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Extra Dimensions
Nick Evans
University of Southampton
We live in 3 space dimensions – why?
Why look for more? Can there be more?
What would they look like?
Motivation - Grand Unification
If two particles collide with Planck Scale energy their gravitational
interaction would be as strong as the other forces – one hopes to unify
all forces into a single simple theory
Kaluza-Klein 5d Theory
m
Linearized field eqn for A is Gauge Theory
Compactification
Distinguish 5th dimension by enforcing
2 PR periodicity – why not? 
Motivation II – String Theory
Open and closed strings
naturally give both gravitons
and gauge bosons –
progress towards a TOE?
String theory’s symmetries are anomalous
unless in 9 space dimensions (and requires
supersymmetry)
Again they may be compactified…. we learn that
extra dimensions are at least a possibility in
quantum theories of gravity…
Kaluza Klein Tower
The wave function of a free particle must be 2 PR periodic
There are a discrete set of states associated with all particles with
energy (masses) in integer steps.
Note Universal XD mimic supersymmetry – every particle has a
massive partner.. BUT of the same spin… spin information will be
crucial! Hep-ph/0205314
Experimental Constraints
Momentum in the 5th direction must be conserved
So KK modes can only be pair produced
They can occur in SM loop diagrams and make contributions to
eg the rho parameter (S,T,U) - hep-ph/0012100
Limit is
1/R ~ 400 GeV +
They could be directly pair produced at the Tevatron 1/R ~ 450 GeV +
(looking like eg massive quark production) - hep-ph/0201300
LHC will probe up to 1/R ~ 3 TeV
Non-Renormalizability
The Standard Model in 5+ dimensions is not a good theory – it becomes
strongly coupled quite quickly as one moves to the UV
0
All gauge fields will
become strongly coupled
by 100TeV
QCD does not make protons at
the 100TeV scale so this must
be forbidden
XD implies new physics close by!
It can bring the GUT scale down
But why no proton decay?
The Lattice/ Deconstruction hep-th/0104005, hep-th/0104035
One can impose a cut off by latticizing the XD eg for the weak SU(2) group
Here U are 3 massless bi-fundamental scalars that couple the SU(2)s on
neighbouring sites – link variables
Interestingly one can manufacture 3 such scalars as the Goldstone bosons of a
symmetry breaking
SU(2) x SU(2) -> SU(2)
So we can view the theory above as a 4d theory with an SU(2) N gauge group and
N-1 Higgs fields - the extra gauge groups put the KK modes in by hand
The Higgs’ vev scale determines the basic
KK mode mass and hence the radius of
the extra dimension… to go back to the
continuum we take it to infinity.
We’re left with a theory with a tower of massive Ws but no higgs.
What about Unitarity? – we must have a higgs to restore Unitarity
with massive Ws!!
Wrong! Extra heavy Ws can restore Unitarity without an explicit
Higgs boson - There’s now a big literature on higgsless models.
hep-ph/0111016, hep-ph/0305237, hep-ph/0312324
Again we can’t actually take the cut off to infinity since:
The low energy gauge coupling after all the breakings is given by
N had better not be too big!
Braneworld – string theory origin
Closed strings can wrap XD there are
states with mass
m T (2 p R )
And KK modes of mass
T-duality -
n/R
change their names!
KK
winding
The open string spectrum on a space of radius R are the
same as open strings on a space of radius R’= 1/T (2 p ) R
If you try to compactify away a space R -> 0 it returns as an infinite R’
Braneworld – string theory origin
Open strings do not have winding modes – so in a theory of both
open and closed strings one can create:
Gauge particles
live on a sub-space
or “brane”
Gravitons live in a higher
dimension “bulk”
“Existence proof” for
such a world
Braneworld
It is possible to isolate some or all of the SM fields on a 3-brane hep-ph/9803315
Gravity lives in higher dimensions – they can’t be flat and infinite since
Surprisingly gravity is very poorly tested at short distances – until recently it’s
effects had not been measured on scales below 1mm – limits now find no
deviation down to 0.1mm – hep-ph/0307248
Remarkably the compactification radius of the bulk could be as big as 0.1mm!!
This has a huge effect on the true Planck Scale
The n=2 case is a dream scenario since:
• gravity would be strong and hence accessible at the LHC…
• it would remove the hierarchy between the weak and Planck scale….
We’re expecting to find a fundamental scalar – the higgs – at the 1 TeV
scale to generate the W and Z masses… but a scalar mass is unstable
to growing to the biggest scale in physics….
[ Incidentally the Cosmological constant is measured to be 1/(1mm) so there was
some hope to find a change in gravity at that scale in any case (hepph/9708329)…. ]
We’re experimentally down to 0.1mm without deviation from Newton’s law so no big break through!
We have learnt that gravity could become strong at any point hence forth in
collider experiments … and extra dimensions could be rather large by
particle physics scales
Braneworld at the LHC
Gravitons can be emitted into the extra dimensions – missing
energy signals!
They may return round the loop and deposit energy on the brane at
a rate of 10 12 Hz
If gravity is strong at 1 TeV particle collisions may form a mini-black
hole – the formation rate is hard to compute without a quantum
theory of gravity though! Presumably they will decay by Hawking
radiation giving a thermal spectrum output.
We would need a complete quantum theory of gravity!
Warped Extra Dimensions – Randall/Sundrum
The metric on a slice of space between two 3-branes is AdS
On the IR brane
The warp factor introduces an exponential hierarchy between the
weak scale and the Planck scale – at expense of explaining
separation in y…
One can look at fluctuations of the metric ie gravitons. It turns out the
localized energy density of the brane binds a massless graviton to the
brane – our gravity… the 5d gravity modes are localized away from our
brane and couple weakly – their masses are 1TeV, 2 TeV etc
In fact even in the limit where the slice is made infinite the coupling
suppression of the 5d continuum only changes the gravitational potential
a little…
AdS/CFT Correspondence
In fact 5d gravity models in AdS space appear to be related to 4d
strongly coupled gauge theories like QCD…
Gauge gravity duality emerged from string theory…. Essentially
one treats the renormalization group or energy scale of “QCD” as a
5th space direction. Couplings and operators run in that direction
and are represented by fields living in the 5th direction
is a scalar in the 5th direction
Eg
->
Eg
-> a graviton in the 5th direction
There are therefore 5d gravity models (toy) of QCD which allow
computations of bound state masses and their interaction strengths
arXiv:0711.4467
The RS models are possibly generated by a 4d gauge theory too – the full
implications for real world gravity remain obscure… we live in dS space!
Summary
• KK & string theory suggest there could be more than 3 dimensions
• The key signature is a KK tower for every particle propagating in the
higher dimensions
• Brane theory means one can pick and choose which fields know about the
extra dimensions
• The number of dimensions does need explaining
• The natural compactification scale for extra dimensions is the Planck
scale… but what if they’re large?
• If the extra dimensions are warped they need not even be compact
• If the Planck scale is brought down to 1 TeV we solve the hierarchy
problem… and possibly study the TOE at the LHC
• However unlikely, we must seek everything possible at LHC