Fakultät für PhysikProf. Dr. Th. Müller, Priv.-Doz. Dr. W. WagnerDr. P. Schieferdecker, Dr. F.-P. Schilling, Dr. J. Wagner-Kuhr 4.12.2008

Computerpraktikum zur Vorlesung

Teilchenphysik für Fortgeschrittene

Reconstructing charm resonances from DELPHI data

compiled and written byZoltán Albrecht, Gary Barker and Markus Moch

1 Introduction

This part of the Computer Praktikum is concerned with observing so-called ‘charm’ orD-mesons i.e. bound states of a charm(anti-charm) quark with another anti-quark(quark).In our case the other quark will either be an ‘up’ quark or a ‘down’ quark. The data wewill use has been taken by the DELPHI detector at the CERN Large Electron-PositronCollider (LEP), see Figure 1.

In the LEP ring, intense beams of electrons and positrons are accumulated, acceleratedand finally made to collide in the region of each of the four detectors. For the data set wewill use, the collision energy is finely tuned to be at the rest mass of the Z0 boson whichsubsequently decays into pairs of leptons (about 30% of the time) or pairs of quarks (about70% of the time). It is the production of quark pairs that we are interested in here andthe process is illustrated in Figure 2. The Z0 is produced at rest in the laboratory frameand decays to a quark-anti quark pair (qq̄) which fly apart back-to-back. As the qq̄ pairseparate, the magnitude of the strong force field between them increases until the storedenergy becomes large enough to create new qq̄ pairs in a process known as fragmentationor hadronisation. The result is that our original quark and anti-quark become bound upinside hadrons which, in turn, form part of a whole cluster or jet of other fragmentationhadrons that approximately follow the direction of the parent quark or anti-quark. It isthese jets of particles, or their decay products, that enter our detector and are measured.

1 INTRODUCTION 2

POINT 4.

LAKE GENEVA GENEVA

CERN Prévessin

POINT 6.

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POINT 2.

CERN

SPS

ALEPH

DELPHI

OPAL

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+e Positron

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Abbildung 1: The LEP collider at CERN is 27km in circumference and about 100m un-derground. This picture illustrates the position of the four detectors; ALEPH, DELPHI,L3 and OPAL.

z0

q−

_e

+e

dah

rons

q

Abbildung 2: The process of e+e− annihilation to form a Z0 boson that then decays intoa qq̄ pair.

2 THE DELPHI EXPERIMENT 3

2 The DELPHI Experiment

2.1 Overview

DELPHI (DEtector with Lepton, Photon and Hadron Identification) is shown in Figure 3.It consists of a collection of sub-detectors arranged as concentric cylinders going to largerand larger radii away from the interaction region, rather like the layers of an onion. Eachsub-detector is optimised to perform a certain task so that, taken as a whole, DELPHIis able to measure the trajectory and 4-vector as well as identify all particles that enterthe detector (the exception to this are neutrinos that interact with matter so weaklythey escape direct detection, but show themselves by the presence of missing energy inreconstructed events).

Abbildung 3: The DELPHI detector showing the layout of the various sub-detectors inboth the Barrel and End-Cap regions.

The electron and positron beams circulate in a high vacuum (10−12 Torr) inside aberyllium beam pipe that goes through the centre of DELPHI. Directly outside the beampipe are the charged particle tracking devices. Closest to the interaction region is theVertex Detector (VD) consisting of solid state silicon-strip devices providing high precisiontrajectory reconstruction of short lived particles. Outside of the VD are a series of lessprecise but large, gas ionisation tracking devices of which the largest is the Time Projection

2 THE DELPHI EXPERIMENT 4

Chamber(TPC) which can track the passage of particles between a radius of about 40cm and 110 cm. All tracking devices sit inside a 1.23 T magnetic field, provided bythe Superconducting Solenoid, oriented along the z-axis. This is important in order tomeasure the momenta of tracks since the momentum in the plane transverse to the z axisis proportional to the measured radius of curvature of the track.

Surrounding the TPC is a Ring Imaging CHerenkov or RICH detector which iden-tifies the type of charged particle by reconstructing the Cherenkov light cone emittedwhen a relativistic charged particle passes through a dielectric medium. At still largerradii from the interaction region come the calorimetry devices for reconstructing particleenergies. Electron and photon identification is provided by the High Density ProjectionChamber(HPC) and the return yoke of the solenoid is instrumented with the Hadron Ca-lorimeter. The very outside layer of DELPHI are the Muon Chambers designed to trackthe passage of the very penetrating muons.

Although we have described here just the cylindrical or Barrel section of DELPHI, asimilar scheme is also followed to instrument the forward regions in End-Caps that canalso be seen in Figure 3. The response of the detector to a typical 2-jet event of the typedescribed in Section 1 is shown in Figure 4. The lines represent reconstructed chargedparticle tracks and the boxes are calorimetry clusters, the size of which is proportional tothe energy deposited by a particle at that point.

2.2 Charged Particle Tracking

The (x, y, z) coordinates or hits recorded by the various tracking detectors as a chargedparticle passes through the detector, are sorted by pattern recognition algorithms into setslikely to have originated from the passage of the same physical particle. The hits are thenfitted to a helix function to give a track described by five free parameters referenced tothe perigee point, that is the point of closest approach of the track to the origin of thecoordinate system. The fitted parameters are:

• � - the x,y distance to the perigee point in the x, y plane (carries a geometrical sign),• z - the z coordinate of the track at the perigee point,• θ - the polar angle of the track,• φ - the azimuthal angle made by the tangent to the track at the perigee point,• 1/r - the inverse of the radius of curvature of the track (signed by the charge of the

particle).

The most precise tracking information and the most important sub-detector for thereconstruction of hadrons containing c and b−quarks, is the VD. There are three layersof silicon strip devices at radii of 6.3. 9.0 and 10.9 cm from the interaction point, givingan intrinsic spatial precision of around 8μm in both the x, y and z planes respectively.

2 THE DELPHI EXPERIMENT 5

DELPHI Interactive AnalysisRun: 26154Evt: 3018

Beam: 45.6 GeV

Proc: 1-Oct-1991

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Abbildung 4: A 2-jet Z0 → qq̄ event as seen in DELPHI.

Figure 5 shows the reconstruction of a Z0 → bb̄ event in the VD. Tracks fitted throughthe detector hits are shown and stars indicate the location of three separate verticesfrom which tracks originate. The lower picture is a blow-up of this central region. Thereconstructed vertex in the middle corresponds to the e+e− interaction point or primaryvertex, where the b and b̄ quark were produced. The tracks coming from this point areeither particles produced in the fragmentation process or decay products of excited states

2 THE DELPHI EXPERIMENT 6

0.0 cm 7.5 c

26024 / 1730 DELPHI

0.0 cm 2.0 c

26024 / 1730 DELPHI

Abbildung 5: A B-meson decay reconstructed in the VD.

that decay very fast by the strong or electromagnetic interaction. The remaining twosecondary vertices correspond to the decay points of the two B hadrons, produced inthe fragmentation process, and the tracks emanating from these points are the B hadrondecay products.

2 THE DELPHI EXPERIMENT 7

12

C F liquid radiator6 14

lightCherenkov

lightCherenkov

electronsphoto

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UV photon detector

5

particle

mirror

Abbildung 6: The operation principle of the DELPHI RICH detectors.

2.3 Charged Particle Identification

Hadrons containing c and b quarks typically decay well before entering the DEL-PHI detector. It is only relatively stable decay products and fragmentation tracks (i.e.e±, μ±, π±, K±, p) that live long enough to register signals in the detector and it is fromthese that we must try and reconstruct what process actually occurred in the event. Inaddition to accurately reconstructing the trajectory of particles through the detector, itis clearly important to be able to identify the type of particle we have detected.

The RICH detectors in DELPHI play a central role in determining the identity ofcharged tracks. The technique used relies on the following dependence between the openingangle (θC) of the Cherenkov light cone emitted by a particle and the mass of the particle(M): cos(θC) = 1/n ·

√1 + M2/p2 where n is the refractive index of medium the particle

traverses. A gas and a liquid medium are used in the DELPHI RICH’s in order to besensitive to a large spread in particle momenta: the liquid is used for the range from 0.7to 8 GeV/c and the gas is used from 2.5 to 25 GeV/c. The basic principle of light conedetection by the RICH is illustrated in Figure 6.

Another powerful tool for particle identification, in the low momentum region, comesfrom sampling the energy loss (dE/dx) as particles traverse the VD and TPC. Thistechnique exploits the differences in the Bethe-Bloch energy loss formula for particles of

2 THE DELPHI EXPERIMENT 8

Abbildung 7: The performance of charged particle identification in DELPHI. The plotsshow how dE/dx information from the TPC together with the liquid and gas RICH de-tectors allow the identification of lepton, pion, kaon and proton states over essentially thefull momentum range of the data.

different mass in the region below the minimum ionizing point. The performance of thismethod together with the RICH’s is shown in Figure 7. As we shall see in the next section,the separation of kaons from pions is particularly important for the identification of heavyquark hadron decays.

Various algorithms have been developed inside DELPHI, e.g. based on neural networks,to combine in an optimal way dE/dx and RICH information for a particle identificationcapability over the full momentum spectrum.

3 D-MESON SPECTROSCOPY - THEORY 9

3 D-Meson Spectroscopy - Theory

D, (D̄) mesons are two-quark states where one of the quarks is a c (c̄) e.g. D+ = cd̄, D− =c̄d, D0 = cū, D̄0 = c̄u. Quantum ChromoDynamics or QCD is thought to be the appro-priate theoretical framework in which to calculate the properties of such states. In practicehowever, the tool of perturbative expansion can often not be used and we must resortto the use of models. Fortunately, many aspects of meson states containing a heavy c orb quark can be understood to some approximation if we consider the heavy quark(anti-quark) to be de-coupled from the light anti-quark(quark) partner. The most extremeform of this is the so-called spectator model in which the heavy quark is considered tobe completely independent of the light quark. Since the dynamics of the light quark areconsidered to have no influence on the heavy partner, this model works best when thedifference in mass between the quark and anti-quark is as large as possible e.g. the spec-tator model is a better approximation to the B system and D system than it is to theK system. In the DELPHI e+e− → qq̄ data set, D mesons can be produced whenever ac quark is present which is the case for e+e− → cc̄ events and in e+e− → bb̄ where theb quark has decayed weakly to a c quark. Charm quarks can in principle be produced inthe fragmentation process, but due to their relatively high mass of 1.1 to 1.4 GeV /c2 theprobability is negligibly small. The fraction of all e+e− → qq̄ events that are cc̄ is about18%, about 22% are bb̄ events with the remainder involving the lighter u, d, s quarks.

Figure 8 shows the expected spectroscopy of D-mesons containing a u or d quarkwhere the vertical axis represents the rest mass and the states are horizontally separatedby their total spin (J = S(intrinsic spin)+L(orbital angular momentum)) and P = parityassignment.

The ground state D and D∗ are well established experimentally and can be consideredas the degenerate levels of the L = 0 state. Representing these states in Dirac notation as| ± 1/2,±1/2 > where the first(second) index is the 3rd component of the c(light) quarkintrinsic spin:

|D〉 = (|+1/2,−1/2〉 − |−1/2, +1/2〉) /√

2

|D∗, +1〉 = |+1/2, +1/2〉|D∗, 0〉 = (|+1/2,−1/2〉 + |−1/2, +1/2〉) /

√2

|D∗,−1〉 = |−1/2,−1/2〉

This notation explicitly illustrates that one would naively expect, from spin countingalone, a production ratio of D : D∗ = 1 : 3. This is in fact true to a good approximationin the B meson system where B∗’s represent 75% of all B + B∗ mesons. In the D-systemhowever the proportion of D∗’s is more like 62% and results from the greater impact ofnon-spectator effects in D-mesons relative to B-mesons.

Of the four L = 1 orbitally excited states only the two narrow states (D1 and D∗2), with

decay widths of about 20MeV , have been clearly observed experimentally. The D∗0 andD∗1 are expected to have much larger decay widths and have not yet been clearly verifiedexperimentally. Figure 8 also illustrates higher mass radial excitations of the ground state,

4 D-MESON SPECTROSCOPY - PRACTICE 10

Me

Abbildung 8: The spectroscopy of D-mesons containing a u or d quark. The shaded are-as show the expected widths of these states. For better clarity the decays of the D1 andD∗2 states into p and ππ are not shown.

labeled D′, D�′, which are expected to decay into D(D∗)ππ and the D�

′has now been

measured in this channel by DELPHI.The decay branching ratios of D∗ mesons are tabulated below: The suppression of

D+ production indicated by the branching ratios in Table 1 result in the following overallproduction branching ratios of D0, D+ from c quarks:

BR(c → D0) � 56%BR(c → D+) � 22%

4 D-Meson Spectroscopy - Practice

Reconstructing D-states from data can be rather difficult in practice because there aretypically many potential kaon and pion candidates per event, and not all of them aredecay products from a D. The general technique of spectroscopy is to form the invariantmass of various combinations of particles that fit the decay pattern of the state you areinterested in i.e. the signal. The correct combinations then form a cluster or peak in theinvariant mass spectrum around the true physical mass of the signal while the wrongcombinations form an underlying spectrum of combinatorial background that, in general,will not peak at the mass value of the signal.

4 D-MESON SPECTROSCOPY - PRACTICE 11

Decay Products RateD∗+ D0π+ 68%

D+π0 31%D+γ 2%

D∗0 D+π− ForbiddenD0π0 62%D0γ 38%

Tabelle 1: The decay branching ratios of charged and neutral D∗ states. The charge con-jugate reactions are assumed in the above numbers. Note that the mass difference betweenD∗0 and D+ is less than a pion mass, making this decay channel kinematically forbidden.

The mass peak can then be fitted, usually with a Breit-Wigner function in combinationwith some parameteristion of the combinatorial background. The Breit-Wigner functionhas the following form,

σ(x) =Γ

2π [(x − x0)2 + (Γ/2)2]and is fully defined by x0, the position of its maximum (about which the distribution issymmetric), and by the full width at half maximum (FWHM) Γ since,

σ(x = x0) = 2 · σ(x = x0 ± Γ/2).Although originally derived to describe the cross-section of resonant scattering, the func-tion is widely used to fit non-interfering resonance peaks where x0 is the resonance massand Γ is the energy width of the resonance. Note that in practice, reconstructed resonancepeaks have widths larger than Γ due to the effects of detector resolution.

Our study of D spectroscopy will centre on the following decay channel:

D∗+ → D0π+∗↪→ K−π+ or K−π+π−π+

This channel has the advantage that a) all the decay products are charged particles sothat a precise momentum measurement can be made, b) the D∗+ is relatively easy toidentify experimentally through the detection of the slow pion labeled as π+∗ . This namederives from the fact that there is very little energy left in the rest frame of the D∗+ afterdecay since,

Q = M(D∗+) − M(D0) = 145.436 ± 0.016 MeV � M(π+) = 139.57 MeV .This means that the decay π+ will have low momentum in the D∗+ rest frame whichmakes it a useful experimental ‘tag’ for the presence of a D∗.

Feynmann diagrams illustrating these decay channels are shown in Figure 9. Note thatthe D∗+ decay shown in Figure 9(a), is a strong interaction process mediated by gluon

4 D-MESON SPECTROSCOPY - PRACTICE 12

K−

K−

D0

π−+π

+π

D0

D0

D*+

+π+π

u−

u−

u−

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u−

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W+

d−

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g

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(c)

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Abbildung 9: Feynmann diagrams for (a) D∗+ decay and D0 decay into the two channelsof interest: (b) D0 → K−π+ and (c) D0 → K−π+π−π+

exchange and happens so fast that the D0 and π+ decay products can be considered asoriginating from the primary vertex. The D0 decay however, involves the much slowerweak interaction resulting in a proper lifetime of τ(D0) = 0.4 ps, cτ = 124μm. The kaonand pion decay products in this case originate from a secondary vertex separated from theprimary vertex which, in principle, could be resolved with the precision tracking capabilityof DELPHI. Branching ratios for the above channels are listed in Table 2.

Channel Branching RatioD∗+ → D0π+ 67.7 ± 0.5%D0 → K−π+ 3.83 ± 0.09 %D0 → K−π+π−π+ 7.49 ± 0.31 %

Tabelle 2: Current world average measurements of branching ratios for the channels ofinterest.