The work strongly suggest that we should have seen
signs from these higher dimensions already, that is,
if they exist. We should have seen the effects directly
via the nature of the electromagnetic radiation
originating from charges moving and accelerating
within these higher dimensions.
The webside page with a layman explanation:
The paper itself requires a graduate physics level for
a good understanding. Care has been taken to make
it as accessible as possible. As often happens, after
finishing the document I found out that much of the
paper isn't really original work. It is mentioned that
a full derivation of the massless propagators in the
space-time domain of any dimension is also given in
reference . Reverences  and  show identical
results as we do. Interestingly, reference 
discusses gravitons instead of photons. The
correspondence is of course the result of both
having massless propagators with the difference
that the gravitons correspond to a spin 2 tensor
field and the photons to a spin 1 vector field.
 S. Hassani, Mathematical Physics, (Springer-
Verlag, New York, 1998) Is mentioned to contain
a complete derivation of the massless propagators
in the space-time domain in any dimensional space.
 D. V. Gal'tsov, Radiation reaction in various
dimensions, Physical Review D 66, 025016 (2002).
 Cardoso et. al. Gravitational Radiation in
Physical Review D 67 064026 (2003). hep-th/0212168