1. Extra Dimensions (XD) Xiaozhou zhou 9/25/2007

2. Outline  What does extra dimension mean?  Why extra dimensions?  Why can’t we see extra dimensions?  The history of extra dimension theories  How to discribe extra dimensions  Modern extra dimension models(ADD and RS)

3. What extra XD?  Extra dimension: Extra degrees of freedom which can be interpreted as space- time coordinates  We only consider space-like XD here, i.e., there are always only one time dimension. Suppose we have one more time-like dimension, invariant mass: We can find some mechanism to get rid of tacyon, but that’s more complicated.

4. Why XD?  The simplest answer: Theorists can always guess anything that’s not ruled out by experiments.  Why isn’t it a waste of life?  It’s required by string theory  It can give rise to some interesting mechanisms which may solve the theoretical problems in Standard Model  It may unify different symmetries in 4D with more fundamental higher dimensional symmetry  It can be a type of emergent phenomenon with some kind of duality (typically AdS/CFT)

5. Why can’t we see XD?  If there are extra dimensions, why can’t we see them? Or more formally, why is our physics all 4 dimensional-like?  There are two ways to “hide” extra dimensions:  Compactification The 4D(3+1) world is infinite, while extra dimensions are compacted with a very small scale which can not be detected by low energy experiments.  Brane-world scenario Our 3D world is localized on some subspace of the whole universe called D-brane. High energy is needed to escape to extra dimensions.

6. The ‘Ancient’ XD theory Kaluza(1919)-Klein(1926) Model  Added one extra spacial dimension (compacted to S1) to Einstein’s General Relativity equation  4D gravity and electromagnitic field are unified in a 5D gravity theory  Gauge symmetry of EM field is interpreted as the traslational symmetry of the 5th dimension. The periodical boundary condition of the 5th dimension gives rise to quantization of charge  A severe problem: the coupling strength of the emergent 4D gravity and EM field are exactly the same, which is conflicting with even common sense  XD idea was discarded for decades until……

7. String Theory  The fundamental blocks in string theory are one dimensional strings  The exact number of space-time dimensions can be calculated by considering the consistence of the theory (Eg, photons should be massless)  Bosonic string(only contains bosons): 26 dimensions  Superstring(with supersymmetry): 10 dimensions  M-theory(with supergravity): 11 dimensions By using topological tools to constrain spacetime structure and compactification, we can get rid of the old problems The big embarrassment: No way to go down!

8. Modern XD theories  With some borrowed ideas from string theory but much closer to the real world.  Effective theory with energy scale higher than Standard Model but much lower than Plank scale (10^19GeV). May be considered as low energy approximation of string theory, or just some independent models with unknown UV completion.  Rich phemomena: Much easier to give predictions to experimental signals within the range of Large Hadron Collider(LHC), which is about several TeV. May be tested soon.  Variace of models: ADD, RS, UED……

9. How to describe XD?  Geometric description of spacetime:  Kaluza-Klein(KK) Decomposition Masses of KK tower:

10. 4D Effective Field Theory  Higher dimension Higher energy scale To go back to 4D effective theory, we need these steps: 1. Throwing away high order modes, only keep the modes with masses in our experimental range 2. Integrating out the extra dimensions 3. Renormalizing the wave function 4. Redefining the coupling constant

11. Large XD  ADD model: Nima Arkani-Hamed, Savas Dimopoulos & Gia Dvali(1998)  Set-up:  Number of extra dimensions: n  Flat extra dimensions compacted as a circle with radius R  Fermions and gauge fields are confined to our 4D world  Gravity can propagate freely in all the dimensions Solving hierachy problem:  Hierachy problem: Why is gravitational interaction much weaker than strong and electroweak interactions?  Solution in ADD: The coupling of gravity is about the same order of oher interactions, but it’s diluted by extra dimensions.

12. Large XD Experimental predictions  Deviation from Newton’s gravity at scale smaller than R How small? n=1 -> R~10^18km (scale of solar system) - ruled out! n=2 -> R~1mm - on the edge of experiment  KK graviton spectrum on the collider small mass splitting(almost continuous spectrum):  Many contraints come from astronomy and cosmology

13. Warped XD  RS model: Lisa Randall & Raman Sundrum (1999)  RS1 Set-up:  One extra dimension with non-trival warp factor  Fermions and gauge fields are on the IR 3-brane  Gravity is in the bulk Solving hierachy problem: The gravity profile is ‘warped down’ at IR brane where it overlaps very little with SM fields

14. Warped XD  AdS/CFT duality RS bulk is a part of space (a spacetime with a negative cosmological constant) AdS/CFT duality: The gravity theory in 5D AdS space is physically equal to 4D comformal gauge field theory The most intriguing feature of RS model  Experimental predictions  Gravitons with TeV scale mass and TeV sccoupling to SM fields  Deviation from 4D energy conservation  Little constraints from astronomy and cosmology

15. Conclusions  Extra dimension is a possible and worthwhile extension of 4D physics  Extra dimension is not a new idea, but the structures and mechnisms are variant from model to model so we still need to find out the most possible one  Extra dimension theories have rich phenomena which can be tested on colliders and astronomical observations  Modern extra dimension models may provide us a ladder from low energy world to string theory or other unknow mother theory