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Unusual Signs in Quantum Field Theory

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Unusual Signs in Quantum Field Theory Unusual Signs in Quantum Field Theory Thesis by D´onal O’Connell In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2007 (Defended May 16, 2007) ii c 2007 D´onal O’Connell All Rights Reserved iii To D´ısa iv The Collar I Struck the board, and cry’d, No more. I will abroad. What? shall I ever sigh and pine? My lines and life are free; free as the rode, Loose as the winde, as large as store. Shall I be still in suit? Have I no harvest but a thorn To let me bloud, and not restore What I have lost with cordiall fruit? Sure there was wine Before my sighs did drie it: there was corn Before my tears did drown it. Is the yeare onely lost to me? Have I no bayes to crown it? No flowers, no garlands gay? all blasted? All wasted? Not so, my heart: but there is fruit, And thou hast hands. Recover all thy sigh-blown age On double pleasures: leave thy cold dispute Of what is fit, and not. Forsake thy cage, Thy rope of sands, Which pettie thoughts have made, and made to thee Good cable, to enforce and draw, And be thy law, While thou didst wink and wouldst not see. Away; take heed: I will abroad. Call in thy deaths head there: tie up thy fears. He that forbears To suit and serve his need, Deserves his load. But as I rav’d and grew more fierce and wilde At every word, Me thoughts I heard one calling, Childe: And I reply’d, My Lord. George Herbert v Acknowledgements I thank my advisor, Mark Wise, for teaching me quantum field theory, for his frequent amusing jokes, and especially for saving projects which seemed to be doomed. I thank him for showing me what it really means to do physics, for sharing his ideas with me and for patiently answering my many questions. I also thank Mark for educating my taste in Chateauneuf du Pape. I thank Martin Savage for teaching me that nuclei are every bit as interesting as the high energy frontier. Martin taught me a useful skill—that is, how to complete papers, and I think it’s fair to say this thesis would not exist if I’d never figured that part out. I also thank him for many memorable and informative conversations. I thank Michael Ramsey-Musolf for his counsel, collaboration, and support over the last few years. I thank him for serving on my defense and candidacy committees. I thank John Preskill for serving on my defense and candidacy committees, Brad Fil- ippone for serving on my defense committee, and Robert McKeown for serving on my candidacy committee. I thank Jiunn-Wei Chen and Andre Walker-Loud for an especially fruitful collaboration and for their friendship. I thank Alejandro Jenkins for collaboration but mainly for many bad conversations in dubious bars. I thank him for continued patience as my time has been absorbed by the completion of this thesis. I claim that, contrary to suggestions in his thesis, it is he who is the real quack. Despite all of these issues, I nevertheless learned a great deal from Alejandro, and I thank him for his sharing his wit, and his knowledge, and also for his friendship. I thank my collaborators Ben Grinstein and Ruth van de Water, who both taught me me a great deal of useful physics. I thank current and former members of my group, Lotty Ackermann, Christian Bauer, Vincenzo Cirigliano, Matt Dorsten, Rebecca Erwin, Misha Gorshteyn, Michael Graesser, vi Moira Gresham, Jennifer Kile, Christopher Lee, Keith Lee, Sonny Mantry, Stefano Profumo, Michael Salem, Sean Tulin, Peng Wang, and Margaret Wessling. I am especially grateful to Michael Graesser, Chris Lee, Sonny Mantry and Michael Salem for helping me out with various aspects of my work over the last few years. But most of all, I thank Lotty for being such a tolerant office mate for several years now. I thank Jie Yang, who has had the misfortune of teaching with me in my last terms at Caltech, for cheerfully bearing more than her fair share of the teaching load. I thank Kris Sigurdson for many memorable times in my early years in Caltech. I fondly remember many trips to the cinema, evenings in bars, and various board games enjoyed in his company; the occasion when Kris upstaged the President of Caltech in the Athenaeum is especially memorable. I thank him, too, for advice on how to make progress in academia. I thank Robert Hodyss, Tristan McLoughlin, Paul O’Gorman, and Christophe Basset for providing excellent company on a great many occasions, without which it would have been difficult to keep at my work when the challenges seemed too great. I thank my apartment mates Jonathan Pritchard, Asa Hopkins, and Ben Toner for not complaining about my antisocial habits too loudly. I thank my friends Igor Bargatin, Michael Boyle, Paul Cook, Patrick Dondl, Michael Edgar, Jeff Fingler, Lisa Goggin, Rosie Jones, Ramon van Handel, Hannes Helgason, Vala Hj¨orleifsd´ottir, Ilya Mandel, Tony Miller, Carlos Mochon, Paige Randall, Nikoo Saber, Paul Skerritt, Graeme Smith, Tristan Smith, Ian Swanson, Tasos Vayonakis, Ketan Vyas and Daniel Wagenaar for many good times. I thank Brega Howley for organizing an excellent Christmas dinner every year, and especially for always making sure to choose a date which suited me. I thank my family: my mother, Jane, Tim, Eoin, Lise, Tadhg, Hilda, Eva, and Sadhbh for putting up with my long absences. I particularly thank Tadhg for awakening my interest in science when I was small. I especially thank Ard´ısEl´ıasd´ottir´ for all her help and support over the last several years. vii Abstract Quantum field theory is by now a mature field. Nevertheless, certain physical phenomena remain difficult to understand. This occurs in some cases because well-established quantum field theories are strongly coupled and therefore difficult to solve; in other cases, our current understanding of quantum field theory seems to be inadequate. In this thesis, we will discuss various modifications of quantum field theory which can help to alleviate certain of these problems, either in their own right or as a component of a greater computational scheme. The modified theories we will consider all include unusual signs in some aspect of the theory. We will also discuss limitations on what we might expect to see in experiments, imposed by sign constraints in the customary formulation of quantum field theory. viii Contents Acknowledgements v Abstract vii 1 Introduction 1 2 Extrapolation Formulas for Neutron EDM Calculations in Lattice QCD 6 2.1 Introduction . 6 2.2 Strong CP-Violation in Chiral Perturbation Theory . 9 2.3 Neutron EDM at Finite Volume . 12 2.4 Neutron EDM in Partially-Quenched QCD . 14 2.5 Conclusions . 20 3 Ginsparg-Wilson Pions Scattering in a Sea of Staggered Quarks 22 3.1 Introduction . 22 3.2 Determination of Scattering Parameters from Mixed Action Lattice Simulations 25 3.3 Mixed Action Lagrangian and Partial Quenching . 29 3.4 Calculation of the I = 2 Pion Scattering Amplitude . 35 3.4.1 Continuum SU(2) . 36 3.4.2 Mixed GW-Staggered Theory with 2 Sea Quarks . 38 3.4.3 Mixed GW-Staggered Theory with 2 + 1 Sea Quarks . 43 3.5 I = 2 Pion Scattering Length Results . 47 3.5.1 Scattering Length with 2 Sea Quarks . 47 3.5.2 Scattering Length with 2+1 Sea Quarks . 48 3.6 Discussion . 49 ix 4 Two Meson Systems with Ginsparg-Wilson Valence Quarks 51 4.1 Introduction . 51 4.2 Mixed Action Effective Field Theory . 55 4.2.1 Mixed Actions at Lowest Order . 56 4.2.2 Mixed Action χPT at NLO . 61 4.2.2.1 Dependence upon sea quarks . 68 4.2.2.2 Mixed actions at NNLO . 69 4.3 Applications . 70 4.3.1 fK /fπ ................................... 71 I=1 4.3.2 KKI = 1 scattering length, aKK ................... 73 I=3/2 4.3.3 Kπ I = 3/2 scattering length, aKπ . 79 4.4 Discussion . 82 5 Minimal Extension of the Standard Model Scalar Sector 85 5.1 Introduction . 85 5.2 Scalar Potential . 87 5.3 Phenomenology . 89 5.3.1 Very Light h− .............................. 90 5.3.2 5 GeV < m− < 50 GeV . 92 5.4 Concluding Remarks . 94 6 The Story of O: Positivity constraints in effective field theories 95 6.1 Introduction . 95 6.2 Superluminality and Analyticity . 96 6.3 The Ghost Condensate . 98 6.4 The Story of O .................................. 99 6.5 The Chiral Lagrangian . 103 6.6 Superluminality and Instabilities . 105 6.7 Conclusions . 107 7 Regulator Dependence of the Proposed UV Completion of the Ghost Condensate 109 x 7.1 Introduction . 109 7.2 Computation . 111 7.3 Conclusions . 115 8 The Lee-Wick Standard Model 116 8.1 Introduction . 116 8.2 A Toy Model . 118 8.3 The Hierarchy Problem and Lee-Wick Theory . 122 8.3.1 Gauge Fields . 123 8.3.2 Scalar Matter . 124 8.3.3 Power Counting . 125 8.3.4 One-Loop Pole Mass . 128 8.3.4.1 The normal scalar . 129 8.3.4.2 The LW-scalar . 130 8.3.4.3 The LW-vector . 131 8.4 Lee-Wick Standard Model Lagrangian . 132 8.4.1 The Higgs Sector . 132 8.4.2 Fermion Kinetic Terms . 136 8.4.3 Fermion Yukawa Interactions . 138 8.5 Conclusions . 139 9 Neutrino Masses in the Lee-Wick Standard Model 141 A Explicit Extrapolation Formulae 148 A.1 mπ and fπ for 2-Sea Flavors .
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