First Measurement of

Charged Current Cross Setions

with Longitudinally

Polarised Positrons at HERA

.

Dissertation

an der FakultatfurPhysik

der Ludwig-Maximilians-UniversitatMunhen

vergelegtvon

Ringaile_ Plaakyte_

aus Litauen

.

1. Gutahter: Prof. Dr. Christian Kiesling

2. Gutahter: Prof. Dr. Martin Faessler

Abstrat

Theanalysis presentedinthisthesis onernsthee +

pdeep inelastiharged

urrent sattering ross setions measured with the H1 detetor at HERA.

Theanalyseddataweretakenintheyears2003-04 ataentre-of-massenergy

p

s=319GeV andomprisetwoperiodsofpositronbeamswithpositiveand

negative longitudinalpolarisation,orresponding to anintegrated luminosity

of 26.9 pb 1

and 20.7 pb 1

, respetively. The rosssetions are measured in

a kinematiregion of four-momentum transfer squared Q 2

>300 GeV 2

and

inelastiityy 60:9.

Fromtheanalyseddatathepolarisationdependeneofthetotalharged

ur-rent rosssetionis determinedforthersttimeat highQ 2

. The resultsare

ompared withpreditionsof the StandardModel and found to be in

agree-ment,i.e. possibleontributionsfromnon-standardright-handedurrentsare

onsistent withzero.

Kurzfassung

Diese Arbeit beshreibt der Bestimmung der Wirkungsquershnitte fur

tief-inelastishe Streuung mittels geladener Strome von Positronen an Protonen

mit dem H1 Detektor am Speiherring HERA. Die hier analysierten Daten

wurdenindenJahren2003-04beieinerShwerpunktsenergievon p

s = 319GeV

aufgenommen. Sie setzen sih aus zwei Perioden mit positiver und

nega-tiverlongitudinalerPolarisationdesPositronenstrahls mitintegrierten

Lumi-nositatenvon 26.9pb 1

und20.7pb 1

zusammen. Die Wirkungsquershnitte

wurdeninderkinematishenRegionQ 2

>300 GeV 2

(Quadrat des

Viererim-pulsubertrags) undy60:9(Inelastizitat)bestimmt.

DieAbhangigkeitdestotalenWirkungsquershnittsvonderPolarisationwurde

erstmaligbeihohem Q 2

gemessen. Die Resultatewerden verglihen mitden

VorhersagendesStandardmodells. Siestimmenmitdiesemuberein,

Introdution i

1 Theoretial Overview 1

1.1 DeepInelastiSattering (DIS) . . . 1

1.1.1 Kinematis ofDISProesses . . . 2

1.1.2 Basis of theCross SetionCalulations . . . 3

1.1.3 Quark Parton Model(QPM) . . . 4

1.1.4 QuantumChromodynamis (QCD). . . 5

1.1.5 Theoretial Aspets ofEletroweak Interations. . . . 9

1.2 Neutraland Charged CurrentCross Setions . . . 15

1.2.1 UnpolarisedLepton-Proton DISCrossSetions . . . . 15

1.2.2 DISCross Setions withPolarised Leptons . . . 19

1.2.3 Radiative Corretions . . . 19

1.3 Bakground Proessesto Charged Current . . . 21

1.3.1 Photoprodution . . . 21

1.3.2 Lepton-Pair Prodution . . . 22

1.3.3 WProdution . . . 23

1.4 Monte Carlo(MC) event generators . . . 24

2 The H1 experiment at HERA 27 2.1 HERAAelerator . . . 27

2.1.1 LongitudinaleBeamPolarisationat HERAII . . . . 30

2.2 H1 Detetor . . . 36

2.2.1 TrakingSystem . . . 38

2.2.2 Calorimetry . . . 40

2.2.3 LuminositySystem . . . 43

2.2.4 Time ofFlight System (ToF) . . . 44

2.3 H1 Trigger System . . . 44

2.3.1 The FirstTrigger Level (L1) . . . 46

2.3.2 The SeondTriggerLevel(L2) . . . 48

2.3.3 The ThirdTriggerLevel(L3) . . . 48

2.3.4 The Fourth/Fifth Trigger Levels (L4/5) . . . 48

2.4.1 Introdutionto Neural Networks . . . 49

2.4.2 H1 Level2 NeuralNetwork Trigger (L2NN) . . . 51

2.4.3 Trainingof NeuralNetworksand theirPerformane . 53 3 Reonstrution of Kinemati Quantities 59 4 Neutral Current Events for the Charged Current Analysis 63 4.1 HadroniEnergy Calibration . . . 64

4.2 InterationVertex Reweighting forSimulatedEvents . . . 69

4.3 Pseudo ChargedCurrent Events(PSCC). . . 70

5 Charged Current Data 73 5.1 ChargedCurrent Seletion. . . 74

5.1.1 Run Seletion . . . 74

5.1.2 Loose Pre-Seletion . . . 76

5.1.3 Final Seletion . . . 77

5.2 EÆienyEstimationUsingPSCC Events . . . 94

5.2.1 TriggerEÆieny . . . 95

5.2.2 Vertex RequirementEÆieny . . . 97

5.2.3 ep BakgroundRejetion EÆieny. . . 103

5.2.4 Non-ep BakgroundRejetion EÆieny . . . 105

5.3 Final ChargedCurrentEventSamples . . . 109

6 Charged Current Cross Setion Measurement 113 6.1 Denitionof (x;Q 2 ) Bins . . . 113

6.2 Cross SetionCalulation . . . 117

6.3 SystematiUnertainties . . . 118

7 Results 121 7.1 PolarisedCCCross Setions . . . 121

7.1.1 PolarisationDependeneof theTotalCCCross Setion122 7.1.2 The Q 2 Dependene ofthePolarisedCCCross Setion126 7.1.3 The x Dependene of thePolarised CCCrossSetion 127 7.1.4 PolarisedDoubleDierentCross Setion. . . 128

7.2 UnpolarisedCCCross Setions . . . 128

7.2.1 The Q 2 Dependene oftheCC CrossSetion . . . 130

7.2.2 The x Dependene of theCCCross Setion . . . 132

7.2.3 UnpolarisedDouble Dierent CrossSetion . . . 133

8 Summary and Outlook 137

A Tables of Results 139

Theunderstandingof thestrutureof matter,its propertiesand interations

is one of the fundamental questions in physis and, onsequently, a driving

foreforpartileresearh.

The experiments of partile sattering, where the result of two body

inter-ationis examined, fora long time have been suessfully used asa tool to

probethestrutureofmatter. Theideaof suhexperimentswasbornbythe

lassial Rutherford experiments [1℄ where a beam of partiles was red

onto a thin gold foil as target. The observed rare large deetions of the

sattered partilesserved asevidene forthesubstrutureof thegoldatoms,

leadingto thedisovery of theatomi nuleus.

Thereare fourknown typesof interations innature, gravitational 1

,

eletro-magneti, weak and strong. In the rst experiments where the elasti

sat-teringof theeletron on stationary targets wasanalysed(e.g. Hofstadter [2℄

in the late 50's), the eletromagneti fore was used to measure the harge

distributionsofthenuleusandoftheproton. Withtheinreaseofahievable

energiesforthebeampartilestherstinelastisatteringexperiments,with

largefour-momentumtransfersquaredQ 2

betweentheinomingandoutgoing

eletron, beame possible. This led to extended studiesof the substruture

of the proton. In 70's the eletron-proton sattering experiments at SLAC

(Stanford Linear Aelerator Center) for therst time showed the so-alled

'saling' behavior, i.e. independene of Q 2

, of the proton struture

fun-tions[3,4℄. Aording toBjorken[5℄,'saling' isexpetedwhen theeletrons

satter elastially on point-like harged spin 1=2 partons inside the proton,

today known as quarks. The Bjorken 'saling' is a basi idea of the parton

model (Quark Parton Model, QPM) introdued by Feynman [6℄ in the 60's.

The interations of quarks are mediated by 'olor' eld quanta, the gluons,

and are desribed within the framework of the theory of strong interations

(QuantumChromodynamis, QCD).

1

Thedevelopmentofweakinterationphenomenologystartedintheearly30's

with Fermi, who suggested a 4-fermion model for neutron deay [7℄. The

greatsuess oftheeletroweak uniationtheorydevelopedbyGlashow[8℄,

Weinberg [9℄ and Salam [10℄ (GWS model) was elebrated in 1983 with the

disovery of the predited heavy W

and Z partiles in pp experiments at

CERN [11,12℄.

Quantum Chromodynamis together with the theory of eletroweak

inter-ationsomposes theStandardModelof elementarypartilephysis.

A uniquepossibilityto explore the struture of theproton and, at thesame

time to probe the theory of eletroweak interations, has been initiated by

theeletron 2

-protonolliderHERA(HadronElektronRingAnlage)atDESY

(Deutshes Elektronen-Synhrotron)in1992. HERAextendedthekinemati

regime inQ 2

by morethentwoordersof magnitudeoftheone ahievableby

theexperimentswhihuse stationary targets(e.g. SLAC).

The resultsof HERAphysis withdata taken untilthe endof theyear 2000

("HERA I") involve tests of the eletro-weak and QCD theories,

investiga-tionsofperturbativeQCDanddirativesattering,photo-prodution,QCD

analysesof jetand searhesfornew partiles.

The possibility to ollide protons with a longitudinally polarised eletron

beam, as well as a signiant inrease of the instantaneous luminosity, was

providedwiththeupgradeofHERAin2003("HERAII"). Thisallowsto

ex-tendthestudiesofHERAIforsuhrarephysisproessesashargedurrent

interations (e

p ! (

)

X) and to perform preise tests of the eletroweak

setorof theStandardModel.

Weak harged urrent proesses with W exhange measured with the H1

detetorat HERAarestudiedinthisthesis. The datataken in2003-04 with

positiveandnegativelongitudinalpolarisationforthepositronbeamareused

tomeasurevariousrosssetionsforhargedurrentinterations. The

polar-isation dependene of the total harged urrent ross setion was measured

for the rst time at high momentum transfer Q 2

. The results of thisthesis

have beenpublished[13℄ inDeember2005.

The thesisonsists of eight hapters. Generaltheory aspets of Deep

Inelas-ti Sattering are disussed in the rst hapter. The HERA ep ollider and

the H1 experiment are desribed next. A shortintrodution to Neural

Net-worksand theirusageforthetriggering ofinlusiveDISevents(e.g. harged

urrents) whih are of partiular importane for the HERA II running, are

2

The"eletron"heregenerallyreferstoeletronsandpositronsunlessexpliitlystated

also disussed in hapter 2. The reonstrution of the event kinematis is

thetopi of hapter 3. Relevant aspets of theneutral urrent measurement

(e

p ! e

X) used in the harged urrent analysis are given in hapter 4.

The seletion of harged urrent events and the ross setion extration are

topis of hapters 5 and 6. The results of the harged urrent measurement

Theoretial Overview

Thisanalysisonernsthestudyofthehargedurrentdeepinelasti

satter-ing(DIS)proessesmeasuredat HERAwiththeH1detetor. Inthishapter

the deep inelasti sattering, DIS kinematis and the ross setions of DIS

proessesare explained. The basiDIStheory aspets,i.e. the quarkparton

model (QPM), thetheory of strong interations (QCD)and theeletroweak

setor of the Standard Model are introdued here as well. The exhaustive

theorydesriptionan bereviewed inothersoures like [14℄.

1.1 Deep Inelasti Sattering (DIS)

Deepinelastisattering(DIS)atHERAistheeletronsatteringonprotons

wheretheeletronhassuÆientenergytointeratwithahargedonstituent

oftheproton(quark)and,asaresult,amultihadroninalstateisprodued.

There are two deep inelasti eletron-proton sattering proesses measured

at HERA: neutral urrent (NC), ep ! eX, and harged urrent (CC),

ep ! X. The exhanged partiles between theeletron and the quark in

neutralurrentreationsarethephoton()andthebosonZ(Figure1.1left).

In theharged urrent sattering proess a harged boson W

is exhanged

(Figure1.1 right). Photonsareinteratingeletromagnetially,thebosonsZ

andW

aremediatorsof theweakfore. Thesattered (anti)neutrino (

)

in

theCCproess isnotdetetedand resultsinan apparent missingtransverse

momentum. However, the missing transverse momentum an be measured

*e (l)*

*+*

*_*

*/Z(q)*

### γ

*+*

*_*

*e (l’)*

*X(P’)*

*p(P)*

*e (l)*

*+*

*_*

*X(P’)*

*p(P)*

*_*

*+*

*W (q)*

*e*

*(l’)*

### ν

*)*

*(*

Figure 1.1: Diagrams of neutral NC (left) and harged CC (right) urrent

deep inelasti sattering proesses. The symbols denote the partiles, the

four-momenta of the partiles aregiven in brakets. Thelabel "X" denotes

the hadroni nal state.

1.1.1 Kinematis of DIS Proesses

Thefour-vetorsofpartilesinvolvedinthesatteringproess(seeFigure1.1)

are:

l- the four-vetor ofinomingeletron,

P - thefour-vetor of theinomingproton,

l 0

-the four-vetor of theoutgoingeletron,

P 0

- thefour-vetor ofthehadroni nalstate X,

q=l l 0

isthefour-momentumarriedbytheexhangedboson.

Thevariablesommonly usedto determine theDISkinematisare:

thefour-momentum transfer squaredq 2

:

q 2

=(l l 0

) 2

q 2

isameasureofthevirtualityoftheexhangedboson,usually

usedasthepositive quantityQ 2

:

Q 2

= q

2

(1.1)

the inelastiity y, whih is the frational eletron energy transfer by

theexhanged bosonto theproton:

y= P q

P l

(1.2)

IntheprotonrestframewhereP =(M;0;0;0),y = (E

e E

0

e )=E

e

as alulated from P q = P(l l 0

) = M(E

e E

0

e ) and

P l = ME

the Bjorken saling variable x:

x= Q

2

2P q

(1.3)

In the "innite momentum frame" (where the transverse

mo-mentum of the interating, i.e. struk quark of theproton an

be negleted)xis the(longitudinal)proton momentum fration

arriedbythestrukquark. Thisframerepresentsthebasiidea

oftheQuarkPartonModel (QPM),i.e. theassumptionthatthe

proton is madeout ofpoint-likeonstituentsor "partons".

All variables introdued are related with the eletron-proton enter-of-mass

energysquared,s=(P+l) 2

:

Q 2

=sxy (1.4)

Aswillbe explainedlater, inQPMthe DISproessis viewed aselasti

sat-tering of theeletron with a parton,therefore any of two of thevariables x,

y and Q 2

are suÆient to desribe the DIS kinematis at xed ep enter of

massenergy p

s.

1.1.2 Basis of the Cross Setion Calulations

InQED,theone-photonexhangeamplitudeisdominatinginthelowest-order

DISproess. The rosssetion ofinelastilepton-nuleonsatteringthusan

befatorisedintoaleptoni(L

)andahadroni(W

)tensor(see,e.g.[14℄):

dL

W (1.5) L

iswellknowninQED.Itisassoiatedwiththeouplingoftheexhanged

photon to the lepton lineand depends only on the four-momenta of the

in-omingand outgoinglepton(see Figure 1.1):

L =2[l l 0 +l l 0 +(q 2 =2)g ℄ (1.6) Here,g

is themetritensor.

ThetensorW

desribesthenuleonvertexand hasthegeneral form

W = W 1 g q q q 2 + W 2 M 2 p

(P q)q q 2 p

(P q)q q 2 (1.7) W 1 and W 2

representthestrutureofthehadronasitis'seen' bythevirtual

In70'stheeletron-protonsatteringexperimentsatSLAC(Stanford

Lin-ear Aelerator Center) [3,4℄ showed that at high Q 2 (Q 2 5 GeV 2 ) the

dependeneoftheinelastiprotonfuntionsonQ 2

vanishesandbeome

fun-tionsofx alone. Thisrepresents thesaling behavior of theprotonfuntions

whihwassuessfullyexplainedbyBjorken[5℄andleadto thepartonmodel

asexplainedinthenextsetion. OftenthefuntionsW

1 andW

2

areexpressed

intermsof the"struture funtions"F

1

(x) andF

2

(x) inthefollowingway:

MW

1 (x;Q

2

)!F

1 (x) (1.8) W 2 (x;Q 2 ) P q

M !F

2

(x) (1.9)

Assumingthattheprotononstituentspartiipatinginthesatteringarespin

1

2

partiles,Callanand Gross showed [15℄:

F

2

(x)=2xF

1

(x): (1.10)

Thisrelationwasonrmedexperimentally[17℄forlowvaluesofQ 2

(and p

s)

andreets thefatthat theharged partonsinside theprotonarry spin 1

2 .

Usingthenotationsin(1.8) and(1.9),thedoubledierentialrosssetionfor

inelastisatteringan beexpressed as:

d 2 dxdQ 2 = 4 2 xQ 4 y 2 2 2xF 1

(x)+(1 y)F

2 (x)

(1.11)

With1.10one anobtainthattherosssetionforinelastisattering (1.11)

an bewrittenas:

d 2 dxdQ 2 = 2 2 xQ 4

(1+(1 y) 2 )F 2 (x) (1.12)

wheretheterm1+(1 y) 2

isalledthe heliityfator Y

+ .

1.1.3 Quark Parton Model (QPM)

The main idea of Quark Parton Model is based on Bjorken saling and its

interpretatation by Feynman [6℄: Deep inelasti lepton-proton sattering is

the sum of inoherent elastilepton-parton sattering proesses. The

orre-sponding ross setion of the whole proess is the sum of all lepton-parton

ross setions. As disussed in previous setion, this senario predits sale

invariane of the proton struture funtions, i.e. their independene on the

kinematisale(Q 2

Theassumptionthatallhadronsonsistofquasi-freepoint-likepartilessoon

wasshown to be inonsistent withthe leptonand nuleonsattering

experi-ments [17℄. Aording to QPM, the proton ismade of two up and one down

quark. Theproton struturefuntionF

2

(x)theniswrittenasthesumof the

quarkdistributionfuntions [16℄:

F

2 (x)=

X

i e

2

i xq

i

(x)=x n

e 2

u

u(x)+e 2

d d(x)

o

Here,e

i

is theharge of thequarki.

However, it was determined experimentally [17℄ that the average total

mo-mentum arriedbyquarksinsidethe protonis:

Z

1

0

xu(x)dx+ Z

1

0

xd(x)dx0:36+0:18=0:54

This result learly suggested that about half of the proton momentum is

arried by eletrially neutral onstituents (the other half is arried by the

harged quarks). In addition,theinterpretationof quarksastheonlyproton

onstituents had to be revised by the experimentally observed saling

vio-lations. The saling violation of the F

2

proton struture funtion measured

experimentally byH1 andsome xed target experimentsasfuntionofQ 2

is

presentedinFigure 1.2.

Nowadays the neutral proton onstituents are known as gluons. The theory

desribingtheinterationsbetweentheprotononstituentsisalledQuantum

Chromodynamis (QCD). As will be shown in the next setion, the saling

violationsareexplainedwithintheframeworkofQCDvia interationsof the

quarksandgluons.

1.1.4 Quantum Chromodynamis (QCD)

Quantum Chromodynamis (QCD) is an important part of the Standard

Model whih desribes the strong interation, one of the four fundamental

fores of nature. It assumes quarks to be elementary elds interating via

masslessspin1bosons,knownasgluons. Gluons arrythequantumnumber

olor whih has three values (red, blue, green) and is desribed by SU(3)

C .

Quarks do not exist free but are bound in olor-singlets (quark-antiquark

pairs,mesons,and three-quark states, baryons)states (onnement).

Interations between quarks and gluons in QCD are desribed by

relativis-ti quantum eld theory with a non-Abelian gauge group SU(3)

C

. This

means,thatthegluonsthemselvesarryolorharge,i.e. areself-interating.

QCDthushasanimportantdiereneomparedtoquantumeletrodynamis

### 10

### -3

### 10

### -2

### 10

### -1

### 1

### 10

### 10

### 2

### 10

### 3

### 10

### 4

### 10

### 5

### 10

### 6

### 1

### 10

### 10

### 2

### 10

### 3

### 10

### 4

### 10

### 5

### Q

### 2

### / GeV

### 2

### F

### 2

### ⋅

### 2

### i

### x = 0.65, i = 0

### x = 0.40, i = 1

### x = 0.25, i = 2

### x = 0.18, i = 3

### x = 0.13, i = 4

### x = 0.080, i = 5

### x = 0.050, i = 6

### x = 0.032, i = 7

### x = 0.020, i = 8

### x = 0.013, i = 9

### x = 0.0080, i = 10

### x = 0.0050, i = 11

### x = 0.0032, i = 12

### x = 0.0020, i = 13

### x = 0.0013, i = 14

### x = 0.00080, i = 15

### x = 0.00050, i = 16

### x = 0.00032, i = 17

### x = 0.00020, i = 18

### x = 0.00013, i = 19

### x = 0.000080, i = 20

### x = 0.000050, i = 21

_{H1 e}

### +

_{p high Q}

### 2

### 94-00

### H1 e

### +

### p low Q

### 2

### 96-97

### BCDMS

### NMC

### H1 PDF 2000

### extrapolation

### H1 Collaboration

Figure 1.2: Struture funtionF

2

as funtion of Q 2

for dierent x regions.

The points represent H1 measurements whih are ompared with the H1 t

forthe proton density funtions[18℄(band). Themeasurementsof two xed

target experiments (BCDMC [19℄ and NMC [20℄) are also inluded in the

graph.

Renormalisation and the running strong oupling onstant

In QED, the eetive eletromagneti oupling onstant depends on the

momentum transfer Q 2

arried by the mediating photon and inreases with

the inrease of Q 2

. This is a onsequene of "vauum polarisation", where

virtuale +

e pairspartiallysreentheharge,similarto adieletrimedium

whih sreenstheeletri harge.

In analogy to QED, the oupling strength

s

in QCD depends on Q 2

ar-riedby themediating boson. The leading order graph inQCD of the gluon

line orreted to vauum polarisation (Figure 1.3 (a)) represents this idea

wherethequark-antiquarkloopleadsto asreeningof theolorharge. The

Figure 1.3 (b). It has been shown [21,22℄, that gluon loops work as

"anti-sreening": With the inrease of Q 2

the seond term "overomes" the olor

sreeningandresultsinadereaseoftheouplingonstantwithinreasingof

Q 2

("running strongoupling onstant"). This is the origin of asymptoti

freedom.

(a) (b)

Figure 1.3: Contribution of the qq loop to vauum polarisation (a), gluon

loop ontribution tovauum polarisation (b).

The running of the oupling onstant

s

between a referene sale and a

givenQ 2

an bewritten[14℄ as:

s (Q 2 )= s ( 2 ) h 1 s ( 2 ) 12

(33 2f)ln Q 2 2 + i (1.13)

Here, the number 33 arises from the gluon loop, f is the number of quark

avours and 2f is the ontribution from the quark pair loops. From the

equation 1.13 itan be seen, thatthe runningof

s

depends on thenumber

ofquarkavoursf,i.e. aslongasf 16

s

willdereasewithinreasingQ 2

.

Sinetheknownnumberofquarkavoursis6,

s

isexpetedto fallwiththe

inreaseofQ 2

.

At suÆiently low Q 2

, the eetive oupling will beome large. It is

us-tomary to denote theQ 2

saleat whih thishappensby 2

QCD

. The strong

oupling

s (Q

2

)an be writtenas:

s (Q 2 )= 12

(33 2f)ln(Q 2 = 2 QCD ) (1.14)

For6 quarkavours

QCD

isapproximatelyequalto 300-500 MeV.

Atlargeenergysales(equivalenttolargeQ 2

)wherethedistanebetweenthe

partonsaresmallowingtotheunertaintypriniple,

s

beomessmallandthe

partonsan move"freely" insidetheproton. Thispropertyis alled

asymp-toti freedom. Similarly, at small Q 2

, i.e. large distanes, the oupling

between the partons inreases and results in onnement, where quarks

Fatorisation

AppliationsofQCDasdisussedupto now,arelimitedtotheshort-distane

region(perturbativeQCDorpQCD 1

).Duetohigherorderorretionsatlong

range (infrared divergenes, i.e. radiation of soft gluons o partons, for

example) it is impossibleto alulate DIS rosssetion. The fatorisation

theoremfatorises theross setioninto a"short distane" omponent (i.e.

elasti eletron-parton sattering), whih is alulable within pQCD, and a

non-perturbative "long distane" omponent (parton distributions), whih

hastobedeterminedexperimentally. Asa result,theinlusivelepton-proton

DISrosssetionisalulableinpQCDwithempiriallyparameterisedparton

densities (ata given referene sale) inside theproton. The partondensities

arealledparton distribution funtions (PDF).

Although the parton distribution funtions annot be alulated, their Q 2

dependene isalulable inpQCD asdesribed bytheDGLAP

(Dokshitzer-Gribov-Lipatov-Alterelli-Parisi)evolution equations[23℄:

q(x;t) t = (t) 2 Z 1 x dy y q(y;t)P qq x y

+g(y;t)P

qg x y (1.15) g(x;t) t = (t) 2 Z 1 x dy y q(y;t)P gq x y

+g(y;t)P

gg x y (1.16)

Here,t=ln(Q 2 = 2 QCD ),P ij

aretheso-alledsplitting funtions. P

ij

desribe

theprobabilityofparton j with momentumfration y to produea parton i

withmomentumfrationx,whentheQ 2

salehanges fromQ 2 toQ 2 +dQ 2 .

Figure 1.4 shows four Feynman diagrams for eah of these elementary

pro-esses.

*P (x/y)*

_{qq}

_{qq}

*q(y)*

*g(y−x)*

*q(x)*

*q(y)*

*q(y−x)*

*P (x/y)*

_{qg}

_{qg}

*g(x)*

(a) (b)

*P (x/y)*

*gq*

*g(y)*

*q(x)*

*q(y−x)*

*g(y)*

*g(x)*

*gg*

*P (x/y)*

*g(y−x)*

() (d)
Figure 1.4: Feynman diagrams for eah of the four splitting funtions P

ij

and the orresponding elementary proesses: (a)-(b) gluon emission by a

quark, ()quark-antiquark reation, (d) gluon emissionby a gluon.

1

PerturbativeQCD is the studyof the theoryof QCD in energyregimes where the

Theeletron-protonsatteringproess anbeviewedasan interationwhere

the eletron ats as a soure of virtual photons, i.e.

p ! X. The

ex-hanged virtual photon (q 2

6=0) an be either transverselyor longitudinally

polarised. In theQPM model, dueto onservation of theheliityand

angu-lar momentum, the interation of an eletron and a longitudinally polarised

virtual photon is not possible. In QCD, due to additional partiles at the

hadroni vertex (the quark an radiate a gluon or a gluon an split into a

quark-antiquarkpair,asshowninFigure 1.4), there isno longerdiÆultyto

onserve theheliityand angular momentum witha longitudinallypolarised

virtual photon. The Callan-Gross relation (1.10) is no longer valid and the

dierene of the two struture funtions F

2

(x) and F

1

(x) is onneted to a

non-zerolongitudinalstruturefuntionF

L :

F

L

(x)=F

2

(x) 2xF

1

(x) (1.17)

InQPM, thelongitudinalstruturefuntionF

L =0.

1.1.5 Theoretial Aspets of Eletroweak Interations

Firstattempts to understand weak interations started in 1933 when Fermi

formulatedthe 4-fermionmodel to desribe theneutron deay:

n!p+e +

e

Fermiproposeda pointinterationLagrangianas follows:

L

F =

G

F

p

2 J

(np)J

(e)= G

F

p

2

(p)

(n)

(e)

()

(1.18)

wherethersturrentisassoiatedwiththetransitionofntop(hadroni

ur-rent),theseond termorrespondsto the(e) pair(leptoniurrent). These

urrentsareoupledwiththeouplingonstantG

F

("Fermionstant")whih

isequalto 1:1610 5

GeV 2

[24℄.

However, the violation of parity (non-invariane of interations under spae

oordinate inversion, or mirror reetion) whih nowadays is a well known

propertyof the weak interations, was learly not built into Fermi's

vetor-vetortheory. Anexampleofaquantityillustratingparityviolation(hanging

sign under spae oordinate inversion) is heliity. Heliity is a projetion of

partile'sspinalong its diretionof motion,asshown inFigure1.5.

The violation of parity was initially proposed by T.D. Lee and C.N. Yang

(1956) [26℄ in K meson deay (then known as the puzzle). One year

### p

### s

### s

_{p}

Figure 1.5: Shemati presentation of dierent heliity states. Heliity is

theprojetion of thepartilespin tothediretion ofmotion: h=~s~p=j~s~pj.

For spin 1

2

partiles heliityaneitherbepositive(+ ~

2

)-the partileisthen

"right-handed" (right), or negative ( ~

2

)-the partile is then "left-handed"

(left).

A generalisation of the Fermi's vetor theory to inlude parityviolation was

independentlyproposedin1958 byFeynman andGell-Mann [28℄ and

Sudar-shanand Marshak[29℄. Theysuggesteda(V A)Lagrangianforweak

inter-ations. Thenotation(V A)meansthatbothvetor andaxialvetor parts

are ontained inthe weak urrent J

. The purely left-handed nature of the

neutrino,violatingparityinamaximalway,isintroduedbythe 1

2

(1

5 )

op-erator. Thustheexpressionforthehadroniweakurrentatthequarklevelis

J

(h)=u

1

5

2 d

C

+::: (1.19)

Here,thefator

yieldsavetor oupling whereas

5

givestheaxial

ve-tor,uistheDiraspinoroftheuquark,andd

C

istheCabibbo-rotatedquark

eldwithharge 1=3 introduedbyCabibbo[30℄ in1963: When thequarks

aregrouped into families(aordingto theirhargesand masses), the quark

transitionsintheweak deays(e.g. d!uinneutron deay)are ourring

not onlywithin a family, but (to lesserdegree) from one family to another.

Theweakforethusouplesnottothequarkpairsbutrathertolinear

ombi-nationsof thephysialquarks, writtenasthesineand osineof theso-alled

Cabibboangle

C :

d

C

=dos

C

+ssin

C

(1.20)

ThetwoomponentquarkspinorwithinorporatedCabibbostrutureofthe

harged urrents usuallyisdenoted as:

q=

u

d

C

L

ThesymbolLindiatesthatonlytheleft-handedpartsofthewavefuntions

enterintothe weaktransitions.

Allthree generations of quarksknown today an be expressed via the 33

Cabibbo-Kobayashi-Maskawa matrix [16℄whihisanextensionoftheoriginal

0 d 0 s 0 b 0 1 A = 0 U ud U us U ub U d U s U b U td U ts U tb 1 A 0 d s b 1 A (1.21)

Theprobabilityfor a transitionof a quark q to aquark q 0

is proportionalto

thesquareof themagnitudeof thematrixelement jU

qq 0j

2

.

TheombinedFeynman, Gell-MannandCabibbotheoriesshowedagood

de-sriptionofthe experimentallymeasuredharged-urrent interationswithin

rst-order perturbation theory. However, at energies of order 1= p

G

F

(or-responding to 300 GeV) violation of the unitarity, i.e. onservation of

probability,isenountered.

Unitaritywassaved(inlowestorder)byintroduinganeweldquantum,the

intermediate vetor boson W. Inontrast to the eletromagnetield

quan-tum(photon), W hasto be harged astheweak urrentsinvolve ahangeof

harge (by one unit) and also has to be massive (aording to the

Yukawa-Wikargument[33℄ that therange of afore is inverselyproportionalto the

mass of the exhanged eld quantum). In the approximation of Q 2

M

W

(hereM

W

isthemassoftheW),theouplingstrengthG

F

isproportionalto

theweak ouplingsonstant g:

G F p 2 = g 2 8M 2 W (1.22)

However, even with W exhange it an be shown that unitarity in weak

interations is not saved at higher orders: The theory turned out as

non-renormalisable(dueto thenon-vanishingmassofW).

Developing the theory of weak interations,in 1958 Bludman[34℄ suggested

theexistene of neutral weak interations with a hargelessmediator,

nowa-days known asthe Z boson. The rst experimental evidene of the neutral

weakinterationswasrevealedin 1973 at CERN[35℄.

Theproperties oftheZ bosonare desribedinthe framework ofeletroweak

interation, established by Glashow in 1961, Weinberg in 1967 and Salam

in 1968 (GWS model) where the eletromagneti and weak interations are

understood astwo aspets of the same interation. The theory inorporates

a new quantum number, weak isospin I, in analogy to isospin of the strong

interations. As disussed above, eah family of quarks (and leptons) form

so-alledweakdoubletsofleft-handedfermionswhihantransformintoeah

other by theexhange of a W. The eletri harge e

f

of the fermionsin the

samefamilydiersbyone unit. Beingadoublet, theweak isospinis I =1=2

and the third omponent is I

3

= 1=2. The right-handed fermions do not

oupleto theW and aredesribedassingleswithI =I

3

=0. Thewholelist

leptons e e L ; L ; L e f = n 0 1 o I = 1 2 I 3 = n + 1 2 1 2 o e R ; R ; R e f

= 1 I =0 I

3 =0 quarks u d 0 L ; s 0 L ; t b 0 L e f = n + 2 3 1 3 o I = 1 2 I 3 = n + 1 2 1 2 o u R ; R ; t R e f

=+2=3 I =0 I

3 =0 d R ; s R ; b R e f

= 1=3 I =0 I

3 =0

Table 1.1: Weakmultipletsof leptonsand quarks. e

f

is theeletriharge,

I istheweakisospin, and I

3

its thirdomponent.

ConservationofI inweakreationsrequiresthreestatesofI

3

( 1;0;+1),i.e.

threebosons. Thesestatesould beassignedto W +

,W andathirdneutral

eldquantum W 0

,also with apurely left-handedoupling. In addition,one

ould postulate a state B 0

, a singlet of the weak isospin I = I

3

= 0. Its

ouplingstrength (g 0

) to the weak neutralurrent doesnot have to be equal

of theone forW

and W 0

(g). The new eldsB 0

and W 0

wereassumed to

arrymass.

EletroweakuniationwassuggestedbyGlashow,introduingtheweak

ana-logto hyperharge 2

Y

W

,usingtheGell-Mann-Nishijimaformula[14℄:

e f =I 3 + 1 2 Y W (1.23)

Here,the eletri harge e

f

isin unitsof e. The leptondoublets inthisase

have Y

W

= 1 whilequark doubletshave Y

W

=+1=3.

The basi idea of the eletroweak uniation is to desribe the neutral

ur-rent mediators (photon and a hypothetial Z) as linear ombinations of B 0

and W 0

suh that one state, the photon, remains massless. This mixing is

expressed asa rotation throughtheweak mixingangle, or Weinberg angle 3 , W : 2

HyperhargeY =B+S whereB isthebaryonnumberandS isthestrangeness.

3

Thereis notheoretialpreditionfor

W

. Thereforeitsvalue hasto bedetermined

experimentally. Thepresentbestestimateofsin 2

ji=sin

W jW

0

i+os

W jB

0

i (1.24)

jZi=os

W jW

0

i sin

W jB

0

i (1.25)

Therelationbetween theWeinbergangle

W

and theweak ouplingsg,g 0

:

tan

W =

g 0

g

: (1.26)

Uniation is expliitly expressed by relating the weak oupling g with the

eletrihargee:

gsin

W

=e (1.27)

Theombination ofa newAbeliangroup U(1)assoiatedwith"weak

hyper-harge"andtheSU(2)groupassoiatedwith"weakisospin"resultedintothe

SU(2)U(1) group of eletroweak interations. Thus the uniation of the

weakand eletromagnetiforeswasestablished. However, themassesof the

weak bosons W

andZ stillhave to beaommodated withinthetheory.

TheW bosonouplesto left-handedleptonsandquarkswithequalstrength,

irrespetive of theirharge. In the oupling of Z the eletri harges playa

role aswell. The ouplingstrength of Z to a fermion f isgiven by

C(f)=

ie

sin

W os

W

(f); (1.28)

(f)=I

3 e

f sin

2

W

(1.29)

Apart from the neutrino, the neutral urrent ouplings, in ontrast to the

harged urrents(see equation 1.19) arenotpure V-Atype, e.g.:

J neutr

(h)=u 1

2

L

(1

5 )+

R (1+

5 )

u+:::

(1.30)

Here, the oeÆients

L and

R

are the left handed and right handed hiral

ouplingsorresponding to (f) inequation 1.29.

The masses of the mediator bosons W

and Z are aquired via the Higgs

mehanism. As has been shown by Peter Higgs in 1964 [36℄, it is possible

to generate masses for bosons (and fermions) without destroying gauge

in-variane (i.e. therenormalisabilityof the theory) by introduinga omplex

salareld,alledtheHiggseld. Themainassumptionofthismehanismis

anon-zerovauumexpetationvaluev fortheHiggseld. Withanon-zero

vauum expetationvalueitispossibleto generatesreeningurrentstogive

The masses of the weak bosons W

and Z are related to the vauum

expetation valuev and ouplingsg and g 0 via: M W = g 2 v M Z = p g 2 +g 0 2 2 v

Using1.26, M

W

and M

Z

arerelated bytheWeinbergangle:

os W = M W M Z (1.31)

Theexperimentallydetermined masses ofthe W

andZ bosonsare [24℄:

M

W

=80:4250:038 GeV M

Z

=91:1880:002 GeV

The vetor and axial vetor terms (

and

5

) in 1.30 lead to the

deni-tions of vetor and axial vetor harges (onstants) v

f

and a

f

for fermions

(Table1.2). The relationsof v

f anda f to L and R

are given by:

v f =2 L (i)+ R (i) a f =2 L (i) R (i) e ; ;

e;; u;;t d;s;b

e

f

0 -1 +2/3 -1/3

L

1/2 1=2+sin 2

W

1=2 2=3sin 2

W

1=2+1=3sin 2 W R 0 sin 2 W 2=3sin 2 W 1=3sin 2 W v f

1 1+4sin

2

W

1 8=3sin 2

W

1+4=3sin 2

W

a

f

1 -1 1 -1

Table 1.2: Couplingonstants of leptons and quarksto the Z boson. e

f is

1.2 Neutral and Charged Current Cross Setions

1.2.1 Unpolarised Lepton-Proton DIS Cross Setions

Inludingthe weak neutralurrent (Z exhange), the neutral urrent

dier-ential rosssetionfor unpolarisedinteratingpartilesis given as[14℄:

d 2 e p NC dxdQ 2 = 2 2 xQ 4 e p NC (1.32)

withtheredued NCrosssetion term

e p NC =Y + ~ F 2 Y x ~ F 3 y 2 ~ F L (1.33) Here,Y

=1(1 y) 2

istheheliityfator(seeequation 1.12), ~ F 2 , ~ F 3 and the ~ F L

are generalised struture funtions. ~

F

L

is thelongitudinal struture

funtionwithaontributiontotherosssetionproportionalto y 2

,therefore

onlyimportantintheveryhighyregion(inaddition, ~

F

L

issmallinthelarge

Q 2

rangeonsideredinthisthesis). The ~

F

L

termwillnotbedisussedfurther

inthisthesis.

~ F 2 and ~ F 3

an be expressed interms of ve struture funtions desribing

the ontributions from pure photon exhange, Z interferene and pure Z

exhange: ~ F 2 =F 2 v e W Q 2 Q 2 +M 2 Z ! F Z 2 +(v 2 e +a 2 e ) W Q 2 Q 2 +M 2 Z ! 2 F Z 2 (1.34) x ~ F 3 =a e W Q 2 Q 2 +M 2 Z ! xF Z 3 2a e v e W Q 2 Q 2 +M 2 Z ! 2 xF Z 3 (1.35)

Here,the purephoton exhange isdesribed byF

2

,pure Z exhange byF Z

2

and xF Z

3

, and Z interferene by F Z 2 and xF Z 3 . v e

is theweak vetor and

a

e

theweakaxial-vetor ouplingoftheeletronto theZ (see equation1.29)

andaregiveninTable1.2. TheWeinbergangle

W

(seeequation1.27)enters

thequantity

W

inthefollowingway:

W = 1 4sin 2 W os 2 W

In the Quark Parton Model (see setion 1.1.3) the struture funtions are

[F 2 ;F Z 2 ;F Z 2 ℄=x

X q [e 2 q ;2e q v q ;v 2 q +a 2 q ℄fq(x;Q 2

)+q(x;Q 2 )g (1.36) [xF Z 3 ;xF Z 3 ℄=x

X q [e 2 q a q ;2v q a q ℄fq(x;Q 2

) q(x;Q 2

)g (1.37)

Thesumsrunoverallquarkavorsq. Dueto thelimitedkinematirangeat

HERAthesumrunsovervequarkavors,i.e. theenterofmassenergy p

s

isnotsuÆient to produea topquark.

The Charged Current ross setion an be presented in a form similar to

theone of NCreations[14℄:

d 2 e p CC dxdQ 2 = G 2 F 2x M 2 W Q 2 +M 2 W 2 e p CC (1.38)

withtheredued CC rosssetion term e

p

CC

,also oftendenoted as~

CC : e p CC =~ CC =Y + ~ W 2 +y 2 ~ W L Y x ~ W 3 (1.39) Here, G F

is the Fermi onstant whih isrelated to the weak oupling g and

eletromagnetioupling e(seeeq 1.27):

G F = g 2 4 p 2 M 2 W = e 2 4 p 2sin 2 W M 2 W (1.40) ~ W 2 , ~ W L and ~ W 3

are struturefuntions analogous to the Neutral Current

ase (see eq. 1.33). Sine theCharged Current interations arepurely weak

proesses,W 2 ,W L andW 3

donotontaineletromagnetiandinterferene

terms. The generalised CC struture funtions for unpolarised interating

partilesan be expressed[37℄ as:

~ W 2 = 1 2 W 2 (1.41) x ~ W 3 = 1 2 xW 3 (1.42)

In the Quark Parton Model W 2 and W 3 (W L

= 0) are sensitive to

dier-enes ofquark andanti-quark distributionsand aregiven by:

e +

: W

+

2

=2x d+s+u+

; xW

+

3

=2x d+s [u+℄

e : W

2

=2x u++d+s

; xW

3

=2x u+ [d+s℄

ThebandtquarksdonotontributetoW

2

andxW

3

beauseoftworeasons:

First,thetquarkistoomassivetobeproduedintheHERAkinematirange,

seondly, the probabilityof the u ! b transition aording to the

Cabibbo-Kobayashi-Maskawa matrix isvery small(U

ub

,seeequation 1.21).

Aording to 1.43, the struture funtion term ~

CC

is related to the quark

densitiesvia

e +

: ~ e

+

p

CC

=x[u+℄+(1 y) 2

x[d+s℄

e : ~ e p

CC

=x[u+℄+(1 y) 2

x[d+s℄

(1.44)

Eletro-weak Uniation

Unpolarised neutral and harged urrent ross setions measured with the

H1 detetor at HERA I as funtion of Q 2

are shown inFigure 1.6. Asseen

inthe piture, NC and CC ross setionsbeome about equalin magnitude

at Q 2

& 10 4

GeV 2

. This follows from the propagator term (1=Q 2

) 2

for NC

and (1=Q 2

+M 2

W )

2

forCC interations introduinga Q 2

dependene whih

isdierent forNCand CCuntilQ 2

.M 2

Z(W) .

Thisso-alled eletro-weak uniation regionis preditedbytheeletroweak

setor of Standard Model as was desribed in setion 1.1.5 (see also

equa-tion1.40).

e +

p and e p Charged Current Cross Setions

Thedierenesbetweene +

p ande pharged urrentrosssetions(see

Fig-ure1.6) arisedueto thefollowingfators:

- the proton is omposed of two u and one d valene quark, e +

probes

thedquarkwhilee probestheuquark(seeeq. 1.43,1.44). Sinethe

ouplingstrengthof theW isthesame forallfermions,thisimpliesa

fatorof two intheeletron-quarkharged urrent interations;

- onservation of angularmomentum impliesa (1 os) 2

dependene

of the sattered for e +

p interations in the enter of mass (CM)

frame(baksatteringisnotallowed). Thedistributionoftheangular

momentumisatforine preations(seeFigure1.7). Conservation

of angular momentum bringsanotherfator of three dierene inthe

e +

p ande prosssetions;

- at high four momentum transfer squared Q 2

, the d quark

distribu-tion as funtion of x falls o faster at large x than the one of the

### 10

### -7

### 10

### -6

### 10

### -5

### 10

### -4

### 10

### -3

### 10

### -2

### 10

### -1

### 1

### 10

### 10

### 3

### 10

### 4

### Q

### 2

### /GeV

### 2

### d

### σ

### /dQ

### 2

### / pb GeV

### -2

### H1 e

### +

### p

### NC 94-00

### CC 94-00

### H1 e

### -

### p

### NC 98-99

### CC 98-99

### √

### s = 319 GeV

### y

### <

### 0.9

### H1 PDF 2000

### Neutral and Charged Current

### H1 Collaboration

Figure 1.6: Cross setions of unpolarised neutral (irles) and harged

ur-rent (boxes) deep inelasti sattering proesses as funtionof Q 2

, measured

with the H1 detetor. The e +

p ollisions are shown as full symbols, e p as

open symbols. Theresults areompared to the Standard Model expetations

using a NLO QCDt (band). From [18℄.

### θ

*e*

### +

*u*

*d*

### J = +1

_{z}

_{ν}

*e*

*u*

*d*

### ν

### θ

### J = 0

_{z}

Figure 1.7: Illustration of the angular dependene in harged urrent

rea-tions (in the CM frame, assuming the z diretion along the e

motion and

being the sattering angle of the ): (left) the angular momentum

ompo-nent J

z

is notonserved if the is sattered bakwards, i.e. the interation

amplitude is proportional to (1 os); (right) the interation amplitude

does not depend on beause J

1.2.2 DIS Cross Setions with Polarised Leptons

After the upgrade of HERA in the year 2000 (details of the upgrade will

be disussed in hapter 2), HERA has the apability to longitudinally

po-larisedthe leptonbeam. ThepolarisationoftheleptonbeamP

e

(eletronor

positron)is denedas:

P e = N R N L N R +N L : (1.45) Here,N R (N L

) isthe numberof right (left)handed leptons inthe beam. In

anunpolarisedbeam,thenumberofleftandrighthandedleptonsisthesame.

The harged urrent rosssetion of polarised leptonswith unpolarised

pro-tonsis preditedbythe StandardModel to have a linear dependene on the

leptonpolarisationP

e

(allvariablesasexplainedineq. 1.38):

d 2 e p CC dxdQ 2

= 1P

e G 2 F 2x M 2 W Q 2 +M 2 W 2 ~ e p CC (1.46)

As seen in the equation above, the total CC ross setion thus is equal to

zerofor"left-handed"(seeFigure1.5)positronsaswellasfor"right-handed"

eletrons.

Unlike for harged urrents, the ross setion for neutral urrent reations

is inuened by the longitudinal lepton polarisation only at high Q 2

. This

dependeneappearsviatheZ exhangeandanbeexpressedinthefollowing

way: d 2 e p NC dxdQ 2 = 2 2 xQ 4 h e p NC +P e e p NCpol i (1.47) Here, e p NCpol

is similar to e

p

NC

but involves polarised struture funtions.

Moredetaileddesriptionan be foundin[38℄.

1.2.3 Radiative Corretions

The ross setions presented in the last two setions are alulated at the

leading-order (LO or Born approximation, O( 2

)). The main ontribution

to higher order proesses omes from additional photon lines, either

inter-nal (virtual)or external (real). The emission of the real photon an hange

theep entre-of-mass energy and thus the event kinematis is also hanged.

Therefore the measuredross setions have to be orreted for higher order

radiative eets (typially of order few perents) denoted by the orretion

termÆ r

d 2 dxdQ 2 = d 2 dxdQ 2 Born 1

1+Æ r

(1.48)

More generally, the radiative orretions an be separated into two

ontri-butions: The ones rising from the eletromagneti and those from the weak

proesses:

1+Æ r

=(1+Æ QED

)(1+Æ weak

) (1.49)

ThealulationoftheradiativeorretionsfortheNCrosssetionsinvolves

asetofFeynmandiagrams, eah beingagaugeinvariant [39℄. Adetailed

dis-ussion aboutthe radiative orretionfor the NC sattering is given in [39℄.

Theseparationofthe QED- weakontributionsfortheCCradiative

orre-tionsisdesribed in[40℄ and willbe shortlydisussedinthissetion.

Radiative orretions for CC sattering

The four leading Feynman diagrams for radiative CC sattering (i.e. with

additionalphotons) areshowninFigure 1.8: The emission of a photonfrom

theinoming eletron (a), from the inomingand outgoingquark (b, )and

from the W (d). Similarly to the real orretions shown in Figure 1.8, the

virtualeletroweakone-loopdiagrams(orretionstotheeW andqq 0

W

ver-ties,self-energyorretionsoftheinvolvedpartiles)alsogiverisetoinfrared

divergenes. Afterombiningreal and virtualorretions,theinfrared

diver-genes aneleah other and the resultingQED orretions on the CCross

setion anbe expressedinthefollowingway [40℄:

d 2 dxdQ 2 = d 2 dxdQ 2 Born (1+e

2 l J l ep +e l e f J int +e 2 f J qua ) (1.50) Here, J l ep ; J int and J qua

are the "leptoni", "interferene" and "quarkoni"

ontributions,eahbeinggaugeinvariant;e

l ande

f

arehargesoftheeletron

and theinomingquark,respetively.

The terms J

int and J

qua

in the numerial programs for the alulation of

DIS ross setions are usually negleted (the results for the individual

on-tributionsareavailablein [41℄, [42℄). However, it is important to notie (for

thestudies presentedin setion 5.2.2) thatthe leptoniontribution J

l ep

in-volves terms ontaining the photon radiation of the inoming eletron

(Fig-ure 1.8 (a)), as well as the terms involving the outgoing quark radiation

(Figure1.8()).

As mentioned above, the CC radiative orretions also ontain purely weak

ontributions. They arisemostly dueto W self-energy terms. The inuene

*q’*

*q*

### γ

### ν

*e*

*W*

(a) (b) () (d)

Figure1.8: Feynman diagrams for radiative harged urrent sattering with

theemissionofthephotonfromtheinomingeletron(a),fromtheinoming

andoutgoing quark (b,),from W (d).

The photon radiationof the inomingeletron isommon for both, CC and

NC,andisusuallyalledInitialState Radiation(ISR).Thephotonradiation

of the outgoing eletron exists only in NC events and is alled Final State

Radiation(FSR).

1.3 Bakground Proesses to Charged Current

Thetypialsignatureofharged urrenteventsismissingtransverse

momen-tum originating from the esaping neutrino. Therefore, the bakground to

harged urrent interations are ep events with an apparent transverse

mo-mentum imbalane usually aused by imperfet detetion of the nal state

partiles. There aretwo reasons whya missing momentum an be measured

in transversely balaned events: Part of the event nal state may esape

detetion, or the measurement of the nal state was inaurate (resolution

eets).

The possiblebakground proesses to CC are photoprodution, neutral

ur-rent events, lepton pairand real W prodution. In thefollowingthese

bak-groundlassesare desribed 4

.

1.3.1 Photoprodution

Photoprodution (p) is a proesses where a quasi-real (i.e. almost on the

mass shell) photon, emitted by the eletron, is interating with the

pro-ton. The ross setion of events with photon exhange depends on the

four-momentum transfer as 1=Q 4

(see 1.32). Therefore, photoprodution is the

dominant proess asQ 2

! 0 (events with Q 2

&few GeV 2

areonsidered as

DISNCevents).

4

The ontributionof NC eventsto CCbakgroundand their rejetion methods are

TwoexamplesofphotoprodutionproessesarepresentedinFigure1.9. The

reation on theleft sideof the Figure 1.9usually isalled the diret proess

(the wholephoton takes part inthehard subproess witha parton from the

proton), while the reation on the right side represents a resolved proess

(the photon ats asa soure of partons, one of whih takes part inthe hard

subproess). Thehadroninalstateinthephotoprodutionproessonsists

of two(or more)jets.

### p

### q

### q

### e

_{e}

### p

### e

_{e}

### q

### q

Figure 1.9: Feynman diagrams for diret (left) and resolved (right)

photo-prodution proesses.

The photoprodution events an "obtain" a missing transverse momentum

in the detetor (i.e. beome bakground to CC) when some hadroni nal

statepartiles(e.g. 's)esapedetetion,and(or)byimperfetmeasurement

of thehadroninalstate dueto limited detetor aeptane.

Photoprodutioninterations arethemainbakgroundto CC events(at low

Q 2

therate of p events isa fewhundredtimes largerthan theCCrate).

1.3.2 Lepton-Pair Prodution

The dominant proess for lepton-pair prodution at HERA is ep ! epl +

l

or ep ! eXl +

l (the pair of leptons mainly originate from photon-photon

interations, photonsbeing radiated bythe eletron and the proton). If the

leptonpairinthenalstateis +

(Figure1.10),theeventmaylooklikethe

CCevent beause muons do notontribute muh to thealorimetri energy.

If thelepton pair onsist of eletrons, they will be measuredin thedetetor

and will usually not ause any missing transverse momentum. The typial

rateof thelepton-pair event prodution proess isabout8eventsper1pb 1

of luminosity(theCCevent rate is about40 events per1 pb 1

### µ

### µ

### e

### _

### +

### p

### X

### e

Figure 1.10: Feynman

dia-gramoftheleptonpair

produ-tion: ep ! e +

X.

1.3.3 W Prodution

AnotherrareproessisrealW

prodution. TheW

isproduedinep ! eWX

or ep ! WX reations. When the boson deays leptonially, the event

nal state may mimi a harged urrent ration dueto the produed . An

example diagram where the W deays into

is shown in Figure 1.11.

TherealbosonprodutionproesshasaverysmallrosssetionatHERA[44℄

(typiallyaboutone event isexpeted per2 pb 1

of luminosity).

### _

### µ

### ν

### µ

### X

### W

### e

### p

### W

### e

Figure 1.11: Feynman

dia-gram for real W

prodution:

1.4 Monte Carlo (MC) event generators

Monte Carlo event generators areprograms to simulate partilereations in

the detetor. MC's are used in various physis analyses as a powerful tool

to estimatedetetor eets, whihannot be determinedfrom thedata. The

mainreasonsforsimulatingspei physisproessesinthisanalysis are:

thedeterminationof thedetetor aeptane,

theontributionof thebakgroundproesses,

thedeterminationof theeÆieny ofseletion uts,

theestimationof radiative eets.

Allthese pointswillbe disussedinthefurther hapters.

The mainstepsto produeMonteCarlo simulatedspei physis proesses

areshortlydesribed below.

Generation of the Spei Physis Proesses

Event generators ontain the Born level QCD matrix elements of hard

pro-esses. The rst step to produe MC simulated proesses is the random

event generation initialised aording to these matrix elements spei for

the physis proess. The next step is simulation of parton showering and

hadronisationproesses.

Parton Showering and Hadronisation

Thereation of hadronistates from theinitialpartonsinvolvestwo distint

proesses,parton showering and hadronisation. Higher orderQCD radiation

is representedby leadinglogarithmi parton showers. In this stage the high

energyprimary partonslose theirenergyradiatingseondary partonswhih,

inturn,produeothers and soon. These branhingsof partonsare done

a-ordingthesplittingfuntionsasdesribedinsetion1.1.4 (Figure1.4). The

proess stops when the energiesof the partons beome too small, i.e. below

1 GeV. At suh small energies the strong oupling,

s

, is too large and

theshoweringproesses annotbe desribedanymore byperturbative QCD.

Atthisstage, thenalstate onsists ofmany"free"partons, i.e. quarksand

gluons. Then hadronisation takes plae whih employs empirial models to

desribe theformationof the hadroni nalstate. Here,olored partons are

boundinto olorlesshadrons.

A shemati view ofthee

p sattering proess illustratingpartonshowering

and hadronisationis given inFigure1.12.

### hadro-p

### e

### e

### ’

### fragmentation

### partons

### hard scattering

### parton shower

### perturbative

### not perturbative

### hadrons

### fragmentation

### H1 - detector

### detector objects

### simulation

### reconstruction

### hadron decay

### hadronisation

Figure1.12: Shemati illustration of the epsattering proess over

pertur-bative (hard sattering) and non-perturbative proesses inluding the

simu-lation of partiles and their reonstrution (reonstrution is desribed in

hapter 3).

nisationproesses. Oftenused(inthisanalysisaswell)aretheColourDipole

Model (CDM) [46℄ for the parton showering and the Lund String F

ragmen-tation [47℄ forhadronisation. Asa result of the generationa list of thenal

state partiles,haraterisedbytheir four-vetors, isreated.

Generators used in the present analysis

ChargedandneutralurrenteventsweregeneratedwithDJANGOH1.2[48℄.

Thelepton-proton sattering inDJANGOH is based on LEPTO[49℄

inlud-ingleading order QED orretions with the HERACLES [50℄ program. The

parton showeringis generated aording to ARIADNE [51℄ usingtheColour

DipoleModel. JETSET[52℄isusedtosimulatethehadronisationproesswith

theLundStringFragmentation model. The parametrisationofMRSH [53℄is

usedfortheprotonPDFandattheanalysislevelisorretedtothe

parametri-sationof H1 PDF2000 [18℄.

Photoprodution proesses are generated with PYTHIA [54℄. The leading

order parametrisation CTEQ[55℄ is usedfor the proton PDF and GRV [56℄

forthephotonPDF.

Leptonpair produtionproesseswere generatedwith GRAPE [57℄.

RealW

produtionmehanismswere generated withthe EPVEC[44℄

Inthisanalysis100000ofgeneratedhargedurrenteventswereused,almost

1millionofNCevents,3millionofphotoprodutioneventsandmore than

400 000 oflepton-pair and W produtionevents.

Detetor Simulation

In this step, whih is the same for all MC generators, the interations of

the generated partiles with the detetor material are simulated. This is

donebypropagatingthegeneratednalstatepartilesusingtheirfour-vetor

information and simulating random interations with the dierent detetor

omponents. The simulationof eletromagneti and hadronishowers inthe

alorimetersis donewitha fast parametrisationprogram H1FAST [58℄. The

detetor response is alulated from the simulated interations, ionisations

and energydeposits.

IntheH1experimentafulldetetorsimulationisreatedwiththeH1SIM[59℄

softwareusingtheGEANT [60℄ program.

Partile Reonstrution

The simulated events are subjeted to thesame reonstrution and analysis

hainasthedata. Therelevantpartilereonstrutionmethodsforthis

anal-ysis(same fordataand MC)are desribedinhapter 3.

MC toData Corretion

After thesimulationand reonstrution steps,some detetor eets maynot

befullymodelledintheMCevents. Therefore,itisimportantattheanalysis

leveltoensurethattheMCsimulationorretlymodelsthedetetorresponse

anddesribesthedataineveryanalysisaspet. Aswillbedesribedindetail

inhapters4-6, inase ofobserved disrepanies,thesimulation isorreted

The H1 experiment at

HERA

TheHERA("HadronElektronRingAnlage")aeleratoristheonlymahine

world-wideollidingeletronswithprotons. Two detetorsto registerep

ol-lisionsweredesigned andbuiltintheendoftheeightieswiththemajortask

istostudythestrutureoftheproton. ThekinematirangeofHERAismore

thantwo ordersof magnitudelarger thantherange aessible sofarinxed

target experiments(see Figure 2.1).

In thishapter a shortdesription of the HERAaelerator and the H1

de-tetor is given. Emphasis is put on the H1 trigger system and the studies

performed toredue theDISevent triggerratesusingtheseondlevelneural

network trigger (L2NN).

2.1 HERA Aelerator

The HERAaelerator (Figure 2.2) is loated at DESY (Deutshes

Elektro-nenSynhrotron),Hamburg. Ithastheirumfereneof6.3kmandseparate

storageringsforeletronsandprotons. Eletronsareaeleratedto27.6GeV

and protonsto 920 GeV (820 GeV before 1998). The energy of theeletron

beam energyis limited by synhrotronradiationwhilethe p beam islimited

inenergybythestrengthof themagnetield ofthesuperondutingdipole

magnets.

The ollisionsat the enter-of-mass energy p

s=318 GeV (301 GeV before

1998)takeplaeintwointerationregionssurroundedbytwolarge

multipur-pose detetors, H1 [61℄ and ZEUS[62℄. In addition, there is one operational

xed-target experiment, HERMES [63℄, where the eletron beam is brought

into ollision with polarisedgas targets in order to study thespin struture

of theproton. The seond xed-target experiment, HERA-B[64℄, where the

proton beam was used to produenal states withb quarks, has been

**x**

**Q**

**2**

** (GeV**

**2**

**)**

**E665, SLAC**

**CCFR, NMC, BCDMS,**

**Fixed Target Experiments:**

**ZEUS**

**H1**

**10**

**-1**

**1**

**10**

**10**

**2**

**10**

**3**

**10**

**4**

**10**

**5**

**10**

**-6**

**10**

**-5**

**10**

**-4**

**10**

**-3**

**10**

**-2**

**10**

**-1**

**1**

Figure 2.1: Kinemati plane (x;Q 2

) of HERA and omparison with some

xed target experiments.

The aelerator rings an store up to 210 partile bunhes for eah of the

eletron and proton beams (due to limitations of the injetion system only

180 olliding bunhes are stored routinely). Eah bunh ontains

approxi-mately10 11

partiles and are separated by 96 nstime intervals. In addition

to the olliding bunhes, there are usually about 10 bunhes in eah beam

without olliding partners (pilot bunhes). Measuring reations indued by

suh eletron orproton pilotbunhesenablesstudyingnon-epindued

bak-ground(forexample,beam-gaseventsoriginatingfrom protonollisionswith

theremaininggas nuleiin thebeam).

HERA was suessfully running from the year 1992 until the end of 2000

("HERA I"). After a longshutdown(lasting fromthe years 2001 and 2002),

theseond, "HERA II",period wasstarted. Two majorimprovementshave

been ahieved during the shutdown. New foussing magnets were installed

insidethedetetors,leadingto signiantinreaseof theinstantaneous

lumi-nosityL,whihis denedas:

L= fN

e N

p

4

x

y h

1

m 2

s i

Figure2.2: Shemati view of the HERA aelerator.

Here, f is thebunh rossingfrequeny, N

e

and N

p

are the numberof

ele-tronsandprotons ineah bunh,and

x and

y

are theGaussiantransverse

beam prolesinthe xand y diretions at theinterationpoint.

Anyphysial ross setion is related with the numberof observed events N

and theintegrated luminosityL as:

=

N

L

Theintegratedluminosityintheequation above isgiven asL= R

L dt. The

integrated luminosityolletedbyH1 for theHERA Iand HERA IIperiods

(until the endof 2005) versus time is shown in Figure 2.3. From the gure

one an see that the installation of the foussing system has improved the

instantaneousluminositybyaboutafator of three.

The seond improvement of HERAIIis the possibilityto longitudinally

po-larise the eletron beam (for physis explanations and onsequenes of

lon-gitudinallypolarised leptons to DIS reations, see setion 1.2.2). The

longi-tudinaleletronbeampolarisationisahieved installingspinrotatorsaround

**Days of running**

**H1 **

**Integrated **

**Luminosity **

**/ **

**pb**

**-1**

**0**

**500**

**1000**

**1500**

**0**

**100**

**200**

**electrons**

**positrons**

**HERA-1**

**HERA-2**

Figure 2.3: H1 integrated luminosity as funtion of time for HERA I and

HERAIIperiods, up to the end of the year 2005.

2.1.1 Longitudinal e Beam Polarisation at HERA II

The eletrons at HERA beome transverselypolarisedthrough the emission

of synhrotron radiation (the Sokolov-Ternov eet [65℄): When eletrons

move inlosed orbits guidedby themagneti dipole elds of a storage ring,

theyemitsynhrotronradiation; averysmallfrationoftheemittedphotons

will ause a spin-ip between "up"and "down" quantum states of the

ele-tronspin. Sinetheprobabilitiesofthe"up-to-down"and"down-to-up"spin

statesaredierent,theinitiallyunpolarisedeletron beambeomespolarised

withtimeaording to:

P(t)=P

max (1 e

t=

) (2.2)

Here, P

max

is a theoretial limit for the maximalpolarisation and is '0:93

(not taking into aount possible depolarisation eets); is the build-up

time. For a storage ring with radius R and eletron energy E

e

, R

2

=E 5

e

whih amounts to 40 minat HERA.

To ahieve longitudinaleletron polarisationfortheepinterationsthe

verti-al polarisationis rotated into longitudinal just before theinterationpoint

and, in order to maintain stable beam polarisation, it must be rotated bak

to vertial immediately after. At HERA II, the hains of speial magnets,

Figure 2.4: Shemati viewof the HERAring after the upgrade.

(theHERMES experiment already had suh spinrotators duringthe HERA

I period. The shemati representation of theHERA ring after the upgrade

withinstalledspinrotators is showninFigure 2.4.

Theahievabledegreeoflongitudinaleletronpolarisationislimitedbymany

fatorswhih have to betakenintoaount buildingup andmaintainingthe

polarisation in the storage ring. For example, the so-alled "spin mathed"

rotator optis is inuened by the magnet alignment and positioning

prei-sion, eld errors and orbit orretions, number of beam position monitors,

et. The tehnique usedfor optimisingthepolarisationin theHERAring is

alledharmoni losed orbit spin mathing and isdesribedin[67,68℄.

The typial degree of polarisation ahieved at HERA II is 40%, asshown

inFigure2.5.

Polarisation Measurement

There aretwo tehniques to measure thepolarisation ineletron sattering:

Moller sattering, ee ! ee, and Compton sattering, e ! e. The rst

method is experimentally simple, but limited to beam urrents I

e

5A.

The seond method is more ompliated to implement but is fast and

au-rate,andisthereforeroutinelyusedinexperimentsoperatingwithhighbeam

urrents.

BothlongitudinalandtransverseeletronpolarisationmeasurementsatHERA

rely on the same physial priniple: Spin dependent Compton sattering of

** 0**

** 10**

** 20**

** 30**

** 40**

** 50**

** 60**

** 70**

**0**

**2**

**4**

**6**

**8**

**10**

**12**

**14**

**16**

**18**

**20**

**22**

**24**

** 0**

** 10**

** 20**

** 30**

** 40**

** 50**

** 60**

** 70**

** HERA-e Polarisation on Tuesday May 04 2004**

**Polarisation [%]**

**Time [h]**

**Transverse**

**Longitudinal**

Figure 2.5: Typial HERA II longitudinal and transverse polarisation as

funtionof time.

The ross setion for the Compton sattering proess, e ! e, is a well

known QEDproess,expressedas [69,70℄:

d 2 dEd = 0

(E)+S

1

1

(E)os2+S

3 P Y 2Y

(E)sin+P

Z 2Z (E) (2.3) S 1 andS 3

arethe linearand irularomponents ofthe photonbeam

polar-isation 1 , P Y and P Z

are the transverse and longitudinal omponents of the

eletron beampolarisation,

i

arealulableterms dependingon thephoton

energy. From theequation above onean see that

- ameasurement ofthe polarisationan beperformedbyswithingthe

sign of S

3

(maximising the irular polarisation S

3

! 1 and thus

minimisingthelinearterms p S 2 1 +S 2 2

!0). Thisresultsinan

asym-metry whihis proportionalto P

z ;

- ifthepolarisationofthe laserbeamis known,thelongitudinal

polar-isation of the eletrons an be determined from the azimuth photon

energy distribution; to determine the transverse polarisation, in

ad-dition to the energy distribution, the azimuthal distribution of the

photonhasto be measured.

Therearetwopolarimeterswhihindependentlymeasurethedegree of

trans-verse and longitudinal polarisation at HERA: The Transverse Polarimeter

(TPOL[71℄) and theLongitudinalPolarimeter(LPOL[72,73℄). TPOL

mea-sures the spatial energy asymmetry (inluding the azimuthal information),

LPOL measurestheenergy asymmetry. As thespinrotatorsystem doesnot

hange the degree of leptonbeam polarisation, the measurements of TPOL

and LPOLmust beonsistent inmagnitude.

1

Generally,todesribelightpolarisationtheStokesvetorSisused. Forthepolarised

lightS= p S 2 1 +S 2 2 +S 2 3