Physics Session Maths Events 2006

1. Physics Session Maths Events 2006 Particle Physics: How to discover a particle and measure its mass? Muge Karagoz Unel & Chris Dennis Oxford University, 27 March 2006

2. Particle Accelerators and Detectors Animation for the ATLAS Experiment at CERN - Geneva

3. How Events are Created in ATLAS:a 14000 GeV proton-proton collider

4. Elementary Particles Known Today Not yet observed • A proton consist of u and d quarks and lots of gluons floating around within it. • Top quark (t) is the heaviest elementary particle observed, with a mass of 175 GeV

5. Resonance: a Particle w/ Mass & Width • A particle with a finite life time, i.e., which can decay, will have a finite width. • An example is the Z boson, a mediator of weak force, discovered in 1983. • Mass(Z) = 91.187 GeV, width(Z) = 2.495 GeV DELPHI experiment @ LEP electron-positron collider

6. We can observe particles (reconstruct them) using their interaction properties with matter and how they behave in magnetic and electric fields • Reconstruction is limited to detector’s accuracy and resolution. This means there is an uncertainty in the energy and momentum measurements. • Therefore, the theoretical width of the particle will be smeared when measured. • In our example, we will use muons to reconstruct our mysterious particle. We will use tracking information for momentum.

7. 2nd muon 1st Muon Simulated Event Display of Mysterious Particle x y Transverse Momentum vector x • Most of the energy given to the muons from the decaying mother particle ended up as transverse momentum.

8. Introduction to Mathematical Concepts: • Einstein’s theory of relativity: E= mc2 (mass is a form of energy). • To measure in an object's frame of reference = to measure velocities relative to that object, which is taken to be stationary. This frame of reference is called the object's rest frame. • The invariant mass is defined as the energy of an object in its rest frame and is the same whatever frame of reference we measure it in. • The total invariant mass of a particle's decay products equals the particle’s invariant mass.

9. Building the Formulae • c (speed of light in vacuum) is usually taken = 1 in calculations, as we like to represent masses in magnitudes of eV  easy comparison with the incoming beam energies. • g is the relativistic correction due to special relativity. Under non-relativistic conditions, g ~ 1. • E and p (px,py,pz) form a 4-vector, for which the total proper length is the invariant mass of the system (mass of decaying particle). • We can measure E and p of the observable particles in the ATLAS detector, so, we can calculate masses of particles!

10. Analysis Tools: Data samples and Histograms • We will use Monte Carlo simulation data of our mysterious particle. • MC data are the outcome of a generation of an interaction and simulation of a detector as it would be in real data. • We will use a data analysis framework called ROOT. It is written in C++. • When we take data, we collect them in structures (ntuples) to easily analyze them. Ntuples contain measured variables per event basis: for ex., event number, measured momentum, # of tracks,.. • We will use an ntuple containing necessary ingredients to calculate our particle’s mass using the invariant mass equation. • We will apply selections on variables and observe the effects of the selections on the measurements. • The effects can be best viewed by plotting. We often use histogramming, where a variable is plotted in bins of some quantity.

11. On with the code and the analysis! • There are 10000 events in the sample (mysterious_particle.root). • The code tries to find 2 muons in each event (numberOfMuons) and calculates the invariant mass (invMass) of the pair, summing over their energy (muonEnergy) and momentum (muonXmomentum, muonYmomentum, muomZmomentum). • If a pair is found, entries are added to the invariant mass plot. • The code fits a function (Gaussian, a good approximation). The fit gives you the mass (Mean) and its width (Sigma) in GeV units. • Transverse momentum of the muons (ptMuon) and the theoretical mass of the mysterious particle will be plotted for you as well. • Your duty is to use the numbers on the plots, to: • Calculate the efficiency of finding muon pairs (Entries). How many are found? • Compare the measured and theoretical mass and widths. • Apply a selection on ptMuon and observe the effects on the efficiency and measured sigma. How much do they differ? • The steps and the formulae you will use are listed on the provided sheets. • You only need to modify analysis_example.C, but feel free to explore other files, too!

12. What have you just “discovered”? • A graviton!! • Graviton is predicted to be massive in theories which predict extra spatial dimensions on top of our ordinary 3d one. • These theories claim to explain why gravity appears to be weak to us, by the help of the Extra Dimensions. • G couple to energy  can be produced in ATLAS and can decay into practically any kind of elementary particles! • If you were looking at real data today, you would have discovered a G of 500 GeV in dimuon decay mode! • More on ED theories: http://www-pnp.physics.ox.ac.uk/~karagozm/ed_guide

13. When you go home… • How many of these 500 GeV-Gravitons we hope to observe after 2 years of data-taking with ATLAS, if ED is correct? • If you are curious, all you need to know: luminosity of the collider integrated in time (), graviton production cross section (the probability of creating one) (), and the efficiency of your detector’s catching it ()! • Here are some numbers: • = 2000 picobarn-1 •  = 15 picobarn •  = 0.5 (50%) • Could you already guess?

14. Final Remarks: • We hope you enjoyed today’s session! • Acknowledgments: • Animations courtesy of the ATLAS Collaboration’s Outreach Committee • Data and event display produced using ATLAS Software • A number of illustrations from FNAL and CERN public pages • A huge thanks to Chris Dennis, John Dale and Graham Lee! • More information: • E-mail: muge.karagoz.unel@cern.ch • Web: www-pnp.physics.ox.ac.uk/~karagozm • Feel free to contact about any questions you may have! Science starts with curiosity and continues by analytical, numerical and empirical scrutiny!