1. Kaluza-Klein Towers Revisited
Jean Nuyts
(Results obtained in collaboration
with Fernand Grard)
University of Mons-Hainaut
November 12, 2007

2. Introduction A) What are Kaluza-Klein towers?
B) Why revisit Kaluza-Klein towers?
C) Hermitic-Symmetric-Self-adjoint Operators
Boundary Conditions
D) Kaluza-Klein towers in 5-dimensional Flat space
E) Warped Kaluza-Klein towers. Singularities
F) Physics: The signature of Kaluza-Klein towers
G) Conclusions

3. What are Kaluza-Klein towers?
It was discovered about 75 years ago that if
one extends the 4-dimensional space-time by
one or more dimensions, the fields living in
the full space (the bulk space) can decompose
in the 4-d subspace (the brane space) as a
superposition of 4-d particles.

4. Why revisit Kaluza-Klein towers?
If a field propagates in a five dimensional space with
one extra variable with a finite range of s
--- xi ,t [-, +] normal space
--- s [0, 2 R] a ribbon
--- s [0  2 R] a cylinder
The natural choice of boundary conditions
are Box or Periodic or Antiperiodic conditions
BUT there are OTHER CHOICES

5. 5-d Flat space Notation x ( = 0, 1, 2, 3), x5 = s dS2 = dxdx – ds2
x  [-,+] , s [0,2R]
5-dim REAL SCALAR MASSLESS FIELD
(    -  s  s )  = 0

6. Elementary Principles
d) The observables are self-adjoint or at least
symmetric. Hence the eigenvalues (when they exist)
are real. The eigenvectors are orthogonal.
(This is precise enough for what follows.)
e) Two symmetric operators A and B can have
common eigenvectors (can be diagonalized
simultaneously) if they commute
[ A , B ] = 0

7. Comment: We have witnessed recently a
series of articles on
« Non-hermitian operators with real
eigenvalues … »
Strangely ….

8. 5-d Flat space The Kaluza-Klein reduction (    -  s  s )y = 0
goes as follows. Take
(x,s) = n [x]n (x) [s]n (s)
then the real scalar massless field equation is
trivially solved by
(     + mn2 ) [x]n (x) = 0
(  s  s + mn2 ) [s]n (s) = 0

9. Flat space For the scalar product is selfadjoint (automatic …)
is not selfadjoint
commutes with

10. 5-d Flat space
is the requested symmetry property.
It implies the Boundary Conditions

11. 5-d Flat space Boundary conditions
The Boundary equation can be solved by
Boundary Conditions BC
Two linear relations among the values of
(0) , (2R) , s(0) , s(2R)
(And the same relations for the ’s).
The are all summarized by the six following cases

12. 5-d Flat space Boundary conditions T = 1
BC1 
BC2 

13. 5-d Flat space Boundary conditions
BC3 
BC4 

14. 5-d Flat space Boundary conditions
BC5 
BC6 

15. 5-d Flat space The Kaluza-Klein solutions
mn2 > (massive 4-d scalars)
[s]n (s) = n sin(mns) + n cos(mns)
mn2 = (massless 4-d scalars)
[s] 0 (s) = A s + B
m-n2 = - hn2 < (4-d scalar tachyons)
[s] -n (s) = -n sinh(hns) + -n cos(hns)

16. Why do these considerations lead in general to Towers when (mn) > 0
5-d Flat space Towers
Why do these considerations lead in general to
Towers when (mn) > 0
There are two relations for each set of
Boundary Conditions (BC1 …. BC6)
--- The first one fixes the relation between
the parameters of the solutions n and n.
--- The second one is then an equation for mn.
Since it involves sin and cos, there are
in general an infinite number of solutions

17. 5-d Flat space Massless scalars, Tachyons
--- To obtain massless particles (Surface)
One of the boundary conditions
leads to a relation between A and B
The second one restricts in general the
parameters to take specific values
--- Since for the tachyons the functions are sinh
and cosh there exists in general at most one
tachyon

18. 5-d Flat space The Kaluza-Klein solutions
Classically the following BC are considered
Box conditions
BC6
Periodic or antiperiodic conditions
BC2 with 3=0, 1=4=
They lead to
Regular towers mn = n / 2R
n integer even and/or odd

19. 5-d Flat space The other Boundary Cases … Example of a mass equation
BC1
Non regular towers

20. 5-d warped space BASIC INGREDIENTS (sign of k !!!)

21. 5-d warped space Symmetry and Commutation
--- Both D and Q2 are symmetric for the
scalar product, but
they do NOT commute
They cannot be diagonalised together
--- The 4-d box = e-2ks D
and Q = e-2ks Q2
commute but e-2ks Q2
is NOT symmetric
--- Hence a Paradox
--- The way out Q’ = L-1QL
L = exp(ks)
makes Q’symmetric by a
non-unitary transformation

22. Section F (5-d warped space) Boundary Conditions
After some algebra one obtains
So that the Boundary Conditions of the warped space
are simply the Boundary Conditions of the flat space
with
T = e4kR

23. 5-d warped space Equations - Form of the solutions
The Kaluza-Klein decomposition goes as follows
Particles
m+n2 > 0
Massless
m0 = 0
Tachyons
m-n2=-h2 < 0

24. 5-d warped space Discussion
Again two relations
for each set of Boundary Conditions
--- one serves to determine the ratio /
--- the second one gives the masses
+++ In general a non-regular tower for m2 > 0
+++ A condition on the parameters for m=0
+++ A lonely tachyon state for m2=-h2 < 0

25. 5-d warped space Physical Considerations
Physical miracle
-- Suppose that there is only one scale in
the theory namely the Planck scale MPL = Tev
-- All the parameters (k,  …) of mass
dimension d take the reduced form
of order 1
-- Then if one takes

26. 5-d warped space Discussion
The Masses comming out in the towers
are of the order of a Tev
This is the famous
Randall-Sundrum discovery
Our contributions :
+++ A detailed study of the self-adjointness of the
operators
+++ All the allowed towers

27. The Signature of the Kaluza-Klein Towers Physics at LHC ?
Hierarchy Question
Why is the Planck Mass Tev
so high compared to the 0.1 Tev seen
in laboratories (fine tuning ?)
Arkani-Hamed, Dimopoulos, Dvali
Randall, Sundrum

28. The Signature of the Kaluza-Klein Towers Randall, Sundrum
and the cross sections may be large one may expect
(at least for non zero mass)
quasi-regular bumps for the KK particles (?) productions at
LHC ????
Caution : This is based on a study of the
massless scalar 5-d case
To be done : the spin 1 and spin 2 cases carefully

29. Kaluza-Klein Towers : Conclusions
--- We have done a very carefull study of
ALL the allowed sets
of Boundary Conditions
for a massless 5-d field leading to
ALL the allowed forms
of the Kaluza-Klein towers for scalars
for
A five-dimensional flat space
A five-dimensional warped space
with singularities

30. Kaluza-Klein Towers : Conclusions
OUR FINDINGS
--- Quasi-regular but not fully regular towers
--- Mass zero at the bottom of the tower for certain special
relations between the parameters appearing in the
Boundary Conditions.
--- These last relations define, in general, a surface
in the parameter space
--- close to one side, a particle of small mass appears
--- close to the other side, a tachyon appears

31. REFERENCES
Grard, F., Nuyts,J.,Elementary Kaluza-Klein Towers revisited,
Phys. Rev.D 74, (2006), hep-ph/
Grard, F., Nuyts, J., Warped Kaluza-Klein Towers revisited,
(to be published in Phys.Rev. D)
Grard, F., Nuyts, J., Warped Kaluza-Klein Towers with singularities,
(in preparation)
Arkhani-Ahmed, N., Dimopoulos, S., Dvali, G., The Hierarchy Problem and
New Dimensions at a Millimeter,
Phys. Lett., B429, 263 (1998), hep-ph/ , SLAC-PUB-7769, SU-ITP-98/13
Randall, L., Sundrum, R., Large Mass Hierarchy from a Small Extra Dimension,
Phys. Rev. Lett.} {\bf{83}}, 3370 (1999), hep-ph/ ,
An Alternative to Compactification,
Phys. Rev. Lett.} {\bf{83}}, 4690 (1999), hep-th/
Rizzo,T.J., Pedagogical Introduction to Extra Dimensions,
hep-ph/ , SLAC-PUB-10753