerning phenomenological implications, since it is the gravitational realm which is most affected by these “brane-world scenarios”, it is not surprising that significant constraints come from the requirement that classical gravity should behave as observed in the regimes we have already explored experimentally. On the quantum-gravity side some constraints also emerge; in particular, interestingly, while more conventional pictures lead to graviton effects that are negligibly small, one finds [3] that certain portions of the parameter space of a given brane-world scenario turn out to be excluded for predicting graviton effects that are inconsistent with data obtained at existing particle colliders. Larger portions of these parameter spaces will be probed at planned colliders, such as LHC at CERN. In this brief note I observe that, in addition to graviton contributions to processes studied at particle colliders, there is another class of quantum-gravity effects which could have important implications for brane-world scenarios. These effects are associated with the rather robust quantum-gravity expectation [9, 10, 11, 12, 13, 14] that physical length scales should not be definable with perfect accuracy, there should be a minimum length uncertainty, and there should be quantum fluctuations of lengths. This is conventionally (and somewhat generically) expressed with formulas of the type ∆R ≥ Lmin, intended to be valid for any physical length scale. I shall argue that, if such quantum limitations on the stabilization of length scales apply to the length scales that characterize the bulk geometry, there might be implications also for observables on the brane where the Standard Model fields reside. In conventional quantum-gravity scenarios [9, 11, 12, 13] Lmin is expected to coincide with LQG, the length scale that characterizes the strength of gravitational interactions (LQG would be given by the Planck length Lp ∼ 10 m in the conventional picture with only 3+1 space-time dimensions, but in the bulk of a brane-world scenario one can have LQG ≫ Lp). In quantum-gravity scenarios based on string theory traditionally there has been the expectation [10] that the measurability bound should be even more stringent: Lmin ∼ Ls > LQG, where Ls is the string length (Ls > LQG in the perturbative regime). More recently the analysis of certain stringy scenarios with several length scales [14] has suggested that in presence of appropriate hierarchies of scales it may be possible to have Lmin < LQG. For example, in the scenario considered in Ref. [14] it appears that Lmin ∼ (MD0Ls) LQG < LQG, where MD0 is the mass of D-particles. For brane-world scenarios in which quantum gravity (possibly in the guise of a string theory) behaves in the bulk in such a way that Lmin ≥ LQG one would find that every given length scale Rbulk characterizing the bulk geometry (e.g., a curvature radius or an overall length of a finite extra dimension) would be affected by a quantum limitation: ∆Rbulk ≥ Lmin ≥ LQG. In the ordinary case, in which LQG ∼ Lp, such quantum limits are very weak for all lengths R that we can access experimentally (extremely small relative uncertainty ∆R/R ∼ LQG/R), but in the bulk of a braneworld scenario they can be significant because LQG ≫ Lp and some of the length scales Rbulk are not much larger than LQG. In the mentioned stringy scenarios with several length scales and an appropriate hierarchy of scales it might be possible to have ∆Rbulk ∼ Lmin < LQG, but values