Extra Dimensions (XD)

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

2. Outline • What doesextra 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 extraXD? • 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: • Throwing away high order modes, only keep the modes with masses in our experimental range • Integrating out the extra dimensions • Renormalizing the wave function • 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 bulkis 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