Report Bachelor Project Physics & Astronomy
Method to determine the width of the
Higgs boson of simulations of
gg → H → ZZ
∗
and gg → ZZ
∗
decay modes
By Ferran Faura Iglesias
Student nr. 11045914
Report Bachelor Project Physics and Astronomy, size 15 EC. Conducted between April 2 2018 and June 27 2018.
University of Amsterdam
Institute for Nuclear and High energy physic
Supervisor
mw. dr. H. Snoek
Second corrector prof. dr. W Verkerke
Summary
The Standard Model of particle physics predicts the width to be 4.03·10−3M eV , however the ATLAS and CMS detectors are not able to measure the width due to their resolution. It is therefore necessary to find an alternative method to determine the width of the Higgs boson. This report describes the test of a method to determine the width. The method consists of the analysis on five simulations of the gg → H → ZZ∗ and gg → ZZ∗ decay processes with different predetermined widths. The ratio (R) of the off-shell and on-shell region of the invariant mass distribution is a sensitive parameter to the width. By finding the R of all simulations and plotting these as a function of the predetermined width the relation
R = 0.017 + 4.369ΓH
was found. This relation was then used to determine the width of a simulation that did not contain a predetermined width. Using the R of this dataset with the equation and solving for ΓH resulted in ΓX = (6.513 ± 0.145)ΓSM, where ΓSM is the predicted value of the width
by the Standard Model. However, the ˜χ2 of this result was 4.24. This suggests that there is an error in the method and therefore additional tests must be performed to optimize the method.
Nederlandse samenvatting
Het Standaard Model voor de deeltjesfysica voorspelt dat de levensduur van het Higgs boson 1.56 · 10−22s is. Echter opereren de ATLAS en CMS detectoren momenteel niet op een resolutie dat deze in staat zijn de levensduur te kunnen meten. Om deze reden is het van belang een alternatieve methode te vinden om de levensduur te kunnen vinden. Dit artikel beschrijft de test van een alternatieve methode. Deze methode is het analyseren van vijf simulaties van processen waarin het Higgs boson wordt gecre¨eerd. Elke simulatie had een vooraf gedefinieerde levensduur. Uit de analyse volgde een grootheid die een functie bleek te zijn van de levensduur. Deze functie kon vervolgens op een databestand gebruikt worden dat een onbekende levensduur bevatte. Dit bestand bevatte tevens de voor de levensduur gevoelige grootheid en hierdoor was het mogelijk om met de functie de onbekende levensduur te vinden. Er zijn echter onzekerheden in het bepalen van deze levensduur die suggereren dat de gebruikte methode momenteel niet compleet is. Er is om deze reden vervolgonderzoek nodig om deze onzekerheden op te lossen.
Contents
1 Introduction 1
2 LHC and the ATLAS experiment 1
2.1 The Large Hadron Collider . . . 1
2.2 The ATLAS Experiment . . . 3
3 The Higgs boson 4 3.1 The Standard Model of particle physics . . . 4
3.2 The Higgs boson . . . 5
3.3 Quantum Interference . . . 6
3.4 Higgs Width . . . 9
4 Analysis 12
5 Conclusion 16
1
Introduction
After finding the Higgs boson in the Large Hadron Collider (LHC) by the ATLAS Collabora-tion and the CMS Experiment [1][2], certain properties of this particle are yet to be confirmed. One of the properties that is still not confirmed through experiment is the lifetime of the Higgs boson. The lifetime is predicted to be 1.56·10−22s[3] according to the Standard Model(SM). It is however, due to the resolution of the detectors available, not yet observed. Other ways must be found to confirm the prediction of the SM or to find proof of other lifetimes that therefore suggest that the SM needs to be extended. This report describes the test of a method to find the width, which is the reciprocal of the lifetime. The importance of finding a valid method, is to eliminate the necessity to construct a new expensive collider that is able to measure on higher resolutions. At least for the purpose of finding the Higgs’ width. The method consists of using five Monte Carlo simulations of gg → H → ZZ∗ and gg → ZZ∗ decay processes. The five simulations differ in predetermined widths. By analyzing the resulting dataset a parameter is found that is sensitive to the width. This parameter is the ratio between the off-shell and on-shell region of the invariant mass distribution. The relation between the ratio and the width results in a function. It might be possible to use this function on a dataset containing an unknown width and therefore be able to extract this width.
Chapter 2 consists of a brief summary of some general information about the LHC and its four main experiments. It elaborates on the ATLAS experiment and its important detection elements. Chapter 3 first gives a basic explanation of the SM and the Higgs boson, whereas it elaborates on quantum interference. Finally, it explains the relevant aspects concerning the Higgs width and the concepts of the method to determine the width used in this research. Chapter 4 describes the analysis done on the five simulations and shows the results of the research. Finally, chapter 5 ends the report with some concluding remarks.
2
LHC and the ATLAS experiment
2.1 The Large Hadron Collider
The LHC is a particle accelerator located at the European Organization for Nuclear Research (CERN) near Geneva, Switzerland. It has a circumference of about 27km and is therefore the world’s largest particle accelerator. The LHC accelerates proton-beams in opposite directions, where superconducting magnets are utilized to bend the trajectories of the beams. This results in proton-proton collisions, or pp-collisions. It is designed to achieve 14 T eV pp-collisions at a luminosity of 1034cm−2s−1. This means that beams with up to 1011 protons collide about 40 · 106 times per second inside the LHC ring[4].
During the first run of the LHC from 2010 to 2013, known as ’Run I’, a peak centre of mass energy,√s, of 8 T eV was achieved[5]. It was during this run that the Higgs boson was discovered. After Run I the LHC got a two year shut down, during which the accelerator got an upgrade in order to reach the centre of mass energy it was designed for. After this shut
probably last until 2019, where it will be shut down again[4].
There are four points of interaction on the LHC ring, which contain the four primary LHC experiments. The two general purpose detectors are ’A Toroidal LHC ApparatuS’ (ATLAS) and the ’Compact Muon Solenoid’ (CMS). These two detectors are build in a way such that they are complementary. Both the experiments discovered the Higgs boson about the same time. Another experiment on the LHC ring, is the LHC beauty (LHCb) that is focused on states involving the bottom quark. The fourth detector is ’A Large Ion Collider Experiment’ (ALICE), which is designed to study ion-ion collisions in the LHC. The large temperatures generated in these collisions replicate early stages of the universe. Figure 1 shows a schematic view of the particle accelators at CERN, but more importantly the LHC and its four primary experiments. The relevant detector in this article is ATLAS and section 2.1 expands further on this experiment.
Figure 1: Schematic view of the complete LHC complex. It shows particle accelerators at CERN, including the Large Hadron Collider and its four primary detectors[5].
2.2 The ATLAS Experiment
The design of the ATLAS detector is strongly influenced by the search of the Higgs boson. It is sensitive to a large variety of signatures, which are signals that indicate that some kind of particle is detected. This means that it is able to measure particles that are products of for example Higgs boson decays. These particles are characterized by their energy, momenta and geometrical properties. By analyzing this information, it’s possible to reconstruct what occurred in the collision. This could be for example the decay of the Higgs boson into other familiar particles. The particles that are truly measured include electrons, muons, hadrons and photons. Neutrino’s are indirectly measured, by calculating the missing transverse momentum pt. Here transverse means the direction perpendicular to the beam axis. The important
parts of the ATLAS detector are methods to measure electrons, photons and hadrons using electromagnetic and hadronic calorimetry. The ability to measure the momenta of muons with high accuracy and track particles with high momenta efficiently. ATLAS has almost full coverage in the azimuthal angle (φ), whereas it has high coverage in pseudo-rapidity (η). The pseudo-rapidity is the spatial coordinate describing the angle of a particle relative to the beam axis and is defined as
η = − ln (tanθ 2), thus making it a function of the polar angle (θ)[4].
Figure 2 shows a schematic view of the plane perpendicular to the beam axis. The AT-LAS detector is composed in such a way that it’s able to determine a pattern in energies and momenta. It is then possible to assign these patterns to certain particles and therefore find the decayed particle. The different patterns are characterized by the type of interaction the particle has with the measurement apparatus. These interactions are the weak, strong or elec-tromagnetic interactions. The weak interactions are not visible in the detector, excluding the spontaneous decays of particles through the weak force. The inner detector, which is directly connected to the beam tunnel as viewed in figure 2, has the purpose to track charged particles created out of the collisions and contributes to the reconstruction of particle trajectories. The mass of the inner detector is relatively low as unwanted collisions with the detector material must be minimized[6].
Surrounding this inner detector are the calorimeters, that are designed to provide data of incident particles in the calorimeter medium. The calorimeters absorb the energy, resulting in an electrical signal that is read out by the computer. The electromagnetic calorimeter, which is closer to the beam axis, measures the energy of photons and electrons. The energy deter-mination is based on the electromagnetic showers created by incident photons and electrons. Discrimination of electrons and photons is possible because the electrons leave a visible tra-jectory in the inner detector. The hadronic calorimeter has its purpose in measuring incident hadrons. The hadrons that interact with the medium through the strong interaction, create a broader shower than the electromagnetic showers. The energy measurement is therefore less precise[6].
Figure 2: A schematic cut-away in the plane perpendicular to the beam axis. The dashed lines indicate that the detector is not able to measure the particle[6].
Ultimately there is the outermost element of the detector, which is the muon spectrome-ter. This part detects charged particles that are noticed by the previous elements, but have not created showers while traversing through all the detection elements. The spectrometers measure the muon’s momenta by bending their trajectories using magnetic fields. By com-paring with the inner detector’s findings, it is then possible to reconstruct and determine that the particle was indeed a muon[6]. After all momenta are measured and calculated, one will notice that there is missing momentum. This is due to the neutrino’s being undetected and, as mentioned before, taking along some part of the pT while escaping the detector.
While the ATLAS detector is outstandingly optimized in finding the Higgs boson’s mass, it lacks precision to determine its width. The resolution of ATLAS is about 1 GeV [7], whereas the width is supposedly in the region of several M eV0s[3]. Therefore it is necessary to use different methods to determine the Higgs’ width.
3
The Higgs boson
3.1 The Standard Model of particle physics
The LHC was designed with the purpose to confirm predictions of the Standard Model or otherwise discover unknown physics, thus expanding the current understanding of particle physics. One example of a prediction that is confirmed by the LHC, is the discovery of the Higgs boson. Figure 3 shows the table of elementary particles, which after the discovery of the Higgs boson is the current understanding of particles and forces. These particles are the constituents of known matter like protons and neutrons. There are three categories of parti-cles, which are the quarks, leptons and the force carriers. Then there are three generations of matter, whereas the first generation quarks and leptons are the building blocks of known atoms. The second and third generations of matter have higher mass and decay through the weak interaction to first generation particles[6]. The SM describes the interactions between
all matter particles as the exchange of force carriers, which are the photon (γ) for electro-magnetic interactions, the W and Z bosons for weak interactions and the gluons for strong interactions. The SM predictions were heavily tested and confirmed, however it was unclear what caused the photon to be massless while the W and Z boson’s did possess mass. This led to the proposition that the Z and W boson’s acquired their mass through spontaneous symmetry breaking of a scalar field. This scalar field is now known as the Higgs Field and the evidence for it is the existence of the Higgs boson[2].
3.2 The Higgs boson
The Higgs boson was discovered by ATLAS[1] and CMS[2] at a mass of about mH = 125
GeV using data of√s = 7 and 8 T eV collisions. The Higgs boson production cross-section is predicted to be around 23(29) and 19(14) pb[2]. The cross-section indicates the probability for a certain interaction to occur. The Higgs boson was found through five different decay modes which are H → ZZ, H → γγ, H → W+W−, H → τ+τ− and H → b¯b. The research described in this article focuses on the H → ZZ, therefore further text will not elaborate on other decay modes. The state will furtherly be denoted as H → ZZ. The final states ZZ in the low-mass region are made up of at least one off-shell Z boson and will be denoted as Z∗. An off-shell particle is the same particle as the on-shell equivalent, except it has no defined mass-state. In this article the off-shell region described, meaning the invariant mass region where only off-shell Higgs bosons appear, is the area > 300 GeV .
Figure 3: A figure of ’The Standard Model of Elementary particles’[8]. All known elementary particles are shown and categorized into groups which are the quarks, leptons, gauge bosons and scalar bosons. The quarks interact through all three forces, which means quarks exchange force carriers with other particles. Leptons do not interact through the strong interaction.
The Feynmann diagram to describe the decay mode relevant in this article is viewed in figure 4(a). Figure 4(b) shows that there are several paths from the same initial state to the same final state. The presence of these different paths creates quantum interference and will be explained in a later section. This interference influences the amount of particles measured by the detection elements.
3.3 Quantum Interference
As mentioned before there are several paths to end up with a certain final state. This final state could be two Vector bosons, like the Z bosons, where the initial states are two gluons. This is illustrated in figure 4, as Feynman diagrams. These diagrams show the same initial and final state, they have however distinct intermediate states. As visible, only diagram (a) in figure 4 shows a creation of an Higgs boson. The Feynman diagrams describe quantum mechanical paths and for this reason they are able to interfere with each other. This means that when this interference is destructive, less events are observed by the detector than when there would only be one path.
Figure 4: Two diagrams of the gg → ZZ∗ interaction[9]. Figure (a) shows the diagram with an intermediate state that contains an on-shell Higgs boson and final states with an off-shell and on-shell Z bosons. Figure (b) is considered the background continuum, where there are no intermediate states with an Higgs boson.
This interference is analogous to the interference of light. The double slit experiment is a useful experiment to show the quantum interference of light when it moves through two slits. If light would be considered to be a particle, then when single photons are incident on a double slit the pattern of figure 5 would be expected. A stream of particles moving through an hole and ending at the detector, as if it where macroscopic bullets. However, when doing the ex-periment an interference pattern is observed as seen in figure 6. This pattern has a sinusoidal function, describing the probabilities of the photon to be observed on that point in space. This suggests that single photons interfere with itself when moving through two separate slits.
Figure 5: Expected pattern of incident photons on a double slit, when assuming photons are particles. The intensity increases from left to right [10].
The analogy becomes clear in this last part. The initial state of the photon, before entering the two slits, is described as a quantum mechanical wave-function. Whereas the final state is the collapse of this wave-function at one point in the detector, where the photon is described as a particle. By moving through the slits the photon takes two different paths simultaneously and therefore two different parts of the wave-function are now able to interfere. This translated to the Feynman diagrams means that all the paths described are taken simultaneously and are therefore able to interfere.
Figure 6: Sinusoidal function that shows the interference pattern of photons moving through two slits. The x-axis represents the position in the detector, whereas the y-axis shows the amount of photons detected[10].
This interference is visible in the data of pp-collisions because, as mentioned before, there are more or less detected events depending on whether it’s constructive or destructive inter-ference. When looking at the gg → ZZ interaction, we know that there are multiple paths to reach the final state. The graph in figure 7 shows how two diagrams similar to the ones in figure 4 interfere. It is clear that the blue marks, which are the points that contain the interference, show that there are less events detected. This means that these two particular paths destructively interfere.
The Feynman diagrams represent complex amplitudes, that when taken the absolute value squared give the probability for the corresponding interaction. When these amplitudes corre-spond to interactions with equal initial and final states, the amplitudes have to be summated before being squared. Equation (1) shows how the interference follows from a mathematical description. |A1+ A2|2= |(a1+ ib1) + (a2+ ib2)|2 = |(a1+ a2) + i(b1+ b2)|2 = (a1+ a2)2+ (b1+ b2)2 = a21+ b21+ a22+ b22+ 2a1a2+ 2b1b2 (1)
Here A1and A2are two arbitrary complex amplitudes, where a and b are the corresponding
real and imaginary components respectively. If compared with equation (2), that shows the description of the different amplitudes without considering interference, it is evident that the final probabilities are not equal.
|A1|2+ |A2|2 = |(a1+ ib1)|2+ |(a2+ ib2)|2
Figure 7: A graph that shows the interference of two different paths of gg → ZZ. The red marks correspond to the separate Higgs creation and background continuum processes. The amplitudes are squared before they are summated. This means that there is no interference. The blue marks correspond to the simultaneous processes of Higgs creation and background continuum. In this case the amplitudes are squared after they are summated. When looking at the Higgs resonance about 126 GeV , it is clear that there are less events detected. This means that the two different paths destructively interfered.
3.4 Higgs Width
The Standard Model is not able to predict the Higgs’ mass, however it does predict the width of the Higgs boson. The predicted value of the 125 GeV Higgs boson is ΓSM = 4.07 · 10−3
M eV with a relative uncertainty of+4.0%−3.9%[3]. The reciprocal of this value indicates the lifetime of the Higgs boson. The value of the width is significantly smaller than the resolution that ATLAS operates on. As mentioned before, other methods must be used to confirm or exclude the predicted width.
This report explains a method tested to find the Higgs’ width, by using Monte Carlo simulations. It is possible to simulate pp-collisions that will only result in the gg → H → ZZ∗ and gg → ZZ∗ decay modes. Figure 8(a) shows a plot of the invariant mass distribution of 4 leptons out of a Monte Carlo simulation of the decay including the Higgs boson. The 4 leptons are the real measured particles by the detection elements because the final states of the decay modes are in both cases ZZ∗ → 4l. The resonance visible is at 126 GeV , however one would expect the resonance to be at 125 GeV . This is due to the program simulating the Higgs boson to have a mass of 126 GeV .
(a) Resonance about 126 GeV . (b) Blue line indicating Breit-Wigner fit.
Figure 8: Plots showing the invariant mass distribution of 4 leptons out of a Monte Carlo simulation including only gg → H → ZZ∗ decay mode. The distribution only shows a narrow region about the 126 GeV , as to illustrate the sharpness of the peak clearly. Figure (b) shows how the Breit-Wigner fit follows the distribution, illustrated with a blue line.
The simulations are based on Standard Model predictions, this suggests that the Breit-Wigner fit as seen in figure 8(b) gives the predicted ΓSM. The Breit-Wigner fit uses the
function as seen in equation (3) [11].
Breit − W igner = 1 (x − m)2+ 1
4g2
(3) The use of a Breit-Wigner distribution is due to the relation between the production cross-section and the invariant mass. Equation (4) [7] shows that this relation is similar to a Breit-Wigner distribution. dσpp→H→ZZ dM4l2 ∼ g2HgggH2 ZZ (M4l2 − m2 H)2+ m2HΓ2H (4) Here σ is the production cross-section of the Higgs boson, M4l is the invariant mass of the 4
leptons, mH is the Higgs’ mass, ΓH is the Higgs’ width and the g’s in the numerator are the
Higgs bosons couplings to the initial and final state. Plotting the results of the simulation and using a fit to calculate the width is the basic principle that explains the method used to find the Higgs’ width.
This exact simulation is not a complete description of the process. It is important to note that when analyzing data from pp-collisions, the Higgs peak will appear different than as seen in figure 8. This is due to the resolution of the ATLAS detector. Figure 9 shows the results of a Monte Carlo simulation that included the detector resolution. It is clear from the figure that the peak is spread out, which results in an uncertainty in determining the width. Notice how the peak no longer obeys a Breit-Wigner relation, as shown more concretely in figure 10, and to properly fit these data points a Gaussian curve is necessary.
Figure 9: Figure shows a plot of a Monte Carlo simulation of the invariant mass distribution of 4 leptons as result of the gg → H → ZZ∗ decay mode. If one compares this plot to 8 it’s noticeable that the peak is less well-defined.
Figure 10: Figure shows the invariant mass region about the Higgs boson peak when consid-ering detector resolution. The blue line indicates the Breit-Wigner fit and shows that this curve does not properly follow the distribution of the invariant mass.
The Gaussian function used is shown in equation (5) [11]. Gauss = exp− 0.5 x − m
s 2
(5) Here x, m and s are arbitrary parameters that are are chosen to properly fit. So by using this Gaussian function to fit, as shown in figure 11, it is possible to calculate the width of the peak. The width of figure 11 using the Gaussian fit results in a value of about 1.35 GeV , which is
not possible now. This suggests that an alternative method must be used to determine the width. In the scope of this report this method is the definition of a variable that corresponds to a relation between number of events (N) and σ. This variable is the ratio between the off-shell and on-shell region of the invariant mass distribution and will further be explained in the following section.
Figure 11: Figure shows the invariant mass region about the Higgs boson peak when consid-ering detector resolution. The blue indicates the Gaussian fit and shows that this curve does properly follow the distribution of the invariant mass.
4
Analysis
As mentioned before, this article explains a method used to determine the width of the Higgs boson. This is done by simulating gluon-gluon fusion processes, using Monte Carlo methods. In particular the gg → H → ZZ∗ and gg → ZZ∗ decay modes are simulated while also including the detector resolution. The gg → ZZ∗ process is added to include the interference that could occur. The idea is to have a dataset based on the SM, that contains an unknown width. By using different simulations with predetermined widths and extracting a relation between the ratio and the widths one could determine the value of the unknown width. This simulation, which will be defined as the ’X-file’, emulates real measurements where the width is also unknown. The ratio (R) is a valid variable that is sensitive to the width, due to the relations of the parameters that make up the ratio. There is a relation between N and σ as seen in equation (6)[12].
N = σLint (6)
Here Lint is the integrated luminosity and is equal over the complete distribution. When
dividing N of the off-shell region with N of the on-shell region, as seen in equation (7), it is possible to eliminate the Lint dependency. This is convenient, due to the unknown values of
R = Nof f Non = σof fLint σonLint R = Nof f Non = σof f σon (7)
The relation between σ and the width ΓH is visible in equation (8)[7], which is the result of
the integration of equation (4).
σi→H→f ∼
g2igf2 ΓH
(8) The different predetermined widths used by the 5 simulations were 1 ΓSM, 2.5 ΓSM, 5
ΓSM, 7.5 ΓSM and 10 ΓSM. Figure 12 shows the 5 different invariant mass distributions,
therefore illustrating the influence of the width. The plots show no noticeable difference in interference.
Figure 12: The resulting invariant mass distributions of the five simulations plotted in the same frame. There is no noticeable influence on the interference by the widths. It appears that the peak about 126 GeV is lower, this is due to the width being larger and the value of the cross-section on that point being smaller. The ’G’ in the legend stands for ΓSM.
The next step is to calculate the integral of the region around the peak of 126 GeV . The value of the integral is determined by using the Gaussian fit, which is shown in figure 13. The figure only shows the fit done on the 1 ΓSM file , however similar calculations are done on
the other datasets. The value of the integral gives the σ of the integrated region, which is σon. Finding σof f is done by summing the bin-entries of the tail of the distribution, this tail
predetermined width. Using a first-order polynomial to fit, results in a linear relation between the parameters that is viewed in equation (9).
R = 0.017 + 4.369ΓH (9)
This fit gives a ˜χ2 of 4.24, which is explicable due to one point not following the first-order polynomial function.
Figure 13: Plot of a narrow region about the 126 GeV peak of the 1 ΓSM simulation. The
blue line indicates the Gaussian curve used to fit the data.
(a) Results of R. Red line indicates polynomial
fit. (b) Results of R with factor 2 re-scaling of errors.
Figure 14: Plot of the results of R as function of the predetermined widths. Figure (a) shows the results with ordinary errors, while (b) shows the same results with re-scaled errors. The red line illustrates the first-order polynomial fit.
The function obtained that describes the relation between R and ΓH, is now used on the
X-file. Unlike the 5 simulations that have σ on the y-axis, the X-file contains N on the y-axis. As seen in equation (7) this is equal and therefore shows the value of R in this analysis.
The ratio obtained from the X-file is now used with equation (9). This results in a width ΓX = (6.513 ± 0.145)ΓSM and is shown in figure 15.
Figure 15: Figure shows the plot of the ratio as function of predetermined width, including the found width of the X-file. The blue dashed line indicates the point of ΓX, while the red
line illustrates equation (9).
The ˜χ2 is significantly larger than 1 and therefore shows that the errors are not well calculated. A ˜χ2 = 1.06 is obtained when manually re-scaling the errors with a factor of 2, as
shown in figure 14(b). It might confirm that there is an error in determining the errors, that could be created in the simulations. To verify this, 50 new simulations of 1 ΓSM are created.
Figure 16 shows the histogram filled with the 50 calculated widths. The mean found is 4.453 with a root-mean-squared of 0.165. Comparing with the values of the original simulation, which were 4.432 and 0.114, shows that the widths and errors are similar. This means that the simulations are probably not creating the wrong errors. There were 3 widths of the 50 simulations that were significantly larger, which is probably due to statistical fluctuations. While the real source of the issue is unknown, statistical fluctuations could be an explanation for R10ΓSM not obeying equation (9).
Figure 16: Histogram filled with the ratios of 50 simulations of 1 ΓSM. The mean is 4.453
and the root-mean-squared is 0.165. Three ratios have been omitted, to be able to present the data. These ratios are significantly larger, due to statistical fluctuations.
5
Conclusion
Determining the ratios of the five simulations results in a function as viewed in equation (9). Using this function with a dataset that contains an unknown width and solving for ΓH
results in ΓX = (6.513 ± 0.145)ΓSM, where ΓSM stands for the predicted width by the SM.
The ˜χ2 found in the analysis is 4.24. This is significantly larger than 1, suggesting that the found error might not be a correct illustration of the real error. To test this, 50 alternative simulations with a predetermined width of 1ΓSM are used to statistically determine a ratio
and error. The ratio and error results of the 50 simulations are 4.453 and 0.165 respectively. Comparing with the values of the original simulation of 1ΓSM, which are 4.432 and 0.114,
shows that the values are similar. There appears to be a statistical fluctuation in the data that is the source of the large ˜χ2.
6
Discussion
The ˜χ2 of 4.24 suggests that the width ΓX = (6.513 ± 0.145)ΓSM is probably not correct.
However, finding a ˜χ2 of approximately 1 does not mean that the resulting width is close to the real width. This is due to the systems that are simulated being idealized situations, where most of the decay modes of the Higgs boson and background processes are omitted. For further research background processes and the other decay modes could be added.
not be correct. To test this 50 simulations were created to check for statistical fluctuations, as illustrated in figure 16. While this analysis indicated that the result of the large ˜χ2 was due to statistical fluctuation, one could argue that 50 simulations are not sufficient to conclude this. This suggests that in further research an analysis of a significant amount of simulations could be done to exclude the incorrect error calculation.
To verify the validity of the research, it could be replicated with particles that have ex-perimentally confirmed widths.
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