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Lecture Notes in Physics 913 Giuliano Panico Andrea Wulzer The Composite Nambu–Goldstone Higgs Lecture Notes in Physics Volume 913 FoundingEditors W.Beiglböck J.Ehlers K.Hepp H.Weidenmüller EditorialBoard M.Bartelmann,Heidelberg,Germany B.-G.Englert,Singapore,Singapore P.Hänggi,Augsburg,Germany M.Hjorth-Jensen,Oslo,Norway R.A.L.Jones,Sheffield,UK M.Lewenstein,Barcelona,Spain H.vonLöhneysen,Karlsruhe,Germany J.-M.Raimond,Paris,France A.Rubio,Donostia,SanSebastian,Spain M.Salmhofer,Heidelberg,Germany S.Theisen,Potsdam,Germany D.Vollhardt,Augsburg,Germany J.D.Wells,AnnArbor,USA G.P.Zank,Huntsville,USA The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new devel- opmentsin physicsresearch and teaching-quicklyand informally,but with a high qualityand the explicitaim to summarizeand communicatecurrentknowledgein anaccessibleway.Bookspublishedinthisseriesareconceivedasbridgingmaterial between advanced graduate textbooks and the forefront of research and to serve threepurposes: (cid:129) to be a compact and modern up-to-date source of reference on a well-defined topic (cid:129) to serve as an accessible introduction to the field to postgraduate students and nonspecialistresearchersfromrelatedareas (cid:129) to be a source of advanced teaching material for specialized seminars, courses andschools Bothmonographsandmulti-authorvolumeswillbeconsideredforpublication. Editedvolumesshould,however,consistofaverylimitednumberofcontributions only.ProceedingswillnotbeconsideredforLNP. VolumespublishedinLNParedisseminatedbothinprintandinelectronicfor- mats,theelectronicarchivebeingavailableatspringerlink.com.Theseriescontent isindexed,abstractedandreferencedbymanyabstractingandinformationservices, bibliographicnetworks,subscriptionagencies,librarynetworks,andconsortia. Proposalsshouldbe sent to a memberof the EditorialBoard, ordirectly to the managingeditoratSpringer: ChristianCaron SpringerHeidelberg PhysicsEditorialDepartmentI Tiergartenstrasse17 69121Heidelberg/Germany [email protected] Moreinformationaboutthisseriesathttp://www.springer.com/series/5304 Giuliano Panico (cid:129) Andrea Wulzer The Composite Nambu—Goldstone Higgs 123 GiulianoPanico AndreaWulzer IFAE DipartimentodiFisicaeAstronomia UniversitatAutònomadeBarcelona UniversitàdiPadova Barcelona,Spain Padova,Italy ISSN0075-8450 ISSN1616-6361 (electronic) LectureNotesinPhysics ISBN978-3-319-22616-3 ISBN978-3-319-22617-0 (eBook) DOI10.1007/978-3-319-22617-0 LibraryofCongressControlNumber:2015951831 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) Preface Half a century after its formulation, the Standard Model (SM) is by now the established theory of Electro-Weak (EW) and Strong interactions, the discovery of the Higgs boson being the most recent of an impressive series of experimental confirmations. Still the SM is not the fundamental theory of Nature, and not just becausenotheorycanbe regardedas“fundamental”innaturalsciences.Concrete reasons to extend the SM are the existence of gravity, for which no complete high-energydescription is available, and other incontrovertibleexperimentalfacts suchasdarkmatter,neutrinomassesandoscillations.Next,thereare a numberof theoreticalissuesbasedon“Naturalness”considerations,amongwhichtheflatness andhomogeneityoftheuniversethatcallsforcosmologicalinflation(whichisalso supportedbyobservations),thestrongCPproblem,and,ofcourse,theNaturalness problem associated with the Higgs boson mass. This latter problem is the main motivationforthecompositeHiggsscenariowhichwewilldescribeinthepresent Notes. Sinceitisnotfundamental,theSMisaneffectivetheory,i.e.apartialdescription of Nature that emerges, under suitable conditions,as an approximationof a more fundamental theory. In this extended theory, the operators in the SM Lagrangian shouldfindtheiroriginasaneffectivedescriptionofthemorefundamentaldynam- icsandtheircoefficients,whicharejustphenomenologicalinputparameterswithin theSM,shouldbecomecalculableprovidingtheexplanationoftheirobservedvalue. UnveilingthefundamentaloriginoftheSMistheultimategoalof“Beyondthe SM” (BSM) physics. Actually in this spirit, the letter “B” of the acronym should betterbereadas“Behind”ratherthan“Beyond”,inthesensethatweareinterested in departuresfromthe SM predictionsonlyto the extentto which they will guide ustowardstheunderstandingofitsfundamentalorigin.Alackofdiscovery,i.e.the exclusion of some hypotheticalalternative model, could be equally or even more helpfulinthisrespect. TheambitiousaimofBSMphysicsshouldnotobscuretheimportantby-products thatemergefromthislineofresearchinthelongpathtowardsitsfinalgoal.First, BSM is of great help in developing a deep understanding of the SM itself, of the surprisingandnon-genericfeaturesthatunderlieits currentphenomenological v vi Preface success, and even to appreciate the true measure of this success. Consider, for instance,theprecisemeasurementsoftheEWbosonspropertiesperformedatLEP in the 1990s. It is impossible to explain why they provided such an important confirmationoftheSMwithoutreferringtothealternativeconstructions,perfectly plausible at that time, which were predicting deviations and were excluded by these measurements. In this respect, BSM physics is of great pedagogical value. Second,BSM is essential to design furtherexperimentaltests of the SM. It offers anassessmentofwhichsectorsofthetheoryarelessaccuratelytested,outliningthe experimentaldirections in which a new physics discovery is more likely to come or,equivalently,thoseinwhichfurthernon-trivialconfirmationsoftheSMcouldbe found.BypurelyworkingwithintheSM,i.e.withoutcomparingitwithalternative models,onecouldonlymeasureitsparameterswithincreasingaccuracyandcheck the statistical compatibility of the overall fit. If the latter program succeeds, we will have established that the SM is one possible viable description of the data, but this will not strengthen our belief that it is really the SM, and not something else, what we are seeing in Nature.Exploringpossible alternativesis essential for thelatterpurpose.Asalternativesonecouldconsideruncontrolledandunmotivated modifications of the SM Feynman rules, which are unfortunatelyoften employed in SM studies, orsensible hypothesesresultingfromBSM speculations.The third by-product of BSM physics is that it stimulates theoretical research in quantum field theory, in a direction that lies in between pure SM phenomenology and abstracttheoreticalspeculations.Beingneithernarrowlydirectedtoasingletheory like the former nor detached from phenomenology like the latter, BSM offers a complementaryviewpoint. Inthisspirit,wewrotethepresentNoteswithathreefoldaim.First,todescribe thecompositeHiggsscenarioinviewofitspossiblerelevanceasthetrueextension of the SM. Namely we will assess, at the best of the present-day theoretical and experimental understanding, how likely it is that a model of this class might be actually realized in Nature. Second, we will identify the most promising possible experimentalmanifestationsofthescenario,outliningrelevantdirectionsforBSM discoveries or SM confirmations. These directions include indirect studies of the Higgs and the top quark couplings and the direct production of new particles with specific features. Third, we will carefully explain the tools that underlie the formulationofthescenarioandthestudyofitsimplications.Someoftheseareold concepts.Someothersarerecentideasormodernrephrasingofoldones.Wethink thatthesewillfindotherapplicationsinthefuture,notonlyinsidebutalsooutside the composite Higgs domain. The material is presented in a pedagogicalfashion. BasicknowledgeofquantumfieldtheoryandoftheSMistheonlyprerequisite. These Notes are organized as follows. The “Introduction” is devoted to the Naturalness problem and to how it is addressed by a composite Higgs. The next three chapters provide a first characterization of the phenomenology in the EW, topandHiggssectorsbyonlyrelyingonsymmetriesandpower-countingestimates. Thisleadstorobustbutsemiquantitativeconclusions,whichshouldbeconfirmedby concretemodels.Aclassofsuchmodels,basedoncollectivebreaking,isintroduced inChap.5.TheyserveasbenchmarksforthedetailedstudyofthecolliderandEW precisionphenomenologypresentedinChaps.6and7,respectively. Preface vii Acknowledgments We thank J.D. Wells (LNP Editor for Particle Physics) for suggesting us to write down a volume on composite Higgs. We learned most of what we know on the subjectbydiscussingandcollaboratingwithC.Grojean,A.Pomarol,andespecially R.Rattazzi.WealsothankB.Bellazzini,J.Serra,G.Dall’AgataandF.Zwirnerfor their comments on the manuscript and R. Contino, L. Merlo, and S. Rigolin for usefuldiscussionsonfour-derivativebosonicoperators. Barcelona,Spain GiulianoPanico Padova,Italy AndreaWulzer Contents 1 Introduction .................................................................. 1 1.1 TheSMIsanEffectiveFieldTheory .................................. 2 1.2 ANaturalElectroweakScale........................................... 6 1.3 DimensionalTransmutation............................................ 11 References..................................................................... 15 2 GoldstoneBosonHiggs...................................................... 17 2.1 VacuumMisalignment.................................................. 17 2.2 TwoSimpleExamples.................................................. 21 2.2.1 TheAbelianCompositeHiggsModel......................... 21 2.2.2 TheMinimalCompositeHiggsModel ........................ 26 2.3 GeneralCCWZConstruction........................................... 32 2.3.1 TheBasicFormalism........................................... 32 2.3.2 GaugeSourcesandLocalInvariance.......................... 39 2.3.3 TwoDerivativeTensorsandResonances...................... 41 2.4 PartialFermionCompositeness ........................................ 45 2.4.1 TheBasicIdea .................................................. 45 2.4.2 HiggsCouplingstoFermions.................................. 53 References..................................................................... 74 3 BeyondtheSigma-Model.................................................... 77 3.1 OneScaleOneCoupling................................................ 79 3.1.1 Large-NPowerCounting....................................... 86 3.2 HigherDerivativeOperators............................................ 91 3.2.1 Orderp4Bosonic ............................................... 92 3.2.2 OrderpFermionic.............................................. 100 3.3 TheCompositeHiggsPotential ........................................ 106 3.3.1 HiggsPotentialCharacterized.................................. 107 3.3.2 HiggsPotentialEstimated...................................... 117 3.3.3 HiggsVEV,MassandTuning.................................. 122 References..................................................................... 132 ix