Basics of Physics in Extra Dimensions

Motivation for extra dimensions

• Hierarchy problem– Vast expanse between the scales of weak interactions and

gravitational interactions is not aesthetic. 16 orders of magnitude.

• Dark Matter– WMAP results indicate that 23% of the mass/energy

required to close the universe is in the form of cold dark matter (CDM).

– Some models naturally provide a candidate for dark matter

Common Factors in XD Models• Small or large, extra dimensions are usually

compactified– radius of compactification ranges from 10-33 cm → 10-1

cm, depending on model and assumptions

• Extra dimensions manifest themselves in (3+1)-dimensions as Kaluza-Klein towers of states– Our universe consists of the zero-modes of SM fields– Higher tower states represent momentum states in the

extra dimensions.

• Gravity propagates in the extra dimensions– Dimensions in which neither gravity nor the SM fields

propagate are irrelevant

Relate XDs to observation

• SM fields are constrained to live on the 3-brane– Charged SM fields cannot travel very far in extra

dimensions (~1 TeV ~ attometer ).– For small XD, this is naturally accounted for– For large XD, methods of constraint must be invented to

keep SM fields within an attometer of the brane

toy derivation of the KK tower of states for a scalar field

• Heuristic derivation showing how mass terms appear – infinite tower of KK modes

• 5th dimension compactified implicitly

• Flat metric

2

2

in2

Rn

n 0

i(m n)2 in2 im2

R R Rn m n m

n 0 m 0 n 0 m 0

n n n nn 0 n 0

2 n

R

2

R

X x , x e

e in e im e

Orbifolds

• An orbifold is the quotient of a space by its symmetries.– Doesn’t necessarily reduce the dimension of the space– Implicitly absorbs redundancies

• S1/Z2 orbifolds are convenient within XD because it provides a mechanism for generating boundary conditions.– The circle maps to an interval, whose endpoints are

identified as the fixed points