La ricerca del bosone di Higgs a LEPGennaio 2008Giacomo Sguazzoni

LEP (Large Electron-Positron) e stato il piu potentecollider elettrone-positrone ad oggi mai costruito. Era ospitatoin un tunnel circolare di circa 27 km di circonferenza che adessoospita LHC.

Perche LEP non e esattamentecircolare?

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L’energia dei fasci. LEP ha lavorato con energie dei fascicomprese tra ∼ 45 GeV e ∼ 100 GeV. Infatti, nella sua primafase sperimentale, dal 1989 al 1995, ha prodotto collisioni e+e−

intorno a√

s ∼ 91.2 GeV per lo studio delle proprieta del bosoneZ. La seconda fase sperimentale di LEP, detta LEP2 dal 1996 al Perche si usa la notazione

√s; cosa

significa?.2000, e stata caratterizzata dal progressivo aumento dell’energiadei fasci fino a oltre 100 GeV per fascio. A LEP2 il programma di

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1

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5

6

10030 300

mH [GeV]

!"2

Excluded Preliminary

!#had =!#(5)

0.02758±0.000350.02749±0.00012incl. low Q2 data

Theory uncertaintymLimit = 144 GeV

http://lepewwg.web.cern.ch/LEPEWWG/

punta e stata la ricerca della particella di Higgs. Prima di LEP,nel 1989, della massa dell’Higgs si sapeva solo che doveva esserepiu grande di ∼ 1 GeV: oggi sappiamo che deve essere maggioredi 113 GeV. Quindi, combinando il limite superiore dai test diprecisione con il limite inferiore dalla ricerca diretta, LEP indicache la massa dell’Higgs del Modello Standard sia compresa in unpiccolo intervallo tra 113 e circa 200GeV.

La luminosita e l’altro parametro fondamentale di un accel-eratore di particelle. Per un qualsiasi processo fisico il ritmo diproduzione degli eventi ad esso relativi e dato da:

dN

dt= σL (1)

1

dove σ e la sezione d’urto del processo. Nel caso di collisionecentrata (head–on) tra le particelle dei due fasci, la luminositaistantanea in ciascun punto di interazione e data da1: 1W.-M. Yao et al., J. Phys. G 33, 1

(2006)

L = fNe+Ne−

4πσhσv, (2)

dove Ne− ed Ne+ sono, rispettivamente, il numero di elettronie positroni per pacchetto, f la loro frequenza di interazione e Perche i ”fasci” non sono continui ma

suddivisi in pacchetti?σhσv e il prodotto delle standard–deviation delle gaussiane cherappresentano la distribuzione di densita delle particelle di unpacchetto lungo le dimensioni trasverse orizzontale e verticale.Dai parametri in tabella si ricava un ambito per le luminosita Ne+ 4.5 · 1011

Ne− 4.5 · 1011

f−1 25 µsσh 2.5− 8 µmσv 200− 300 µm

istantanee compreso tra 1031 cm−2s−1 e 1032 cm−2s−1.La luminosita integrata L,

L =∫ t2

t1

L dt =N

σ, (3)

rappresenta la luminosita accumulata in un certo intervallo ditempo. L’unita di misura conveniente per LEP e pb−1 (picobarninversi). LEP ha avuto eccellenti prestazioni in termini diluminosita istantanea e cio ha permesso, nella fase due, diraccogliere una luminosita integrata pari a ∼ 700 pb−1 per √

s[ GeV] L[ pb−1]130-136 10

161 10172 10183 50

189-192 200196 80

200-202 120205-207 220

esperimento, suddivisa fra le varie energie.A LEP operavano quattro esperimenti: ALEPH, DELPHI,

OPAL e L3. Tutti erano esperimenti cosiddetti omni-purpose,cioe disegnati per studiare tutti gli aspetti delle interazioni e+e−

e quindi piuttosto simili, caratterizzati da elevata ermeticita, per

perche l’ermeticita e importante?

coprire la maggior percentuale possibile dell’angolo solido totale,e per collezionare, di ciascun evento, la massima quantita diinformazioni.

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ALEPH (Apparatus for LEp PHysics), prendiamolo ad esempioe descriviamolo con qualche dettaglio.

Si riconoscono due sottosistemi complementari aventi simmetriacilindrica rispetto all’asse dei fasci: il sottosistema piu interno eprincipalmente dedicato alla determinazione della traiettoria delleparticelle cariche (tracking) tramite i rivelatori di tracciatura che,grazie alla presenza di un grosso magnete superconduttore cheli avvolge, sono in grado di misurare con precisione l’impulsodelle particelle cariche. Il sottosistema piu esterno e compostoesclusivamente di apparati (calorimetri) dedicati alla misuradell’energia della particelle e alla loro identificazione. La filosofiadi realizzazione di questi sottosistemi e opposta: i rivelatori ditracciatura devono perturbare il meno possibile il moto delleparticelle di cui misurano la traiettoria. Viceversa i calorimetrifermano completamente le particelle (misura distruttiva) affinchel’energia, depositata completamente nel volume sensibile deldispositivo, sia accuratamente misurata.

I rivelatori di tracciatura che si incontrano procedendo radial-mente verso l’esterno a partire dal punto di interazione sono iseguenti:

• VDET, rivelatore di vertice;

• ITC, camera a fili interna (Inner Tracking Chamber);

• TPC, camera a deriva (Time Projection Chamber).

I rivelatori di calorimetria, elencati con lo stesso criterio, sono:

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• ECAL, calorimetro elettromagnetico (ElectromagneticCalorimeter);

• HCAL, calorimetro adronico (Hadronic Calorimeter).

Nella parte piu esterna troviamo le camere per la rivelazionedi muoni (Muon Chambers) che, dato il piccolo rapporto E/m,oltrepassano quasi indisturbati i rivelatori piu interni.

Tra i due calorimetri e presente il solenoide del magnetesuperconduttore che, raffreddato ad elio liquido e percorso dauna corrente di 5000 A, fornisce un campo magnetico uniformedi 1.5 T parallelo all’asse dei fasci per una precisa misura delmomento.

Molto vicino alla beam–pipe, ad una distanza di ±2.5 m dallazona di interazione, sono collocati i monitor di luminosita che,dalla misura del rate dei processi Bhabha a piccolo angolo la cuisezione d’urto e ben nota in QED, permettono di misurare laluminosita. Perche?

4

La misura dell’impulso trasverso delle particelle cariche

MAGNET COILWIRE

CHAMBERS

INNER

FIELD CAGE

WIRE CHAMBER SUPPORTS

CENTRAL MEMBRANE

OUTER

FIELD CAGE

4710 cm

3600 cm

(diameter)

si basa sulla presenza del campo magnetico. Il metodo e ladipendendenza della risoluzione in impulso dalle grandezze ingioco e descritto in appendice a pag. 19.

In ALEPH il cuore dell’apparato tracciante era costituito dallaTPC, una camera a deriva cilindrica di ∼ 4.7 m di lunghezza,∼ 3.6 m di diametro esterno e ∼ 0.6 m di diametro internoche garantiva fino a circa 20 punti di misura (vd. figura)con risoluzione σrφ ∼ 180 µm e σz ∼ 800 µm per traccia. Larisoluzione ottenuta su 1/pT era 10−4( GeV/c)−1 ⊕ 5 × 10−3/pT

(con pT in GeV/c). Il termine indipendente da p e dovuto allo

Il termine ∝ 1/p e compatibile con iparametri riportati?

scattering multiplo.

Di che evento si tratta? Perche latraccia sulla sinistra ha meno puntidell’altra?

Lo scattering multiplo deteriora la risoluzione di un apparato

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nucleus impact parameters a single large angle scatter is possible. This is described by Rutherford

scattering and the angle ! is given by

d

d

"

!#

sin /$

1

24

Larger impact parameters are more probable and the scattering angle will be smaller as the

nuclear charge is partly screened by the atomic electrons. Hence in a relatively thick material

there will be a large number of random and small deflections. This is described by multiple

Coulomb scattering. The relative probability of scattering as a function of scattering angle is

illustrated in Fig. 5.

1

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scattering angle !

rela

tive

in

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sity

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multipleCoulombscattering

singlescattering

events

!0

% = 2.25

% = 1.80

% = 0

With vertex point

10–1

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101 102 103

pT

GeV

Mo

me

ntu

m R

eso

lutio

n &

p/p

Figure 5: The relative probability of scattering as a Figure 6: The estimated momentumresolution

function of scattering angle. In CMS as a function of pt at various %.

The r.m.s. of the scattering angle is given by

!'

0

0

13 6

. (

MeV

pcZ

L

Xinc

where L is the thickness of the material in the tracker and X0 is the radiation length of the material,

Both are measured in m. The apparent sagitta due to multiple scattering is given by

sL

ms =!

0

4 3

If the extrapolation error from one measuring plane to the next is larger than the point resolution,

i.e. !0&r > "x, then the momentum resolution will be degraded. The relative momentum resolution

due to multiple scattering is then given by

s

s

dp

p B LXs

BL

p

ms

ms

. sin .

= ( =0 051 0 3

80

2

ce (3)

Hence the relative momentum resolution is independent of p and is proportional to 1/B. For

example dp/p ( 0.5% for argon gas with L=1m and B=1T. The estimated momentum resolution

tracciante specialmente a piccoli impulsi. Questo e dovuto allecollisioni successive della particella carica incidente con i nucleiatomici del materiale che attraversa. Sperimentalmente si osservache la distribuzione dell’angolo di scattering di una particellache attraversa uno strato di materiale sufficientemente spessoe costituita da una parte centrale gaussiana che risulta dallasuccessione di urti e di una coda a grandi angoli che mostra iltipico andamento alla Rutherford (∝ 1/ sin4 θ) in quanto dovutaa singoli urti

θ0 ∼14 MeV

βpcZinc

√x

X0. (4)

5

Il segnale del bosone di Higgs nelle collisioni e+e− eprincipalmente dovuto al cosiddetto processo di Higgs-stralunghe+e− → Z∗ → ZH.

perche la Z e virtuale? Perche unprocesso analogo con un γ al postodella Z∗ non esiste?

La sezione d’urto di produzione dipende dalla massa dell’Higgse dall’energia del centro di massa. In particolare, per una fissatamassa dell’Higgs mH, si ha un vistoso incremento della sezioned’urto di produzione non appena si supera la soglia

√s ∼ mZ+mH.

A cosa e‘ dovuto il vistoso picco intornoa 90 GeV?

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mH = 50 GeV/c2

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Centre-of-mass energy, !s (GeV)

Hig

gs p

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ctio

n cr

oss s

ectio

n (p

b)

Al di la della dipendenza dalla massa dell’Higgs e da√

s,la sezione d’urto di produzione e comunque dell’ordine del pb.Questo da’ l’idea di quanti eventi possono essere prodotti a LEPcon luminosita integrate dell’ordine di qualche decina di pb−1 peresperimento al di sopra dell’energia di soglia.

Il decadimento del bosone di Higgs nel range di massaaccessibile a LEP avviene principalmente in bb. Perche e il canale bb il canale di

decadimento privilegiato?

Standard Model

bb_

!+!

-

cc_

gg

BR(H)

W*W

*

Z*Z

*

mH

[GeV]

10-2

10-1

1

60 70 80 90 100 110 120

Quindi le caratteristiche peculiari degli eventi e+e− → HZ →bbff, che nel loro insieme sono dette segnatura (parola gergalebrutta che deriva dall’inglese signature, firma), sono: un sistemabb con massa invariante compatibile con la massa dell’Higgs e

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un sistema ff con una massa invariante compatibile con la massadella Z. Dato che la Z decade in gran parte in adroni, il canale BR della Z

canale BR [%]νν 20.0`+`− 10.3

e+e− 3.4µ+µ− 3.4τ+τ− 3.4

hadrons 69.9qq 42.8cc 12.0

bb 15.1

principe per la ricerca del bosone di Higgs e il canale cosiddettoin quattro jet e+e− → HZ → bbqq che ha luogo nel 70% deicasi. Semplificando, con jet si intende l’insieme di particellericonducibili ad uno stesso partone, i quark nel nostro caso,confinate in uno stretto cono angolare. Di solito si ha a che farecon jet nel caso di stati finali adronici a elevata molteplicita diparticelle prodotte.

Il fondo al segnale che interessa e dovuto a molti processiprevisti nel modello standard che hanno uno stato finale, cioelo stato “stabile” che siamo in grado di osservare direttamente

In che senso si intende l’aggettivostabile?

nel rivelatore, simile al segnale che stiamo cercando. In molticasi questa somiglianza si ha solo in recessi assai improbabilidello spazio delle fasi a disposizione del processo; ma la scarsaprobabilita puo venir compensata dalle copiose sezioni d’urto diproduzione con il risultato che il fondo non puo essere trascuratonel confronto numerico con gli eventi di segnale attesi.

Il b-tagging e, nel caso della ricerca del bosone di Higgs, latecnica che permette di ridurre al minimo il fondo ovvero didiscriminare efficacemente i possibili eventi di segnale da quellismaccatamente di fondo. Questa tecnica permette di capire seun jet e stato prodotto dal decadimento di un quark b e risultaprovvidenziale nel caso in esame proprio perche il bosone di Higgsdecade quasi sempre in una coppia di quark di tipo b. Viceversaquark di tipo b sono presenti molto piu raramente negli stati finalidei processi di fondo.

I quark b prodotti dal decadimento dell’Higgs danno luogoalla formazione di b-adroni, ovvero adroni che contengono quarkb. La ricostruzione esclusiva sarebbe il modo piu diretto diidentificare adroni con b: si sceglie un canale di decadimento,si cercano nell’evento i prodotti di questo decadimento e ci siassicura (p.e. ricostruendo la massa invariante) che siano venutida un adrone con b. Questo metodo permetterebbe di selezionarecampioni molto puri, ovvero con fondo molto piccolo, ma e

7

anche molto inefficiente e nel nostro caso non e appropriato datoche dobbiamo confrontarci con segnali piccoli. Un metodo piuefficiente e alternativo, detto inclusivo, consiste nello sfruttarecaratteristiche generali a tutti gli adroni con b senza che sianecessario ricostruirli.

La vita media e la peculiarita fisica degli adroni contenenti quark Vita media in psdi alcuni b-adroniB± 1.65B0 1.55B0

s 1.49Λb 1.23

b che si sfrutta maggiormente nel b-tagging. Grazie anche al boostrelativistico, un adrone con b prodotto da Higgs ha una lunghezzadi decadimento di qualche millimetro. Prendiamo ad esempio unHiggs di mH ∼ 100 GeV; assumendo di essere vicino alla sogliacinematica, il bosone di Higgs viene prodotto praticamente ariposo nel sistema di riferimento del laboratorio; nel decaderei due quark b acquistano un’energia pari a mH/2 ∼ 50 GeV,

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xwd

B

1/N

dN

/dx

wd

B

Data

Peterson

Kartvelishvili

ALEPH

energia che in media conservano al 70% adronizzando in adronicon B (a lato la distribuzione di xB = EB/Eb). Questo avvienein conseguenza della fisica che governa il meccanismo dellaframmentazione. Dato che EB ∼ 35 GeV � mB ∼ 5 GeV,la lunghezza di decadimento tipica e λ = βγcτ = cτp/mB ∼cτE/mB ∼ 3 · 108ms−1 · 1.5 ps · 35 GeV/5 GeV ∼ 3 mm.

Riassumendo, nel processo e+e− → HZ → bbqq, dato chesia l’Higgs che il bosone Z hanno una vita media praticamentenulla, molti adroni hanno di fatto origine nel punto di interazioneo vertice primario dove cioe e avvenuta l’annichilazione tral’elettrone e il positrone incidenti. Le tracce delle particellecariche dei prodotti di decadimento degli adroni non b, a piccolavita media, sono registrate dall’apparato tracciante e, estrapolateall’indietro, permettono di ricostruire il vertice primario che variada evento ad evento in quanto, come detto, i pacchetti di elettronie positroni che collidono hanno dimensioni finite.

Anche gli adroni con b hanno origine in questo punto maThe Higgs particle in the standard model: experimental results from LEP 523

B B

VertexPrimary

VertexSecondary

VertexSecondary

Figure C.1. Schematic of b hadron decay vertices.

of the much heavier t quark.) Because of this large mass of the b quarks, the b hadrons arealso relatively heavy. For example, the masses of the b mesons B± and B0 are both about5.279 GeV/c2 [12].

The presence of the other hadrons in the production process makes it difficult to determinethe mass of the b hadron directly without exclusive construction. However, the large mass ofthe b hadrons plays a fundamental role in the other features that can be utilized directly.

Lifetime. In spite of their large masses, the b hadrons have unexpectedly long lifetimes.Some examples of lifetimes are [12]

B± : 1.653 ± 0.028

B0 : 1.548 ± 0.032

B0s : 1.493 ± 0.062

!b : 1.229 ± 0.080

all in units of 10!12 second. It implies, for example, that a 50 GeV B± travels an average of4.7 mm before decaying. Each of the four experiments at LEP has a vertex detector capableof ascertaining flight distances of this magnitude.

A number of hadrons originate at the interaction point, where the incoming electron andpositron annihilate. Most of the decay products of these non-b hadrons can be traced backto this common point, known as the primary vertex. The b hadron also originates at thisprimary vertex, but it travels a few millimetres to a new point before decaying. The decayproducts which originate from this new point can be reconstructed into another vertex, knownas the secondary vertex. Figure C.1 shows a schematic view of this process. It is the abilityto distinguish secondary vertices from the primary vertex that makes lifetime important fortagging b jets.

The distance of closest approach between a line and a point is called the impact parameter.Tracks which do not pass through the primary vertex will have large impact parameters, whilethose which do pass through the primary vertex will have impact parameters close to zero. Theimpact parameter is a distance, but the fact that charged particles belong to jets which havea direction allows the impact parameter to be given a sign. Tracks whose points of closestapproach to the primary vertex are ‘downstream’ from the primary vertex are given a positivesign, while those whose points of closest approach are ‘upstream’ are given a negative sign, asshown in figure C.2. Jets with large, positive impact parameter tracks are identified as b jets.It is also useful to determine approximately the positions of the secondary vertices.

Decay products. Approximately 20% of the decays of the b hadrons produce an electron

percorrono qualche millimetro prima di decadere. Le tracce delleparticelle cariche di ciascun adrone con b, estrapolate all’indietro,si incontreranno (entro le risoluzioni sperimentali) in un altrovertice detto vertice secondario.

Nell’ottica di un approccio inclusivo pero, piu che la ri-

~1/!

"

d0

costruzione di un vertice secondario, si puo valutare quantociascuna traccia ricostruita sia compatibile o meno con un parti-cella che abbia avuto origine nel vertice primario. A tale scoposi misura il parametro di impatto, ovvero la minima distanza trala traccia estrapolata e il vertice primario. Le tracce che nonhanno origine da quest’ultimo avranno un parametro di impattosignificativo mentre quelle originatesi nel vertice primario avrannoun parametro di impatto compatibile con zero. Il parametrodi impatto, indicato normalmente con d0, puo essere facilmentestimato: dato che nel decadimento di una particella relativistical’apertura angolare dei prodotti di decadimento e dell’ordinedi 1/γ, si ha semplicemente d0 = λ/γ = βcτ . Il parametro diimpatto e praticamente un’invariante relativistico. Nal caso degli

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adroni con b, d0 ∼ 0.5 mm.Pur essendo tecnicamente una distanza, al parametro di

524 P A McNamara III and S L Wu

Jet Direction

I.P. TrackNegative

I.P. TracksPositive

Figure C.2. Impact parameter definition, showing two positive impact parameter tracks and onenegative impact parameter track. The tracks are extrapolations of charged particle trajectories backto the region of the primary vertex. The tracks shown are all travelling to the right.

or a muon. Because of the large mass of the b quark, in the rest frame of the b hadron, themomentum of the electron and muon is large in these decays. When the b hadron has a largemomentum itself, as for those produced in the decay H ! bb, the electron or muon oftenhas a large momentum transverse to the direction of motion of the b hadron. These leptonswith large transverse momenta are characteristic of the decays of b hadrons, and can be usedin the identification of b jets. Because such high transverse momentum electrons and muonscan be found in less than 20% of all b hadron decays, this information cannot be requiredwhile maintaining a high efficiency. However, when it exists it can significantly enhance theb-tagging performance.

These are the most salient features of the b jets. There are many other features, such asthe jet shape. By themselves, they are less powerful for the purpose of b-tagging; however,since they are independent of the variables described above, they play an important role inenhancing the discriminating power of the b-tag.

A major task is to combine the discriminating powers of all these and other variables. Thereare several ways to carry out this combination, including for example the method of maximumlikelihood. Another way is to use a neural network, as already described in appendix B.

Appendix D. Statistics and likelihood ratio

Given an observation, one would like to know if a given signal has or has not been observed.One would like to be able to draw conclusions about various hypotheses in a systematic mannerbased on the collected data.

The LEP Higgs results are interpreted in a frequentist framework. In the frequentistframework, the results are quantified in terms of the probability of obtaining a given data setunder specific signal hypotheses. The frequentist makes this type of statement in order to avoidthe use of prior probabilities. In order to make a statement about the posterior probabilities ofa set of signal hypotheses after an observation, one must know, or make assumptions about, therelative probabilities of the signal hypotheses before an observation is made. The frequentistargues that this is a subjective choice and that results expressing the probability of a givensignal depend strongly on the manner in which the choice is made.

In order to quantify the consistency of an observation with a given hypothesis, theobservation is reduced to a test-statistic Q which has a range of possible values of varyingconsistency with the hypothesis. This test-statistic can be any of a variety of functions which

impatto di una traccia viene assegnato un segno confrontando laconfigurazione della traccia stessa rispetto al vertice primario inrelazione alla direzione del jet (che da una stima della direzione divolo del quark che l’ha originato) cui quella traccia appartiene. Letracce il cui punto di massimo avvicinamento al vertice primarioe a valle rispetto alla direzione del jet hanno un parametro diimpatto positivo. Altrimenti il parametro di impatto e negativo.L’esistenza di tracce con parametro di impatto negativo e un puroeffetto strumentale che deriva dalla risoluzione finita dei nostririvelatori. Il criterio di b-tagging richiede tracce con parametri diimpatto elevati e positivi, ovvero associabili a jet originatesi daquark b.

Questa tecnica di b-tagging e applicabile se il nostro apparatoe in grado di distinguere vertici secondari rispetto al verticeprimario o di misurare il parametro di impatto. Le lunghezze didecadimento e il parametro di impatto stesso sono dell’ordine deimillimetri o frazioni di millimetro; e quindi evidente che questatecnica sperimentale non sarebbe fruibile utilizzando la sola TPC,la cui risoluzione spaziale e dello stesso ordine di grandezza. Tuttigli esperimenti di LEP erano quindi dotati di un rivelatore aquesto dedicato detto rivelatore di vertice.

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Il rivelatore di vertice di ALEPH, VDET, consisteva di !

z

due strati approssimativamente cilindrici di sensori a Silicio amicrostrisce doppia faccia, coassiali con l’asse dei fasci, posti a∼ 6 cm e ∼ 11 cm di raggio, per una lunghezza totale di ∼ 40 cmdistribuita simmetricamente rispetto al punto di interazione. Laposizione in cui la particella carica attraversa i sensori di silicioera ricostruita con una precisione di ∼ 10 − 20 µm. Questaelevatissima risoluzione spaziale insieme alla posizione favorevoledel rivelatore di vertice rendeva possibile il b-tagging con ilparametro di impatto.

La risoluzione sul parametro di impatto puo essere facilmente

1 2

d

PV

stimata approssimando con un segmento rettilineo la traccia inprossimita del punto di interazione. Se σ1 e σ2 sono le risoluzionispaziale sui due strati, da semplici considerazioni geometriche siricava che la risoluzione sul parametro di impatto e:

σgeomd0

∼ σ1r2

r2 − r1⊕ σ2r1

r2 − r1. (5)

Oltre ai termini puramente geometrici occorre tener conto delloscattering multiplo nel materiale del primo strato sensibile,

σmsd0∼ r1θ0, (6)

che da un contributo inversamente proporzionale all’impulso dellaparticella. Per limitare lo scattering multiplo era cura realizzare irivelatori di vertice in modo da minimizzare il materiale passivo.Il rivelatore di vertice di Aleph aveva uno spessore totale dicirca 1.5%X0. La risoluzione sul parametro di impatto era Questa risoluzione e compatibile con i

numeri relativi alla geometria e allecaratteristiche del VDET?

σd0 ∼ (30 + 70/pT ) µm con pT in GeV. L’effetto della vita mediasulla distribuzione del parametro di impatto e ben visibile, inquesto caso su leptoni da B.

ALEPH Coll., Updated measurement ofthe average B hadron lifetime,Phys.Lett.B295 (1992) 174-186.

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La variabile di b-tagging propriamente detta si costruiscepero tendendo conto che la risoluzione sul parametro di impatto diuna traccia e variabile; si definisce la significativita Sd0 = d0/σd0

su cui l’effetto della vita media lunga e altrettanto visibile.

ALEPH Coll. A Measurement of Rb

using a Lifetime-Mass Tag, Phys. Lett.B 401 (1997) 150-162.

Le tracce con S < 0 forniscono una misura diretta della funzione Se gli errori fossero strettamentegaussiani R(|S|) avrebbe la forma

|ST |e−|ST |2/2σ2(perche?); in realta, a

causa di consistenti code non gaussianenegli errori dovute agli algoritmi ditracciatura, un buon fit si ottieneaggiungendo alle componente gaussianadue esponenziali decrescenti del tipo|ST |e−|ST |/λ. Inoltre viene definitauna R(|S|) specifica a seconda delnumero di hit nel VDET.

di risoluzione del parametro di impatto R(|S|). Una volta notaquesta funzione, ottenibile facendo un fit opportunamentenormalizzato alla parte negativa della distribuzione di S, perciascuna traccia con ST si puo calcolare un livello di confidenzaPT (ST ) che quella traccia provenga dal vertice primario:

PT (ST ) ≡∫ −∞

−|ST |R(x)dx. (7)

Per le tracce con ST < 0, PT (ST ) si definisce −PT (−ST ). PT (ST )si comporta come un livello di confidenza: una distribuzionepiatta tra −1 e +1 dovuta alle tracce che provengono dal verticeprimario con un picco a valori positivi piccoli dovuto alle tracceche hanno origine da particelle a vita media lunga.

PT

No

. tr

ack

s/1

00

00

0

2

4

6

8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

11

Nel nostro caso siamo interessati a capire se un intero jet sia omeno riconducibile ad un quark di tipo b; il livello di confidenza diciascuna traccia nel jet si combina allora in un livello di confidenzadel jet nel suo insieme, ovvero il livello di confidenza Pjet chetutte le N tracce del jet provengano dal vertice primario:

Pjet = ΠN−1∑k=0

(− lnΠ)k

k!, Π ≡

N∏n=1

PTn . (8)

Matematicamente Pjet e la probabilita che N tracce provenientidal vertice primario si combinino per produrre il valore osservatodi Π o qualsiasi altro valore meno probabile. Pjet e visibilein figura (NB PH va inteso come Pjet). Una discussione sulladerivazione di Pjet si trova in appendice a pag.20.

ALEPH Coll. A Measurement of Rb

using mutually exclusive tags, Phys.Lett. B 401 (1997) 163-175.

p p!

Ad esempio, il taglio − log10 Pjet > 3 permette di selezionarejet da b con efficienza dell’ordine del 40% e una purezza superioreall’80%.

ALEPH Coll. A precise measurement ofΓZ→bb/ΓZ→hadron, Phys. Lett. B 313(1993) 535-548.

ALEPH

Oltre alla vita media, gli adroni con b hanno altre carateristiche,meno peculiari, ma comunque utili per migliorare la nostra

12

capacita di distinguere jet da b da jet da quark leggeri. Inparticolare:

1. la massa invariante del jet, dato che il quark b ha∼ 4 GeV di massa, sensibilmente maggiore dei quark u,d, s e c; questo si riflette nella massa invariante del jetoriginato dal quark b;

2. la presenza (in circa il 20% dei casi) di un leptone conelevato pT rispetto alla direzione del jet. Come mai? Elevato quanto?

Tutte queste informazioni vengono combinate in modo ottimalein un’unica variabile η da una rete neurale (neural net, NN); lavariabile η ha un picco a η = 1 per i b-jet (vedi in figura a sinistral’istogramma ombreggiato) e permette di migliorare, anche senon di molto, le prestazioni nella selezione dei b-jet rispettoall’utilizzo della sola informazione dal parametro di impatto cherimane comunque determinante (figura di destra).

ALEPH Coll. Search for the neutralHiggs bosons of the MSSM in e+e−

collisions at√

s from 130 to 172GeV,Phys. Lett. B 412 (1997) 173-188.

0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

!

Events (a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(1-!udscg)jet

! b je

t

(b)

Pjet

Neural Network

La selezione per la ricerca di e+e− → HZ → bbqqad ALEPH e descritta con qualche dettaglio nell’appendice apag. 21. La selezione consiste prima di una serie di tagli per L’esercitazione proposta sulla ricerca di

Higgs a LEP consiste nel disegnare unaselezione a valle di questa“preselezione” contro gli eventi a “2fermioni”.

eliminare eventi a “2 fermioni” con emissione di un γ di statoiniziale. Per raggiungere il piu arduo obbiettivo di eliminare ilrestante fondo a “4 fermioni” o qqgg, molto piu simile al segnale,viene di nuovo utilizzata una rete neurale basata, tra le altrevariabili, sul b-tag di ciascun jet. Con i tagli scelti l’efficienza diselezione sul segnale e tipicamente del 50% e il fondo rimanentedominante e dato dal processo piu simile al segnale che stiamocercando ovvero produzione di una coppia ZZ data la sezioned’urto di produzione elevata e il BR Z → bb non trascurabile. Irisultati della selezione sono discussi in appendice C.

L’interpretazione statistica dei risultati si basa sullaAttenzione a non confondere QmH conla statistica di test tmH utilizzatanell’esercitazione; entrambe sonolikelihood ratio ma mentre tmH edefinito sul singolo evento (e segnale oe fondo?) QmH e definito sull’interoinsieme di eventi osservati (l’insiemedegli eventi osservati e compatibile conla presenza del solo fondo o, inalternativa, con la presenza del segnaleinsieme al fondo?)

costruzione di un likelihood ratio cosı come ci insegna il lemma diNeyman-Pearson, ovvero il rapporto tra due densita di probabilita

13

calcolate nel punto dello spazio dei parametri osservato:

QmH ≡e−(s(mH)+b) (s(mH)+b)N

N !

N∏i=1

[g(di|Hsig(mH)) + g(di|Hbkg)]

e−b bN

N !

N∏i=1

g(di|Hbkg)

(9)dove:

- N e il numero di eventi osservato;- s(mH) e il numero di eventi di segnale attesi ad una data

massa mH;- b e il numero di eventi di fondo attesi;- l’evento i-esimo e identificato da un vettore di di variabili

discriminanti; nel caso particolare della ricerca dell’Higgs di =(mrec

H , NNoutput)i dove mrecH e la massa ricostruita dell’Higgs Vedi gli appunti sull’esercitazione per

maggiori dettagli su mrecH ; sempre

nell’esercitazione NNoutput e sostituitodal b-tagging che, pero, sappiamoessere l’ingrediente fondamentale dellaNN stessa.

e NNoutput e lo scalare che risulta dall’applicazione della reteneurale, normalmente compreso tra 0 e 1 e vicino a 1 per unevento molto simile al segnale;

- g(d|Hsig(mH)) e g(d|Hbkg) sono rispettivamente le dis-tribuzioni di probabilita in funzione di d nell’ipotesi di segnaleHsig(mH) e nell’ipotesi di fondo Hbkg, cioe la frazione di eventiattesi rispettivamente di segnale e di fondo in un dato bin localedi d.

La scrittura di QmH puo essere semplificata:

QmH = e−s(mH)

[1 +

s(mH)b

]N N∏i=1

[1 +

g(di|Hsig(mH))g(di|Hbkg)

]. (10)

Normalmente si usa −2 ln QmH perche la produttoria si trasforma Il fattore −2 viene aggiunto perconvenzione data l’analogia tra−2 ln QmH e la distribuzione di χ2.

in una piu comoda sommatoria:

−2 ln QmH = 2{

s(mH)−N ln[1 +

s(mH)b

]+

−N∑

i=1

ln[1 +

g(di|Hsig(mH))g(di|Hbkg)

]}. (11)

I termini della sommatoria sono veri e propri pesi che vengono L’attento lettore osservera che ilrapporto g(di|Hsig(mH))/g(di|Hbkg)e esattamente la statistica di test tmHutilizzata come variabile discriminantenell’esercitazione.

assegnati a ciascun evento.L’osservazione da fare e che, a valle della preselezione, non

si applica piu alcun taglio tale che gli eventi che passano iltaglio sono considerati segnale e quelli che non lo passano sonoconsiderati fondo ovvero, in altre parole, i primi hanno peso 1,i secondi peso 0. Il metodo utilizzato, invece, assegna a ciascunevento un peso tanto maggiore quanto e piu simile al segnale.In questo senso il metodo utilizzato e piu sofisticato del metodotradizionale.

La figura sotto riporta la distribuzione di −2 ln Q115GeV relativaa tutti e quattro gli esperimenti LEP ottenuta con metodi MonteCarlo e la linea verticale indica il valore osservato (mH e sceltovicino al limite in massa, pari appunto a 115 GeV ).

14

LEP Collaborations, Search for thestandard model Higgs boson at LEP,Phys. Lett. B 565 (2003) 61.

0

0.02

0.04

0.06

0.08

0.1

0.12

-15 -10 -5 0 5 10 15

-2 ln(Q)

Pro

bab

ilit

y d

ensi

ty

Observed

Expected for background

Expected for signalplus background

LEPm

H = 115 GeV/c

2

(a)

0

0.01

0.02

0.03

0.04

0.05

-60 -40 -20 0 20 40 60

-2 ln(Q)

Pro

bab

ilit

y d

ensi

ty LEPm

H = 110 GeV/c

2

(b)

0

0.05

0.1

0.15

0.2

0.25

-6 -4 -2 0 2 4 6

-2 ln(Q)

Pro

bab

ilit

y d

ensi

ty LEPm

H = 120 GeV/c

2

(c)

Figure 6: Probability density functions corresponding to fixed test masses mH, for the backgroundand signal plus background hypotheses. The observed values of the test statistic!2 ln Q are indicatedby the vertical lines. The light shaded areas, 1! CLb, measure the confidence for the backgroundhypothesis and the dark shaded areas, CLs+b, the confidence for the signal plus background hypothesis.Plot (a): test mass mH = 115 GeV/c2; (b): mH = 110 GeV/c2; (c): mH = 120 GeV/c2.

18

Ovviamente ogni ipotesi di massa ha un valore osservato di−2 ln QmH ; questo e rappresentato nelle figure sottostati (pertutto LEP e solo per ALEPH) in funzione dell’ipotesi di massamH

-10

-5

0

5

10

15

20

25

100 102 104 106 108 110 112 114 116 118 120

mH

(GeV/c2)

-2 l

n(Q

)

ALEPH(a)

-10

-5

0

5

10

15

20

25

100 102 104 106 108 110 112 114 116 118 120

mH

(GeV/c2)

-2 l

n(Q

)

DELPHI(b)

-10

-5

0

5

10

15

20

25

100 102 104 106 108 110 112 114 116 118 120

mH

(GeV/c2)

-2 l

n(Q

)

L3(c)

-10

-5

0

5

10

15

20

25

100 102 104 106 108 110 112 114 116 118 120

mH

(GeV/c2)

-2 l

n(Q

)

OPAL(d)

-30

-20

-10

0

10

20

30

40

50

100 102 104 106 108 110 112 114 116 118 120

mH

(GeV/c2)

-2 l

n(Q

)

LEP four-jet(e)

-30

-20

-10

0

10

20

30

40

50

100 102 104 106 108 110 112 114 116 118 120

mH

(GeV/c2)

-2 l

n(Q

)

LEP all except four-jet(f)

Figure 2: Observed and expected behaviour of the test statistic!2 ln Q as a function of the test massmH when the combination procedure is applied to subsets of the LEP data. Plots (a) to (d): data setsfrom individual experiments; (e): the four-jet final state and (f): all except the four-jet final state,with the data of the four experiments combined. The same notation as in Figure 1 is used.

14

-30

-20

-10

0

10

20

30

40

50

106 108 110 112 114 116 118 120

mH

(GeV/c2)

-2 l

n(Q

)

Observed

Expected for background

Expected for signal plus background

LEP

Figure 1: Observed and expected behaviour of the test statistic !2 ln Q as a function of the testmass mH, obtained by combining the data of the four LEP experiments. The full curve represents theobservation; the dashed curve shows the median background expectation; the dark and light shadedbands represent the 68% and 95% probability bands about the median background expectation. Thedash-dotted curve indicates the expectation for !2 ln Q, given the signal plus background hypothesis,when the signal mass on the abscissa is tested. For the expected behaviours, the medians of thesimulated distributions are shown.

13

dove sono anche riportati i valori mediani attesi per l’ipotesi disolo fondo (linea blu con le bande che rappresentano gli intervallia 1σ - verde - e a 2σ - giallo) e per l’ipotesi segnale piu fondo(linea marrone). Il valore osservato dal solo esperimento ALEPHmostra un eccesso intorno a mH ∼ 115 GeV.

La presenza o meno di un eventuale segnale ad una datamassa mH si ricava dal valore di 1 − CLb(mH), l’area gialla nel

15

plot delle distribuzioni di −2 ln Q115GeV, ovvero l’integrale delladistribuzione di probabilita di −2 ln QmH nell’ipotesi di solo fondoda −∞ al valore osservato. Se l’ipotesi solo fondo e corretta1 − CLb(mH) e uniformemente distribuito tra 0 e 1 con valore Per convenzione

1− CLb(mH) = 2.7 · 10−3 e1− CLb(mH) = 5.7 · 10−7 indicanorispettivamente un eccesso a 3σ e 5σrispetto al numero di eventi di fondoattesi.

mediano pari a 0.5. In presenza di segnale ad una data massa,1− CLb(mH) e molto piccolo.

La figura sotto mostra 1 − CLb(mH) per tutto LEP, conle convenzioni analoghe a quelle della figura precedente, daconfrontarsi con il valore ottenuto dal solo esperimento ALEPHche mostra un eccesso a 3σ.

10-3

10-2

10-1

1

80 85 90 95 100 105 110 115 120

mH

(GeV/c2)

1-C

Lb

3!

2!

ALEPH (a)

10-3

10-2

10-1

1

80 85 90 95 100 105 110 115 120

mH

(GeV/c2)

1-C

Lb

3!

2!

DELPHI (b)

10-3

10-2

10-1

1

80 85 90 95 100 105 110 115 120

mH

(GeV/c2)

1-C

Lb

3!

2!

L3 (c)

10-3

10-2

10-1

1

80 85 90 95 100 105 110 115 120

mH

(GeV/c2)

1-C

Lb

3!

2!

OPAL (d)

10-3

10-2

10-1

1

80 85 90 95 100 105 110 115 120

mH

(GeV/c2)

1-C

Lb

3!

2!

LEP four-jet (e)

10-3

10-2

10-1

1

80 85 90 95 100 105 110 115 120

mH

(GeV/c2)

1-C

Lb

3!

2!

LEP all except four-jet (f)

Figure 8: The background confidence 1! CLb as a function of the test mass mH for subsets of theLEP data. The same notation as in Figure 7 is used. Plots (a) to (d): individual experiments; (e):the four-jet and (f): all but the four-jet final state, with the data of the four experiments combined.

20

10-3

10-2

10-1

1

80 85 90 95 100 105 110 115 120

mH

(GeV/c2)

1-C

Lb

3!

2!

LEP

Observed

Expected for signal plus background

Expected for background

Figure 7: The background confidence 1 ! CLb as a function of the test mass mH. Full curve:observation; dashed curve: median expected background confidence. Dash-dotted line: the medianexpectation for 1! CLb, given the signal plus background hypothesis, when the signal mass on theabscissa is tested; the dark and light shaded bands represent the 68% and 95% probability bandsabout the median expectation. The horizontal solid lines indicate the levels for 2! and 3! deviationsfrom the background hypothesis (see the Appendix for the conversion).

19

Nell’approccio frequentista, il limite in massa si ricava dal con-fidence level dell’ipotesi segnale + fondo, CLs+b(mH), l’area verdenel plot delle distribuzioni di −2 ln Q115GeV, ovvero l’integraledella distribuzione di probabilita di −2 ln QmH nell’ipotesi disegnale piu fondo dal valore osservavto a +∞. L’ipotesi segnalead una data massa mH piu fondo si puo considerare esclusase CLs+b(mH) e inferiore a 5%. Tuttavia questa procedurapuo portare a situazioni patologiche: in presenza di una sot-tofluttuazione del fondo sufficientemente grande potrebberoessere escluse erroneamente masse per le quali non dovrebbeesserci sensibilita a causa del piccolo numero di eventi disegnale attesi. Il problema si aggira utilizzando il rapportoCLs(mH) ≡ CLs+b(mH)/CLb(mH) per il quale vale sempreCLs(mH) > CLs+b(mH) dal momento che CLb(mH) e sem- Con l’aggettivo conservativo si

qualificano di solito quelle scelte dimetodo che sono “al ribasso” in terminidi prestazioni ma che assicuranomaggiori garanzie in termini diaffidabilita del risultato.

pre minore di uno. Si considerano escluse le masse per cuiCLs(mH) < 0.05 e stavolta il limite ottenuto e conservativo.

16

In virtu dell’eccesso osservato il limite in massa risulta legger-mente piu piccolo del limite aspettato

La riga orizzontale rappresenta il valoreCLs = 0.05 al di sotto del quale lacorrispondente mH e esclusa.

10-6

10-5

10-4

10-3

10-2

10-1

1

100 102 104 106 108 110 112 114 116 118 120

mH

(GeV/c2)

CL

s

114.4115.3

LEP

Observed

Expected forbackground

Figure 9: The ratio CLs = CLs+b/CLb for the signal plus background hypothesis, as a function ofthe test mass mh. Solid line: observation; dashed line: median background expectation. The darkand light shaded bands around the median expectation for the background hypothesis correspond tothe 68% and 95% probability bands. The intersection of the horizontal line for CLs = 0.05 with theobserved curve is used to define the 95% confidence level lower bound on the mass of the StandardModel Higgs boson.

21

E interessante il plot del peso ln(1 + s/b) per gli eventi chemaggiormente contribuiscono a −2 ln QmH (in figura quellorelativo a ALEPH); per ogni valore di mH ciascuno di questieventi ha un peso tanto maggiore quanto piu assomiglia (intermini statistici) ad un Higgs di quella massa.

0

0.5

1

1.5

2

2.5

100 105 110 115 120

mH

(GeV/c2)

ln(1

+s/

b)

1

2

3

5

6

8 15

11

0

0.5

1

1.5

2

2.5

100 105 110 115 120

ALEPH

0

0.5

1

1.5

2

2.5

100 105 110 115 120

mH

(GeV/c2)

ln(1

+s/

b)

16

10

12

0

0.5

1

1.5

2

2.5

100 105 110 115 120

DELPHI

0

0.5

1

1.5

2

2.5

100 105 110 115 120

mH

(GeV/c2)

ln(1

+s/

b)

4

9

14 17

0

0.5

1

1.5

2

2.5

100 105 110 115 120

L3

0

0.5

1

1.5

2

2.5

100 105 110 115 120

mH

(GeV/c2)

ln(1

+s/

b)

7

13

0

0.5

1

1.5

2

2.5

100 105 110 115 120

OPAL

Figure 3: Evolution of the event weight ln(1 + s/b) with test mass mH for the events which havethe largest contributions to !2 ln Q at mH = 115 GeV/c2. The labels correspond to the candidatenumbers in the first column of Table 2. The sudden increase in the weight of the OPAL missingenergy candidate labelled “13” at mH = 107 GeV/c2 is due to switching from the low-mass to high-mass optimization of the search at that mass. A similar increase is observed in the case of the L3four-jet candidate labelled “17” which is due to the test mass dependent attribution of the jets to theZ and Higgs bosons.

15

Quanto l’approccio statistico sia piu potente dell’approcciotradizionale (un evento pesa 1 o 0 a seconda che passi o meno unacerta selezione) lo si intuisce se si prova a fare un plot della massaricostruita mrec

H per gli eventi che superano una certa selezione

17

“tight” cioe abbastanza stretta per ottenere un campione a altapurezza:

La selezione puo essere, per esempio,un taglio sul NNoutput.

0

5

10

15

20

25

0 20 40 60 80 100 120

mH

rec (GeV/c2)

Ev

ents

/ 3

GeV

/c2

LEP Loose!s– = 200-209 GeV

Data

Background

Signal (115 GeV/c2)

Data 119

Backgd 116.5

Signal 10

all ! 109 GeV/c2

17

15.8

7.1

0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

mH

rec (GeV/c2)

Ev

ents

/ 3

GeV

/c2

LEP Tight!s– = 200-209 GeV

Data

Background

Signal (115 GeV/c2)

Data 18

Backgd 14

Signal 2.9

all ! 109 GeV/c2

4

1.2

2.2

Figure 5: Distributions of the reconstructed Higgs boson mass, mrecH , obtained from two selections with

di!erent expected signal purities. The histograms show the Monte Carlo predictions, lightly shadedfor the background, heavily shaded for an assumed Standard Model Higgs boson of mass 115 GeV/c2.The points with error bars show the data. In the loose and tight selections the cuts are adjusted insuch a way as to obtain, for a Higgs boson of mass 115 GeV/c2, approximately 0.5 or 2 times moreexpected signal than background events when integrated over the region mrec

H > 109 GeV/c2. In thesearches where the event selection depends on the test mass (see the Appendix), its value is set at115 GeV/c2. 17

E evidente che e molto difficile apprezzare la presenza o menodi un picco di segnale (vedi l’istogramma rosso che corrispondealla simulazione di un Higgs di 115GeV). Infatti assegnandoa ciascun evento semplicemente un peso 0 o 1, non si fa altroche scartare informazione preziosa che viene invece utilizzata Nell’esercitazione ci mettiamo in una

situazione piu semplice e quindiutilizzeremo il metodo tradizionale purtagliando su tmH che e comunque unlikelihood ratio definito sul singoloevento.

al meglio utilizzando il likelihood ratio. Questo e stato moltoimportante per la ricerca di Higgs a LEP e lo e, in generale, inanalisi difficili e vicino al limite statistico.

18

Appendice A: la misura dell’impulso trasverso

T.S. VirdeeExperimentaltechniques, (1998),prepared for 1998European School ofHigh-Energy Physics.

6

! "

! ! !

int

/ /

.

.

= # $ %

=

R A where R A fm

A mb

2 2 3 1 3

0 7

1 2

35infact = whereinel 0 0

In dealing with hadrons it is convenient to measure the depth and radial extent of the resulting

cascades in terms of interaction length (&int) which is defined as

&!

int

int

/ = #

A

NA

A

1 3

The values of the above mentioned parameters for various materials are listed in Table. 1.

3. EXPERIMENTAL MEASUREMENTS: MEASUREMENT OF MOMENTUM

Consider the motion of a charged particle in a uniform solenoidal magnetic field (Fig.4). The

radius of curvature, r, is given by:

rp

B

T

.=

0 3

where r is measured in m, B is the magnetic field strength measured in T and pT is the momentum

perpendicular to B and measured in GeV/c.

COIL

COIL

AB

L

C

Sagitta, s

'

r

pt

p

B

JV_294

Figure 4: The trajectory of a charged particles in a magnetic field.

The angle ' is given by

sin '

2 2=

L

r

If r >> L then'

'2 2

0 3

.$ ( $

L

r

BL

pT

Therefore the sagitta, s, is given by

7

s r r

r

r BL

pT

cos( / )

.

= !

" ! !#

$%

&

'(

)

*+

,

-.

= "

/

/

/

2

1 11

2 4

8

0 3

8

2

2 2

As an example s " 3.75 cm for pT=1 GeV/c, L=1m and B=1T. Suppose the sagitta is measured

using points A, B and C. Then

s xx x

ds dxdx dx

dx x

ds xx x

x

BA C

BA C

i

s

( )

( )( ) ( )

( )

= !+

0 = ! !

"

( ) = + + = 1

2

2 2

4 4

3

2

2 22 2

2 2

assuming is the independent single point error2

22 2

2 2

where 2s is the error on the sagitta. The relative momentum resolution can now be estimated as:

dp

p s s

dp

p

p

BL

T

T

s x

T

T

xT

( / )

.

( )

= =

=

2 2

2

3 2

3

2

8

0 32

2

Hence the momentum resolution will degrade linearly with increasing pT but will improve for

higher field and larger radial size of the tracking cavity. The latter improvement is quadratic in L!

The next question that can be asked is the arrangement of N measuring points. Uniform spacing

is best for minimizing the effect of multiple scattering (see 3.1.2) and the resolution is given by

dp

p

p

BL N

T

T

x T .

=+

2

0 3

720

42

For example, dpT/pT " 0.5% for pT = 1 GeV/c, L = 1m, B = 1T, 2x = 200 µm and N=10. For the

best momentum resolution N/2 points should be grouped at the centre and N/4 points at the two

ends of the track. Then

dp

p

p

BL N

T

T

x T .

=2

0 3

2562

leading to an improvement in the momentum resolution by a factor of 0.6.

However in a real tracker the errors due to multiple scattering need to be included.

3.1 Multiple Scattering

The electric field close to an atomic nucleus may give a large acceleration to a charged particle.

This will result in a change of direction for a heavy charged particle (m>me). For small particle-19

Appendice B: la combinazione della probabilita di tracciain un campione di molte tracce

D. Brown, M. FrankTagging b hadronsusing track impactparameters, ALEPH92-135 (1992).

20

Appendice C: l’analisi per la ricerca di e+e− → HZ → 4jetad ALEPH

In questa appendice e descritta con qualche dettaglio l’analisiper la ricerca di e+e− → HZ → 4jet ad ALEPH basata su reteneurale.

P. A. McNamara III andSau Lan Wu The Higgsparticle in the standardmodel: experimentalresults from LEP, Rep.Prog. Phys. 65 (2002)465-528.

The Higgs particle in the standard model: experimental results from LEP 481

or

H ! ! +!" and Z ! qq.

Adding up these two contributions, the total branching ratio for this process is 8%.The sum of the branching ratios for (a)–(d) is 82%, only slightly less than the sum of the

branching ratios for H into bb and ! +!". From this detailed list of decay products, it is seenthat, except for the second process under (d), there are always two hadron jets coming fromH ! bb. It is therefore most important to distinguish b jets from the jets due to other quarks orhard gluons. Because of the long lifetime of hadrons containing a b quark, the decay verticesof these b hadrons are separated from the point of annihilation of the incoming electron andpositron. This separation can be detected by using a vertex detector which is located very closeto the beam pipe (see section 3.2). These jets due to the b quark can be further identified bycombining information from several variables which can distinguish between jets from lightquarks and those from b quarks. At LEP, this combination is most commonly done with neuralnetworks. The identification of jets due to b quarks is discussed in detail in appendix C.

Each of the four collaborations at LEP—ALEPH, DELPHI, L3 and OPAL—studies eachof these four final states (a), (b), (c) and (d). The methods of analysis used by these fourcollaborations are quite similar, but not identical. Rather than describing each experiment’sanalysis for each final state, the analyses for one or two experiments will be highlighted whendescribing each of the four above final states. The chosen analyses will be used to illustratethe main features in the study of that particular final state.

4.2. Four jets

We begin the descriptions of the analyses with the decay to four jets, the most powerful finalstate. The analyses of the ALEPH and OPAL experiments will be used to illustrate the keyfeatures of the analysis. The choice of ALEPH is natural because, among the four experiments,only ALEPH has several golden events, i.e. clean events that fit the process

e+e" ! ZH ! four jets. (19)

The OPAL analysis, while based largely on the same information about the four-jet final state,uses different analysis techniques.

ALEPH. In ALEPH, the identification of four-jet events coming from the process (19) isbased on neural network (NN) techniques. More precisely, two neural networks are used, oneto identify b jets as discussed above and also in appendix C, and one to distinguish signal dueto the process (19) from background events due to other processes.

A neural network’s primary purpose, as discussed in appendix B, is to distinguish betweenclasses of events, such as signal and background. While a neural network can be trained todistinguish signal from all other types of events, most events are clearly not consistent withprocess (19) and can be cleanly removed without sacrificing any signal. In other words, asubsample of events with a four-jet topology is first selected, and the neural network is used toidentify those events in the subsample which are most consistent with the signal interpretation.The events selected by the neural network are then combined to compute a likelihood for thesignal hypothesis, and a likelihood for the no-signal hypothesis. The ratio of these likelihoodsis used to determine if the experimental data are consistent with the signal hypothesis or theno-signal hypothesis.

The selection of this subsample of events is to be described first.

21

Evento adronico(elevata molteplicita ditracce cariche, energiavisibile > 0.1

√s).

4 jet

No fotoni energetici(contro e+e− → Zγcon γ visibile).

No sbilanciamento inpz (contro e+e− → Zγcon γ invisibile).

Rete neurale (NeuralNet, NN) contro i fondirimanenti (WW,Z → bbgg, ZZ).

482 P A McNamara III and S L Wu

Events which are consistent with the topology (19) must have four hadronic jets, andtherefore a large number of charged and neutral particles. To be included in the subsample,events are required to have at least eight well-measured charged tracks, and the sum of thecharged track energies must be at least 10% of the initial e+e! centre-of-mass energy

"s [29].

This requirement removes events of no interest to this analysis, in which little or no energygoes into the production of quark jets.

To select events which are consistent with a four-jet topology, the charged tracks andneutral calorimeter objects are grouped into four jet clusters. This clustering is accomplishedwith the DURHAM jet finding algorithm [30] which first defines all particles as clusters, theniteratively combines pairs of clusters based on their ‘closeness’ to one another, until only fourcomposite objects remain. This closeness is determined by a test variable

yij = 2(1 ! cos !ij ) min(Ei, Ej )

Evis

where Ei and Ej are the energies of the clusters i and j , !ij is the angle between them, andEvis is the visible energy measured in the detector. When the event has been clustered intofour jets, the two ‘closest’ jets are required to be sufficiently ‘far’ away from one another to beconsidered two separate jets. This is accomplished by requiring that y34, the y value associatedwith clustering the event into three jets, be above 0.004 [29].

At this stage, a large background still remains. Of this remaining background, a significantfraction comes from the process

e+e! # Z"

followed by

Z # qq

which may include the radiation of one or more hard gluons whose decay products can beclustered into their own jet(s). This process has a sizable cross section, even though it involvesthe initial-state radiation of a photon. Such a photon is often produced close to the beamdirection, and may or may not be observed directly in the detector. If the photon is observedin the detector, then this background event can be removed by requiring simply that, in anyjet, the fraction of energy which comes from electromagnetic sources (identified electrons andelectromagnetic calorimeter clusters) must be less than 80%, and each jet is required to haveat least one charged track [29].

If the photon is sufficiently close to the beam direction and hence not observed, then thesebackground processes can be removed by introducing a cut on the momentum imbalance.More precisely, the requirement is on the sum pz of the momenta of the observed particlesalong the beam direction z

|pz| < 1.5c(mvis ! 90 GeV/c2) (20)

where mvis is the invariant mass of the observed particles [31].A large number of background events still satisfy the loose selection described above. The

distribution of the sum of the jet b-tag neural network outputs #1, #2, #3 and #4, for the eventsselected by the neural network is shown in figure 7. At this stage, the dominant backgroundis from WW production, which also produces four-jet events. These events can be removedby requiring the event to contain b jets, as the W is expected to decay to b only rarely. Eventsfrom Z # qq which remain have satisfied the four-jet requirement, implying that one or morehard gluons were radiated in these events, and have been clustered into their own jets. Themain differences between bbgg and the signal are in the event kinematics and the jet properties,so these must be examined in detail. Further background comes from the pair production of

22

Input della NN:Variabili di b-tagging

The Higgs particle in the standard model: experimental results from LEP 483

Figure 7. Sum of the jet b-tag values for the ALEPH four jet events. Data are shown as points, andbackgrounds are shown as shaded histograms. A 114 GeV Higgs signal with arbitrary normalizationis shown as an empty histogram.

Z bosons. In some of these events, one of the Z bosons decays to bb, but the ZZ cross sectionis much lower than the HZ cross section for Higgs masses near mZ . For heavier Higgs bosons,the H can be distinguished from the Z by its mass.

The neural network used to distinguish signal events from background events has 19 inputvariables, one hidden layer containing 30 nodes, and a single output node, indicating whetherthe event is signal-like or background-like [32]. For the four-jet analysis, each of the jetsmay be paired with any other, giving three possible pairs of di-jets. Each di-jet can then beconsidered as either the Z or the H boson decay products, giving a total of six possible pairingchoices. The neural network is trained by exposing it to all six possible pairing choices foreach background event, but only the correct pairing for signal events. This enables the networkto both discriminate between signal and background, and to select the correct pairing choice inthe signal. The neural network is designed to be only weakly dependent on the centre-of-massenergy, and is most efficient close to the kinematic limit. The neural network used to identifysignal events in data collected in 2000 is the same network used to identify signal in 1999.

The variables used by the neural network can be loosely divided according to their functionin either suppressing background or selecting the correct signal pairing combination. Threeb-tagging variables are used to select H ! bb events. They are

1. min(!3, !4)

2. (1 " !3)(1 " !4)

3.!4

i=1 !i

where !i is the b-tag neural network output value for jet i, and 3 and 4 are the indices for theassumed Higgs jets. The first two variables will both reject events with no b jets, and help toselect the pairing. The third b-tag variable allows the network to identify bbbb events.

23

Input della NN:configurazione dei jet.

Quanto l’evento nonassomiglia a un eventoback-to-back?

Input della NN: variabilicinematiche.

Input della NN: variabilicontro eventi con jet dagluoni (il jet da gluoneha una massa invariantemaggiore perche vienedalla frammentazionedei due quark prodottidal gluone).

484 P A McNamara III and S L Wu

The four-jet preselection is fairly loose, so information about the quality of the four-jetevents is added to the neural network in the form of the variables

4. y34, the DURHAM [30] transition value, discussed above5. !, the sum of the four smallest di-jet angles6. B, an event broadening variable, computed by dividing the event into hemispheres defined

by the thrust axis, and computing

Bhemi =!Ntracks

i=1 |pti |!Ntracks

i=1 |pi |for each hemisphere, where pi and pti are the momentum and momentum perpendicularto the thrust axis for track i. B is the minimum of the two hemisphere values [33].

These variables also work to suppress topologies typical of qq events with gluons, where thejets produced by gluons typically have less energy than those produced by quarks, and areoften close to the quark jets.

Kinematically, events from the signal will be distinguishable from the backgroundsbecause of the masses of the di-jet pairs. If the analysis selection is to be made only weaklydependent on the Higgs mass, that mass cannot be included in the neural network. In theproduction of a Higgs near the kinematic limit, however, where the H and Z bosons areproduced nearly at rest, the jets from either boson will be nearly back-to-back, with energiesclose to half the mass of the mother particle. This leads to the use of variables which rely onthe distinctive kinematics of an HZ event near threshold, such as

7. m12, the mass of the Z di-jet8. min(cos("ij ) + cos("kl)) for any permutation of ijkl

9, 10. Emin, Emin2, the two lowest jet energies11. Emax , the maximum jet energy12, 13. the decay angles of the H and Z.

The energy variables introduce a small energy dependence to the analysis, but it is found tobe minimal. They are primarily useful for selecting events near threshold, and ensuring thatthe event does not have two low-energy jets more consistent with qqgg production than signalproduction. The decay angles contribute weakly to selecting the correct pairing choice, butalso help to distinguish signal from qq background.

In addition to the kinematic and b-tag information, the weak discrimination between quarkand gluon jets can be used to reject events from bbgg. Gluon jets typically have larger invariantmasses than light quark jets. Therefore, they look more like b-jets without secondary verticesthan like light quark jets. Jet shape variables which serve either as anti-gluon tags or assupplemental b-tags for bbbb events include

14, 15. boosted sphericity of lowest-sphericity jets16, 17. multiplicity of tracks with rapidity greater than 1.6, in the Z jets18, 19. jet masses of two lowest-mass jets.

The neural network output value is shown in figure 8. Events with a neural network valuegreater than 0.763 are accepted by the analysis. If more than one di-jet combination in an eventpasses this selection, the combination with the largest neural network output value is chosen.

In data collected in 2000, ALEPH selected a sample of 52 four-jet Higgs candidates, whileexpecting 47.2 candidates from the non-Higgs backgrounds [34]. The neural network doesmake some use of quantities like the jet energies and jet angles which one would expect to besomewhat correlated with the mass, but as the mass is one of the more powerful discriminants

24

The Higgs particle in the standard model: experimental results from LEP 485

Figure 8. Neural network distribution of ALEPH events after preselection. The data are shown aspoints, and the backgrounds as shaded histograms. A 114 GeV/c2 signal is shown with arbitrarynormalization as a clear histogram.

between signal and background, it should be exploited more fully. Instead of using it inthe selection, it is incorporated into the analysis as a discriminant in the likelihood ratio, asdiscussed in appendix D.

Ideally, the mass for a signal would be a delta function at the true Higgs mass, withbackground uniformly distributed over a wide range of masses. An accumulation at a specificmass would then be an indication of the presence of signal. In practice, the Higgs massreconstruction has a resolution, and can have a long tail extending to lower values of mass.Techniques which can either reduce the size of the tail or improve the core resolution willimprove the sensitivity of the analysis directly.

The main contribution to the tail is the selection of the wrong di-jet pairing. As a result,the choice of pairing dependent variables in the neural network is crucial. For a given wrongpairing choice, however, little can be done to improve the situation.

The main contribution to the core resolution is mismeasurement of the jet energies andangles. Missing particles such as neutrinos and calorimeter mismeasurement lead to anuncertainty in the jet energy which translates directly into Higgs mass resolution. To reducethe impact of this resolution, a kinematic fit is performed on the jet energies and angles, usingthe constraints that the total energy of the event must be twice the beam energy and that thetotal event momentum must be zero [35].

This fit introduces an anticorrelation between the mass of the Higgs di-jet and the massof the Z di-jet, as the fit forces the sum of their energies to be fixed. ALEPH exploitsthis correlation by defining the reconstructed Higgs mass as mH = m12 + m34 ! mZ . Bysignificantly improving the Higgs mass resolution, this mass definition increases the sensitivityto the signal. Figure 9 shows the distribution of this mass variable for events selected by theneural network.

To further improve the sensitivity to Higgs, ALEPH uses the 19-variable neural networkoutput as a second discriminant [29]. This allows events with extremely high neural network

25

486 P A McNamara III and S L Wu

a)ALEPH Data

Background

Signal

0123456789

10

50 60 70 80 90 100 110 120 130

Reconstructed Mass (GeV/c2)

Eve

nts/

(4 G

eV/c

2 )

b)ALEPH Data

Background

Signal

00.5

11.5

22.5

33.5

44.5

5

50 60 70 80 90 100 110 120 130

Reconstructed Mass (GeV/c2)

Eve

nts/

(4 G

eV/c

2 )

Figure 9. Reconstructed mass distribution for events selected by the ALEPH four-jet analysisidentified as (a) 2b events and (b) 4b events. The data are shown as points with error barsrepresenting the statistical error associated with a measurement of that number of events. Thebackground is shown as a light shaded histogram. A 114 GeV Higgs signal has been added to thebackground as a darkly shaded histogram.

output values to be treated as more signal-like than those with low neural network values. Ascandidates with lower neural network output values are de-emphasized, the choice to rejectevents with neural network output values lower than 0.763 is primarily for simplicity, asin the calculation of the likelihood ratio, events with lower neural network values wouldbe assigned weights very close to zero, and would be expected to be primarily frombackground processes.

ALEPH computes the probability densities for signal and each background type separatelyfor each discriminant. Two-dimensional probability densities are constructed as the productof the one-dimensional probability densities, and different signal or background types are thensummed to obtain the total signal or background probability density. The probability densityfunctions are constructed in this manner to reduce uncertainties due to limited statistics, whichcan be difficult to overcome in a multidimensional space. The discriminants are, however,known to be somewhat correlated, so a correction is applied to the final result to compensatefor this correlation [34].

The probability densities for signal and background are combined with the signal andbackground expectations to compute event weights for each candidate selected by the neuralnetwork. Three four-jet candidates are found to have especially high weights. These candidateshave been given labels (a)–(c). All of these events were collected at centre-of-mass energiesgreater than 206 GeV and have large neural network output values [34].

The first of these candidates, labelled (a), was collected at centre-of-mass energy of206.6 GeV. It is a four-b candidate with reconstructed mass 110.0 GeV/c2. Three of the fourjets are well b-tagged, and secondary vertices can be discerned in them, as seen in figure 10.The fourth jet has a b-tag value of 0.214. With the chosen pairing, the jet with lowest b-tagvalue is considered to be one of the Higgs jets. The probability of obtaining such a low jetb-tag in a 4b event is 19%.

The most likely background for a 4b event comes from the ZZ process, which can bedistinguished from the signal only on the basis of the mass. To verify that this event is not aZZ event with a bad pairing assignment, a kinematic fit including the Z boson width and massresolution is performed on each of the possible pairing choices. In the pairing most compatiblewith this hypothesis, the fit obtains large Z boson masses of 98.9 and 101.6 GeV/c2.

26

The Higgs particle in the standard model: experimental results from LEP 487

ALEPH!=180

!=0 ("!43 )*SIN(!)

x xx

! x!

!x

x

!

!

! !x

!

!xx

!!

!

x

xx

x

x

!! !

x

x!

xx

!

x

x

x

! !

x!

x!

!

x

!!

x!

x! !

xx

!

!

x

ooo

o

oooo

o ooo

o

o o

o

o

o

o

o

o

o

o

o

o

oo

o

oo

o

o o

P>.20 Z0<5 D0<2 NT=4 man.cutYX

0 1cm

0 1cm

Ko

Figure 10. ALEPH four-jet Higgs candidate (a), with reconstructed mass 110.0 GeV/c2 [34].

The second candidate, labelled (b), was collected at centre-of-mass energy 206.7 GeV.Similar to candidate (a), it is a 4b candidate, and all four jets are well b-tagged. It has a recon-structed mass of 112.9 GeV/c2. Secondary vertices can be distinguished, as seen in figure 11.

Also visible in the upper right corner of figure 11 is a 22 GeV energy deposit in the small-angle SIlicon CALorimeter (SICAL). This energy was probably deposited by a beam-relatedparticle unconnected to the remainder of the event. When this energy deposit is included, themeasured energy in the event is 252 GeV, too far above twice the beam energy to be explainedby detector resolution effects, and the event has a total momentum which is in the direction ofthis 22 GeV energy deposit. Further, the energy is deposited in a region in the SICAL in whichthe majority of beam-related backgrounds are observed. If the energy deposit in the SICALwere removed, the reconstructed mass would shift from 112.9 to114.5 GeV/c2, and the neuralnetwork output for the event would shift from 0.997 to 0.998.

As with candidate (a), the most probable non-signal explanation for this event is ZZproduction. When the fit to the ZZ hypothesis is applied, the best fit gives Z boson masses of94.0 GeV/c2 and 97.3 GeV/c2.

The third ALEPH four-jet candidate (c) was also collected at centre-of-mass energyof 206.7 GeV. It has a reconstructed mass of 114.3 GeV/c2. Unlike the previous events,candidate (c) is a 2b candidate, and both of the H jets are very well b-tagged. The secondaryvertices of these jets can be seen in figure 12, along with a muon from the semileptonic decayof one of the b hadrons. This event is very well measured, with a missing momentum of13.8 GeV/c which travels along the centre of the jet containing the muon. This suggeststhat the missing momentum is the result of the neutrino produced in association with themuon in the decay of the b hadron. The two non-b-tagged jets have an invariant mass of92.1 GeV/c2, which is consistent with the production of a Z boson. If the b-tagged jets are

27

488 P A McNamara III and S L Wu

ALEPH!=180

!=0 ("!43 )*SIN(!)

!xx

xx

!

x

x

! !

x

!x

x !!

x!xxx

x

! xx

x!

!!x !

!

x!

!

x! xx

!

!!

x!

!

!

!

x

ooo ooooo

oo

ooo

o

oo

oo oooo

ooooo

o

oo

oo

o

oooo

oo

o

o

o

P>.20 Z0<5 D0<2 NT=4 man.cutYX

0 !0.1cm 0.1cm

0 1cm

Figure 11. ALEPH four-jet Higgs candidate (b), with reconstructed mass 112.9 GeV/c2 [34].

combined with the missing energy, the resulting particle has a mass of 114.4 GeV/c2, verysimilar to the 114.3 GeV/c2 obtained with the reconstructed mass algorithm described aboveand inconsistent with the ZZ hypothesis.

OPAL. While neural networks are used to perform the Higgs selection in the majority of theLEP searches, they are not the only multivariate analysis tool used. The OPAL experiment usesa likelihood method both to identify jets from b decays and to select Higgs candidates [36].

Like the ALEPH analysis, the OPAL four-jet selection is carried out in two steps. To rejectevents from easily separable backgrounds and to retain only four-jet events, a preselection isapplied to the collected data. After this preselection, a likelihood for the event to come fromsignal events is constructed, and events with sufficiently high likelihood values are retained.

To select hadronic events, OPAL rejects events with fewer than seven electromagneticclusters and five charged tracks. To reduce the background contribution from two-photoninteractions, energy in the electromagnetic calorimeter is required to be greater than 14% ofthe centre-of-mass energy. This background is further reduced, along with background frombeam–gas and beam–wall interactions by requiring that the energy deposited in the detectorshould not be strongly imbalanced along the beam axis [37].

To reject events from radiative returns to the Z resonance, OPAL constructs an effectivecentre-of-mass energy

!s " after initial-state radiation. This

!s " is constructed by first

attempting to reconstruct up to two initial state photons escaping undetected along the beamwith kinematic fits imposing four-momentum conservation. The

!s " variable is then computed

from the jet directions produced by a fit excluding both photons identified in the detector andany reconstructed photons [37]. To remove events in which this centre-of-mass energy isconsistent with the Z,

!s " is required to be at least 150 GeV [38].

The Higgs particle in the standard model: experimental results from LEP 489

ALEPH!=180

!=0 ("!!47)*SIN(!)

x

!

x

x

x!

x

x

x

!

x

!!

x

x

!

x!

!

x

x

!!

!

x

!

x

!x

!x

x

!

o

oo

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oo

oo

ooo

oo

ooo

o

oo o

o

o

ooo

oo

oo

o

o

ooo

P>.20 Z0<5 D0<2 NT=4 man.cutYX

0 !0.3cm 0.3cm

0 1cm

µ

Figure 12. ALEPH four-jet Higgs candidate (c), with reconstructed mass 114.3 GeV/c2 [34].

The event is then forced to cluster into four jets using the DURHAM algorithm [30]. Toimprove the clustering of the event into jets, objects more consistent with a jet other than theone they are clustered into are reassociated to the closer jet [38]. Events which are inconsistentwith a four-jet interpretation are removed by requiring that each jet contain at least two chargedtracks, and that the DURHAM ‘distance’ between the third and fourth jets y34 be greater than0.003. In addition, the event shape C-parameter [39], which is similar to sphericity, is requiredto be greater than 0.25.

The OPAL b-tag algorithm for distinguishing b jets from light quark jets is based on acombination of neural network and likelihood techniques. A likelihood technique is used tocombine: (a) a high-pT lepton tag; (b) a neural network trained to combine lifetime-basedtagging variables; and (c) a second neural network trained to combine tagging variables basedon jet kinematics [38]. Figure 13 shows the distribution of the OPAL b-tagging variable.

While events selected by these preselection cuts will be broadly consistent with theexpected signal, stronger requirements on the kinematics of the events can be imposed withoutsignificantly reducing the signal efficiency. A kinematic fit requiring energy and momentumconservation is applied to the jets, and the !2 probability of the fit is required to be at least10!5 [38]. To select jet pairings kinematically consistent with the signal hypothesis, a kinematicfit is performed requiring both energy and momentum conservation and that the pair of jetsassumed to be from the Z has the mass of the Z [40]. Jet assignments which have!2 probabilitiesgreater than 10!5 are retained by the analysis [38].

After the preselection, a likelihood is used to select signal candidates [40]. To computethe likelihood, first the probability for a single variable i to be of a single class j is computedas

The Higgs particle in the standard model: experimental results from LEP 527

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