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Superradiant Black Holes and the Abelian Higgs Model

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Superradiant Black Holes and the Abelian Higgs Model Towards Extra-Terrestrial Particle Detectors: Superradiant Higgs Fields on Black Hole Spacetimes by Luke Edward Sellers An honors thesis presented for the partial fullfillment of the degree of Bachelor of Science Department of Physics University of California, Santa Barbara Santa Barbara, CA June 2021 This honors thesis is hereby approved by, Professor Sathya Professor Douglas Eardley Guruswamy Signature Signature Date Date 1 Acknowledgements My education was starkly interrupted due to medical complications, which kept me out of school for an entire year. During this time, I was unable to use my hands and arms properly, so my submission of this thesis marks a full recovery that seemed doubtful for quite some time. I would therefore like to take this opportunity to, in no particular order, thank those who made it possible for me to return to physics under these exceptional circumstances. First, I would like to thank my academic advisor and long-time professor, Sathya Guruswamy. Her instruction was deeply influential on the way that I think about physics, and her guidance and unwavering support throughout my return to school was indispensable. Second, I would like to thank my first research advisor, Professor Carlos Garcia- Cervera. Our work together saw me rapidly progress in the field of computational physics, the tools of which were pivotal for the completion of this study. Third, I would like to thank Noah Lebovic and Toolchest for collaborating with me and brainstorming ways that parallel cloud computing can increase the efficiency of computational physics. The projects that we worked on allowed me to collect numerical results at a rate that would otherwise not be possible on my local machine, thereby vastly increasing the efficiency of the numerical aspects of this study. Fourth, I would like to thank my research and thesis advisor, Professor Douglas Eardley. His knowledge and guidance were instrumental both for zeroing in on a thesis project and for accumulating the knowledge required for completing this study. I very much enjoyed our weekly meetings, and I appreciate all of the time 2 that he dedicated to this project. Fifth, I would like to thank the Shirley Ryan Ability Lab, where I completed a month long pain management program designed to get me back to school. The knowledge that I acquired during their tutelage has been indispensable in returning to my normal activities. Finally, I would like to thank Sara Sterphone for her continued support while I was away from school, and furthermore for assisting me with my return. Sara always offered a warm presence that made me feel like a priority even while I was away. When I returned to UCSB, Sara continued to check in on me to make sure that I was handling my symptoms. She was a wonderful person and she will be sorely missed. 3 Abstract In this report, we complete a numerical and analytical study of the evolution of the Abelian Higgs model on black hole spacetimes with an emphasis on conditions that would lead the onset of a superradiant instability. We study the Higgs field in particular to assess the plausibility of probing electro-weak physics that has thus far eluded experimental detection at the LHC. Our study involves the full non-linear evolution of a classical Higgs field for two different systems. The first is the Klein-Gordon equation in Minkowski spacetime with a tunable damping/anti-damping term to emulate a superradiant source. We analyze this system in order to assess whether or not superradiance persists in a system experiencing exponential growth when an inertial, nonlinear potential is present. We then move on to the evolution of the Higgs field on a Kerr background in Boyer-Lindquist and φ~ coordinates. Our study of this system informs us whether the nonlinearity of the Higgs potential will eclipse the macroscopic process through which the linear field would otherwise acquire an exponential growth rate. We fur- thermore comment on a plausible classical symmetry-breaking mechanism in which fields with a zero vacuum expectation value acquire a radially-dependent expectation value within the ergoregion of the Kerr spacetime. Finally, we apply the results of our classical evolutions to the quantum me- chanical picture via the classical field approximation and draw conclusions on the ensuing particle physics of the Higgs-Kerr system. We then compare these results to the collisions at the LHC and ultimately conclude whether these Higgs-Kerr systems provide a viable alternative to the LHC for encountering electro-weak physics. 4 Contents 1 Motivation: The State of Physics at the LHC 8 1.1 Introduction . 8 1.2 The Abelian Higgs Mechanism . 8 1.3 The Hierarchy Problem . 9 1.3.1 Basic Overview . 9 1.3.2 Solutions and Detection Status . 11 1.4 Contents: Looking Elsewhere . 12 2 Superradiance 14 2.1 Introduction . 14 2.2 Superradiance and Dissipation . 14 2.2.1 Rotational Superradiance . 15 2.3 Black Hole Superradiance . 16 2.3.1 Kerr Metric Review . 17 2.3.2 Penrose Process . 18 2.3.3 Kerr Superradiant Condition . 19 2.4 Pertinent Literature Results . 20 2.4.1 Black Hole Perturbation Theory . 20 2.4.2 Black Hole Instabilities . 21 3 Higgs Evolution: Superradiant Model in Minkowski Space 26 3.1 Introduction . 26 3.2 Equations of Motion . 27 5 3.3 Numerical Methods . 28 3.3.1 Finite Differences . 29 3.3.2 Code Tests . 31 3.4 Examining Superradiance . 35 3.4.1 Checking Γ(bt) .......................... 35 3.4.2 Testing the Higgs . 37 3.5 Conclusion . 38 4 Higgs Evolution: Kerr Spacetime 40 4.1 Introduction . 40 4.2 Boyer-Lindquist Evolution . 42 4.2.1 Equations of Motion . 43 4.2.2 Boundary Conditions . 46 4.2.3 Numerical Methods . 50 4.2.4 Energy Conservation . 52 4.2.5 Spontaneous Symmetry Breaking Mechanism . 56 4.2.6 Growth in the Massive Linear Regime . 59 4.2.7 Higgs Evolution . 60 4.3 φ~ Coordinates . 61 4.3.1 Comments on (2 + 1)D Evolutions . 62 4.3.2 Evolution in (3 + 1)D ...................... 63 4.4 Conclusions . 64 5 Discussion: Observational Signatures 65 5.1 Introduction . 65 5.1.1 Assessing Black Hole Candidates . 65 5.2 Particle Physics Signatures . 68 5.2.1 Classical Field Approximation . 69 5.2.2 Higgsenova . 70 5.3 Conclusion: Comparison with the LHC . 70 6 6 Conclusion 72 7 Chapter 1 Motivation: The State of Physics at the LHC 1.1 Introduction In 2012, the detection of a ∼ 125 GeV Higgs candidate at the LHC provided ex- perimental verification for the final piece of the Standard Model (SM) of particle physics [1]. The Higgs boson is of particular interest due to its role as the catalyst for Spontaneous Symmetry Breaking (SSB). In the context of particle physics, SSB is the mechanism through which particles acquire a mass. This is the reason, for example, why the W ± and Z bosons are observed to have a non-zero mass, but the photon, which does not interact with the Higgs, does not. This mass asymmetry would seem arbitrary for a fundamental theory of physics. However, at energy scales above the SSB threshold (∼ 1 TeV), the weak-nuclear bosons and the photon are each massless, thereby assuming a more unified orientation in the theory. 1.2 The Abelian Higgs Mechanism Though not the full non-abelian Glashow-Weinberg-Salam theory of weak interac- tions (for details, see section 20.2 of [2]), we can see how a spin-1 gauge boson acquires a mass through the Abelian Higgs Model (AHM). The Lagrangian for the 8 AHM is [2] 1 L = − F F µν + D φDµφ∗ − V (φ) (1.1) 4 µν µ where Dµ = @µ + ieAµ is the usual covariant derivative for U(1) gauge invariance, and λ V (φ) = jφj4 + µ2jφj2 (1.2) 2 2 p 2 2 has the minimum V = 0 at φ = 0 if µ ≥ 0 or jφj = µ /λ = φ0 if µ < 0. For µ the former case, there is no AµA term, so Aµ is a massless particle in the theory. For the latter case, φ is said to take on a Vacuum Expectation Value (VEV). If we expand the degrees of freedom of φ 2 C about this VEV like 1 φ ! φ0 + p (φ1 + iφ2) (1.3) 2 then the covariant derivative term takes the form 1 1 p D φDµφ∗ ! @ φ @µφ∗ + @ φ @µφ∗ + 2eφ A @µφ + e2φ2A Aµ + ::: (1.4) µ 2 µ 1 1 2 µ 2 2 0 µ 2 0 µ 1 µ 2 2 and we see that Aµ now has a mass term 2 mAA Aµ with mA = 2e φ0. Since Fµν = @µAν − @νAµ only depends on derivatives of Aµ and V (φ) depends only on φ, this new mass term will not be canceled by other contributions. We have therefore recovered a massive Aµ particle by breaking the symmetry of V (φ). For the sake of simplicity, we will limit our considerations of the Higgs field to the AHM for the entirety of this thesis. 1.3 The Hierarchy Problem 1.3.1 Basic Overview Although the Higgs mechanism provides a more unified way of viewing the members of the SM, it comes at the cost of introducing what is known as the Hierarchy 9 Problem. The Hierarchy problem is related to the way that masses are treated in a quantum field theory. In most theories of physics, there is no distinction between the mass pa- rameters that appear in the equations of motion and the observable mass. However, in quantum field theory, the mass parameter that appears in the Lagrangian 1 L = m φ2 (1.5) m 2 0 is known as the 'bare mass' and need not represent the physical mass m of the φ particle.
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