qcd uncertainties in charged and neutral current … 1 w e us 'renormalized' her in th...

UNIVERSITA DEGLI STUDI DI MILANO

FACOLTA DI SCIENZE E TECNOLOGIECorso di Laurea Triennale in Fisica

QCD UNCERTAINTIES IN CHARGEDAND NEUTRAL CURRENT DRELL-YAN

PROCESSES AND THE W MASSDETERMINATION

Relatore:Alessandro Vicini

Candidato:Guido BrunieraMatricola n. 827916

Codice PACS 12.38.-t

Anno Accademico 2015-2016

Summary

The purpose of this thesis is to study the neutral and charged current Drell-Yan processes (whichare respectively the production of a lepton pair and of a lepton and a neutrino, each with hightransverse momentum), in view of the precise measurement of the W boson’s mass MW . In fact,since the value MW is predicted by the Standard Model, a precise measurement of such quantitycan contribute to test its validity.

The lepton pair’s transverse mass and the charged lepton’s tranverse momentum distributionsare considered in the analysis, which makes use of the Template Fit procedure to extract thevalue of MW .The theoretical uncertainty upon MW deriving from a variation of the QCD scales, namely therenormalization, factorization and resummation scales, is analysed both separately in neutral andcharged current processes and in the ratio W/Z. We discuss the QCD uncertainties separatelywith and without taking into account the correlation among the distribution bins through thecalculus of the covariance matrix, to be used then in the fit procedure. The interesting featureis that all such correlations are computed by averaging over the scale combinations.

In Chapter 1 we present the issue and the goal we aim for, which is a quantitative estimate ofthe impact of the QCD scale choice on the value MW .

In Chapter 2 we present the Drell-Yan process, both according to the parton model and alongwith QCD corrections. We then introduce the studied variables, their features and how they aresensitive to QCD scales. In particular how the arbitrary choice of QCD scale values results in asystematic theoretical error in determining the W boson mass.

In Chapter 3 we show how the value of MW is extracted from real data and which procedure wefollowed to estimate the systematic uncertainty deriving from the choice of QCD scale values.We will end by presenting which variables we used in the Template Fit procedure and how wecan account for the presence of correlation among different bins and observables.

In Chapter 4 we first present the simulation settings and codes, and then show the results obtainedfrom the analysis of distributions given by simulations. Basically we will show for both W− andW+ data how a particular choice of QCD scale values results in a shift in the determination ofMW , considering either all bins independent or correlated.

In Chapter 5 we briefly summarize our work and the obtained results, suggesting some futuredevelopments.

Contents

1 Introduction 1

2 The Drell-Yan process 32.1 The Drell-Yan mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 W and Z boson production . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 QCD and perturbative corrections to Drell-Yan model . . . . . . . . . . . . . . . 7

2.2.1 LO and NLO QCD perturbative corrections . . . . . . . . . . . . . . . . . 72.2.2 W and Z transverse momentum distributions . . . . . . . . . . . . . . . . 9

2.3 The studied observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 W and Z transverse mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 The Xm⊥ and Xp⊥ observables . . . . . . . . . . . . . . . . . . . . . . . . 122.3.3 The ratio observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.4 The normalized observables . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Scale uncertainties on Xm⊥ and Xp⊥ . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 W boson’s mass determination 183.1 The Template Fit technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 Analysis of real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.2 Comparison of theoretical distributions . . . . . . . . . . . . . . . . . . . . 19

3.2 The template fit analysis of W and Z simulations . . . . . . . . . . . . . . . . . . 233.2.1 The set of pseudo-data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 The choice of templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3 Observables included in the fit . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 The least square method in MW determination . . . . . . . . . . . . . . . . . . . 27

4 Simulations and results 294.1 The simulation settings and the procedure . . . . . . . . . . . . . . . . . . . . . . 294.2 Analysis of distributions for W− . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Analysis of distributions for W+ . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Conclusions and perspectives 42

Bibliography 43

Chapter 1

Introduction

Through the centuries and the years, physicists have studied nature in order to understand it,starting from the macroscopic world down to the microscopic one. If we once thought that atomswere the smallest and simplest bricks of nature’s puzzle, we then discovered that they werecomposed by protons, neutrons and electrons. And now we know that there exist even smallerparticles, called quarks, that constitute all hadrons, among which we find protons and neutrons.In such investigation of nature that aims at the microscopic world populated by particles, it thenbecame relevant the construction of apparatus and experiments which enable physicists to carryout their research studies, such as the CERN Large Hadron Collider (LHC).

Actually, the Drell-Yan process was studied at LHC, and it allowed physicists to test the validityof quantum chromodynamics (QCD) theory. The QCD corrections to such process agreed withdata coming from experiments, thus validating the theoretical predictions of QCD. Moreover,such theory predicted a specific description of protons’ nature, parametrized by the PartonDistribution Functions (PDFs), and Drell-Yan experiments contributed to test such definitionof the proton. The very important feature that allowed such test is that Drell-Yan final statedoes not depend on QCD, and the analysis of its kinematics carries information about the QCDbehaviours which involve only quarks and gluons, sensitive to the strong interaction, and affectingthe initial state only.It then seems that the first interest in studying the Drell-Yan process is related to the investigationof the nature of the proton and to tests about the validity of QCD predictions. Actually, theincreased power of colliders, especially since they began to be built not as linear colliders but ascircular ones, allows nowadays precise measurements of particle properties, such as their mass.In such frame the Drell-Yan process acquires a new relevance as it describes the production of thevector gauge bosons Z andW±, first discovered in 1983. Since the final state does not depend onQCD, it provides information about the physics underlying the preavious step of such productionprocess and it allows a sharper measurement of the mass of the gauge bosons, with a relativeerror of about 10−4.

The measurement of the mass of the W boson, MW , is a test of the Standard Model itself: thetheoretical prediction is given once the Standard Model Lagrangian has been initialized usingother parametres than MW itself. As a consequence the Standard Model provides a quantitativeprediction for such value, which has to be compared to the values coming from experiments. Anexample of the comparison between theoretical predictions as given by different theories (Stan-dard Model, SUSY, ...) is depicted in Fig. 1.0.1b.

2

– 2–

0

0.25

0.5

0.75

1

80.2 80.3 80.4 80.5 80.6

Entries 0

80.2 80.6MW[GeV]

ALEPH 80.440±0.051

DELPHI 80.336±0.067

L3 80.270±0.055

OPAL 80.415±0.052

LEP2 80.376±0.033χ2/dof = 49 / 41

CDF 80.389±0.019

D0 80.383±0.023

Tevatron 80.387±0.016χ2/dof = 4.2 / 6

Overall average 80.385±0.015

Figure 1: Measurements of the W-boson massby the LEP and Tevatron experiments.

and Bose-Einstein correlations (BEC) between quarks from dif-

ferent W’s (8 MeV) are included. The mass difference between

qqqq and qqℓνℓ final states (due to possible CR and BEC effects)

is −12±45 MeV. In a similar manner, the width results obtained

at LEP have been combined, resulting in ΓW = 2.195 ± 0.083

GeV [1].

The two Tevatron experiments have also identified common

systematic errors. Between the two experiments, uncertainties

due to the parton distribution functions, radiative corrections,

and choice of mass (width) in the width (mass) measurements

are treated as correlated. An average W width of ΓW = 2.046±0.049 GeV [2] is obtained. Errors of 20 MeV and 7 MeV

accounting for PDF and radiative correction uncertainties in this

width combination dominate the correlated uncertainties. At

the 2012 winter conferences, the CDF and D0 experiments have

December 18, 2013 12:01

(a) MW value as given by different experiments.

1.2 Electroweak precision physics 27

All MSSM points included in the results have the neutralino as LSP and the sparticle masses pass thelower mass limits from direct searches at LEP. The Higgs and SUSY masses are calculated using FeynHiggs

(version 2.9.4) [121, 122, 123, 124, 125]. For every point, it was tested whether it is allowed by direct Higgssearches using the code HiggsBounds (version 3.8.0) [126, 127]. This code tests the MSSM points againstthe limits from LEP, Tevatron and the LHC.

The results for MW are shown in Fig. 1-8 as a function of mt, assuming the light CP -even Higgs h in theregion 125.6 ± 0.7(3.1) GeV in the SM (MSSM) case. The red band indicates the overlap region of the SMand the MSSM. The leading one-loop SUSY contributions arise from the stop sbottom doublet. Howeverrequiring Mh in the region 125.6 ± 3.1 GeV restricts the parameters in the stop sector [128] and with it thepossible MW contribution. Large MW contributions from the other MSSM sectors are possible, if eithercharginos, neutralinos or sleptons are light.

The gray ellipse indicates the current experimental uncertainty, whereas the blue and red ellipses shows theanticipated future LHC and ILC/GigaZ precisions, respectively (for each collider experiment separately) ofTable 1-12, along with mt = 172.3 ± 0.9 (0.5, 0.1) GeV for the current (LHC, ILC) measurement of the topquark mass. While, at the current level of precision, SUSY might be considered as slightly favored over theSM by the MW -mt measurement, no clear conclusion can be drawn. The smaller blue and red ellipses, onthe other hand, indicate the discrimination power of the future LHC and ILC/GigaZ measurements. Withthe improved precision a small part of the MSSM parameter space could be singled out.

168 170 172 174 176 178mt [GeV]

80.30

80.40

80.50

80.60

MW

[GeV

] MSSM

MH = 125.6 ± 0.7 GeVSM

Mh = 125.6 ± 3.1 GeV

MSSMSM, MSSM

Heinemeyer, Hollik, Stockinger, Weiglein, Zeune ’13

experimental errors 68% CL / collider experiment:

LEP2/Tevatron: todayLHCILC/GigaZ

Figure 1-8. Predictions for MW as a function of mt in the SM and MSSM (see text). The gray, blue andred ellipses denote the current, and the target LHC and ILC/GigaZ precision, respectively, as provided inTable 1-12.

In a second step we apply the precise ILC measurement of MW to investigate its potential to determineunknown model parameters. Within the MSSM we assume the hypothetical future situation that a lightscalar top has been discovered with mt1

= 400 ± 40 GeV at the LHC, but that no other new particle hasbeen observed. We set lower limits of 100 GeV on sleptons, 300 GeV on charginos, 500 GeV on squarks ofthe third generation and 1200 GeV on the remaining colored particles. The neutralino mass is constrained

Community Planning Study: Snowmass 2013

(b)MW value theoretical predictions and its dependenceon the mass of top quark.

Figure 1.0.1: MW value report.

Up to now the theoretical prediction of MW carries an error of about 10 MeV, while the com-monly accepted value as coming from experiments has an error of about 15 MeV, as it is shownin Fig. 1.0.1a. Any decrease of such error would contribute to improve the test goodness.

In our work we studied the theoretical systematic error affecting the experimental measurementof MW , deriving from the choice of three QCD parametres, namely the renormalization, factor-ization and resummation scales, which enter the Drell-Yan process description. Eventually, weaimed at giving a quantitative estimate of such theoretical uncertainties.

Chapter 2

The Drell-Yan process

In this section we introduce and present the Drell-Yan process and its relationship with the ZandW boson production, first in the parton model and then with perturbative QCD corrections.We then present the studied observables in the analysis of this process, which will also be usedfurther on. We end this chapter by illustrating how QCD affects the simulations of Drell-Yanevents and how it influences the theoretical prediction of the mass of the gauge bosons W andZ.

2.1 The Drell-Yan mechanism

The Drell-Yan process consists in the production of a lepton pair l+l− with a large invariantmass-squared, M2 = (pl+ + pl−)2 � 1 GeV2, in quark-antiquark annihilation. This process isalso characterised by being completely inclusive for the presence of any other particle in the finalstate.

302 9 The production of vector bosons

HfqtelJ

Fig. 9.1. Lepton pair production in the Drell-Yan model.

absorbed into 'renormalized', scale-dependent parton distributions.1 Thekey point is that all collinear divergences appearing in the Drell-Yan cor-rections can be factored into renormalized parton distributions in thisway. Factorization theorems show that this is a general feature of inclu-sive 'hard-scattering' processes in hadron-hadron collisions [2]. We willsee explicitly how this happens at O(as) in the following section. Takinginto account the leading corrections, Eq. (9.1) then becomes

°AB = J2 J (9.2)

A final step is to take into account the finite contributions left behindafter the singularities have been factored into the parton distributions.These constitute a genuine O(as) perturbative correction to the crosssection, sometimes called the 'K-factor':

°AB = I dxldx2 fq(XUM2)fq{X2,M2)q J

X [<70 + a ax + a2 a2 + . . . , (9.3)

with a = as(M2)/27r. The structure displayed in Eq. (9.3) is completely

general, applicable to a wide variety of hard-scattering processes, includ-ing jet and heavy flavour production described in Chapters 7 and 10respectively. Most of the important hard-scattering processes have been

1 We use 'renormalized' here in the sense of Chapter 4, i.e. having combined collineardivergences with an unphysical bare distribution to produce a finite, physical partondistribution.

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Figure 2.1.1: Lepton pair production in the Drell-Yan model.

In the naive parton model, the cross section σAB for producing such a pair in the collision ofproton beam A and proton target B is obtained by simply weighting the subprocess cross sectionσ for qq → l+l− with the parton distribution functions (PDFs) fq(x) and fq(x) extracted from

2.1. The Drell-Yan mechanism 4

deep inelastic scattering, and summing over all quark-antiquark combinations in the beam andtarget:

σAB =∑q

1∫0

dx1

1∫0

dx2fq(x1)fq(x2)σqq→l+l−+X , (2.1.1)

where X stands for any additional particle in the final state. At lowest order in perturbationtheory, and assuming M � MZ

1, the cross section is given by the annihilation process, qq →γ∗ → l+l−. This lowest-order cross section for quark-antiquark annihilation into a lepton pairvia an intermediate off-mass-shell photon is easily obtained from the e+e− → qq:

σ(q (p1) q (p2)→ l+l−

)=

4πα2

3s

1

NQ2q , (2.1.2)

where s = (p1 + p2)2, α is the fine-structure constant, Qq is the quark electric charge and N isa colour factor which in this case equals 3 2.In general the incoming quark and antiquark have a spectrum of collision energies

√s, and so it

is more appropriate to consider the differential cross sections. Now, let

M2 = q2 (2.1.3)

be the square of the invariant mass of the Drell-Yan pair. We parametrize the longitudinalmomentum q using the rapidity, Y , of the virtual photon:

q0 = M coshY (2.1.4)

where q0 is measured in the pp centre of mass frame. We will express the longitudinal fractionsof the quarks, and hence the Drell-Yan cross section in terms of the observables M2 and Y . Inthe pp centre of mass frame, the proton momenta take the explicit form

P1 = (E, 0, 0, E), P2 = (E, 0, 0,−E) ,

where E satisfies s = 4E2 3 . Ignoring their small transverse momenta, we can write the quarkand antiquark momenta as x1 and x2 times these vectors, so that

q = x1P1 + x2P2 = ((x1 + x2)E, 0, 0, (x1 − x2)E) . (2.1.5)

By computing the invariant square mass of this vector, we find

M2 = x1x2s. (2.1.6)

Then, by comparing (2.1.4) with (2.1.5), we find out

expY =

√x1

x2(2.1.7)

By inverting (2.1.6) and (2.1.7) we can determine x1 and x2 in terms of M2 and s:1MZ is the Z boson mass.2N = 3 derives from an average over colour orientations of the initial-state quarks. It means that only when

the colour of the quark matches the colour of the antiquark can the annihilation into a colour singlet final statetake place.

3Remind that s = (P1 + P2)2, where P1 and P2 are the 4-momenta of the two protons.


x1 =M√seY , x2 =

M√se−Y . (2.1.8)

Using relations (2.1.6) and (2.1.7) we can now convert the integral over x1 and x2 in (2.1.1) intoan integral over the parametres M2, Y of the produced leptons. The Jacobian of the change ofvariables is easily computed:

∂(M2, Y

)∂ (x1, x2)

=

∣∣∣∣x2s x1s1

2x1− 1

2x2

∣∣∣∣ = s =M2

x1x2.

Therefore, the cross section for lepton pair production becomes:

d2σ

dM2dY

(pp→ l+l−

)=∑q

fq(x1)fq(x2)4πα2

3M2s

1

NQ2q , (2.1.9)

where x1 and x2 are given by Eq. (2.1.8). By such a change of variables we have written theDrell-Yan cross section in terms of variables carrying information completely derivable from deepinelastic scattering. These new variables are either completely invariant under longitudinal boost(such as M2) or merely additive (such as Y ) (which means that their differential is invariant).

Such a cross section does exhibit a scaling property: if we consider the differential lepton pairmass distribution only, the subprocess cross section for producing a lepton pair of mass M is

dσ

dM2=σ0

NQ2qδ(s−M2) , (2.1.10)

where we have defined

σ0 =4πα2

3M2. (2.1.11)

Reminding that s = x1x2s, substituting Eq. (2.1.10) into Eq. (2.1.1) gives the parton-modelcross section for this process:

dσ

dM2=

1∫0

dx1

1∫0

dx2

∑q

{fq(x1)fq(x2) + (q ↔ q)} × dσ

dM2

(qq → l+l−

)

=σ0

N

1∫0

dx1

1∫0

dx2δ(x1x1s−M2)×[∑

q

Q2q {fq(x1)fq(x2) + (q ↔ q)}

]. (2.1.12)

(Following the usual convention, the sum here is over quarks only and the additional qq contri-butions are indicated explicitly.)

In the parton model, the distribution functions fi(x) are indipendent of M2, and so the lep-ton pair cross section multiplied by M4 exhibits scaling in the variable τ = M2/s:


M4 dσ

dM2=

4πα2

3Nτ

1∫0

dx1

1∫0

dx2δ(x1x1 − τ)×[∑

q

Q2q {fq(x1)fq(x2) + (q ↔ q)}

]

=4πα2

3NτF(τ) . (2.1.13)

The Drell-Yan cross section data do indeed exhibit scaling behaviour to a good approximation[1], thus confirming the parton model. Actually such scaling property relies upon the assumptionthat the parton can have only limited transverse momentum k⊥ relative to the direction of theparent hadron in the infinite momentum frame. We can then generalize the idea of a partonprobability distribution to take this "intrinsic" transverse momentum into account:

dξf(ξ)→ d2k⊥dξP (~k⊥, ξ) , (2.1.14)

with∫d2k⊥P (~k⊥, ξ) = f(ξ). When the hard scattering scale in an inclusive process (such

as Drell-Yan’s) is very large, it may be a reasonable approximation to neglect the intrinsick⊥ altogether, which corresponds to P (~k⊥, ξ) = δ(~k⊥)f(ξ). Note that this approximation isimplicit in the representation used in (2.1.5) for the incoming parton (q and q) momenta in thederivation of the Drell-Yan cross section given above. This automatically implies that the leptonpair has zero transverse momentum. In fact experiments show that Drell-Yan lepton pairs havea distribution in transverse momentum, and this can be used to infer the distribution in thepartons’ intrinsic k⊥. Named ~p⊥ the transverse momentum of the lepton pair, it is found that atsmall p⊥, k⊥ distribution is Gaussian with a very good agreement [2], and the magnitude of thecorresponding intrinsic transverse momentum is indeed of the order of a typical hadronic massscale. A simple model is given by P (~k⊥, ξ) = h(~k⊥)f(ξ), which gives the following distributionin the transverse momentum of the lepton pair

1

σ

d2σ

dp2⊥

=

∫d2k⊥1

∫d2k⊥2δ

(2)(~k⊥1 + ~k⊥2 − ~p⊥)h(~k⊥1)h(~k⊥2) , (2.1.15)

where σ ≡ d2σ/dM2dY as defined above.However, there is an excess of events at large transverse momentum. This is evidence for a newperturbative mechanism for generating large p⊥: the 2 → 2 scattering processes qq → γ∗g andqg → γ∗q. Historically, this meant that quarks could "hard-scatter" strongly, generating thetransverse momentum of the lepton pair by emitting a gluon with large transverse momentum.This feature will be further discussed in Section 2.2

2.1.1 W and Z boson production

A striking feature coming from Drell-Yan cross section experimental data at√s = 1.8 TeV ([3, 4])

is the appearence of the Z resonance at M ∼ MZ . At such high energies, the qq → γ∗ → l+l−

contribution must be supplemented by the additional contribution from Z exchange. In practice,lepton pair production with M ∼ MZ is analysed in terms of the production cross section forZ bosons (qq → Z), multiplied by the branching ratio for decay into the leptonic final state(Z → l+l−).This actually led to consider as Drell-Yan processes two other processes: the production of thevector boson W and the production of the vector boson Z, both particles first discovered in 1983

2.2. QCD and perturbative corrections to Drell-Yan model 7

at CERN pp collider [5].The W and Z decay widths are small (ΓW = 2.08 GeV and ΓZ = 2.50 GeV in the StandardModel) compared to their masses, and so it is sufficient to consider the production of effectivelystable particles, multiplying the cross sections by the appropriate final-state branching ratios. Atleading order, the qq → W, Z subprocess cross sections are readily obtained from the couplingsof the gauge bosons to fermions:

σqq′→W =

π

3

√2GFM

2W

∣∣Vqq′∣∣2 δ(s−M2W )

σqq→Z =π

3

√2GFM

2Z

(V 2q +A2

q

)2δ(s−M2

Z) , (2.1.16)

where GF is the Fermi constant, MW andMZ are respectevelyW boson’s and Z boson’s masses,Vqq′ is the appropriate CKM matrix element and Vq and Aq are respectively the vectorial andaxial-vectorial coefficients of bosons’ coupling to fermions.The Z boson actually decays into two leptons Z → l+l−, and evidence of this particle was foundin studying the invariant mass distribution, that could be reconstructed from the 4-momenta ofthe leptons, which are easily detected. The same could not be done for the W boson: as it is anelectrically charged particle, it decays W+ → l+ν or W− → l−ν. Thus, since neutrinos cannotbe detected, we are unable to reconstruct the square invariant mass distribution, yet we can useother variables built from the transverse momentum of charged lepton and neutrino (computedby difference from lepton’s one), such as the transverse mass distribution, as we will explain inSection 2.3.

2.2 QCD and perturbative corrections to Drell-Yan model

In this section we present the O (αS) and the O(α2S

)QCD corrections to the Drell-Yan cross

section, with particular relevance to the first one. We will discuss also other QCD features, suchas renormalization and factorization scales. We will end by presenting some features about thetransverse momentum of the Drell-Yan pair and about the resummation scale.

2.2.1 LO and NLO QCD perturbative corrections

We now present the O (αS) corrections to the parton model Drell-Yan cross section, where αS isthe strong coupling constant. We begin by considering the parton-level Drell-Yan cross sectionfor the leading-order process q(p1) + q(p2)→ l+l−:

M4 dσ

dM2=

4πα2

3NτF(τ) , (2.2.1)

where, from Eq. (2.1.10),

F = Q2qδ(1− τ) (2.2.2)

for a single flavour of quark with charge Qq. In general, F will have a perturbative seriesexpansion in powers of the strong coupling constant:

F = F0(τ) +αS2πF1(τ) + . . . . (2.2.3)


The contributions at O (αS) concerns: (1) virtual gluon corrections to the leading-order contri-bution

(Fqq,V1

), (2) real gluon corrections from q(p1) + q(p2)→ γ∗ + g(k)

(Fqq,R1

), and (3) the

quark-gluon scattering process q(p1) +g(p2)→ γ∗+ q(k)(Fqg1

)together with the corresponding

qg contribution.When the bare parton distributions, which must be convoluted with the Fqq1 and Fqg1 functions,are replaced by scale-dependent "renormalized" distributions, the collinear singularities (whichderive from processess (1)-(2) and (3)) exactly cancel. Actually this is the verification at O (αS)of a factorization theorem [6], which states that for a general class of inclusive "hard-scattering"processes in hadron-hadron collisions all collinear singularities at every order in perturbationtheory can be absorbed into universal parton distributions. The remaining finite perturbativecorrections modify the leading-order parton-model cross section.To conclude this short discussion over O (αS) corrections, we highlight that their magnitudemainly depends on the lepton pair mass and on the overall collision energy.The O

(α2S

)corrections to the Drell-Yan cross section can also be computed, but their expres-

sions are too lenghty to present. These next-to-leading order contributions are generally smallerthan the O (αS) corrections, the exact values depending on the lepton pair mass, the collisionenergy, and the type of beam and target.We remark that the O (αS) perturbative QCD corrections to the W and Z cross sections arethe same (for a virtual photon of the same mass) — the gluon is "flavour blind" and couplesin the same way to the annihilating quark and antiquark. The O

(α2S

)corrections are also the

same, apart from an internal quark-triangle graph contribution to Z production only which is,however, numerically small [7].

Beside these perturbative corrections, there are two other QCD features that must be takeninto account. The first one refers to proton PDF: the Parton Distribution Functions are notindependent of the hard scattering process collision energy, as the parton model would grant. Asa consequence, the proton PDF will have a further dependence:

fi(x, µ2F ) , (2.2.4)

where µF is named factorization scale.Note that the magnitude of µF should be of the same order of the energy at which the processoccurs.The name "factorization scale" is due to the fact that in a way it tells us the limit beyond whichwe are no longer able to resolve the inner structure of the proton, thus factorizing its description.The second feature involves the strong coupling constant αS . As a measurable quantity, αSdepends on the energy scale of the process, schematically:

αS = αS(µ2R) , (2.2.5)

where µR is named renormalization scale.Again, the magnitude of µR should be of the same order of the energy at which the processoccurs.


2.2.2 W and Z transverse momentum distributions

Like Drell-Yan lepton pairs, even most W and Z bosons (collectively denoted by V ) are pro-duced with relatively little transverse momentum, i.e. p⊥ � MV . However, part of the totalcross section corresponds to the production of large transverse momentum bosons. The relevantmechanisms are the 2 → 2 processes qq → V g and qg → V q. The diagrams are identical tothose for large p⊥ direct photon production; by computing the annihilation and Compton matrixelement, one finds:

∑∣∣∣Mqq′→Wg∣∣∣2 = παS

√2GFM

2W

∣∣Vqq′∣∣2 8

9

t2 + u2 + 2sM2W

tu,∑∣∣∣Mgq→Wq′

∣∣∣2 = παS√

2GFM2W

∣∣Vqq′∣∣2 1

3

s2 + u2 + 2tM2W

−su , (2.2.6)

where GF is the Fermi constant, Vqq′ is the appropriate CKM matrix element, MW is the massof the W boson, s is the square invariant mass, t and u are defined as follows:

t = −1

2s(1− cos θ∗) ,

u = −1

2s(1 + cos θ∗) , (2.2.7)

where θ∗ is the scattering angle as measured in centre-of-mass frame.Similar results hold for the Z boson.The transverse momentum distributions dσ/dp2

⊥ are obtained by convoluting these matrix ele-ments with parton distributions. The next-to-leading-order (O

(α2S

)) perturbative corrections,

obtained from real emission processes like qq → V gg and virtual (loop) corrections to the leading-order processes, have also been calculated [8].The poles at t = 0 and u = 0 in the matrix elements (2.2.6) cause the leading-order theoreticalcross section to diverge as p⊥ → 0. The leading behaviour at small p⊥ comes from the emissionof a soft (kµ → 0) gluon in the process qq → V g. Schematically (with M = MW or MZ)

dσR

dp2⊥

= αS

(A

ln(M2/p2⊥)

p2⊥

+B1

p2⊥

+ C(p2⊥)

), (2.2.8)

where A and B are calculable coefficients and C is an integrable function. However, that thecomplete O (αS) (real plus virtual) correction to the total cross section is finite.If we consider the emission of multiple soft gluons, the leading contributions at each order havethe form

1

σ

dσ

dp2⊥' 1

p2⊥

[A1αS ln

M2

p2⊥

+A2α2S ln3 M

2

p2⊥

+ · · ·+AnαnS ln2n−1 M

2

p2⊥

+ . . .

], (2.2.9)

where the Ai are calculable coefficients of order unity. The higher-order terms in the series areevidently important when

αS ln2 M2

p2⊥> 1 . (2.2.10)


Taking into account the relative magnitude of the An coefficients, this corresponds to p⊥ valuesless than 10 — 15 GeV.Fortunately, the leading logarithms in Eq. (2.2.9) can be resummed to all orders in perturbationtheory. The resummed double leading logarithms give

1

σ

dσ

dp2⊥' d

dp2⊥

exp

(−αS

2πCF ln2 M

2

p2⊥

), (2.2.11)

which vanishes at p⊥ = 0. However, note that the production of a W or Z boson with p⊥ ≈ 0does not require that all emitted gluons are soft, merely that their vector transverse momentumsum is small. In particular, the double-leading-logarithm result (2.2.11) omits contributions fromthe multiple emission of soft gluons with k⊥i ∼ p⊥ and

∑i~k⊥i = ~p⊥. Such additional contribu-

tions "fill in" the dip at p⊥ ' 0 predicted by (2.2.11).In Fig. 2.2.1 we report the graphic of dσ/dpW⊥ as obtained from a computer numerical simulation.

0

50

100

150

200

0 5 10 15 20 25

dσ

dpW ⊥

(pb/G

eV)

pW⊥ (GeV)

pW−⊥ distribution as

given from a simulation

MC

Figure 2.2.1: pW−⊥ distribution as computed by a numerical simulation.

The resummed series (2.2.11) solve the divergence problem at small p⊥ and it well parametrizesthe region on the left of the peak in Fig. 2.2.1. However it does not work at large p⊥, which isthe region on the right of the peak in Fig. 2.2.1. In such limit there is no divergence problem,and the perturbative results are reliable. Then, one way to account for both these behavioursis to convolute them into a single formula by introducing a third scale µQ, named resummationscale.This scale has to account for these two behaviours in different regions, and it appears, for instance,in the exponential (2.2.11) as:

exp

(−αS

2πCF ln2

µ2Q

p2⊥

). (2.2.12)

2.3. The studied observables 11

Note that although we derived these results for W bosons, the same discussion also holds for Zbosons.

2.3 The studied observables

In this section we will take a look at the distributions we will use to analyse the Drell-Yan process,describing their features and explaining the choice made.

2.3.1 W and Z transverse mass

In the rest frame of the decaying W 4, the energy of the charged lepton (which we shall assumefor definiteness to be an electron) is simply MW /2, and this can be utilized to make a precisionmeasurement of the W mass. Because the neutrino is not directly detected in the experiments,it is not possible to reconstruct the W rest frame precisely. However the transverse momentumof the electron, which is invariant under longitudinal boosts, also carries information on MW .If we compute the angular distribution of the electron in the W rest frame, we find:

1

σ

dσ

d cos θ∗=

3

8

(1 + cos2 θ∗

), (2.3.1)

where we have averaged over W+ and W− production and θ∗ is the angle between the chargedlepton direction and the incoming beam. If we assume that theW has zero transverse momentum,then cos θ∗ is given in terms of the transverse momentum (p⊥e) of the electron by

cos θ∗ =

(1− 4p2

⊥es

) 12

(2.3.2)

so that

1

σ

dσ

dp2⊥e

=3

s

(1− 4p2

⊥es

)− 12(

1− 2p2⊥es

) 12

, (2.3.3)

and such change of variables introduce a Jacobian factor that diverges as p⊥e →√s

2 The processis then resonant at s 'M2

W , and the distribution (2.3.3) is strongly peaked at p⊥e = MW /2 (theJacobian peak), where we have set s = M2

W , and it can therefore provide an accurate measurementofMW . In reality, the square-root singularity of the distribution is somewhat smeared out by thefinite width and non-zero transverse momentum of theW bosons. Information from the neutrinotransverse momentum can also be taken into account, thus further sharpening the dependenceof the distribution on MW . Note that the Jacobian factor appears when changing variable fromthe scattering angle in the centre-of-mass frame, to the transverse momentum of the electron; itis actually an integration factor affecting all those variables built from p⊥e.Identifying the magnitude and azimuthal angle of the missing transverse energy with the neutrinotransverse momentum, one can define the transverse mass,

M2⊥ = 2 |p⊥e| |p⊥ν | (1− cos ∆φeν) . (2.3.4)

4From now on we will talk about the W boson for ease; the same considerations, unless explicitly stressed,hold for the Z boson


At leading order (i.e. qq′ → W → eν) and in the absence of any quark transverse momentum,we have |p⊥e| = |p⊥ν | = p∗, ∆φeν = π, and so M⊥ = 2 |p⊥e|. The transverse mass distributiontherefore also has a Jacobian peak, at M⊥ = MW . In terms of measuring MW , the advantageof using the transverse mass distribution is that it is less sensitive to the transverse momentum(pW⊥ ) of the W boson. If pW⊥ is small, the transverse momenta of the leptons in the laboratoryand W centre-of-mass frames are related by a simple Galilean transformation:

p⊥e = p∗ +1

2pW⊥

p⊥ν = −p∗ +1

2pW⊥ . (2.3.5)

It is straightforward to show that, to leading order in pW⊥ , Eq. (2.3.4) is unchanged by such atransformation.

Note that the transverse mass can be completely reconstructed during experiments, since bothlepton’s and neutrino’s transverse momentum can be observed (the first one directly, the sec-ond one by difference), which allows physicists to be sensitive to MW in the transverse massdistribution because of the Jacobian enhancement at W resonance.

2.3.2 The Xm⊥ and Xp⊥ observables

Using the lepton transverse momentum (p⊥l) and the transverse mass, we can build two newadimensional variables defined as follows:

XVm⊥

=M⊥MV

(2.3.6)

XVp⊥

=2p⊥lMV

. (2.3.7)

As we see, XVm⊥

is merely the transverse mass normalized to the nominal value ofMV5: therefore

its distribution has a Jacobian peak at XVm⊥' 1. On the other hand XV

p⊥is twice the transverse

momentum normalized to the nominal value ofMV . Again it shows a Jacobian peak at XVp⊥' 1.

In Fig. 2.3.1 and Fig. 2.3.2 we report the graphics of dσ/dXWm⊥

and dσ/dXWp⊥

as obtained froma computer numerical simulation.The two distributions are affected by QCD, although at different levels. Since XV

m⊥refers to a

property of the lepton pair it is less sensitive to how we describe the fall-back against a gluon:therefore it will be less sensitive to the resummation scale, whereas it will be more sensitive to thefactorization scale, which concerns the description of protons. On the other hand, XV

p⊥is more

sensitive to the way we describe the gluon emission, thus being more sensitive to the resummationscale. It will also be very sensitive to the proton description, then to the factorization scale. Onthe contrary the renormalization scale affects the whole process and therefore both variables areinfluenced by its value.

Along with variables XVm⊥

and XVp⊥

, we have considered in our analysis the pV⊥ variable, whosefeatures have already been presented in Section 2.2.2 together with its distribution’s shape (Fig.

5Remind that V stands for both Z and W .


0

2000

4000

6000

8000

10000

12000

14000

16000

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

(pb)

XWm⊥

XW−m⊥ distribution as

given from a simulationMC

Figure 2.3.1: XWm⊥

distribution as computed by a numerical simulation.

0

1000

2000

3000

4000

5000

6000

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

(pb)

XWp⊥

XW−p⊥ distribution as

given from a simulationMC

Figure 2.3.2: XWp⊥

distribution as computed by a numerical simulation.

2.2.1). As it is a property of the gauge boson, it is correlated to the pair description, and thusit may be convenient to include it in the set of studied observables. It is deeply affected by thechoice of the resummation scale, that is to say our description of the fall-back against a gluon,and by the choice of the factorization scale. These features will be shown in Fig. 2.4.2.


2.3.3 The ratio observables

Other variables may be defined to study the Drell-Yan W production. For instance, for eachchosen variable we define the ratios W/Z. They are important because QCD treats W and Zboson production in the same way, and therefore the dependence on QCD scales is almost thesame for both processes. If a certain choice of QCD scales induces an effect on the shape ofa W observable distribution, a similar effect will be seen in the corresponding Z distribution.As a consequence, the ratio observables should be more stable with respect to the QCD scalechoice, due to relative compensations which occurs when taking the ratio, yet they will still carryinformation about the W boson and its mass.Again, the studied observables are XW/Z

m⊥ = XWm⊥

/XZm⊥

and XW/Zp⊥ = XW

p⊥/XZ

p⊥. Since they are

adimensional and peaked at 1, the ratio observables are somehow expected to be rather flat,which would imply that W and Z production occurs in the same way. However the differencesbetween the two processes result in distortions of the plot, as shown by the graphics in Fig. 2.3.3.

0

1

2

3

4

5

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

/dσ

dX

Z m⊥

XWm⊥

XW−m⊥ ratio distribution

as given from a simulation

MC

(a) XW/Zm⊥ .

0

1

2

3

4

5

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

/dσ

dX

Z p⊥

XWp⊥

XW−p⊥ ratio distribution

as given from a simulation

MC

(b) XW/Zp⊥ .

Figure 2.3.3: Ratio distributions as computed by a numerical simulation.

We see that the two distributions are not flat, as a consequence of the difference between theprocesses involved.

Also for the pV⊥ variable we have considered the ratio observable, pW⊥ /pZ⊥.

2.3.4 The normalized observables

In addition to bare absolute distributions, we can also introduce normalized distributions, labeledwith an overbar, defined as follows:

X =1

σ

dσ

dX, (2.3.8)

where X is the considered observables, and σ is the integral norm of the differential cross sectionwith respect to X, defined as:

σ =

Xmax∫Xmin

dσ

dXdX . (2.3.9)

2.4. Scale uncertainties on Xm⊥ and Xp⊥ 15

Xmin and Xmax mark the range within which we want to normalize the variable X.The measurement of the value ofMW is somehow identified with the position of a resonance peakof certain distributions, or, in other words, with a particular shape of such distributions. Thisnormalization procedure allows us to remove the normalization uncertainties and let us studythe shape uncertainites only.We can still talk about normalized distributions also for the ratio observables introduced inSection 2.3.3. In such case, the normalization (always computed through (2.3.9)) happens afterthe definition of the ratio. In other words, we first take the ratio and generate the new observables,and then we normalize them. The normalized ratio observables are not computed as ratio betweentwo normalized observables themselves.

2.4 Scale uncertainties on Xm⊥ and Xp⊥

As we said in Section 2.2, the three QCD scales are needed to describe the Drell-Yan process ofW and Z production. Their values are usually expressed as a multiple ξ of the typical energyscale of the process (i.e., MW for W boson production while MZ for Z boson production):

µR = ξRMV

µF = ξFMV

µQ =xQ2MV = ξQMV . (2.4.1)

However, the choice of such scales is arbitrary, but, if we could compute the cross section toall orders in perturbation theory, its value σ would be independent of the scale choice. Sincewe cannot treat the Drell-Yan process perturbatevely to all orders and our computations areworked out up to a truncated order (the second order) in perturbation theory, we have a residualdependence on renormalization, factorization and resummation scales. The subjectivity we havein their choice leads to an uncertainty in the calculus, as we see from the different shapes of XW

m⊥and XW

p⊥variable plots in Fig. 2.4.1 and, for pW⊥ variable, in Fig. 2.4.2.


2000

4000

6000

8000

10000

12000

14000

16000

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

(pb)

XWm⊥

ξF = 1.00, ξQ = 0.50 ξR = 0.25ξR = 0.50ξR = 1.00ξR = 2.00ξR = 4.00

(a) XWm⊥ : fixed values of ξF and ξQ.

1000

2000

3000

4000

5000

6000

7000

0.6 0.7 0.8 0.9 1 1.1 1.2dσ

dX

W p⊥

(pb)

XWp⊥

ξF = 1.00, ξQ = 0.50ξR = 0.25ξR = 0.50ξR = 1.00ξR = 2.00ξR = 4.00

(b) XWp⊥ : fixed values of ξF and ξQ.

2000

4000

6000

8000

10000

12000

14000

16000

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

(pb)

XWm⊥

ξR = 1.00, ξQ = 0.50 ξF = 0.25ξF = 0.50ξF = 1.00ξF = 2.00ξF = 4.00

(c) XWm⊥ : fixed values of ξR and ξQ.

1000

2000

3000

4000

5000

6000

7000

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

(pb)

XWp⊥

ξR = 1.00, ξQ = 0.50ξF = 0.25ξF = 0.50ξF = 1.00ξF = 2.00ξF = 4.00

(d) XWp⊥ : fixed values of ξR and ξQ.

2000

4000

6000

8000

10000

12000

14000

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

(pb)

XWm⊥

ξR = 1.00, ξF = 1.00ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(e) XWm⊥ : fixed values of ξR and ξF .

1000

2000

3000

4000

5000

6000

7000

8000

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

(pb)

XWp⊥

ξR = 1.00, ξF = 1.00 ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(f) XWp⊥ : fixed values of ξR and ξF .

Figure 2.4.1: W− numerical simulation: XWm⊥

and XWp⊥

plot shape distortions due to differentchoices of QCD scales.


0

50

100

150

200

250

0 5 10 15 20 25

dσ

dpW ⊥

(pb/G

eV)

pW⊥

ξF = 1.00, ξQ = 0.50

ξR = 0.25ξR = 0.50ξR = 1.00ξR = 2.00ξR = 4.00

(a) pW⊥ : fixed values of ξF and ξQ.

0

50

100

150

200

250

0 5 10 15 20 25

dσ

dpW ⊥

(pb/G

eV)

pW⊥

ξR = 1.00, ξQ = 0.50

ξF = 0.25ξF = 0.50ξF = 1.00ξF = 2.00ξF = 4.00

(b) pW⊥ : fixed values of ξR and ξQ.

0

50

100

150

200

250

0 5 10 15 20 25

dσ

dpW ⊥

(pb/G

eV)

pW⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(c) pW⊥ : fixed values of ξR and ξF .

Figure 2.4.2: W− numerical simulation: pW⊥ plot shape distortions due to different choices ofQCD scales.

As we see, in each histogram there is a band of values, as a consequence of a different choice ofparametres ξR, ξF and ξQ. This means that the theoretical prediction depends on the choice ofQCD scales.In such distributions we can observe a peak that is interpreted in light of the presence of a Wboson6, and whose position tells us something about its mass MW . An accurate comparisonbetween the theoretical prediction and data allows the determination of its value. In order towork out such comparison we use theoretical numerical simulations, for which we have to make achoice for the values of the QCD scales. As a consequence, the resulting MW value will dependon these scales and it will be influenced by their values. The choice of QCD scales is then asource of theoretical systematic error upon the value MW .

6We refer to the W boson only, nevertheless the same considerations hold for the Z boson.

Chapter 3

W boson’s mass determination

In this section we first present how the value of the W boson’s mass can be extracted from realdata, and then we will present our way of working, which is set in a purely theoretical frame. Wewill therefore present how we may estimate the impact of QCD scale choice on MW ending withsome considerations about the fit procedure in presence of correlation and the variables includedin such fit.

3.1 The Template Fit technique

In this section we first present how real data are analysed in order to extract the value of aparametre involved in the studied process. Then we discuss how we can estimate in a purelytheoretical frame the distortions due to different QCD scale choices usingMW as quantifier, thushaving also an estimate of the derived systematic uncertainty upon such value.

3.1.1 Analysis of real data

In reality, the mass of the W boson can be extracted from experimental data which usually referto observables such the transverse mass or the charged lepton transverse momentum. Usingdetectors, physicists are able to acquire these data, but a great role is played by the detectorsthemselves. As a consequence, physicists have to know the detectors’ properties, their responseto a stress and their limits. Detectors have to be calibrated in order to be accurate, which meansthat they have to respond in a specific way to a specific stress. Moreover they have to be efficient:as a matter of fact any detector is not able to detect everything which means that some signalsmight be lost or slightly distorted, which unavoidably results in distortions in the shape of datahistograms. All these effects have to be somehow accounted for when simulating the events inorder to compare the simulation data to the experiment’s ones. Since the simulation data will beused to fit (as we will later explain) the experimantal data, a large part of such simulations willbe dedicated to the so called detector simulation, that is to say, after the event generations, thetheoretical simulation emulates the detectors’ behaviour thus making the simulation data moresimilar to real ones.

The detector simulation is very important within the theoretical simulation of the experiments,and its implementation is specific of the chosen experiment and its features (it is quite obvious

3.1. The Template Fit technique 19

that the detectors vary from one experiment to another). The reason of the importance of atheoretical simulation relies on the tecnique used to extract the value of a parametre (such asMW ) from the esperimental data. It is commonly employed the Template Fit procedure.First we create a large number of histograms by means of the numerical simulation, each corre-sponding to a different value of the parametre we would like to measure. These histograms arecalled templates and they are to be compared to real data obtained from experiments: the bestestimates of the parametre is given by the template which maximizes the agreement with thedata set. In order to "compute" such agreement the least square method is broadly used, relyingon the chi square evaluator, based on the following formula:

χ2 =

Nbins∑k=1

(Ok − Tk)2

σ2Ok + σ2

T k

, (3.1.1)

where Ok stands for the value of the k-th bin of the data histogram, while Tk stands for the valueof the k-th bin of the selected template histogram; σ refers to the error on the value of the k-thbin.Clearly, since it is χ2 > 0 the perfect agreement would correspond to χ2 = 0. As this is highlyunrealistic, we will have that the lower the χ2 value is, the higher the agreement will be. Bycomputing the χ2 value for all the templates, the criteria which will select one template amongall of them is given by the χ2 minimization: the value of the parametre which best estimates thephysical quantity we want to measure will be the one referring to the template that best fits thedata minimizing the χ2.We remark that the formula (3.1.1) relies upon the strong assumption that all bins are notcorrelated to one another, in other words they can be considered indipendent. Such choice mightnot be the correct one, and a more precise analysis should also investigate the presence of acertain degree of correlation, as we will explain in Section 3.3.

3.1.2 Comparison of theoretical distributions

In a purely theoretical frame, we are interested in exploring how much a distribution generatedfor fixed values of QCD scales differs from a reference distribution with other fixed values ofQCD scales (both these distributions are calculated using the same value of MW = M

(0)W ). This

is achieved using the least square method, and the W mass in order to estimate the "distance"between the two distributions. Templates are generated for different values of MW , using thesecond choice of QCD scales. Then the two distributions are fitted, and the template corre-sponding to the minimium chi square will provide the value of MW which best emulates thebehaviour of the different QCD scale choice for both distributions. We expect that M (2)

W = M(0)W

since the corresponding distributions have been generated using the same QCD scale choice, yetgenerally M (1)

W 6= M(0)W . The difference between the M (1)

W and M (2)W will then tell us how the two

distributions differ from one another.To better explain such method we now give an example. Suppose that we generated twoXW

m⊥dis-

tributions: distribution number 1 using the QCD scale choice ξR = 2.00, ξF = 0.50, ξQ = 1.000,and distribution number 2 with the choice ξR = 1.00, ξF = 1.00, ξQ = 0.500, both using thevalue MW = 80.398 GeV. We now want to estimate how much they differ. We then generatetemplates using the QCD scale choice ξR = 1.00, ξF = 1.00, ξQ = 0.500 for different values ofMW (for instance 80.298 GeV ≤MW ≤ 80.498 GeV, with a step of 2 MeV). We now fit the twodistributions: the second one will exactly correspond to the template with MW = 80.398 GeV,


the second one istead will maximize the agreement with a different template. Let it be the tem-plate with MW = 80.420 GeV. Then the "distance" between the two distributions is given bythe difference of the MW values obtained from the fit: (80.420− 80.398) GeV = 22 MeV.Ideally, we would like to fit the real data, instead we introduce MW to estimate in a purely the-oretical frame the difference between the two hypothesis. Using this way of working we can givean estimate of the theoretical systematic uncertainty that affects the experimental measurementof MW due to the arbitrary choice of QCD scales.This way of working has been adopted for each QCD scale combination, and the obtained set ofdata (125 distributions for each chosen variable) is referred to as the set of pseudo-data.To sum up, we have a set of pseudo-data and we have generated a set of reference theoreticaldistributions for different values of MW : we expect that if any distribution is distorted withrespect to the reference one, the extracted value of MW by means of Template Fit will changethus undergoing a shift; the spread of all the shifts from the set of pseudo-data will give us anestimate of the difference between the two distributions and of the theoretical uncertainty derivedfrom the QCD scale choice.We can actually appreciate such distortions in Fig. 3.1.1 and for normalized distributions in Fig.3.1.2. We highlight how the distortions referring to mere ξR and ξF variations are suppressedonce the distributions have been normalized, since these scales are more strictly related to thenormalization of the distributions themselves.


0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

/dσ

dX

ref

m⊥

XWm⊥

ξF = 0.50, ξQ = 1.000

ξR = 0.25ξR = 0.50ξR = 1.00ξR = 2.00ξR = 4.00


0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

/dσ

dX

ref

p⊥

XWp⊥

ξF = 0.50, ξQ = 1.000

ξR = 0.25ξR = 0.50ξR = 1.00ξR = 2.00ξR = 4.00


0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

/dσ

dX

ref

m⊥

XWm⊥

ξR = 1.00, ξQ = 1.000

ξF = 0.25ξF = 0.50ξF = 1.00ξF = 2.00ξF = 4.00


0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

/dσ

dX

ref

p⊥

XWp⊥

ξR = 1.00, ξQ = 1.000

ξF = 0.25ξF = 0.50ξF = 1.00ξF = 2.00ξF = 4.00


0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

/dσ

dX

ref

m⊥

XWm⊥

ξR = 1.00, ξF = 0.50

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000


0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

/dσ

dX

ref

p⊥

XWp⊥

ξR = 1.00, ξF = 0.50

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000



and XWp⊥

plot shape distortions due to differentchoices of QCD scales, resulting in the ratio with respect to the ξR = 1.00, ξF = 1.00, ξQ = 0.500distribution (labeled as ref distribution).


0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

/dσ

dX

ref

m⊥

XWm⊥

ξF = 0.50, ξQ = 1.000

ξR = 0.25ξR = 0.50ξR = 1.00ξR = 2.00ξR = 4.00


0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

/dσ

dX

ref

p⊥

XWp⊥

ξF = 0.50, ξQ = 1.000

ξR = 0.25ξR = 0.50ξR = 1.00ξR = 2.00ξR = 4.00


0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

/dσ

dX

ref

m⊥

XWm⊥

ξR = 1.00, ξQ = 1.000

ξF = 0.25ξF = 0.50ξF = 1.00ξF = 2.00ξF = 4.00


0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

/dσ

dX

ref

p⊥

XWp⊥

ξR = 1.00, ξQ = 1.000

ξF = 0.25ξF = 0.50ξF = 1.00ξF = 2.00ξF = 4.00


0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

/dσ

dX

ref

m⊥

XWm⊥

ξR = 1.00, ξF = 0.50

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000


0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

/dσ

dX

ref

p⊥

XWp⊥

ξR = 1.00, ξF = 0.50

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000



and XWp⊥

plot shape distortions due to differentchoices of QCD scales, resulting in the ratio with respect to the ξR = 1.00, ξF = 1.00, ξQ = 0.500distribution (labeled as ref distribution). We notice that the shift distortions due to the variationof ξR or ξF only are suppressed, and the distribution shapes almost coincide.The distributions have been normalized in the range [0.6, 1.2].

3.2. The template fit analysis of W and Z simulations 23

3.2 The template fit analysis of W and Z simulations

In this section we describe the features of the template fit analysis of W and Z simulations,presenting the pseudo-data, the templates and the observables included in the fit.

3.2.1 The set of pseudo-data

As we said in Section 2.2, in studying the Drell-Yan process W and Z production there arethree unphysical parametres we introduce to make our description of the process realistic. Aswe said in Section 2.4, the renormalization, factorization and resummation scale are expressedas a multiple ξ of MV (cfr. (2.4.1)). These scales are usually made to vary within the rangeof values {1/2, 1, 2} (the range of values for ξQ is instead {1/4, 1/2, 1} because of the 1/2factor in (2.4.1)). Since our aim is to quantify how much a distortion given by a certain choiceof these parametres results in a shift in the measure of MW , we decided to use the wider range{1/4, 1/2, 1, 2, 4} (the half of values for ξQ). Thus we generated 53 sets of pseudo-data, onefor each combination of the three scales, wich simulate the real data.The pseudo-data have been generated by means of numerical simulation for both W and Zprocesses, and they have then been used also to generate the 125×125 ratio sets of pseudo-data.

3.2.2 The choice of templates

The Template Fit procedure in studying the Drell-Yan process has been applied to the analysisof two different types of histograms, referring to variables XV

p⊥and XV

m⊥. As we said at the

end of Section 2.2, the first one is more sensitive to QCD scales, whereas the second one isless. The corresponding W -template histograms have been generated for different values of MW ,ranging from 80.198 GeV to 80.598 GeV for XW

p⊥with a step of 4 MeV, and from 80.298 GeV to

80.498 GeV for XWm⊥

with a step of 2 MeV. The difference between the two steps is due to thedifferent sensitivity of templates during the fitting procedure. This choice allows us to be moreprecise in the extraction of MW from the fitting procedure.As these templates have a fixed choice of QCD parametres (the central choice), the simulation runwas unique. The procedure used to create histograms for different values of MW is a reweightingprocedure: the propagator term referring to W production has been multiplied and divided bythe factor (

s−MW20

)2+ Γ2

WMW20(

s−MW21

)2+ Γ2

WMW21

, (3.2.1)

where ΓW is the decay width of the W boson, and the pedices 0 and 1 in MW refer to theprevious choice of MW value (the 0) and the new one (the 1).In this way, we account for the new choice ofMW using an exact procedure to generate templates.Such procedure does not lead to errors, altough it may lead to a certain degree of correlationamong all templates.

In Fig. 3.2.1 we have reported the ratio of different templates with respect to the MW =80.398 GeV-template (central choice1). We can actually appreciate the difference in these sets ofdata through the distortion of the corresponding plot. Moreover, we remark that the distortions’magnitude for XW

p⊥is about 0.1% whereas for XW

m⊥is about (0.5− 1)%.

1When talking about templates, the central choice refers to the template generated using MW = 80.398 GeV.


0.998

0.999

1

1.001

1.002

1.003

1.004

0.6 0.7 0.8 0.9 1 1.1 1.2

ratio

XWp⊥

templates ratio withrespect to MW = 80.398 GeV

MW = 80.386 GeVMW = 80.394 GeVMW = 80.398 GeVMW = 80.402 GeVMW = 80.410 GeVMW = 80.418 GeV

(a) XWp⊥ templates.

0.99

0.995

1

1.005

1.01

1.015

1.02

0.6 0.7 0.8 0.9 1 1.1 1.2

ratio

XWm⊥

templates ratio withrespect to MW = 80.398 GeV

MW = 80.388 GeVMW = 80.394 GeVMW = 80.398 GeVMW = 80.402 GeVMW = 80.408 GeVMW = 80.418 GeV

(b) XWm⊥templates.

Figure 3.2.1: Template ratio for variables XWp⊥

and XWm⊥

; W− channel.

Templates have not been generated for the Z boson variables. In any case we considered asZ-templates the set of data computed using the central choice of QCD scales.

3.2.3 Observables included in the fit

As we said in Section 2.3, we considered in our analysis the lepton pair transverse mass distri-bution and the charged lepton’s transverse momentum distribution for both W and Z bosons,conveniently normalized to the nominal value of MV so as to show the Jacobian peak at aboutXVk ' 12. Moreover we also considered the pZ⊥ observable.

It is quite natural to consider the W observables since they have a straightforward sensitivityto MW , however we can also account for Z data. We can consider Z observables because theZ process description has a very low sensitivity to MW , which fits as an overall normalizationfactor3. From now on we will ignore such dependence.In spite of the absence of sensitivity to MW , it is important to include Z observables in order toreduce the overall uncertainty due to QCD scales. Moreover, QCD is "flavour blind" and thenit treats W and Z production in a similar way, notwithstanding the differences between the twoprocesses and of the two particles: W and Z processes show similar behaviours when we applythe same QCD variation. It is then reasonable to expect that a combination (not necessarily alinear one) of W and Z data might minimize the QCD scale dependence.The first naive combination we decided to use is the ratio W/Z, as forementioned in Section2.3.2: the ratio histograms are built by assigning to their k-th bin the ratio between the value ofk-th bin of W ’s and Z’s histograms. The error upon such ratio is propagated starting from theerror on W and Z values.This simple combination should reduce the shape distortions due to the choice of QCD scale, aswe can actually see in Fig. 3.2.2 and, for normalized distributions, in Fig. 3.2.3. We notice thatthe similar behaviour of W and Z processes under QCD effects flattens the shapes of the ratiodistribution plots.

2Remind that both XVp⊥ and XV

m⊥ are dimensionless by construction.3MW cannot fits in Z propagator as it only accounts for the double contribution of Z exchange and of a photon

exchange.


0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

Z m⊥

/dσ

dX

ref

m⊥

XZm⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(a) Z: XZm⊥ .

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

Z p⊥

/dσ

dX

ref

p⊥

XZp⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(b) Z: XZp⊥ .

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

/dσ

dX

ref

m⊥

XWm⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(c) W−: XWm⊥ .

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

/dσ

dX

ref

p⊥

XWp⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(d) W−: XWp⊥ .

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W/Z

m⊥

/dσ

dX

m⊥

ref

XWm⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(e) ratio: XW/Zm⊥ .

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W/Z

p⊥

/dσ

dX

p⊥

ref

XWp⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(f) ratio: XW/Zp⊥ .

Figure 3.2.2: Xm⊥ and Xp⊥ plot ratio with respect to the central choice distribution (labeled asref distribution) for Z, W and ratio observables. The values of ξR and ξF are both set to 1.00,while ξQ varies within its whole range.


0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

Z m⊥

/dσ

dX

ref

m⊥

XZm⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(a) Z: XZm⊥ .

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

Z p⊥

/dσ

dX

ref

p⊥

XZp⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(b) Z: XZp⊥ .

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W m⊥

/dσ

dX

ref

m⊥

XWm⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(c) W−: XWm⊥ .

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W p⊥

/dσ

dX

ref

p⊥

XWp⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(d) W−: XWp⊥ .

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W/Z

m⊥

/dσ

dX

ref

m⊥

XWm⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(e) ratio: XW/Zm⊥ .

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.6 0.7 0.8 0.9 1 1.1 1.2

dσ

dX

W/Z

p⊥

/dσ

dX

ref

p⊥

XWp⊥

ξR = 1.00, ξF = 1.00

ξQ = 0.125ξQ = 0.250ξQ = 0.500ξQ = 1.000ξQ = 2.000

(f) ratio: XW/Zp⊥ .

Figure 3.2.3: Xm⊥ and Xp⊥ plot ratio with respect to the central choice distribution (labeled asref distribution) for Z, W and ratio observables. The values of ξR and ξF are both set to 1.00,while ξQ varies within its whole range. The normalization range is [0.6, 1.2].

3.3. The least square method in MW determination 27

For the ratio graphics of Fig. 3.2.2 and Fig. 3.2.3, we used the same scale choices for bothW and Z, because we want to implement the similarity of the two processes and use it to ourbenefit. Nevertheless the ratio W/Z generally gives us 125× 125 possible combinations of QCDscales, 125 for the W boson and 125 for the Z boson, that can be exploited. However the lackof similarity which occurs when considering different QCD scale choices for W and Z can leadto increase the QCD uncertainty and thus the shift in the extracted MW value.

3.3 The least square method in MW determination

As we suggested at the end of Section 3.1.1, the fitting procedure can be generalized in orderto account for the correlation between bins. In order to include the correlation we can build amatrix which describes the covariance with respect to QCD scale variations in our pseudo-dataset. The new feature is that all averages are arithmetical and made over the scale combinations;this follows from the subjectivity in the QCD scale choice, thus in the equivalence among thedata set. Briefly, the whole procedure can be presented as follows:

• we select the variables we want to include in the fit (for instance XWp⊥

and XZp⊥

, each takenwith a fixed QCD scale combination), and create an array by juxtaposing the data valueof the observables. We will call such vector d;

• we compute the covariance matrix Cov: for each pair of values at positions i and j in thearray d we get the corresponding bins, and create two vectors, we will name b(i) and b(j),containing all values of such bins for each scale combination. The element Covij is thentheir covariance, computed as:

Covij =

(1

Nscales

Nscales∑k=1

b(i)k b

(j)k

)−(

1

Nscales

Nscales∑k=1

b(i)k

)(1

Nscales

Nscales∑k=1

b(j)k

)4; (3.3.1)

• we compute the inverse matrix Cov−1 of Cov;

• we create new vectors for each choice k of templates for both variables by juxtaposing thesingle templates, just as in the first point. We will name these arrays t(k);

• for each template vector t(k) we take the difference with the data vector, element by element,thus constructing a new vector, which we will name O(k): O(k)

i = di − t(k)i ;

• we are now able to compute the χ2 value for each template k as a quadratic form of Cov−1

operating onto O(k) in the following way:

χ2k = O(k)T ·Cov−1 ·O(k) =

Nscales∑i=1

Nscales∑j=1

Cov−1ij O

(k)i O

(k)j ; (3.3.2)

• the best estimate of the MW value will be given by the template which will minimize theχ2 value among all the values computed with the formula (3.3.2).

4Nscales is the number of QCD scale combinations that are considered.

3.3. The least square method in MW determination 28

Such way of proceeding is highly general and allows to account for multiple distributions to whichconstrain the Template Fit. Moreover it accounts for the correlation among the observables andamong bins and it does not merely consider the statistical error upon the distribution k-th valueitself.

In our analysis we considered up to three set of observable combinations altogether. For bothXWp⊥

and XWm⊥

, they are: W data only, W data plus Z data and W data plus Z data plus pZ⊥data. We expect that by adding more observables we should notice a general improvement ofthe fit.

We eventually remark that our definition of covariance is not a "statistical" one, as it does notrely upon a statistical set of data and that the definition of the product average value suppose acommon variation of the QCD scales, assuming they are perfectly correlated.

Chapter 4

Simulations and results

In this section we present the adopted way of working, the simulation settings and codes, andthe final results we achieved through the analyis of W and W/Z distributions and through theanalysis of correlated observable distributions.

4.1 The simulation settings and the procedure

The code used to implement all simulations can be found in the POWHEG BOX package [9],which, among all its features, allows the user to simulate the Drell-Yan processes of W and Zproduction. The implementation of the code relies on MonteCarlo methods and parton showercorrections, implementing all LO and NLO QCD corrections we discussed in Section 2.2 [10].The code is written in Fortran and it relies upon the library LHAPDF in order to handle PDFs.All our simulations have been worked out using POWHEG only, without implementing the partonshower pythia. This is an unphisical choice, yet it allowed a strict and complete explorative studyof Drell-Yan processes, retaining the full dependence on QCD scales.

The settings of the simulation can be initialized in the file powheg.input, which allows the user toset various parametres. First of all one has to decide the kind of colliding hadrons: we consideredpp collisions only. Then one has to choose the V boson decay mode (whether electronic, muonicor tauonic). In all our simulations we opted for the electronic decay mode, which means that,for instance, the neutral current Drell-Yan process with Z production will be: qq → Z → e+e−.Afterwards, the PDF sets for both hadrons have to be set. Using LHA numbering we choose thePDF set NNPDF3.0.The last parametres to be set are the values of QCD scales, discussed at the end of Section 2.2.We explored all 125 combinations of the three scales in our simulations in order to observe thedifferent behaviours in the process description.The last two fields we accounted for in file powheg.input are the ones that allows the userto initialize the random seeds for MonteCarlo event simulations and the number of events wewant to be simulated. We simulated 100 million Drell-Yan events for each choice of the QCDscales, splitting each simulation into 20 job runs of 5 million events per run, thus speeding upthe computation. At the end of the 20 runs, we avereged bin by bin the obtained histograms toget final histograms corresponding to the 100 million simulation, and the corresponding errorswere propagated. We highlight that the random seeds were exactly the same for each of the 125

4.1. The simulation settings and the procedure 30

simulations: they just changed among the runs, yet they were generated once and only once.This way of proceeding was adopted for Z production and for both W− and W+ productions.The above method was used to simply obtain the bare W and Z distributions. In order togenerate the ratio observable distributions, we calculated all the 125× 125 histogram ratios, soas to account for all combinations of W and Z data. For such distributions, the k-th bin’s valuewas calculated as ratio between W histogram’s k-th value and Z histogram’s k-th value, and thecorrresponding error propagated through the ratio, as we already mentioned in Section 3.2.3.For every simulation, the acceptance cuts are set as follows: an event is accepted if pl⊥ ≥ 25 GeV,where pl⊥ is the transverse momentum of the charged lepton, and |η| ≤ 2.5, where η is the pseudo-rapidity.

Along with the distributions (our pseudo-data), we generated the templates. The W templatesrefer to the central choice ξR = 1.00, ξF = 1.00 and ξQ = 0.50 for QCD scales, but just like thethe pseudo-data they were generated the same way, as average over 20 histograms obtained from5 million event runs. The ratio templates W/Z were calculated as ratio between the W templateand the Z distribution referring to QCD choice ξR = 1.00, ξF = 1.00 and ξQ = 0.50.For each W pseudo-data we found the template minimizing the χ2, while for the ratio pseudo-data we fitted only the distributions corresponding to the diagonal choices1 for the QCD scales,which means that both W and Z starting data were obtained with the same choice of QCDparametres. The Template Fit did not considered the whole bin range of variables XV

m⊥and

XVp⊥

as given from the simulation (which is [0.6, 1.2]) yet a sub-interval [0.9, 1.1] ([0.9, 1.11]when considering the correlation among bins) in order to be more sensitive to the Jacobianpeak description and not to be too much influenced by the distributions’ tails, which are morefluctuating as due to MonteCarlo limits.The aim of our analysis is to estimate how much the choice of QCD scales affects the extractedvalue of MW . In other words, we want to compensate for the distortions given by the arbitrarychoice of such scales with a shift inMW value, which gives us an estimate of the QCD theoreticaluncertainty upon the value of the W boson mass.

The whole procedure may be summarized in the following way:

• generation of pseudo-data for all combinations of QCD scales using the value MW =M refW = 80.398 MeV for variables XV

m⊥, XV

p⊥and pZ⊥; generation of templates using the

central choice scale combination and different values of MW , for variables XWm⊥

and XWp⊥

;

• for each set of pseudo-data, we calculate the χ2 value for each template (both in the classicalway and using the covariance matrix method presented in Section 3.3);

• for each pseudo-data we select the minimum χ2 and thus the corresponding template andMW value. Such value will be our best estimate of MW for the given choice of QCD scales;

• we calculate the shifts MW −M refW and plot them against the combinations of QCD scales,

numbered from 1 to 125.

We remark that beside POWHEG we have written ex novo all code we needed, using C++language and bash scripts. The routines used to compute the inverse of the covariance matrix

1The name "diagonal choices" refers to the fact that we can see the 125× 125 pairs of W and Z combinationsas entries of a matrix. As a consequence the diagonal choices correspond to the choice of the same QCD valuesfor W and Z, that is to say the "diagonal elements" of the matrix.

4.2. Analysis of distributions for W− 31

are given by the LAPACK package, written in Fortran but which can be used also with C++through an interface.

4.2 Analysis of distributions for W−

We now present the results of the fit for the W− production.

The first thing we checked is the behaviour of MW shifts when we fix two scales to a certain pairof values and let the third one free to vary. The trends are shown in Fig. 4.2.1, where we cansee how the shifts in MW responds to the variation of a single QCD scale.We notice how as ξR increases, the shifts turn from positive to negative, almost changing onlythe sign but not their absolute values, while as ξF increases, also the shifts turns from negativeto positive. In the end, as ξQ increases, XW

m⊥remains unaffected, whereas for XW

p⊥the shifts are

negative and their absolute values decrease.


-30

-20

-10

0

10

20

30

0 1 2 3 4 5

(MW

−80.3

98

GeV

)(M

eV)

ξR

ξF = 1.00, ξQ = 0.500

MW from W-fitMW from W/Z-fit

(a) XWm⊥ : fixed ξF and ξQ while ξR is free.

-200

-150

-100

-50

0

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−80.3

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)(M

eV)

ξR

ξF = 1.00, ξQ = 0.500


(b) XWp⊥ : fixed ξF and ξQ while ξR is free.

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ξF

ξR = 1.00, ξQ = 0.500MW from W-fit

MW from W/Z-fit

(c) XWm⊥ : fixed ξR and ξQ while ξF is free.

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ξR = 1.00, ξQ = 0.500


(d) XWp⊥ : fixed ξR and ξQ while ξF is free.

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)(M

eV)

ξQ

ξF = 1.00, ξF = 1.00MW from W-fit

MW from W/Z-fit

(e) XWm⊥ : fixed ξR and ξF while ξQ is free.

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20

40

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)(M

eV)

ξQ

ξF = 1.00, ξF = 1.00 MW from W-fitMW from W/Z-fit

(f) XWp⊥ : fixed ξR and ξF while ξQ is free.

Figure 4.2.1: W− Template Fit results: relationship between QCD single scale variations andMW shifts.


Since we aim not only at giving an estimate of the maximum positive and maximum negativeshift inMW , but also at seeing the patterns along which the shifts place themselves, we developeda compact way of representing the fit results, in order to be able to compare the spread of MW

value against all choices of QCD scales, for both W and W/Z. The graphics are organized inthe following way: on the x -axis the QCD scale combinations are represented, numbered from1 to 125. Within the 125 values, there are five groups of 25 values corresponding to increasingvalues of ξR; within each of these groups there are again five groups of 5 values, corresponding toincreasing values of ξF . At last, these five values correspond to increasing values of ξQ. On they-axis there is the difference MW − 80.398 GeV which identifies the shift in MW value as givenby the Template Fit procedure. This representation allows us to show all shifts in a single plots.

In Fig. 4.2.2a and Fig. 4.2.2b, we can see the spread of MW value for both W and ratioobservables when considering either Xm⊥ or Xp⊥ for all scale combinations.

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Xm⊥ fit results MW from W-fitMW from W/Z-fit

(a) Xm⊥ .

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scales

Xp⊥ fit results MW from W-fitMW from W/Z-fit

(b) Xp⊥ .

Figure 4.2.2: Template Fit results for W−

In Fig. 4.2.3 the fit results for normalized distributions are presented: the patterns that can beseen in Fig. 4.2.2 and the difference between W and ratio plot patterns are generally flattened.


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Xm⊥ fit results

(normalized distributions)


(a) Xm⊥ .

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0 20 40 60 80 100 120

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)(M

eV)

scales

Xp⊥ fit results

(normalized distributions)


(b) Xp⊥ .

Figure 4.2.3: Template Fit results for W−: normalized distributions, with normalization range[0.9, 1.1].

Considering the plots in Fig. 4.2.2 and in Fig. 4.2.3, we notice that for the ratio observablesthe MW spread is lower than for W observables, thus confirming that there might be some QCDcompensation when taking the ratio W/Z. On the contrary the W observables have a greaterspread in MW values, especially if we consider XW

p⊥.

In the following table we report the maximum positive and negative shifts in MW value for alldistributions.

variable spread (without normalization) [MeV] spread (with normalization) [MeV]

1 XWm⊥

(−78 / + 26) ±8

2 XW/Zm⊥ (−6 / + 4) ±4

3 XWp⊥

±200 (−200 / + 140)

4 XW/Zp⊥ (−96 / + 24) (−76 / + 24)

By comparing row by row the shifts given by normalized distributions with those given by bareones, we notice that the spread is reduced. On the other hand, by comparing the first row withthe third one, or the second one with the fourth one, we see that the Xm⊥ observables have lessspread in MW values. Even, if we consider XW

p⊥, the spread covers the whole template range,

and as a consequence it does not provide any particular information. It tells us that when fittingXWp⊥

from W data to extract the W boson mass, we need to account for a shift error in MW ofabout ±200 MeV (according to our choice of templates).From such values we can infer that it is generally more convenient to use the XW

m⊥observable

in order to extract MW by means of a Template Fit procedure, yet in order to further reducethe theoretical error in W− mass determination, deriving from a certain choice of QCD scalefor templates, it is better to use the ratio W−/Z observable which allows only about a ±5 MeVerror to be accounted for.

The results exposed up above have all been obtained using the Template Fit procedure withoutconsidering any correlation among bins. Actually, in our analysis we also used the procedureexplained in Section 3.3, applied to mere W data, W plus Z data and pZ⊥ data in addition to W


and Z data, for both XWm⊥

and XWp⊥

observables (and their normalized counterparts). We remarkthat, in such last case, the template distribution for pZ⊥ refers to the central choice of QCD scalevalues.For mere W distributions we expect the shifts to be smaller than those computed considering allbins as independent, since we are now considering their mutual correlation. We also reasonablyexpect that, by constraining the fit also to Z distributions and then to pZ⊥ distribution as well,the spread in MW values will be reduced. The fit results are shown in Fig. 4.2.4 and Fig. 4.2.5.

-10

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GeV

)(M

eV)

scales

Xm⊥ fit results(covariance method)

MW from fit - W onlyMW from fit - W + Z, diagonal choices

MW from fit - W + Z + pZ⊥, diagonal choices

(a) Xm⊥ .

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Xp⊥ fit results(covariance method)



(b) Xp⊥ .

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)(M

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scales

Xm⊥ fit results (covariancemethod), normalized distributions



(c) Xm⊥ .

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GeV

)(M

eV)

scales

Xp⊥ fit results (covariancemethod), normalized distributions



(d) Xp⊥ .

Figure 4.2.4: Template Fit results using the covariance matrix forW−. Only the diagonal choicesare plotted.The normalization range for Xm⊥ and Xp⊥ is set to [0.9, 1.11], while for pZ⊥ it is set to [0, 8.75].

In Fig. 4.2.4 we can see that by considering more observables in the fit, i.e. adding more infor-mation to the analysis, we lower the spread in MW . The first thing that stands out is that themere W distributions when fitted considering the covariance between each pair of bins lead to aspread much lower than the one computed not considering their correlations, as we preaviouslydiscussed. Even, if we consider the Xm⊥ case, the systematic uncertainty given by the arbitrarychoice of QCD scales is almost completely suppressed forW only andW +Z distributions, and itis completely suppressed when adding pZ⊥ to the fit. On the contrary, the Xp⊥ observable is more


sensitive to QCD scales, therefore the spread in MW values is decreased, however not suppressedeven though we added XZ

p⊥and pZ⊥ to tighter constrain the fit.

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scales


MW from fit - W + Z all scalesMW from fit - W + Z + pZ⊥ all scales

MW from fit - W + Z, diagonal choicesMW from fit - W + Z + pZ⊥, diagonal choices

(a) Xm⊥ .

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W−

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)(M

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scales




(b) Xp⊥ .

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MW from fit - W + Z, all scalesMW from fit - W + Z + pZ⊥, all scales


(c) Xm⊥ .

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)(M

eV)

scales

Xp⊥ fit results (covariance method), normalized distributions



(d) Xp⊥ .

Figure 4.2.5: Template Fit results using the covariance matrix for W−. All 15625 choices areplotted, along with the diagonal choices.The normalization range for Xm⊥ and Xp⊥ is set to [0.9, 1.11], while for pZ⊥ it is set to [0, 8.75].

The graphics in Fig. 4.2.5 contain all 15625 scales combinations (125 for W and 125 for Z2, andwe can then desume that the diagonal choices of QCD scales seem to be the ones which higherdecrease the MW shifts. A non-diagonal choice leads to destructive behaviours which make thefit worse, increasing instead of reducing the systematic uncertainty upon the extracted W bosonmass.

In the following table we report the spread width for W -only fits and all diagonal choices W +Zand W + Z + pZ⊥ fits made by means of the covariance matrix:

2The scale choice for pZ⊥ is the same of the considered Z observable, since they come from the same simulation.

4.3. Analysis of distributions for W+ 37

variable included spread (without normalization) spread (with normalization)observables [Mev] [Mev]

1 Xm⊥ W only (−0 / + 2) (−0 / + 2)2 Xp⊥ W only (−20 / + 12) (−20 / + 12)3 Xm⊥ W and Z 0 04 Xp⊥ W and Z (−4 / + 12) (−4 / + 20)5 Xm⊥ W , Z and pZ⊥ 0 06 Xp⊥ W , Z and pZ⊥ (−0 / + 4) ±4

The first thing we remark is that all spread are much lower than those obtained considering thebins as independent of each other. In such case accounting for the correlation helps in reducingthe impact of QCD scale choice.Again we notice that Xm⊥ has a very low spread that is completely canceled already with theaddition of the Z observable. Xp⊥ instead has a residual shift inMW value even after the additionof Z variable and pZ⊥. We also highlight that the normalization does not reduce the MW spreadas it does when considering all bins independent: as we can see from the table, the spreads ofvalues with or without normalizing the distributions are almost the same.

4.3 Analysis of distributions for W+

We now present the results of the fit for the W+ production.

Since W+ and W− bosons are not produced the same way, we do not expect the same results forW+ as we found for W−, however being one boson the anti-particle of the other, we reasonablyexpect the W+ results to have similar behaviours.

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Xm⊥ fit results MW from W-fitMW from W/Z-fit

(a) Xm⊥ .

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250

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scales

Xp⊥ fit results MW from W-fitMW from W/Z-fit

(b) Xp⊥ .

Figure 4.3.1: Template Fit results for W+

In Fig. 4.3.1 we have plotted theMW spread against the QCD scale combination for bothW andratio observables, considering Xm⊥ and Xp⊥ . Just like the W− boson, we see the Xm⊥ patternbeing more defined than the Xp⊥ one and again for W variables the spread of MW values forXWp⊥

covers the whole template range. The ratio observables show again a lesser spread.


In Fig. 4.3.2 we can appreciate the MW spread plotted against the QCD scale combination fornormalized distributions. We see again how the normalization flattens the differences in shape ofW ’s and ratio’s patterns, making them more similar to one another. Besides, it generally reducesthe spread of MW values.

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eV)

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Xm⊥ fit results(normalized distribution)


(a) Xm⊥ .

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Xp⊥ fit results(normalized distribution)


(b) Xp⊥ .

Figure 4.3.2: Template Fit results for W+: normalized distributions, with normalization range[0.9, 1.1].

We remark that all plots in Fig. 4.3.1 and Fig. 4.3.2 have been created following the sameprescriptions used for the W− graphics, Fig. 4.2.2 and Fig. 4.2.3.In the following table we report the maximum positive and negative shifts in MW value for alldistributions.

variable spread (without normalization) [MeV] spread (with normalization) [MeV]

1 XWm⊥

(−100 / + 34) (−8 / + 10)

2 XW/Zm⊥ (−8 / + 4) (−4 / + 6)

3 XWp⊥

±200 (−200 / + 148)

4 XW/Zp⊥ (−116 / + 44) (−68 / + 52)

The same considerations made for the W− boson holds here for the W+ results. We only remarkthat for W+ the spread in MW value is generally greater than W−’s one.

We now present the analysis results obtained using the Template Fit procedure considering thecorrelation among bins. We followed the same logic order employed in W− analysis: we appliedthe procedure explained in Section 3.3 to mere W data, W plus Z data and pZ⊥ data in additionto W and Z data, for both XV

m⊥and XV

p⊥observables (and their normalized counterparts). The

results are shown in Fig. 4.3.3.


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eV)

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(a) Xm⊥ .

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(b) Xp⊥ .

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(c) Xm⊥ .

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)(M

eV)

scales

Xp⊥ fit results (covariancemethod), normalized distributions



(d) Xp⊥ .

Figure 4.3.3: Template Fit results using the covariance matrix forW+. Only the diagonal choicesare plotted.The normalization range for Xm⊥ and Xp⊥ is set to [0.9, 1.11], while for pZ⊥ it is set to [0, 8.75].

Again we notice that by considering more observables in the fit we lower the spread inMW . Alsothe same considerations already presented for W− boson hold now for W+ boson. However weremark that for XW

m⊥variable the spread is completely suppressed only when adding XZ

m⊥and

pZ⊥ to the fit, whereas in the W− case we had a null shift of MW already when adding only XZm⊥

.

In Fig. 4.3.4 the plots for all 15625 combinations are portrayed. Again we see how non-diagonalchoices actually make the fit worse, in the same way as for W−.


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(a) Xm⊥ .

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(b) Xp⊥ .

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(c) Xm⊥ .

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)(M

eV)

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Xp⊥ fit results (covariance method), normalized distributions



(d) Xp⊥ .

Figure 4.3.4: Template Fit results using the covariance matrix for W−. All 15625 choices areplotted, along with the diagonal choices.The normalization range for Xm⊥ and Xp⊥ is set to [0.9, 1.11], while for pZ⊥ it is set to [0, 8.75].

In the following table we report the spread width for W -only fits and all diagonal choices W +Zand W + Z + pZ⊥ fits made by means of the covariance matrix:

variable included spread (without normalization) spread (with normalization)observables [Mev] [Mev]

1 Xm⊥ W only ±2 ±22 Xp⊥ W only (−16 / + 36) (−20 / + 36)3 Xm⊥ W and Z (−0 / + 2) ±24 Xp⊥ W and Z (−12 / + 8) (−8 / + 12)5 Xm⊥ W , Z and pZ⊥ 0 06 Xp⊥ W , Z and pZ⊥ ±4 (−4 / + 12)

We notice that the spread decreases as we add up other variables, even though the absolutespread band is wider than in the W− case: if we consider the W -only case, for instance, theshifts are greater. However the reduction of the systematic uncertainty is relevant, especially ifcompared to the results obtained without accounting for the correlation.


We end highlighting that the normalization does not reduce the MW spread as it does whenconsidering all bins independent: as we can see from the table, the spread of values with orwithout normalizing the distributions are almost the same.

In the following table we summarize the results obtained for W− and W+ by comparing theirspread in MW , with and without considering the correlation among bins:

without correlation:W− spread [MeV] W+ spread [MeV]

XWm⊥

(−78 / + 26) (−100 / + 34)

XW/Zm⊥ (−6 / + 4) (−8 / + 4)XWm⊥

±8 (−8 / + 10)

XW/Zm⊥ ±4 (−4 / + 6)XWp⊥

±200 ±200

XW/Zp⊥ (−96 / + 24) (−116 / + 44)XWp⊥

(−200 / + 140) (−200 / + 148)

XW/Zp⊥ (−76 / + 24) (−68 / + 52)

with correlation:included variables W− spread [MeV] W+ spread [MeV]

Xm⊥ W only (−0 / + 2) ±2Xm⊥ W only (−0 / + 2) ±2Xp⊥ W only (−20 / + 12) (−16 / + 36)Xp⊥ W only (−20 / + 12) (−20 / + 36)Xm⊥ W and Z 0 (−0 / + 2)Xm⊥ W and Z 0 ±2Xp⊥ W and Z (−4 / + 12) (−12 / + 8)Xp⊥ W and Z (−4 / + 20) (−8 / + 12)Xm⊥ W , Z and pZ⊥ 0 0Xm⊥ W , Z and pZ⊥ 0 0Xp⊥ W , Z and pZ⊥ (−0 / + 4) ±4Xp⊥ W , Z and pZ⊥ ±4 (−4 / + 12)

Chapter 5

Conclusions and perspectives

In our work we tried to elaborate in a quantitative way the role of a theoretical uncertaintysource (the arbitrary choice of QCD scales) in the prediction of Drell-Yan observables and inthe extraction of the mass of W boson MW . The observable distributions have been treatedconsidering the bins both independent and correlated to each other with respect to QCD scalevariations. In the second case we developed a procedure to exploit this additional information.The correlation has been introduced by means of the covariance matrix calculated with respectto the scale variation, and used in the fitting procedure.In our analysis we observed how Xm⊥ is more stable (in other words, less sensitive) to the scalevariation, whereas Xp⊥ is less. We considered a wider range for QCD scales than the usual onein order to better investigate the issue, and for each choice we extracted the corresponding shiftin MW value. The confirmation of such results would still make possible a precise measurementof the mass of theW boson, with the better awareness of the presence of a systematic theoreticalerror.

This study will have to be repeated, first including the parton shower corrections through pythia,which have not been implemented since they are very CPU demanding and we preferred to focuson the orderliness yet retaining the full dependence on QCD scales. Moreover, the statisticalproperties should be further investigated and the inclusion of other variables that would improvethe fit should be explored. At last, the systematic errors that might have been introduced usingour definition of covariance should be investigated.

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[3] CDF collaboration, F. Abe et al., Phys. Rev. D49 (1994) 1.

[4] CDF collaboration, A. Bodek, Proc. Int. Workshop on Deep Inelastic Scattering and RelatedPhenomena, Rome, April 1996.

[5] UA1 collaboration, G. Arnison et al., Phys. Lett. B122 (1983) 103;UA2 collaboration, G. Banner et al., Phys. Lett. B122 (1983) 476.

[6] J.C. Collins and D.E. Soper, Ann. Rev. Nucl. Part. Sci. 37 (1987) 383, and referencestherein.

[7] K. Hikasa, Phys. Rev. D29 (1984) 1939;D.A. Dicus and S.S.D. Willenbrock, Phys. Rev. D34 (1986) 148.

[8] P. Arnold and M.H. Reno, Nucl. Phys. B319 (1989) 37; ibid. B33O (1990) 284(erratum);R. Gonsalves, J. Pawlowski and C.-F. Wai, Phys. Rev. D40 (1989) 2245.

[9] S. Alioli, P. Nason, C. Oleari, and E. Re, A general framework for implementing NLOcalculations in shower Monte Carlo programs: the POWHEG BOX, JHEP 1006 (2010)043, [1002.2581]

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