Table Of ContentLecture Notes in Physics 913
Giuliano Panico
Andrea Wulzer
The Composite
Nambu–Goldstone
Higgs
Lecture Notes in Physics
Volume 913
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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
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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
