Beyond the Standard Model: The Pragmatic Approach to the Gauge Hierarchy Problem A dissertation presented by Rakhi Mahbubani to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts May 2006 c 2006 - Rakhi Mahbubani All rights reserved. Thesis advisor Author Nima Arkani-Hamed Rakhi Mahbubani Beyond the Standard Model: The Pragmatic Approach to the Gauge Hierarchy Problem Abstract The current favorite solution to the gauge hierarchy problem, the Minimal Su- persymmetric Standard Model (MSSM), is looking increasingly fine tuned as recent results from LEP-II have pushed it to regions of its parameter space where a light higgs seems unnatural. Given this fact it seems sensible to explore other approaches to this problem; we study three alternatives here. The first is a Little Higgs theory, in which the Higgs particle is realized as the pseudo-Goldstone boson of an approximate global chiral symmetry and so is naturally light. We analyze precision electroweak observables in the Minimal Moose model, one example of such a theory, and look for regions in its parameter space that are consistent with current limits on these. It is also possible to find a solution within a supersymmetric framework by adding to the MSSM superpotential a λSHuHd term and UV completing with new strong dynamics under which S is a composite before λ becomes non-perturbative. This allows us to increase the MSSM tree level higgs mass bound to a value that alleviates the supersymmetric fine- tuning problem with elementary higgs fields, maintaining gauge coupling unification in a natural way Finally we try an entirely different tack, in which we do not attempt to solve the hierarchy problem, but rather assume that the tuning of the higgs can be explained in some unnatural way, from environmental considerations for instance. With this philosophy in mind we study in detail the low-energy phenomenology of the minimal extension to the Standard Model with a dark matter candidate and gauge coupling unification, consisting of additional fermions with the quantum numbers of SUSY higgsinos, and a singlet. iii Contents Title Page . i Abstract . iii Table of Contents . iv Citations to Previously Published Work . vi Acknowledgments . vii Dedication . viii 1 Introduction and summary 1 2 Precision Electroweak Observables in the Minimal Moose 4 2.1 The Theory . 7 2.2 The Gauge Boson Sector . 8 2.3 Non-linear Sigma Model Sector . 11 2.4 Plaquette Terms . 11 2.5 Electroweak Symmetry Breaking . 12 2.6 Fermion Sector . 13 2.7 Higgs Sector . 15 2.8 Results . 15 2.9 Summary and Discussion . 18 3 The New Fat Higgs: Slimmer and More Attractive 22 3.1 Constructing a Model . 23 3.1.1 Details of the Model . 25 3.1.2 Conformality and Confinement . 26 3.2 Analysis . 28 3.2.1 λ and the Higgs Mass Bound . 28 3.2.2 Gauge Coupling Unification . 31 3.2.3 Phenomenology . 33 3.3 Summary and Discussion . 34 4 The Minimal Model for Dark Matter and Unification 37 4.1 The Model . 39 4.2 Relic Abundance . 41 4.2.1 Higgsino Dark Matter . 43 iv Contents v 4.2.2 Bino Dark Matter . 44 4.3 Direct Detection . 46 4.4 Electric Dipole Moment . 46 4.5 Gauge Coupling Unification . 49 4.5.1 Running and matching . 51 4.6 Summary and Discussion . 57 5 Conclusion 59 Bibliography 60 A The Workings of a Top Seesaw 68 B The Neutralino Mass Matrix 69 C Two-Loop Beta Functions for Gauge Couplings 70 Citations to Previously Published Work Chapters 2-4 are lifted almost entirely from the following papers: \Precision electroweak observables in the minimal moose little Higgs model", C. Kilic and R. Mahbubani, JHEP 0407, 013 (2004),[arXiv:hep-ph/0312053]. \The new fat Higgs: Slimmer and more attractive", S. Chang, C. Kilic and R. Mahbubani, Phys. Rev. D 71, 015003 (2005),[arXiv:hep-ph/0405267]. \The minimal model for dark matter and unification”, R. Mahbubani and L. Senatore, Phys. Rev. D 73, 043510 (2006),[arXiv:hep-ph/0510064]. Electronic preprints are available on the Internet at the URL http://arXiv.org vi Acknowledgments Thanks first and foremost to Nima Arkani-Hamed who enriched my years as a graduate student with little nuggets of physics wisdom and endless amusing anecdotes about physicists past and present. I feel enormously privileged to have been his student. I also owe an huge debt of gratitude to Jay Wacker, without whose canny combination of cajolery and threats I would probably still be advisor-less. Thanks to my husband Will for his constant love and support and for bolstering my confidence when I most needed it. Thanks to my friends for keeping me sane; to Can, whose Daily Bad Joke will be sorely missed; and especially to Shiyamala, who taught me how to keep being me, and imparted to my daily routine some much-needed feminine frivolity. vii To my parents and teachers Who made me believe that nothing was impossible viii Chapter 1 Introduction and summary Over the last few decades the search for physics beyond the Standard Model (SM) has largely been driven by the principle of naturalness, according to which the parameters of a low energy effective field theory should not be much smaller than the contributions that come from running them up to the cutoff. This principle can be used to constrain the couplings of the effective theory with positive mass dimension, which have a strong dependence on UV physics. Requiring no fine tuning between bare parameters and the corrections they receive from renormalization means that the theory must have a low cutoff. New physics can enter at this scale to literally cut off the high-energy contributions from renormalization. Applying this principle to the SM means that despite spectacular agreement with current experimental data, this theory is widely held to be incomplete due to an instability in its Higgs sector; radiative corrections to the Higgs mass suffer from one-loop quadratic divergences leading to an undesirable level of fine-tuning between the bare mass and quan- tum corrections. This suggests the emergence of new physics at energy scales around a TeV, which will be investigated in the near future with direct accelerator searches. The electroweak sector of the SM has been probed to better than the 1% level by precision experiments at low energies as well as at the Z-pole by LEP and SLC. The data obtained can also severely constrain possible extensions of the SM at TeV energies [1, 2, 3, 4]. Supersymmetry (SUSY) provides arguably the most attractive solution for this hi- erarchy, since it comes with gauge coupling unification as an automatic consequence. How- ever its simplest implementation, the Minimal Supersymmetric Standard Model (MSSM), is looking increasingly fine-tuned as recent results from LEP-II have pushed it to regions of 1 2 Chapter 1: Introduction and summary parameter space where a light higgs seems unnatural.1 This is problematic for the MSSM since SUSY relates the quartic coupling of the higgs to the electroweak gauge couplings, which at tree level bounds the mass of the lightest higgs to be less than that of the Z. Radiative corrections can help increase this bound, with the largest contribution coming from the top yukawa, giving 2 2 2 3 4 2 mt~ mh0 mZ + 2 ht v log 2 (1.1) ≈ 8π mt for large tan β. Since this effect is only logarithmic in the stop mass however, consistency with the LEP-II mass bound requires the stops to be pushed up to at least 500 GeV. At 2 the same time radiative corrections to mHu are quadratic in the stop mass 2 3 2 Λ δmHu 2 mt~ log (1.2) ≈ −4π mt~ There is therefore a conflict between our expectation that the stop is heavy enough to significantly increase the higgs mass through radiative corrections and yet light enough to cut off the quadratic divergence in a natural way.2 Requiring consistency with LEP-II results therefore forces us to live with a fine tuning of a few percent. This `little hierarchy' problem [8, 9] raises some doubts about the plausibility of low energy SUSY as an explanation for the smallness of the higgs mass. In this dissertation we attempt to take an entirely open-minded approach to this long-unresolved problem and study solutions that are distinct from the MSSM both in philosophy and low-energy phenomenology. Each of the following three chapters comes at the problem from a different direction, and can be read relatively independently of the others. Their contents are summarized as follows: Investigation of precision electroweak observables in a particular solution to the hi- • erarchy problem via the Little Higgs mechanism, in which the higgs is taken to be a pseudo-goldstone boson of some larger global symmetry, with a mass that is protected from large radiative corrections. Study of the NMSSM with a composite scalar higgs as a way to alleviate the super- • symmetric 'little hierarchy' problem. 1See references [5, 6] for further discussion. 2A recent paper [7] attempted to resolve this conflict by suppressing the size of radiative corrections to 2 mHu from the stop. Chapter 1: Introduction and summary 3 Comprehensive analysis of the low-energy phenomenology of the most minimal non- • supersymmetric theory that has a good dark matter candidate and gives rise to gauge coupling unification. We do not assume that the gauge hierarchy is solved in this model; rather we entertain the possibility that a small higgs mass is selected for in the string theory landscape by reasoning similar to that used by Weinberg in his anthropic bound for the cosmological constant.