On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

22 Chapter 1. Introduction: Problems beyond the Standard Model
more precise, consider an extension of Minkowski spacetime M 4 by n compact extra
dimensions S n ,
M = M 4 × S n . (1.43)
These extra dimensions are assumed to be only accessible by gravity, while the SM
is confined to a domain wall or brane with a thickness δ – in contrast to the volume
of the extra dimensions, which is called bulk. The radii of these extra dimensions are
considered small enough to have escaped detection so far, but still large compared to
the Planck length. In spherical coordinates it follows for the spatial infinitesimal line
element (assuming all radii R equal for simplicity)
ds 2 = gijdx i dx j = dr 2 + r 2 dΩ 2 2 + R 2 dΩ 2 n , (1.44)
so that � |g| ∼ R n r 2 , with g ≡ det gMN and M, N = 0, 1, . . . , n + 4. The Poisson
equation for the gravitational potential of a point-like source, measured at r ≫ R
then reads
which is solved by
∂i
�� |g| gij∂j
�
V (r) = R n � � 2
∂r r ∂r V (r) = δ(r) (1.45)
V (r) ∼ − G4+n
R n r
= −G4
r
, (1.46)
where G4+n denotes the gravitational constant in 4+n dimensions and G4 = G4+n/R n
the effective four-dimensional gravitational constant. The weakness of gravity, or
equivalently the size of the effective 4D Planck mass
M 2 Pl ∼ 1/G4 = R n M n+2
Pl(4+n) , (1.47)
is attributed to the number and size of additional compact extra dimensions, so that
the fundamental constant M Pl(4+n) might be of the order of the weak scale. For
experiments at macroscopic distances, r ≫ R, the existence of compact extra dimensions
manifests itself as a seemingly extreme weakness of gravity, while at a more
fundamental level it can be explained by the field lines of gravity escaping in new
spatial directions. At distances r < R however, the radial coordinate of the compact
dimensions in (1.45) is treated on the same footing as the one corresponding to the
infinite dimensions. As a consequence one would expect a radical modification to
the inverse square law once experiments are able to resolve the compact dimensions.
This of course depends on their size, which on the other hand is roughly fixed by the
requirement to solve the hierarchy problem. The hierarchy between the weak and the
Planck scale is explained for the following arrangements of number and radii of extra
dimensions (assuming M Pl(4+n) = 1 TeV)
Number n 1 2 3 . . .
Radius R ∼ 10 13 m ∼ 10 −9 m ∼ 10 −11 m . . .