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Mass Determination of New Particle States Mass Determination of New Particle States Mario A Serna Jr Wadham College University of Oxford arXiv:0905.1425v1 [hep-ph] 9 May 2009 A thesis submitted for the degree of Doctor of Philosophy Trinity 2008 This thesis is dedicated to my wife for joyfully supporting me and our daughter while I studied. Mass Determination of New Particle States Mario Andres Serna Jr Wadham College Thesis submitted for the degree of Doctor of Philosophy Trinity Term 2008 Abstract We study theoretical and experimental facets of mass determination of new particle states. Assuming supersymmetry, we update the quark and lepton mass matri- ces at the grand unification scale accounting for threshold corrections enhanced by large ratios of the vacuum expectation value of the two supersymmetric Higgs fields vu=vd ≡ tan β. From the hypothesis that quark and lepton masses satisfy a classic set of relationships suggested in some Grand Unified Theories (GUTs), we predict tan β needs to be large, and the gluino's soft mass needs to have the opposite sign to the wino's soft mass. Existing tools to measure the phase of the gluino's mass at upcoming hadron colliders require model-independent, kinematic techniques to determine the masses of the new supersymmetric particle states. The mass deter- mination is made difficult because supersymmetry is likely to have a dark-matter particle which will be invisible to the detector, and because the reference frame and energy of the parton collisions are unknown at hadron colliders. We discuss the current techniques to determine the mass of invisible particles. We review the transverse mass kinematic variable MT 2 and the use of invariant-mass edges to find relationships between masses. Next, we introduce a new technique to add additional constraints between the masses of new particle states using MT 2 at different stages in a symmetric decay chain. These new relationships further constrain the mass differences between new particle states, but still leave the absolute mass weakly determined. Next, we introduce the constrained mass variables M2C;LB, M2C;UB, M3C;LB, M3C;UB to provide event-by-event lower-bounds and upper-bounds to the mass scale given mass differences. We demonstrate mass scale determination in realistic case studies of supersymmetry models by fitting ideal distributions to sim- ulated data. We conclude that the techniques introduced in this thesis have precision and accuracy that rival or exceed the best known techniques for invisible-particle mass-determination at hadron colliders. Declaration This thesis is the result of my own work, except where reference is made to the work of others, and has not been submitted for other qualification to this or any other university. Mario Andres Serna Jr 4 Acknowledgements I would like to thank my supervisor Graham Ross for his patient guidance as I explored many topics about which I have wanted to learn for a long time. For the many hours of teaching, proofreading, computer time and conversations they have provided me, I owe many thanks to my fellow students Tanya Elliot, Lara Anderson, Babiker Hassanain, Ivo de Medeiros Varzielas, Simon Wilshin, Dean Horton, and the friendly department post-docs James Gray, Yang-Hui He, and David Maybury. I also need to thank Laura Serna, Alan Barr, Chris Lester, John March Russell, Dan Tovey, and Giulia Zanderighi for many helpful comments on my published papers which were forged into this thesis. Next I need to thank Tilman Plehn, Fabio Maltoni and Tim Stelzer for providing us online access to MadGraph and MadEvent tools. I would also like to thank Alex Pinder, Tom Melina, and Elco Makker, with whom I've worked on fourth-year projects this past year, for questions and conversations that have helped my work on this thesis. I also would like to thank my mentors over the years: Scotty Johnson, Leemon Baird, Iyad Dajani, Richard Cook, Alan Guth, Lisa Randall, Dave Cardimona, Sanjay Krishna, Scott Dudley, and Kevin Cahill, and the many others who have offered me valuable advice that I lack the space to mention here. Finally I'd like to thank my family and family-in-law for their support and engaging distractions during these past few years while I've been developing this thesis. I also acknowledge support from the United States Air Force Institute of Technology. The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the US Government. Contents 1 Introduction 1 2 Mass Determination in the Standard Model and Supersymmetry 6 2.1 Past Mass Determination and Discovery of New Particle States . .7 2.2 Dark Matter: Evidence for New, Massive, Invisible Particle States . 16 2.3 Supersymmetry: Predicting New Particle States . 21 3 Predictions from Unification and Fermion Mass Structure 35 3.1 Supersymmetric Thresholds and GUT-Scale Mass Relations . 37 3.2 Updated fits to Yukawa matrices . 43 4 Mass Determination Toolbox at Hadron Colliders 49 4.1 Mass Determination Challenges at Hadron Colliders . 49 4.2 Invariant Mass Edge Techniques . 53 4.3 Mass Shell Techniques . 56 4.4 Transverse Mass Techniques . 57 5 Using MT 2 with Cascade decays 62 5.1 MT 2 and MCT in this context . 62 5.2 Application to symmetric decay chains . 64 6 The Variable M2C : Direct Pair Production 67 6.1 An improved distribution from which to determine MY .............. 68 6.2 Using MT 2 to Find M2C and the max MT 2 Kink . 70 6.3 Symmetries and Dependencies of the M2C Distribution . 72 6.4 Application of the method: SUSY model examples . 76 7 The Variable M2C : Significant Upstream Transverse Momentum 82 7.1 Upper Bounds on MY from Recoil against UTM . 84 7.2 Modeling and Simulation . 87 7.3 Factors for Successful Shape Fitting . 89 7.4 Estimated Performance . 99 8 The Variable M3C : On-shell Interlineate States 105 8.1 Introducing M3C .................................... 106 8.2 How to calculate M3C ................................. 108 8.3 Factors for Successful Shape Fitting . 111 8.4 Estimated Performance . 117 9 Discussions and Conclusions 122 i A Renormalization Group Running 124 A.1 RGE Low-Energy SU(3)c × U(1)EM up to the Standard Model . 125 A.2 RGE for Standard Model with Neutrinos up to MSSM . 126 A.3 RGE for the MSSM with Neutrinos up to GUT Scale . 128 A.4 Approximate Running Rules of Thumb . 129 B Hierarchical Yukawa couplings and Observables 130 C Verifying MT 2 in Eq(5.4) 133 D Fitting Distributions to Data 134 E Uniqueness of Event Reconstruction 137 F Acronyms List 139 ii List of Figures 2.1 The states of the SU(3) 10 of J P = 3=2+ baryons in 1962. Also shown is the SU(3) 27......................................... 11 2.2 One Loop Contributions to the Higgs mass self energy. (a) Fermion loop. (b) Higgs loop. 16 2.3 Direct dark matter searches showing the limits from the experiments CDMS, XENON10, and the DAMA signal. Also shown are sample dark-matter cross sections and masses predicted for several supersymmetric models and universal extra dimension models. Figures generated using http://dmtools.berkeley.edu/limitplots/. ................... 20 2.4 One Loop Contributions to the supersymmetric Higgs Hu mass self energy. (a) top loop. (b) stop loop. 26 2.5 Gauge couplings for the three non-gravitational forces as as a function of energy scale for the (left) Standard Model and (right) MSSM. 29 3.1 Updates to the top-quark mass, strong coupling constant, and bottom-quark mass are responsible for the quantitative stress of the classic GUT relation for yb=yτ .. 36 3.2 Updates to the strange-quark mass and Vcb are responsible for the quantitative stress of the Georgi-Jarlskog mass relations and the need to update values from Ref. [1]. 36 4.1 (Left:) Two body decay and it's associated mll distribution. (Right:) Three body decay and it's associated mll distribution. 54 4.2 Cascade decay from Z to Y to X ending in N and visible particle momenta α1, α2, α3.......................................... 56 4.3 Events with the new state Y is pair produced and in which each Y decays through a two-body decay to a massive new state X and a visible state 1 and then where X subsequently decays to a massive state N invisible to the detector and visible particles 2. All previous decay products are grouped into the upstream transverse momentum, k...................................... 56 4.4 Events with the new state Y is pair produced and in which each Y decays through a three-body decay to a massive state N invisible to the detector and visible particles 1, 2, 3, and 4. All previous decay products are grouped into the upstream transverse momentum, k................................ 60 4.5 The max MT 2(χ) shows a kink at the true MN and MY . For this simulation, mY = 200 GeV, and mN = 99 GeV. 61 5.1 Shows constraints from max MT 2 used with different combinations as described in eqs(5.7,5.8,5.9) and the max m12 described in Eq(5.11). Intersection is at the true mass (97 GeV; 144 GeV; 181 GeV) shown by sphere. Events include ISR but otherwise ideal conditions: no background, resolution, or combinatoric error. 65 iii 6.1 The distribution of 30000 events in 5 GeV bins with perfect resolution and no background. The three curves represent MY = 200 GeV (dot-dashed), MY = 150 GeV (dotted) and MY = 100 GeV (solid) each with M− = 50 GeV. Each distribution cuts off at the correct MY ........................
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