of the Proton Structure Function F
2
at Low Q
2
and Very Low x
at HERA
by
of the Proton Structure Function F
2
at Low Q
2
and Very Low x
at HERA
DISSERTATION
zur Erlangungdes Doktorgrades
des Fachbereichs Physik
der Universitat Hamburg
vorgelegt von
Ulrich Fricke
aus Hamburg
Hamburg
Prof. Dr. F.-W. B uer
Gutachterder Disputation: Prof. Dr. E.Lohrmann
Prof. Dr. J. Meyer
Datumder Disputation: 15.12.1999
Dekan des
Fachb ereichsPhysikund
Vorsitzenderdes
The kinematic region covered bythe two HERA exp eriments H1 and ZEUS in the measurement of
thetotalvirtualphoton-proton( p)crosssection p tot
andtheprotonstructurefunction F
2
hasb een
signi cantly extended since the start of datataking in 1992. In 1995 the two exp eriments extended
their kinematic acceptance to prob e the transition region b etween the regime of p erturbative QCD
(pQCD,Q 2
1:5GeV 2
) andthephotopro duction region(Q 2
0 GeV 2
). Byshifting theinteraction
p oint in b oth exp eriments the lower limit in Q 2
was extended down to 0.6 GeV 2 . To access even lowervalues ofQ 2 (0.11{0.65GeV 2
),theZEUS BeamPip eCalorimeter(BPC)wasinstalled in 1995.
To furtherextend thekinematic acceptance of theBPC and decreasethe systematicuncertainties, a
new detector, the Beam Pip e Tracker(BPT), wasinstalled in front of the BPC in 1997. It consists
of two silicon microstrip detectors and is lo cated b etween the BPC and the interaction p oint. By
makinguseoftheBPT,themainsystematicuncertainties relatedtotheBPC(energycalibrationand
alignment), tophotopro duction background,and totheuncertaintyoftheinteraction p oint p osition,
werereducedsigni cantly. Thetotalsystematicerrorwasreducedbyroughlyafactoroftwotothree.
The kinematic region was extended towardslower values of Q 2
and towardslower and higher values
ofx. Atverylowvalues ofQ 2
andverylowvaluesofxthemeasurementwasextendedintopreviously
unexplored areas, while thenew data at highx allows forthe rst time a comparison with the data
from the xed-target exp eriment E665. p tot and F 2
have b een measured in inelasticneutral current
scattering,e +
p!e +
X,using theZEUSdetector atHERA.The analysiscoversthekinematic region
for0:045Q 2 0:80GeV 2 and 310 7 x10 3
. This corresp onds toa rangein the
p
center-of-mass energy of 25 W 281 GeV. The data is compared to various mo dels for the low x and
lowQ 2
region. It canb ewell describ ed bya phenomenologicalmo del basedon Reggetheoryand the
Generalized Vector Dominance Mo del. Deviations of the data from this mo del and the comparison
topredictionsfromother mo delsindicate thatthee ectsof p erturbative QCDarealreadypresent at
Q 2 aslowas0.5GeV 2 . Zusammenfassung
DerMeb ereichderb eidenHERA-Exp erimente,H1undZEUS,zurBestimmungdestotalen
Wirkungs-querschnitts
p
tot
und derProton-StrukturfunktionF
2
konnteseitdemAnfangderDatennahme
deut-lich vergroertwerden. Beide Exp erimentedehnten ihrenMeb ereich zu kleinen Werten von Q 2
aus,
um die
Ub ergangsregion zwischen dem Wirkungsb ereich der p erturbativen QCD (pQCD, Q 2
1:5
GeV 2
) und dem Photopro duktionsb ereich (Q 2
0 GeV 2
) zu untersuchen. Durch Verschiebung des
WechselwirkungspunkteskonntedieuntereGrenzeinQ 2
biszu0.6GeV 2
ausgedehntwerden. Umauch
denBereichvon0:11Q 2
0:65GeV 2
zuuntersuchen,wurde1995dasZEUS-Strahlrohrkalorimeter
(BPC) installiert. Zur weiteren Ausdehnung des Meb ereiches und zur Verb esserung der
Mege-nauigkeit wurde 1997 ein weiterer Detektor, der Beam Pip e Tracker (BPT), vor dem BPC
instal-liert. Dieser auszweiSiliziumstreifend etektoren b estehende Detektor wurdezwischen dem BPCund
demWechselwirkungspunkteingebaut. DurchseineVerwendungkonntendiedominierenden
Unsicher-heitenderMessung,gegeb endurchdasBPC(EnergiekalibrationundPositionierung),dieAbschatzung
desPhotopro duktionsuntergrundesund dieBestimmung des Wechselwirkungspunktes umeinen F
ak-tor zwei bis drei verringert werden. Der Meb ereich konnte zu kleineren Werten von Q 2
, sowie zu
kleineren und groeren Werten von x, ausgedehnt werden. Wahrend die ersten b eiden
Erweiterun-gen inunerschlossene Bereiche gehen,erlaubt dieletztereeinen Vergleich mit DatendesExp eriments
E665. p tot andF 2
wurdeninderReaktione +
p!e +
X mitdemZEUS-Detektorb eiHERAgemessen.
Die hier b eschrieb ene Analyse wurde im Bereich 0:045Q 2 0:80 GeV 2 und 310 7 x 10 3
durchgef uhrt. Dies entspricht einem Bereich der
p-Schwerpunktsenergie von 25 W 281 GeV.
DieErgebnisse wurdenmit Vorhersagenf ur denBereich von kleinenx und Q 2
verglichen. DieDaten
konntendurcheinphanomenologischesMo dellgutb eschrieb enwerden. Abweichungenvonden
Vorher-sagen dieses sowie anderer Mo delle lassen vermuten, da der Ein uder p erturbativen QCD b ereits
Michaela
and
1 Introduction 1
2 Theoretical background 3
2.1 A brief history of lepton-nucleon scattering . . . 3
2.2 The lowQ 2 and very lowx region . . . 4
2.3 De nition of the kinematicvariables. . . 5
2.4 Structure functions . . . 7
2.5 Virtual photon-proton scattering . . . 8
2.6 The Quark Parton Mo del (QPM) . . . 10
2.7 The QuantumChromo dynamics (QCD) . . . 11
2.7.1 Factorization . . . 13
2.7.2 The DGLAP equations . . . 13
2.7.3 The BFKLequation . . . 15
2.7.4 The CCFM equation . . . 16
2.7.5 Saturation . . . 16
2.8 The transition region . . . 16
2.8.1 Vector Dominance Mo del. . . 17
2.8.2 Regge theory . . . 19
3 HERA and DIS experiments 21 3.1 Deep Inelastic Scattering (DIS) . . . 21
3.2 HERA designand exp eriments. . . 23
3.3 Structure function measurementsat HERA . . . 24
3.4 Reconstruction of kinematicvariables at HERA . . . 24
4 The ZEUS detector at HERA 31 4.1 The maindetector . . . 32
4.1.1 The Central TrackingDetector . . . 33
4.1.2 The uraniumcalorimeter . . . 34
4.2 Proton and neutron detectors . . . 36
4.3 The luminositydetector and electrontaggers . . . 36
4.4 The ZEUStrigger and data acquisitionsystem . . . 37
4.5 Eventreconstruction and analysis . . . 38
5 The Beam Pipe Calorimeter and Beam Pipe Tracker 41 5.1 BPC design . . . 41
5.2 BPC readout and trigger . . . 44
5.5 Commissioning of the BPT. . . 48
5.6 BPT data quality monitoring . . . 50
5.7 MC simulationofBPC andBPT . . . 51
6 Detector studies 53 6.1 Intro duction . . . 53
6.2 BPC p ositionreconstruction . . . 53
6.3 BPC timeandshower width reconstruction . . . 55
6.4 Preliminary alignmentof the BPC. . . 56
6.5 BPT track reconstruction . . . 59
6.6 BPT vertexreconstruction . . . 59
6.7 Alignmentof the BPT . . . 61
6.8 BPT eÆciency. . . 63
6.9 BPT p ositionand angular resolution . . . 66
6.10 BPC energyreconstruction and calibration . . . 68
6.10.1 Energy reconstruction . . . 68
6.10.2 Energy calibration . . . 69
6.10.3 Estimationof the BPC energy non-linearity . . . 72
6.11 BPC ducialarea . . . 76
7 MC generation 79 7.1 Signal events . . . 79
7.2 Mo di cations to RAPGAP . . . 82
7.3 Mixing of DJANGOHand RAPGAP events . . . 82
7.4 Background MCevents . . . 85
8 EÆciency and data quality studies 87 8.1 Intro duction . . . 87
8.2 Vertex and b eamtilt . . . 87
8.3 BPC timing . . . 91 8.4 BPT eÆciency. . . 92 8.5 BPC triggereÆcicency . . . 92 9 Event selection 97 9.1 Trigger selection. . . 97 9.2 Reconstruction . . . 99
9.2.1 Reconstructionof BPC and BPT quantities . . . 99
9.2.2 Reconstructionof the hadronic nal state. . . 99
9.2.3 Reconstructionof kinematicvariables . . . 99
9.3 Background reduction . . . 100
9.4 Analysis cuts . . . 101
9.5 E ects of the selectioncuts . . . 112
9.6 Comparison of data and MC . . . 112
10 Extraction of p tot (W 2 ;Q 2 ) and F 2 (x;Q 2 ) 121 10.1 Intro duction . . . 121
10.2 Binning ofthe data . . . 121
10.3 Determinationof p tot and F 2 . . . 127
10.3.1 Treatment ofBPC andBPT eÆciency . . . 129
10.3.2 Treatment of (F L ). . . 130
10.3.3 Treatment ofthe radiative correction . . . 130
10.3.4 Unfolding of p tot and F 2 . . . 134
10.4 Evaluation of the systematic uncertainties . . . 136
10.4.1 Systematicerrors related to the p ositron identi cation . . . 136
10.4.2 Systematicerrors related to the mainZEUSdetector . . . 137
10.4.3 Systematicerrors related to the MC eventsimulation . . . 138
10.4.4 Othersources of systematic uncertainties . . . 138
10.5 Results on p tot and F 2 . . . 140 11 Results 147 11.1 Intro duction . . . 147
11.2 The functionalform of p tot used inthe unfolding . . . 147
11.2.1 The Q 2 -dep endence of p tot . . . 148
11.2.2 The W-dep endence of p tot . . . 153 11.3 Mo dels for p tot and F 2 and inthe lowQ 2 and very low xregion . . . 154
11.3.1 Abramowicz,Levin,Levy, Maor (ALLM, ALLM97) . . . 154
11.3.2 Adel,Barreiro, Yndurain (ABY) . . . 154
11.3.3 Badelek,Kwiecinski(BK) . . . 155
11.3.4 Cap ella,Kaidalov, Merino, Tran-Than-Van (CKMT,CKMT98) . . . 155
11.3.5 D'Alesio,Metz,Pirner (DMP) . . . 155
11.3.6 Desgrolard, Lengyel, Martynov (DLM) . . . 156
11.3.7 Donnachie, Landsho (DL,DL98) . . . 156
11.3.8 Golec-Biernat,W ustho (GBW) . . . 157
11.3.9 Haidt(HAIDT) . . . 157
11.3.10Martin, Ryskin,Stasto (MRS) . . . 157
11.3.11Schildknecht,Spiesb erger (SCSP) . . . 158
11.4 Comparison of F 2 to various mo dels . . . 158 11.5 Slop e of F 2 . . . 161 12 Conclusions 165 A Bin de nitions 169 B BPC trigger cuts 172
2.1 Lowest orderFeynmandiagram describing unp olarized ep scattering . . . 5
2.2 Resolution of the proton substructure . . . 7
2.3 Exp ectedb ehaviour of F 2 for a certainsubstructure of the proton . . . 12
2.4 Range of validity for various evolutionequationsin the (x Q 2 )-plane . . . 14
3.1 Kinematiccoverage inthe (x Q 2 )-plane of xed-target and HERA exp eriments . 21 3.2 Aerialview ofthe DESY lab oratory . . . 22
3.3 The HERA accelerator complex . . . 23
3.4 Isolinesof the primarymeasuredvariables. . . 25
3.5 Schematicsof the nal state inneutral currentep scattering. . . 26
4.1 The mainZEUSdetector alongthe b eamdirection . . . 31
4.2 The mainZEUSdetector p erp endicular to the b eamdirection . . . 32
4.3 Layout of a CTDo ctant . . . 33
4.4 Layout of a FCALmo dule . . . 34
4.5 Lo cation of ZEUSdetectors in p ositive Z-direction . . . 35
4.6 Lo cation of ZEUSdetectors in negativeZ-direction. . . 36
4.7 Schematicdiagram ofthe ZEUStrigger, data acquisitionsystem,and software . . 38
5.1 BPC mo dules and mo di ed b eampip e . . . 41
5.2 CAD drawing of the BPC mo dules . . . 42
5.3 BPC triggercon guration in1997 . . . 44
5.4 The BPT in1997 . . . 46
5.5 Determinationof BPT delaytime . . . 49
5.6 Determinationof BPT threshold . . . 51
6.1 Resolution and bias of the BPC p osition reconstruction . . . 55
6.2 BPC alignmentusing elastic QED Compton events. . . 57
6.3 BPT strip clustering. . . 58
6.4 Comparison ofMC and data: Reconstructed Z-vertex . . . 60
6.5 BPT vertexresolution . . . 61 6.6 BPT eÆciency . . . 64 6.7 BPT eÆciencycorrectionvs XCUT and X BPC . . . 65 6.8 BPT noise . . . 66
6.9 Angular resolution of BPT and BPC. . . 67
6.10 BPC energy uniformity . . . 70
6.11 BPC energy scale . . . 71
6.12 BPC radiation dose pro le and accumulateddose . . . 73
6.13 Schematicsof 60 Co scans of the BPC . . . 74
6.16 Determinationof the BPC ducialarea . . . 77
7.1 Di ractiveand non-di ractive eventpictures . . . 79
7.2 Generated x, y,and Q 2 distributionsfor b oth MCsamples . . . 81
7.3 max distributions fordata and MC . . . 83
7.4 Parametrizationof the di ractivefraction . . . 84
8.1 Mean X-, Y-, and Z-vertexin the data as afunction of the run numb er . . . 88
8.2 Determinationof the p ositron b eamtilt . . . 89
8.3 Positronb eamtilt as a functionof the run numb er. . . 90
8.4 BPC timing . . . 91
8.5 BPT eÆciencyas a functionof the run numb er . . . 93
8.6 FLTand SLTeÆciency . . . 94
8.7 TLT eÆciencyin the low y region . . . 95
9.1 BPC triggercon guration used for the analysis . . . 98
9.2 Comparison ofMC and data: Low y region (partone) . . . 102
9.3 Comparison ofMC and data: Low y region (parttwo) . . . 103
9.4 Comparison ofMC and data: Mediumy region (partone) . . . 104
9.5 Comparison ofMC and data: Mediumy region (parttwo) . . . 105
9.6 Comparison ofMC and data: Highy region(part one). . . 106
9.7 Comparison ofMC and data: Highy region(part two) . . . 107
9.8 Comparison ofMC and data: Low y (prescaled) region(part one) . . . 108
9.9 Comparison ofMC and data: Low y (prescaled) region(part two) . . . 109
9.10 Comparison ofMC and data: ISR region (partone) . . . 110
9.11 Comparison ofMC and data: ISR region (parttwo) . . . 111
9.12 Reduction of b eam-related background by the BPT . . . 114
9.13 Reduction of photopro duction background by the BPT . . . 115
9.14 Estimationof the photopro duction background . . . 116
9.15 Backgroundestimationusing the BPT hitmultiplicity . . . 118
9.16 Backgrounddistribution in the (x Q 2 )-plane. . . 119 10.1 Resolution in yand Q 2 . . . 122 10.2 Selected(y Q 2 )-bins . . . 123 10.3 Fractional y and Q 2 resolution p er (y Q 2 )-bin . . . 124 10.4 Migrationof y and Q 2 inthe (x Q 2 )-plane . . . 125
10.5 Bin quality factors for each(y Q 2 )-bin . . . 126
10.6 BPC triggereÆciencyp er (y Q 2 )-bin . . . 129
10.7 Angular distribution ofISR and FSR photons . . . 131
10.8 Migrationof events due to ISR . . . 132
10.9 F 2 as determinedinthe ISR bins . . . 133
10.10 Individual systematicerrors of F 2 for all bins (part one) . . . 139
10.11 Individual systematicerrors of F 2 for all bins (part two) . . . 140
10.12 p tot as a functionof Q 2 for xedvalues of W . . . 141
10.13 p tot as a functionof W 2 for xed values of Q 2 . . . 142 10.14 F 2 (x;Q 2 ) as a function of x for xedvalues of Q 2 (Q 2 >0:20 GeV 2 ) . . . 143 10.15 F 2 (x;Q 2 ) as a function of x for xedvalues of Q 2 (Q 2 0:20 GeV 2 ) . . . 144 11.1 Extrap olation of p tot to Q 2 =0GeV 2 . . . 149
11.2 Comparison of 0 to direct measurementsat Q 2 = 0 GeV 2 and results of ts to
the W-dep endence of p 0 . . . 150 11.3 R= p L = p T as a function ofQ 2 . . . 152 11.4 Comparison ofF 2 to several mo dels . . . 159 11.5 Comparison ofF 2 intermsof 2
=bin to various mo dels . . . 160
11.6
e
as a function of Q 2
3.1 HERA parameters . . . 24
5.1 BPC p erformancesp eci cations . . . 43
5.2 Sp eci cations of the BPT silicon microstripdetectors . . . 48
6.1 Parametersused inthe BPC p osition reconstruction for data and MC . . . 54
6.2 BPC alignmentin 1997 . . . 56
6.3 BPC and BPT alignmentin 1997 . . . 62
6.4 Maskeddead and noisyBPT strips . . . 63
9.1 E ects of the selectioncuts for data . . . 112
9.2 E ects of the selectioncuts for MC . . . 113
10.1 Summaryof bin quantities . . . 126
10.2 Results on the estimationof F 2 inthe ISR region. . . 134
10.3 Results on the measurementof F 2 and p tot (part one, Q 2 >0:25 GeV 2 ) . . . 145
10.4 Results on the measurementof F 2 and p tot (part two,Q 2 0:25 GeV 2 ) . . . 146
11.1 Parametersofthe functionalformof p tot used in the unfolding . . . 148
11.2 Extrap olated crosssection at Q 2 = 0 GeV 2 . . . 151 11.3 W-dep endence of p tot extrap olated to Q 2 = 0GeV 2 . . . 153 11.4 Comparison ofF 2 to various mo dels . . . 161 11.5 e as obtained froma t to the BPT and E665 data . . . 162
A.1 Bins of the ISR regionin the (y Q 2 )-planeused to estimatethe uncertainty due to radiativecorrections . . . 169
A.2 Bins inthe (y Q 2 )-plane used to extract F 2 and p tot (part one, Q 2 >0:25 GeV 2 ) 170 A.3 Bins inthe (y Q 2 )-plane used to extract F 2 and p tot (part two,Q 2 0:25 GeV 2 ) 171 B.1 1997 BPC FLTand SLT trigger cuts . . . 172
Introduction
What isthe structure of matter? This isone of the oldestquestions askedby men. Starting in
the 18th centurygreatprogress has b een madeto answerthis question. Deep erunderstanding
intheb ehaviourof chemicalelementsleadto theintro ductionof thep erio dictableof elements
by Dmitri Mendelejew in 1869. In 1897 the electron was discovered by Thompson. Exp
eri-ments by Rutherford together with Geiger and Marsden (1909{1911) revealedthat the atoms
consist of a tiny p ositively charged nucleus of less than 20 fmdiameter made out of protons
and neutrons and electronscirculating around it. Withthe discoveryof the neutron by James
Chadwick in 1932 all constituents of the atom were discovered. However, analysis of cosmic
rays and the data fromthe rst particle acceleratorslead to the discovery of several hundreds
of hadrons by the 1960s. The substructure of the hadrons rst prop osed by Gell-Mann and
Zweig in 1964 [Ge64, Zw64] and later by Feynman [Fe69] was exp erimentally con rmed by
the rst inelastic electron-proton scattering exp eriments at the Stanford Linear Accelerator
Center (SLAC) and later by several xed-target exp eriments. In the Standard Model of
par-ticlephysics as we know it to day, all matter is comp osed of the fermions,leptons and quarks,
which interact through gauge elds with each other via the exchange of gauge bosons. The
predictions of the Standard Mo del are in go o d agreement with exp erimentalresults. Several
op en questions remain ab out the structure of matter. The existence (or non-existence) of the
Higgs b oson which is p ostulated by the Higgs mechanism is one of these questions as is the
existenceofsup ersymmetricparticles. Furthermore,there isstilltheop en questionif thereisa
substructureofleptonsand quarks. Futureaccelerators likethe LargeHadron Collider(LHC)
proton-proton collidercurrentlyb eing built at Conseil Europeenp our laRechercheNucleaire
(CERN)or the prop osedlinear acceleratorslikeTESLA willincreasetheexp erimentally
acces-siblearea to address these and other questions.
TheHadron-Electron-Ring-Anlage(HERA)isthe rstelectron-protoncollider. Itislo catedat
the DESY lab oratory (Deutsches Elektronen-Synchrotron) in Hamburg, Germany. It follows
thetraditionofelectron-protonscatteringexp erimentsat SLACandseveral xed-targetexp
eri-ments,whichhavecontributedsubstantially to the exp erimentalcon rmationof the Standard
mo del. Withacenter-of-massenergy of 300 GeV,the HERA colliderwas able to access anew
kinematicregion to explore the structure of the proton. The kinematicregion covered by the
two HERAexp eriments,H1and ZEUS,inthemeasurementofthe total virtualphoton-proton
( p) cross section p tot
and the proton structure function F
2
has b een signi cantly extended
since the start of data taking in 1992. One of the surprising resultsfrom HERA was the
con-tinuingriseof thetotalvirtual photon-proton ( p) crosssection p tot
withincreasingsquare of
the p center-of-massenergy W 2 for Q 2 1:5 GeV 2
, whichis well describ ed by p erturbative
QCD(pQCD).Incontrastto theseresults,theriseoftotal crosssectionforrealphoton-proton
scattering p tot (Q 2 0 GeV 2 ) as a function of W 2
is less strong. This is not well describ ed
by pQCD, but shows go o d agreement with mo dels within the framework of non-p erturbative
b etween the regime of p erturbative QCD (pQCD, Q 2
1:5 GeV 2
) and the photopro duction
region(Q 2
0GeV 2
). Byshifting the interaction p ointinb oth exp erimentsthe lowerlimitin
Q 2
was extended down to 0.6 GeV 2
. To access even lower values of Q 2
(0.11{0.65 GeV 2
), the
ZEUS Beam Pip e Calorimeter (BPC) was installed in 1995. From the analysis of the shifted
vertexandthe ZEUSBPCdataitwasconcludedthatthedataiswelldescrib edbypQCDdown
to Q 2 1:0 GeV 2 . At lower values of Q 2
the data is in go o d agreement with a description
basedon Regge theory and the GeneralizedVector DominanceMo del. The transition b etween
b oth regimes was found to b e smo oth.
To further extend the kinematicregion covered by the BPC and decrease the systematic
un-certainties, a new detector, the Beam Pip e Tracker (BPT), was installed in front of the BPC
in 1997. It consists of two silicon microstrip detectors and is lo cated b etween the BPC and
theinteractionp oint. Makinguse oftheBPT, themainsystematicuncertaintiesrelatedto the
BPC(energy calibrationand alignment),the amount ofphotopro duction background,and the
determination of the interaction p oint were reduced by roughly a factor of two to three. The
kinematicacceptance was extendedtowards lower values of Q 2
and towards lower and higher
valuesofx. Thetwoformerextensionsgointopreviouslyunexploredareas,whilethe latterone
resultsinoverlapwithdatafromthe xed-targetexp erimentE665. Theanalysispresentedhere
isbased on 3:9pb 1
of data takenduring 1.5 months fromthe 1997 HERA run. Presented are
themeasurementsoftheproton structurefunctionF
2
andthetotalvirtualphoton-proton ( p) crosssection p tot
inthe transitionregionfromdeep inelasticscatteringto thephotopro duction
regime. Thedataisfrome +
pscatteringatacenter-of-massenergyof300GeVusingthe ZEUS
BPC and BPT. The kinematicregion covered in terms of the momentumtransfer Q 2
ranges
from 0:045 to 0:80 GeV 2
. Bjrkenx ranges b etween 310 7
and 10 3
. This corresp onds to a
range inthe
p center-of-massenergy of 25W 281 GeV.
The rst part of this thesis covers the theory describing inelastic lepton-proton scattering
es-p ecially at low values of Q 2
and Bjrkenx (chapter2). This includesa discussion on various
asp ectsof the physicsrelatedto thetransition regionsuchas abriefintro duction into the V
ec-tor Dominance Mo del and Regge theory. Chapter 3 describ es design and p erformance of the
electron-proton collider HERA. Several metho ds to reconstruct the relevant kinematic
quan-titiesfrom the data are discussed and comparedin termsof resolution. This is followed by a
description of the ZEUSdetectorin chapter4, whichconcentrates on the comp onents used in
this analysis. BPCand BPT willb e describ edinmoredetailin chapter5as these are the two
mainZEUScomp onentsused.
Data selection,the generation of simulatedevents,and systematic checks are describ ed inthe
second part of this thesis. Several detector studies like reconstruction, alignment,calibration,
and the estimationof the BPCand BPT eÆciencyare describ edin chapter6. The next
chap-terdetailsthe generationof signal andbackground(MC) eventsused forthese studiesand the
extractionofF 2 and p tot
. Severaltime-dep endentquantitieslikethep ositionoftheinteraction
p ointand theelectronb eamtilt,whichin uencethe dataselection,are discussedinchapter8.
The eventselection including background rejection and an estimationof the amount of
back-ground in the nal data sampleis describ ed inchapter9.
The third part of this thesis covers the extraction of the total virtual photon-proton cross
section
p
tot
and the proton structure function F
2
(chapter 10) and a discussion on the
inter-pretation of the obtained results (chapter 11). Chapter 10 gives a detailed discussion on the
determinationof systematicuncertainties. The resultson p tot and F 2
arecomparedto various
Theoretical background
2.1 A brief history of lepton-nucleon scattering
Scattering exp eriments played a pivotal part in the development of the Standard Mo del of
particle physics as we know it to day. They can b e divided into two categories, xed-target
exp eriments,in which a high energy particleis scattered on a stationary target, and colliding
b eam exp eriments, where b oth the projectile and the target particle are moving. In order to
prob e the structure of matter,electronsor muonshave oftenb een used as projectileparticles,
with targets of hydrogen, deuteriumor bare protons. Information ab out the structure of the
target particle can b e derivedfrom the measured angular and energy distribution of the
scat-teredand (or)the target particleor ofparticlespro ducedinthe interaction. The square ofthe
center-of-mass energy in a xed-target exp eriment is prop ortional to the b eam energy of the
projectile particles. In acollidingb eam exp erimentsigni cantly larger values of the
center-of-mass energy are p ossible since the square of the center-of-massenergy is directlyprop ortional
tothepro ductoftheprojectileandtargetb eamenergies. The rstmathematicaldescriptionof
theangular distributionofthe scatteredprojectileparticlewas madebyRutherfordinorderto
describ e the results of the scattering of -particles on a thin sheet of gold foil. His ansatz was
validfornon-relativisticspin-lessp oint-likeprojectilesand(heavy)targets. Inordertoprop erly
describ e later scattering exp erimentsit had to b e mo di ed. Mottextended the description to
includerelativistic particleswith spin 1
2
[Mo29 ]. However,since neither an extendedstructure
nor the anomalous magnetic momentof the target was incorp orated, his ansatz still failed to
describ e the scattering of electrons o protons. Finally in 1950, Rosenbluth included the spin
1
2
ofthe targetand projectileparticlesand the nite sizeand the anomalousmagneticmoment
of the target inhis calculationof the cross sectionof elastic electron-proton scattering [Ro50 ].
The Rutherford [Cl86] exp eriment (1909{1911 ) led to the conclusion that atoms are made
out of a tiny p ositive charged nucleus of less than 20 fm diameter, which is surrounded by
electrons. In the early1950s Hofstadter conducted scattering exp erimentsof electronso
pro-tons [Ho53, Ho55 , Ho57 ], which revealed for the rst time evidenceof an extended structure
of the proton. In the 1960s, electron-proton scattering exp eriments at SLAC con rmed the
earlier results [Pa68 , Bl69] and found that the two structure functions W
1
and W
2
[Dr64 ],
which describ e deep inelastic electron-proton scattering, only dep end on the Bjrken scaling
variable x (scaling). This implied that the electrons are scattered on free p oint-like charges.
Two mo dels were develop ed to describ e the structure of the proton and other hadrons. The
Quark Model was develop ed indep endently by Gell-Mann and Zweig [Ge64 , Zw64 ] to explain
Gell-Mann'sand Ne'eman'sprop osed classi cationof observedhadrons known as the Eightfold
spin-1
2
particlesthat made up the proton in b oth mo dels, quarks and partons are identical. In
orderto resolvetheinconsistencyofthe QuarkMo delwiththe Pauliexclusionprinciple,itwas
suggested that quarks carry an additional quantum numb er called colour [Gr64 ], which was
exp erimentallycon rmed. Sinceno evidenceforthecolour-charged hadrons hadb eenfound, it
was concludedthat observedhadrons arecoloursinglets. Thegaugetheory Quantum
Chromo-dynamics(QCD)ofthestronginteractionsingeneral[Fr73,Gr73 ,We73 ]wasabletoexplainthe
exp erimentalresult,that no exp erimenthas found free quarks(quark con nement). The QCD
isanon-Ab eliangaugetheory,whichrequiresthe interactionsamongitsgaugeb osons, the
glu-ons. At short distances,the quarks are quasi-free (asymptotic freedom),but at large distances
the interactionb ecomes stronger due to the interactions amongthe gluons,whichpreventsthe
observation of free quarks. The resulting prediction from QCD, that the scaling b ehaviour of
thedeep-inelasticstructurefunctionsW
1
andW
2
islogarithmicallybroken,wasexp erimentally
observedat the FermiNational AcceleratorLab oratory (FNAL)in1974 [Fo74]. A numb er of
xed-target exp eriments have b een carried out at CERN, DESY, FNAL, and SLAC in order
to obtain more information ab out the substructure of the nucleons. The HERA accelerator
at DESY is the rst colliding b eam exp eriment using electron and proton b eams. With a
center-of-massenergy of300GeV,the kinematicregioncoveredbythe twoHERAexp eriments
H1 and ZEUS extends several ordersof magnitude b eyond that of xed-target exp erimentsin
terms of larger values in Q 2
and much lower values in the Bjrken scaling variable x. The
analysis ofthe data at mediumand high Q 2
[Ai96,De96 ],together with the study ofthe total
virtual photon-proton ( p) cross section p tot
and the proton structure function F
2
at lowQ 2
and verylowx [Br97]have resultedin fascinating and encouraging results.
2.2 The low Q
2
and very low x region
Asurprising early observation at HERA was the rapid riseof the proton structure functionF
2
with decreasing x for xed value of Q 2
even for Q 2
as low as 1.5 GeV 2
[Ai96], [De96 ]. This
translates to a strong rise of the total virtual photon-proton cross section p tot with the p
center-of-massenergyW. Sincethetotalcrosssectionforrealphoton-protonscattering
p
tot was
measuredto haveamuchslowerrise withW, the questionarose whereand howthe transition
b etween the two regions takes place. In 1995 the kinematicregion covered by the two HERA
exp eriments H1 and ZEUS was extended in order to prob e the transition region b etween the
regimeofp erturbativeQCD(pQCD,Q 2
1:5 GeV 2
)and thephotopro duction region(Q 2
0
GeV 2
). Byshiftingtheinteractionp ointinb othexp eriments,thelowerlimitinQ 2
wasextended
downto0.6 GeV 2
. ToaccessevenlowervaluesofQ 2
(0.11to0.65GeV 2
),theZEUSBeamPip e
Calorimeter (BPC) was installed in 1995. The results from 1995 ([Br97], [Br98a]) con rmed
the earliermeasurements. Itwas found thatthe data was welldescrib edby p erturbativeQCD
(pQCD) down to Q 2 =1GeV 2 . Atlower values of Q 2
the data was b est describ ed by amo del
based on Regge theory (see section 2.8.2) and the Generalized Vector Dominance Mo del (see
section 2.8.1). The data suggest that there is a transition region b etween the two domains,
which extends up to approximately Q 2
= 1 GeV 2
, and that the transition is smo oth. Several
questions remain ab out how the transition towards Q 2
= 0 GeV 2
takes place and how to
describ e the data. The rise of F
2
with decreasing x is exp ected to stop at a certain Q 2 as F 2 (x ! 0;Q 2
! 0) should b e zero. Even for the lowest value of Q 2 = 0:11 GeV 2 , F 2 was
stillfound to b e increasing towards lowervalues of x. The
pcross sectionas measuredfrom
the BPC in 1995 were extrap olated to Q 2
= 0 GeV 2
using the Regge and GVDM motivated
e (k
µ
)
e,
ν
e
(k
µ
/
)
γ
*
, Z
0
, W
±
(q
µ
)
P (p
µ
)
X (p
µ
/
)
Figure 2.1: Feynman diagram describing unp olarized ep scattering to lowest order
inp erturbationtheory [Su98]. Inthecase ofNCevents,theexchanged gaugeb oson
is eithera virtualphoton
or aZ 0
b oson and the nal state leptonis an electron.
For CC events the nal state lepton is a neutrino due to the exchange of aW +
or
W b oson.
extrap olated cross sections at Q 2
= 0 GeV 2
and the H1 and ZEUS measurements of the p
cross section. More precise data is needed to investigatethe problems describ ed ab ove and to
comparethe data to new or up dated mo delsfor this kinematicregion. A further extension of
the kinematicacceptanceat lowQ 2
and verylowx isdesirabletowards lowervaluesof Q 2
and
highervaluesof x. Theformeronewillallowthe studyoftheb ehaviourof p tot andF 2 closerto
the photopro duction regime,whilethe latter one results inan overlap with the regioncovered
by the xed-target exp eriment E665. An extension upwards from Q 2
= 0:65 GeV 2
together
with a reduction of the systematic uncertainties is also desirable for a detailed study of the
transition towards the region of pQCD. The motivation of the analysis presented here was to
addressthe problemsdescrib ed ab ove.
2.3 De nition of the kinematic variables
Thissectiongivesashortintro ductiontothekinematicvariablesusedtodescrib ethescattering
ofunp olarizedelectronsonunp olarizedprotons. Thenatural systemofunitsisusedthroughout
this thesis, i.e. h =1 and c= 1. Furthermore, the term`electron' willb e used as a synonym
forb oth electronsand p ositrons unless explicitlystated otherwise.
The Feynman diagram to rst order in p erturbation theory for the scattering of unp olarized
electronson unp olarizedprotons isshown in gure2.1. Electronsandprotons withinitial
four-momentumk =(E e ; ~ k e )and p=(E P ; ~
P) resp ectivelyinteract via the exchangeof a Standard
Mo delelectroweakgauge b oson:
e(k)+P(p) !l (k 0
)+X(p 0
) (2.1)
Ignoringinitialand nalstateradiationfromthe lepton,the nalstate consistsofthescattered
leptonl (k 0 =(E 0 l ; ~ k 0 l
)) and the hadronic nalstate system X (p 0 =(E X ;p~ X )). In the case of a 0
leptonisaneutrinoduetotheexchangeofaW +
orW b oson. TheHERAcolliderexp eriments,
H1and ZEUS,cannot detectthe neutrino directly,but areable to measuredirectlythe energy
anddirectionofb oththescatteredlepton(inthecaseofNCeventsonly)andthehadronic nal
state system. For xed b eam energies, as in the case of the HERA collider, two indep endent
variables are suÆcient to de ne the unp olarized inelastic ep event kinematics. Dep ending on
the kinematicregioncovered, the detectors used, and whether NCor CC eventsare analyzed,
oneofseveraloptionsofhowtoreconstructthesevariablesischosen. Thesewillb ediscussedin
moredetail in section 3.4. The following variables provide a relativistic-invariant formulation
of the unp olarized inelastic epevent kinematics:
s = (k+p) 2 '4E e E P (2.2) Q 2 = (k k 0 ) 2 = (p p 0 ) 2 = q 2 Q 2 s (2.3) x = Q 2 2(pq) 0x1 (2.4) y = pq pk 0y 1 (2.5) W 2 = (p+q) 2 =(p 0 ) 2 =m 2 p + Q 2 x (1 x) W m p (2.6) t = (p p 0 ) 2 (2.7)
For the 1997 HERA running p erio d a p ositron b eam of E
e
= 27:5GeV and a proton b eam of
E
P
= 820GeV were used. The resulting center-of-mass energy p
s is 300GeV neglecting the
electronand protonmasses. Q 2
isthenegativesquare ofthe momentumtransferq anddenotes
the virtuality of the exchanged gauge b oson, i.e. Q 2
= 0 corresp onds to real photon-proton
scattering. x is the Bjrken scaling variable interpreted in the Quark Parton Mo del as the
fraction of the proton momentum carried by the struck parton. y is the fraction of energy
w.r.t. the initial electron energy transferred b etween the lepton and hadronic system in the
proton rest frame. W 2
isthe square of the invariantmass of theproton gaugeb oson system. t
isthe four-momentumtransfer at theproton vertex. Ignoring the electronand proton masses,
x,y,Q 2
, and s are relatedthrough the following relation:
Q 2
'sxy (2.8)
The momentumtransfer q = p
Q 2
can b e related to the wavelength of the virtual b oson
through Heisenb erg's uncertaintyprinciple:
= 1 j~q j 2m p x Q 2 (2.9)
In order to resolve objects of size , has to b e smaller than . At lowQ 2
the resolution is
smallandthesubstructureoftheproton is`visible'(see gure2.2). AthigherQ 2
theresolution
2
proton
proton substructure
QCD Compton
BGF
increasing resolving power Q
Figure2.2: Resolution of the proton substructureas a function ofQ 2
[Qu96].
for the reaction ab ove can b e describ ed in terms of proton structure functions and in terms
of the scattering of virtual photons o protons. The following sections give an overview of
b oth approaches. Compared to the single photon exchange, the exchange of the heavy Z 0 (m Z 0 =91:2GeV) and W (m W
=80:2 GeV)b osons is kinematicallysuppressed by a term
Q 4 =(Q 2 +M 2 Z 0 ;W ) 2
[In87]. Sincethecontributionfrom(
Z 0
)-interferenceisalsosuppressed
by a factor Q 2 =(Q 2 +M 2 Z 0
) [In87], the single
exchange is dominant at low Q 2
. Since only
NC events at low Q 2
were used in this analysis, the following discussion will b e restricted to
the case of NC scattering through a virtual photon as the exchanged gauge b oson, to lowest
orderin p erturbation theory.
2.4 Structure functions
The concept of structure functions is one of the main to ols to explore the structure of the
nucleusingeneral[Ha84 ]. Inthe singleb oson exchangeapproximationthe crosssectioncan b e
factorizedinto aleptonic tensor L
and a hadronic tensor W d L W (2.10)
Neglecting the electron mass the leptonic tensor has b een calculated from Quantum
Electro-dynamics (QED)to b e L = 1 2 Tr (6k 0 6k )=2(k 0 k +k k 0 + q 2 2 g ) (2.11) whereg
isthe metrictensor. It issymmetricin and. The detailof the interaction at the
hadronic vertex and hence the substructure of the proton which takes part in the interaction
are parametrized by the hadronic tensor W
. The most general form of W
, taking into
account Lorentz-invariance and the symmetryof L
in and , is [Ha84] W = W 1 g + W 2 m 2 p p + W 4 m 2 q q + W 5 m 2 (p q +q p ) (2.12)
The scalars W
i
dep end on q and pq. Four-current conservation can b e used to reduce the
numb erof indep endentscalars W
i
. Usually,W
4
and W
5
are chosen to b e replacedby:
W 4 =( pq q 2 ) 2 W 2 + m 2 p q 2 W 1 W 5 = pq q 2 W 2 (2.13) Thefunctions W i;i=1;3
dep end on twoLorentz-invariantvariables,whichinthis caseare chosen
to b e = pq m p and Q 2
. The b ehaviour of these functions as a function of and Q 2
re ect the
dynamics of the strong interaction. Usually the three functions are transformed into proton
structure functions F i;i=1;3 : F 1 (x;Q 2 ) = m p W 1 (;Q 2 ) (2.14) F 2 (x;Q 2 ) = W 2 (;Q 2 ) (2.15) F 3 (x;Q 2 ) = W 3 (;Q 2 ) (2.16)
Usingthe proton structure functionconvention, thedouble-di erentialdeep-inelastic NCBorn
e
p!e
X cross sectioncan b e writtenas
d 2 NC (e p) dxdQ 2 ! Born = 2 2 xQ 4 [Y + F 2 (x;Q 2 ) y 2 F L Y xF 3 (x;Q 2 )] (2.17) F L = F 2 2xF 1 Y = 1(1 y) 2 F 3 (x;Q 2
)describ es the parityviolation contributiondue to (
Z 0
)-interferenceand issmall
in the low and mediumQ 2
range. Neglectingthe contribution from F
3 (x;Q
2
) one obtains the
following expression forthe Born cross section intermsof y and Q 2 : d 2 NC (e p) dydQ 2 ! Born = 2 2 Y + yQ 4 F 2 y 2 Y + F L ! (2.18)
The three proton structure functions are de ned with resp ect to the Born cross section. As
also higher order QED corrections contribute to the measured cross section, a correction has
to b e applied in the extraction of F
2
from the data. This is usually parametrized by a QED
radiativecorrection factor Æ
r (y;Q
2
)to the Born cross section:
d 2 NC (e p) dydQ 2 ! Meas = d 2 NC (e p) dydQ 2 ! Born [1+Æ r (y;Q 2 )] (2.19)
2.5 Virtual photon-proton scattering
The deep inelastic scattering of electronso protons by the exchange of a virtual photon can
b e viewed as the scattering of virtual photons o the proton. If the lifetime of the virtual
section p
tot
for the scattering of virtual photons o protons [Dr64,Ha63 ,Gi72]. This leads to
the following requirement:
x r 1+ 4m 2 p x 2 Q 2 2r p m p (2.20) where r p 5GeV 1
is the radius of the proton. Since virtual photons may b e b oth
longi-tudinally and transversely p olarized, the total virtual photon-proton cross section is de ned
as p tot p T + p L (2.21) where p T and p L
are the cross section for the scattering of transverse and longitudinal
p olarized virtual photons o a proton resp ectively. Using the proton structure functions F
1
and F
2
and Hand's convention [Ha63 ] for the de nition of the ux factor K of the virtual
photons, the twocross sectionsare givenby:
p T = 4 2 m p K Hand F 1 (2.22) p L = 4 2 K Hand " 1+ Q 2 4x 2 m 2 p ! 2xm p Q 2 ! F 2 F 1 m p # (2.23) 4 2 m p K Hand F 2 2x F 1 = 4 2 m p K Hand F L 2x (2.24) K Hand = Q 2 2m p = Q 2 2m p 1 x x
Equation 2.24 is onlyvalid if Q 2 =4x 2 m 2 p
is signi cantly larger than 1. F
L
is referred to as the
longitudinalstructure functionb ecause of the relationshipto the longitudinalcross section
L
in equation 2.24. Since b oth p L and p T
are required to b e greater or equal to 0, F
L is
b ound to b e in the range of 0 F
L F
2
. The total virtual photon-proton cross section from
equation2.21 can b e writtenas
p tot p T + p L = 4 2 Q 2 (1 x) 0 B @1+ 1 Q 2 4m 2 p x 2 1 C A F 2 (x;Q 2 ) (2.25)
In the case of HERA equation 2.24 is valid, and in this analysis x is much smaller than 1.
Therefore,equation 2.25 can b e simpli edto
p tot 4 2 Q 2 F 2 (x;Q 2 ) (2.26)
Rewritingthe Born cross sectionfrom equation 2.18 intermsof
T and L yields d 2 NC (e p) dydQ 2 = ( p T + p L )= ( p tot ) e (2.27) (y) = 2(1 y)=(1+(1 y) 2 ) Photon Polarization 2 2 2
where ( p tot ) e = p T + p L
is called the e ective
p cross section. For the BPC and BPT
data (y) has a value of (0.31{0.99) dep ending on y. Because the center-of-mass energy at
HERA is xed, (y) cannot b e varied indep endently of x and Q 2
. The measured quantity is
the e ective cross section p T + p L . For the extraction of ( p tot ) [F 2
] one needs to assume
the value of(y)[F
L
]for eachbin. This isdone byrewriting equation2.27 usingthe ratio R of
thelongitudinaland transversecross sectionR = p L = p T =F L =2xF 1
andassumingacertain
mo delfor the b ehaviour ofR.
d 2 NC (e p) dydQ 2 = p T (1+R) (2.28) The contribution of F L
to the di erential ep cross section increases for y ! 1. In the case
of HERA it can b e determined if d 2
=dxdQ 2
is measured at xed values of x and Q 2
, but
at di erent center-of-mass energies s. This can b e done by either varying the energies of the
electronand/ortheprotonb eamorbyusingradiativeeventsatreducedcenter-of-massenergies
due to initial state radiation [Bo99e][Ke98].
2.6 The Quark Parton Model (QPM)
Two mo delswere develop ed to describ e the structure of the proton and other hadrons, F
eyn-man's parton model [Fe69] and the Quark Model. The latterone was develop ed indep endently
by Gell-Mannand Zweig[Ge64,Zw64 ] to explainthe classi cationof observedhadrons known
as the Eightfold Way, which had b een prop osed by Gell-Mann and Ne'eman. In the parton
mo del the proton consists of quasi-free p oint-like objects. Each so-called parton i carries a
fraction
i
p of the proton momentump (0
i
1). The inelastic ep cross section is given
by the incoherent sum of quasi-elastic electron parton scattering. If the partons were indeed
p oint-like,one would exp ect that even with increasing momentumtransfer Q 2
no new details
would b e visible. In 1968 Bjrken predictedthe b ehaviour of the structure functions for the
high energy limitof Q 2 !1, ! 1, but x= Q 2 2m p
nite. His prediction that the structure
functions would dep end only on a dimensionless scaling variable x was con rmed by SLAC
exp eriments. F 1 (x;Q 2 )!F 1 (x) (2.29) F 2 (x;Q 2 )!F 2 (x) (2.30)
Inthe in nitemomentumframeof theproton, the scalingvariable xcan b e interpretedas the
fractional momentum
i
of the struck quark. Neglecting the parton mass m
x
and the proton
mass, four-momentumconservation impliesfor this fraction:
m 2 p = (p+q) 2 = 2 p 2 Q 2 +2pq =m 2 x Q 2 +2pq ! = 1 2pq (m 2 p m 2 x +Q 2 ) Q 2 2pq (2.31)
In 1969 Bjrken and Paschos suggested that the elementary p oint-like spin-1
2
particles that
made up the proton in b oth mo dels,quarks and partons were identical,thus the name Quark
that only one parton takes part in the interaction. The probability that an additional parton
takespart inthe interaction issuppressedby thegeometricalfactor1=(r 2 p Q 2 ),wherer p isthe
radius of the proton. The QPM relates the structure functions F
1
and F
2
to the sum of the
parton distributionfunctions xf
i
(x) weighted by the square of their electric charge e
i
inunits
of the proton charge e.
F 2 (x) = 1 2 X i e 2 i f i (x) (2.32) F 1 (x) = 1 2x F 2 (x) (2.33)
Equation 2.33 is known as the Callan-Cross relation[Ca69 ] and was approximatelycon rmed
by SLAC exp eriments. It impliesthatF
L
or,intermsof virtualphoton-proton scattering
p
L
is 0. The predicted fractional charge of the quarks was con rmedusing neutrino and electron
nucleonscatteringdata andthe p ostulatednumb erofthreevalence quarksinthe proton (uud)
and neutron (ddu) using the Llewellyn-Smithsum rule [De75]. Although the QPM was very
successful in explaining some of the early ep results, some problems of this mo del b ecame
apparent. One prediction fromthe QPM mo delwas that the sum of the resp ectiveintegrated
distribution functionsxf
i
(x)should b e equalto unity:
1 Z 0 dxx X i f i (x)=1 (2.34)
The exp erimentalvalue of the sum in equation 2.34 was approximately 0:5. The conclusion
was that ab out half of the momentum of the proton is carried by neutral particles [Ab83].
Also the fact that no free quarks were observed (quark con nement)could not b e explained.
Both problems weresolved by the formulation of a eld theory of the strong interaction, the
Quantum Chromo dynamics (QCD), which in the asymptotic limit Q 2
! 1 repro duces the
QPM.
2.7 The Quantum Chromodynamics (QCD)
The Quantum Chromo dynamics (QCD) is the gauge theory of the strong interaction. It was
develop edat theb eginningofthe 1970s. Theadditionalquantumnumb ercolourofthe quarks,
intro duced to solve the inconsistency of the Quark Mo del with the Pauli exclusion
princi-ple[Gr64 ], was found to b e the colour charge of QCD. Three colour states were needed: `red'
(r), `green' (g), and `blue' (b). The three coloured quarks of one avour form a triplet. The
gaugeb osons ofQCDaretheeightgluons,whichcarryacombinationofcolourandanti-colour.
In 1979 theywereexp erimentallyobserved through three-jet eventsat the PETRA colliderat
DESY [Wu84]. In contrast to the QED, QCD is a non-Ab elian gauge theory, which is based
on a SU(3) gauge group. Therefore, the gluons are able to interact with each other, which is
a fundamentaldi erence b etweenQCD and QED. In the case of QED, the e ectivecoupling,
i.e. the e ective charge decreases for small momentum transfers (large distances), while for
QCD it is the other way around. This allows the description of two rather di erent exp
x
F
2
1/3
x
F
2
1/3
sea
valence
x
F
2
1/3
Figure2.3: Exp ectedb ehaviourofF
2
foracertainsubstructureoftheproton[Su98 ].
In the case of onlythree valence quarks (left),F
2
would have a single p eak at 1/3.
For three b ound valence quarks (middle) the distribution is smeared. If also QCD
dynamicsistakenintoaccount,thedi erentcontributionstoF
2
fromseaandvalence
quarks haveto b e separated.
QCD coupling constant
s (Q
2
) dep ends on the numb er of quark avours n
f
and a free scale
parameter and isgivenin the leadingorder approximation by the followingformula:
s (Q 2 )= 4 (11 2n f =3)ln ( Q 2 2 ) (2.35)
has b een measuredto b e (100 300) MeV [Ba96]. For large Q 2
,
s
is smalland the quarks
are quasi-free and can b e describ ed by p erturbative calculations. In the case of low Q 2
,
s
b ecomes large and it isexp ected that p erturbative calculations are not valid b eyond a certain
minimumQ 2
. The dynamics of the parton distributions inside the proton are given by three
reactions: gluon splitting (g ! gg), quark-gluon radiation (q ! qg), and pair pro duction of
so-calledsea quarks (g !qq ). The exp ectedqualitativeb ehaviour of F
2
as a functionof x for
di erentparton comp ositions of the proton is picturedin gure 2.3. In the case of onlythree
valence quarks without Fermi motion one would exp ect the proton momentumto b e equally
dividedb etweenthem,i.e. F
2
wouldhaveasinglep eakat1=3andequation2.34wouldb evalid.
For valencequarks b ound by gluon exchangea somewhat smeared distributionis exp ected. If
the whole QCD dynamics are included, F
2
is exp ected to rise at low x. This is b ecause the
low x region is p opulated by gluons and sea quarks and the quark density is large. Because
the resolution increases with Q 2
, more quark-antiquark pairs originating from gluons can b e
resolved at higher Q 2
. Therefore, the rise of F
2
at low x for xed Q 2
is exp ected to increase
with Q 2
. The large x region is dominated by the valence quarks. With increasing Q 2
decreases due to gluon radiation. The resulting logarithmic dep endence of F
2
on Q at xed
x is referred to as scaling violation. Both the scaling violation and rapid rise of F
2
at small
x have b een measured by the HERA exp eriments H1 and ZEUS ( [De93 ], [Ab93]). Another
e ectof the quark-antiquarkpair-pro duction via gluons isthat, contrary to the QPM, quarks
can havetransversemomentum. Therefore,theycan coupleto longitudinallyp olarized virtual
photons and the Callan-Gross relation 2.33 isno longervalid.
2.7.1 Factorization
IntheframeworkofQCD,hadron-hadronandlepton-hadronscatteringaredescrib edintermsof
interactionsb etweenthe quarksandgluonsofone hadronwith thosefromtheotherhadron, or
theleptonresp ectively. Twoingredientsareneededtocalculateforexampletheepcrosssection.
The interaction b etween the virtual photon and a quark with a given momentumfraction in
the proton isa short-range pro cess and can b e calculated usingp erturbative calculations. The
probability to nd aparticular quark havinga momentumfraction b etween x and (x+dx) is
a long-range pro cess. It cannot b e calculated in p erturbative QCD (pQCD). The separation
of the scattering pro cesses in short-range and long-range physics is called factorization. An
additional scale, the factorization scale
F
, has to b e intro duced. In pQCD the calculation of
self-energy diagrams such as gluon splitting into a quark-antiquark pair or the recombination
ofthe pair into agluon yieldsdivergentintegrals. By intro ducing the renormalizationscale
R
the divergence is absorb ed into the de nition of the long-range parton distribution functions.
Onlymomentalessthan
R
are integrated over. Severalrenormalizationschemesare used, for
examplethe minimal subtraction scheme (MS) or the deep inelastic scattering (DIS) scheme.
For the latter one, the structure function F
2 (x;Q 2 )is givenas F 2 (x;Q 2 )= n f X i e 2 i h xq i (x;Q 2 )+xq i (x;Q 2 ) i (2.36) wheren f
isthenumb erofquark avoursandq
i andq
i
arethequarkandanti-quarkdistribution
functionsofthe hadron resp ectively. Theyare pro cessindep endent. Thequark andanti-quark
distributionsand the gluondistributionfunction g
i (x;Q
2
)mustb e determinedexp erimentally.
However,if theyare known at one particularvalue ofQ 2
they can undercertain conditions b e
calculated for other regions. This is done usingthe DGLAP, BFKLor CCFMequations. The
DGLAPequationsallowonetodeterminethepartondistributionsfor xedxatanyvalueofQ 2
iftheyare knownat aparticularvalueQ 2
0
. TheBFKLequationscanb eused todo itthe other
wayaround. Attemptshaveb eenmadeto achieveauni edBFKL/DGLAPdescription[Kw97 ].
TheCCFM equations[Ca90 ]werederivedinorder tob e able to evolvetheparton distribution
inb oth x and Q 2
. Figure2.4 shows the domainsof the DGLAP, BFKL,and CCFMevolution
equations. The three sets ofequations are discussed inthe following sections.
2.7.2 The DGLAP equations
The Dokshitzer-Grib ov-Lipatov-Altarelli-Parisi (DGLAP) equations [Al77, Gr72 ] are a set of
(2n
f
+1) coupledintegro-di erentialequations. Theycan b eused to determinethe quarkand
gluon distribution functions for any value of Q 2
if they are known at one particular value Q 2
0
withintherangeofapplicabilityofpQCD.TheDGLAPequationsarederivedbyrequiringthat
b oth F
1
and F
2
as measurable quantities,should not dep end on the choiceof the factorization
scale
F
. Starting from the requirement 2 F (dF i (x;Q 2 )=d 2 F ) = 0 (i = 1;2), the DGLAP
the parton density `visible'at a certain x and Q 2
. Increasing Q 2
leads to a b etter
spatialresolution. Smallervaluesinx yieldan increaseintheparton densitydriven
bythe gluon density. Athigh parton density saturation isexp ectedto diminishthe
rise of F
2
with decreasing x. The `critical'line indicates the transition region into
the region of high parton density where saturation and shadowing is exp ected to
dominate. The DGLAP equation allows the evolutionin Q 2
for xedx, the BFKL
equationtheevolutioninxfor xedQ 2
. TheCCFMequationsdescrib eanevolution
in b oth x and Q 2
.
the dominant contribution at large x and large Q 2
were summed to all orders and all others
neglected. Theremainingtermshavethe form n s (lnQ 2 ) n
. Therefore,the DGLAP equations
are onlyvalid as long as the impactof the neglectedtermsis small,whichisexp ected for
s (Q 2 )ln(Q 2 )O (1) s (Q 2 )ln 1 x 1 (2.37)
The DGLAP equations forthe quark,anti-quark, and gluondistributions are givenby:
dq i (x;Q 2 ) dlnQ 2 = s (Q 2 ) 2 Z 1 x dz z q i (z;Q 2 )P q q x z +g(z;Q 2 )P q g x z (2.38) dq i (x;Q 2 ) dlnQ 2 = s (Q 2 ) 2 Z 1 x dz z q i (z;Q 2 )P q q x z +g(z;Q 2 )P q g x z (2.39) dg(x;Q 2 ) dlnQ 2 = s (Q 2 ) 2 Z 1 x dz z " n f X (q i (x;Q 2 )+q i (x;Q 2 ))P g q x z +g(z;Q 2 )P g g x z # (2.40)
ij
i with momentumfraction x originatingfrom aparton of typ e j having amomentumfraction
z when the scale changed from Q 2 =GeV 2 to Q 2 =GeV 2 +dln(Q 2 =GeV 2 ). Up to now they
are calculated up to next-to-leading order (NLO) and can b e found in [Gu80 , Fu82 ]. If the
quark,anti-quark,andgluondistributionsareknown atastartingscaleQ 2
0
theycanb eevolved
usingthe equations (2.38{2.40). Underthe assumption that the contributionsfrom quarksare
negligible at low x it is p ossible to extract the gluon density directly from a measurement of
F
2
. Using the metho d prop osed by Prytz in leading order [Pr93] and the DGLAP equations
the following relation b etweenF
2 and g(x;Q 2 ) is derived: dF 2 (x;Q 2 ) dln Q 2 5 s (Q 2 ) 9 2 3 xg(2x;Q 2 ) (2.41)
TheDouble Logarithmic Approximation (DLLA)can also b eused to estimatethe gluon
distri-butionatlowvaluesofx,wheretheLLAapproximationusedtoderivetheDGLAPequationsis
notvalid. Leadingtermsin(ln 1
x
)accompaniedbyleadingtermsin(ln Q 2
)areincluded,which
results inthe gluon distribution b elow,which isnumericallycompatiblewith x 0:4 [Le97]. xg(x;Q 2 ) exp v u u u t 2 4 48 11 2 3 n f ln 0 @ ln Q 2 2 ln Q 2 0 2 1 A ln 1 x 3 5 (2.42) s (Q 2 )lnQ 2 1; s (Q 2 )ln 1 x 1; s (Q 2 )ln Q 2 ln 1 x O (1)
Equation 2.42 violates unitarity in the limit x ! 0, whichis also true for the solution of the
BFKLequation2.43discussedinthenextsection. Themo delofsaturationinwhichthegrowth
of the gluon and sea quark density at low x is comp ensated by quark-antiquark annihilation
and gluon recombinationis discussedbrie y insection2.7.5.
2.7.3 The BFKL equation
Work done by Balitzky,Fadin, Kuraev,and Lipatov resultedin the BFKLevolutionequation.
Thisequationprovidesanevolutioninxfor xedvaluesofQ 2
[Ba78]fortheunintegratedgluon
distributionf g (x;k 2 T ). k 2 T
is the square of the transversemomentumof the gluons. Incontrast
to the DLLA approximation the BFKLevolutionschemeprovidesaway to sumup allleading
termsinln 1
x
. TheBFKLequationaccordingto [As94]andtherelationshipof theunintegrated
to integrated gluondistribution are givenin equation 2.43 and2.44 resp ectively.
x @f g (x;k 2 T ) @x = 3 s k 2 T Z 1 0 dk 0 2 T k 0 2 T 2 4 f g (x;k 0 2 T ) f g (x;k 2 T ) jk 0 2 T k 2 T j + f g (x;k 2 T ) q 4k 0 4 T +k 4 T 3 5 Kf g (2.43) s ln(Q 2 ) 1; s ln 1 x =O (1) xg(x;Q 2 ) = Z Q 2 o (dk 2 T =k 2 T )f g (x;k 2 T ) (2.44)
K is the BFKL kernel. The solution of 2.43 is dominated by the largest eigenvalue of the
kernel K resultinginthe following x and Q 2
dep endencefor F
2 [As94a]: F 2 (x;Q 2 ) (Q 2 ) 1=2 x ; = 3 s 4ln 2 (2.45)
2
given by the Froissart b ound [Fr61] b ecause F
2
is related to the total cross section of virtual
photon-proton scattering p tot by equation2.25: p tot = 4 2 Q 2 F 2 m 2 (ln s s 0 ) 2 (2.46) m
is the mass of the charged pion and s
0
a scale factor, whichhas to b e determinedexp
eri-mentally. Becauseof the limitgivenby equation 2.46there must existsomemechanismwhich
damp ens the rise of F
2
at low x. Two mo dels of such a mechanism are brie y discussed in
section2.7.5. A mo di edversionof the BFKLequations takes into account the recombination
ofgluons(gg !g)asonemechanismtodamp entheriseofF
2
. Theansatzprop osedbyGrib ov,
Levin,and Ryskin includes non-lineartermsinto equation 2.43:
x @f g (x;k 2 T ) @x =K f g 81 2 s (k 2 T ) 16R 2 k 2 T [xg(x;k 2 T )] 2 (2.47) 2.7.4 The CCFM equation
The equationprop osed by Catani,Ciafaloni,Fiorani, and Marchesini (CCFM)isbased on the
coherent radiation of gluons. In the limit of low x ! 0 the CCFM equation is equivalent to
the BFKL equation, while for x ! 1 it repro duces the DGLAP equations. CCFM based MC
generators did archieve a reasonably go o d description of the F
2
data from HERA, but until
recently failed to describ e the pro duction of forward jets at HERA, which is b elieved to b e
a go o d signature for parton dynamics at low x. In [Ju99 ] the results of a mo di ed version
of the MC generator based on CCFM were found to b e in go o d agreement with F
2 data for 5 10 6 < x < 0:05 and 3:5 < Q 2 < 90 GeV 2
and cross section for forward jet pro duction.
Whether this improved mo del is able to provide a go o d description of the F
2
data for lower
and higher values of Q 2
remainsto b eseen.
2.7.5 Saturation
Itisexp ectedthattheriseofthequarkandgluondensityatlowxstopsataacertainx
min (Q
2
),
b ecause of quark-antiquark annihilation and recombination of gluons. x
min is exp ected to dep endon Q 2 , b ecauseat lowQ 2
the resolving p owerof the virtual photon islowcomparedto
higherQ 2
and lesspartons can b e seen. This is indicatedby the `critical line'in gure 2.4. It
has b eenestimated in[Le97 ] that recombination of gluonsresult in saturation if
xg(x;Q 2 ) r 2 p r 2 g (Q 2 ) 5GeV 1 2 Q 2 6Q 2 (2.48) r p
is the radius of the proton ( 1 fm) and r
g (Q
2
)= 2=Q the gluon radius at a certain value
of Q 2
. So far the gluon densities derivedfrom HERA have b een well b elow this limitand no
signsofsaturation haveb eenobserved. Inamo delprop osed byMueller[Mu90],the saturation
starts in smalllo calized areas of the proton, the hot spots. This would result in saturation at
loweroverall gluon densities.
2.8 The transition region
The main motivation for the measurement presented in this thesis was to further expand the
measurement[Br97]. Thisisdoneto studythe transitionfromtheregionofpQCDat Q 1:0
GeV 2
to the photopro duction limit (Q 2 0 GeV 2 ). For Q 2 ! 0 GeV 2
the virtual
photon-proton cross section
p
tot
approaches the cross section for real photon-proton scattering p
tot .
As real photons can onlyb e transverselyp olarized, p L has to vanish at Q 2 =0. p tot = lim Q 2 !0 h p T i = lim Q 2 !0 " 4 2 Q 2 F 2 (x;Q 2 ) # (2.49)
Two constraints for the structure functions F
2 and F L in the limit of Q 2 ! 0 GeV 2 can b e
derived from the ep cross section in terms of structure functions. The hadronic tensor W
fromequation2.12 rewrittenin termsof F
1
and F
2
neglectingthe contribution ofF
3
at lowQ 2
exhibits twosingularities. Since b oth F
1
and F
2
are physicalquantities, the singularities have
to b e canceledby imp osingthe following conditions on F
1 and F 2 : F 2 =O (Q 2 ) F L =F 2 2xF 1 =O (Q 4 ) (2.50)
As exp ectedfromthe b ehaviour of the strong coupling constant (equation2.37), pQCD is not
abletodescrib ethedatadowntoQ 2
0GeV 2
. AspQCDwasfoundonlytoworkab oveQ 2
=1
GeV 2
[Br97],non-p erturbativeconceptshaveto b eusedtodescrib e p tot andF 2 intheregionof lowQ 2
. Mostof the phenomenologicalmo delsused to describ e the transition region arebased
on the concepts of the Vector Dominance Model (VDM) [Sa60 ] and/or Regge theory [Co70 ].
These concepts willb e discussed in the next section. A description of the various mo delsand
parametrizations,whichare comparedto the results of this analysisis givenin section11.
2.8.1 Vector Dominance Model
The Vector Dominance Mo del (VDM) is based on the phenomenological observation that
photon-hadron interactions exhibit striking similarities to hadron-hadron interactions. In the
VDM the photon is a sup erp osition of the bare photon j i
bare
and a hadronic comp onent
j i
hadronic
. The latter one is given by a uctuation of the photon into a quark-antiquark pair
with the samequantumnumb ers(J PC =1 , Q=B =S =0): j i=j i bare +j i hadronic (2.51)
TheVDMmakestheassumptionthatthephoton-hadron interactionisgivenbytheinteraction
of the hadronic comp onents of the photon. Furthermore, it assumes that the photon only
uctuatesintothethreelightestvectormesons( 0
,!,and ),whichallhavethesamequantum
numb ersasthe barephoton. TheVDMansatzisonlyvalidifthe uctuationtime
f
,whichcan
b e estimatedusing the uncertaintyprinciple,is large compared to the interaction time[Le97 ].
f
can b e estimatedfrom the energy di erence E b etweenthe mass of the vectormeson m
V
and the momentumof the bare photon and is givenby:
f 2 m 2 V +Q 2 (Q 2 0) (2.52)
Notethatequation2.52isvalidforb othvirtualandreal(Q 2
=0GeV 2
)photons. Infact[Ab95]
in the limit of x ! 0, even virtual photons at high Q 2
can uctuate into qqpairs and
f is givenby: f 1 (2m x) (2.53)
valid. j i
hadronic
isthen givenby:
j i hadronic / X V= 0 ;! ; 4 r V (1+Q 2 =m 2 V ) ! jVi (2.54)
Thesumextendsoverthethreelightestvectormesons. Inthe frameworkofthe VDMthecross
sections for transversely and longitudinally p olarized photons, p T (W;Q 2 ) and p L (W;Q 2 )
from equation 2.21, are related to the total cross sections of transversely and longitudinally
p olarizedvector mesons scattering o protons at Q 2 =0 GeV 2 : p T (W;Q 2 ) = 0 @ X V= 0 ;! ; 4 r V (1+Q 2 =m 2 V ) 2 + p T;C 1 A Vp T (W) (2.55) p L (W;Q 2 ) = 0 @ X V= 0 ;! ; 4 r V (1+Q 2 =m 2 V ) 2 V Q 2 m 2 V + p L;C 1 A Vp T (W) (2.56)
W is the center-of-mass energy of the ( p)-system as de ned inequation 2.6. The p ossible
di erencein Vp T and Vp L at Q 2
=0istakeninto accountby thefactors
V
,whichareexp ected
to b e within 0 V 1 [Ba92]. p T;C and p L;C
were not included in the original VDM,
but are added to account for higher mass states than the three used vector mesons in the
extension of this mo del discussed b elow. The coupling constants r
V
have b een determined
exp erimentally in p and e +
e reactions [Ba92]. The measurements con rm that the VDM
ansatz is valid. However, several exp erimental results from inelastic ep scattering were not
repro duced by the VDMmo del as discussedab ove. Itwas found that the three lightest vector
mesonsonlycontributeat approximately78 %ofthe total crosssection(r
0 =0:65, r ! =0:08, r
= 0:05). The generalized vector dominance model (GVDM) [Sa72 ] is an extension of the
VDM. It includes not only the three lightest vector mesons but all higher mass states [Sa72 ].
A simpleextension of the VDM is to include the additional term p T;C ( p L;C ) to equation 2.55
(2.56) to take into accountthe contribution from highermass states. A simpleansatz ofthese
terms[Sa72 ]is also used in the analysis presented here (seechapter11):
p T;C = 4 r C (1+Q 2 =m 2 0 ) (2.57) p L;C = 4 r C C " m 2 0 Q 2 ln 1+ Q 2 m 2 0 ! 1 (1+Q 2 =m 2 0 ) # (2.58)
In the most general form of the GVDM, the equations 2.55 and 2.56 are mo di ed by taking
into accountthe diagonal approximation of the transversephoton absorptioncross section:
p T (W 2 ;Q 2 ) = Z m 2 0 dm 2 T (W 2 ;m 2 ) (1+Q 2 =m 2 ) 2 (2.59) p L (W 2 ;Q 2 ) = Z m 2 0 dm 2 T (W 2 ;m 2 ) (1+Q 2 =m 2 ) 2 Q 2 m 2 (2.60) T (W 2 ;m 2 ) = (1=4 2 ) e + e (m 2 ) hadr (W 2 ;m 2 ) (2.61)
The sp ectral weight-function
T
isprop ortional to the cross sectionse + e ! m V and m V p! m V 0 p,wherem V and m V 0
arevector mesonstates withdi erentmasses. The VDMisincluded
inthe GVDMas the sp ecial cases of
T (W 2 ;m 2 )= P V (4 r V )Æ(m 2 m 2 ) T (W).
