Bounds on the Effective Theory of Gravity in Models of Particle Physics and Cosmology PDF
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Springer Theses Recognizing Outstanding Ph.D. Research Michael Atkins Bounds on the Effective Theory of Gravity in Models of Particle Physics and Cosmology Springer Theses Recognizing Outstanding Ph.D. Research For furthervolumes: http://www.springer.com/series/8790 Aims and Scope The series ‘‘Springer Theses’’ brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent fieldofresearch.Forgreateraccessibilitytonon-specialists,thepublishedversions includeanextendedintroduction,aswellasaforewordbythestudent’ssupervisor explaining the special relevance of the work for the field. 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Michael Atkins Bounds on the Effective Theory of Gravity in Models of Particle Physics and Cosmology Doctoral Thesis accepted by University of Sussex, Brighton, UK 123 Author Supervisor Dr. Michael Atkins Dr. XavierCalmet Department of Physicsand Astronomy Department of Physicsand Astronomy Universityof Sussex Universityof Sussex Brighton Brighton UK UK ISSN 2190-5053 ISSN 2190-5061 (electronic) ISBN 978-3-319-06366-9 ISBN 978-3-319-06367-6 (eBook) DOI 10.1007/978-3-319-06367-6 Springer ChamHeidelberg New YorkDordrecht London LibraryofCongressControlNumber:2014939057 (cid:2)SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. 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While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Supervisor’s Foreword ItisagreatpleasuretowritethisforewordforMichaelAtkins’thesis.Michaelwas my first Ph.D. student at the University of Sussex and I was very fortunate to be able to recruit a student of this quality having just arrived at Sussex. Michael’s Ph.D. work has been exceptional and fully deserves to bepublished asa Springer Thesis. This research took place at an exciting time for the development of the subject.Ononehand,thediscoveryoftheHiggsbosonattheCERNLargeHadron Collider(LHC)hasprofoundimplicationsforparticlephysics.Ontheotherhand, the data from the Planck satellite are very much compatible with the simplest inflationary models. Michael’s work connects these different fascinating topics: The Higgs boson, besides being responsible for the spontaneous symmetry breaking of the electroweak symmetry of the standard model of particle physics could also be the inflaton and thus responsible for the early universe exponential inflation. The standard model has now passed the last test for its validity with the discovery of the Higgs boson, which resembles very much the Higgs boson predicted by the standard model of particle physics. It could seem that we have now a complete description of particle physics. However, Planck data confirms whatweknewfromothersources,namelythatacrucialingredientisnotdescribed by the standard model: there is no candidate within this model for cold dark matter. SincetheLHCexperimentshavenotprovideduswithanyempiricalclueofthe energy scale at which the standard model might breakdown, it is worth investi- gatingformalrequirementsemergingfromaunificationofthestandardmodeland of general relativity. This was the topic of Michael’s thesis. Using effective field theories techniques combined with unitarity considerations, Michael has investi- gatedwhatcouldbelearnedinthemostconservativemannerfromtheunification of these two models. Inhisthesis,Michaelhasconsideredwhetherthereisaboundonthenumberof fields that can be coupled to general relativity in a consistent manner. This is related to the question of perturbative unitarity. Unitarity is one of the very few theoretical tools available to theoretical physicists to probe the energy scale at whichamodelissupposedtoloseitsvalidity.Perturbativeunitarityconsiderations doleadabound,notonlyonthenumberoffieldscoupledtogravitybutalsoonthe non-minimal coupling of the Higgs boson which could be problematic for Higgs v vi Supervisor’sForeword inflation models. Going beyond the standard techniques based on perturbative unitarity bounds, Michael has studied the resummation technique proposed by JohnDonoghueknownasself-healingwhichcouldexplaintheparadoxicalresults obtainedusingboundsderivedfromtherequirementofperturbativeunitarity.This work has been extended to models with more than four dimensions. One of the highlights of this work has been the derivation of the first experi- mentalboundeveronthenon-minimalcouplingoftheHiggsbosontospace–time curvature which is important as such since this is a new fundamental constant of nature. The experimental bound is also important for models where the Higgs boson plays the role of the inflaton, which is responsible for the very early expansion of our universe. Michael has been an ideal Ph.D. student, being able to grasp ideas fast and develop them into interesting results. I am confident that he could have had a successfulacademiccareer,butMichaelhastoomanyintereststosettledowninto oneestablishedcareerpath.Iwishhimtheverybestonwhateverpathhiseclectic taste will take him. Brighton, March 2014 Dr. Xavier Calmet Abstract Theeffectivetheoryofgravitycoupled tomatterrepresentsafullyconsistentlow energytheoryofquantumgravitycoupledtotheknownparticlesandforcesofthe standardmodel.Inrecentyears,thisframeworkhasbeenextensivelyusedtomake physical predictionsof phenomena inhigh energy physics and cosmology. Inthis thesisweusetheoreticaltoolsandexperimentaldatatoplaceconstraintsonvarious popular models which utilise this framework. We specifically derive unitarity boundsingrandunifiedtheories,modelsoflowscalequantumgravity,modelswith extradimensionsandmodelsofHiggsinflation.Wealsoderiveaboundonthesize oftheHiggsboson’snon-minimalcouplingtogravity.Thisrepresentsanimportant areaofresearchbecauseithelpsustobetterunderstandthetheoriesandmodelsthat manyphysicistsarecurrentlyworkingonandcruciallyitcaninformuswherewe canreliablyusetheeffectivetheoryapproachandwhereitbreaksdown. vii Acknowledgments First and foremost I would like to thank my supervisor Xavier Calmet, for all his help, support and encouragement throughout my Ph.D. I would also like to thank alltheothermembersofthefacultywithwhomIhavesharedvaluablediscussions, inparticularDavidBailin,StephanHuber,SebastianJaeger,MarkHindmarshand Daniel Litim. I am very grateful to have shared my time at Sussex with some wonderful fellow students with whom I have enjoyed learning, researching and socialising with—I wish them all the best of luck in the future. I would also like to thank all my friends and family who have supported me throughout my Ph.D. and who have provided me with welcome breaks and distractions from my work. Particular thanks go to all my housemates over the yearswhohaveputupwithmymoods,listenedtomyrantsandprovidedmewith deliciousmeals.Finally,IwanttosayahugethankyoutomypartnerIsy,whohas supported and encouraged me throughout and has inspired me to complete this project. ix Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Effective Theory of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Unitarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Unitarity of a Superposition of States . . . . . . . . . . . . . . 8 1.2.2 Example: Unitarity of WW Scattering . . . . . . . . . . . . . . 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Unitarity of Gravity Coupled to Models of Particle Physics. . . . . . 15 2.1 Unitarity of Linearised General Relativity . . . . . . . . . . . . . . . . 15 2.1.1 j ¼ 2 Partial Wave Amplitude . . . . . . . . . . . . . . . . . . . 16 2.1.2 j ¼ 0 Partial Wave Amplitude . . . . . . . . . . . . . . . . . . . 17 2.2 Unitarity of Models of Particle Physics . . . . . . . . . . . . . . . . . . 17 2.3 Running of the Planck Mass and Renormalisation Group Improved Unitarity Bound . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1 Model with Large Number of Fields. . . . . . . . . . . . . . . 22 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Unitarity of Models with Extra Dimensions. . . . . . . . . . . . . . . . . . 25 3.1 Extra Dimensions and Kaluza-Klein Modes . . . . . . . . . . . . . . . 25 3.1.1 Extra Dimensional Models as Effective Theories with a Low Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.2 Kaluza-Klein Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.3 Partial Wave Amplitude for KK Graviton Exchange. . . . 31 3.1.4 Width of KK Gravitons. . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Unitarity of KK Graviton Resonances . . . . . . . . . . . . . . . . . . . 32 3.2.1 Sum of Breit-Wigner Resonances . . . . . . . . . . . . . . . . . 32 3.2.2 Beyond the Breit-Wigner Approximation. . . . . . . . . . . . 34 3.3 Unitarity in the ADD Model. . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.1 Introduction to the ADD Model . . . . . . . . . . . . . . . . . . 36 3.3.2 Unitarity in the ADD Model . . . . . . . . . . . . . . . . . . . . 38 3.3.3 KK Sum and Unitarity in the Zero Width Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.4 KK Sum and Unitarity Including Breit-Wigner Width. . . 42 3.3.5 Summary of the Unitarity Bounds. . . . . . . . . . . . . . . . . 44 xi
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