Andy Yen
Mentor: Harvey Newman
Co-Mentors: Marat Gataullin, Vladimir Litvine
California Institute of Technology , Pasadena, CA, 91125
Abstract
One of the strengths of the physics program at the Large Hadron Collider (LHC) is the sensitivity to numerous beyond the Standard Model (SM) physics scenarios. Of the various beyond SM hypotheses, superstring theory is an appealing alternative because of its ability to incorporate quantum gravity and solve the hierarchy problem by exploiting the extra degrees of freedom gained by extending into extra spacetime dimensions. Recent theoretical work has demonstrated that the string mass scale (M
S
) can be as low as 1 TeV meaning that a series of resonances can occur and string effects can be seen at LHC energies through a modification of the SM γ + jet amplitude due to the non-SM scattering process gg→gγ.
In this paper, we present the first search strategy to probe for low mass strings using the Compact Muon
Solenoid (CMS) detector. The search method uses a cut-based selection which takes advantage of the high precision electromagnetic calorimeter in CMS to cleanly identify the highly energetic photon that characterizes this signal topology. The results demonstrate that with just 100 pb -1 of integrated luminosity, it is possible to probe for deviations from the SM due to TeV-scale strings at the 5σ level for M
S
up to 1
TeV.
1

1 Introduction
When the first physics runs commence in 2010, the LHC will be the first accelerator to probe deep into the multi-TeV scale. With it comes the potential to revolutionize our current understanding of elementary particles. While extraordinarily accurate, the best current model, the Standard Model (SM) of Particle Physics, is far from complete as there are numerous observed physical phenomena which is it unable to explain
1
. As many theoretical papers have emphasized, there are plenty of reasons to believe in the existence of new physics beyond that of the current SM [2,3]. One such alternate model is superstring theory. Superstring theory attempts explain all the fundamental forces and particles of nature by modeling them as vibrations of tiny supersymmetric strings. By utilizing extra degrees of freedom in the form of extra spacetime dimensions, superstring theory also attempts to incorporate quantum gravity to create a grand unified theory. Superstring theory became connected to particle phenomenology with the development of the concept of the D-brane, a class of extended objects upon which open strings can end with Dirichlet boundary conditions. It has recently been demonstrated that signals for superstring theory may be detected at the LHC provided the fundamental mass scale is sufficiently low [1-2].
The mass scale (M
S
) of these fundamental strings may be as low as a couple of TeV provided that spacetime extends into large extra dimensions [4]. Coincidentally, this extension into extra dimensions also provides a mechanism for solving the hierarchy problem, i.e. why the weak force is 10
32
times stronger than gravity [3]. M
S
gives a lower bound on the collider center of mass energy (√s) above which Regge-like resonances can occur. The “new physics” signal results from a scattering process that occurs on the (color) U(3) stack of D-branes. Of much more consequence is the fact that the amplitudes of this scattering are “model independent”: that is, the calculations yield the same results for arbitrary compactifications of superstring theory ranging from four to ten dimensions, including those that break symmetry [5]. Hence, the results that can be obtained will be very general and valid for a wide range of superstring scenarios.
2 The LHC and CMS Detector
The LHC, which was first turned on in September 2008, is the most powerful particle accelerator ever built. Measuring twenty-seven kilometers in circumference and situated 50-150 meters underground, the LHC with its 14 TeV center of mass energy proton-proton collisions will considerably extend the range of Higgs masses that can be probed, reaching well beyond the exclusion limits set by the earlier colliders such as the Tevatron. One of the two general purpose detectors at the LHC which is collecting and analyzing data from the produced collisions is the
Compact Muon Solenoid (CMS). CMS consists of an all-silicon tracker, a high precision electromagnetic calorimeter (ECAL) composed of lead tungstate crystals, a hadron calorimeter
(HCAL), a four Tesla superconducting solenoid, and arrays of muon chambers (Figure 1).
1 For instance, the SM does not explain dark matter/energy, oscillations of neutrino and quark favors, and why there are three generations of matter [1].
2

Figure 1 : Cutaway view of the CMS detector showing its main components.
The most important detector components needed for this study are the tracker, the ECAL, and the HCAL. A brief overview of these subdetectors is given below.
2.1 Tracker
The CMS tracker is 5.5 meters in length and 1.1 meters in radius. Its innermost layer consists of silicon pixel detectors capable of precise three dimensional measurements while its outer layers consist of silicon strip detectors capable of two dimensional measurements. In terms of pseudorapidity
2 , the CMS tracker extends in the region |η| < 2.5 where η = 
 ln tan

 


and θ is the polar angle relative to the beam axis. Only charged particles leave tracks in the tracker.
These tracks are reconstructed using the method described in detail in [6]. Here, a brief overview of the reconstruction method is given. The first step in track reconstruction is seed generation using the cluster-driven pixel-seed finding strategy. Seed generation starts with a supercluster in the ECAL (covered in Section 2.2) which is then propagated backwards through the magnetic field to the innermost layers of the tracker. There, within a loose ΔR window 3 , the pixel detectors are scanned for a hit. If a hit is found, it is then propagated outwards to the next pixel layer where another hit is searched for. A seed is created when two compatible hits are found within the pixel detectors. From that point onwards, subsequent hits are looked for in the next innermost layers of the tracker and so on until the last layer of the tracker is reached. The
2 In particle physics, pseudorapidity (η) is used instead of the polar angle θ because a Lorentz boosts leads to the addition to η of a constant that is angle-independent. As a result, the difference in η between two particles is independent of the Lorentz boosts along the beam axis. η = ∞ corresponds to the beamline while η = 0 corresponds to θ = 90°.
3 𝛥𝑅 = √𝛥𝜂 2 + 𝛥𝜑 2 where φ is the azimuthal angle.
3

number of hits required to constitute a track can be varied but in general, a larger number of hits will lead to better track reconstruction. The final step in track reconstruction is performing a fit of the track to estimate the key parameters of the particle such as the location of its interaction vertex and its momentum. Normally, this fit is performed using the Kalman Filter
4
, but for electrons, due to the non-Gaussian fluctuations induced by bremsstrahlung emission (See Section
2.2), a Gaussian Sum Filter (GSF) is used [7].
2.2 ECAL
The CMS ECAL is composed of 75,848 lead tungstate crystals completely surrounding the tracker. The ECAL is divided up into two sections, the barrel region (|η| < 1.5) and the endcap region (1.5 < |η| < 3). Due to initial miscalibration, the ECAL is expected to have a precision of around 1.5% at LHC startup; however within a month of data taking, the ECAL is expected to be calibrated to its design energy resolution of approximately 0.5% for electrons and photons with transverse momentum ( p
T
) > 100 GeV [8]. Here and in the following, the transverse direction is defined with respect to the collision axis. At energies above 100 MeV, electrons and positrons traveling through dense material lose energy through the bremsstrahlung process, radiating photons as a result of the Coulomb interaction with the electric fields of the atomic nuclei [9].
Similarly, photons interact with matter by converting through the process of electron-positron pair production. Both pair production and bremsstrahlung processes produce secondary photons and electrons which can also interact with matter leading to a chain reaction called an electromagnetic shower.
The electromagnetic showers are reconstructed using superclustering algorithms. The superclustering algorithms start by identifying a seed which is a crystal that has an energy above a certain threshold. Starting from the seed crystal, the algorithm searches in all directions in the
η-φ plane and scans through all the crystal energies until it sees a rise in crystal energies or until crystal energies drop below a certain threshold, allowing it to start or terminate the collection of energies encountered while scanning to form a supercluster. Through use of the superclustering algorithms, almost all the energy from photons and electrons can be collected in ECAL superclusters, with the relative energy resolution increasing at higher energies. This is true even for photons which convert early or electrons which shower extensively in the material in the tracker. The only exceptions are very low energy electrons (or converted photons) which spiral in the magnetic field and never fully reach the ECAL.
2.3 HCAL
The CMS HCAL consists of four sections. The barrel HCAL (HB) surrounds the barrel region of the ECAL and the endcap HCAL (HE) surrounds the endcap region of the ECAL, providing combined coverage over the region |η| < 3. In addition, there are two forward calorimeters (HF) extending out to |η| = 5, and an outer hadron calorimeter (HO) located outside the solenoid which provides extra containment to keep very high energy jets from reaching the muon chambers.
Electrons and photons deposit almost all of their energy in the ECAL while hadrons deposit most of their energy in the HCAL. Thus, the hadronic jets are measured using both the HCAL and
4 The Kalman and Gaussian Sum Filters are recursive filters that are used to estimate the state of a dynamic system from a series of incomplete or noisy measurements. More details can be found in [6] and [7].
4

ECAL subdetectors. Compared to the high resolution ECAL, the CMS HCAL has much less granularity and lower resolution.
3 Low Mass Strings at the LHC
Low mass strings will lead to the following scattering process (at the parton level): gg→gγ. This is a “new physics” signal which will show up as a non-SM contribution to the SM process pp→γ+jet. This SM background will primarily consist of the processes gq→γq and qq̅ →γg. The string scattering process is illustrated in Figure 2.
Figure 2: Open string disk diagram for gg→gγ scattering. The dots represent vertex insertions of gauge bosons on the boundary of the world sheet [5].
Due to the high M
S
associated with this scattering process, the characteristic signature of this process will be a quite distinctive isolated hard photon. Theoretical predictions indicate a minimum p
T
cut of 250 GeV can be used. This signal can be even better identified by searching for the energetic jet which is also produced. Figure 3 shows the deviation from the SM pp→γ+jet cross section predicted by the low mass strings model. It is immediately apparent that even for a low M
S
value of 1 TeV, a significant deviation is not expected until approximately p
T
~250 GeV.
5

Figure 3: The SM QCD pp→γ+jet cross section compared to the cross section with string effects taken into account for M
S
= 1 TeV.
There are two general approaches which can be used to verify the existence of these low mass string effects. One is to simply look for an excess of high p
T
direct photon events that also contain a tagged jet. A significant excess over SM predictions would be indicative of a conclusive signal. Another method is the so called “bump-hunting” approach. In this approach, one would reconstruct the invariant mass of the high p
T
photon and corresponding jet and plot the resulting distribution for the events which pass the selection. Then, one would try to spot resonant behavior (bumps) similar to the ones shown in Figure 4. If significant bumps are found, then a discovery can be claimed. Furthermore, due to the unique angular distributions inherent to the D-brane model, TeV-scale resonances associated with strings can be differentiated from other possible beyond-the-SM scenarios [5].
Figure 4: dσ/dM vs M
γ+jet plotted for SM QCD background and string signal + SM background [5].
6

The theoretical string amplitude is given by Equation 1 shown below
(1) where s , t and u are the Mandelstam variables, g s
is the strong coupling constant and μ is defined as
(2)
C(N) =
2(𝑁
2
−4)
𝑁(𝑁 2 −1)
is a constant parameter and Q
2
=
1
6 𝜅 2 𝑐𝑜𝑠 2 𝜃 𝑤
where N=3 (the number of D branes needed to generate the eight gluons of the SM), 𝜃 𝑤
is the Weinberg angle while 𝜅 2
is a model dependent mixing factor. For our analysis, we assume a minimal model [11] and take 𝜅 2
=0.02. The string amplitude possesses poles at n = 𝑠/𝑀 2 𝑠
where n is an integer (see Ref. 2 for details). This leads to a series of resonances which can be seen in the plot of the differential cross section of string events (Figure 5).
Figure 5: Differential cross section of string events for M
S
=1 and 2 TeV [10].
As these resonances are expected to show up in the high invariant mass region where there is very little Standard Model background, bump-hunting is a promising approach for an early discovery. Because of difficulties in establishing the theoretical widths Γ for higher values of n , we focus on the first peak ( n = 1) in this initial analysis. The amplitude for this peak is given by
|𝑀(𝑔𝑔 → 𝑔𝛾)| 2 ~ 𝑔
4
𝑄
2
𝐶(𝑁)
𝑀
4 𝑠 where the spin dependent widths are 𝛤 𝐽=0
{
(𝑠−𝑀
2 𝑠
𝑀
8 𝑠
) 2 +(𝛤 𝐽=0 𝑀 𝑠
) 2
+ 𝑡
4
+𝑢
4
(𝑠−𝑀
2 𝑠
) 2 +(𝛤 𝐽=2 𝑀 𝑠
) 2
}
= 0.75𝛼 𝑠
𝑀 𝑠
and 𝛤 𝐽=2 = 0.45𝛼 𝑠
𝑀 𝑠
( 𝛼 𝑠
(3)
is the strong coupling constant).
7

4 Data Samples
As actual experimental data will not be available until spring 2010, simulated data was produced using Monte Carlo methods. The early running condition of 10 TeV center of mass energy was used. Signal events were generated using a custom parton level event generator that was interfaced with PYTHIA 6.4 [12] for the hadronization. Only the first ( n = 1) string resonance was simulated and the CTEQ6D parton distribution function was used. As the background contribution falls steeply with the photon p
T
, we restricted our signal production to the kinematic region of p
T
>250 GeV. These events were then propagated through the CMS detector using
CMSSW 3.2.1 [13]. The signal data samples are summarized in Table 1.
Sample κ 2
String M
S
=1TeV 0.02 min p
T
250 GeV
σ (pb)
3.97
# Events
10075
Table 1: String signal data sample.
In addition to the SM Photon+Jet irreducible background, there is also a significant contribution from SM QCD Dijet events where one Jet is misidentified as a photon. Although the fake rate is expected to be low in CMS, the sheer size of the QCD Dijet cross section (10^5 pb) makes this a significant background. In developing the selection and analysis, a variety of Photon+Jet and
QCD samples with various p
T
min values were used. These samples are from official Summer
2008 production using PYTHIA 6.4 and CMSSW 2.2.x and are summarized in Table 2.
Sample
Photon+Jet
Photon+Jet
Photon+Jet
Photon+Jet
QCD Dijet
QCD Dijet
QCD Dijet
QCD Dijet p p p p p p p p
T
T
T
T
T
T
T
T
>170
>300
>470
>800
>170
>300
>470
>800
No. of Events
1M
300k
100k
100k
1.5M
1M
3M
3M
Equivalent Luminosity (pb
-1
)
1.94x10
4
7.15x10
4
2.21x10
5
5.00x10
6
2.40x10
1
2.73x10
2
9.50x10
3
2.50x10
5
Table 2: Background Samples
5 Photon Reconstruction
Due to the high resolution CMS ECAL, after calibration, high p
T
photons (with p
T
>100 GeV) are expected to be reconstructed with a resolution of 0.5%. In our analysis, we utilized the photon objects reconstructed by CMSSW 3.2.1. Our signal sample had a mean photon p
T
of approximately 400 GeV. The energy resolution and reconstruction efficiency of these photons were checked to verify that the photon objects offer adequate performance. Reconstruction efficiency was determined by evaluating how many Monte Carlo (MC) simulated photons have a matching photon candidate within dR<0.05; the efficiency was found to be 98.0%. Energy resolution was determined by fitting the energy response (Reconstructed energy/MC energy) with a Crystal Ball fit with two polynomial tails [18]. As shown in Figure 6, the reconstructed energy resolution is very close to the CMS ECAL design resolution, where the upper part of the
8

resolution curve corresponds to unconverted photons, as well as photons which convert but deposit relatively little energy in the tracker.
Figure 6: Photon energy distribution. The data are fitted to a Crystal Ball function [18] showing that photons which leave relatively little energy in the tracker have close to the design resolution.
6 Jet Reconstruction
Due to poor HCAL granularity and intrinsic fluctuations in hadronic showers, jet energy resolution is much worse compared to the photon energy resolution. Thus, the invariant mass resolution of the photon+jet system will be dominated by the jet resolution. Since early discovery of low mass strings is dependent on a bump-hunting search strategy, achieving good jet reconstruction is crucial for this analysis. A brief description of the jet reconstruction algorithms and energy corrections used in this analysis is given below.
6.1 Jet Algorithms
Jet reconstruction works similarly to photon reconstruction in that HCAL/ECAL clusters above a certain threshold are designated as seeds and all energy deposits within a certain ΔR cone are associated with the jet. This is the approach used in the Iterative Cone algorithm, one of the most basic and most common jet clustering algorithms [14]. The clustering is done iteratively over all possible seeds until only stable cones are left. A cone is defined to be stable if its geometric center agrees with the sum of its constituent four vectors (within some tolerance).
However, jets are much more complicated than photons and can often consist of dozens of different particles. This makes the reconstruction much more complicated and more susceptible to error. These errors fall into two main categories known as collinear unsafety and infrared unsafety. In the case of collinear unsafety, the most energetic particle in the jet gets split in the reconstruction into two nearly collinear particles. Given that jets are often composed of numerous unstable hadrons, this situation can occur quite often when hadrons decay in the detector. As demonstrated in Figure 7, collinear splitting can change the final state and lead to mis-reconstruction of the jet.
9

Figure 7: Collinear unsafe jet reconstruction. When the most energetic seed is reconstructed as two separate energetic objects, the reconstruction results change.
In the case of infrared unsafety, the jet reconstruction algorithm is not stable when soft particles
(from initial state radiation, final state radiation, or pileup) are added into the event. Although it is very fast, the Iterative Cone algorithm is neither infrared safe nor collinear safe. As shown in
Figure 8, the high rate of misreconstruction leads to poor Jet energy resolution and response.
Figure 8: Energy Resolution of Iterative Cone Jets is approximately 12.3%.
While the poor jet energy response can be corrected by applying Jet energy scale corrections, the substantial amount of integrated luminosity required to perform this calibration makes this unsuitable for an early discovery search. In addition, energy scale corrections do nothing to improve resolution.
Our studies of Jet reconstruction performance found that much improved performance could be achieved by utilizing the sisCone5PFJet object in CMSSW 3.2.1, the most recent version of the
CMS reconstruction and simulation software that was available at the time of the study [13].
This jet object combines the so-called sisCone jet reconstruction algorithm with Particle Flow energy corrections. The sisCone algorithm is both collinear and infrared safe so it is able to
10

achieve much better resolution; a description of the algorithm is given in Ref. 15. The “Particle
Flow” (PF) algorithm attempts to reconstruct all particles in an event using information from all of the CMS subdetectors [16]. By incorporating the extra information contained in other subdetectors, particularly the tracker, the PF algorithm is able to quite accurately individually identify all the particle types within the event and apply specialized energy corrections for each type of particle. Because the specific type of particle is known, more accurate energy corrections can be applied. These specific corrections can also be much more accurately derived using either Monte Carlo or data driven methods with a lower amount of integrated luminosity.
6.2 Jet Correction
Careful analysis of the sisCone5PF Jet energy response revealed a fairly significant low energy tail which is larger than one might expect for a collinear-safe reconstruction algorithm. Analysis revealed that for approximately 13.5% of the events, there was a secondary jet located within
ΔR<1 of the original sisCone jet. This indicates that the sisCone jet algorithm is suffering from some form of collinear unsafety due to the fact that the original sisCone algorithm was designed for Jets with energy under 100 GeV while the gluon jets from the string signal have an average p
T
of approximately 400 GeV.
To correct this problem, an extra jet correction was implemented on top of the sisCone and PF algorithms. The new custom correction was applied only to events where a secondary jet was found within ΔR<1 of the selected jet candidate. The four vector of the corrected jet was obtained by taking the vector sums of the momentum of the two jets. The corrected η and φ are obtained by taking the p
T
weighted average of the η and φ of the two jets (including a correction for the discontinuity in the φ coordinate). The improvements in jet reconstruction are demonstrated in Figures 9 and 10. As we can see, for the jets which undergo this extra collinear correction, the improvements in ΔR between reco and gen jets and energy resolution are significant.
Figure 9: ΔR and Energy response of Jets before and after collinear correction.
11

Figure 10: Overall jet energy response before and after the collinear correction (applied to approximately
13.5% of the events) showing the diminished low energy tail.
When applying the collinear correction, care must be taken to avoid “overcorrection”. If the minimum jet p
T
threshold is too low, numerous soft objects categorized as jets will be included in the analysis and possibly counted as secondary jets. This will lead to jets without collinear splitting having soft jets added to them yielding an unphysical energy response significantly greater than one.
6.3 Jet Reconstruction Results
After applying the collinear correction to the sisCone5PFJets, a jet reconstruction efficiency of
90% was attained, where the efficiency is defined as the percentage of MC Jets that have a reconstructed jet within a ΔR cone of 0.5. As demonstrated in Figure 11, the mean relative jet energy response of 0.935 and energy resolution of 6.86% are a large improvement over the
Iterative Cone results shown in Figure 8. Evaluating the central peak widths using standard
Gaussians, we find an improvement of 35.31% over the Iterative Cone results, with roughly a quarter of that improvement coming from the collinear correction. These results, along with the
ΔR differences between generated jets and reconstructed jets, are better than the results obtained in all previous dedicated jet reconstruction analyses done in CMS [17].
12

Figure 11: Jet energy response and resolution.
7 Selection Algorithm
We restrict our analysis to |η|<2.5 as there is no tracking information beyond this geometric region. An analysis of the event kinematics demonstrated that on the reconstruction level, there is little discernible difference between the signal and the irreducible SM Photon+Jet background.
Thus, the primary goal of the selection is removing the reducible QCD dijet background which is best accomplished by improving the jet fake rejection rate. A cut-based selection was used which cut on the following variables:

Jet p
T

Photon p
T
 Δφ between photon and jet: The photon and jet are produced as a result of a scattering process so they are expected to be produced back to back. Thus, the difference in phi between photon and jet should be approximately π.

Photon Tracker Isolation: Since photons do not leave tracks unless they convert early in the tracker, most photon superclusters should have zero associated tracks. On the other hand, jets often contain charged particles which can leave tracks in the vicinity of the π° or η-meson faking a photon.

Photon ECAL Isolation: The ECAL isolation variable is calculated by summing up the energies deposited in individual crystals within a ΔR<0.4 cone around the center of a supercluster and then subtracting out the raw energy of the supercluster. Photons produced from the born and box processes should be well isolated, while jets fakes will have other associated particles which will lead to more energy deposited that is not picked up by the supercluster (leading to a higher value of the ECAL isolation variable).

Photon H/E: H/E is calculated by summing up the energies deposited in the HB and HE regions of the HCAL within a ΔR < 0.25 cone around the reconstructed position of an
ECAL supercluster, and then dividing by the total energy of the supercluster. The HCAL isolation is implemented to reject showers with leakage into the HCAL and prevent jets from being misidentified as photons because jets deposit most of their energy in the
HCAL rather than the ECAL.
13


Photon R9: The R9 shower shape variable is defined as the sum of the energies collected in a central 3x3 ECAL crystal matrix around the center of a supercluster divided by the total energy of the supercluster. For a supercluster created by a single photon, almost all of the photon energy will be collected in the 3x3 matrix. On the other hand, superclusters produced by π° jet-fakes will typically have a lower percentage of their total energy collected within the central 3x3 crystal matrix as other components of the jet may leave energy outside of the central 3x3 but still within the supercluster.
These cuts are optimized to maximize the discovery potential. Table 3 summarizes the final cut values that were used in this analysis.
Cut
Jet p
T
Photon p
T
ΔR between Jet and Photon
ECAL Isolation (Solid Cone)
Track Isolation (Solid Cone)
H/E
Value
> 125 GeV
> 175 GeV
> 2.5
< 40 GeV
< 3 GeV
< 0.05 GeV
R9 > 0.94
Table 3: Optimized selection cut values.
In each event, the photon and jet candidates are selected by taking the highest p
T
jet or photon contained in the event. As energy loss in reconstructed jets is greater than losses in reconstructed photons, a lower value is used for the Jet p
T
cut. For an event to pass the selection, it must contain at least one photon and one jet candidate passing the above selection criteria. Figure 12 shows the distributions of two of the cut variables for signal and backgrounds. There are no significant differences between signal and SM Photon+Jet background (as expected), but the cuts are very effective in rejecting QCD Dijet events.
Figure 12: H/E and R9 distributions for signal and background. Because of the large size of the QCD sample, the QCD distributions are rescaled by factors of 0.1 and 0.01 in the above plots.
The selection efficiencies obtained (given in Table 4) show that the QCD Dijet is almost entirely suppressed but because of the large cross section, QCD still contributes a large fraction of the total background events (Table 5).
14

Sample
String ( M
S
= 1 TeV)
Photon+Jet
QCD Dijet
Selection Efficiency
68.1%
45.4%
0.01%
Table 4: Selection Efficiency for Signal and Background
Sample
String ( M
S
= 1 TeV)
Number of Events for L = 100 pb
-1
268
Photon+Jet
QCD Dijet
2076
613
Cumulative Background 2690
Table 5: Expected number of events passing selection for 100pb -1 of integrated luminosity.
The invariant mass of the photon and jet is reconstructed both at the generator level and reconstruction level. The results are shown in Figure 13. As we can see, the first string resonance has a theoretical width of approximately 2.78%. After reconstruction, a mass resolution of 6.12% is achieved which is quite good given the CMS HCAL limitations.
8
Figure 13: The distributions of the generated (left) and reconstructed (right) Photon+Jet invariant mass, for the string resonance corresponding to the case of M
S
= 1 TeV.
Statistical Analysis
A log-likelihood ratio is used to calculate the statistical significance of the signal peak over the background and estimate the integrated luminosity required before a 5 sigma discovery can be claimed. The likelihood ratio λ is calculated where λ is the ratio of the probability of observing the given distribution when there is both signal and background, to the probability of observing it when there is only background. This allows us to evaluate the obtained invariant mass distribution using both the signal plus background hypothesis and the background-only hypothesis, and quantitatively determine which one is more probable. Treating each bin in the distribution as a single Poisson counting experiment, the log-likelihood ratio is given by
15

(4) where the sum is over a mass window containing the string resonance peak [19]. s i
, b i
and n i represent the number of signal, background and signal plus background events in the i th
bin. The log-likelihood ratio can be related to the statistical significance by 𝑆 = √⟨2 𝑙𝑛𝜆⟩ . The probability that a claimed five sigma signal is caused by a local fluctuation of the background is approximately 2.9·10
-7
.
Because there was not enough statistics in the QCD Dijet background sample, the QCD invariant mass background distribution was smoothed by fitting the available data to the functional form 𝑓(x) = 𝑝
0
/x 𝑝
1 where p reduced chi-squared (χ
2
0
and p
1 are constants determined by the fit. The result of this fit has a
/ndf) of 0.924 and is shown in Figure 14. Figure 15 shows the final signal+background and background only distributions used in calculating the log-likelihood ratio.
The ratio was calculated in the mass window from 700 GeV to 1300 GeV containing the first string resonance.
Figure 14: QCD Invariant Mass Fit
Figure 15: Signal+Background and Background only invariant mass distributions for 1 fb -1 of integrated luminosity.
16

The results of the log-likelihood fit demonstrated that with just 100 pb
-1
of integrated luminosity, it is possible to make a low mass string discovery with a significance of 9.84. Furthermore, a 5 sigma discovery can be made with just 25.8 pb -1 of integrated luminosity.
9 Conclusion
These results constitute the first analysis on the search for low mass strings at the LHC. An effective cut-based search strategy for low mass strings using the CMS detector has been developed. Additionally, a method for improving the reconstruction of high energy jets in CMS has been demonstrated. In this paper, we have shown quite conclusively that with under 100pb
-1 of integrated luminosity, it is possible to discover the existence of low mass strings associated with superstring models with M
S
up to 1 TeV and mixing factor κ
2
as low as 0.02. Thus, the low mass string scenario is a promising new physics channel for early LHC running.
10 Acknowledgement
First of all, I would like to thank Professor Harvey B. Newman for providing me with the opportunity to do this study. Without your support and encouragement, this research would not have been possible. Secondly, I would like to thank Dr. Marat Gataullin and Dr. Vladimir
Litvine for helpful suggestions and invaluable assistance with the data generation. I would also like to thank Dr. Arunava Roy of the University of Mississippi and Professor Haim Goldberg of
Northeastern University for assistance on the more theoretical aspects of this project. Lastly, I would like to thank the Caltech Student-Faculty Programs Office for coordinating the SURF program and the Rose Hills Foundation for generously providing funding for this project.
17

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Collider . Phys. Rev. Lett. 101 , 241803 (2008).
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