The Search for Large Extra Dimensions using
Dijet Production from pp Collisions at 1.8 TeV
Dale Stentz
Coe College, Cedar Rapids, IA 52402
Iowa State University, Ames, IA 50011
Abstract
The search for extra dimensions has been a topic of great interest and has been
investigated with a variety of methods and techniques of analysis.[1,2,3,4] The
existence of Large Extra Dimensions (LED) can be determined given that (at
~ Mew or greater) gravity and its mediator, the graviton (spin 2), can access
these extra dimensional manifolds. Atwood has developed a model using
hadron colliders and the cross-sections for a 2 T 2 hadronic dijet process.[2]
We propose to use this model and to make a best fit as well as to establish
bounds using Ms, the Planck energy scale for when quantum gravity causes a
noticeable change from the SM, and n, the number of compacted extra
dimensions.
Introduction
Popular string theories predict a 10 + 1 dimension space-time with the extra
dimensions creating compacted manifolds. However, the size of these manifolds would
be on the order of 10-35 m, and the energy needed for experimental test is significantly
far out of the range of even very imaginative future colliders. The model proposed by
Arkani-Hamed, Dimopoulos, and Dvali (ADD) introduces the possibility for extra
dimensions (ED) to exist in order to solve the hierarchy problem of the physics at two
very different energy scales. The first scale is the current experimental scale where
physics is dictated by the SM (~Mew, the electro-weak scale), and the other scale is at
the Planck scale (MPl).[1] In addition, ED may also help explain losses in transverse
momenta and monojet events.[1,2,3,4]
The strength of gravity is about 1037 times weaker than the weak nuclear force.
At some point on the energy scale, as most physics and especially Grand Unified
Theories seem to indicate, all of the forces must be “unified.” This will certainly
happen around the Planck energy scale (MPl » 1017 GeV). Although the three strongest
forces appear to be coming closer in relative strength with increasing energy, an
extrapolation of the coupling constants for these forces using the SM will not cross at
the same point.[5] However, supersymmetry and other related theories such as
Technicolor, which have be cleverly devised in part to solve this (7σ) problem, seem to
predict that they will cross together with additional physics beyond the SM.[1,5]
On the other hand, if unification is to occur, at some point gravity must make a
tremendous climb in strength to be unified with the other forces. On a seemingly
unrelated note, gravity has not been measured much lower than about 1 cm. To a large
extent the SM ignores gravity and presumably gravity has 1/r potential at high energies
(a.k.a. small distances). However, as pointed out by ADD, the fact that physics has had
the notion of gravity being unchanged through approximately 33 orders of magnitude is
quite “remarkable.” Another thing to note is that super symmetry and other “beautiful”
theories have not been totally successful in their efforts to solve the hierarchy problem
and other problems related to SM.
To resolve this ADD has suggested that at some energy scale (say in the TeV
range) gravity will increase in strength such that it will be comparable to the relative
strength of the other fundamental forces at that energy scale. This is due to ED that can
be accessed by gravitons but not any other part of matter (from the SM). This is
reasonable since SM gauge forces have been clearly measured and explained (or nearly
so) to the electro-weak scale (Mew ~1 TeV). The energy at which this will occur is
hypothesized as being close to the electro-weak scale. This energy, Ms, is therefore the
energy at which quantum gravity can “affect” physics of the SM at a detectable and
comparable level. The end result is that ED may help solve the hierarchy problem by
suggesting that there is new physics after Ms.
As the theory goes, there could be an n = 2 ED (where 3+1+n is the total number
of dimensions) almost as large as 1 mm. Since this radius or size is much bigger than
the Planck unit of distance or the approximate radii given in superstring theories, these
ED will be hereafter described as Large Extra Dimensions (LED). Newtonian physics
would change and the “new” gravitational force would be ~ 1/r4. The gravimetric
potential for a LED with radius R (as given by ADD) would be,
V (r ) ~
m1 m2 1
for r ` R
M sn + 2 r n +1
The radius of these LED is given by,[1,2]
M Pl2 ~ M sn + 2 R n
For the case of n = 1, R is approximately 1011 m which is clearly ruled out by
astronomical observations.[1,2] In addition, there is some indication that n = 2 may be
ruled out for a variety of reasons. [2,3] Regardless, there is a possibility that we have ndimensional compacted manifolds that would be a possible solution or partial solution
to the hierarchy problem and may also explain not fully understood physics phenomena
and new physics which may be just on the horizon.[1]
The Model
Measuring and determining the possibility for LED is not complicated. All one
has to do is find suitable processes or mechanisms in which gravity could have an effect
via its mediator: the graviton. We consider a case that has already been theoretically
explored and incorporate this theoretical model to determine the values of Ms and n.
This involves hadron collisions ( pp ) with dijet production.[2] In this case, virtual
gravitons are desired as they are practically independent of n while real graviton
processes have cross-sections proportional to something like 1 / M sn + 2 . Atwood points
out that for n = 2 there can be unmistakable evidence of LED by real gravitons due to an
easily observable monojet with missing energies. However, such investigation is
troubled by lack of faith and resource in data with monojets. We bound Ms and n by
hypothesizing that jets could be formed by virtual graviton exchange rather than just
gluon exchange with collisions.
There are seven distinct hadronic processes. They are:
qq → q′q ′
qq′ → qq′ or qq ′ → qq ′
qq → qq
qq → qq
gg ↔ qq
qg → qg
gg → gg
With the effective luminosity for a particular process i and sub-process j, we have,
1
Li (τ ) j = ∫
τ
f a ( x) f b ( x / τ )
s
dx where τ ≡
x
s0
The PDFs are given by fa and fb where a or b is the ‘index’ that describes that particular
sub-process. For example, for the process qq → qq we could have the sub-process of
uu → uu . As a result, fa would be the structure function for the up quark while fb
would represent the structure function an anti-up quark. It is important to quickly note
that we are only concerned with combinations not permutations. It is easy to show that
switching fa and fb (i.e. u u → u u ) does not make a difference since τ = x1 x2 where xi is
the momentum fraction of parton i. As a result, these cases are ambiguous. The
effective luminosity for an entire process is simply,
L i (τ ) =
∑L
i
(τ ) j
all j
The differential tree-level hard cross-section for a given process i with respect to z (cos
θ where θ is scattering angle) is given by,[2]
3
 πα s2

2πα s F s0τ
8πF 2 s0 τ 3
dσ i


= (κ s )i 
(
)
+
(
)
h
z
g
z
fi ( z) −
i
i
8
4

2
M
M
dz
s
τ
s
s
 0

where f i ( z ), g i ( z ), hi ( z ), and κ i are functions of z that relate to the cross-section (see
Appendix A),
  s 0τ 
 Ln 2 
F ≡   Ms 
 2
 n − 2
for n = 2
and
for n > 2
z = cosθ =
t −u
s
Unfortunately, Atwood’s model may not give a reliable fit since data[6] shows
that NLO-QCD Monte Carlo simulations will vary almost 10 fold at high τ with respect
to LO calculations. As a result, we made a substitution for the first term and an
“adjustment” for the second term.
8π F 2 s03 τ 3 ~
dσ i
= J i′( z ) − Γi ( z ) +
hi ( z )
dz
M s8
~
hi ( z ) = (κ s )i hi ( z )
where
In order to get a cross-section relation with τ as our variable we integrate over a
region of z. The functions f(z) and g(z) are not modeled exact as some processes create
~
a singularity at –1 and 1. However, the math modeling of hi ( z ) does not have this
problem. As a result we integrate over all of z. Thus, we have
dσ i (τ )
dz
dz
−1
1
σ i (τ ) = ∫
Differential cross-section in respect to τ for the process i is simply
dσ i
≡ Li (τ ) σ i (τ )
dτ
For Γi = 0 we have the following:
dσ i
8π F 2 s03 τ 3 ~
= Li (τ ) ∫ J i′(τ ) dz + Li (τ ) ∫
hi ( z ) dz
dτ
M s8
−1
−1
1
1
The effect of Γi = 0 is not only helpful in making the model a little simpler but
reasonable as well. In general, the interference term is relatively small in comparison to
the complete gravity term. For a
s0 = 1800 GeV collision the dominate process is
qq → qq which is 0 for different flavors and a negative value for the same flavor.
Therefore, the obtain value for Ms will most likely slightly underestimate Ms in
comparison to the “real” value. Let
J (τ ) ≡ ∑ Li (τ )
all i
z0
∫ J i′(τ ) dz and A(τ ) ≡ ∑ Li (τ )
− z0
all i
8π F 2 s03 τ 3 ~
∫ M s8 hi ( z ) dz
− z0
z0
Here we have defined J(τ) as the accurate NLO-QCD-MC event simulation data from
the program JETRAD.[6, 7, 8] The total differential cross-section with τ is defined as
Ω(τ ) ≡
dσ
dσ
= ∑ i = J (τ ) + A(τ )
dτ all i dτ
As a result, we can use Ω(τ), an equation defined for our convenience, with parameters
n and Ms and fit the model with the data.
Fitting of Ω (ττ) with D∅
∅ data
The fitting was done by finding the minimum value for χ2 using a program and a
discrete set of modeling data. The actual modeling data was generated in steps of 25
GeV from 500 GeV to 50 TeV and whole values of n from 3 to 7. This was done with
Mathematica as it could do numerical integration and handle large arrays quickly and
very accurately. To actually calculate the effective luminosity we used the CTEQ4(M)
structure functions.[9]
We used the D∅ data from Run 1b ( s0 = 1.8 TeV and an integrated
luminosity of 91.9 ± 5.6 pb-1). For each event, the total transverse energy, HT, was
calculated as
n
H T ≡ ∑ ETj and ETj = E j sin θ
j =1
where n is the number of jets and ETj is the transverse energy of jet j.[9] Unfortunately,
while HT is a robust quantity which is experimentally pleasing to find, our gravity term
in is terms of τ. However, we can convert data with HT into data using τ by defining a
relation between the two.
1  H T 
Let τ = 
s0  sin θ 
2
dσ
dN dσ dH T
=
dτ dH T dN dτ
dH T s0 sin θ
=
dτ
2 HT
∴
dσ  dN
=
dτ  dH T
2
and
dσ 1
=
dN L
 1  s0 sin θ
 
 L  2 H T
2




where L is the integrated luminosity and sin θ is the average value for sin θ. As a
result, given order pairs with HT and dN/dHT it is possible to translate them to τ and
dσ/dτ, respectively. (Of course, this could be done by translating τ into terms of HT as
well.)
We used JETRAD to show that this relationship was correct. To find HT we
simply added all the transverse energies of the jets. As for τ, we calculated the product
of x1 and x2. The plot of this data is seen in figure 1 (see Appendix B for the subroutine
used to acquire this data using JETRAD). The data supports the model for a conversion
with sin θ = 0.912582. It should be noted that there is a very elegant way to convert
data from τ to HT using physical and mathematical relationships and a little bit of
trickery. This is beneficial since the JETRAD model data and the D∅ data are in terms
of HT. This procedure will be look at in great detail at a later date. We have included
the derivation and the procedure necessary for this in Appendix C.
Model: s = A(HT)
2
2
2
s (TeV )
1
χ = 0.04575
A = 1.200763444
± .002379380
0.1
0.01
JETRAD event
Fit for <sin θ>
0.001
10
100
1000
H T (GeV)
Figure 1: Correlation between HT and τ, s/s0, in order to find sin θ
Results
After converting the D∅ data, we used the fitting program with the generated
model data to determine the best fit for Ms and n. Every value of n had a true minimum
and the 10 best plots had equal χ2 values to 2 decimal places. The best fit was with n =
3 and Ms = 2200 GeV with χ2 7.38. However, the “best” fit was not really a good fit.
For the 12 data points (10 degrees of freedom) this corresponds to a confidence level
(CL) of about 69%. A plot of n = 3 with Ms = 2200 with the data is seen in figure 2.[7]
dσ/dτ (pb)
10
3
10
2
Best Fit Model
D0 Data
Model: J(τ) + A(τ)
n = 3 and
M s = 2.2 TeV
10
1
0.10
0.15
0.20
τ
0.25
0.30
0.35
Figure 2: Best fit model with D∅
∅ data
We have the best fit according to our model. Originally, we wanted to apply a
95% CL in order to apply bounds to Ms for each value of n. However, this CL
translates into a χ2 value of 3.94, which is smaller than the best fit. Therefore, we much
reject the model and, as such, draw no limits or conclusions to the existence of LED.
We plotted the values of χ2 as a function of Ms and n in order to illustrate the
circumstance (see Figure 3).
10
10
2
χ (Ms)
10
10
5
n=3
n=4
n=5
n=6
n=7
95% CL
4
3
2
10
1
750
1000
1250
1500
1750
2000
2250
M s (GeV)
Figure 3: χ2 as a function of Ms for different n with 95% CL
Conclusions and future work
Large extra dimensions may or may not exist. Nevertheless, their sci-fi like
nature appeals to our desire to understand the universe as a means to answer the
questions that we ask every day. Curiosity about the universe is an innate intuitive
ability that we as humans possess naturally. Understanding the universe or even a small
piece of it is an unnatural ability that can be achieved only by problem solving. This is
the case with ED theory, as it attempts to solve the hierarchy problem, an unknown
circumstance brought about by two very different energies levels.
Currently, there is no outstanding evidence to suggest that supersymmetry,
Technicolor, ED, and especially superstring theory are correct. In fact, we know that
despite the reliability of the SM there are still inconsistencies with it, and this research
is clearly evidence for that. Simply put, “All models are wrong, but some are
useful.[10]” However, we negate this by problem solving and finding more useful and
meaningful models. We think of what might be a reasonable theory and develop it if
there is merit to it. We use this theory and get it to hopefully work by problem solving
and experimenting. Regardless of whether or not we succeed immediately, we learn
and try to solve the problems of today with the dream of tomorrow’s answers.
The theory developed by ADD is relatively flexible in comparison to other
solutions of the hierarchy problem and physics above the current obtainable energy
scale.[11] Although its versatility makes it difficult to immediately disprove, the LHC
will provide strong bounds or possible proof or disproof about the existence of extra
dimensions. If extra dimensions existed at 2 TeV for n = 4, then (at τ ≈ 0.4) there
would be on the order of 106 difference in the cross-section between this type of extra
dimension model and the SM.[2] Even Ms = 6 TeV the cross-section would still be
about 10 times larger then what the SM predicts.[2]
The best fit obtained by this model would suggest a “large” 3 dimensional
manifold with radial dimension of ≈ 2.7Å. First of all, gravity cannot be detected
through normal experimental means at the energy/range tested in this research. It is
quite feasible that particle accelerators will replace torsion balances and other
“classical” gravitational experimental apparatuses as physics looks at gravity at its
unexplored range. In addition, ADD points out that for large n (n ≥ 5) , the range is on
the order of the weak scale or smaller, and therefore there is no large hierarchy.
Future work will be preformed to try to get a better model for the effect of LED.
In addition, the SM, which has been a very durability model over the years, needs to be
“tweaked” from a theoretical standpoint so that it is in better agreement with the data.
From an experimental level, data detection will become better and, as a result, errors
may be reduce and also show that the SM does indeed fit the experimental results. It is
even quite possible, that we are seeing the small effects of new physics, which may or
may not be explained by other models. The answer will not come easily. However, it
will come only with great effort and our ability to problem solve. In regardless, chances
are that through 14 orders of magnitude we will “discover” new physics. From there we
will be bombarded by new problems with solution that await us in the future.