Universita degli Studi di Milano-Bi o a
F
a olt
a di S ienze Matemati he, Fisi he e Naturali
|
E ole Polyte hnique, Palaiseau
The CMS Ele tromagneti Calorimeter
for the Higgs Boson Sear h
H ! ZZ
()
! 4e at the LHC
Coordinatore: prof. Claudio Destri
Tutori: dr. Yves Sirois
prof. Tommaso Tabarelli de Fatis
Tesi diDottorato di
Federi o Ferri
Matri ola R00281
S uoladiDottorato di S ienze
for the Higgs Boson Sear h
H ! ZZ
()
! 4e at the LHC
Coordinatore: prof. Claudio Destri
Tutori: dr. Yves Sirois
prof. Tommaso Tabarelli de Fatis
Tesi sostenuta il giorno 10 gennaio 2006 presso l'Universita degli Studi di
Milano-Bi o a difronte aduna ommissione omposta da:
dr. Yves Sirois
prof. TommasoTabarelli de Fatis
dr. PhilippeBlo h (referee)
prof. Ezio Meni hetti (referee)
Tesi diDottorato di
Federi o Ferri
Introdu tion 1
1 Standard Model Physi s (and Beyond) 3
1.1 GeneralCon epts . . . 3
1.1.1 Lo al Invarian e . . . 4
1.1.2 Spontaneouslybroken symmetries . . . 7
1.2 The SU(2) L U(1) Y Model . . . 10
1.2.1 The Gauge Se tor . . . 11
1.2.2 Fermions . . . 13
1.2.3 AnomalyCan ellation . . . 15
1.2.4 The HiggsBoson . . . 17
1.2.5 StandardModelHiggsProdu tioninp p ollisions . 23 1.3 GoingBeyond . . . 28
1.3.1 Supersymmetry . . . 31
1.3.2 Extra-dimensions . . . 34
Bibliography . . . 38
2 The CMS Dete tor at LHC 39 2.1 The LargeHadron Collider . . . 39
2.2 The Compa t Muon Solenoid . . . 42
2.2.1 Magnet . . . 45
2.2.2 Tra ker . . . 45
2.2.3 Ele tromagneti Calorimetry . . . 49
2.2.4 Hadron Calorimetry . . . 64
2.2.5 Muons System . . . 66
2.2.6 The Triggerand DataA quisition System . . . 68
3 ECAL: Test Beam Studies 71
3.1 ExperimentalSetup . . . 73
3.2 Test BeamResults . . . 75
3.3 Studieson theAmplitudeRe onstru tion . . . 77
3.3.1 NoiseMonitoring . . . 77
3.3.2 Amplitude Re onstru tionintheTime Domain . . . . 78
3.3.3 Amplitude Re onstru tioninCaseofSaturated Signals 81 3.3.4 Amplitude Re onstru tionintheFrequen yDomain . 86 3.4 Con lusions . . . 90
Bibliography . . . 91
4 Ele tron Re onstru tion in CMS 93 4.1 Ele tron Propagationtowards theCalorimeter . . . 94
4.2 Ele tron Candidates . . . 101
4.2.1 Calorimetri Re onstru tion. . . 101
4.2.2 Ele tron GSFTra kRe onstru tion . . . 104
4.3 EnergyMeasurementsand Ele tron Classi ation . . . 106
4.4 EnergyS ale Corre tion . . . 112
4.5 Energy-MomentumCombination . . . 118 4.6 E e tson theH !ZZ () !4ephysi s . . . 123 4.7 Con lusions . . . 125 Bibliography . . . 128 5 The H !ZZ () !4e hannel 129 5.1 Signaland Ba kgroundDe nition. . . 130
5.2 Signaland Ba kground: Generationand Simulation . . . 132
5.2.1 Signal . . . 133
5.2.2 Ba kgrounds . . . 135
5.3 EventsTriggerand Presele tion . . . 139
5.4 SignalRe onstru tionand Ba kgroundReje tion . . . 140
5.4.1 O -line re onstru tion . . . 141
5.4.2 InternalBremsstrahlung . . . 142
5.4.3 Vertexing . . . 145
5.4.4 Isolationand Ele tron Identi ation . . . 147
5.4.5 Kinemati s ofthe re onstru ted events. . . 149
5.5 Signi an eDe nition . . . 151
5.6 SequentialSele tionAnalysis . . . 152
5.7.1 Neural Network Stru ture . . . 155
5.7.2 Multi-LayerPer eptrons . . . 158
5.7.3 Neural Networks andProbability . . . 158
5.7.4 Neural Network analysis . . . 161
5.8 Con lusions . . . 168 Bibliography . . . 170 Con lusions 171 A knowledgements 175 Resume i Riassunto v
The Standard Model of ele troweak intera tions shows an in rediblygood
agreementbetweentheoryandexperiments. However,itdoesnotyetgivean
answerto a numberof fundamental questions, namely themost important
ofall: theoriginofparti lemass. Oneoftheme hanismsproposedtojustify
massive parti les (andsoto explainthebreakdownof theSU(2)
L
U(1)
Y
symmetry group,upon whi h theStandard Model theoryis built) is based
on as alar eldwhi hwillmanifestitself throughamassive s alarparti le
alledHiggsboson,whi hremainstobefound. Extensivedire tandindire t
sear hesforthisparti leshavebeen arriedoutattheLEP2experimentand
have xeda lowerbound(m
H >114:4 GeV = 2 at 95% C.L.) and anupper bound (m H < 237 GeV = 2
) to the mass of the Higgs boson, indi ating a
valueof114 GeV = 2
at 95%C.L. asthebest t to theexperimentalvalues.
The work presented inthisthesishas been arried outinthe ontest of
the Compa t Muon Solenoid (CMS) ollaboration. CMS is one of thetwo
generalpurposeexperiments(in onjun tionwith ATLAS)whi harebeing
installed at the Large Hadron Collider (LHC) at CERN, along with two
experiments dedi ated to thephysi s of the b quark (LHCb) and to heavy
ions (ALICE). LHC is a proton-proton ollider with a nominal energy of
14TeV= 2
inthe enterof massand anominalluminosityof10 34
m 2
s 1
and willallow tosear h fortheHiggsboson inthefullrange oftheallowed
masses. The golden hannel for the dete tion of what is favoured to be a
\light" Higgs boson is via its de ay into two photons, whi h will provide
forthesignala leanexperimentalsignatureoverthehadroni ba kground.
However,ade ay hannelthatisremarkablyimportantnotonlyforthe
pos-sibilitytodete tthisparti lebutalsoforthedeterminationofitsproperties
(e.g. spin, CP, ouplings to gauge fermions et .) is the one in whi h the
Higgsbosonde aysintoapairofele tronsandpositronsviaanintermediate
state of two Z bosons (H !ZZ ()
! 4e). In this ontext, the
ex ellent energyand angularresolutions.
Thisthesishasfo used onthe hara terization oftheCMSele tromagneti
alorimeter,bothwithtestbeamdataandwithsimulation,andonthestudy
of the expe ted performan e of the CMS dete tor for the dis overy of the
Higgsboson inthe hannelH !ZZ ()
!4e.
Afteratheoreti aloverviewoftheStandardModel( hapter1) theLHC
olliderand theCMSdete tor willbepresented( hapter 2).
Chapter3illustratestheresultsoftestbeamstudiesdevotedtotheanalysis
of theele troni noiseand of the signal amplitude re onstru tion from the
readoutofthe ele tromagneti alorimeter.
The detailed simulation of the CMS dete tor allows for the study of the
ele tronre onstru tion insideCMS,whi hhas tofa ethe strongsolenoidal
magneti eld (4 T) inside CMS and the tra ker material in front of the
alorimeter. Theresultsare presentedin hapter 4.
Chapter5 shows the analysisof theexpe ted CMSperforman eforthe
se-quentialdete tionte hniqueoftheHiggsbosoninthe hannelH!ZZ ()
!
4e,usingbothastandardsele tion anda NeuralNetwork approa hto
Standard Model Physi s
(and Beyond)
The theoreti al pathtowards a uni edtheory of weak and ele tromagneti
intera tions beganin1933 when Fermiproposedhis theory ofthe de ay.
It took more than four de ades to rea h what is now alled the Standard
Modelofele troweakintera tionswhi h,togetherwiththeQuantum
Cromo-Dynami s,providesat present themosta uratedes riptionofthree ofthe
known intera tionof Nature withelementaryparti lesand elds.
Theaiminthis hapterisnottogivea ompleteandexhaustive
des rip-tion of the theory, but rather to illustrate its basi prin iples, their
onse-quen esand thefundamentalquestionsthat arestillnotfully answered.
1.1 General Con epts
TheFeynman{Gell-ManLagrangiandes ribingweakV Aintera tion
pro- esses at low energy is manifestlynon-renormalizable, sin e it ontains
op-eratorswithmassdimensionof6(whileane essary onditionforthe
renor-malizability is the presen e of operators with mass dimension less than or
equalto4). ForexampletheLagrangiandes ribingthenu leon de ayand
themuon de ay isgiven by
L= G p 2 p (1 a 5 )ne (1 5 ) e G p (1 5 )e (1 5 ) e ; (1.1)
where
,
5
are Dira matri es, a ' 1:23 is a onstant determined
ex-perimentally. Remarkably,the oupling onstantsforthetwopro esses,G
and G
, are equal and usually denoted by G
F
, alled Fermi onstant and
roughlyestimated by((~ ) 3
=300 GeV ) 2
.
Apossibleremedy to thenon-renormalizabilityistheintrodu tionof a
me-diator for the point-like weak intera tion providing a term in the matrix
elementsto xthequadrati divergen esinhigherorderperturbative
al u-lations. Thismediatormustbeamassive ve tor(toexplaintheshortrange
oftheweakfor e) and exist innature intwo harged states (toexplainthe
harge- hanging manifestations of the weak intera tions). For a omplete
an ellation of all the divergen es at all orders, a neutral ve tor boson is
alsorequired.
To satisfy all the requirements in a oherent and onsistent way, three of
what Weinberg alls \good ideas"('t Hooft, 2005) are needed: the quark
model, the idea of gauge (or lo al) symmetry and that of spontaneously
broken symmetry. In whatfollows,theattentionwillbe fo usedonthelast
two aspe ts.
1.1.1 Lo al Invarian e
Sin e its rst formulation in Maxwells equations whi h unify ele tri and
magneti intera tions (1864), the on ept of gauge invarian e has held an
in reasinglyimportantroleinthedes riptionofNatureanditsfundamental
intera tions. Thefreedomof hoosingmanypotentialsto des ribethesame
physi s an in fa t be reformulated in terms of a gauge symmetry in the
Lagrangian. Su ha reformulation leadsto onserved harges (viaNoethers
theorem) and to other important onsequen es su h as theintrodu tionof
new eldsand intera tions into thetheory.
In ele trodynami s, forexample, requiringthefermion free-parti le
La-grangian L free = (i m) (1.2)
tobeinvariant underalo alU(1)symmetry !e iq (x)
suggests a
redef-initionof thederivative
(so alled ovariant derivative)as
D +iqA (x); (1.3) where A
is a new ve tor gauge eld. Provided that the gauge eld A
transformsas
theobje t D behavesinfa t asthe eld under aU(1) phaserotation:
D !e iq (x)
D : (1.5)
The newinvariant Lagrangian Lbe omesthen
L=L free q A ; (1.6)
where the lastterm ouples the eld A
to : a new eld (identi edwith
thephoton) hasappearedin thetheory.
To obtain the omplete QED Lagrangian it suÆ es to introdu e a kineti
termforthe eldA
,thatisalo allyinvarianttermdependingonthe eld
and its derivativesbut noton . It an be shown(see for example(Peskin
and S hroeder, 1995)) that out of the four possible ombinations onlyone
ful lsthene essaryrequirementsofrenormalizabilityofthetheoryandgood
behaviourunderdis retesymmetries:
F = A A : (1.7)
It willbeusefulin thefollowingto noti ethatF
an be rewrittenasthe
ommutator betweentwo ovariant derivatives:
[D ;D ℄=[ ; ℄+iq([ A ℄ [ A ℄) q 2 [A ;A ℄ =iq( A A ); (1.8) that is [D ;D ℄=iqF : (1.9)
The ompleteQEDLagrangian is then
L QED =L free q A 1 4 F F : (1.10)
ItmustbestressedthatA
isamassless eld: amasstermwouldbeinfa t
proportionalto A
A
,thusviolating thegauge invarian e:
A A !(A )(A )6=A A : (1.11)
Yang and Mills proved that when when the symmetry group is
non-Abelian, the onstru tion of the theory follows the same prin iples (Yang
and Mills, 1954). The physi al onsequen es are howeverdi erent and are
onsideredthe invarian eunder a lo altransformation of the SU(2) group
(whi hintheoriginalYang-Millspaperwassupposedtobetheisotopi spin
for a doublets of Dira elds, the proton and the neutron). If the eld
transformsas (x)!G(x) (x)e i i (x) i 2 (x); (1.12) where i
are the group generators, then the ovariant derivative takes the
form D igB ; B b i i 2 ; (1.13) b i
being three ve tor elds, one for ea h generator of the gaugesymmetry
group.
To assurethelo alinvarian e, B
musttransform a ordingto
B (x)!G(x) B (x)+ i g G y (x): (1.14)
FollowingbyanalogytheAbelian ase, thekineti termforb i
anbefound
and the Lagrangian ompleted. Indeed, onsidering a eld-strength tensor
builtupwith the ommutator betweentwo ovariant derivativesone nds
[D ;D ℄= igF i i 2 ; (1.15) with F i = b i b i ig b i ;b j : (1.16)
Using the Pauli's matrix identity to simplify the kineti term (F
)
2
and
expandingthe ovariant derivative,the Yang-Mills Lagrangian be omes
L YM =L free g 2 b i i 1 2 trF F : (1.17)
As for theAbelian ase, the symmetry ompletely di tates the form of
theintera tion, therebyleadingto a ri hers enario.
Inadditiontothegaugebosonpropagator andto the ouplingofthegauge
eldsto thefermions, thetheory hasthree- and four-gauge-bosons verti es
( g. 1.1),as a onsequen e of the non linear term inF
. These new
self-intera tions for the (massless) gauge bosons exist even without fermions,
while Abelian gauge theories without fermioni elds are free (i.e.
non-intera ting)theories.
Theprin ipleoflo alinvarian eisa onsistentwayto havemasslessve tor
bosons andidatesintothetheory: inorder to be usedtodes ribetheweak
intera tions,however, they must a quire a mass, hen e requiringa
sponta-neous breakingof the symmetry. The me hanism by whi h thissymmetry
U(1)
(QED)
!
Photon Propagator
SU(2) !
Gauge FieldPropagator
3 GaugeBoson Vertex
4 GaugeBoson Vertex
Figure 1.1: Examples of ouplings pres ribed by an Abelian gauge
symmetry(U(1))andanon-Abelianone(SU(2)).
1.1.2 Spontaneously broken symmetries
If atheory is des ribedby aLagrangian invariant under a given symmetry
butitsphysi alva uumisnot,thenthesymmetryissaidtobespontaneously
broken.
ThereareinNatureseveralo urren esofspontaneoussymmetry
break-ing. Aferromagneti systemis a anoni al example. Above the Curie
tem-peraturethe magneti dipole moments show a rotational SO(3) symmetry
with all the dipoles randomly oriented in a three dimensional spa e
(para-magneti phase). The introdu tionof an external magneti eld expli itly
breaks this SO(3) symmetry down to SO(2) by for ing the spins to be
alignedalongaprivilegeddire tion(paralleltothe elditself). Turningthe
eldo restores theoriginalsymmetry.
Thesystembehavesdi erentlywhenitstemperatureisbelowtheCurie
tem-perature. The lowest energy on guration orresponds to a parallel
align-ment of the magneti dipoles: there is a non-zero magnetization along a
preferred dire tion even in absen e of external elds(i.e. of expli it terms
in the Lagrangian breaking its symmetry). The SO(3) symmetry is then
spontaneously broken down to SO(2) by the system's ground state, whi h
\ hooses"one parti ular on gurationamong in nitepossibilities(the
(a) (b)
Figure1.2: FormforthepotentialV(
)ofequation1.19depending
onthesignof 2
: negative(a)andpositive(b).
hasbeen hosen,it an notbe hangedunlessan amount ofenergyis
intro-du edintothesystemforea hof thedipoles,inorderto reorientthem ina
di erentdire tion.
Thesimplestexampleofspontaneoussymmetrybreakingin eldtheory
is realized with dis rete symmetries (namely parity). It shall be however
dis ussed the slightly more advan ed example of a omplex s alar theory
invariantunder aglobal U(1) symmetry.
Thestarting Lagrangianis oftheform
L= ( ) V( ); (1.18)
wherethee e tive potentialV( )is hosen as V( )= 2 + 2 ( ) 2 ; >0: (1.19)
Two ases, dependingonthe signof 2
,are onsidered( g. 1.2).
If 2
<0,thesymmetryisexa t andthere existsa uniqueva uumstate for
thetheory,at h i=0.
On the other hand, if 2
> 0 (whi h also means that an no longer be
interpretedasamassforthe eld )theva uumstateisin nitelydegenerate
forallthe on gurationssatisfying
j j= 2 1=2 v: (1.20)
Choosing one of them spontaneously breaks the U(1) symmetry. The
theoriginalsymmetry arepreserved.
Byexpli itly hoosinga va uum on gurationwithonlya real part
h i
0
=v; (1.21)
itis possibleto expand aboutthisgroundstate byde ning
(x)=v+ 1 p 2 ( 1 (x)+i 2 (x)); (1.22) with 1 and 2
reals alar elds.
The potentialthenbe omes
V( )= 4 2 + 1 2 2 2 1 +O( 3 i ) (1.23)
and,omittingthe onstantterms,theLagrangian anthereforebeexpressed
as L= 1 2 ( )( )+ 1 2 ( )( )+ 2 2 1 : (1.24) The eld 1
has a quireda massm
1 = p 2 while 2 ismassless.
It is possible to get the avour of this e e t by looking at theform of the
potential( g. 1.2): the mass termfor
1
is a onsequen e of the restoring
for e against radial os illations, while the symmetry under U(1) rotations
that the Lagrangian still exhibits means that no restoring for es against
angularos illationsexist,therebyallowinga massless
2 eld.
The appearan e of massless s alars when a global ontinuous
symme-try is spontaneously broken is a onsequen e of a general theorem known
as Goldstone's theorem. The number of new massless parti les (so alled
Goldstone bosons) inthetheory is relatedto thedegrees of freedomof the
symmetry group: a rotation in N dimensionsis des ribed by N(N 1)=2
parameters, ea h of them orresponding to a ontinuous symmetry.
Af-ter a spontaneous breakdown of the O(N) symmetry to an O(N 1),
there are still (N 1)(N 2)=2 unbroken symmetries. The number of
masslessGoldstone bosons orrespondingto thebroken symmetries isthen
N(N 1)=2 (N 1)(N 2)=2=N 1. It istrivialto verify thatinthe
previousexamplethisleadsto exa tlyone Goldstone boson.
One an nowaskswhat happensrequiringU(1) to be alo alsymmetry
inthepreviousexample. Thederivationofthepotential(1.19) isstillvalid,
di erent result be ause of the di erent kineti term due to the ovariant
derivative 1
. One infa tobtainsthat
(D )(D ) = 1 2 ( 1 ) 2 + 1 2 ( 2 ) 2 + p 2qvA 2 +q 2 v 2 A A +O((A ; 1 ; 2 ) 3 ): (1.25)
The lasttermis simplya massterm forthegauge boson A
whi h is
pro-portionalto theva uum expe tation valuev of the eld (m= p
2 qv).
This(mira ulous!) interplaybetweenlo alinvarian eandspontaneous
sym-metrybreaking, rstnoti edbyHiggs(Higgs,1964), allowsto re on ilethe
problemsasso iatedwiththedes riptionoftheweakintera tions. Theneed
of massive gauge bosons is satis ed byrequiring the theory to ful ll (very
elegant) lo al symmetry prin iples at the pri e of introdu ing new elds
subje ted to appropriatee e tive potentials (whi his lesselegant,indeed).
In a ertain way themassless gauge bosons \eat" theGoldstone s alarsto
get one more degree of freedom, the transversely polarizedstate properof
masslessparti les. 1.2 The SU(2) L U(1) Y Model
The Standard Model of ele troweak intera tions uni es weak and
ele tro-magneti intera tions. Itisagaugetheorywithexa tsymmetrieswhi hare
spontaneouslybroken. ProposedindependentlybyWeinberg, Glashowand
Salam ((Weinberg, 1967), (Glashow, 1961)), the Standard Model was
for-mulatedonthebasisofthelargestpossiblesymmetrygroupasso iatingthe
leptons(SU(2)U(1)) asinferred byexperimentalresultsat that time. It
ledtotheuni ationofweakand ele tromagneti intera tions,respe tively
theSU(2)and U(1)sub-groups.
Summarized below are the main experimental fa ts explained by the
theory,asoutlinedby(Renton, 1990):
leptons and quarksarehalf-spinparti les;
when weak harged urrent intera tions o ur (mediated by W
ex- hange) leptonsand quarks ome inweak isospindoublets;
harged urrent intera tions appear to be purely left-handed (V A
is a hiral theory) and to violateC and P maximally,while(almost)
onserving CP;
1
L Y
leptonsand quarks ome inthree generations;
harged leptonandquarkmasses substantiallyin reasefromone
gen-erationto thenext, whileneutrinosarevery light parti les;
inadditionto harged urrents,therearetwokindsofneutral urrents:
one oupling toall quarksandleptons(mediatedbyZ ex hange)and
theother ouplingonlytoele tromagneti hargedparti les(mediated
by ex hange);
short-rangeweak intera tionsaremediatedbythree massive parti les
(W
; Z,withmassmO(100GeV = 2
))whilein nite-range
ele tro-magneti intera tions aremediatedby one masslessboson( ).
Notallofthesefa tswereknownwhenthe rstpapersbyWeinberg,Glashow
and Salamwere published. The presen eof a weak neutral urrent,for
ex-ample, wasone of themostsu essfulpredi tion ofthe theory.
1.2.1 The Gauge Se tor
Imposingthelo alinvarian eofthetheory undera SU(2)U(1)
transfor-mationgivesfour(massless)gauge elds, three orrespondingto theSU(2)
symmetry(W i
, i=1;2;3) and one to theU(1) (B
). They appearinthe
de nitionofthe ovariant derivative
D = igW i i ig 0 Y 2 B ; (1.26) where g and g 0
are the oupling onstants of the SU(2) and U(1) groups
respe tively 2 , i i
=2 are the generator of SU(2) and Y is a quantum
numberusually alled weak hyper harge.
Followingtheformalismoutlinedinthepreviousse tion,as alarHiggs eld
is introdu ed into the theory in order to give a mass to the weak gauge
elds. TheU(1)symmetry,whi h orrespondstothemasslessphoton,must
however notbe broken.
The simplest hoi e for is adoubletrepresentationof SU(2):
0 + ! : (1.27) 2
Givingto a harge 1=2 under U(1), its omplete SU(2)U(1) transfor-mationbe omes ! e i i i e i =2 : (1.28)
If a quiresa va uum expe tation valueof theform
h i= 1 p 2 0 v ! ; (1.29)
where and arerealnumbers. thenh i isnotinvariantunder anyof the
originalfour generators. It is invariant,however, under the transformation
orresponding to 1 = 2 = 0 and 3
= , i.e. the linear ombination
Q=( 3
+Y=2) orresponding totheele tri harge. Three massive bosons
a quirethereforeamassviatheGoldstones alarsasso iatedwiththethree
broken symmetries, butthephoton remainsmassless.
By evaluating the kineti term for (D
)
D
, it is possible to gure out
fromthe masstermstheW
bosons asthelinear ombination
W = 1 p 2 W 1 iW 2 ; (1.30)
andtheneutralve torbosonZ andtheele tromagneti ve tor potentialA
as Z = 1 p g 2 +g 02 gW 3 g 0 B A = 1 p g 2 +g 02 g 0 W 3 +gB : (1.31)
Themasses fortheweakgauge bosons are
m W =g v 2 ; m Z = p g 2 +g 02 v 2 (1.32)
By de ningthe Weinberg angle as themixing angle between (W 3
;B) that
gives (Z ;A),the followingrelationsare obtained:
Z A ! = os# W sin# W sin# W os# W ! W 3 B ! ; (1.33) with os# W = g p g 2 +g 02 ; sin# W = g 0 p g 2 +g 02 (1.34) Rewriting D
as a fun tion of the gauge bosons mass eigenstates would
L Y
leadtotheimportantrelationbetweentheele tri hargeeandthe oupling
onstantsg,g 0 : e=gsin# W : (1.35)
Moreover themasses of Z and W are notindependent:
m W =m Z os# W : (1.36)
Three free parameters of the gauge se tor hen e exist: the two oupling
onstants g and g 0
and the va uum expe tation value v of the Higgs eld.
Theseparameters are usuallyexpressedusingtheele tromagneti oupling
onstant
e.m.
,the Fermi onstant G
F
and themassof theZ boson, whi h
are measuredwith a very high a ura y (Eidelman et al., 2004). The
ou-pling onstant e.m. = gg 0 4 p g 2 +g 02 = 1 137:03599911(46) (1.37)
isdeterminedfromtheanomalousmagneti momentofele tronsandpositrons,
G F (~ ) 3 = 1 p 2v 2 =1:16637(1)10 5 GeV 2 (1.38)
from themuon de ay,and
m Z = v 2 p g 2 +g 02 =(91:18760:0021) GeV = 2 (1.39)
from theZ-lineshapes anat LEP1.
1.2.2 Fermions
Ifjustone familyofquarksandleptonsis onsidered(e.g. (e;
e
),(u;d)) 3
in
thedes riptionoftheele troweakpro esses,therepresentationsofSU(2)
L
U(1)
Y
assignedto thefermionsmustpreserve the hiralnature oftheweak
harged urrent intera tions and the oupling of ele tromagnetism to
left-and right-handedfermions. Theserequirementsleadto
3
Thegeneralizationtotheothertwofamiliesoffermions(;
),( ;
),( ;s),(b;t)is
L L = e L e L ! =P L e e ! (2; 1) e R =P R e (1; 2) Q L = u L d 0 L ! =P L u d 0 ! 2; 1 3 u R =P R u (1; 4 3 ) d 0 R =P R d 0 (1; 2 3 ) (1.40) whereP L = 1 5 2 andP R = 1+ 5 2
aretheproje tionoperatorsonorthogonal
eli ity states, and the last olumn represents the quantum numbers
orre-spondingto therepresentations of SU(2)
L
U(1)
Y .
FromtheGell-Mann{NishijimarelationQ=
3
+Y=2 it anbenoti edthat
an eventual right-handed neutrino
R
, singlet of the gauge group, would
have vanishing both harge and weak hyper harge. This neutrino would
therefore not intera t ele troweakly and onlyindire t measurements ould
proofits existen e.
Inthe expressiongiven above, downquarks ome witha \ 0
": quark mass
eigenstates,infa t, donot oin ide withweak intera tion eigenstates. The
latterare a linear ombination ofthe masseigenstates throughtheunitary
mixingmatrix 0 B d 0 s 0 b 0 1 C A = 0 B V ud V us V ub V d V s V b V td V ts V tb 1 C A 0 B d s b 1 C A ; (1.41)
whi his generallyreferred to astheCabibbo-Kobayashi-Maskawa matrix.
Itishasbeenshownexperimentallythatfermionsaremassiveparti les 4
.
However, a masstermof thegeneri form
m = m( L R + R L ) (1.42)
would break the gauge invarian e in the Lagrangian (
L and
R
belong
to di erent representation of SU(2) and have di erent U(1) harges) and
is therefore not allowed. Notwithstanding thisunpleasant feature of hiral
Lagrangians, it ispossibleto buildamass term withthe helpof the Higgs
eld. The masstermforthe leptonsis
L Yukawa = X i=e;; i L i L e i R +h. . ; (1.43) 4
Re entresultsfromneutrinoos illationexperimentsseemtoindi atenon-zeromasses
L Y
where
i
are new dimensionless parameters of the theory. Repla ing the
eld byits expe tationvalue yields
L Yukawa = X i=e;; i v p 2 e i L e i R +h. . : (1.44)
It follows that themass forthe leptoniis proportionalto its Yukawa
ou-plingto theHiggs:
m i = 1 p 2 i v: (1.45)
Pro eedinginthe same way forthequarkmass terms,one obtains
L Yukawa = X i=d;s;b i v p 2 d i L d i R +h. . + X i=u; ;t i v p 2 u i L u i R +h. . ; (1.46)
and forthemassof thequarki
m i = 1 p 2 i v: (1.47)
An additional ompli ation for quarks, whi h is not made expli it here, is
that the Yukawa ouplings involve mass eigenstates. To have the
orre-sponding expression in terms of the weak eigenstates base, the
Cabibbo-Kobayashi-Maskawa matrix elements (1.41) must be properly introdu ed,
inorder to passfrom themasseigenstates to the weakintera tionones.
1.2.3 Anomaly Can ellation
Even ifa theory is renormalizable(and non-Abeliangauge theories are, as
demonstrated by 't Hooft ('t Hooft, 1976)) there an be urrents whose
onservation (throughgauge invarian e+Noether's theorem) holdsat tree
levelbutisviolatedin rstloopdiagrams. Anexamplesofsu ha urrentis
givenin gure1.3: allthedivergen es omingfromthese loopsmust an el
outto give a nitetheoryat all perturbativeorders.
It an be shownthat
A ab /tr h 5 a n b ; oi =A ab + A ab ; (1.48)
wherethe tra e istaken overall thefermion familiesand inthelast
equiv-alen e thefa tor
5
hasbeenexpli itly set equal to 1 for left- and
Figure 1.3: Example of a triangle anomaly. In this ase the
axial-ve tor urrentisrepresented.
For a theory whi h equally ouples left- and right-handed fermions, the
an ellation omes automati ally, sin eA ab
+ =A
ab
. Indeed, the Standard
Modelisa hiraltheory,andthisautomati an ellationdoesnottakepla e.
It an be shown, however, that the only anomaly inthe theory is
propor-tionalto tr h f a ; b gY i = 1 2 Æ ab X fermion doublets Y: (1.49)
Using the Gell-Mann{Nishijima relation, the ondition for the absen e of
anomalies an beexpressed asafun tion ofele tri harge:
Q=Q R Q L = X right-handed doublets Q X left-handed doublets Q (1.50)
Considering a single fermion generation in the Standard Model, one
left-handedleptondoublet orrespondsto one left-handedquarkdoublet,while
right-handeddoublets areabsent. This translatesin
Q= Q L =1+ 2 3 1 3 = 1 3 ; (1.51)
whi hmeansanomalieshavebeenintrodu edintothetheory. Bysupposing,
however that quarks ome with an additional three- avoured harge with
respe ttotheleptons,assuggestedbythestrongintera tiontheory,afa tor
3, whi h orresponds to the three di erent possible \strong harges" the
L Y thenbe omes Q= Q L =1+3 2 3 1 3 =0; (1.52)
sotheanomalies an el(withinea h singlefermion generation).
Given the Standard Model of ele troweak intera tion, an indi ation for a
des riptionofstrongintera tionshasbeenfoundasa onditionforits
renor-malizability.
1.2.4 The Higgs Boson
It has been shown in the previous se tions how the Higgs eld give mass
to gauge bosons and fermions, but this is not the only onsequen e of the
introdu tion of a s alar eld into the theory. As demonstrated in se tion
1.1.2 a newmassives alar parti leisexpe ted to appear.
Toseehowthis anhappenwithintheStandardModel,one anparametrize
the expansion of the Higgs eld about its ground state in the following
way(so alledunitarygauge):
(x)= 1 p 2 U(x) 0 v+H(x) ! ; (1.53)
whereU(x)isageneraltransformationofSU(2)toprodu ethemostgeneral
double- omponent spinor and H(x) is a real eld su h that hh(x)i = 0.
U(x) analwaysbeeliminatedfrom theLagrangianbyagauge
transforma-tion soitwillnotbe onsideredinthe followingdis ussion.
Oneseeks to writeexpli itlyin termof theexpansion1.53 all thepie es of
theStandardModel Lagrangian ontaining theHiggs eld . The e e tive
Lagrangian for and theYukawa ouplingsto thefermions.
The usualform of theLagrangian for is
L H =(D ) y (D ) V( y )= =(D ) y (D )+ 2 y ( y ) 2 ; (1.54)
wherethepotentialrea hesa minimumat
v 2 1=2 : (1.55)
Plugging in thepotentialyields
L V = 2 H 2 vH 3 1 4 H 4 = = 1 m 2 H H 2 r m H H 3 1 H 4 ; (1.56)
Figure 1.4: Feynman diagrams and rules for the intera tion of the
L Y
(a)
(b)
Figure 1.5: (a)Bran hingratioforH de ayfor avarietyof hannels
as afun tion of the Higgs boson mass. (b) Total de ay width of the
The eld H is thereforea massive s alar,witha massgiven by m H = p 2v; (1.57)
andis alledHiggsboson.
Thekineti term inL
H
written intermsof gives
L K = 1 2 ( H) 2 + m 2 W W + W + 1 2 m 2 Z Z Z 1+ H v 2 : (1.58)
Finally,the Yukawa Lagrangian produ es forea h fermion f a term of the
form L f = m f ff 1+ H v (1.59)
Anillustrationof theHiggsboson ouplingstothegaugebosonsandto the
fermions (and the ubi and quarti self-intera tion ouplings) is given in
gure 1.4. As the asso iated Feynman rules show, the ouplings are
om-pletelydeterminedbythemasses of theparti lesinvolved andbytheweak
intera tion oupling onstants. In parti ular, ouplings to W
and Z are
proportionalto mass of the gauge bosons squared, whilefor fermions
ou-plingsare dire tlyproportionalto thefermions' mass. The oupling to the
gluonsandtothephotonsviafermioni loopisalsointeresting. Indeed,due
toits mass,thetquarkgivesthedominant ontribution. The rst oupling
relationisparti ularlyimportant forthe Higgsboson produ tionpro esses
at hadron olliders. Onthe other hand, these ond oupling relations
pro-videone ofthe leanest signatures forexperimental dete tion.
Adetailedviewofthebran hingratiosforthedi erentde ays oftheHiggs
bosonisgivenin gure1.5(a) asafun tionofm
H
. Asa onsequen eofthe
linearHiggs oupling to thefermion masses, form
H <2m
W
the dominant
hannelis H !bb,whi h orrespondsto thede ay intheheaviest fermion
kinemati ally a essible. Beyond the threshold for the produ tion of two
gauge boson H ! WW ()
and H ! ZZ ()
be ome dominant be ause of a
fa torm 3 H =m 2 W ;Z
inthepartialwidth. Thetotal de aywidthoftheHiggs
boson asa fun tionof theHiggs massis given in gure 1.5(b): the
asymp-toti behaviour isproportionalto m 3
H .
Existing Constraints on m
H
Althoughthe Higgsmass isa free parameter ofthe StandardModel,there
aretheoreti alargumentsofinternal onsisten yofthetheorygiving
L Y
Figure 1.6: Theoreti allimitsontheHiggsbosonmassassumingthe
validityoftheStandardModeluptoas ale.
have been arried out.
ByassumingtheStandardModelto bevalidatleastupto a ertainenergy
s ale , a lower bound for m
H
omes from the requirement for the
sym-metry breaking to a tually o ur. This transposes into the ondition for
the potential V(h i) <V(0), that is equivalent to >0 at all s ales. On
the other hand, sin e perturbative orre tions to the Higgs self intera tion
terms make in reasing with energy, requiring to keep nite up to the
s ale translates inanupperboundform
H
. Thesetwo theoreti al limits
are shown in gure 1.6. From what on erns theexperimental onstraints,
resultsofdire tsear hesatLEPIIareshownin gure1.7: valuesform
H up
to 114:4 GeV= 2
are ex luded. Indire t onstraints based on the
require-ment thatallthemeasurementsofele troweakobservables(e.g. asymmetry
measurements, mass for W
, top quark mass et .) be onsistent allow to
ex ludeaHiggsmassgreaterthan237GeV = 2
at95%C.L.. Thebest tfor
all these measurements gives the value m
H =114 +69 45 GeV = 2 at 95% C.L. ( gure 1.7) assumingm top =1784 GeV = 2 .
However, indire t onstraints on Higgsboson masshave a limited
Figure1.7: ExperimentallimitsontheHiggsbosonmass omingfrom
dire tsear hesatLEP(theex ludedregionisshadowed)and 2
result
ofa tonele troweakobservablesassumingm
H
L Y
Figure1.8: One-standard-deviation(39.35%)un ertaintiesinm
H asa
fun tionofm
t
forvariousinputs,andthe90%CLregion( 2
=4:605)
allowed by all data.
s (m
Z
) = 0:120 is assumed ex ept for the ts
in luding theZ-lineshapedata. The95%dire t lowerlimitfrom LEP
2isalsoshown.
onlylogarithmi allyonm
H
,whilefermionsgive ontributionsquadrati ally
dependent on m
f
. It turnsoutthat, be ause of the largemass for thetop
quark ( omparable to the predi ted Higgs mass), un ertainty on the top
mass an sensiblyshift the onstraints on m
H
, as illustrated in gure 1.8,
inwhi hthedependen e ofele troweak observablesonm
H
and m
t
is made
expli it.
1.2.5 Standard Model Higgs Produ tion in p p ollisions
The des riptionof theintera tionof two protonsis based,withintheQCD
framework, on the parton model approximation. This onsists in
onsid-ering the in oming beam of hadrons equivalent to a beam of onstituents
( alled partons and identi ed with quarks and gluons) whose momentum
distributionsinsidethehadronis hara terized bypartondensityfun tions
(pdf) f
i
(x;). The probabilityto ndtheparton i arryinga fra tion
be-tween x and x+dx of the initial momentum p of the hadron is given by
dxf
i
(x;), where is the typi al energy s ale of the pro ess. The pdf's
do not depend on the parti ular pro ess onsidered are and are therefore
Figure1.9: Representationoftheimprovedpartonmodelformula(eq.
1.60).
upon theenergys ale oftheintera tion.
The general expressionfor the produ tion ross se tion of some nal state
with high invariant mass from the intera tion of two protons beams with
momenta p
1
and p
2
( gure 1.9) an then be expressed by the so alled
improved parton modelformula:
(p 1 ;p 2 )= X i;j Z dx 1 dx 2 f i (x 1 ;)f j (x 2 ;) ij (x 1 p 1 ;x 2 p 2 ; s ();): (1.60)
The ross-se tionforthemostimportantpro essesatLHCisshownin gure
1.10.
In p-p ollisions, the dominant Higgs produ tion me hanism over the
entire mass range a essible at LHC (see hapter 2) is via gluon fusion
(gg ! H), where the Higgs ouples to the gluons through a heavy quark
loop ( gure1.11).
The leading ontribution to the loop omes from the top quark. The
otherquarks ontributetotheloopbyafa toratleastsmallerbyO(M 2 b =M 2 t )
be ause of theform ofthe Higgsboson ouplingto thefermions.
As summarized in (Del Du a, 2003) QCD orre tions at the Next to
LeadingOrder(NLO) have been omputedand showan in reaseof theLO
rossse tion by 10-80%, therebyleading to a signi ant hange of the
the-oreti al predi tions. NNLO al ulations have re ently be ome available in
L Y
Figure1.10: Cross-se tionforthemostimportantpro essatLHCasa
fun tionofthe enterofmassenergy. Therateofeventsperyearisalso
reportedontherights ale,assumingforLHCanintegratedluminosity
of100fb 1
(HighLuminosityphase).
Figure 1.11: Gluonfusionpro ess fortheHiggsboson produ tionin
Figure 1.12: Higgs produ tion via gluon fusionin pp ollisionsat a
enter of mass energy of 14 TeV= 2
as afun tion of the Higgs mass.
The produ tion rate has been omputed in the large m
top
limit, to
leading order, NLO and NNLO a ura y. The shaded bands display
therenormalisation
R
andfa torisation
F
s alevariations. Thelower
ontours orrespondto R =2m H and F =m H
=2, while theupper
ontoursto R =m H =2and F =2m H .
L Y
Figure1.13: Weakbosonfusion(WBF)pro essfortheHiggs
produ -tioninhadroni ollisions.
thegluons by an e e tive oupling (valid ifthe Higgsmassis smaller than
thethresholdforthe reation ofatopquarkpair). Itisexpe tedto
approx-imatethefullmassiveres alingfa tor within10%upto 1TeV = 2
, overing
the entire Higgs mass range a essible at LHC. NNLO orre tions display
an in rease of about 15% at m
H
= 120 GeV = 2
with respe t to the NLO
evaluation. Figure 1.12 shows the e e t of the higher order orre tions to
theHiggstotal rossse tion via gluonfusion.
These ondlargestprodu tionme hanismfortheHiggsbosonisviaweak
bosonfusion (WBF, qq ! qqH), where the Higgs is radiated o theweak
bosonex hangedin thet- hannelbetween thetwo in omingquarks ( gure
1.13). Sin e thedistributionfun tions ofthe in omingvalen equarkspeak
at values of the momentum fra tions x0:1-0:2, the two outgoingquarks
arenaturallyhighlyenergeti . Theythereforehadronizeintotwojetswitha
largerapidityintervalbetweenthem,typi allyat forward-ba kward
rapidi-ties. Anotherinterestingpropertyis theabsen e ofhadroni produ tionin
therapidityintervalbetweenthetwojets,sin ethe olourlessweak
intera -tion bosonex hanged between the in omingquarks auses gluon radiation
to o urs only as bremsstrahlung o the quark legs. This features an be
used to distinguish WBF Higgs produ tion from gluon gluon fusion. NLO
orre tions in
s
to the WBFprodu tionpro ess have been omputedand
foundto bemodest (ontheorder of 5-10%)(Puljak,2000).
The ross-se tionsfor thetwo produ tion pro esses illustratedabove along
withminorpro essessu hasHiggsstrahlungorttasso iatedprodu tionare
σ
(pp
→
H+X)
[
pb
]
√
s = 14 TeV
M
t
= 175 GeV
CTEQ6M
gg
→
H
→
Hqq
_
’
→
HW
_
→
HZ
gg,qq
_
→
Htt
_
M
H
[
GeV
]
0
200
400
600
800
1000
10
-4
10
-3
10
-2
10
-1
1
10
10
2
Figure 1.14: Cross-se tionfortheStandard Model Higgsprodu tion
atLHC.
1.3 Going Beyond
Despitethein rediblygoodagreementbetweenStandardModelpredi tions
and experiments (for an example of some observablessee the gure 1.15),
there are both on eptual problems and phenomenologi al indi ations of
newphysi s beyond it.
Parti lemassandquantumnumberssu hastheele tri harge,weakisospin,
hyper hargeand oloursarenotexplainedbytheStandardModel. F
urther-more,there is no reasonwhy leptons and quarks ome in di erent avours
andwhytheirele troweak intera tionmixinsu hape uliarway. Isthisan
indi ationtowardsmoreelementary onstituentsofmatterthanquarksand
leptons?
After the extension of the Standard Model, based on experiments, to the
group SU(3)SU(2)U(1) in order to in lude the strong intera tions
(SU(3) group), one is also tempted to in lude gravity in the same way.
However, typi al energy s ales for quantum gravity are of the order of
M P 1= p G N 10 19 GeV = 2
, seventeen ordersof magnitudehigher than
thetypi alele troweak intera tions. CantheStandardModelwithoutnew
physi sbevalidupto su hlargeenergies? Thisappearsunlikely,sin ethere
are no indi ationsin the Standard Model of why the typi al weak s ale of
masses issosmallrelativelyto thePlan kmassM
P
(hierar hyproblem).
Measurement
Fit
|O
meas
−
O
fit
|/
σ
meas
0
1
2
3
0
1
2
3
∆α
had
(m
Z
)
∆α
(5)
0.02758
±
0.00035 0.02767
m
Z
[
GeV
]
m
Z
[
GeV
]
91.1875
±
0.0021
91.1874
Γ
Z
[
GeV
]
Γ
Z
[
GeV
]
2.4952
±
0.0023
2.4965
σ
had
[
nb
]
σ
0
41.540
±
0.037
41.481
R
l
R
l
20.767
±
0.025
20.739
A
fb
A
0,l
0.01714
±
0.00095 0.01642
A
l
(P
τ
)
A
l
(P
τ
)
0.1465
±
0.0032
0.1480
R
b
R
b
0.21629
±
0.00066 0.21562
R
c
R
c
0.1721
±
0.0030
0.1723
A
fb
A
0,b
0.0992
±
0.0016
0.1037
A
fb
A
0,c
0.0707
±
0.0035
0.0742
A
b
A
b
0.923
±
0.020
0.935
A
c
A
c
0.670
±
0.027
0.668
A
l
(SLD)
A
l
(SLD)
0.1513
±
0.0021
0.1480
sin
2
θ
eff
sin
2
θ
lept
(Q
fb
) 0.2324
±
0.0012
0.2314
m
W
[
GeV
]
m
W
[
GeV
]
80.425
±
0.034
80.389
Γ
W
[
GeV
]
Γ
W
[
GeV
]
2.133
±
0.069
2.093
m
t
[
GeV
]
m
t
[
GeV
]
178.0
±
4.3
178.5
Figure 1.15: Comparison of the measurements withthe expe tation
oftheSM al ulatedforthe veSMinputparametervaluesinthe
min-imum of the global 2
of the t (The ALEPH, DELPHI, L3, OPAL,
SLD Collaborations, the LEP Ele troweak Working Group, the SLD
Ele troweakand HeavyFlavourGroups,2005). Thepull ofea h
mea-surement is reported aswell. The dire ted measurements of m
W and
W
me hanism for the ele troweak symmetry breaking, is not satisfa tory as
well. Loop orre tionsto theHiggsmassarequadrati allydivergent,giving
riseto theso- allednaturalityproblem.
If the Standard Model is not the fundamental theory, it will be valid up
to a ertain energy s ale . This limit an be viewed as a ut o whi h
parametrizesour ignoran e onthe new physi sthat willmodifythe theory
atlargeenergys ales. Itistheninterestingtolookattherelevantquantities
of theStandard Model upon the ut o s ale, requiringthat no
\unnat-ural"dependen e on arise. For what on erns the Higgs mass, in order
not to ex eed the limits indi ated by dire t and indire t sear hes must
be small, of the order of O(1 TeV= 2
), but annot be too smallsin e new
physi s hasnotbeendete tedat the present experiments.
Moreover, another unsatisfa tory theoreti al aspe t ofthe StandardModel
is the number of arbitrary parameters. These in lude three independent
gauge ouplings, apossibleCP-violatingstrong-intera tionparameter, two
independent masses for weak bosons, six quark and three harged-leptons
masses,threegeneralizedCabibboweak-mixinganglesandtheCP-violating
Kobayashi-Maskawa phase.
On the other hand, from the experimental side there is a strong eviden e
of neutrino os illations, implying massive neutrinos and the violation of
the family leptonnumber (and at least nine more arbitrary parameters in
the Standard Model to a ommodate these e e ts). Dire t measurements
ofneutrino masses, mainlyfrom -de ayexperiments, have imposedupper
limitsfromO(1eV = 2
)fortheele tronneutrinotoO(10 2
eV= 2
)forthetau
neutrino,whi h are roughly tenorder of magnitudes lessthan theheaviest
fermionmass(m t O(10 2 GeV = 2
)). Althoughthereare nosymmetriesin
thetheoryprote tingneutrinosfromhavingamass(e.g. amasslessphoton
isimposedbytheU(1) gaugesymmetry,relatedto the ele tri harge
on-servation), the me hanism to give su h a mass is not trivial. If a Yukawa
oupling via Higgs boson is invoked, a right-handed neutrino must be
in-trodu ed into theStandardModel,unless theun on rmedhypothesis that
neutrinos are Majorana parti les is true. A right-handed neutrino in the
Standard Model, however, should be neutral both to ele tromagneti and
weak harge, from the onstraintsimposed byLEP on thenumber of
neu-trino families((LEP Ele troweak Working Group, 1999)). Thus it will be
asinglet of SU(2)U(1), withthe right ofan additional Dira mass term
intheLagrangian that willbe totallyun onstrained. So more ompli ated
Figure 1.16: One-loop orre tions to the Higgs boson mass due to
fermioni (a)orbosoni (b)degreesoffreedom.
Model should be introdu ed. The so alled \see-saw" me hanism is the
mostpopularalternative, whi h ombines left- and right-handedneutrinos
inDira andMajoranamassterms,inorderto justifysu hsmallmassesfor
theneutrinos.
Two are thepossibleextensions of the StandardModel that willbe brie y
onsideredinthefollowing: supersymmetryand extra-dimensions.
1.3.1 Supersymmetry
Mainlymotivatedto stabilizetheHiggsmassquadrati divergen es,
super-symmetry onsists in assuming the existen e of a symmetry Q that
trans-formingfermionstobosonsandvi eversa. Forea hfermioninthetheoryis
thenintrodu edanewbosonand,byanalogy,to ea hbosonsisasso iateda
fermion. Thishasan immediate onsequen eon theone-loop orre tionsto
theHiggsmass( gure 1.16). Infa t, termdueto fermioni degrees of
free-domenterswithanoppositesignwithrespe tto orre tions duetobosoni
degrees of freedom. If
f
and
s
are the Higgs ouplings to fermions and
bosonsrespe tively,theone-loop orre tionm 2
H
totheHiggsmassbe omes
proportionalto m 2 H /( s f ) 2 +O( 4 ): (1.61)
Forsuitablevaluesof the oupling onstantsthequadrati divergen es
dis-appear, leavingonlylogarithmi divergen es.
Ina supersymmetri StandardModelea h fermionis then oupledto a
bo-sonin a supersymmetri multiplet, alled\supermultiplet": to ea h lepton
is asso iated a so alled \slepton", a \squark" to ea h quark. In thesame
spin 0 spin 1/2 spin 1 SU(3) C SU(2) L U(1) Y ~ u L ; ~ d L u L ;d L 3 2 + 1 3 ~ u R u R 3 1 + 4 3 ~ d R d R 3 1 2 3 ~ ;e~ L ;e L 1 2 1 ~ e R e R 1 1 2 H + u ;H 0 u ~ h + u ; ~ h 0 u 1 2 +1 H 0 d ;H d ~ h 0 d ; ~ h d 1 2 1 ~ g g 8 1 0 ~ w ;w~ 0 W ;W 0 1 3 0 ~ b 0 B 0 1 1 0
Table1.1: Parti le ontentofasupersymmetri StandardModel.
gaugesupermultiplet.
In the simplest extension of the Standard Model ( alled Minimal
Super-symmetri Standard Model) the Higgs se tor is omposed by two s alar
doublets,withtheirfermioni partners. Intable1.1thelistoftheStandard
Model parti les and theirsupersymmetri partners (\superpartners") with
thequantum numbersof ea hsupermultipletisgiven asa referen e.
Inorderto implementthebaryon(B)andlepton(L)number onservation,
anew onserved quantum number alled R -parityisde ned as
P
R =( )
3(B L)+2S
; (1.62)
whereB = 1=3 for quarksand squarks and 0 otherwise, L= 1 for leptons
and sleptons,0 otherwise, and S is theparti lespin. P
R
is equalto +1for
standardparti leswhileit takesthe value 1forsuperpartners.
Some onsequen es of theR -parity onservationare:
the lightestsupersymmetri parti le(LSP) withP
R
= 1 isstable;
supersymmetri parti les de ay into states with an odd number of
superpartners;
(a) (b)
Figure 1.17: Evolution of theele troweak, strong,and gravitational
oupling onstantswiththeenergys aleoftheintera tionsfor (a)the
StandardModelalone and(b) theMinimalSupersymmetri extension
oftheStandardModel(MSSM).
The rstthingtobenoti edisthatsuperpartnersofstandardparti les(e.g.
a s alarele tron) withthe same masswould have already beendete tedin
experiments. Sin e none of them has been observed so far, despite
exten-sive sear hes at ollider ma hines, the supersymmetry must be broken in
a realisti theory. However, the feature of having
f =
s
to all orders in
perturbationtheory,that an elthedivergen esof theHiggsmass,mustbe
preserved inthebroken theory.
Theme hanismbywhi hthesupersymmetryisbrokenisthemaindiÆ ulty
inbuildingasupersymmetri extensionoftheStandardModel. Twoarethe
mainsolutionsproposed.
The rst one onsists in introdu ing a so- alled soft breaking term in the
StandardModelLagrangian,thatisthemostgeneralsupersymmetri
break-ingtermpreserving
f =
s
. Thisparametrizeourignoran eofthebreaking
me hanismwiththeintrodu tionof105freeparametersintothetheorythat
an be redu ed by further assumptionsbased on experimental onstraints
(e.g. absen eofFlavourChangingNeutral Currentpro esses,CP violation
et .).
The se ond me hanism involves gravity and is generally referred to as the
gravity-indu ed supersymmetry breaking (mSUGRA). It is the results of
someunderlyingme hanismthatbreaksthesymmetryataverylarges ale,
presumably ompatiblewith thePlan kmasss ale.
An en ouraging indire teviden e of supersymmetry is that theuni ation
Model tout- ourt, as it is shown in gure 1.17. Moreover,
supersymme-try is predi tedas a natural onsequen e by mostof the attempt to build
grand-uni edtheories(e.g. stringtheories).
1.3.2 Extra-dimensions
Ase ondpossibleextensionoftheStandardModelisbasedon
phenomeno-logi al theories involving the gravitational intera tion. The general idea
behind these theory is to solve the hierar hy problembringing the gravity
down to the weak intera tion s ale, obtaining the observed Plan k mass
s aleasaresultsofa(4+n)-dimensionalworld. Inour4-dimensionalspa e
gravitywouldappearweak,asfor elineswouldes apeinextradimensions.
Thestartingpointistheobservationthatele troweakintera tionshavebeen
probedat distan es 1 E.W. =m 1 W
whilegravitationalfor eshavebeen
in-vestigated only to distan es of the order of 1 m, whi h is 33 orders of
magnitudegreaterthantheintrinsi energys aleofgravity,givenbym 1
P .
Theassumptionthatgravityat 1 mwouldbethesameatm 1
P
isthen
not ompletelyjusti ed. Changes ould happen inbetween.
Theproposedtheories an bemainly dividedinto two lasses,a ording to
thekindofextra dimensionproposed:
at ompa ti edextradimensions;
warped extradimensions.
Ea hof thetwo previous ategories an be dividedintwo groups:
gravitationalextradimensions: onlythegravitational elds an
prop-agate inextra dimensions;
universal extra dimension: Standard Model elds and gravitational
elds an propagate inextradimensions.
In the following the prin ipal ideas behind at ompa ti ed and warped
gravitationalextradimensionswillbebrie y illustrated.
As enarioproposedbyArkani-Hamed,Dimopoulos,Dvali(Arkani-Hamed
etal.,1998)isthatinadditiontothespa e-timedimensionswelivein,there
aren ompa tspatialdimensionsofradiusRa essibletothegravitybut
nottotheotherthree fundamentalfor es. StandardModelparti les annot
freely propagate in 4+n dimensions but would be lo alized on the
extra-dimensionsaregravitons.
The onsequen eisthatinourworldgravitymanifestsitselfasanextremely
weakfor e,withtypi alintera tionenergiesoftheorderofthePlan ks ale,
despiteinthefull(4+n)dimensionsthey aresuppressattheweak
intera -tion s ale.
Atdistan esr R thegravitationalpotentialbetweentwomasses m
1 and
m
2
ismodi eda ording to theGauss'slawin(4+n)dimensions:
V(r) m 1 m 2 m n+2 P(4+n) 1 r n+1 ; r R : (1.63)
On the other hand, when the distan e between the two masses is mu h
greater than R , then their gravitational ux lines an no longer penetrate
insidetheextra dimensions,and theusual1=r potentialisobtained:
V(r) m 1 m 2 m n+2 P(4+n) 1 r ; r R : (1.64)
The e e tive 4-dimensionalm
P isthen given by m 2 P m n+2 P(4+n) R n : (1.65) By assuming that m P(4+n) is of the order of m W and by demanding R to
be su h that the observed m
P
is reprodu ed, the following value for R is
obtained: R10 30 n 17 TeV = 2 m W 1+ 2 n : (1.66)
For n = 1 this will imply R 10 13
m, so deviation from Newton's law
shouldappear at solarsystem distan es. However, if n2 su h deviation
wouldappearonlybelow1 mm,thatisdistan esnotyetprobedby
experi-ments.
A ording to thismodel, thephenomenology of theStandard Model is
en-ri hed with a graviton and all its Kaluza-Klein ex itations re urring on e
every 1=R , perextradimensionn.
Adi erentmodel(RandallandSundrum,1999)isbasedonthe
hypoth-esisoftheexisten eofatleastoneextradimensiona essibletogravityand
thatthemetri ofthespa e-time dis riminatesbetweenthetraditionalfour
oordinatesand theadditionalones.
spa e has the usual \ at" metri multiplied by a \warp" fa tor rapidly
hangingasa fun tionof theadditional oordinate:
ds 2 =e 2kr dx dx +r 2 d 2 (1.67)
wherek isas aleoftheorderofthePlan ks ale,r
isthe ompa ti ation
radius, isthe oordinateof theextra dimension(0 < <) and is the
usualMinkowski'smetri tensor.
Thegravity s ale,whi his at theele troweak s ale,is given by
=m P e kr (1.68) where 1 TeV= 2 an be obtained with kr 11;12. Massive
Kaluza-Kleinex itations ofgravitons appearwith amassgiven by
m n =kx n e kr =x n k m P ; (1.69) where x n is the n th
root of the Bessel fun tion of order 1 (x
n
= 3:8317;
7:0156; 10:1735 forn=1; 2; 3).
The ouplingofthegravitonto theStandardModelparti leisproportional
to1=
. Thegravitonmassisdeterminedbytheratiok=
. Thesearethe
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The CMS Dete tor at LHC
Inspiteoftheremarkablea ura yinthedes riptionofelementaryparti les
intera tions, theStandardModel does notyet give an answerto a number
of fundamental questions (see hapter 1). Building upon its past strength
ofunderstandingopen problems,thephysi s ommunityhasfo usedits
at-tentionto hadron olliders,whi hareparti ularlysuitableforthedis overy
ofnew physi s. TheLarge Hadron Collider(LHC)at CERN,presently
un-der onstru tion,representsthenewgenerationofhadron ollidersandwill
undoubtedlyhelpto llgapsinourknowledge.
Afterabriefdes riptionofthema hine,the hapter willfo usonCMS,one
of the two general purpose dete tors (ATLAS and CMS) whi h will be
in-stalledat LHCalongwithtwoexperimentsspe i allyorientedtobphysi s
(LHCb)and heavyions physi s(ALICE).Parti ularemphasiswillbe given
to the mainfo usof this thesis,the te hni alaspe ts and expe ted
perfor-man esof theele tromagneti alorimeter of theCMSdete tor.
2.1 The Large Hadron Collider
The LHCwillprovideproton-proton ollisionat a enterof massenergy of
14 TeV = 2
(7+7). The available energy forthe intera tions of the proton
elementary onstituents willthen rea h the TeV range,whi his aboutone
order of magnitude greater thanthe typi al LEPand Tevatron intera tion
energies.
The LHC will be pla edin the already existent 26:7 km long LEP tunnel
and is supposed to start its a tivity in2007. Sin e ollisions willo ur
Figure2.1: LayoutoftheLargeHadronColliderwiththefour
exper-imentsthatwillbelo atedatea h intera tionpoint.
two di erent magneti eld on gurations are required. Super ondu ting
dipoles operating at 1:9 K will provide a 8:4 T magneti eld. Boosts
will be given by 400 MHz super ondu ting radiofrequen y avities with a
voltage ranging between 8 and 16 MV. The hannels for the two beams
a eleration willbe insertedina single ryostat.
Protons will be delivered to LHC by an upgrade of the CERN existing
fa ility. Thisfa ilitywillbringtheprotonsto theinje tionenergyintoLHC
of450GeVinfoursteps( gure2.1): theLINACwillbringthemto50MeV ,
theBooster willfurther a elerate them up to 1:4 GeV , thePS to 25 GeV
andtheSPSwillinje tthemintotheLHCattheirinitialenergyof450GeV
aftera nala eleration step..
Thebun hes, withanominalnumberof10 11
protonsea h,willhave avery
smalltransverse spread(
x
y
15 m) and will be7:5 m long in the
beam dire tions at the ollision points. A summary of the main te hni al
parametersof LHC isgiven intable 2.1.
Theluminosityofana eleratorthat ollidesbun hesofn
1 andn 2 parti les at afrequen yf isgiven by L=f n 1 n 2 4 ; (2.1)
Parameter Value
Cir umferen e[km℄ 27
Numberofmagnet dipoles 1232
Dipolar magneti eld[T℄ 8:386
MagnetTemperature [K℄ 1:9
Beamenergy[TeV= 2 ℄ 7 Nominalluminosity[ m 2 s 1 ℄ 10 34
Protonsperbun h 1:0510 11
Bun h spa ing[m℄ 7:48
Bun h timeseparation[ns℄ 24:95
Transverse beamsize I.P.[m℄ 200
rmsbun hlength[ m℄ 7:5
Crossingangle [rad℄ 210 4
Beamlifetime [h℄ 7
Luminositylifetime [h℄ 10
Table2.1: Mainte hni al parametersoftheLargeHadronCollider.
where
x and
y
representtheGaussianbeampro leintheplaintransverse
to thebeamaxis.
The nominal LHC luminosity is L = 10 34 m 2 s 1 and orresponds to an
integrated luminosity over one year of LHC runningof 100 fb 1
. Thiswill
value be rea hed after an initial phase at 10 33
m 2
s 1
(so alled \low
luminosity"phase) whi hwillbemainlydedi atedtotunethedete tor
per-forman es,to sear h fornew parti lesand to studythequark bphysi s.
TherequirementsontheLargeHadronCollider reate several hallenges
from the experimental point of view. The need of high statisti s to dete t
rare pro ess requires very high luminosity,with the onsequen es of a high
event rate dueto ommon QCDpro essesand an extremelydense parti le
environment.
Indeed, the total p-p ross se tion at the LHC energy is estimated to be
100 mb (Eidelman et al., 2004), whi h, given the ma hine parameters,
impliesan average of about 20 p-p intera tion per bun h rossing, 10 9
in-tera tions per se ond. A strong online event sele tion is therefore needed
in order to redu e the event rate at around 10 2
Hz , 7 orders of magnitude
less, whi h orresponds to the maximum data storage rate rea hable with
to distinguishevents belongingto di erentbun h rossings, whi h are
sep-arated onlyby25 ns.
Regarding the hallenge given by the parti le density, a typi al minimum
bias ollision at LHC will produ e on average 5:5 harged parti les with
mean transverse momentum around0:5 GeV = and 8primary photonsper
unit of pseudorapidity. An interestingevent, whi h typi ally ontains high
p
T
leptons,highE
T
hadronjets,b-jets,largemissingtransversemomentum,
willalwaysbesuperimposedonthispile-up. Dete torsmusthen ehave ne
granularity in order to separate parti les very lose in spa e by means of
sophisti atedre onstru tionalgorithms.
Moreover,toextra tasmu hinformationaspossiblefromaninteresting
sig-nal,multi-purposedete tors shouldalso ful llthefollowingrequirements:
fullhermeti itytoallowforana uratemeasure ofthemissing
trans-verseenergyandmomentum( omingfromalmostnon-intera ting
par-ti les, likeneutrinosorsupersymmetri neutralinos);
apabilitytore onstru tleptonsinawiderangeoftransversemomenta
and rapidity(to re onstru tgauge bosons,tag b-jets et .);
apabilitytore onstru t hargedtra kswithagoodpre isionontheir
transverse momentum andimpa t pointposition(toeÆ iently
re on-stru t and tagB parti lesand );
apability to re onstru t hadron jets from QCD pro ess and heavy
parti les de ays.
A very high parti le ux traversing ea h omponent of the dete tor also
impose restri tive requirements on the material that an be used for the
dete tor onstru tion: the best results will be obtained with the optimal
ompromisebetweendete torperforman eandparti leradiationresistan e.
2.2 The Compa t Muon Solenoid
Inorderto satisfythepreviousbasi requirements,CMShasoptedfora
ompa tdete tor ina solenoidalmagneti eld oaxial withthebeam-line.
Thephilosophyadopted forthedete tor designhasbeen:
Figure2.2: ThreedimensionalrepresentationoftheCMSdete tor.
ii) thebest alorimetry ompatible withi);
iii) ahigh quality entraltra king to a hievebothi)and ii);
iv) ahadron alorimetrywitha 4 solid angle overage;
v) a nan iallya ordabledete tor.
The apparatus exhibits a ylindri alsymmetry around thebeam dire tion
and dete tors are installed following an onion-like stru ture of onse utive
layers in the entral region(barrel) and several disks inthe forwardregion
(end aps). As hemati view oftheCMS dete torisgiven in gure2.2and
a longitudinal view of one quarter of the dete tor in gure 2.3. The full
lengthis 21:6 m, thediameter is15 m, for a total weight of 12500 t and
an average densityof3:3g m 3
.
Tra kingand alorimetrysub-dete torsare pla edinsidethe
super ondu t-ing solenoidwhilethe muon system is integrated in thereturn yoke of the
magneti eld.
In the following dis ussion, the di erent dete tor omponents will be
