Download Extra Dimensions, no kidding

Extra Dimensions, No Kidding
by
(pseudonym) Nina Windhausen
G7
Submitted for the
BOWDOIN PRIZE
For Essays in the Natural Sciences
Harvard University
Epitome:
Extra spatial dimensions are lifting the Standard Model of particles and their interactions
as well as quantum field theories with gravity in, to a new level. Elegantly and simply
formulated they are begging researchers to look for them. Which we do.
Word Count:~2,500
If I stopped you on the street one day and told you that there is a whole universe,
about one-millimeter away from the world we know, you would think I am a very
drunk or crazy person. Independent of one’s degree of creative insanity or
imagination, the above statement is true. Or to be more pragmatic we should say
that we have not searched for a whole new world a millimeter away from us. Please
be attentive and I will try to explain:
From the 5th dimension of the 1920’s to the dth dimension of the 1990’s
75 years ago Gunar Nordstrom, Theodor Kaluza and Oscar Klein independently put
together a 5 dimensional Langrangian (the equation of motion, say) that was describing
5-dimensional gravity. As opposed to 4-dimensional gravity which is what we see and
feel and describe using Einstein’s general relativity (where gravity is the result of spacetime curvature) or Newton’s Law. So they did do this, and then they “compactified” the
5th dimension. What does this mean? It means they curled up the 5th dimension on a
circle until it becomes small enough that essentially is not measurable and hence the
effects of this curled up extra dimension are not observed at the every day energies and
distances . The remarkable result of Kaluza and Klein was that, from the 5-dimensional
gravity, they derived the 4-dimensional gravity plus electromagnetism (photons that is).
When they saw what they found I think we can assume that they must have
independently thought that there was a mistake in the calculation because this seemed far
too dramatic, and then, they must have independently jumped up and down because they
found no mistake.
The obstacle was however that our knowledge of the physics laws that describe
elementary particles and their interactions, neither experimentally nor theoretically was
advanced enough to be able to make something of Kaluza and Klein’s [1] finding.
A couple of years ago Nima-Arkani Hamed, Savas Dimopoulos and Gia Dvali [2]
worked through the same idea. Namely that there can be more than 3 spatial dimensions
and the extra dimensions can be curled up and thus compactified around circles (figure 1)
so that we cannot feel them and the compactification radius size is small enough that we
have not measured them. The authors pointed out that we have not measured the
gravitational strength in distances less than a millimeter!
Energy desert
Figure 1. The extra dimensions
Can be compactified around
circles. When you look at them
from very far away these circles
will be reduced to points.
Weak (us- now)– 1000
times the proton
mass(103 GeV)
Planck (the beginning) –
1019 GeV
Space Oasis
Figure 2. The “hierarchy of scales” (energy
desert) problem (up) and (down) the new
view of the scales in the compactified extra
dimensions world
When we read their paper on Nov 98, myself and a fellow grad student at the time, we
were stunned with the fact that there might be a hidden interaction at the range of the
Van-der-Waals and Casimir forces which are well known to scientists for quite some
time As far as experiment is concerned these extra dimensions are there and
experimentalists cannot wait to measure them.
The paradigm of extra dimensions does much more that excite our imagination. It solves
the so called “hierarchy of scales” problem. Simply stated the problem is that if gravity
is unified with the rest of the forces it seems that this can only be done in energies close
to the very beginning of the universe. The energy scale that we live today (called weak in
figure 2) is many orders of magnitude away from the energy scale where the strength of
the forces of nature is comparable (called Planck in figure 2; this is actually the inverse of
Newton’s gravitational constant in the units we measure mass, energy and inverse
length). The energy distance between these seemingly fundamental scales, the Weak and
the Planck, can be physics unpopulated this is why we call it “energy desert”. If we view
this picture from the extra dimensions point of view then the higher dimensional Planck
scale can be as low as the Weak scale. The “energy desert” picture and the multidimensional space-representation in figure 2 noted as the “space oasis” seem to be
related with some sort of “duality” or equivalence. This “space oasis” is described in
words as follows:
 The world we are able to see and measure today is confined on a 4 dimensional wall.
 Gravity exists in a 4+d “bulk” where d is the number of extra dimensions and “bulk”
is the spatially enriched volume.
 Gravity feels weak to us on the wall because of the enormity of the bulk volume.
 Gravitons, the gravity mediators, live in the “bulk” as momentum tower of states that
we call Kaluza Klein states.
 Each Kaluza Klein graviton state couples to our wall with 4- dimensional Planck
suppressed strength.
 The number of Kaluza Klein states is huge; for a process involving energy E the
number goes as ~(ER)d where R is the compactification length and d is the number of
extra dimensions.
 The sum over all Kaluza Klein states is not 4 dimensional Planck suppressed but
4+d Planck suppressed therefore weak scale suppressed so we do have sizable
reactions to measure.
Kaluza-Klein states? Cannot picture it
If a spatial dimension is periodic (circle) then the momentum along his dimension is
quantized as shown in equation (1). d is the number of extra dimensions and R is the
compactification radius of the extra dimensions
d
(1)
R
From our lower dimensions point of view this momentum
states that live in the higher dimension look like massive
particles with mass given by pression (2). m0 is the 0th
mass-less state, which lives on the 4-dimensioanl wall.
p
m 2  m02 
d2
R
Figure 3. Kaluza-Klein
momentum tower of states
4
R
2
R
3
R
(2)
2
p
1
R
 R
0
When the compactification radius becomes very large then these momentum states form a
continuum tower know as Kaluza-Klein tower of states as shown in Figure 3.
How is this all connected with strings?
It is well known by now that the Standard Model of the particles and their interactions is
very fine and describes everything very well. It is also well known that the Standard
Model has aspects that we do not yet understand. One particular problem is a combined
description of everything that is Standard Model with gravity, the lack of it, that is. For
several decades, scientists have tried to quantize Einstein’s gravitation and they were
always unsuccessful ending up in non-renormalizable theories plagued with divergencies
(i.e we could predict and measure nothing based on these). Some fifteen years ago string
theories appeared as candidates for the solution to these problems. And then they
reappeared as one unified string theory now also known as M-theory. The thinking about
particles and interaction within string theory is really elegant and mostly natural and
geometric. Up to now we treat elementary particles as point-like objects, and a point, a
size-less point in the space-time sheet is a rather weird occurrence and a very successful
approximation as we have discovered in the past thirty years of particle physics
experiments. String theory considers the fundamental forms to be extended objects
instead and it describes a real multidimensional world having a size, measured in the
Planck scale unit of about R=10-35 m. In the vacuum state, one time dimension and three
spatial dimensions expand to the size we observe. Each resulting tower of particles
includes a mass-less state and its excitations(these are again the Kaluza-Klein tower of
states) whose masses are set by the size of the compact dimensions. The Standard
Model particles are the mass-less states that acquire their small observed mass from the
breaking of electroweak symmetry and supersymmetry. But there is no compelling
reason[3] why all dimensions should be either the size of the Planck scale or the size of
the observed universe - it could happen that dimensions of the order of the weak scale
(TeV) arise in determining the ground state. In such a theory the quarks and leptons are
to be identified with states that have no Kaluza-Klein excitations while the gauge bosons,
the forces mediators, have a tower of excitations of mass d/R where d is the number and
R is the size of the extra dimensions. The interactions of the excitations with quarks,
leptons and gauge bosons, is fixed by the theory. So in this case not only gravitons but
also the mediators of the other forces (for example the photons and the W and Z bosons )
have Kaluza Klein tower of states in the “bulk”. By allowing the Planck or string scale
to be at the TeV regime a brand new field is born namely string phenomenology. Up to
now string theory was predicting the existence of gravity but not much more. Now and in
the next ten years we will be looking for stringy effects, Kaluza Klein gravitons as well
as Kaluza Klein bosons in high energy colliders. Which brings us to the very exciting
experimental aspect of extra dimensions.
You got to measure it to believe it : examples
So what would we need to measure and how, in order to discover this extra space in
which we are embedded? One category of experiments is those measuring Newton’s
Law at distances less than a millimeter. These are table-top Cavendish [3]– type
experiments. Currently one is carried out at the University of Colorado by J.Price and
collaborators and it involves mechanical oscillators as test masses [4] and another at
Stanford by A. Kapitulnik and collaborators[5]. They plan to measure the gravitational
strength down to 100 microns and with better than 10% precision thus being sensitive to
many of the predictions of this new age multidimensional low scale quantum gravity.
These experiments are due to report results in the summer of 2000.
The experimental particle physics community is thrilled that the extra dimensions world
might be detectable at particle collisions. Within the toroidal extra dimensions scenario
there is two kind of searches that you can do to detect effects from gravity living in the
“bulk”. One type is the "direct kind": direct means that there is a reaction between
Standard Model particles and in the final products of the reaction there is an actual
Kaluza Klein graviton emitted to the bulk together with some other standard model
stuff. Take for example the standard model reaction of quark+antiquark to graviton+
gluon: this is the case of direct graviton emission-- the graviton is the missing energy and
Figure 5. Kaluza Klein graviton direct
emission simulation. The segmented flat
sheet is a cylindrical detector viewed
unfolded. The tower sticking up is a
deposition of energy color-coded
indicating the nature of particles from
the collision that fell on that bin. The
height of the tower indicates the
magnitude of the energy that was
deposited. [14]
the jet in the detector is the gluon in this example. In Figure 5 there is a simulation of
such an event. The Kaluza Klein graviton cannot be detected as it “escapes” in the bulk,
but the gluon of the final state has a large energy and shows in the detector as a single
tower of energy. The missing energy in the event accounts for the lost in the extra space
graviton. It is estimated that with the imminent collider run we can probe effective
Planck scales up to 1500 GeV with direct graviton emission processes at Tevatron’s
Collider Detector at Fermilab and D0 detectors. A similar reaction that involves
collisions of electrons with positrons at the Large Electron Positron collider at the Center
for European Nuclear Research outside of Geneva in Switzerland , where the final state
is again a Kaluza Klein graviton escaping in the bulk and a photon, or a Kaluza Klein
graviton and a Z boson gives an estimate of the energy probe of low scale quantum
gravity and excludes effective Planck scales (the Planck scale of the higher dimensions
theory) from 500 GeV to 1000 GeV for 2 to 6 extra dimensions[6].
The other type of searches is a reaction between standard model particles with final states
being standard model particles. So, where is the graviton involved ? There are quantum
fluctuations in this reaction that can involve gravitational interaction, so although the
graviton is not directly produced is affecting the observables of the standard model final
states. These are called "indirect searches" or searches where there is a "virtual graviton
exchange". For this kind of searches there is a lot of theoretical input and room for
interpretation. However if we do see departure from the standard model predictions in
different distributions this will give us a clue. If we quantify a reaction involving proton
antiproton collisions with two leptons in the final state there are three parts that
contribute: one is the Standard Model part which is shown in Figure [7]5a). The x and y
axis are showing the invariant mass of the final lepton pair and the angular distribution
and the z axis represents the relative number of events observed. In figure 5b) we see the
interference term between the Standard Model process and the gravitational interaction
contributed from the Kaluza Klein graviton exchange and Figure 5c) shows only the pure
Kaluza Klein graviton exchange contribution. Comparing the z axis upper value of the
three parts we see that the Standard Model term is carrying the maximum weight and
therefore we have to be very careful and have large statistics data samples in order to
uncover the effects of the Kaluza Klein graviton exchange.
a)
b)
Variations on a theme: Up to infinite (number) infinite (size) extra dimensions
Not only there are extra compactified dimensions scenarios but the phenomenology in the
past year is so rich that we progressed from many large size extra dimensions to one
infinite extra dimension[8] to one infinite extra dimension, our wall and the Planck wall
where gravity is localized [9] to infinite infinite extra dimensions and the embedded walls
intersecting each other[10] to neutrino Kaluza Klein states in the bulk [11]to one large
longitudinal extra dimension[12] to time extra dimension[13]and probably some
framework was submitted in the particle physics archives as I am writing this.
Cosmologists and astrophysicists are also working hard to look for evidence of extra
dimensions in the evolution of cosmos and using available observations.
Epilogue
If someone stops you on the street and tells you there is a whole universe, less than onemillimeter away from the world we know, you may say “Yes, probably yes, until by
measurement we exclude it, Yes”
Bibliography
[1]Kaluza Th.Sitzungsber. Press.Akad.Wiss.Math K1 (1921) 966
Klein O. Z.Phys. 37 (1926) 895
[2] The Hierarchy Problem and New Dimensions at a Millimeter, Nima Arkani-Hamed,
Savas Dimopoulos, Gia Dvali, Phys.Lett. B429 (1998) 263-272
[3]Weak Scale Superstrings, J.Lykken, hep-th/9603133
[4] Experimental Status of Gravitational-Strenght Forces in Sub-Centimeter Regime,
J.C.Long, H.W.Chan, and J.C.Price,1999, hep-ph/9805217
[5] A. Kapitulnik and T.Kenny, NSF proposal 1997
[6]Search for Extra Dimensions in Boson and Fermion Pair Production in e+eInteractions at LEP, The L3 Collaboration, hep-ex/9910056
[7] Drell-Yan and Diphoton production at Hadron Colliders and Low Scale Gravity
Model, Kingman Cheung , Greg Landsberg, hep-ph/9909218
[8]An alternative to Compactification, Lisa Randal ,Raman Sundrum, hep-th/9906064
[9]The Shape of Gravity, J.Lykken, L. Randall, hep-th/9908076
[10]Infinitely Large New Dimensions, Nima Arkani-Hamed, Savas Dimopoulos, Gia
Dvali, Nemanja Kaloper, hep-th/9907209
[11]Neutrino Masses from Large Extra Dimensions, Nima Arkani-Hamed, Savas
Dimopoulos, Gia Dvali, John March-Russell, hep-ph/9811448
[12]Direct collider signatures of large extra dimensions I. Antoniadis, K.Benakli and
M.Quiros hep-ph/9905311
[13]Constraints on extra Time Dimensions G. Dvali, G. Gabadadze and G.Senjavonic
[14]”Experimental Signal from Higher Dimensions”, Vistas on XXI Century Particle
Physics,M.Spiropulu, http://www.hep.physics.wisc.edu/aspen2000/Aspen_Program.htm