Facoltà di Scienze Matematiche, Fisiche e Naturali

Ricerca del decadimentoX(3872)→ J/ψω

nell’esperimento LHCb al CERNTesi di Laurea Magistrale in Fisica

Candidato: Relatore:Lorenzo Capriotti Roberta Santacesaria

Correlatore:Antonio Augusto Alves Jr.

Anno Accademico 2013/2014

Contents

1 Exotic charmonia: theoretical and experimental status 81.1 Exotic quarkonia in the charm sector . . . . . . . . . . . . . . 91.2 X(3872) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Experimental status . . . . . . . . . . . . . . . . . . . 101.2.2 Theoretical models . . . . . . . . . . . . . . . . . . . . 131.2.3 The J/ψω decay channel . . . . . . . . . . . . . . . . . 15

2 The LHCb experiment at CERN 172.1 Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . 172.2 LHCb detector . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Vertex Locator (VELO) . . . . . . . . . . . . . . . . . 202.2.2 Ring-Imaging Cherenkov Detectors (RICH) . . . . . . 202.2.3 Tracking System . . . . . . . . . . . . . . . . . . . . . 212.2.4 Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.5 ECAL and HCAL . . . . . . . . . . . . . . . . . . . . . 242.2.6 Muon system . . . . . . . . . . . . . . . . . . . . . . . 262.2.7 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Analysis tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.1 Stripping and reconstruction . . . . . . . . . . . . . . . 272.3.2 Monte Carlo simulations . . . . . . . . . . . . . . . . . 28

3 Data Analysis 293.1 Data samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Event selection . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 Stripping cuts . . . . . . . . . . . . . . . . . . . . . . . 303.2.2 Decay reconstruction . . . . . . . . . . . . . . . . . . . 313.2.3 Preselection cuts . . . . . . . . . . . . . . . . . . . . . 32

3.3 Monte Carlo sample . . . . . . . . . . . . . . . . . . . . . . . 353.4 Multivariate analysis . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.1 Input variables and training phase . . . . . . . . . . . . 423.5 B+ mass fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5.1 Application phase of MVA . . . . . . . . . . . . . . . . 483.6 Background subtraction: sPlot technique . . . . . . . . . . . . 50

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3.6.1 Application of sPlot technique . . . . . . . . . . . . . . 503.7 J/ψω mass spectrum . . . . . . . . . . . . . . . . . . . . . . . 523.8 Resonances contribution . . . . . . . . . . . . . . . . . . . . . 533.9 Hypotheses for the excess of events at 5350 MeV/c2 . . . . . . 573.10 Efficiency corrections . . . . . . . . . . . . . . . . . . . . . . . 613.11 J/ψω mass resolution . . . . . . . . . . . . . . . . . . . . . . . 70

4 Results 724.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Appendices 76

A The sPlot technique 77A.1 Preliminary step: total correlation . . . . . . . . . . . . . . . . 78A.2 The sPlot formalism . . . . . . . . . . . . . . . . . . . . . . . 78

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List of Figures

1.1 Charmonium spectrum . . . . . . . . . . . . . . . . . . . . . . 101.2 Peak observed by Belle (2003) . . . . . . . . . . . . . . . . . . 111.3 Result of the X(3872) particle angular analysis from CDF . . . 111.4 Determination of X(3872) quantum numbers: BaBar analysis

on the left, LHCb analysis on the right . . . . . . . . . . . . . 121.5 Distribution of the test statistic t for the simulated experiments

with 2−+ and 1++ at LHCb . . . . . . . . . . . . . . . . . . . 121.6 Simulation of the prompt production cross section as a function

of the relative centre of mass momentum of the system D0−D∗0(the allowed phase space region is highlighted) . . . . . . . . . 13

1.7 Spectrum of [cq][cq] states according to the tetraquark model . 141.8 Corrected M(J/ψω) distribution for B+ (top) and B0 (bottom)

decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 The CERN accelerator complex . . . . . . . . . . . . . . . . . 182.2 Polar angles of the b- and b-hadrons calculated with PYTHIA 192.3 LHCb detector, lateral view (non-bending plane) . . . . . . . . 192.4 VELO system configuration and VELO sensors . . . . . . . . 202.5 RICH detectors layout . . . . . . . . . . . . . . . . . . . . . . 212.6 Inner Tracker layout . . . . . . . . . . . . . . . . . . . . . . . 222.7 Trigger Tracker layout . . . . . . . . . . . . . . . . . . . . . . 232.8 Tracking system layout . . . . . . . . . . . . . . . . . . . . . . 232.9 LHCb magnet, perspective view . . . . . . . . . . . . . . . . . 242.10 Calorimeters segmentation: ECAL on the left, HCAL on the

right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.11 Resolved π0 reconstruction efficiency for the channel B0 →

π+π−π0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.12 Muon system configuration . . . . . . . . . . . . . . . . . . . . 26

3.1 B mass spectrum after the application of preselection cuts . . 353.2 Dalitz plot for K+ω squared mass vs J/ψω squared mass,

generator level (MC) . . . . . . . . . . . . . . . . . . . . . . . 363.3 B+ mass spectrum (MC), before and after fixing the π0 mass . 383.4 π0 mass spectrum (MC) . . . . . . . . . . . . . . . . . . . . . 39

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3.5 B+ mass vs π0 mass (MC) . . . . . . . . . . . . . . . . . . . . 393.6 ω mass (MC) . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.7 Dalitz plot for K+ω squared mass vs J/ψω squared mass, after

the reconstruction and preselection phases (MC) . . . . . . . . 403.8 Input variables, with signal and background distributions . . . 433.9 Correlation matrices for signal and background samples . . . . 443.10 MVA methods performance (ROC curve) . . . . . . . . . . . . 443.11 PLE method response . . . . . . . . . . . . . . . . . . . . . . 453.12 B+ mass spectrum, after the cut on rL . . . . . . . . . . . . . 453.13 B+ mass spectrum, before and after the cut on rL . . . . . . . 463.14 B+ mass fit, without the bump contribution, for rL > 0.65 . . 473.15 B+ mass fit, with the bump contribution, for rL > 0.65 . . . . 473.16 Statistical significance (blue histogram), signal efficiency

(magenta histogram) and background rejection (blackhistogram) as a function of rcutL (left axis for the significanceand right axis for the rest) . . . . . . . . . . . . . . . . . . . . 49

3.17 Fit on B+ mass for both rcutL , with fit results . . . . . . . . . . 503.18 ω invariant mass distribution after the rL cut . . . . . . . . . 513.19 ω invariant mass signal distribution (background subtracted) . 513.20 π0 mass signal distribution (background subtracted) . . . . . . 523.21 J/ψω mass, MVA_LL > 0.65 . . . . . . . . . . . . . . . . . . 533.22 J/ψω mass, MVA_LL > 0.15 + M(ω) cut . . . . . . . . . . . 533.23 J/ψππ and Kπ invariant mass, MVA_LL > 0.65 . . . . . . . 543.24 J/ψππ and Kπ invariant mass (background subtracted),

MVA_LL > 0.65 . . . . . . . . . . . . . . . . . . . . . . . . . 543.25 J/ψω mass without Ψ(2S) and K∗(892) contributions,

MVA_LL > 0.65 . . . . . . . . . . . . . . . . . . . . . . . . . 553.26 J/ψω mass without Ψ(2S) and K∗(892) contributions,

MVA_LL > 0.15 + M(ω) cut . . . . . . . . . . . . . . . . . . 553.27 M(J/ψω) vs M(Kω), MVA_LL > 0.65 . . . . . . . . . . . . . 563.28 M(J/ψω) vs M(Kω), MVA_LL > 0.15 + M(ω) cut . . . . . . 563.29 M(J/ψω), MVA_LL > 0.65, after cut on M(Kω) . . . . . . . 573.30 M(J/ψω) , MVA_LL > 0.15 + M(ω) cut, after cut on M(Kω) 573.31 B+ mass vs (J/ψKππ) mass . . . . . . . . . . . . . . . . . . . 583.32 B+ mass vs (K J/ψ) mass . . . . . . . . . . . . . . . . . . . . 593.33 B+ mass vs (J/ψππ) mass . . . . . . . . . . . . . . . . . . . . 593.34 B+ mass vs (K π) mass . . . . . . . . . . . . . . . . . . . . . . 603.35 B+ mass vs (K ω) mass . . . . . . . . . . . . . . . . . . . . . . 603.36 Effect of the cut flow on the J/ψω mass distribution (black

histogram before the application of the cuts, red histogram after) 633.37 Effect of the cut flow on the J/ψω mass distribution (black

histogram before the application of the cuts, red histogram after) 64

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3.38 Efficiency of every step as a function of M(J/ψω). Asymmetricerrors are computed according to the Agresti-Coull formula [48]. 65

3.39 Total efficiency distributions for both MVA cut values . . . . . 663.40 Fit on the linear component of the total efficiency for both MVA

cut values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.41 Fit on the M(Kω) cut efficiency for both MVA cut values . . . 673.42 Total efficiency distributions, data from MC and functional form 683.43 M(J/ψω), MVA_LL > 0.65, efficiency corrected . . . . . . . . 693.44 M(J/ψω) , MVA_LL > 0.15 + M(ω) cut, efficiency corrected 693.45 δM for Mreco(J/ψω) ∈ [4100, 4150] MeV/c2 . . . . . . . . . . . 703.46 J/ψω mass resolution . . . . . . . . . . . . . . . . . . . . . . . 713.47 J/ψω mass, threshold region . . . . . . . . . . . . . . . . . . . 71

4.1 Corrected J/ψω mass with the new MC normalization,MVA_LL > 0.65 . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Corrected J/ψω mass with the new MC normalization,MVA_LL > 0.15 + M(ω) cut . . . . . . . . . . . . . . . . . . 73

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Introduction

In this thesis, a search for the decay X(3872) → J/ψω within the decaychannel B+ → K+(J/ψ → µ+µ−)(ω → π+π−(π0 → γγ)) is shown. Theanalysis is performed on data collected by the LHCb experiment at theLarge Hadron Collider at CERN during 2011 and 2012. This channel hasbeen studied by the BABAR experiment at the SLAC National AcceleratorLaboratory in Menlo Park, in California (USA), resulting in a limit on thebranching fraction: BR(X(3872)→ J/ψω) > 1.9%.This thesis is structured as follows:

• in Chapter 1, an introduction on exotic quarkonia is presented, focusingon the X(3872) since the first evidence at the Belle experiment in 2003.A report on the theoretical and experimental status follows, along withthe importance of the specific decay that has been the object of thisanalysis.

• in Chapter 2, a description of the LHCb detector is given, from thephysics reasons behind the choice of its geometry to the technicalspecifications of every subdetector. The analysis strategy followed inthe LHCb experiment is explained in a general outline.

• in Chapter 3, a detailed and stepwise exposition of the analysisworkflow is presented: the processes of event reconstruction, the eventselection, the statistical background subtraction and the analysis ofheavy resonances contributions are described. The last part of thechapter covers the computation of the efficiency correction function andthe J/ψω mass resolution.

• in Chapter 4, the results of this analysis are shown, along with a shortsummary and the forecast for LHC Run-II.

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Chapter 1

Exotic charmonia: theoretical andexperimental status

Since the proliferation of discoveries of new strongly interacting subatomicparticles, it arose an urge to create a model for the purpose of categorizing theobserved states and eventually predicting new ones. Fermi and Yang, in 1949,thought of describing all the resonances as proton-neutron bound states, andSakata [1] extended the model in 1956 in order to include the strange particles.Although these models gave wrong predictions regarding the baryons, and weretherefore unable to explain the totality of hadrons, they were the starting pointfor Gell-Mann and Zweig who, independently, began to develop what we knowtoday as the “Constituent Quark Model” [2].CQM was born in 1961 and it’s based on the SU(3) formalism: the fundamentalrepresentation is composed by three elementary particles (up quark, downquark and strange quark) and the antifundamental representation by theirantiparticles. Mesons and baryons are described as a tensor product of theserepresentations. Using the Kronecker decomposition, they can be expressedas follows:

Mesons: 3⊗ 3 = 1⊕ 8

Baryons: 3⊗ 3⊗ 3 = 1⊕ 8⊕ 8⊕ 10(1.1)

or, in other words, a meson is a [qq] bound state and a baryon is a [qqq] boundstate, where q = (u, d, s). Among the generators of the SU(3) algebra, the socalled Gell-Mann matrices, only two commute between them, and they arerelated to isospin I and hypercharge Y : these two quantities can label eachparticle in each meson and baryon multiplet. With this model, Gell-Mannand Ne’eman were able to predict, in 1962, a new particle which would havecompleted the baryons decuplet, the Ω−, and it was discovered two yearslater at Brookhaven National Laboratory [3]. CQM remains a valid effectivetheory for classifying hadrons, even after the introduction of heavy quarks(c, b, t).

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However, recently discovered states seem not to fit into the conventionalmesons and baryons model: in particular, their internal structure might bedifferent from [qq] and [qqq]. Actually, other combinations are in principleallowed: multiquarks (such as tetraquark or pentaquark), mesonic molecules,and, given the non Abelian nature of Quantum Chromodynamics (QCD),even gluons can contribute explicitly to this structures with hybrid states,glueballs, and others. Since none of these states have been observed for a longtime after the theories were first published (around mid-1960s), it was chosento label every non [qq] or [qqq] state as exotic state.

1.1 Exotic quarkonia in the charm sectorDifferent exotic meson candidates exist. In the light quark sector, for example,the f0(980) and the a0(980) are considered to be strong candidates for KKmolecules. It is very difficult, however, to disentangle these states from thevery dense background of non exotic states, as it would require enormous datasample and a very refined data analysis. The charmonium sector ([cc] states),instead, provides a cleaner environment, given by the large difference betweenheavy quark masses (unlike the light quark sector), and can profit from a widerange of detailed studies on the excited charmonium spectrum.In Figure 1.1 the charmonium spectrum is shown [4], where the solid lines areCQM predictions, the shaded lines are the observed conventional charmoniumstates, and the red dots are the exotic candidates placed in the most probablespin assignment column. Various DD mass thresholds are also shown.The charmonium states shown in Figure 1.1 are described as acharm-anticharm pair bound by a short distance force dominated by asingle gluon exchange, plus a linearly increasing confining potential. Theenergy levels are then computed by solving a non-relativistic Schrödingerequation with that potential, while the splitting within multiplets iscaused by a spin-dependent correction of order (v/c)2. This QCD-basedphenomenological description has been particularly efficient in describingthe observed charmonium states until 2003, when the Belle Collaborationobserved for the first time a narrow peak in the J/ψπ+π− invariant massspectrum at 3872 MeV/c2 [5]. Since the observation of the X(3872) by theBelle Collaboration, various other neutral exotic charmonium states have beenobserved by different collaborations, all of them decaying into a charmoniumstate (mostly J/ψ) or a DD∗ pair.Hints of three charged structures were also observed by Belle:Z(4430)−, Z1(4050)−, Z2(4250)−, between 2007 and 2008 [6, 7]. Thesemesons are particularly interesting since, being charged, their minimal quarkcontent is necessarily exotic: |ccdu〉. The Z(4430)− resonance has been

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Figure 1.1: Charmonium spectrum

recently confirmed by the LHCb Collaboration, demonstrating its tetraquarknature [8].

1.2 X(3872)

1.2.1 Experimental status

The X(3872) was first seen in 2003 by the Belle Collaboration at KEK, inJapan, as a narrow peak in the J/ψπ+π− invariant mass distribution inthe decay B+ → K+π+π−J/ψ. It is shown in Figure 1.2. After the Belleannouncement, it was observed also by the CDF [9] and D0 [10] Collaborationsat the Fermilab Tevatron in Chicago, and was confirmed also by the BaBarCollaboration at the SLAC National Accelerator Laboratory in California [11].Both BaBar and Belle observed the decay X(3872) → γJ/ψ, which indicatesthat the X(3872) has positive charge conjugation, C = +1 [12, 13]. Thissuggested that the dipion system from the decay X(3872)→ J/ψπ+π− comesfrom a ρ meson. This idea was indeed confirmed by CDF, with the implicationthat the X(3872) cannot be an excited charmonium state because that decaywould violate isospin conservation [14].Moreover, BaBar observed evidence for the decay X(3872)→ J/ψω at a rate

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Figure 1.2: Peak observed by Belle (2003)

comparable to that of J/ψππ [15]

BR(X(3872)→ J/ψω)

BR(X(3872)→ J/ψππ)= 0.8± 0.3 (1.2)

suggesting that X(3872) is a mixture of I = 0 and I = 1 states.An angular analysis of the final state particles, performed by CDF in thechannel X(3872) → J/ψπ+π− rules out all the JPC assignments except for1++ and 2−+, as can be seen in Figure 1.3 [16].

Figure 1.3: Result of the X(3872) particle angular analysisfrom CDF

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The BaBar analysis of the decay X(3872) → J/ψω favoured the JPC = 2−+

hypothesis with a confidence level of CL = 68% over the 1++ hypothesis,without ruling it out (CL = 7%). The LHCb Collaboration, however,determined in 2013 that the quantum numbers of X(3872) are 1++, with 8.2σrejection of the 2−+ hypothesis from the channel X(3872)→ J/ψππ [17].

(a) BaBar 3π mass distribution (b) LHCb X(3872) helicity angledistribution

Figure 1.4: Determination of X(3872) quantum numbers:BaBar analysis on the left, LHCb analysis on the right

Figure 1.5: Distribution of the test statistic t for the simulatedexperiments with 2−+ and 1++ at LHCb

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1.2.2 Theoretical models

Several models have been proposed to explain the nature of X(3872) [18, 19].The more plausible are:

• Excited charmonium state: the observed decay in J/ψ implies that theX(3872) must contain a cc pair. The cc state with JPC = 1++ is theχ′c1 (with spectroscopic term symbol 23P1) and has not been observedyet. The predicted mass is M(χ′c1) = 3950 MeV/c2. However, the decayX(3872) → J/ψρ violates isospin conservation in the hypothesis of apure charmonium state.

• Mesonic molecule [20]: the X(3872) mass is very close to the sum ofthe masses of the D0 and D∗0 mesons. This led to the speculation thatthe X(3872) could be a D0 − D∗0 loosely bound state, with very smallbinding energy (E0 ≈ 0.25 MeV), which implies a molecule radius ofabout R ≈ (2MDE0)−

12 = 8 fm. Such a large radius allows the molecule

constituents to decay autonomously and, since the D0 − D∗0 moleculewavefunction is expected to contain an admixture of J/ψρ and J/ψω, itcould explain the isospin violation.However, the small binding energy allows a small range of relative centreof mass momentum for the molecule to be formed in hadronic colliders.A simulation has been performed [21], in which the integrated promptproduction cross section is shown as a function of the relative momentumof the centre of mass: in the phase space region that would allow theformation of a D0− D∗0 molecule with a binding energy of E0, the upperbound on the cross section is found to be σthprompt = 0.085 nb, about40 times smaller than the lower bound on the CDF experimental crosssection (σCDFprompt = 3.1 nb). This result is shown in Figure 1.6.

Figure 1.6: Simulation of the prompt production cross sectionas a function of the relative centre of mass momentum of thesystemD0−D∗0 (the allowed phase space region is highlighted)

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• Tetraquark model [22]: this model considers 4 quark states of the form[cq][cq]. Due to the X(3872) mass, q = u, d. The two neutral flavoureigenstates are:

Xu = [cu][cu]

Xd = [cd][cd]

while the neutral mass eigenstates are described by a mixing angle:[Xlow

Xhigh

]=

[cos θ sin θ− sin θ cos θ

] [Xu

Xd

]with a mass difference of ∆M = (7 ± 2)/ cos(2θ) MeV/c. The mixingangle can be determined by the ratio of 3π to 2π decay rates and it’sfound to be θ ≈ 20 from Belle data. Summarizing, the tetraquark modelpredicts a neutral exotic partner for the X(3872) with a mass differenceof ∆M = 8±3 MeV, along with other partners according to the spin-spininteractions between diquarks (which can be scalar or vector).Moreover, only one of the two mass eigenstates (Xlow and Xhigh) issupposed to be produced in B+ decays, while the other is supposedto be produced in B0 decays. This means that the X(3872) seen inB+ decays and the one seen in B0 decays are different states with massdifference ∆M . Belle determined a mass difference between the X(3872)produced in charged versus neutral B decays of ∆M = 0.9 ± 0.9 MeV,which is consistent with zero. Furthermore, no other state in the [cq][cq]spectrum has been observed.

Figure 1.7: Spectrum of [cq][cq] states according to thetetraquark model

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Other models have been suggested (cusp close to M(D0D∗0) threshold,dynamically generated resonance, hybrid ccg resonance) but they presentnumerous gaps with experimental data and are regarded as very unlikelymodels with respect to the aforementioned ones.

1.2.3 The J/ψω decay channel

As previously specified, the decay channel X(3872) → J/ψω was first seenby BaBar in 2010, both in B+ and B0 decays in X(3872)K+ and X(3872)K0,respectively. The number of observed signal events is 21.1± 7.0 for B+ decayand 5.6±3.0 for B0 decay, leading to a combined signal consisting of 34.0±6.6events. The probability to be an upward background fluctuation is 3.6×10−5,corresponding to a significance of 4.0σ for a normal distribution. In figure1.8, the efficiency corrected M(J/ψω) distribution is shown for charged (top)and neutral (bottom) B decays, along with the result of the fit. A secondpeak, at about 3920 MeV/c2, can be observed and corresponds to the decayY(3940)→ J/ψω: the Y(3940) meson, subsequently called X(3915), has beenan exotic candidate until 2012, when BaBar measured its quantum numbersand identified it with the excited charmonium state χc0(2P) [23].

Figure 1.8: Corrected M(J/ψω) distribution for B+ (top) andB0 (bottom) decays

It must be noticed that the decay X(3872)→ J/ψω is subthreshold, sinceM(X(3872)) < M(J/ψ) + M(ω). The mass difference is 9 MeV/c2, and this

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implies that only a low mass ω contributes to this decay.The BaBar analysis result for the charged B decay channel is:

BR(X(3872)→ J/ψω)× BR(B+ → X(3872)K+) = (6± 2± 1)× 10−6

which, using the PDG limit of BR(B+ → X(3872)K+) [42], gives the followinglimit:

BR(X(3872)→ J/ψω) > 0.019

As shown in Section 1.2.1, BaBar used this decay channel to measure theX(3872) quantum numbers. The assignment JPC = 2−+ is found to befavoured by the data. LHCb, two years later, measured JPC = 1++ in theJ/ψππ channel with a 8σ significance.As already stated, the final states J/ψρ and J/ψω have different isospin(respectively, I = 1 and I = 0). The ratio (1.2) implies a mixture of states withdifferent isospin: it is possible, then, that the particle decaying in J/ψρ may bedifferent from the particle decaying in J/ψω. A measurement of this channelis therefore required, in order to search for the isospin partner predicted bythe tetraquark model, to obtain a better measurement on the branching ratioand possibly to repeat the angular analysis already performed on the J/ψππchannel, to verify whether the two aforementioned decays come from the sameexotic state or not.

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Chapter 2

The LHCb experiment at CERN

LHCb is, along with ATLAS, CMS and ALICE, one of the main experimentsat CERN (Conseil Européen pour la Recherche Nucléaire), in Geneva. Thepurpose of the experiment is to analyse data from proton-proton collisionsinside the LHC accelerator (Large Hadron Collider), focusing on the decaysof heavy mesons (composed by charm and bottom quarks), to explore thedifference between matter and antimatter and to search for rare decays andrare particles. LHCb is located at the Intersection Point 8 (IP8), one of thefour points along the LHC circumference where protons collide; from 1989 to2000, IP8 was the location of the DELPHI experiment.

2.1 Large Hadron ColliderThe particle accelerator LHC [24] is situated inside a 26.7 km circumferencetunnel (built from 1983 to 1988 and formerly used for LEP - Large ElectronPositron collider) at a depth of about 100 m underground, between Switzerlandand France, near the city of Geneva. It’s the most powerful particle acceleratorever built, operating at a centre of mass energy of 7 TeV from 2010 to 2011, 8TeV during 2012 and 13 TeV from 2015, when the machine will be restartedafter an almost two years period of stop (Long Shutdown 1). The acceleratorcomplex at CERN, shown in Figure 2.1, made by 4 smaller accelerators, injectsthe proton flux into the LHC pipes after a stepwise preliminary accelerationup to 450 GeV. LHC is composed of two rings (beam pipes) where the protonbunches travel in opposite directions in ultra-high vacuum. The accelerationis given by a system of superconducting magnets placed alongside the rings,with an operating temperature of -271.25 C (1.9 K). The protons are thenforced to collide in four points (IP1 - ATLAS, IP2 - ALICE, IP5 - CMS, IP8 -LHCb) with a crossing rate of 40 MHz, that is 25 ns between two consecutivecollisions.

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The nominal peak luminosity is L = 1034 cm−2 s−1; the LHCb experiment wasdesigned to work at about 1/50 of such luminosity, to avoid misreconstructionof secondary vertices, even if during operation this experiment has proved tobe able to run at luminosity as high as 1033 cm−2 s−1.

Figure 2.1: The CERN accelerator complex

2.2 LHCb detectorThe LHCb detector is a single-arm spectrometer with an angular acceptancefrom ≈10 mrad to 300 mrad in the bending plane and to 250 mrad in thenon-bending plane [25, 26]. Its geometry is very different from the “barrel plusendcaps” design which characterizes the other large detectors at CERN, andit is due to the fact that at high energy the b-hadrons (and b-hadrons) areproduced mainly in the forward region. In Figure 2.2 the distribution of thepolar angles of b- and b-hadrons is shown: they are mainly produced at verysmall angles with respect to the beam direction and in the same hemisphere.The LHCb detector is composed by several subdetectors: from the nearestwith respect to the interaction point (vertex locator) to the furthest (muondetector). Each subdetector has a unique design and is optimized to measure

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Figure 2.2: Polar angles of the b- and b-hadrons calculatedwith PYTHIA

a different physical quantity. In Figure 2.3 the geometry of the detector andthe position of each subdetector are shown.

Figure 2.3: LHCb detector, lateral view (non-bending plane)

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2.2.1 Vertex Locator (VELO)

The first subdetector, starting from the interaction point, is the Vertex Locator(VELO) [27]. Displaced secondary vertices are a distinctive feature of b-hadrondecays, and a precise measurement of track coordinates close to the interactionpoint is a fundamental requirement for the LHCb experiment. The VELOsystem consists in 25 silicon stations placed along the beam direction, eachone composed by one left module and one right module. Each module iscomposed by two measuring sensors: a radial one (R sensor) and an angularone (Φ sensor). The azimuthal coverage is about 182 for each sensor, givinga small overlap between the right and left modules in order to simplify therelative alignment and guarantee a full azimuthal acceptance. The VELOsystem configuration and a schematic view of the sensors are shown in Figure2.4.The VELO system is able to measure tracks in the full LHCb angularacceptance; in addition to that, the backwards hemisphere is also partlycovered (in order to enhance the resolution on the primary vertex) and thetwo most upstream modules are used as a pile-up veto counter for the L0trigger.The errors on the primary vertex arises mainly from the number of tracksproduced in a pp-collision. For an average event, the resolution in thez-direction is 42 µm and 10 µm perpendicular to the beam. The resolutionon the decay length ranges from 220 µm to 370 µm, depending on the decaychannel.

Figure 2.4: VELO system configuration and VELO sensors

2.2.2 Ring-Imaging Cherenkov Detectors (RICH)

Hadron identification in LHCb is achieved with a high performanceRing-Imaging Cherenkov system, composed by two detectors aiming atdifferent momentum ranges [28]. RICH1, located upstream of the magnet,identifies low momentum particles (from 1 GeV/c up to about 60 GeV/c)combining silica aerogel and C4F10 gas radiators with a polar angle acceptance

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from 25 to 300 mrad. RICH2, located downstream of the magnet and thetracking stations, has a more limited angular acceptance (from 15 to 120 mradin the horizontal plane and from 15 to 100 mrad in the vertical plane); it coversthe high momentum range, from about 15 GeV/c up to about 100 GeV/c, usinga CF4 radiator. Both RICH1 and RICH2 layouts are shown in Figure 2.5.Cherenkov light is focused onto the photon detector planes using tiltedspherical mirrors and secondary plane mirrors, in order to reflect the image outof the spectrometer acceptance. The baseline photon detectors are multianodephotomultiplier tubes (MaPMT). The anodes are arranged in an 8 × 8 arrayof pixels, each 2 mm × 2 mm, separated by 0.3 mm gaps.

(a) RICH1 (b) RICH2

Figure 2.5: RICH detectors layout

2.2.3 Tracking System

The tracking system, consisting of 4 stations (TT, T1, T2, T3) between thevertex detector and the calorimeters, provides efficient reconstruction andprecise momentum measurement of charged tracks, track directions for ringreconstruction in the RICH and information for L1 and higher level triggers[29, 30]. Each tracking station (except for TT) is composed by an InnerTracker (IT), located in an elliptical shaped region around the beam pipe, andan Outer Tracker (OT) which covers most of the acceptance. In the regioncovered by the IT the particle flux is about 20 times bigger than the flux inthe OT region, so IT requires a finer granularity and a different technology tohandle a greater flux.The first station, the Trigger Tracker (TT) is located between RICH1 and the

21

magnet. Due to its reduced dimensions it contains only IT modules. Theother stations are located between the magnet and RICH2.The silicon tracker system (composed by TT and IT) uses about 270000 siliconmicrostrips detectors with a strip pitch of 198 µm for the IT and 183 µm forthe TT. To improve track reconstruction, the detectors are composed by fourlayers arranged in an x-u-v-x geometry, in which the strips are vertical in thefirst and in the last layer, whereas the other two (u,v) layers are rotated bystereo angles of ±5C, providing the sensitivity in the vertical direction. InFigure 2.6 the IT (around the beam pipe) is shown, and in Figure 2.7 the TTis shown.The Outer Tracker is composed by an array of straw tubes modules. Eachmodule consists of two panels and two sidewalls, which form a mechanicallystable and gas-tight box, and contains up to 256 straw tubes, with an innerdiameter of 4.9 mm, filled with a mixture of argon (70%) and carbon dioxide(30%) gas, which guarantees a fast drift time and a sufficient drift-coordinateresolution (200 µm). Like the silicon tracker, also OT modules are composedby four layers arranged in x-u-v-x geometry. The whole tracking system layoutis shown in Figure 2.8.

Figure 2.6: Inner Tracker layout

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Figure 2.7: Trigger Tracker layout

Figure 2.8: Tracking system layout

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2.2.4 Magnet

The LHCb experiment utilizes a dipole warm magnet to bend the tracks ofcharged particles in order to measure their momentum with a good resolution[31]. The magnet consists of two trapezoidal coils bent at 45C on the twotransverse sides and placed mirror-symmetrically. The bending power, givenby the integrated magnetic field, is 4 Tm, enough to measure momenta ofcharged particles up to 200 GeV/c within 0.5% uncertainty. A perspectiveview of the LHCb magnet is given in Figure 2.9.

Figure 2.9: LHCb magnet, perspective view

2.2.5 ECAL and HCAL

The LHCb calorimeter system [32] is utilized for the identification of hightransverse energy hadrons, electrons and photons candidates. It measurestheir energy and position and selects candidates for the Level-0 trigger.The structure consists of a single-layer preshower detector (PS), made by14 mm thick lead plates and 10 mm square scintillators, followed by anelectromagnetic calorimeter (ECAL) and a hadronic calorimeter (HCAL). Ascintillator pad detector (SPD) is located before the PS.The ECAL submodule is constructed from 70 layers, consisting of 2 mm thicklead plates and 4 mm thick polystyrene-based scintillator plates. The lengthcorresponds to 25 X0. The electromagnetic calorimeter is designed to giveadequate granularity and energy resolution for π0 reconstruction: for Pt < 2GeV/c neutral pions are mainly reconstructed combining two separate clustersin the ECAL (resolved π0), while for greater transverse momenta the twophotons form a single cluster (merged π0). The resolved π0 efficiency (definedas the number of resolved neutral pions identified in a mass window of ±30MeV/c2 over the number of resolved and merged π0 in acceptance) for thechannel B0 → π+π−π0 is found to be between 40-48% for Pt < 2 GeV/c

24

and drops linearly for greater transverse momenta, reaching ≈ 5% for Pt = 6GeV/c, as can be observed in Figure 2.11 [33].The HCAL consists of 16 mm thin iron plates inter spaced with 4 mm thickscintillating tiles arranged parallel to the beam pipe. The length correspondsto 5.6 λI . Figure 2.10 shows the segmentation of the sections of both ECALand HCAL.

Figure 2.10: Calorimeters segmentation: ECAL on the left,HCAL on the right.

Figure 2.11: Resolved π0 reconstruction efficiency for thechannel B0 → π+π−π0

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2.2.6 Muon system

The muon system for the LHCb experiment [34] consists of five trackingstations placed along the beam axis, the first (M1) in front of the calorimetersand the other four, inter spaced with three iron filters, downstream thecalorimeters. The configuration of the stations is shown in Figure 2.12.The muon stations are equipped with Multi Wire Proportional Chambers(MWPCs) except for the inner region of M1, equipped with Gas ElectronMultiplier (GEM) chambers, more appropriate for the greater particle flux.The efficiency for each chamber is required to be high enough to achieve a95% trigger efficiency on the trigger algorithm, which requires a quintuplecoincidence in a time window smaller than 25 ns. The inner and outer angularacceptances of the muon system are 20 (16) mrad and 306 (258) mrad in thebending (non-bending) plane, similar to that of the tracking system. Thisprovides a geometrical acceptance of about 20% for muons from b decaysrelative to the full solid angle. Each station is subdivided in four regions withdimensions and logical pad size which scales a factor of two from one regionto the next one. Since the muons energy decreases with the distance fromthe beam axis, the multiple scattering effect in the absorber increases in thesame direction, limiting the spatial resolution of the detector, therefore thegranularity of the detector varies accordingly.

Figure 2.12: Muon system configuration

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2.2.7 Trigger

The LHCb experiment was designed to operate at a nominal averageluminosity of 2× 1032 cm−2s−1, about 1/50 of the nominal peak luminosity ofLHC: this implies a smaller number of interaction per bunch crossing and abetter vertex reconstruction, a fundamental requirement for B-physics. Therate of useful p-p interaction is 10 MHz, and needs to be reduced at about 2kHz in order to be stored and analysed offline: in order to achieve that, twotrigger selections are used [35]. The Level-0 trigger (L0) is embedded in theelectronics and reduces the event rate at 1 MHz combining informations fromVELO, ECAL, HCAL, PS, SPD and the muon system. L0 allows to discardmost of the events that are not interesting for the physical analysis.The High Level Trigger (HLT) is software-based and reduces the event rate at2 kHz, with full event reconstruction based on particle identification, tracksmeasurement, vertex reconstruction and impact parameter measurement.

2.3 Analysis tools

2.3.1 Stripping and reconstruction

Data selected by the trigger (both L0 and HLT) are stored and analysedoffline by a first set of algorithms to provide a pre-categorization of candidatesaccording to each specific analysis need. This procedure combines allinformations from all subdetectors, identifies particles and associates themwith tracks, vertices (built from the crossing point of two or more tracks) andenergy deposits.At this stage, events are flagged according to the category they belong to,checking the presence of particular features: for example, the DiMuon streamincludes all the events with at least a couple of high transverse momentummuons, with opposite charges. This procedure is called “Stripping”.Each of these streams contains several StrippingLines, a loose selection(with standard requirements) of typical decays, which allows to search fora particular decay channel in a subset of the whole data. An example: it’spointless to search for the decay B+ → K+ω(J/ψ → µ+µ−) in events thatdon’t contain at least a couple of muons with opposite charge, from the samevertex and with an invariant mass not very far from the J/ψ nominal mass. Astandard set of minimal cuts is therefore defined for each StrippingLine. Thegroup of C++ LHCb libraries used to reconstruct tracks, to apply calibrationsand finally provide well identified and momentum measured particles is calledBrunel [36]. The software tool used to reconstruct the relevant physicalprocesses, i.e. typically full b or c decay chains, is called DaVinci [37]. DaVinciis also used for the complete event reconstruction and the selection peculiarto the specific analysis under development.

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2.3.2 Monte Carlo simulations

Simulated datasets can be generated with the Monte Carlo (MC) methods toextract certain parameters useful for the analysis, to have a prior knowledgeof the trend of interesting physical quantities and to train the tools that arelater used on data. MC data can be obtained with the simulation applicationGauss [38]. It generates a physical process of interest through the PYTHIA[39] generator package, which takes into account the physics inherent thep-p interaction and all the known theoretical models in order to simulatecorrectly the collision, the hadronization process and the consequent B decaysof interest. The detector response is then simulated with the Geant4 [40]package, taking into account a detailed description of the detector geometryand the response of each subdetector. All the LHCb software operates withinthe Gaudi framework [41].

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Chapter 3

Data Analysis

In this chapter detailed descriptions of the B+ → K+ωJ/ψ decay channelreconstruction and selection are shown.The analysis strategy consists of the following sequential processing steps,taking as input data selected by the DiMuonJpsi2MuMuDetached stripping line,described in detail in Section 3.2.1:

• First of all, a reconstruction of the decay chain is performed, combiningall reconstructed particles in order to list all the possible signalcandidates for each event with specific requirements. Cuts in this phaseare very loose. This first step is required to extract all the possiblecandidates of interest in the whole stripped dataset, and it is describedin Section 3.2.2.

• Once the events are reconstructed, a preselection is performed in orderto remove a large fraction of obvious sources of background, accordingto LHCb fiducial cuts, which are known to be highly efficient on signalevents and assure a significant background suppression. The preselectionphase is described in Section 3.2.3.

• The preselected events are then processed through MultivariateAnalysis algorithms (MVA), which optimally combine signal-backgrounddiscriminating information and allow to perform a final selection in orderto enhance the signal to background ratio. This process is described inSection 3.4.

• The obtained B+ → K+(J/ψ → µ+µ−)(ω → π+π−π0) mass spectrum isnow analyzed. In order to eliminate the residual background, it is fittedwith a Probability Density Function (PDF) that takes into account thesignal and the background contribution, in order to obtain the functionalform for each component. With these as input, the sPlot techniqueis applied: it consists of a statistical background subtraction throughcomputation of proper weights. In this way, signal and background

29

distributions for J/ψω mass, which is the variable of interest for thisanalysis, are obtained separately. The B+ mass fit and the applicationof the sPlot technique are described, respectively, in Section 3.5 andSection 3.6.

• To correct the obtained J/ψω mass spectrum for the effect of the analysiscuts, a Monte Carlo sample is generated and analysed with the same setof cuts as real data. From this sample, the efficiency curve as a functionof M(J/ψω) is obtained and applied to real data. The same MC sampleis also used to train the aforementioned MVA. The efficiency correctioncomputation is detailed in Section 3.10.

3.1 Data samplesIn this analysis the full 2011+2012 data sample is used, corresponding to anintegrated luminosity of about 1 fb−1 (2011) + 2 fb−1 (2012) of p-p collisiondata, recorded with the LHCb detector at a centre of mass energy of 7 TeVfor 2011 data and 8 TeV for 2012 data.A fully simulated Monte Carlo data sample is produced, consisting of 2million events for the 2011 data taking conditions and 4 million events forthe 2012 data taking conditions. A Phase Space (PHSP) model is appliedto the simulation, in order to obtain a uniform phase space coverage for theefficiencies study. The complete decay chain is:

B+ → K+(J/ψ → µ+µ−)(ω → π+π−(π0 → γγ)) (3.1)

and the complex conjugate.Detailed informations about the MC dataset will be given in section 3.3.

3.2 Event selection

3.2.1 Stripping cuts

The StrippingLine used is the so called StrippingFullDSTDiMuonJpsi2MuMuDetachedLine, which attempts to select J/ψ → µ+µ− candidates, comingfrom a B decay and thus forming a vertex detached from the primary vertex,combining two muon tracks with opposite charge and moderate transversemomentum.Some standard cuts are applied. First of all, in order to obtain a good muontrack, a cut on the track χ2/Ndof is applied, where Ndof is the number ofdegree of freedom of the track fit. The particle identification algorithm mustassert that the selected track is indeed associated to a muon: this implies a cuton the difference of the logarithm of the likelihood fit of the muon hypothesis

30

with respect to the pion hypothesis (Delta Log-Likelihood, DLLµ−π). The twomuon tracks need to originate from the same vertex, and this is obtained witha cut on the J/ψ Decay Vertex (DV) χ2/Ndof .Since the B+ travels for a measurable distance before decaying, the J/ψ isrequired to be detached from the Primary Vertex (PV): a cut on the DecayLenght Significance (DLS) is performed, which is defined as the distancebetween the reconstructed µ+µ− vertex and the PV, divided by its error.When multiple PVs are present in a single p-p collision, only the best PV (interms of reconstruction quality) is considered for this computation.Finally, it is required to the invariant mass of the two muons not to be outside a200 MeV/c2 mass window centred at the J/ψ nominal mass (from the ParticleData Group [42]). A summary of the stripping cuts is provided in Tab 3.1.

Candidate Variable Cut

µ

Pt >550 MeV/cDLLµπ >0.0

Track χ2/Ndof <5.0

J/ψ

DV χ2/Ndof <20DLS >3.0M(µµ) ∈ [2996.916 , 3196.916] MeV/c2

Table 3.1: Stripping line selection cuts

3.2.2 Decay reconstruction

Each J/ψ candidate in the stripped dataset is combined with two pions withopposite charge, a neutral pion and a charged kaon. Reconstructed pions(both charged and neutrals), kaons and muons are extracted from the LHCbStandardParticles repository: it contains particle candidates reconstructedwith standard loose cuts.The standard algorithms used in the reconstruction are, in particular,StdLoosePions, StdLooseKaons, StdLooseMuons, StdLooseResolvedPi0:

• In StdLoosePions, StdLooseKaons and StdLooseMuons a cut on thetransverse momentum of reconstructed particle is applied (Pt > 200MeV/c), along with a cut on the χ2 of the fit on the charged track ImpactParameter (IP) in order to remove a significant fraction of backgroundcomposed by prompt particles (χ2(IP ) > 4.0). Notice that the strippingcut on muons Pt is higher.

31

• In StdLooseResolvedPi0 a photon pair with Pt(γ) > 200 MeV/c andM(γγ)∈ [85, 185] MeV/c2 is combined to form a π0. Each photon pair isrequired to be resolved, i.e. to form two well separated clusters in ECAL(the distance between two impact points must be greater than one cellsize).

An additional cut is imposed on the track fit χ2 of pions to improve theirreconstruction. They are then combined to reconstruct an ω: a cut on theirinvariant mass is applied (M(π+π−π0)∈ [390, 1000] MeV/c2) along with a cuton the χ2 of the vertex fit. The mass window is large enough to include alsothe decay B+ → J/ψK+(η → π+π−π0).The stripping selection of J/ψ candidates is improved with a tighter cut onthe mass (it’s also asymmetrical, to account for the radiative tail) and a cuton the transverse momentum.To reconstruct a B+ candidate, J/ψ, ω and K+ candidates are combined,requiring to have an invariant mass in a 1000 MeV/c2 window centred inthe B+ nominal mass and to form a vertex with a good χ2. In addition toJ/ψ detachment and K-π non-zero IP requirements, a further restriction isdirectly applied on each B+: candidates with measured lifetime less than 0.20ps are discarded, in order to reduce the large combinatorial background fromparticles produced in primary p-p interactions. Finally, a cut on B+ transversemomentum is applied. A summary of the reconstruction cuts is provided inTab 3.2.During this phase, the J/ψ mass is constrained to its nominal value for theevents that pass the reconstruction cuts: since the J/ψ decaying in two muonshas a very low misidentification rate, its width can be removed, thus improvingthe B+ mass resolution.

3.2.3 Preselection cuts

After the reconstruction algorithm, a cut based selection is performed in orderto obtain a significant background rejection retaining a large fraction of signal.

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Candidate Variable Cut

B+

Pt > 1.5 GeV/cDV χ2/Ndof < 15.0

τ > 0.20 psM(KωJ/ψ) |MB −MPDG

B | < 500 MeV/c2

J/ψ

Pt > 1.5 GeV/c2

DV χ2/Ndof < 20M(µµ) ∈ [3040 , 3140] MeV/c2

ω

Pt > 1.0 GeV/c2

DV χ2/Ndof < 20M(π+π−π0) ∈ [390 , 1000] MeV/c2

π±Pt > 200 MeV/c2

IP χ2/Ndof > 4.0Track χ2/Ndof < 4.0

K+ Pt > 200 MeV/c2

IP χ2/Ndof > 4.0

π0 Pt > 200 MeV/c2

M(γγ) ∈ [85 , 185] MeV/c2

γ Pt > 200 MeV/c2

Table 3.2: Reconstruction cuts. The last 4 rows include the StandardParticlescuts.

The cuts are LHCb fiducial cuts, known to be highly efficient on signal events,and they are reported in Table 3.3. In most cases they are simply a hardeningof the reconstruction cuts, but other variables are also considered.DDLKπ is the Delta Log-Likelihood of the kaon hypothesis with respect to thepion hypothesis: it’s a variable provided by the LHCb identification system,it’s positive for a well reconstructed pion and negative for a well reconstructedkaon, and a proper cut reduces the probability of K − π misidentification.A ghost track is defined as a pseudo-random combination of hits, and it’scharacterized by having a low probability of χ2 from the track fit and missinghits. A ghost track therefore is not associated to any real track and it’s merelya result of hit combination mistakes. A specialized algorithm determines aprobability for each track to be a ghost.

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Candidate Variable Cut

B+

Pt > 2.0 GeV/cDV χ2/Ndof < 9.0

τ > 0.25 ps

J/ψ

Pt > 2.0 GeV/cDV χ2/Ndof < 20

M(µµ) ∈ [3040 , 3140] MeV/c2

ω

Pt > 2.0 GeV/cDV χ2/Ndof < 10M(π+π−π0) ∈ [390 , 1000] MeV/c2

µ±

Pt > 1.0 GeV/cIP χ2/Ndof > 9.0

Track χ2/Ndof < 4.0DLLµπ > 0.0

π±

Pt > 200 MeV/cIP χ2/Ndof > 9.0

Track χ2/Ndof < 4.0DLLKπ < 0.0

K+

Pt > 200 MeV/cIP χ2/Ndof > 9.0

Track χ2/Ndof < 4.0DLLKπ > 0.0

π0

Pt > 200 MeV/cM(π0) ∈ [85 , 185] MeV/c2

CL > 0.01All tracks Track Ghost Prob. < 0.4

Table 3.3: Preselection cuts

The Confidence Level (CL) cut assures a good quality particle ID of thephotons. The variable ranges from 0 to 1 and is computed from informationabout the cluster size, the shower shape and the energy deposit. CL on π0 is

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the product of the two photons CLs.The cut on ω mass has not been touched, in order to be able to see also the ηcontribution.The signal efficiency of the preselection cuts set, computed from the fullreconstructed Monte Carlo dataset, is found to be 37.0%.In Figure 3.1 the (K+J/ψω) mass spectrum, after the application of thepreselection cuts, is shown. A peak is visible in correspondence of the B+

nominal mass: in order to enhance the signal to background ratio, i.e. toremove a significant fraction of the background maintaining a high number ofsignal events, the dataset will be processed through a multivariate analysis, aswill be shown in Section 3.4

Figure 3.1: B mass spectrum after the application ofpreselection cuts

3.3 Monte Carlo sampleA Monte Carlo dataset is necessary in order to obtain simulated informationabout different aspects of the analysis, such as signal efficiency of the selectionprocedure, signal distribution for different variables and so on. The sampleused in this analysis is composed by 6 million events (2 million for 2011 and 4million for 2012 data taking conditions).As already stated in section 3.1, the decay (3.1) has been generated followingthe PHSP (PHase SPace) model, according to which each B+ particle decaysin a three body state (J/ψK+ω) without any intermediate resonance. As anexample of the flat shape of the phase space model, the Dalitz plot for theJ/ψω and K+ω systems invariant squared masses is shown in Figure 3.2. Agenerator level dataset has been used to produce this plot, thus before any

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analysis cut.

Figure 3.2: Dalitz plot for K+ω squared mass vs J/ψω squaredmass, generator level (MC)

A set of acceptance and kinematic cuts is applied at generator level to produceonly events that can be triggered and reconstructed by the reconstructionsoftware, before submitting the simulated dataset to the full Geant4 detectorsimulation. These cuts avoid waste of computational resources to simulatetracks which do not have any chance to be reconstructed, because theyare either outside the LHCb geometrical acceptance or below the minimummomentum threshold required for efficient reconstruction.In Tab. 3.4 and Tab. 3.5 a summary of the acceptance and kinematic cuts isprovided: y = − log

[tan( θ

2)]is the pseudorapidity, while θ is the polar angle.

The generated sample is then submitted to the same reconstruction chainas real data, and for each reconstructed event also true generated quantitiesare available for efficiency studies. Like in real data, also in the MC datasetthe J/ψ mass is constrained to its nominal value during the reconstructionprocedure.The following plots are generated from the MC sample after the reconstructionand preselection cuts. In Figure 3.3 the B+ mass spectrum in shown, in black:an asymmetric tail is visible on the right side of the spectrum. This can bepartly explained by the tail for high masses on the π0 mass distribution thatcan be observed in Figure 3.4, while the correlation plot is shown in Figure3.5. The tail on π0 mass distribution is due to the highly energetic neutral

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Candidate Variable Cut

J/ψ y ∈ (1.8, 4.5)

µ, π±, K+ θ ∈ (0.005, 0.400) rad

γ|PxPz| < 4.5

12.5

|PyPz| < 3.5

12.5

Pz > 0 MeV/c

Table 3.4: Acceptance cuts at generator level

Candidate Variable Cut

J/ψ Pt > 1000 MeV/cπ±, K+ Pt > 150 MeV/c

µPt > 500 MeV/cP > 6000 MeV/c

γ Pt > 150 MeV/cB+ τ > 0.0001 ns

Table 3.5: Kinematic cuts at generator level

pions decaying in two photons with a very small angle, which may releasepart of their energy in the same calorimeter cells. The high mass tail is anindication that the correction to take into account this superposition is notproperly done. This is a known problem and a new set of algorithms andvariables are being developed to overcome it. Note that this behaviour in the π0 mass distribution is observable in real data, as will beseen in Figure 3.20.In order to reduce this effect, a new variable for the B+ mass is defined asfollows:

M(B+)π0 fixed = M(B+)−M(π0) + MPDG(π0) (3.2)

where M(π0)PDG is the nominal π0 mass taken from PDG. The B+ massdistribution after the neutral pion mass fixing is shown, in red, in Figure 3.3,

37

where the distribution looks narrower and more symmetric. A residual of thetail is still present, even after the fixing procedure.By fitting the B+ mass spectrum with a Gaussian function, it can be measuredthat the width of the B+ mass peak improves from σMC = 31.4 MeV/c2 toσMC = 23.5 MeV/c2 when the fixing procedure in applied.In Figure 3.6 the three pions invariant mass spectrum is shown, and, asexpected, the same tail on the right side of the distribution can be observed.Figure 3.7 shows the Dalitz plot for (K+ω) invariant squared mass vs (J/ψω)invariant squared mass. Since all these plots are generated from a simulateddataset that has been submitted to the reconstruction and preselection chain,as described in Sections 3.2.2 and 3.2.3, the typical flat shape of the phasespace model, see Figure 3.2, cannot be observed any more.

Figure 3.3: B+ mass spectrum (MC), before and after fixingthe π0 mass

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Figure 3.4: π0 mass spectrum (MC)

Figure 3.5: B+ mass vs π0 mass (MC)

3.4 Multivariate analysisAfter the reconstruction and the cut based selection, the dataset is ready tobe processed through multivariate classification methods: they are algorithmswhich combine information from a set of input variables in order to separatethe signal from the background components. These techniques need to be

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Figure 3.6: ω mass (MC)

Figure 3.7: Dalitz plot for K+ω squared mass vs J/ψω squaredmass, after the reconstruction and preselection phases (MC)

trained on samples of signal events and background events. For the signal,MC events are used, while for the background real data in B+ mass sidebands(|M −MPDG

B+ | > ±150 MeV/c2) are taken. This preliminary training phasedetermines the mapping function that will later classify events in real data.

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The classification is based on an output variable, computed for each event,in which signal and background distributions must be well separated. Thechosen method is the Projective Likelihood Estimator (PLE): the method ofmaximum likelihood consists of building a model out of probability densityfunctions (PDF) that reproduces the input discriminating variables for signaland background. For each event, the likelihood for being of signal type iscomputed as a product of all signal PDFs evaluated at the measured variablevalue, and conveniently normalized. Since a factorization of PDFs will takeplace, this method is only valid with the assumption of uncorrelated (or looselycorrelated) variables.For an event i, a Likelihood ratio rL(i) is defined as:

rL(i) =LS(i)

LS(i) + LB(i)(3.3)

where:

LS(i) =Nvar∏k=1

pkS(xk(i)) (3.4)

LB(i) =Nvar∏k=1

pkB(xk(i)) (3.5)

where pkS and pkB are the signal and background PDFs for the kth input variablexk, and Nvar is the number of input variables. Each PDF is normalized as:∫ ∞

−∞pkS,B(xk)dxk = 1 (3.6)

for ∀k ∈ [1,Nvar].Since the analytic form of the PDFs is generally unknown, this methodempirically approximates the PDF shapes from the training data, buildinga model for each variable as a result of an interpolation of polynomialspline functions of second degree to the signal and background distributionhistograms. Then, for each event, rL(i) is computed at the measured variablevalue.It can be shown that in absence of model inaccuracies, i.e. if the inputvariables present no correlations and the PDFs are built correctly from theinterpolations, the Likelihood ratio provides the best signal from backgroundseparation. This, along with the computational speed of both the training andthe application phases, are the reasons of the choice of this method.The selection is accomplished using the Toolkit for Multivariate Data Analysis(TMVA) [43], provided by the ROOT framework [44].

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3.4.1 Input variables and training phase

The choice of the input variables for a multivariate analysis must be dictatedby two main conditions: it is compulsory to find a set of uncorrelated variables(the more they are correlated, the worse the performances will be) which canguarantee a good discrimination between signal and background distributions.The variables chosen for this analysis are the following (all the logarithms aretaken to smooth the distributions):

• log(minPV [χ2IP (B+)]): the logarithm of the minimum reduced χ2 of

the B+ Impact Parameter with respect to all Primary Vertices in anevent. A signal B+ is expected to come from a PV, and so the IP (i.e.the perpendicular distance between the B+ reconstructed momentumvector and the PV) should be close to zero, with a good χ2. The signaldistribution for this variable presents indeed a peak at zero, far belowthe background distribution peak.

• log(minPV [χ2IP (K+), χ2

IP (π+), χ2IP (π−)]): the logarithm of the minimum

IP reduced χ2 with respect to all PVs and with respect to all chargedhadrons in the final state. Since K and π must come from a detachedvertex, their IP χ2 is peaked at a far greater value than zero, while thebackground is composed mainly of prompt particles with low IP χ2.

• min[cos(θxyJ/ψ,K), cos(θxyJ/ψ,π+), cos(θxyJ/ψ,π−)]: the minimum of the cosine ofthe angle between the J/ψ momentum and the momentum of one ofthe hadrons in the final state, in the transverse (xy) plane. Particlesdecaying from a B are supposed to have a smaller angle between theirtransverse momenta with respect to prompt particles, since the latterlack of a transverse component from the pp interaction.

• Vertex χ2/Ndof (B+): the reduced χ2 of the decay vertex fit of the B+.This variable has a signal peak at 1, corresponding to good reconstructedvertices from the charged tracks of muons, pions and the kaon.

• Pt(π0): the π0 transverse momentum. Background π0s are characterizedby low Pt, while signal ones present a peak at over 1 GeV/c and a longerdistribution tail towards high transverse momenta.

Figure 3.8 shows all the input variables for signal and backgrounddistributions. In Figure 3.9 the correlation matrices for signal and backgroundevents are shown: they are negligible for the signal sample and minor forthe background sample, so that the uncorrelated variables hypothesis canreasonably be considered satisfied. The variables are nevertheless linearlydecorrelated before the beginning of the training, diagonalizing the correlationmatrix and creating another set of variable with no real physical meaning, butwith the same per event method response. The performance of the MVA,

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computed on a MC sample independent with respect to the one used forthe training phase, is represented, in Figure 3.10, by the Receiver OperatingCharacteristic (ROC) curve, where the background rejection (1 - backgroundefficiency) is plotted as a function of signal efficiency, for three different MVAmethods (Fisher discriminants, Boosted Decision Tree and Likelihood). Sincethe performance are very similar, the Likelihood method was chosen because,as already stated, it can be shown that, in the hypothesis of uncorrelatedvariables, this method provides optimal signal to background separation. InFigure 3.11 the Likelihood Ratio distribution for both signal and backgroundis shown.

Figure 3.8: Input variables, with signal and backgrounddistributions

43

(a) Signal sample (b) Background sample

Figure 3.9: Correlation matrices for signal and backgroundsamples

Figure 3.10: MVA methods performance (ROC curve)

3.5 B+ mass fitAs an example of how MVA works, in Figure 3.12 the B+ mass spectrum isshown, corresponding to a cut on rL > 0.65, which is one of the cuts thatwill be used later in the analysis. The comparison of this figure with Figure3.1 shows the remarkable discriminating power of this technique: the twodistributions are directly compared in Figure 3.13. In principle the cut valuefor rL could be determined by choosing a point on the ROC curve of Figure3.10; however, since real data may not be well reproduced by MC events, we

44

Figure 3.11: PLE method response

prefer to choose the optimal cut value by estimating the number of signal andbackground events entirely on real data. This is achieved by fitting the B+

mass distribution, in order to extract the number of signal and backgroundevents as a function of the cut on rL.

Figure 3.12: B+ mass spectrum, after the cut on rL

The unbinned, extended likelihood fit is performed with the RooFit Toolkitfor Data Modeling [45]. The total PDF of the fit was firstly chosen to be asum of a double gaussian for the signal and an exponential function for thecombinatorial background. The result is shown in Figure 3.14. As clearly

45

Figure 3.13: B+ mass spectrum, before and after the cut on rL

visible, the fit does not describe well the B+ mass peak as it seems that acomponent at around 5350 MeV/c2 (about 70 MeV/c2 higher than the B+

nominal mass) prevents the fit from following closely the B mass lineshape.Different hypothesis have been tested to explain this excess of events, listedand described in section 3.9, without reaching any definitive conclusion, and itwas chosen to include it in the fit, describing it with another gaussian function.The total PDF, therefore, is defined as:

pTOT (x) = C0[Nsigpsig(x) + Nbkgpbkg(x)] =

C0[Nsig(CS1 G

S1 (x) + CS

2 GS2 (x)) + Nbkg(C

B1 G

B(x) + CB2 E

B(x))](3.7)

where:

GSi (x) =

1

σSi√

2πe−

(x−MB+)2

2(σSi)2 (3.8)

GB(x) =1

σB√

2πe−

(x−Mbump)2

2(σB)2 (3.9)

EB(x) = eτx (3.10)

where σS,Bi are the widths of the gaussians, MB+ is the mean value of boththe signal gaussians which has been fixed to the B+ nominal mass, Mbump isthe mean value of the background excess at arount 5350 MeV/c2, τ is theslope of the exponential function, CS,B

1 , CS,B2 are the relative yields of the

signal or background components, C0 is a suitable normalization constantand Nsig and Nbkg are the signal and background yields for the whole fit.In Figure 3.15 the fit using the total PDF (3.7) is shown. Adding the bumpcontribution to the PDF improves the width of the peak from 26.3 MeV/c2

46

to 23.2 MeV/c2 and the reduced χ2 from 1.25 to 0.98. Notice that, as told insection 3.3, the B width from MC is σMC = 23.5 MeV/c2.

Figure 3.14: B+ mass fit, without the bump contribution, forrL > 0.65

Figure 3.15: B+ mass fit, with the bump contribution, forrL > 0.65

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3.5.1 Application phase of MVA

The optimal cut on rL is chosen through the observation of the signal efficiencyand background rejection distributions as a function of rL. These are obtainedfrom the estimated number of signal and background events as follows:

εsig =N∗sig(rL > rcutL )

N∗sig(rL > 0)(3.11)

Rbkg = 1− εbkg = 1−N∗bkg(rL > rcutL )

N∗bkg(rL > 0)(3.12)

Where N∗sig and N∗bkg are the numbers of signal and background events in aninterval which contains 95.5% of the signal PDF integral centred in the B+

nominal mass, in a subset of the preselected events given by the conditionrL > rcutL . The condition rL > 0 means that no cuts on rL have been applied,since the range of this variable goes from 0 to 1.To compute εsig and Rbkg as in (3.11) and (3.12), the following relationsbetween Nsig,bkg and N∗sig,bkg have been used:

N∗sig = Nsig ×∫ MB++ρ

MB+−ρpsig(x)dx (3.13)

N∗bkg = Nbkg ×∫ MB++ρ

MB+−ρpbkg(x)dx (3.14)

where the interval [MB+ − ρ,MB+ + ρ], which can be computed numerically,is such that: ∫ MB++ρ

MB+−ρpsig(x)dx = 0.9550 (3.15)

Equations (3.13), (3.14) and (3.15) rely on the fact that psig and pbkg arenormalized to 1 in the whole mass range.In Figure 3.16 the plots for the signal efficiency (magenta histogram) and thebackground rejection (blue histogram) are shown, along with the statisticalsignificance distribution (black histogram).The green dotted line indicates the point of maximum statistical significance,found at MVA_LL = rcutL = 0.9. The significance is referred here to the B+

signal peak, but we are interested in a presumably small subset of such signalcontaining the intermediate X(3872) resonance. Therefore, since the numberof signal candidates is very low here, a decision was made to go on with theanalysis by trying two looser alternative selection criteria:

• Apply a relatively hard cut on the MVA, rcutL = 0.65, with a signalefficiency of 47% and a background rejection of 93%, just before a dropof signal efficiency for higher values of rcutL (red line on the right, Figure3.16).

48

• Apply a looser cut, rcutL = 0.15, with a signal efficiency of 67% and abackground rejection of 76% (red line on the left, Figure 3.16). In orderto recover background rejection, an additional cut, around the ω mass,is applied on the three pions invariant mass, M(ω) ∈ [700, 850] MeV/c2.Notice that, being X(3872)→ J/ψω a subthreshold decay, only an ω withlow mass contributes.

In Figure 3.17 the fit on B+ mass is shown, along with the single componentsof the total PDF, for both the selection criteria.

Figure 3.16: Statistical significance (blue histogram), signalefficiency (magenta histogram) and background rejection(black histogram) as a function of rcutL (left axis for thesignificance and right axis for the rest)

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Figure 3.17: Fit on B+ mass for both rcutL , with fit results

3.6 Background subtraction: sPlot techniqueThe sPlot technique [46] is a statistical tool dedicated to the analysis of adata sample consisting in several sources of events, that are only signal andbackground events for the purpose of this analysis, merged into a single sample.Combining information from a set of variables for which the distributionsfor signal and background events are known (discriminating variables), thistechnique allows to compute, for each event and for each source of events,a particular weight corresponding to the likeliness that the event is of signaltype or background type. This weights, called sWeights, are then applied toanother set of variables, called control variables, in order to obtain the signaland background spectra separately. For this analysis, we use the B+ massspectrum as the only discriminating variable. See Appendix A for a moredetailed description of the sPlot formalism.

3.6.1 Application of sPlot technique

Giving as input to the sPlot technique the B+ mass fit models shown inFigure 3.17, it is possible to plot any desired control variable distribution(in our case, J/ψω mass, π0 mass, ω mass and so on) of the signal or thebackground by weighting each event with the corresponding weight, Wsig

50

or Wbkg, respectively. All the variables are considered after the MVA basedselection, as explained in section 3.5.1.The computation of sWeights is performed with the RooStats framework foradvanced statistical analysis [47], built on the RooFit toolkit.

Figure 3.18: ω invariant mass distribution after the rL cut

In Figure 3.18 the ω mass distribution after the MVA selection only isshown, for MVA_LL > 0.65, while in Figure 3.19 the background subtracteddistribution of the same variable is shown. The η and ω peaks are clearlyvisible, respectively, at 547 MeV/c2 and 782 MeV/c2.

Figure 3.19: ω invariant mass signal distribution (backgroundsubtracted)

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In Figure 3.20 the π0 mass signal distribution is shown, for MVA_LL > 0.65.The asymmetry can be partly due to the imperfect model with which the Bmass is fitted, since it can influence the computation of sWeights. Furthermore,as already told in Section 3.3 and observed in MC data, see Figure 3.4, the π0

mass distribution exhibits a tail for high masses due to the imperfect treatmentof superposition of clusters of the two photons in the calorimeter.

Figure 3.20: π0 mass signal distribution (backgroundsubtracted)

3.7 J/ψω mass spectrumIn Figure 3.21 and Figure 3.22 the (J/ψω) invariant mass signal distribution isshown, for both MVA selections. The red histogram is the Monte Carlo phasespace distribution after the application of the same reconstruction, preselectionand MVA cuts as real data, and it is normalized to the integral of the dataJ/ψω mass spectrum. The Monte Carlo distribution does not describe well thedata, since a huge excess of events is present at about 4600 MeV/c2. Noticethat a similar behaviour was already observed by BaBar[15]. To investigate thereasons behind this excess, a search for intermediate resonances contributionshas been performed, as can be seen in the next section.

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Figure 3.21: J/ψω mass, MVA_LL > 0.65

Figure 3.22: J/ψω mass, MVA_LL > 0.15 + M(ω) cut

3.8 Resonances contributionGiven the discrepancy between the data distribution and the phase spaceMonte Carlo in the J/ψ invariant mass distribution, a possible contributiondue to intermediate resonances was searched for. The presence of resonantstates in the decay can indeed reflect into peaking structures also in the J/ψωmass spectrum.The most obvious resonances, Ψ(2S) and K∗(892), can be observed clearly inthe dataset before background subtraction, as shown in Figure 3.23.

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Figure 3.23: J/ψππ and Kπ invariant mass, MVA_LL > 0.65

Figure 3.24: J/ψππ and Kπ invariant mass (backgroundsubtracted), MVA_LL > 0.65

In Figure 3.24 the same distributions, after the application of sPlot, are shown.The Ψ(2S) and K∗(892) peaks are not completely removed but the numberof events cannot justify the excess observed in the J/ψω mass spectrum.Nevertheless, a cut is applied on both J/ψππ and Kπ invariant mass after thebackground subtraction to remove the residual peaks, excluding the followingranges: 3670 < M(J/ψππ) < 3700 MeV/c2, 880 < M(Kπ) < 910 MeV/c2.The resulting J/ψω mass spectra are shown in Figure 3.25 and 3.26.An interesting structure can be observed when plotting (Kω) invariant massversus (J/ψω) invariant mass after having applied the background subtractiontechnique. It is shown in Figure 3.27 and Figure 3.28, emphasised by a blackcircle, at around M(Kω)≈1400 MeV/c2.The most probable hypothesis is that the structure comes from the decayB+ → K+

1 (1270)J/ψ, with K+1 (1270) → K+ω. The product of branching

fractions is:

BR(B+ → K+1 (1270)J/ψ)× BR(K+

1 (1270)→ Kω) ≈ 2× 10−5

and it is more than three times bigger than our searched signal:

BR(B+ → X(3872)K+)× BR(X(3872)→ J/ψω) ≈ 6× 10−6

54

and one order of magnitude smaller than B+ → K+J/ψω branching ratio:

BR(B+ → K+J/ψω) ≈ 3× 10−4

Figure 3.25: J/ψω mass without Ψ(2S) and K∗(892)contributions, MVA_LL > 0.65

Figure 3.26: J/ψω mass without Ψ(2S) and K∗(892)contributions, MVA_LL > 0.15 + M(ω) cut

55

Figure 3.27: M(J/ψω) vs M(Kω), MVA_LL > 0.65

Figure 3.28: M(J/ψω) vs M(Kω), MVA_LL > 0.15 + M(ω)cut

In order to try to reduce the K+1 (1270) contribution, a cut on (Kω) invariant

mass M(Kω) > 1700 MeV/c2 has been applied, corresponding to the redhorizontal line observable in Figure 3.27 and Figure 3.28. The agreementbetween data and phase space Monte Carlo improves significantly after thiscut, as shown in Figure 3.29 and Figure 3.30. Notice that the same cuts onJ/ψππ, Kπ and Kω invariant masses have been applied also in the MonteCarlo sample. With these cuts, an excess of events can be observed in theX(3872) region, particularly with MVA_LL > 0.15 and M(ω) cut.

56

Figure 3.29: M(J/ψω), MVA_LL > 0.65, after cut on M(Kω)

Figure 3.30: M(J/ψω) , MVA_LL > 0.15 + M(ω) cut, aftercut on M(Kω)

3.9 Hypotheses for the excess of events at 5350MeV/c2

Before to proceed with the analysis flow, a description is given in this sectionof the tests done on data to try to find and interpretation of the backgroundcomponent at arount 5350 MeV/c2 in the B+ mass spectrum.The first hypothesis tested assumes that the excess could be caused by a

57

misreconstructed B decay with a relatively high branching fraction wherea light particle is missing or misidentified. The channel we analysed isB+ → J/ψK+π+π−, with a branching fraction which is about 2.5 timesbigger than BR(B+ → K+J/ψω). If this was the origin of the excess, theunderlying hypothesis would be that a random π0, wrongly associated withthis decay, could produce an accumulation of events in the excess region.In Figure 3.31 the B+ mass spectrum versus the J/ψKππ invariant massspectrum is shown. The B+ → J/ψKππ peak is clearly visible in the latterspectrum, but the contribution of this decay channel to our spectrum is clearlylocated at high B+ masses, i.e. for M(B+)J/ψKω > 5450 MeV/c2. Nonetheless,this contribution has been removed before performing the fit on the B+ massdistribution described in Section 3.5.

Figure 3.31: B+ mass vs (J/ψKππ) mass

Other tests were made to check whether inclusive decays containing knownresonances can be invoked as an explanation for the excess. The B+ masshas been plotted as a function of (K J/ψ), (J/ψππ), (Kπ) and (Kω) systemsinvariant masses, shown, respectevely, in Figures 3.32, 3.33, 3.34 and 3.35.The first spectrum presents no clear structures, as expected since there arenot known resonances with this final state. In the others, the contributionsof, respectively, Ψ (2S), K∗(892) and K∗1(1270) can be observed. However, noclear correlation is found between these three resonances and the bump at5350 MeV/c2.

58

Figure 3.32: B+ mass vs (K J/ψ) mass

Figure 3.33: B+ mass vs (J/ψππ) mass

59

Figure 3.34: B+ mass vs (K π) mass

Figure 3.35: B+ mass vs (K ω) mass

Other hypotheses are currently being tested, such as the possibility of amisidentification between a pion and a kaon or other heavy resonancescontributions, with possibility of interference. Moreover, other tests are beingcarried on by analysing the MC events with B → J/ψX inclusive decays. Theresults, due to data treatment time requested, are incompatible with beingpresented in this thesis.

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3.10 Efficiency correctionsIn order to take into account every loss of signal candidates, an efficiencystudy has been performed for each analysis step. The efficiency of every stepis computed, for each bin of the true, i.e. generated, value of the J/ψω systeminvariant mass MJ/ψω, as the ratio between the number of events after theselection and the number of events before the selection, in a proper MC sample.Obviously, the selection is performed on MC events in an identical manner asreal data, and this is done by applying the analysis cuts on reconstructedquantities, i.e. the variables of interest after the simulation of the detectorresponse.The calculation starts from a MC sample generated in the LHCb angularacceptance. The efficiency of the acceptance is not considered, since this cutis already intrinsically present on data, and we are not interested in absolutemeasurements of physical processes.What matters for data correction is the total efficiency distribution; however,to check the effect of the single analysis step, the efficiency distributions havebeen computed for each one of them:

• Generator level step: as told in section 3.3, a set of kinematic cuts atgenerator level have been performed in order to accelerate the simulationprocess. The MC sample used for this computation is limited to thegenerator phase (it does not include the detector simulation) and itis composed by 40000 events in the full spectrum, 20000 in the rangeM(J/ψω) < 4400 MeV/c2 and 3000 more in the range M(J/ψω) < 3950MeV/c2. The choice to expand the statistics in the low mass range is dueto the fact that in the kinematic threshold region, which is the interestingregion for this analysis, a very low number of events is present, and thisleads to a big statistical error on the computation of the efficiency.The generator level efficiency is computed as:

ε1(MJ/ψω) =Nkin+acc(MJ/ψω)

Nacc(MJ/ψω)(3.16)

• Reconstruction step: for this step the fully simulated Monte Carlodataset (≈ 6 × 106 events) is used. The reconstruction efficiency iscomputed as:

ε2(MJ/ψω) =Nreco(MJ/ψω)

Nkin+acc(MJ/ψω)(3.17)

• Preselection step: for this step the fully reconstructed Monte Carlodataset is used. The preselection efficiency is computed as:

ε3(MJ/ψω) =Npresel(MJ/ψω)

Nreco(MJ/ψω)(3.18)

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• MVA step: computed from the fully preselected Monte Carlo dataset.In this step, the two different selections are taken into account with twodifferent efficiencies. The first one, corresponding to MVA_LL > 0.65,is merely the fraction of events that pass this MVA cut. The second one,corresponding to MVA_LL > 0.15 + M(ω) cut, is the fraction of eventsthat pass the MVA selection and the cut on the three pions invariantmass around the ω peak, as described in section 3.6. The efficiencies forthe MVA cuts are computed as:

ε0.654 (MJ/ψω) =

NMVA_LL>0.65(MJ/ψω)

Npresel(MJ/ψω)(3.19)

ε0.154 (MJ/ψω) =

NMVA_LL>0.15 + M(ω)cut(MJ/ψω)

Npresel(MJ/ψω)(3.20)

• M(Kω) cut: computed from the fully preselected Monte Carlo datasetafter the MVA step. The efficiency is computed as:

ε0.655 (MJ/ψω) =

NM(Kω)>1700 MeV/c2(MJ/ψω)

NMVA_LL>0.65(MJ/ψω)(3.21)

ε0.155 (MJ/ψω) =

NM(Kω)>1700 MeV/c2(MJ/ψω)

NMVA_LL>0.15 + M(ω)cut(MJ/ψω)(3.22)

• Ψ(2S) and K∗(892) cut: computed from the full preselected and MVAselected Monte Carlo dataset, after the cut on M(Kω). The efficienciesfor these cuts are computed as:

ε6(MJ/ψω) =Nno resonances(MJ/ψω)

NM(Kω)>1700 MeV/c2(MJ/ψω)(3.23)

The total efficiency is then computed as the product of the six efficiencies1:

ε0.65(MJ/ψω) =6∏i=1

ε0.65i (MJ/ψω) (3.24)

ε0.15(MJ/ψω) =6∏i=1

ε0.15i (MJ/ψω) (3.25)

In Figure 3.36 and Figure 3.37 the effect of the six cuts on the signal J/ψωmass distribution is shown, while the plots of efficiencies are in Figure 3.38.In Figure 3.39 the total efficiency distributions are shown.

1Obviously the efficiencies ε1, ε2 and ε3 are the same for the two MVA selections becauseapplied before it, for this reason the corresponding suffices have been omitted.

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To overcome the problem of the very low number of Monte Carlo events in thethreshold region after all the analysis steps, and to obtain a smooth efficiencycorrection function, a fit to the efficiency distributions has been done and anextrapolation has been performed in this region. Of course the only way toobtain a correct distribution is to enlarge the simulated dataset in the regionof interest. A new MC sample is in fact being produced.

(a) Generator level cuts (b) Reconstruction cuts

(c) Preselection cuts

Figure 3.36: Effect of the cut flow on the J/ψω massdistribution (black histogram before the application of the cuts,red histogram after)

63

(a) MVA_LL > 0.65 cut (b) MVA_LL > 0.15 + M(ω) cut

(c) M(Kω) cut, MVA_LL > 0.65 (d) M(Kω) cut, MVA_LL > 0.15 + M(ω) cut

(e) Ψ(2S) and K∗(892) cuts, MVA_LL > 0.65 (f) Ψ(2S) and K∗(892) cuts, MVA_LL > 0.15+ M(ω) cut

Figure 3.37: Effect of the cut flow on the J/ψω massdistribution (black histogram before the application of the cuts,red histogram after)

64

(a) Generator level step efficiency (b) Reconstruction step efficiency

(c) Preselection step efficiency (d) MVA step efficiency

(e) Ψ(2S) and K∗(892) cuts efficiency (f) M(Kω) cut efficiency

Figure 3.38: Efficiency of every step as a function ofM(J/ψω). Asymmetric errors are computed according to theAgresti-Coull formula [48].

65

(a) Total efficiency, MVA_LL > 0.65 (b) Total efficiency, MVA_LL > 0.15 + M(ω)cut

Figure 3.39: Total efficiency distributions for both MVA cutvalues

Fitting the total efficiencies of Figure 3.39 with a polynomial function is veryhard, since they are decreasing in both the left and the right extremes, andpolynomials of different orders return functions with different behaviours nearthe threshold region. In order to try to reduce the possibility of a wrongextrapolation, and given the simulation at our disposal, it was chosen to obtainthe total efficiency distribution by fitting the positive slope component and thenegative slope component separately. The first one is composed by the productof the first four steps (from generator level to MVA selection) and the Ψ(2S)and K∗(892) cuts. It is shown in Figure 3.40 and it has been fitted with alinear function .The second one, reported in Figure 3.41, composed only bythe M(Kω) cut efficiency, has been fitted with a 4th order polynomial. Thetotal efficiency function is then obtained by multiplying, for each bin, the twofitting functions; therefore, the total efficiency distribution is described by a5th order polynomial.The fits results for the linear components of the total efficiencies are thefollowing:

f 0.65lin = −0.0064 + 1.70× 10−6x (3.26)

f 0.15lin = −0.0067 + 1.77× 10−6x (3.27)

In order to extrapolate the shape of the efficiency function near threshold, thefit has been performed in a limited range, observable in Figure 3.40, assumingthat the whole efficiency spectrum has got a linear functional form. Thisassumption seems not very likely for high masses but it is acceptable, since theexact shape for high masses is not so important for our analysis. Moreover,the linear fit is conservative in the threshold region, meaning that the firsttwo bins, excluded from the fit, have a higher mean value with respect to

66

the fitting function in both cases. The parametrized efficiency, therefore, isunderestimating the real efficiency, and thus cannot generate structures in theJ/ψω mass spectrum.

(a) Linear fit, MVA_LL > 0.65 (b) Linear fit, MVA_LL > 0.15 + M(ω) cut

Figure 3.40: Fit on the linear component of the total efficiencyfor both MVA cut values

(a) Polynomial fit, MVA_LL > 0.65 (b) Polynomial fit, MVA_LL > 0.15 + M(ω)cut

Figure 3.41: Fit on the M(Kω) cut efficiency for both MVAcut values

For the 4th order polynomial fits to the M(Kω) cut efficiencies, shown inFigure 3.41, the results are the following:

f 0.65pol = −1257 + 1.22x− 0.000445x2 + 7.18× 10−8x3 − 4.34× 10−12x4 (3.28)

f 0.15pol = −1130 + 1.11x− 0.000404x2 + 6.55× 10−8x3 − 3.94× 10−12x4 (3.29)

The total efficiency function is obtained multiplying the two components:

ftot = flin · fpol (3.30)

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In Figure 3.42 the total efficiency distribution is shown for both MVA cutvalues, along with the 5th order polynomial defined in (3.30).This function is used to correct the J/ψω spectrum in real data in order to takeinto account all the effects of the analysis chain. The real data distribution isdivided by (3.30) and, in this way, the resulting spectrum allows to estimatethe branching fraction of the decay of interest, as will be shown in Chapter 4.

(a) Total efficiency, MVA_LL > 0.65 (b) Total efficiency, MVA_LL > 0.15 + M(ω)cut

Figure 3.42: Total efficiency distributions, data from MC andfunctional form

In Figure 3.43 and Figure 3.44 the efficiency corrected J/ψω mass spectraare shown. The number of events of these plots are an estimate of the realnumber of B+ → K+J/ψω events, both resonant and non resonant, generatedfrom p-p collisions in the LHCb acceptance during 2011 and 2012 data takingperiod. The excess of events in the threshold region is emphasized, especiallyfor MVA_LL > 0.15 + M(ω) cut.It must be noticed that the choice of normalizing the MC distribution to thewhole spectrum is not correct, since the events in the X(3872) and χc0(2P)peak must be excluded from the computation of the normalization factor inorder to obtain a correct estimate of the number of non resonant events underthe peak. The correct normalization range is obtained from a study of theJ/ψω mass resolution, as will be shown in the next section.

68

Figure 3.43: M(J/ψω), MVA_LL > 0.65, efficiency corrected

Figure 3.44: M(J/ψω) , MVA_LL > 0.15 + M(ω) cut,efficiency corrected

69

3.11 J/ψω mass resolutionThe J/ψω mass resolution must be studied, in particular in the thresholdregion, in order to set the correct normalization range for the MC distributionand, given the little mass difference between the X(3872) and the χc0(2P),to understand whether the two peaks in the J/ψω mass distribution can beresolved or not in LHCb. The X(3872) and χc0(2P) masses are very close,M(X(3872)) = 3871.68± 0.17 MeV/c2 and M(χc0(2P)) = 3918.4± 1.9 MeV/c2,with a mass difference of ∆M ≈ 47 MeV/c2. Assuming that two peaks canbe resolved if the distance between their mean values is more than 4σ, thisimplies that the maximum resolution allowed to observe two well separatedpeaks is in our case σmax ≈ 12 MeV/c2.The mass resolution has been computed from the MC sample after theapplication of all the analysis cuts. For each event contained in a certain binof the reconstructed J/ψω mass distribution the following quantity is defined:

δM = Mreco(J/ψω)−MTRUE(J/ψω)

The difference δM between the reconstructed J/ψω mass and the true (i.e.generated) mass is well described by a gaussian distribution centred at zero,while its standard deviation represents the mass resolution that we want toobtain.Given the poor statistics near threshold, in order to have a sufficient numberof events for each gaussian fit, if a bin in the reconstructed J/ψω massdistribution contains less than 150 events it is merged with the following binuntil reaching this value. In Figure 3.45, an example of the δM distribution isgiven, for Mreco(J/ψω) ∈ [4100, 4150] MeV/c2.

Figure 3.45: δM for Mreco(J/ψω) ∈ [4100, 4150] MeV/c2

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Figure 3.46: J/ψω mass resolution

In Figure 3.46, the mass resolution distribution is shown, as a function of J/ψωinvariant mass. The distribution has been fitted with a linear function andthe fit results give, for M(J/ψω) = M(X(3872)) and M(J/ψω) = M(χc0(2P)):

σX(3872) ≈ 17.7MeV/c2 σχc0(2P) ≈ 18.5

They are, therefore, greater than the maximum value stated before in thissection, σmax ≈ 12 MeV/c2. This means that we should not able to observetwo well separated peaks for the two aforementioned resonances in the J/ψωspectrum. In Figure 3.47, the J/ψω spectrum in the threshold region is shown,before the application of the efficiency correction, with a bin size matched tothe mass resolution. The X(3872) and χc0(2P) peaks are visible with one bineach, with only one bin between them. However, because of the poor statisticsin this region, we chose to enlarge the binning and to observe a single peak.

Figure 3.47: J/ψω mass, threshold region

71

Chapter 4

Results

The efficiency corrected J/ψω spectrum, computed in Section 3.10, representsan approximation of the real signal distribution, therefore it can be used toobtain an estimate of the number of B+ → K+J/ψω decays, both resonantand non resonant, produced in p-p interaction inside the LHCb geometricalacceptance during 2011 and 2012 data taking periods. However, in order toestimate the number of signal events, the correct normalization range for theMC distribution must be set. Having computed the J/ψω mass resolution,the lower limit can be set far from the signal region, i.e. to M(χc0(2P)) +5σχc0(2P) ≈ 4010 MeV/c2. In Figure 4.1 and Figure 4.2 the efficiency correctedJ/ψω mass distributions are shown, where the MC distributions have beennormalized in the range M(J/ψω) > 4010 MeV/c2. In this way, the part ofthe MC distribution that lies under the X(3872) and χc0(2P) peak representsthe correct fraction of non resonant B+ → K+Jψω events.Since X(3872) and χc0(2P) peaks are not distinguishable in our J/ψω massspectrum, because of the poor resolution and the low statistics, from thedistributions at our disposal we can estimate the sum of branching fractions[BR(B+ → K+X(3872))×BR(X(3872)→ J/ψω) + BR(B+ → K+χc0(2P))×BR(χc0(2P)→ J/ψω)]. In fact, it can be noticed that:

Npeak

NB+→K+J/ψω

≈BRX(3872) + BRχc0(2P)

BR(B+ → K+J/ψω)(4.1)

where:

Npeak = Ndata(MJ/ψω < 4010 MeV/c2)− NMC(MJ/ψω < 4010 MeV/c2) (4.2)

NB+→K+J/ψω = Ndata(MJ/ψω > 4010 MeV/c2) + NMC(MJ/ψω < 4010 MeV/c2)(4.3)

BRX(3872) = BR(B+ → K+X(3872))× BR(X(3872)→ J/ψω) (4.4)

BRχc0(2P) = BR(B+ → K+χc0(2P))× BR(χc0(2P)→ J/ψω) (4.5)

72

Figure 4.1: Corrected J/ψω mass with the new MCnormalization, MVA_LL > 0.65

Figure 4.2: Corrected J/ψω mass with the new MCnormalization, MVA_LL > 0.15 + M(ω) cut

From the distributions in Figure 4.2, thus for MVA_LL > 0.15, the followingvalues have been obtained:

Npeak = 133953± 36421 (4.6)

NB+→K+J/ψω = 555233± 45031 (4.7)

The PDG value for BR(B+ → K+J/ψω) takes into account both resonant andnon resonant contributions. After the subtraction of the resonant, i.e. X(3872)

73

and χc0(2P) contributions, the obtained value for the non resonant one is:

BR(B+ → K+J/ψω) = (2.84+0.61−0.33)× 10−4 (4.8)

using this value for BR(B+ → K+J/ψω), the sum of X(3872) and χc0(2P)branching fractions can be obtained:

BRX(3872) + BRχc0(2P) = (6.9+2.5−2.1)× 10−5 (4.9)

which is consistent within 2σ with the BaBar result:

BRX(3872) + BRχc0(2P) = (3.6+0.9−0.7)× 10−5 (4.10)

The errors in (4.9) are only statistical, and we ignore the systematic errors.

4.1 SummaryIn summary, in this thesis the decay B+ → K+X(3872), with X(3872)→ J/ψωhas been analysed. We observe an excess of events in the J/ψω invariant massdistribution over a non resonant Monte Carlo distribution, in the kinematicthreshold region, corresponding to the sum of the contributions of the X(3872)and the χc0(2P). Due to lack of statistics and poor resolution we are able toonly estimate the sum of the branching fractions [BR(B+ → K+X(3872))×BR(X(3872)→ J/ψω) + BR(B+ → K+χc0(2P))× BR(χc0(2P) → J/ψω)],normalized to non resonant BR(B+ → K+J/ψω), from an estimate of theratio between the number of events in this excess and the whole phase spacedistribution. The results are compatible with existing publications.This analysis has several limitations that can be overcome in the near future.The current dataset is being reprocessed by refitting all decay tracks withthe π0 and B+ masses constrained to their nominal values and requiringthe pointing of the B+ to the primary vertex. This procedure will increasesignificantly the mass resolution and will allow to discard a large fraction ofthe misreconstructed B decays that we are currently treating as signal.The simulated dataset has proven to be insufficient, especially in the regionof interest. A request for another 6 million Monte Carlo events has alreadybeen submitted, but this time a cut on J/ψω invariant mass is applied atgenerator level, M(J/ψω) < 4200 MeV, in order to enlarge the statistics in thethreshold region and improve significantly the precision on the computationof the efficiency.A new set of variables is being developed to adjust the π0 reconstruction, astold in Section 3.3, to overcome the problem of the bad treatment of the energyof the two photons released in the same calorimeter cells, and will be releasedin the next version of the reconstruction software.

74

We observed that adding a gaussian contribution, centred at around 5350MeV/c2 in the B+ mass PDF, significantly improves the fit. This excess ofevents is in the B+ mass spectrum, described in Section 3.5, has been theobject of some tests, shown in Section 3.9, aimed at understanding its nature,without any definitive conclusion. The study of this excess is going on, in orderto find a model with which to improve the B mass description. A better fitmeans that the background subtraction can be computed with more precision,resulting in a reduction of the error in the sum of branching fractions (4.9).Another study is trying to understand whether the phase space model describescorrectly the real data or not. Besides the K+

1 (1270) resonance, in fact,other excited K states may be contributing to the shape of the J/ψω massdistribution.We are confident that, in addition to the improvements listed above, withthe increase of the statistics accumulated with the LHC Run-II which, from2015 to 2018, will triple the delivered luminosity along with an increase ofthe centre of mass energy from 8 to 13 TeV, we will be able to improve themeasurement of BRX(3872). This should also allow us to perform an angularanalysis to determine the X(3872) JPC quantum numbers in the J/ψω decaychannel, to verify whether this particle and the particle decaying in J/ψππ arethe same state or not, and therefore to make progresses on the comprehensionof the nature of this exotic state.

75

Appendices

76

Appendix A

The sPlot technique

The sPlot technique [46] is a statistical tool dedicated to the analysis of a datasample consisting in several sources of events merged into a single sample. Thebasic assumption is that each event is composed by a set of variables of twodifferent kind: discriminating variables and control variables. Information isgathered, for each event, through each discriminating variable, for which thedistributions for all the sources of events are known; the distributions for thecontrol variables are then reconstructed from this information, without makinguse of any a priori knowledge on them.In this analysis only two sources of events are considered: signal events andbackground events.A maximum log-Likelihood fit is performed on the discriminating variableaccording to signal and background PDFs, using the following:

L =N∏i=1

(Nsig

Nsig + Nbkg

psig(yi) +Nbkg

Nsig + Nbkg

pbkg(yi)

)(A.1)

logL =N∑i=1

log [Nsigpsig(yi) + Nbkgpbkg(yi)]−N log(N) (A.2)

where:

• Nsig and Nbkg are the signal and background yields

• yi is the set of discriminating variables for the event i

• psig(yi) and pbkg(yi) are the signal and background PDFs for thediscriminating variables

• N = Nsig + Nbkg is the total number of events

77

A.1 Preliminary step: total correlationLet’s start with the hypothesis of total correlation between the discriminatingvariable y and the control variable x, or in other words, x = x(y). In thiscase, while performing the fit, a knowledge of the x distributions is used, andtherefore it cannot be considered a true control variable.Once the yields have been extracted, a weight can be defined as following:

wsig(yi) =Nsigpsig(yi)

Nsigpsig(yi) + Nbkgpbkg(yi)(A.3)

and similarly for the background. This weight can be used to determine abinned, background subtracted distribution for x. For the j-th bin:

NsigMsig(xj)δxj =∑

|xi−xj |<δxj

wsig(yi) (A.4)

where Msig(xj) is the signal distribution in the j-th bin, xj is the central valueof the bin and δxj its width. The sum runs over all the events for whichxi = x(yi) lies in the j-th bin. The left term of (A.4) is, trivially, the integralof the j-th bin.It can be shown that the distribution Msig reproduces, on average, the truedistribution Msig. Taking the continuous limit:⟨

NsigMsig(x)⟩

=

∫[Nsigpsig(y) + Nbkgpbkg(y)] δ(x(y)− x) wsig(y) dy

= Nsig

∫psig(y) δ(x(y)− x) dy

= NsigMsig(x)

(A.5)

Therefore, the weights defined in (A.3) provide a direct estimate of the x(signal and background) distributions. Of course this formalism is valid onlyin the hypothesis of total correlation between the discriminating and controlvariable. This means that the PDFs of the variable x enter implicitly in thedefinition of the weights and, as a result, the M distribution are biased in away that is not trivially decipherable.In the next section, the more interesting case where x and y are uncorrelatedis shown.

A.2 The sPlot formalismWhen the two variables x and y are uncorrelated, the total PDFs factorizeinto the product:

psig(x, y) = Msig(x)psig(y)

pbkg(x, y) = Mbkg(x)pbkg(y)(A.6)

78

In that case the last passage of (A.5) is no longer valid:⟨NsigMsig(x)

⟩=

=

∫ ∫[NsigMsig(x)psig(y) + NbkgMbkg(x)pbkg(y)] δ(x− x) wsig(y) dxdy

=

∫[NsigMsig(x)psig(y) + NbkgMbkg(x)pbkg(y)]

Nsigpsig(y)

Nsigpsig(y) + Nbkgpbkg(y)dy

= Nsig

∑J

MJ(x)

(NJ

∫pJ(y)psig(y)

Nsigpsig(y) + Nbkgpbkg(y)dy

)6= NsigMsig(x)

(A.7)

where J ∈ [sig, bkg]. The correction term is indeed not identical to theKronecker symbol δJ,sig. It can be shown that it would be, if only thediscriminating variable y were totally discriminating, but in this case a simplecut on y could separate perfectly the signal and background distributions onx.Instead, when y is not totally discriminating, a simple redefinition of theweights allows to overcome the problem in (A.7). It can be noticed thatthe correction term on (A.7) is related to the inverse of the covariance matrix,which is defined as follows:

C−1IJ =

∂2(− log(L))

∂NI∂NJ

=N∑i=i

pJ(yi)pI(yi)

(Nsigpsig(yi) + Nbkgpbkg(yi))2(A.8)

where I, J ∈ [sig, bkg]. When averaging the previous equation, replacing thesum over events with an integral:⟨

C−1IJ

⟩=

∫ ∫ ∑K

NKMK(x)pK(y)pI(y)pJ(y)

(Nsigpsig(yi) + Nbkgpbkg(yi))2dxdy

=

∫ ∑K

NKpK(y)pI(y)pJ(y)

(Nsigpsig(yi) + Nbkgpbkg(yi))2dy

∫MK(x)dx

=

∫pI(y)pJ(y)

Nsigpsig(yi) + Nbkgpbkg(yi)dy

(A.9)

where K ∈ [sig, bkg] and remembering that, by hypothesis, MK(x) isnormalized to 1. Confronting (A.7) and (A.9) the following are obtained:⟨

MI(x)⟩

=∑J

MJ(x)NJ

⟨C−1IJ

⟩(A.10)

NIMI(x) =∑J

〈CIJ〉⟨MJ(x)

⟩(A.11)

79

Therefore, the true distribution M(x) can be reconstructed using a linearcombination of wI(y) (see (A.4)), with the covariance matrix elements CIJ ascoefficients. A new set of weights, called sWeights, is then defined as:

WI(yi) =

∑J CIJpJ(yi)∑K NKpK(yi)

(A.12)

80

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