In the ADD model with n extra dimensions of radius R, the fundamental gravitational scale M_D is related to the 4D Planck mass M_Pl by M_Pl^2 ~ M_D^{2+n} * R^n. If M_D is set to ~1 TeV to solve the hierarchy problem, what is the required size R of the extra dimensions for n = 2?
AAbout 10^{-35} meters (Planck length)
BAbout 1 millimeter — this is shockingly large and would produce deviations from Newtonian gravity at sub-millimeter distances; experiments testing the gravitational inverse-square law at short distances have verified it down to about 50 micrometers, constraining but not yet ruling out n = 2 with M_D = 1 TeV
CAbout 1 nanometer
DAbout 1 Angstrom (atomic scale)
For n = 2 and M_D = 1 TeV: R ~ M_Pl^{2/n} / M_D^{(2+n)/n} ~ (10^{19}/10^3)^1 * 10^{-19} m ~ 10^{-3} m = 1 mm. For n = 1, R ~ 10^{13} m (excluded by solar system gravity). For n = 6, R ~ 10^{-14} m (too small for current gravity tests but accessible at colliders through Kaluza-Klein graviton production). The n = 2 case is particularly interesting because the predicted extra dimension size is near the frontier of gravitational tests.
Question 2 Short Answer
The Randall-Sundrum (RS) model uses a single extra dimension with a warped (anti-de Sitter) metric. The hierarchy is generated by an exponential 'warp factor' e^{-k*r_c*pi}, where k is the AdS curvature and r_c is the size of the extra dimension. How does this solve the hierarchy problem?
Think about your answer, then reveal below.
Model answer: In the RS model, there are two 3-branes (4D surfaces) at the boundaries of the warped extra dimension: the 'Planck brane' where gravity is strong and the 'TeV brane' where the Standard Model fields live. The warped metric rescales all mass parameters on the TeV brane by the factor e^{-k*r_c*pi}. If k*r_c ~ 12, then e^{-12*pi} ~ 10^{-16}, and a fundamental mass parameter of order the Planck scale on the Planck brane becomes of order the TeV scale on the TeV brane. The hierarchy is generated geometrically — no fine-tuning is needed, just a moderate value of k*r_c that can be stabilized by the Goldberger-Wise mechanism. The RS model predicts spin-2 Kaluza-Klein graviton resonances that could be produced at the LHC.
The RS model is elegant because it generates an exponentially large hierarchy from a modestly sized extra dimension (r_c ~ 10 times the Planck length). It also has deep connections to the AdS/CFT correspondence: the RS model can be interpreted as a dual description of a strongly coupled 4D conformal field theory that confines near the TeV scale.
Question 3 Multiple Choice
Both ADD and RS extra dimension models predict signatures at the LHC. The LHC has searched extensively for these signatures. What is the current experimental status?
AExtra dimensions have been discovered at the LHC
BNo evidence for extra dimensions has been found — LHC searches for Kaluza-Klein gravitons (as resonances in dilepton or diphoton spectra for RS, or as missing energy from graviton emission into the bulk for ADD) have set lower limits on M_D of 5-10 TeV (ADD, depending on n) and on the mass of the first RS graviton of 2-5 TeV (depending on the coupling k/M_Pl)
CThe LHC energy is too low to test extra dimension models
DExtra dimensions have been confirmed by gravitational experiments
The non-observation of extra dimension signatures at the LHC means that if extra dimensions exist with the properties predicted by ADD or RS, the fundamental scale must be above the LHC's current reach. For ADD, the mono-jet + MET search (graviton emission) gives M_D > 5-10 TeV for n = 2-6. For RS, the dielectron and diphoton resonance searches give m_{G*} > 2-5 TeV for k/M_Pl = 0.01-0.1. These limits significantly constrain but do not eliminate extra dimension models, as the fundamental scale could be somewhat above the LHC energy reach.