UNIVERSIT.. AT BONN Physikalisches Institut

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1 UNIVERSI.. A BONN Physikalischs Institut Masurmnt of bauty and charm cross sctions in photoproduction using dcays into lctrons with ZEUS at HERA by Markus Jüngst Photoproduction of havy quarks in vnts with two jts and an lctron associatd with on of th jts has bn studid with th ZEUS dtctor using th data rcordd in th yars 996 and 6 7, corrsponding to an intgratd luminosity of L = pb and L = 9 pb, rspctivly. h fractions of vnts containing b quarks, and also of vnts containing c quarks, wr xtractd from a liklihood fit using variabls snsitiv to lctron idntification as wll as to smilptonic dcays. For th scond data-st th list of variabls snsitiv to th flavour could b xtndd using th additional information from th microvrtxdtctor. otal and diffrntial cross sctions for bauty and charm production wr masurd for both data-sts and compard with nxt-to-lading-ordr QCD calculations and Mont Carlo modls. Postal addrss: Nußall D-55 Bonn Grmany BONN-IR-- Bonn Univrsity Fbruary ISSN-7-874

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3 UNIVERSI.. A BONN Physikalischs Institut Masurmnt of bauty and charm cross sctions in photoproduction using dcays into lctrons with ZEUS at HERA by Markus Jüngst Disr Forschungsbricht wurd als Dissrtation von dr Mathmatisch-Naturwissnschaftlichn Fakultät dr Univrsität Bonn angnommn und ist auf dm Hochschulschriftnsrvr dr ULB Bonn onlin lktronisch publizirt. Rfrnt: Korrfrnt: Prof. Ian C. Brock Dr. Jürgn Krosbrg Angnommn am:..9 ag dr Promotion: 4..

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5 Contnts Introduction Havy Quark Production at HERA 5. Kinmatics of Lpton Nuclon Intractions Modl Considrations h Quark-Parton Modl Quantum Chromodynamics Evolution of Parton Dnsitis Photoproduction Dirct and rsolvd procsss Nxt-to-Lading Ordr Procsss Evnt Gnrators Excitation procsss Smilptonic Dcays of Havy Hadrons HERA Masurmnts Bauty in Photoproduction h ZEUS Dtctor at HERA. h HERA Collidr h ZEUS Dtctor h Micro-Vrtx Dtctor h Cntral Dtctor h Uranium-Scintillator Calorimtr I

6 II Contnts..4 Luminosity Monitor Background Rjction riggr and Data Acquisition rack Finding and Evnt Rconstruction 8 4. rack Rconstruction de/dx Masurmnt HERA I Primary Vrtx Rconstruction MVD Information Bam-spot HERA II Primary Vrtx Rconstruction Impact Paramtr Dcay Lngth Hadronic Final Stat Rconstruction Jt Rconstruction Rconstruction of Kinmatic Variabls Evnt Display Evnt Slction 5 5. Data and Mont Carlo Sts Onlin Slction Offlin Slction Evnt Slction Elctron Slction Convrsion Findr HERA II Extnsion Control Distributions Signal Extraction 7 6. Liklihood Function Abundancs Probability Dnsity Functions

7 Contnts III 6.. Elctron Idntification Dcay Idntification Variabl Dscription st Function de/dx Liklihood Summary Liklihood Prformanc 9 7. de/dx Hypothsis Particl Sampls Sparation Powr Corrlations Control Plots Liklihood Variabls Kinmatic Variabls Altrnativ Variabls Cross Sction Dtrmination 6 8. Fitting HERA I HERA II Accptanc Corrctions Systmatic Studis 9. Uncrtaintis Enrgy Scal Elctron Background x γ Rwighting Liklihood Variabls riggr Corrction Luminosity Uncrtainty Consistncy Chcks

8 IV Contnts 9.. Slction Cuts Liklihood Dscription Fit Rang Vrtx Dscription Evnt Yild Summary Cross Sctions and Comparison to hortical Prdictions 5. Visibl Cross Sctions Nxt-to-lading Ordr Prdictions Hadronisation Corrctions Diffrntial Cross Sctions Comparison to Othr Masurmnts Conclusions 7 A Control Plots 9 B Accptancs 5 C Fit Dtails 55 List of Figurs 6 List of abls 66 Bibliography 67

9 Chaptr 6 Signal Extraction In this chaptr th mthod usd for signal xtraction is prsntd. o diffrntiat th rlativly small signal from th larg light flavour background a liklihood mthod has bn dvlopd that has bn maximisd in th sparation powr. Whil powrfull, limitations of th mthod includ imprfctions in th simulation, which rsult in systmatic ffcts in th fitting mthod. hrfor carful chcks of th variabl dscription in both th signal as wll as in th background rgion is ncssary bfor combining th availabl variabls to on global discrimination variabl. 6. Liklihood Function h aim of th liklihood function is to combin th information of svral input variabls into on discriminating variabl without losing too much information. Svral variabls hav bn studid and a slction has bn mad in ordr to gt a larg discrimination powr undr controllabl systmatic uncrtaintis. h, whr α is th particl abundanc and P j i is th probability dnsity function for variabl j and for particl undr study i. In this analysis th main particl typs undr invstigation ar ±, π ±, K ± and p/ p. It was found that aftr th cuts th muon contribution was ngligibl and thrfor thy ar not listd hr. In addition th lctrons can furthr b sub-classifid into lctrons from smilptonic b dcays, b, from smilptonic c dcays, c, and from othr contributions, o. Sinc th hard subprocss is of intrst to compar with thortical calculations, procsss whr a charmd hadron is producd in a b b-vnt, calld cascad dcays (b c ), ar countd as bauty signal. Studis also showd that th distributions of th discriminating variabls and, thrfor also of th liklihood, ar vry similar to liklihood function for hypothsis i is givn by L i = α j P j i 7

10 7 Chaptr 6. Signal Extraction th othr bauty vnts, which justifis this tratmnt. Similar argumnts ar valid for vnts whr a τ is producd smilptonically and thn dcays into an lctron (b τ ), or for lctrons from J/Ψ. In th lattr cas th vnt finding fficincy is wors, as th nutrino signatur is diffrnt. In contrast to th HERA I analysis, whr ths two procsss wr tratd as background, thy wr includd in th signal for th HERA II analysis incrasing th cross sction by a fw prcnt. h rlativ contributions of all ths dcays ar listd in abl 6.. How likly it is to idntify a particl dpnds on th rlativ abundanc of this Dcay PYHIA6. PDG8 PDG/PYHIA b X (dirct).5.86 ±.5. ±. b c + X.9.8 ±.9.89 ±. b c X.4.6 ±.5.4 ±.6 b J/Ψ X.4.8 ±..99 ±. b τ ± X.7.4 ±.4.6 ±.6 otal indirct.. ±.5.9 ±.4 otal (dirct + indirct).7. ±.6.97 ±. abl 6.: Branching ratios for B dcays into lctrons from PYHIAand from th Particl Data Book for th LEP B hadron mix (.4B + +.4B +.4B s +.86Λ b ) []. particl typ, and for a givn st of discriminating variabls, on th probability to gt this variabl combination. his probability is givn by th product of th probability dnsity functions of ach variabl. h usag of liklihood ratios for th particl idntification maks th mthod indpndnt of th normalisation of th probability dnsity functions. 6. Abundancs h first contribution to th liklihood function is th particl abundanc, α. h prior abundanc was takn from th Mont Carlo simulation and chckd with th postrior probability which wr dtrmind during th masurmnt. As th rlativ abundancs vary ovr th kinmatic rgion of intrst, th intrprtation of th liklihood as a finding probability can b improvd by dtrmination of th abundancs including kinmatic dpndncis. hrfor th particl abundancs for lctrons, pions, protons and kaons hav bn xtractd in a -dimnsional grid of η and p of th tracks. It was not possibl to find an analytical function which dscribs th shaps of th abundancs for all particl typs, thrfor th

11 6.. Abundancs 7 abundanc has bn dtrmind in ηp -bins. A small binning is dsirabl as larg stps in th rlativ abundancs would rsult in binning ffcts in th liklihood distributions. o avoid statistical fluctuations bins with low statistics hav bn combind aftrwards. Figur 6. shows th absolut particl abundanc for th four givn particl typs whr th ntris hav bn scald with th bin siz. ± π ± p (GV).5.5 η p (GV).5.5 η p/p ± K p (GV).5.5 η p (GV).5.5 η Figur 6.: Particl abundancs for, π, p/ p and K binnd in η and p. h rlativ abundancs for b, c and o ar binnd in p, bcaus no clar η- dpndnc was found. Figur 6. shows th abundanc for th thr typs. For low momnta th dominant sourcs ar th lctrons from photoconvrsions and from Dalitz dcays, bcaus thy hav a much softr momntum spctrum than th lctrons originating from a smilptonic dcay of a havy flavour hadron. Du to th largr quark mass th momntum spctrum of th b is highr than c making it asir to idntify lctrons with larg momnta. It was studid that for a slction Which is from first principl no problm if th Mont Carlo dscribs th data prfctly.

12 + slb + slc + ot Chaptr. 6. Signal Extraction P (GV) P (GV) P (GV) ± b P (GV) ± c P (GV) ± o P (GV) Figur 6.: Dcay abundancs for lctrons from smilptonic dcays from b quarks, from c quarks, and from othr sourcs as a function of p. of DIS vnts th scattrd lctrons hav a significant ffct on th spctrum of th background lctrons, rsulting in a hardr momntum spctrum. If th contribution of lctron and positron running is not balancd, th abundancs and PDFs hav to b xtractd sparatly for positivly and ngativly chargd tracks. For this analysis it has bn vrifid, that th DIS background is small and dos not hav a significant ffct on th cross sction dtrmination. 6. Probability Dnsity Functions h scond ingrdint for th liklihood function ar th probability dnsity functions, PDFs. hy rflct th rlativ probability to find a particl at a givn valu of a slctd quantity. hs PDFs ar xtractd from th Mont Carlo simulation and hav to b chckd with th data aftrwards. Hnc it is ncssary to hav a rasonabl dscription of th signal as wll as of th background rgion. Som mor dtaild studis will b givn in Chaptr 7. h product of svral PDFs allows th information from diffrnt variabls to b combind into on singl quantity. 6.. Elctron Idntification For th lctron idntification th following variabls hav bn found to hav good sparation powr and wr usd as input variabls for th liklihood: de/dx, th spcific nrgy loss pr unit distanc; E CAL /p trk, th ratio of nrgy dpositd in th calorimtr to th track momntum masurd in th CD;

13 6.. Probability Dnsity Functions 75 E EMC /E CAL, th fraction of nrgy dpositd in th lctromagntic part of th calorimtr; d cll, th dpth of th nrgy dposit in th calorimtr. Enrgy Loss du to Ionisation Combind with th track momntum, th de/dx information can b usd to idntify particls, as th avrag nrgy loss pr distanc dpnds on th momntum and th particl mass (s Figur 4.). h information gaind from th calibration and tst sampls hav bn unfoldd into rlativ abundancs to idntify a particl of a givn class. h output of this probability is implmntd in th common ZEUS softwar so that th usr can pick up th rquird information and combin it with th analysis-spcific PDFs and th variabls usful for his application. Figur 6. shows th output of th four particl typs for th sampl aftr prslction. All four distributions ar wll dscribd, so that it can b usd as an input for th liklihood tst function. h influnc of th rmaining discrpancis, spcially in th lctron and kaon cass, was studid by varying th PDF in data and Mont Carlo. h distribution has also bn valuatd as a function of th kinmatic variabls p and η at diffrnt slctions. h influnc of ths uncrtaintis on th cross sction dtrmination is dscribd in Chaptr 9. Calorimtr Enrgy ovr rack Momntum h ZEUS calorimtr was optimisd for th accurat dtrmination of jt nrgis in th rgion of < E jt < GV. For lowr nrgis thr ar dviations in th compnsation, i.. th nrgy rspons diffrs for hadrons and lctrons at th sam nrgy. As also th rsolutions in th CAL diffr for hadrons and lctrons at th sam nrgy, th nrgy distributions vary for th diffrnt particl typs. In th low momnta rgion th momntum rsolution of th CD is bttr than th CAL nrgy rsolution. By taking th ratio of nrgy and momntum masurmnts, E CAL /p trk, th scal dpndnc of th rlativ uncrtainty is rducd and hnc this variabl can b usd for particl idntification. Originally E CAL /p trk was only intndd to b usd for th idntification of antiprotons [5]. Protons and antiprotons bhav diffrntly whn thy intract with th dtctor matrial. Owing for th annihilation nrgy, th distribution of E CAL /p trk is shiftd to largr valus for th antiprotons. For rlativly light particls th E CAL /p trk ratio should b clos to on. h nrgy in th calorimtr

14 76 Chaptr 6. Signal Extraction Entris 5 4 Entris PDF PDF π Entris 5 4 Entris PDF p PDF K Figur 6.: Distribution of a givn particl and its kinmatic quantitis to b an lctron, pion, kaon or proton. h yllow histogram is th Mont Carlo distribution whil th points rprsnt th data. o xtract th rlativ probability a hypothsis will b normalisd by th sum of all hypothss. is now incrasd by GV du to th rcombination with th dtctor matrial. his ffct is visibl in Figur 6.4, whr th maximum is shiftd to for antiprotons. Diffrnt bhaviour of th dtctor rspons for protons and antiprotons obsrvd has bn obsrvd at ZEUS, but th sparation usd for th particl idntification has its origin in th mass dpndnt shift of th cntral valu [6]. h valus of E CAL /p trk movs to lowr valus for th havir particls, whil th lightr particls ar shiftd to high valus. In Figur 6.(a) th normalisd distributions ar shown for lctrons, pions, kaons, protons and antiprotons. For th first thr cass th sampls of particls and antiparticls ar combind, as no significant diffrnc was obsrvd. E CAL /p trk was th only variabl which had to b tratd in this way, as all othr

15 a) b) 6.. Probability Dnsity Functions 77 c) Figur 6.4: Mont Carlo distribution of E CAL /p trk for pion, protons and antiprotons. h antiprotons ar shiftd to highr valus du to th annihilation procss in th calorimtr. h rsult is takn from [5], whr a sampl of 7 singl track vnts with a flat momntum distribution in [.75,.5] was usd to invstigat th diffrnt dtctor rsponss of protons and antiprotons with rspct to th pion background. variabls wr consistnt for positivly and ngativly chargd particls. With th ctromagntic clls highr clustrd prcision (DCA< of th cm) HERA around II data th low valus of E CAL /p trk showd som imprfctions M n rgy for tracks of th having clustrd small clls; nrgis. d) /P, hs candidats did not affct th final lls; c) fraction of th clustrd clls rsult, to th but track to improv momntum. th background dscription candidats with E CAL <.5 GV wr not considrd to hav a rasonabl dscription. h quality of th variabl dscription was tstd using calibration sampls of th de/dx simulation and using particl nrichd sampl as dscribd in th nxt Chaptr. Figur 6.5 shows a comparison btwn data and Mont Carlo for particl sampls usd in a DIS analysis [7]. h distributions ar shown for pion, proton and lctron sampls, whr th lft figurs ar for th ngativly 9 chargd and th right figurs for positivly chargd tracks. In th cas of lctrons and pions no clar charg dpndnc is obsrvd, but for protons and antiprotons th distributions diffr. Fraction of Enrgy in th l. Calorimtr For th HERA I analysis th fraction of calorimtr nrgy dpositd in th lctromagntic part, E EMC /E CAL, was usd as an ingrdint for th liklihood whras in th latr dvlopmnts th PDF was rplacd by a hard cut and by d cll, a partly corrlatd variabl. As th showr signatur diffrs btwn lctrons, muons and hadrons (S Chaptr..), variabls snsitiv to th showr shap can b usd for th lctron idntification. By construction dsign th fraction of th nrgy dpositd in th innrmost layr of th calorimtr should b clos to on for lctrons. Figur 6.6 shows th shap of this variabl for lctrons, muons and hadrons. Most of th lctrons hav an EMC-fraction clos to on, whr th hadrons dposit a significant fraction of thir nrgy in th hadronic calorimtr,

16 78 Chaptr 6. Signal Extraction Figur 6.5: Comparison of data and Mont Carlo for E CAL /p trk distributions for π ± from K s, p p from λ, ± from γ convrsions, and NC [7]. h uppr four figurs show th sampls using only th ngativly chargd tracks and th lowr thr figurs th sampls using th positivly chargd tracks. For protons and antiprotons a diffrnt bhaviour is obsrvd.

17 6.. Probability Dnsity Functions 79 which is rflctd in lowr valus of E EMC /E CAL. A bttr sparation is achivd for th muons in th sampl, whos nrgis ar split btwn th calorimtr and th muon systm. P π µ EMC E /E CAL Figur 6.6: Normalisd distribution of E EMC /E CAL for lctrons (blu), pions (rd) and muons (black). Most of th lctrons hav an EMC-fraction clos to on, whr th hadrons dposit a significant fraction of thir nrgy in th hadronic calorimtr. h muons, which ar minimal ionising particls, can pass th calorimtr, and dposit part of thir nrgy in th muon chambrs, giving a man valu of E EMC /E CAL around.4. Rlatd variabls ar th opning angl of th showr, dnotd by r cll, th radius of th con containing th clls associatd to th EFO objct, th numbr of clls clustrd togthr, n cll, or th cll dpth, d cll, th cntr-of-gravity for th summd clls in th CAL. h drop of E EMC /E CAL as an input variabl for th liklihood in th HERA II analysis mainly had two rasons. Firstly th gomtry cuts wr improvd rducing th sparation powr of th EMC-fraction. Scondly th obsrvd dficits in th dscription of E CAL /p trk hav bn found to b corrlatd to th EMC-fraction. Evn if this not fully undrstood background was rducd by claning cuts, cross-corrlation of systmatic ffcts btwn liklihood variabls should b avoidd. Dpth of Enrgy Dposit In th HERA I analysis it was alrady studid whthr th position of th EFO, which is th cntr-of-gravity in th calorimtr, can b usd to sparat lctrons and hadrons furthr [4]. h dpth of th nrgy dposit in th CAL was studid sparatly for th diffrnt calorimtr rgions, split into transvrs and paralll

18 8 Chaptr 6. Signal Extraction componnts rlativ to th surfac at th impact point. h dscription of this variabl has bn improvd by additional claning cuts, and spcially by tightr gomtry cuts. h distribution of d cll is shown in Figur 6.(a). Evn if th sparation powr is small compard to th de/dx it has bn usd as it is xpctd to hav a small systmatic uncrtainty. h variabl is quit stabl against uncrtaintis in th Mont Carlo dscription as th distribution is vry similar for all background particls, so that a good dscription in lctron and background nrichd sampls indicats th robustnss against changing contributions of th background. 6.. Dcay Idntification For th dcay idntification th following thr variabls wr found to hav good sparation powr and ar usd as input variabls for th liklihood: p rl, th transvrs nrgy of th lctron candidat with rspct to th associatd jt; φ, th azimuthal angl btwn th lctron candidat and th missing transvrs momntum vctor; δ IP, th signd impact paramtr significanc of th lctron candidat with rspct to th bamspot. Sparation from th Jt Du to th largr mass of th B hadron th dcay products originating from it hav mor phas-spac accssibl, and thrfor th spctrum of lctrons from smilptonic b dcays hav a hardr spctrum than particls from light flavour dcays. h spctrum is masurd as th transvrs momntum of th lctron candidat rlativ to th associatd jt, p rl. In this contxt th jt axis rprsnts th dirction of th original mothr particl in which rfrnc systm th transvrs momntum should b rconstructd. h transvrs momntum is thn calculatd using th th following quation: = p jt p, (6.) p jt p rl whr p jt and p ar th momntum vctors of th jt and th lctron candidat. Schmatically th rconstruction of this variabl is shown in Figur 6.7. his variabl, which is th most important variabl for th havy flavour sparation

19 6.. Probability Dnsity Functions 8 Jt Axis p t rl +/ sl. p Figur 6.7: Schmatic viw of th rconstruction of p rl has alrady bn usd in prvious havy flavour analyss using both muons and lctrons as smilptonic lpton particls (.g. [8, ]). h diffrnt spctra of b, c and o ar shown in Figur 6.(a). In contrast to th good sparation of th bauty signal du to th hardr spctrum, th variabl is not abl to distinguish btwn charm and light flavour. h fact that th spctrum of th charm signal is vn lightr than for th light flavour contribution is undrstood as a consqunc of th highr track multiplicitis in th charm jts. his multiplicity shifts th p rl to lowr valus. Rconstruction of th Nutrino o bnfit from th knowldg that th vnts of intrst contain an undtctd nutrino th missing nrgy distribution was studid. Mor information can b drawn from a combination of th dirction and th siz of th missing nrgy vctor. As th z-componnt of th nrgy sum is not wll masurd and th Lorntz boost of th hadronic four momntum is not known, th missing transvrs momntum vctor has bn dtrmind in th xy-plan to: p miss = p, (6.) whr th sum runs ovr all EFO objcts. his vctor rprsnts th dirction of th scapd nutrino from th smilptonic dcay. Prvious studis hav shown that thr is a good corrlation btwn th dirction of th nutrino and p miss vn if mor soft nutrinos ar prsnt in th rsult [4]. o us th dcay signatur of th nutrino th opning angl btwn p miss and th lctron momntum, φ was usd. By using th lctron as a rfrnc systm it is a complmntary variabl to th p rl which is highly corrlatd to th angl btwn th lctron and th jt dirction. Figur 6.8 shows schmatically th rconstruction of φ. h normalisd distribution for b, c and o is shown in 6.(b). Du to its snsitivity to th prsnc of a nutrino originating from a smilptonic dcay, φ has a similar rspons for bauty and charm, but a diffrnt bhaviour for light flavour

20 8 Chaptr 6. Signal Extraction P miss φ jt jt +/ SL Figur 6.8: Schmatic sktch of φ, th diffrnc of th azimuthal angl btwn th lctron and th nutrino, rprsntd by th p miss -vctor. vnts. h asymmtry in th light flavour contribution is causd by th track multiplicity. h chargd track multiplicity is not balancd and as p miss points on avrag to th sid with lss tracks, mor candidats ar found having a larg angl to p miss. Combining p rl and p miss th liklihood is snsitiv to both th bauty and th charm contributions. Using Liftim Information For th HERA II analysis th improvd tracking using th MVD information givs th possibility to dtrmin th track displacmnt with rspct to th primary intraction point with much highr prcision. his analysis bnfits from th liftim information by using th signd impact paramtr significanc, δ IP, as dfind in Equation 4.. h impact paramtr for light flavour vnts, xcpt for particls originating from K L or Λ dcays, is only tratd by rsolution ffcts, and should thrfor b consistnt with zro. As th rsolution ffcts ar not shifting th IP to any favourd dirction, th IP distribution should b symmtric for th light flavour contribution. h normalisd distribution of δ IP is shown in Figur 6.9. h light flavour background is symmtrically distributd around zro, whr both havy flavour contributions hav positiv tails. Du to th longr liftim of B hadrons th sparation works bttr for b than for c. his variabl, which is only accssibl for th HERA II analysis, hlps to improv th ovrall sparation, vn if th sparation is not as good as for p rl or φ. Corrctions Espcially for th HERA I analysis dviations in th p rl and φ distributions hav bn obsrvd. Studis showd, that th discrpancy is visibl in th background but not in th signal nrichd rgion. Figur 6. shows th ratio of data to Mont

21 6.. Probability Dnsity Functions 8 P - - b X c X bkg δ IP Figur 6.9: Probability dnsity functions for δip givn for b (blu), c (grn) and o (rd). h points show th binnd distribution xtractd from th Mont Carlo. h background valus ar distributd symmtrically around zro, whr th havy flavour contributions hav positiv tails. h sparation from th light flavour background is bttr for smilptonic dcays from bauty. Carlo as a function of p rl and φ in a background dominatd sampl. h pur background sampl to calculat ths corrctions has bn xtractd by cutting on th tst function, which is dscribd at th nd of this Chaptr. h fractions of b and c hav bn found to b ngligibl aftr this slction. With ths factors th charm and th light quark p rl distributions wr corrctd for. For low valus of p rl th corrction is vry small, at p rl =.5 GV, whr th purity of th b contribution is largst, it is of th ordr of 5 % and incrasd up to factors of for vry high valus of p rl. h rgion of physical intrst for this analysis is btwn.5 < p rl < (s Chaptr 7). In th HERA II data st th discrpancy at high valus of p rl is rducd significantly. h corrction was found to b of th ordr of 5 % and raisd to %. In cas of th variabl φ th corrction was of th ordr of 5 %. h corrction has also bn applid, but th ffct was much smallr than for th p rl corrction. h dficit in th dscription of p rl and th ncssity to corrct for it was on of th main systmatic uncrtainty sourcs for th HERA I analysis. For th HERA II analysis th contribution to th total systmatic is substantially rducd du to th th bttr dscription of this variabl. In th cas of p rl th corrction has also bn calculatd for ach bin of th variabls for th final diffrntial cross sction. As an xampl th dviation in th Mont Carlo rlativ to data in bins of p is shown in Figur 6. For low valus of p rl, whr th dviation is small, no dpndnc on p was obsrvd, in contrast to th rgion of highr valus, whr th discrpancis can fluctuat and

22 84 Chaptr 6. Signal Extraction #Datn / #MC.5 #Datn / #MC p rl (GV) miss φ (p,p ) zufo Figur 6.: Ratio of data to Mont Carlo distribution calculatd from a background nrichd sampl to corrct th PDF of p rl and φ [4]. p p rl Figur 6.: Dviation of th Mont Carlo distribution rlativ to data in bins of p rl and p. h dviation as a function of p had to b chckd, in ordr to tst th systmatic ffct of th corrction dpnding on th bins of th final diffrntial cross sctions. should b takn into account in th calculation of th systmatic uncrtaintis. 6.. Variabl Dscription Lik in th tratmnt of abundancs, whr a fin binning with small fluctuations was aimd for, th shap of th variabls usd in th liklihood should b wll dscribd. A rough binning has th advantag of high prcision for th avrag ovr a bin, but can caus problms in cas of migrations at th bin dgs. In addition, larg stps of liklihood probabilitis at th bin dgs should b avoidd, as this would driv to sparations in liklihood rsponss of candidats, narby in phas spac, but with valus around ths dgs. o avoid such ffcts, in th first stp th distributions wr xtractd using a binning which is wid nough to avoid

23 6.. Probability Dnsity Functions 85 statistical fluctuations. In th scond stp a smooth distribution was achivd, which dscribs th shap of th distribution. hs distributions wr found by fit-functions or diffrnt smoothing procdurs. Fit with Function Whrvr it is possibl to dfin an analytical function to fit th PDF distributions this function was usd to dfin th PDFs. Such a function with rasonabl numbr of fit paramtrs and good dscription of th points was found for p rl and φ. Figur 6.(a) shows th binnd distribution for bauty, charm and background particls xtractd from th Mont Carlo including th lin which rprsnts a convolution of a Gaussian and a Landau function fittd to th points. h fit curv dscribs th points vry wll and has also bn chckd with a finr binning in rgion of high statistics. For φ a polynomial of fourth ordr has bn usd as a fit function. h fit curv and th binnd distributions ar shown in Figur 6.(b). In this cas th curvs dscrib th points vry wll again. It was found that th influnc on th cross sction by rplacing th binnd histogram with th smooth curv is ngligibl. h stability is not rflctd in a bttr dscription, but thr ar lss stps in th control plots. Smoothing For th variabls E CAL /p trk and d cll no xplicit function could b found to dscrib all distributions. In most of th cass it was possibl to dfin a function which dscribs th distribution for on particl typ, but not on function for all of thm. h smoothing routins implmntd in th softwar packag ROO [9] wr not sufficint; thrfor a combination of a B-Splin and a Bézir curv was finally usd [, ]. h B-Splin is dfind by: n P (t) = N(i, k, t) P i, (6.) i= whr P i ar th start valus for bin i of th histogram with n bins, and N(i, k, t) is dfind by: N(i,, t) = if(τ[i] t τ[i + ]) N(i,, t) = ls t τ[i] N(i, k, t) = N(i +, k, t) τ[i + k ] τ[i] τ[i + k] t + N(i, k, t), (6.4) τ[i + k] τ[i + ]

24 86 Chaptr 6. Signal Extraction P b X c X bkg rl p (GV) (a) p rl PDF P...8 b X c X bkg mis φ(p (b) φ PDF Figur 6.: Probability dnsity functions for p rl (a) and φ (b) givn for b (blu), c (grn) and o (rd). For φ th zro is supprssd on th y-axis. h points show th binnd distributions xtractd from Mont Carlo. h lins show th fittd distributions which wr finally usd to calculat th probabilitis. whr τ[i] ar th knot points sprad ovr th rang undr considration. In th first stp intrmdiat knot points wr calculatd for th PDFs. h valus P (t) wr xtractd using a splin up to ordr k =. It was chckd that highr ordrs did not improv th dscription of th input histogram any mor. In th scond stp ths points wr smard using a Bézir curv which is dfind by: with th cofficints: n P (s) = B(i, n, s) P i, (6.5) i= ( ) n B(i, n, s) = s i ( s) n, (6.6) i,)

25 6.. Probability Dnsity Functions 87 whr P i ar th n input points P (t) from th B-Splin. In Figur 6. th smooth curvs, P (s), ar compard with th points, which rprsnt th binnd Mont Carlo distributions, P i, for th diffrnt particl sorts.

26 88 Chaptr 6. Signal Extraction P.4 ± π ± ± K p p CAL E /p trk (a) PDF for E CAL /p trk P ± π ± ± K p/p (b) PDF for d cll (cm) d cll Figur 6.: Probability dnsity functions for E CAL /p trk (a) and d cll (b) givn for b (blu), c (grn) and o (rd). h points show th binnd distributions xtractd from th Mont Carlo. h lins show th smoothd distributions which wr finally usd to calculat th probabilitis. For both E CAL /p trk (Figur 6.(a)) and d cll (Figur 6.(b)) th points ar wll rproducd by th smooth curvs. 6.4 st Function Combining ths PDFs with th rlativ abundancs th liklihood function can b computd for various hypothss. In this considration for th hypothsis of th diffrnt particl sorts (, π, K, p) and in th cas of th lctrons, for th diffrnt dcay typs. In th cas of a hypothsis for lctrons from a spcific dcay typ, th abundanc is a product of th lctron and th corrsponding dcay abundanc. In ordr to intrprt th liklihood as a probability and to hav a dirct

27 6.4. st Function 89 comparison of th goodnss of th candidats, th liklihood ratio (tst function ) is usd, whr th hypothsis is dividd by th sum of all diffrnt hypothss. h following thr tst functions ar of main intrst:, th tst function for bing an lctron, b, th tst function for bing an lctron from a smilptonic b dcay and c, th tst function for bing an lctron from a smilptonic c dcay. h first on is givn by: with = L L + L π + L π + L K + L p (6.7) L = α (η, p ) P (de/dx) P (E CAL /p trk ) P (d cll ) (6.8) and by adding th dcay abundanc and th thr dcay variabls b (and analogously c ) is givn by: b = L b L b + L c + L o + L π + L K + L p (6.9) with L b = L α b (p ) P b (p rl ) P b( φ) P b (δ IP ) (6.) h bauty hypothsis was usd to xtract th bauty and charm fractions and th lctron and charm hypothss wr usd to xtract signal nrichd sampls at diffrnt slction stags to control th signal and background dscription de/dx Liklihood h most important contribution to th lctron idntification is th de/dx PDF. o study th sparation powr of this variabl, on can dfin a tst function with de/dx as th only input variabl. Figur 6.4 shows th prformanc of th liklihood, using only th de/dx probability dnsity functions (s Figur 6.) as input for th liklihood ratio: de/dx = P (de/dx) (6.) P π (de/dx) + P (de/dx) + P p (de/dx) + P K (de/dx) h lft figur shows th distribution of th ln de/dx for th lctron candidats dfind in th vnt slction (dropping th cut of ln de/dx < ) split into th four rlvant particl typs. h signal fficincy and background supprssion can b calculatd from th normalisd distribution of th tru lctrons and th fak lctrons in Figur 6.4 (lft). By cutting at ln de/dx < th fficincy for th lctron slction is 98% with a background supprssion of 6%.

28 9 Chaptr 6. Signal Extraction Entris π p P (%) bkg K (%)..5. non ln ln Figur 6.4: de/dx liklihood hypothsis tst for, p, K, π (lft) and th normalisd distribution for and non-. h lin indicats th cut at de/dx <, whr th signal fficincy and background rjction is shown in th inlay plot. 6.5 Summary Combining all ingrdints, th liklihood provids on singl powrful variabl to sparat th havy flavour vnts from th light flavour contribution. h distribution of this variabl was usd to dtrmin th fractions of lctrons originating from smilptonic dcays. In Figur 6.5 th liklihood tst function is shown for th bauty (uppr) and th charm (lowr) hypothss. Both distributions, b and c, sparat th corrsponding signal from th background. For th final cross sction dtrmination only th bauty hypothsis has bn usd. h charm tst function was usd to xtract charm nrichd sampls, which ar vry important to study systmatic ffcts from imprfctions in th signal dscription. In th following chaptr studis of th liklihood prformanc including control plots ar prsntd. h studis ar important to undrstand th influnc of a singl variabl to th liklihood and to b abl to stimat th systmatic uncrtainty du to th liklihood mthods. Systmatic ffcts can b causd by bad dscription of input variabls to th liklihood or, mor subtl, by ffcts on th liklihood by diffrncs in th corrlation btwn th variabls. h unfolding of th cross sctions is dscribd in Chaptr. During th optimisation procss of th signal dtrmination th input of th liklihood was varid using altrnativ variabls for th dcay idntification. Som xampls of ths variabls ar listd in th nxt chaptr.

29 6.5. Summary 9 DAA 6-7 b X c X bkg - - ln Figur 6.5: st function rspons for th bauty hypothsis, b, (uppr Figur) and charm hypothsis, c (lowr Figur). h histograms show th Mont Carlo split into thir contributions from b (blu), c (grn) and light flavour background (yllow). h distribution is compard to data (points). For th final cross sction dtrmination only th bauty hypothsis has bn usd, whr th charm hypothsis is a vry usful tool to xtract charm nrichd sampls for systmatic studis.

30 Chaptr 7 Liklihood Prformanc In this chaptr som studis to undrstand th prformanc of th liklihood and th influnc of th diffrnt variabls ar discussd, with mphasis on th sparation powr of th de/dx masurmnt. Control plots will giv an imprssion of th signal and background dscription by th Mont Carlo. At th nd som altrnativ variabls, tstd for th signal xtraction, ar prsntd. 7. de/dx Hypothsis As xplaind in Chaptr 6 th variabl de/dx is not only abl to distinguish btwn lctrons and background particls, but can also b usd for othr particl hypothss. Figur 7. shows th liklihood tst function using only th de/dx as input variabl for th lctron hypothsis, de/dx, as wll as for pion, kaon and proton hypothsis for th diffrnt particl typs. For th following studis th candidats wr slctd with th cuts listd in abls 5. and 5.4 without ln de/dx <. h uppr lft figur shows th sparation for th lctron hypothsis, which is abl to distinguish btwn lctrons and pions or kaons. h proton distribution has two paks, on in th pur background rgion and on clos to th lctron pak, rsulting in a sparation limitd to th subsampl of th right pak. h origin of th doubl-pak structur ar th crossing points of th bands in th (de/dx, p)-plan. In th following th influncs on th kinmatic distributions of th rmaining particls ar dscribd. h uppr right and lowr lft figurs show that it is also possibl to mak pion and proton nrichd sampl. Du to th ovrlaying structur of th de/dx bands th possibility to obtain a sampl with rasonabl kaon purity is limitd. 9

31 7.. de/dx Hypothsis 9 P..8.6 π p P K ln de/dx - π - ln de/dx P p - ln de/dx P ln K de/dx Figur 7.: Normalisd distributions of th liklihood tst function for, π, p/ p and K using th lctron (uppr lft), pion (uppr right), proton (lowr lft) and kaon (lowr right) hypothss. 7.. Particl Sampls Figur 7. shows a scattr plot of de/dx vs. momntum for all lctron candidats. At this slction stag th band structur for protons with high de/dx at low momnta is alrady rducd by th prslction cuts. Figur 7.: Scattr plot of man nrgy loss, de/dx, vs. track momntum, p, for all lctron candidats. By cutting on ln x de/dx < (.5) a mdium (highly) nrichd sampl, for particl sort x, can b obtaind. In Figur 7. ths sampls ar shown for an lctron,

32 94 Chaptr 7. Liklihood Prformanc pion, proton and kaon nrichd rgion for th lowr cut on th lft sid and th hardr cut on th right sid. Figur 7.: Scattr plot of man nrgy loss, de/dx, vs. track momntum p. h mdium (lft) and highly (right) nrichd rgions wr xtractd by cutting on th diffrnt liklihood hypothss as dscribd in th txt. h sampls shown ar th lctron (blu), pion (rd), proton (grn) and kaon (black) sampls. For th mdium nrichd cas th sampls still hav ovrlapping rgions, whil thy ar clarly sparatd in th highly nrichd sampl. h liklihood valu dpnds on th sparation of th bands, whr crossing-points lad to indistinguishabl rgions. h liklihood cuts caus disconnctd subparts of th particl bands. For xampl th proton candidats ar split into thr rgions: th low momntum and high de/dx rgion, th high momntum and low de/dx rgion and th rgion in th gap of th pion and th lctron band in th intrmdiat rgion. h right figur also shows that it is possibl to idntify lctrons down to.9 GV. Additionally it shows that it is complicatd to idntify lctrons in th rgion around. GV, and that a hard cut on th lctron liklihood would dpopulat this rgion. 7. Sparation Powr o compar th sparation powr of th othr variabls with th on of th de/dx tst function, th tst function has bn computd aftr rplacing th de/dx PDFs by th PDFs of th othr variabls and combining probabilitis with th abundancs as dscribd in th prvious chaptr. In Figur 7.4 th hypothsis tst function is shown for th six diffrnt input variabls. h liklihood distribution is shown for th whol Mont Carlo sampl (yllow) and for th bauty signal (black). Figur 7.4(a) illustrats th importanc of th de/dx for th lctron idntification.

33 7.. Sparation Powr ln de/dx - ln CAL E /p trk (a) st function using de/dx (b) st function using E CAL /p trk ln dcll - ln p rl (c) st function using d cll (d) st function using p rl ln φ - ln δ IP () st function using φ (f) st function using δ IP Figur 7.4: st function shown for th complt Mont Carlo sampl (yllow) and for th bauty signal (black). h tst functions hav bn computd by using th abundanc plus a signal additional input variabl.

34 96 Chaptr 7. Liklihood Prformanc h larg light flavour background is shiftd towards valus abov. With dcrasing sparation powr also E CAL /p trk (7.4(b)) and d cll (7.4(c)) hlp to sparat th lctron and non-lctron contributions. h thr dcay variabls do not distinguish btwn lctrons and background particls, but nrich th havy flavour contribution. Espcially for th p rl variabl (s Figur 7.4(d)) th shap of th liklihood distributions diffr. h rang of th tst function and th sparation is lowr for φ (7.4()) and δ IP (7.4(f)), but both variabls contribut to th final sparation of b, c and th light flavour background. h study of th corrlations btwn th variabls and thir influnc on th systmatic uncrtainty was important for th slction of th liklihood variabls. 7. Corrlations As mntiond in Chaptr 6 it is complicatd to quantify th total systmatic uncrtainty for corrlatd variabl, whil not much additional sparation powr can b obtaind from highly corrlatd variabls. In Figur 7.5 th corrlation matrix for som variabls undr study is shown. h variabls usd for th HERA I analysis ar highlightd by th magnta box. By this studis of th corrlations NClls Radius CllDpth PtRl dphi EovP EMCf Ddx - - Ddx EMCf EovP dphi PtRl NClls Radius CllDpth Figur 7.5: Corrlation matrix for possibl variabls in th liklihood. h variabls usd for th HERA I analysis ar markd by th magnta box. h othr thr variabls ar highly corrlatd and ar also connctd with th EMC-fraction. h corrlation studis wr prformd using th softwar packag MVA [] th cut at E CAL >.5 GV (s Sction 5..) was drivn to stabilis th slction and rduc th imprfction in th E CAL /p trk simulation. As a consqunc of

35 7.4. Control Plots 97 dropping th EMC fraction from th liklihood for th HERA II analysis, it was studid which of th thr additionally shown corrlatd variabls could improv th lctron sparation. h cll radius, r cll, was found to b not wll simulatd and to b too snsitiv to dtctor gomtry. h distribution is a suprposition of svral stp-functions, whr th stp siz and width dpnds on th calorimtr block siz for a givn η rgion. Also th numbr of hit clls, N cll, was xcludd bcaus this intgr variabl showd som migrations which ar not simulatd by th Mont Carlo. hrfor d cll was finally chosn to improv th lctron idntification. 7.4 Control Plots h tst function with diffrnt hypothss uss th following variabls: de/dx, E CAL /p trk, d cll, p rl, φ and δ IP. With ths tools of diffrnt hypothss, it was possibl to chck th quality of th data dscription. Only wll dscribd input variabls can b usd to hav sam liklihood rspons for data and Mont Carlo. In th following som xampls of chcks for th background and signal dscription ar shown. h distributions of all candidats quantifis th quality of th background dscription as th slction at this stag is clarly backgrounddominatd. Additionally th distributions wr chckd for signal nrichd rgions, whr cuts on th liklihood hypothsis hav bn usd to nrich th sampls with bauty or charm vnts. In th following som control plots ar shown. h background nrichd plots corrspond to all candidats and th signal nrichd plots to candidats with ln b <.5 for bauty and ln c < 5 for charm. Additional plots can b found in Appndix A. Control plots for th HERA I analysis can b found in [6]. h cuts for th slction hav bn varid to scan various lvls of puritis and idntify rgions of imprfctions to dcid whthr thy influnc th rsults Liklihood Variabls h first variabls to chck ar th liklihood input variabls. o b snsitiv to th signal to background sparation powr, th distributions must b rasonably wll dscribd for both th signal as wll as for th background contributions. Diffrncs in th shaps for diffrnt slction stags show th rgions of snsitivity as wll as th influnc of th variabl to th liklihood. In Figur 7.6 p rl, which is th main variabl for th bauty sparation, is shown as an xampl for th liklihood variabls. h distribution is rasonably wll dscribd for all thr slctions. Uncrtaintis du to rmaining discrpancis wr studid by a

36 de/dx (mip) CAL E /p trk Chaptr 7. Liklihood Prformanc de/dx (mip) de/dx (mip) 5 5 d 8 cll (cm) 6 5 variation of th Mont Carlo distribution as dscribd in Chaptr 9. h rgion of 5 intrst for th charm-nrichd plots is clarly sparatd from th rgion for th bauty-nrichd plot. his rflcts CAL th probability to idntify th diffrnt sourcs at a givn valu of p rl E /p trk using its PDF (s Figur 6.(a)) d cll (cm) CAL trk rl p (GV) de/dx (mip) E /p Data (6-7) PYHIA φ (scald) b X c X bkg rl p (GV) Data (6-7) PYHIA (scald) b X c X E bkg CAL /p trk 5 5 d cll (cm) φ φ rl d cll p (cm) (GV) rl p (GV) Figur 7.6: Distribution of th liklihood variabl p rl 4. h thr histograms show th distribution for all candidats Data (6-7) which is background dominatd (uppr Data lft), (6-7) for bauty nrichd (lowr lft) PYHIA and charm (scald) nrichd (lowr right). h Mont PYHIA Carlo (scald) is subdividd into th bauty signal (blu), b th Xcharm signal (grn) and th background b contributions X 8 (yllow). h thr contributions c Xwr scald by th fit rsult (s Chaptr c ). X 6 bkg. bkg. 4 A similar ffct can b sn for th variabl δ IP which is shown in Figur 7.7. h thr figurs show th distribution for a background dominatd rgion and for bauty and charm nrichd rgions, φ whr th ngativ sid of th distribution is highly supprssd aftr th liklihood cuts. h variabl is wll dscribd at all thr slction stags. Small discrpancis wr found whn studying th control plots in bins of th kinmatic variabls of th track. h ffct is most pronouncd for tracks going to th forward or backward rgions in th uppr half of th dtctor. Figur 7.8 shows δ IP in bins of p and η, whr a shift is visibl in th lowst lft and right figur. h ffct of this shift on th final cross sction dtrmination was not vry larg, nvrthlss it would b an important stp in th furthr incras in prcision of th HERA II rsults to find a corrction procdur.

37 de/dx (mip) E CAL /p trk Control Plots d cll (cm) Entris Entris φ δ IP Entris rl p (GV) Data (6-7) PYHIA (scald) b X c X bkg δ IP δ IP Figur 7.7: Distribution of δ IP for background (uppr lft), bauty (lowr lft) and charm (lowr right) nrichd sampls. h sum of th Mont Carlo contributions from bauty (blu), charm (grn) and background (yllow), scald with th factors as dscribd in Chaptr, dscrib th data distribution (points) in all thr slctions.

38 Chaptr 7. Liklihood Prformanc Entris -.5 < η < -.5 π < φ < π/ Entris -.5 < η <.5.5 < η <.5 π< φ < π/ π< φ < π/ Entris δ IP δ IP δ IP Entris -.5 < η < -.5 π/ < φ < Entris -.5 < η <.5 π/ < φ < Entris.5 < η <.5 π/ < φ < δ IP δ IP δ IP Entris -.5 < η < -.5 < φ < π / < Entris -.5 < η <.5 < φ < π / < Entris.5 < η <.5 < φ < π / < δ IP δ IP δ IP Entris -.5 < η < -.5 π/ < φ < π Entris -.5 < η <.5 π/ < φ < π Entris.5 < η <.5 π/ < φ < π δ IP δ IP δ IP Figur 7.8: Distribution of δ IP in diffrnt bins of p and η (as dnotd in th figurs) for th background nrichd sampl. Othr dtails as in caption of Figur 7.7. Excpt for th last row, whr a closr viw showd a small shift for th forward and backward tracks, no major discrpancis ar obsrvd.

39 7.4. Control Plots 7.4. Kinmatic Variabls A mor indirct influnc of th slction is visibl in th kinmatic track variabls. hs do not ntr dirctly into th liklihood but ar of larg intrst as 5 thy dfin th kinmatic phas spac undr study. In Figur 7.9 th transvrs momntum of th lctron candidat, p 5, is shown for th thr diffrnt slc- 8 tions. h signal 6 lctrons clarly hav a hardr spctrum than th background 5 4 particls. As th momntum ntrs th liklihood only indirctly via th rlativ 5 abundanc and via th de/dx PDF, th sparation is not that clar as in th cas of p rl If th momntum would de/dx dirctly (mip) ntr th liklihood th rgion of low CAL E /p trk momnta would b dpopulatd making it impossibl to masur th production 5 7 rats at ths nrgis d cll (cm) p (GV) rl N Cand p (GV) φ p (GV) φ Data (6-7) PYHIA (scald) b X c X bkg Data (6-7) PYHIA (scald) b X c X bkg. η p (GV) η η Figur 7.9: Distribution of th kinmatic variabl p. h thr histograms show th Data (6-7) 45 Data (6-7) 4 distribution for all candidats which is background dominatd 5 5 PYHIA (scald) (uppr lft), for bauty PYHIA (scald) nrichd (lowr lft) and charm nrichd (lowr right). h b Mont XCarlo is subdividd into th bauty signal 5 (blu), th charm signal (grn) and thc background X contributions 5 (yllow). h thr contributions wr scald by th fit rsult (s bkg. Chaptr ) b X c X bkg φ φ

40 Chaptr 7. Liklihood Prformanc 7.5 Altrnativ Variabls In th prvious chaptr som altrnativ variabls for th lctron idntification wr sktchd brifly. During th HERA II analysis th studis wr focusd on th improvmnt of th de/dx dscription and on diffrnt variabls using th liftim information. h variabls not usd in th liklihood hypothsis ar still usful for systmatic studis providing indpndnt tchniqus for th signal nrichmnt. For anothr analysis [] th variabl φ was rplacd by p miss,, th componnt of th missing transvrs momntum paralll to th lpton dirction. Figur 7. shows th control plot and th normalisd distributions of p miss, on a logarithmic (lft) and linar (right) scal. h data distribution is wll dscribd Numbr of Entris 4 Numbr of Entris ptmiss * v (GV) ptmiss * v (GV) Arbitrary Units - - Arbitrary Units ptmiss * v (GV) ptmiss * v (GV) Figur 7.: Distribution of th transvrs momntum componnt paralll to th lctron dirction, p miss,. h uppr row shows th control plot for all candidat in Mont Carlo (yllow) and data (points), whr th lowr row shows th PDF for b (blu), c (grn) and bkg. (rd). o compar th dscription in both th maximum as wll as in th tails, th plots ar shown on linar and logarithmic scal. by th Mont Carlo. Similar to φ, this variabl is abl to distinguish btwn th havy and th light flavour contributions. h sparation powr was found to b a littl bit bttr, but in photoproduction vnts th systmatic uncrtainty du to this variabl is largr than for φ so that this variabl was only usd as a systmatic cross chck.