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This is to certify that the

dissertation entitled

INCLUSION OF TEVA'IRON Z DATA INTO GLOBAL
NON-PERTURBATIVE QCD FI’ITII‘B

presented by

Freddie J. Landry

has been accepted towards fulfillment
of the requirements for ’

Pn.D degree in M1—

 

    

Major professor

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Date

 

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6/01 c:/CiRC/DateDue.p65-p. 1 5

INCLUSION OF TEVATRON Z DATA INTO GLOBAL
NON-PERTURBATIVE QCD FITTING

By

Freddie J. Landry

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

200 1

ABSTRACT

INCLUSION OF TEVATRON Z DATA INTO GLOBAL NON-PERTURBATIVE QCD
FITTING

By

Freddie J. Landry

Fits are performed to transverse momentum distributions of Drell-Yan data sets with
non-perturbative functions in the Collins-Soper—Sterman (CSS) resummation formalism.
Fits are made to various global sets of data containing results from both fixed target and
collider experiments. Several functional forms in impact parameter space (b) of the
non-perturbative function are also tested and compared. The best fit to the data is
achieved with the form, —glb2 — g2b21n[%;]— glg3b21n(100xaxb) with the values for

the parameters, gl = 0.2138}, g2 = 0,6838%, g3 = 4.60333.

To my Mom and Dad

iii

ACKNOWLEDGMENTS

There are many people I would like to thank for helping me throughout my
graduate career. First, I would like to thank my advisor Chip for seeing me through this.
He was helpful at finding solutions when I reached a roadblock and he was always
willing to make time for me out of his busy life as chairman and physicist.

Thanks are also due to Dylan for letting me work with him on the final pieces of
his analysis. He was also a good sounding board for our fitting work on more than one
occasion.

The members of the fitting gang, C.-P., Chip, and Glenn are a good group of guys
to work with and were always willing to help this greenhom understand the theory, no
matter how many times I asked.

From my DO days, my trailermates Ashutosh, Ian, Eric, and Rich stand out for
helping me learn the ropes there.

While on the TRGMON project, Philippe taught me all about triggering and how
to really program.

Working with Sandor on the ZCD was always interesting and good for a few
laughs. He showed me the part of high energy physics outside of the cubicle and away
from the computer screen. In the process we got our hands dirty and played with a lot of
interesting toys.

My companions at MSU, Remo, Mike, and N arlock, also get a thanks. I learned a
lot about computers, beer, and presweetened cereal. And they learned to fear my
shotguns.

Finally, I’d like to thank my family, or la familia, those living and those we won’t

forget, for the support and for the good times we’ve had.

iv

TABLE OF CONTENTS

1 Theory ......................................................................................................................... 2
1.1 The Standard Model ................................................................................................ 2
1.2 The Drell-Yan Process ............................................................................................ 5

1.2.1 The Naive Process ........................................................................................ 5
1.2.2 The Lowest Order Nonzero ....................................................................... 11
1.2.3 Higher Order Processes .............................................................................. 18
1.2.4 Resummation Calculation .......................................................................... 23

2 Fermilab and the DO Detector ...................................................................................... 28
2.1 The Accelerator at Fermilab ................................................................................. 28
2.2 The DO Detector ................................................................................................... 31

2.2.1 Central Detectors ........................................................................................ 34
2.2.2 Calorimeter ................................................................................................. 38
2.2.3 Muon Detector ........................................................................................... 41
2.2.4 Triggering System ...................................................................................... 42

3 Refinement of DO Z Data ............................................................................................ 44
3.1 Introduction ........................................................................................................... 44
3.2 Studies With Fast Monte Carlo ............................................................................. 44

3.2.1 Examination of Kinematic Variables ......................................................... 45

3.2.2 Studying Contributions to Smearing Resolution ....................................... 50

3.3 Development of Fast Smearing Program .............................................................. 55

4 Non-Perturbative QCD Fits .......................................................................................... 60
4.1 Method of Fitting .................................................................................................. 60
4.2 Experiments Included in Fits ................................................................................ 61
4.2.1 R209 ........................................................................................................... 62

4.2.2 CDF ............................................................................................................ 63

4.2.3 E288 ........................................................................................................... 64

4.2.4 E605 ........................................................................................................... 64

4.3 Choice of Non-Perturbative Functions ................................................................. 65
4.4 Preliminary Fits ..................................................................................................... 66
4.4.1 Duplication of Original LY Fitting ........................................................... 66

4.4.2 Simultaneous Fitting of All Parameters ..................................................... 68

4.5 Inclusion of Tevatron Run 1 Z Data ..................................................................... 76
4.5.1 Gauss 1 Fit ................................................................................................. 77

4.5.2 LY and DWS Parameterization Fits ........................................................... 78

4.5.3 Gauss 3 Fit ................................................................................................. 78

4.6 Reintroduction of Drell-Yan Data ......................................................................... 80
4.6.1 Fitting with Higher Mass E605 .................................................................. 80

4.6.2 Bringing E288 Back into Fits ..................................................................... 83

5 Conclusion .................................................................................................................... 90
A The Level 1 Trigger Monitor Program ......................................................................... 93
A] Introduction .......................................................................................................... 93
A2 Method of Information Gathering and Computing .............................................. 94

vi

A.3 Implementing User Defined Integration Times ................................................... 95

A31 Method Of Calculating Over Larger Integration Times ........................... 95

A32 User Interface ............................................................................................ 95

A33 Behavior Under Special Conditions .......................................................... 95

A4 Testing and Implementation of New TRGMON ................................................. 96

B The Z Calibration Detector ........................................................................................... 97
B.1 Introduction .......................................................................................................... 97
B.2 Design of Detector ................................................................................................ 98
B.3 Acquisition and Selection of Data ...................................................................... 100
B.4 Calibration Method ............................................................................................. 101
B5 Results ................................................................................................................ 106

C Reprint of Preliminary Fits Paper .............................................................................. 109
Bibliography .................................................................................................................... 121

vii

LIST OF TABLES

Table 1.2: The quarks. The electric charge is given in units of the charge of the electron.
......................................................................................................................... 3
Table 1.3: The four known forces and their particle mediators. ........................................ 4

Table 4.1: Experiments included in global fits with some properties for the data used in

the fits. ........................................................................................................... 62
Table 4.2: Fit C values using experimental data from R209, E288, and CDF Z. ............ 68
Table 4.3: Values for Fit D for three parameterizations. ................................................. 69
Table 4.4: Data set of fit including extended E605 data set, Fit E ................................... 70
Table 4.5: Fit results for Fit E. ......................................................................................... 70
Table 4.6: Fit for LY parameterization excluding E288, Fit F. ....................................... 71
Table 4.7: Fit for DWS parameterization excluding E288, Fit F ..................................... 72
Table 4.8: Results for global fit including ...................................................................... 76
Table 4.9: Fit H results ..................................................................................................... 81
Table 4.10: Data set for Fit 1. ........................................................................................... 84
Table 4.11: Fit I results. ................................................................................................... 84
Table 4.12: Summary of fit results ................................................................................... 87
Table B. 1: Slopes from linear fits to Zflbe, vs 2.7. distributions of module pairs. ......... 107
Table B.2: Intercepts from linear fits to Zflbe, vs 2% distributions of module pairs. ..... 108

viii

LIST OF FIGURES

Figure 1.1: Feynman diagram of virtual photon decaying into a lepton pair ..................... 6
Figure 1.2: A typical example of a quark antiquark collision producing a photon with a
nonzero p, through the radiation of a gluon ................................................. 11
Figure 1.3: Feynman diagrams showing typical processes which produce pr. The upper
diagram is an initial radiation or “Annihilation” process and the lower
diagram shows a “Compton” process. ......................................................... 13
Figure 1.4: Some higher order Feynman diagrams of Drell Yan production. ................. 19
Figure 1.5: Decay of a photon into a quark antiquark pair with a subsequent gluon
emission ........................................................................................................ 20
Figure 2.1: The system of accelerators at Fermilab used to accelerate protons and
antiprotons. .................................................................................................... 29
Figure 2.2: The DO Detector showing each subdetector. ................................................ 32
Figure 2.3: Representation of the DO coordinate system. ............................................... 33
Figure 2.4: Cutaway view of the Central Detectors ......................................................... 34
Figure 2.5: The VTX chamber with the layout of its wires. ............................................ 35
Figure 2.6: Cross section of a TRD chamber ................................................................... 36
Figure 2.7: Cutaway view of the CDC showing the wire placement in each cell. .......... 37
Figure 2.8: The three modules which make up each of the Forward Drift Chambers..... 38
Figure 2.9: Cutaway view of the calorimeter ................................................................... 39
Figure 2.10: Representation of a typical calorimeter cell. ............................................... 39

Figure 2.11: A portion of the calorimeter with pseudo-projective towers at different 77. 40

ix

Figure 2.12: Schematic view of the different trigger levels ............................................. 42
Figure 3.]: Dependence of the absolute resolution on the Z mass and rapidity. ............. 47

Figure 3.2: Absolute resolution dependence on the Z pr and the z of the interaction
vertex ............................................................................................................. 48

Figure 3.3: Absolute resolution dependence on the energy and pseudorapidity of the
highest pT electron ........................................................................................ 49
Figure 3.4: Absolute resolution dependence on the p, of the highest pT electron. .......... 50
Figure 3.5: Effect of EM calorimeter energy and position resolutions on the Z pT
absolute resolution. The absolute resolution is plotted as a function of Z p7. 53
Figure 3.6: Z pT absolute resolution dependence on track resolution and the underlying
event contribution. The absolute resolution is plotted as a function of Z p7. 54

Figure 3.7: Effect of z vertex resolution on Z pT absolute resolution. The absolute

resolution is plotted as a function of Z p, ...................................................... 55
Figure 3.8: Special input distribution to CMS. ................................................................ 57
Figure 3.9: Fit parameters plotted versus pT ..................................................................... 58
Figure 3.10: CMS smeared and SMEAR_PT smeared distributions. ............................. 59
Figure 4.1: Contour of x2+1 for 2 parameter fit, Fit F ..................................................... 73

Figure 4.2: Contours for determining uncertainties for Fit F with the LY

parameterization. These plots were made with fixed g3, g2, and g1

respectively at their values at the minimum .................................................. 74
Figure 4.3: Plots of LY and DWS parameterization curves superimposed with data.

Note that the NORM values given here are applied to the theoretical curves

and are therefore reciprocals of the values given in Table 4.6 and Table 4.7

....................................................................................................................... 75
Figure 4.4: Gauss 1 fit distribution for E605. .................................................................. 77

Figure 4.5:

Figure 4.6:
Figure 4.7:
Figure 4.8:
Figure 5.1:

Figure B.l:

Figure B.2:
Figure B.3:

Figure B.4:

Figure B.5:
Figure B.6:

Figure B.7:

LY function and 2 parameter Fit G compared to each other. Note that the

NORM values given here are applied to the theoretical curves and are

therefore reciprocals of the values given in Table 4.8. ................................. 79
Plots of Fit H results compared to Data ......................................................... 82
Fit I Plots compared to Data. ......................................................................... 85
Fit I Plot of E288 compared to fit curves. ..................................................... 86

Uncertainty contour and two dimensional projections for the Fit I result

with the Gauss l parameterization. ............................................................... 91
A Muon Detector track extrapolated to the CDC does not align with the 2:
positions of the delay line hits. ...................................................................... 98
Diagram of a ZCD module and two connecting modules. ........................... 99
A representation of the placement of one of the ZCD modules over a CDC
sector. .......................................................................................................... 100
Placement of the ZCD module pairs in relation to the CDC sectors. The

numbering of the module pairs is shown as well as the CDC sectors covered

by the ZCD. ................................................................................................. 102
Extrapolation of CDC track out to the ZCD module to obtain Zm ............. 103
Typical Fits to AZ for four fibers. ............................................................... 104
Examples of linear fits to Zfibwvs. Za for modules 1-4. ............................ 105

xi

INTRODUCTION

This analysis is centered on achieving the best prediction of transverse
momentum distributions of Drell-Yan (DY) type events, particularly in the low p7. region
(pr << Q). Here, DY type events are defined as the production of W, Z, or photons
resulting directly from hadron-hadron collisions. The theoretical prediction is made
through the use of a non-perturbative QCD formalism, which includes a parameterized
function. The parameters for this function are set by a fit to data. This analysis seeks to
improve the prediction of DY type distributions by both increasing the data available for
fitting through supplemental analysis of Tevatron Z data, and by performing global fits.

There are several benefits to finding which non-perturbative function or functions
provide a good fit to pT distributions of DY type events. Finding such a match will verify
the concept of universality, the idea that one non-perturbative function can give an
accurate prediction of the cross section distribution for the production of any mediator
boson, at any center of mass energy. A benefit to producing more accurate theoretical W
and Z p, distributions is increasing the precision of the W mass measurement. Finally,
gaining a better measurement of the low p7. region will provide a comparison tool for any
future theory which seeks to predict non-perturbative QCD.

The discussion begins in Chapter 1 with an overview of the current status of
particle physics. Then the history of the calculation of Drell-Yan type events will
presented with some detail. Chapter 2 discusses the Fermilab accelerator which was used
by several experiments included in the fitting. Also included is a description of the DO
detector which produced the Z p, data analyzed in Chapter 3 . A description of the
numerous fits to global data sets and their results are given in Chapter 4. The conclusion

is presented in Chapter 5.

Chapter 1

Theory

1.1 The Standard Model

Elementary particle physics deals with the fundamental particles and forces which
make up everything in the universe. Some properties of these particles are listed in tables
on the following pages and were obtained from the Particle Data Group [1]. All forms of
matter are made of two classes of spin y, particles, leptons and quarks. These two
classes of particles are listed in Table 1.1 and Table 1.2.

In addition to the above classes of particles there are the mediators of the four
fundamental forces. Leptons and quarks interact with each other through the exchange of
these mediator particles, which is the mechanism by which particles exert forces on each
other. All of these mediators are spin 1 except for the graviton, which is expected to have

spin 2. Table 1.3 lists the forces and their mediator bosons.

 

 

 

 

 

 

 

Particle Mass Charge
(MeV/cz) (lei )

electron, e 0511 -1
electron neutrino, Ve 0.0 0
muon, u 105.7 -1
muon neutrino, v“ 0.0 0
tau, 1: 1777.1 -1
tau neutrino, V»; 0.0 0

 

 

 

 

 

Table 1.1: The leptons. The electric charge is given in units of the charge
of the electron.

 

 

 

 

Particle (thil’S/scz) (3321.?

down, d 3-9 -1/3
up, u 1.5-5 +2/3

strange, s 60-170 -1/3

 

charm, c (1.1-1.4)x103 +2/3

 

bottom, b (4.1-4.4)x103 -1/3
top, t 174.3x103 +2/3

 

 

 

 

 

 

Table 1.2: The quarks. The electric charge is given in units of the charge
of the electron.

 

 

 

 

 

Force Particle Mass (GeV/cz)
Strong gluon 0
Electromagnetic photon < 2x10'25
Weak W,Z 80.33,91.187
Gravity graviton 0?

 

 

 

 

 

Table 1.3: The four known forces and their particle mediators.

With the exception of gravity, quantum theories have been developed since the
early part of the 20th century which describe how each of these forces act on elementary
particles. These theories taken together comprise the Standard Model, which to the best
of our present knowledge, describes most of the behavior of the particles in their
interactions through three of the four forces. Quantum Chromodynamics, or QCD, is the
theory which describes processes involving the strong force. Quantum Electrodynamics,
QED, describes interactions involving the electromagnetic force. The formalism of
Glashow, Weinberg, and Salam [2] combines the electromagnetic and weak processes as
different aspects of the same force, the electroweak force.

The Standard Model describes many of the processes observed in high energy
physics. However, there are some fundamental questions which it leaves unanswered.
For example it doesn’t explain why the elementary particles have the particular mass
values that they do. It also doesn’t provide a quantum theory for gravity. There is also
reason to believe that at high enough energies that all four forces are unified into one
force, an idea bolstered by the verification of electroweak theory [3][4] This hypothesis
however, is outside the predictive power of the Standard Model. Finally, as will be
shown in the next section, there are also kinematic regions in which the Standard Model

can’t give us a complete calculation of particle dynamics.

1.2 The Drell-Yan Process

In high energy experiments involving collisions between quarks and antiquarks,
the final state sometimes includes a photon, W, or Z in a process called a Drell-Yan
process. This section, based on [5], [6], and [7], will step through the various techniques
used to calculate cross sections for such processes. Although the derivations are shown
for only the photon, the calculations are essentially the same for the W and Z with a

change of some constants.

1.2.1 The Naive Process

As a first step, the cross section calculation will be shown for what is known as
the naive Drell-Yan process for the production of virtual photons. The cross section for
such a process was first calculated by Drell and Yan [8]. It’s called “naive” because the
assumption is made that the initial partons, (the quarks, antiquarks, and gluons which
constitute hadrons), involved in the reaction have no momentum transverse to the
momentum vectors of the hadrons in the center of mass frame of the initial hadrons. This
of course means that the photon would then have no transverse momentum in this frame
either. This process was recognized experimentally when the photon decayed into lepton
pairs [9]. A Feynman diagram for such a process is shown in Figure 1.1.

Before calculating the cross section, it is helpful to introduce some commonly
used kinematic variables. Here, the values for these variables are defined in the center of
mass frame of the hadrons with the z axis along the hadron momentum vectors. The four
vector momenta of the hadrons are written below where the masses have been neglected

because the virtual photon’s mass is much larger:

P.” = (PI). P)

- (1.1)
P; = (P,0,-—P).

 

(1,1 [3

Figure 1.1: Feynman diagram of virtual photon decaying into a lepton
pair.

The four vector momenta for the initial partons have a similar form:

p5 =(pf.,5,pf.)
pi," = (pZ,O,-pi§).

The parton momenta can be expressed in terms of their parent hadrons also:

#1.. l1
pa _xaPA

u- u
Pb-‘beB-

The x variables represent for each parton its fraction of the parent hadron’s momentum.

Therefore, x can range in value from 0 to l. The photon’s momentum four vector is

PV“ =(EV,6,PVZ).

 

The difference in the longitudinal momenta of the partons is an important quantity in this
process and leads to the definition of the Feynman x, x F:

pr =pf. -pf, =P(x.. -xb)

_PL_
xF=——-xa—xb.

In addition there are three invariants referred to as the Mandelstam variables. For the

hadron level process they are
s E (P: + P532 = 4P2
z 2(1); — PA“)2 = Q2 —- xbs

uE(P¢’—P§’)2=Q2—xas.

Here Q is the photon mass. The corresponding variables at the parton level are given by

Another often used kinematic variable is defined below,

2
r a 9—.
3
Using the relation, Q2 = xaxbs , it becomes
T = xaxb .

Rapidity, defined below, is also an important invariant:

yE—l—lnE+pZ. (1-2)
2 E‘Pz

 

 

For the photon in this process, the rapidity is given by

1 xa

When Drell and Yan calculated the cross section, they considered the calculation

in two parts as given by:

0(1):? —> 7* —> W) = de.dx.f(x.>f(xb)6(qc7 —> y” —+ N'). (1.4)

One part, the integration of the f(x) functions, takes into account that the partons are
constituents of hadrons. The functions, fix), are called parton distribution functions
(PDF). They give the probability for a parton to have the momentum fraction x. The
integration is over all possible values of the x’s. The other part, the 6' factor, is the cross
section for the Drell-Yan process at the parton level. This elementary process is
calculated using the following equation where T is the matrix element calculated using

the Feynman calculus:

lTl2

615' = ——.—.—
(1n1t1al flux)

-(phase space of final states). (15)

It is usually of more interest to calculate and observe differential cross sections,

such as _d_0'_, where 03 is the solid angle of the lepton labeled 3 in Figure 1.1. In order
3
to calculate —, the same calculation of Equation 1.5 is used and then the kinematic
3
quantities for all final state particles are integrated over with the exception of Q3.

The phase space for the final state particles are:

 

- - 2 4 d3 d3
p(p3,p4)=E 7:; ”—35- (1.6)

__54 + _ _
271.6 (P1 P2 P3 P4)2E3 2E4

 

1 2d an , g g _.
= (2”? p34ng4 3 53(p1+ p2 — p3 — p4)5(El + E2 _ E3 __ E4)d3p4,

 

Next, the phase space is integrated over all kinematic quantities except £23. The

integration over d3p4 leads to the constraint, fl, = i5, + [32 — E3, and therefore

 

E4 = \/(P1 + ['52 — [33 )2 + mi . The phase space factor is now

~ 1 PgdP3dQ3
= 6
pm) (2202 413,15,

 

(a+a—g—a)

Integrating this over dp3 gives the density of states in terms of the initial total momentum

and energy, Pr and El. respectively, as

1 P351923 I
Q = * °
F’( 3) 167132lEipgz-Pr'P3E3HE3=E,-—E4

In the center of mass frame, [3,- = O, and the phase space factor simplifies to

1 P3
—_dQ . 1.7
l67r2 E,- 3 ( )
The matrix element factor for d6 is given by Equation 1.8. eq is the electric

charge of the quark, e is the charge of the electron and myis the mass of each final state

lepton.

24

e e
ITI2 = ”q I, 4 '32[(P1 'P3)(P2'P4)+(Pl -p4)(p2 -p3)+m§(pl ‘P2)] (1-8)
4(P1 +P2)

 

This equation is greatly simplified in the center of mass frame and when
conservation of energy and momentum are taken into account. Under these requirements

the following relations hold true and Equation 1.9 gives the simplified matrix element:

24

|T|2 = __eq_e__4 . 32[2(1~:2E'2 + pzp’z cos2 9 + m§E2)] (1.9)
4(Pi + P2)
P 5 P1 ‘ ‘Pz
P, 5 P3 — ”P4
E a E1 = E2 (since m1 = m2)
E’ E E3 2 E4 (since m3 = m4)

 

The initial flux factor from Equation 1.5 is 4\/ (P1 ‘P2 )2 - mlzmQ2 , which in the
center of mass frame simplifies to pl ~ p2 = E2 + p2. Also, in the approximation that the
masses of the initial quarks can be neglected relative to the energy, E, the flux factor
simplifies to SE.

Combining all three factors and simplifying leads to the following:

d6 eze4 2
= ‘1 ZAP—3-1+c0526+ l— & sin29. (1-10)

 

Where 3 is defined as E}. Using the relation, mg? = E32 — pg , the differential cross

section can be written in terms of the mass:

A 2 4 2 2
e e
do = ‘1 2, 1-1’3-351+cos29+ Tl sin26. (1.11)

In the limit where the mass of particle 3 is much smaller than its energy, the cross section

 

simplifies to:

A 2 4
do = eqezA[l+c0529]. (1.12)
dQ3 647: s

 

The term in brackets is characteristic of processes involving spin X2 initial and final state

particles mediated by a spin 1 particle.

10

gluon

 

beamline axis ‘ q

 

 

Figure 1.2: A typical example of a quark antiquark collision producing a
photon with a nonzero pT through the radiation of a gluon.

1.2.2 The Lowest Order NonzerOpT Process

With the initial partons travelling along the beamline in opposite directions it
would be expected that the resulting photon would not have any pr, that is, a momentum
transverse to the beamline. But measurements show a significant number with a nonzero
p7. This is due to gluon radiation from the initial partons, which is not taken into account
by the process shown in Figure 1.1. The differential pT cross section for this process can
be calculated perturbatively as a series in increasing powers of as. Here, the differential
cross section will be derived to order as, or in other words to first order in the series.

Figure 1.2 gives an example of a process in which p7. is generated by the emission
of a single gluon resulting in the photon having an equal, but opposite p, to the gluon.

This process is an example of one of the lowest order processes which produce photons

ll

with transverse momenta. The example of Figure 1.2, often called the “Annihilation”
process and another such process of the same order, the “Compton” process, are shown as
Feynman diagrams in Figure 1.3. In calculating cross sections to first order in 065, both
types of processes must be included.

For the first order processes, it is interesting to examine the kinematic variables
and note the differences from the naive process. In order to simplify calculations, the
center of mass frame of the colliding hadrons is used. The four momenta of the initial
hadrons and partons remain identical to the naive Drell-Yan process. The general four

momentum for any of the final state particles can be written as

k'“ = (EDI—(.Ti’klii)'

I

Here, the momentum vector is represented by a component transverse to the initial
hadrons, Pro and parallel to them, km- The Mandelstam invariants with respect to any of
the final state particles can be calculated in terms of the four momenta. The s invariant is

straightforward and is the same value as in the naive model.
5 = 4P2

The other variables are calculated below:

t=(k“ «115)2 =m2 —2k-PA =m2 —2P(E—k”)
12:01:“ 405‘)2 =m2 ——2k-P =m2 —2P(E+k”).

12

 

 

\s“
gm.
9 \

Figure 1.3: Feynman diagrams showing typical processes which produce
p,. The upper diagram is an initial radiation or “Annihilation” process
and the lower diagram shows a “Compton” process.

13

The equations for t and u can be subtracted from each other to solve for k",

2 _ 2 _
k” = [[2151-0701] _ [m4—Pi—t-HP = (x1 — x2)P. (1.13)

Where x, and x2 have been defined as

xsz-u_E+k"
1 4P2 — J;
x _m2—t_E—kll
2‘ 4102 ‘ fi'

 

 

Notice that instead of the longitudinal momentum of the photon being represented by
(xa — xb)P as in the naive picture, it is now equal to Equation 1.13. There are other
relations that bear the same form as their naive counterparts, yet reveal the differences
between the naive model and first order p7. process. For instance, a variable called the

transverse mass is defined by

2_ _ 2 2
mT=xlx25—m +k ,

whereas m2 = xaxbs in the naive model. Also, instead of rapidity following the relation

of Equation 1.3, the rapidity is now

Finally, from the relation, 5 + f + L? = Q2 , the x variables can be related to each other by

xa

b
xb — x2 xa — X!
With these kinematic variables and the relations between them, the cross section can now
be calculated.
The differential cross section for each of the two processes is given by the

following equation:

d30' _

 

(136 z" -+ *X —> Fax
2 _2deadxbf(xa)f(xb) (J 7 2 ). (1.14)
depTdy iJ depTdy
(136 . . . . . .
Here, —— IS the d1fferent1al cross sectlon for the product1on of a lepton pair
dePTd)’

mediated by a virtual photon, with partons of flavors i and j. The sum is over the
different quark flavors. x and fix) are the same momentum fractions and PDF’s used in
Equation 1.4. The cross sections for the Compton and Annihilation processes will be
calculated separately and then added together to get the total first order differential cross
section.

Labeling the Compton process ab——> Vd, the cross section for the intrinsic process

can be written as

Ezld’dptfiv,fid>
i f
41907}

 

d6“:

where d is the final state quark and Vis the photon. The phase space for this reaction is

d3PV d3Pd '
(27:)3 2EV (2n)32Ed

 

dp = (2fl)454(Pa + Pb — Pv — Pd)

15

Since it is the distribution of Vthat is of interest and not quark d, the cross section can be
integrated over all values of pd to give

2er .3

2 ——”l . (1.15)

d0 = Jj'dxadxbf(xa)f(x xb)m6(pd) EV

Pd =Pa +Pb‘Pv

The remaining delta function can be handled by first transforming it into a

function of xa and xb. From the relation
2 _ A . A 2 _
pd—s+t+u-Q —(x..x,,— xxl xbx2—1)s, (1.16)

the delta function can now be expressed as 6( f (xa)) which allows for the following

property to be used:

—1

wheref(x,-) = O.

 

15(f(x))dx=25(x—xr)%

 

x=xi

Now Equation 1.15 can be written in the following way:

ZZ|TI25(xa — x5)

I f
Ev-di_— - flfidx dxbf(x M)f( xb)647t2EaEbs(xb — x1) '

 

Where x5 is the root of Equation 1.16. Using the general cross section for two to two

body interactions,

91: ‘ifll
d? 1671162

and the relation

E d3cr_1 do __1 do

3 (1.17)
dP’ ZWT dPTd)’: 71' dPT d)’

 

16

and integrating over xa yields

d6 1 R xRxb d6-
——-—- = — dx a . .
dQdegdy EJ bf(xa )f(x,,) (xb — x,) (1'1sz

 

dc; can be calculated by using the result from the well-known original Compton

 

scattering [7], with the substitution of a photon with mass as a final particle

(y+e—>y*+e),

 

 

£_27ra2 f 3‘ 212Q2
d? a2 '

Converting this expression to the q + g ——> 7* + q —> FF + q reaction gives

 

A 2 2 A A A
doc _ a 0656., —t2 — 52 + Zqu
21?sz 9Q2§2 a? '

This result can also be used in obtaining the same differential cross section for the

Annihilation process using crossing symmetry,

 

 

616,1 __8_a2ase§ £2+a2+2§Q2
dfdQZ 27 Q? a? '

These two results can then be inserted into the expression for the total cross section,

d0(AB —> Fax
szdpidy

xfxb (16',‘
x, — x1) (11ng2

 

 

)= i—debf(xf)f(xb)(

R A
xa xb dO'C

+%J‘dxb[f%(xf)f%(xb) +f‘%1(xf)f%(xb)](xb _ x1) dfsz'

 

l7

The behavior of this cross section at low pT can be studied by changing the cross

section expressions through the relation

For instance, the Annihilation result becomes

 

 

2 2 2 2
do 8 05 06 e 1 1 1' x
fi—=— qu Tidxbf(Xf)f(xb)—— 1+ 2 — 1i ,
do dprdy 27 no :2. (x. —x.> (xfxb) 2w.
and taking this result in the limit p; -—> 0 gives
4 2
—_2dO’2 oc ——aS -—1§- n(-Q—2]. (1-18)
dQ dprdy 3% pr pr

So this result diverges in the limit that pr becomes zero, which is obviously an unphysical

result.

1.2.3 Higher Order Processes

As stated earlier, the above calculation takes into account only the most simple of
processes resulting in p, for the photon. In general, much more complicated processes,
multiple gluon emission for example, can occur in vector boson production. Figure 1.4
shows two examples of higher order processes.

The perturbative calculation for the cross section of the multiple gluon emission
process can be done by first looking at the decay, "f —> q'q’g, Figure 1.5. Notice that this
process is related to both the Annihilation and Compton processes through crossing

symmetry.

18

 

 

q 'Y ‘1 %Q "Y
mg g

8 E 8

 

 

 

 

'QI

Figure 1.4: Some higher order Feynman diagrams of Drell Yan
production.

The photon decay cross section can be analyzed by first defining the function S
as the probability that the quark is diverted by an angle less than 9 from its initial
direction due to the gluon emission. T is then defined as the probability that the quark is

diverted by an angle greater than 6. S and Tare then related by the following equation:

S+T=L 01%

It is convenient to define some kinematic variables for the process before
performing any calculations. Let x, and x2 be the fractions of the photon’s four
momentum carried by the antiquark and quark respectively. Note that x2 is the
momentum fraction carried by the quark after it has emitted the gluon. The momentum
fraction carried by the gluon, x3, is then equal to l— xl — x2. The quantity z is the ratio of
the quark’s energy before and after the gluon emission and the variable j is just the ratio

of the invariant mass of the quark and the gluon to the photon mass:

 

_E,,
11+ P 2 _
j=(P2 2P3) =Q(122x1)=1_2xl.
Q Q

19

g
I

 

Y9

Figure 1.5: Decay of a photon into a quark antiquark pair with a
subsequent gluon emission.

If 6 is taken to be small, then the following approximation can be made:

k, 2k,

65tan6=— —
kn Q

Hi

(1.20)

k1 represents the quark’s transverse momentum needed to achieve the angle 6 and kll is
the momentum in the direction of the initial quark’s trajectory. The approximation,

11 5 g, is only good if the gluon’s momentum component in that direction is small and if
k7. is small. The small angle approximation used above also requires that k, be small.

Tcan be related to the cross section as a differential. of the quark’s p,,

Q2
— 1 do
2 _ 4 I2 _ 2
T(kT) — I“; aWkT Wth6 00 — 3qu'

This equation can be solved for the differential cross section by differentiating by k%:

1 do dT dS

__=-_=_, 1.21
0'0 (1k; 4k; 21k; ( )

So by calculating Tand differentiating as above, the p7. differential cross section can be

obtained.

20

Tcan be calculated using the double differential cross section for the

f -—> q§gprocess. This is given by

32055 x? +x§
37: (1—2x,)(1—2x,)'

d0 = :3ij
dxldx2

 

 

Or rewriting this in terms of the kinematic variables relating to the quark, j and z, the

cross section becomes

 

do 4Q3ae2as j 1+ 22 16
. = q + 0 + .
dde 71' l—z 1(1—2) l—z

From the restrictions imposed by the approximation in Equation 1.20, the gluon energy
must be small and therefore z~1. By definition, j is $1 and is therefore also small. The
middle term in the cross section then should dominate and the cross section then can be

approximated by

_1_ do _2fl l+z2
0'0 djdz LPA 37: j(l—z)

 

This is called the Leading Pole Approximation (LPA). Equation 1.20 can be rewritten as

 

1.249292.
T 4

Defining a new variable, 2 to be l-z, and using the relationship

2
g=z(l—z)j

for the p, of the quark, enables one to put limits on the allowable values for j and 'z' in the

integration of T:

21

2 2
PT . _. 6
——=zl—z 52 >—.
Q2 ( )J J 4

Tcan now be written as

 

230:5 1+z2 ._2as 1 dj 1 2d?
T(6)LPA: 7r”(1—z)j dsz-—flj’%—.J‘gi?-

The solution for Tis then

2
map—£1352 dfln 9
37! T] 4j

2a 92
mm = 7‘17)

The “LDLA” subscript refers to the fact that this result is the “Leading Double Log

Approximation.” The solution forS then follows from Equation 1.19:

Now, either S or T can be inserted into Equation 1.21 to obtain the differential

cross section,

_1_ do _4_a_S 1 Q2
2" —2 _1" ‘2— -
0'0 dkT 37: kT kT

Note that it matches the form of the first order Drell-Yan calculation for the triple
differential cross section of the photon in Equation 1.18. The “1” subscript has been
added to S and Tto point out the fact that the calculation is for a single gluon emission

and therefore the cross section calculation is only to first order in 055.

22

This cross section can be generalized to include multiple gluon emissions. Let S,
be the probability of the quark being diverted by n independent gluon emissions by an
angle less than 6 from its initial trajectory. The “initial trajectory” in this instance means

the quark’s trajectory prior to any gluon emissions. Then S" is given by

S=i——%ln2fi .
" n! 37: Q2

This can be summed over It to take into account all possible number of emissions to give

S can be used again to solve for the differential cross section. Since kT is an arbitrary

value though, p, can be substituted for k, to arrive at

2 2
Ligzfizflsiz “(QTJCXP _Zfl‘iin2[flrz_] . (1.22)
0'0 dPT dPT 371' PT PT 375 Q

This relation is the same as for the single gluon case, except for the exponential
factor. This exponential factor, called a Sudakov Form Factor, represents the multiple
gluon processes in the calculation. Note that it eliminates the divergence as p, ——> 0 that

was present in the first order result.

1.2.4 Resummation Calculation

The cross section calculation of the previous subsection is an example of
resummation, a technique of summing the contributions from processes of all orders of as
to obtain a finite cross section. This technique was pioneered by [10]. However, the
derivation in the previous subsection is not a complete solution. The approximations

made only take into account soft gluon emissions, (gluon pr << photon pr).

23

As a result, instances in which gluons are emitted with a p, on the order of the photon’s
p7. are not properly considered, and vector momentum conservation is not handled
properly.

To properly take into account all of these factors and yield a more accurate cross
section, refinements to the resummation technique were developed [1 l][l2]. This
calculation begins by looking at the differential cross section as a perturbative series.

The expression below shows the dominant contributions to the series,

2 2 2
192. 6c 9—g—ln(Q—2]|:vl + v2a51n2[—Q-2—)+ v3a§ ln4[—Q—2-] + . (1.23)
dPT PT PT PT PT

There is a practical problem with this series in that as pr —> 0 the series does not
converge due to the overall factor of i2 and the logarithmic terms. Because this is a
series in as In2 [Q—ZZJ, the series as is fannot converge at small pr, (Pr << Q) even if as is
small. This diverpgience can be cured by reordering the terms that are at least as singular
as —1—2 to produce a converging series. After the resummed part of the perturbative series
is cglculated, the remaining terms of the perturbative series can be calculated and added
to the resummed part to obtain the full cross section calculation.

The resummation calculation begins by writing the singular terms of the

perturbative series as

do a °° 2""1 n 111(sz
—~— v a, 1n —— .
d1)? Pi Eng?) "m Pi

This series can be written the following way (suppressing the coefficients):

lz[aS(L + 1) + (21";(1.3 + L2 + L + 1) + oc§i(L5 + ...) + -..] (124)
PT

24

 

2
TheL term is shorthand for ln[Q—2]. The series can be rearranged into the form:
. PT

do
dpidy

 

~ %(()7521 +oz§Z2 +...). (1.25)
T

Where the terms have the following form:
a521~ aS(L+1)+a§'(L3 + 1.2)+oz§(L5 + L4)+...
ail2 ~ a§(L+1)+a§(L3 + L2)+...

a§Z3 ~a§(L+1)+... (1.26)

Now instead of a series increasing by a factor of aSL2 for each succeeding term, there is
a series in 065- This means that the convergence of the series depends only on as. Note
that the highest power logs for each power of as in ale, aSL +a§L3 + agL5 + ..., arejust
terms of the LDLA result, Equation 1.22, in series form.

There are several formalisms for doing such resumming [13][14] However the
one used in this analysis is given by Collins, Soper, and Sterman (CSS) [12]. This
formalism has been used to study the production of single [15][l6][17][18] and double
[19] weak gauge bosons as well as Higgs bosons [20]. The formalism describes the triple

differential cross section as

do

°° n .13
mxjo dzbe” W(b,Q,xa,xb)+Y(pT,Q,x,,,x,,). (1.27)

Here, the resummed piece is the integral over the W function. The qunction is
comprised of the terms of the perturbative series not included by the resummed piece.
Specifically, it is evaluated as the full perturbative series (non-resummed) minus the
terms as singular as i, the “asymptotic piece.” The integration variable, I) is the impact

T
parameter and the Fourier conjugate to pT. Small [7 values mean large p, values.

25

 

The W function is explicitly written as

WW4)“-Zi-‘fJ-ié’ffaw’) ,(x,,sze-s<b>,
i a A

B

where S has the form,

S=j%[Aln[%2]+B],

and A and B are series with constants A}. and B 1. respectively,

21:20:94,. B=2a§3j.
1' 1'

These constants can be determined by expanding S order by order in (15 and comparing to
the corresponding terms in the perturbative calculation. At present, the A J. and B].
coefficients have been calculated up to j=2 by Davies,Webber, and Stirling (DWS) [21].

A correction must be applied to Equation 1.27 though because as blows up when

 

b > and therefore the integration over the full integration range from 0 to 00 cannot

QCD
be performed. In order to make a smooth limit on the integration, a quantity labelled b. is

defined as

b

b* =
,/1+(b/bm,,)2

 

and is substituted for b into W.
Now however, something must recover the range not integrated over by using b...

This is done phenomenologically by the substitution,

W(b) = W(b.)WN”(b), (1.28)

26

where W” P (b) is a function that covers the non-perturbative range. CSS [12] showed

that this non—perturbative function has the generic form

W”” = exp[—hl (wa) — h, (xb,b) — h,(b)1n(Q2)], (1.29)

where the h’s are to be measured.
A simplified form for WNP was first tested by DWS [21] using the Duke and
Owens PDF [22],

WNP = exp[—glb2 _ g2b21n[2%;]]’ (1.30)

where the g parameters were constants determined by a fit to data. Their results were
combined with a next-to-leading-order calculation [23] by Arnold and Kauffman [15]
providing the first complete prediction for hadron collider DY data with the CSS
formalism.

Another specific form for the function was given by Ladinsky and Yuan in 1994
[24] and also fit to data. This non-perturbative function differs from the DWS function in

that it has three g parameters and a dependence on 1',

WNP : exp[—glb2 _ gzbz “1(32—0] - glg3bln(100xaxb )]. (1.31)

27

Chapter 2

Fermilab and the DO Detector

2.1 The Accelerator at Fermilab

The accelerator located at Fermi National Laboratory in Batavia, Illinois is a
facility which accelerates protons and antiprotons to near light speed. The accelerator
can operate in two modes. In one mode beams of protons and antiprotons are collided.
The other operating mode extracts a beam of protons to external stationary targets. The
analysis in the next chapter was done for data produced from proton-antiproton collisions
and therefore the collider mode of the accelerator will be described.

The Fermilab accelerator is actually a system of separate accelerators [25]. The
beam is accelerated in stages, analogous to an automobile engine with gears, with a
different accelerator handling each stage. Figure 2.1 shows a diagram of the location of
the accelerators relative to each other.

The process of creating high energy beams of protons and antiprotons begins in a
Cockcroft-Walton accelerator. Here, electrons are added to hydrogen atoms to produce
singly charged negative ions. These ions are accelerated to an energy of 750,000 electron

volts (eV) by applying a positive voltage.

28

 

PBar Lina

Debuncher PreAcc

   
   
   
 
  
    
   
  

Booster

Tevatron Extraction

for Fixed Target Experiments
1 >

PBar Injection

MR P Injection
Tevatron

PBar
Target

Main Ring
Tevatro
RF

. CDF

Main Ring RF

Tevatron
Injection

DO detector

Figure 2.1: The system of accelerators at Fermilab used to accelerate
protons and antiprotons.

29

 

The accelerated ions then pass into a linear accelerator or linac. The linac is made
up of five tanks containing drift tubes connected end to end in a line approximately 500
feet long. The ions are accelerated through the tubes by an oscillating electric field which
only acts on the ions when they are in the gaps between the tubes. The system is timed so
that the ions are in the gaps only when the electric field is pointing in the direction to
accelerate the ions toward the next stage. The linac accelerates the ions to 400 million
electron volts (MeV) which are then passed through a carbon foil. The foil strips away
all the electrons leaving only bare protons to pass on to the next stage.

The next accelerator, the Booster, is a synchrotron. Synchrotrons are ring-shaped
accelerators in which charged particles are kept travelling in the ring with dipole bending
magnets. The acceleration is done by cavity electric fields, which increase the particle’s
energy with each revolution. The Booster is 500 feet in diameter and 20 feet below
ground. The protons travel around the Booster about 20,000 times until they attain an
energy of 8 GeV. When the protons leave the Booster they are injected into the Main
Ring in bunches.

The Main Ring, like the Booster, is a synchrotron. However,’the Main Ring is
four miles in circumference and contains a thousand dipole, quadrapole, and higher order
magnets for bending and focusing. The beam in the Main Ring is made up of bunches
because the acceleration is provided by RF cavities.

From the Main Ring, the protons will take one of two paths. They are either sent
into the Tevatron accelerator or extracted from the Main Ring to produce antiprotons.
Protons bound for the Tevatron make revolutions in the Main Ring until accelerated to
150 GeV in energy. Those to be used to produce antiprotons are raised to an energy of
120 GeV.

To make antiprotons, protons are extracted from the Main Ring and sent to collide
into a nickel target. Among the products of the collisions are antiprotons, which are

produced in bunches since the protons arrive at the target in bunches.

30

 

It takes about a million protons colliding with the target to produce twenty antiprotons.
These antiprotons are magnetically selected and sent to the Debuncher Ring which has a
rounded, triangular shape. When the antiprotons come out of the target, their energy
distribution is too widely spread. The Debuncher makes the energy distribution more
compact by a process of stochastic cooling. The process involves analyzing the
antiprotons for deviations from an ideal orbit and applying a corrective “kick” to
minimize these deviations. The roughly triangular shape more easily allows the signal
for the kick to arrive before the beam. After this step, the antiprotons are sent to the
Accumulator Ring for storage until enough antiprotons have accumulated. Once enough
of them are produced, they are injected into the Main Ring and passed into the Tevatron.
The final acceleration stage is the Tevatron where the collisions between protons
and antiprotons take place. The Tevatron is a synchrotron ring located directly under the
Main Ring in the same tunnel. The Tevatron uses magnets with superconducting coils to
accelerate protons and antiprotons to 0.9 TeV, thus its name. Both protons and
antiprotons are simultaneously accelerated in the ring, traveling in opposite directions
until they attain maximum energy and then continue to travel through the ring at that
energy. The proton and antiproton beams consist of six bunches each. Once both beams
are at maximum energy, the counter rotating bunches are focused to produce collisions in

the centers of two detectors, one of which is the DO Detector.

2.2 The DO Detector

The DO Detector is one of two collider detectors in the Tevatron ring. The
detector was designed to be nearly hermetic in detecting almost all particles resulting
from a collision, (neutrinos being the notable exception.) It was also designed to
measure three important properties of detected particles. The three properties are the

energy, the trajectory vector, and the particle’s identity.

31

Muon Chambers ‘

 

 

 

 

 

 

\.
/11

 

 

 

 

 

 

 

Calorimeters Tracking Chambers

Figure 2.2: The DO Detector showing each subdetector.

 

 

 

Figure 2.3: Representation of the DO coordinate system.

In order to fulfill these separate but equally important goals the DO Detector is
made up of several components, each of which is optimized to measure a certain aspect
of a particle. These subdetectors are described separately in the following sections where
[26] has been used as the main reference.

DO uses a coordinate system which has its origin at the nominal beam collision
point. The z axis is aligned along the central axis of the beam pipe with z increasing in
the direction of the proton beam. The y axis points straight up. With 6 defining the usual
azimuthal angle, it is often more convenient instead to use pseudorapidity, 77, defined in
Equation 2.1. In the limit that a particle’s mass is much less than its total energy, 77

approaches the rapidity given in Equation 1.2.

77 =-ln tan 2 (2-1)

33

Interaction

 

 

 

 

 

 

 

 

Poi“
ll \ III
II 1 Ill
11 1 III
1 :111
‘-' 1 i a
fi
Fri 1
“ET

 

 

 

 

 

.——_ ’14”

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r " " a
(DO

Central Drift Vertex Drift

Transition Forward Drift
Chamber Chamber Radiation Chamber
Detector

Figure 2.4: Cutaway view of the Central Detectors.

2.2.1 Central Detectors

The primary task of the central detectors is to accurately measure a particle 5
track Wthh is the path of a particle leaving the collision. The central detectors are the
Vertex Dr1ft Chamber (VTX), the Transition Radiation Detector (TRD), the Central Dr1ft
Chamber (CDC), and the Forward Drift Chambers (FDC). The VTX, TRD and CDC are
contained w1thin a cylinder with the beamline at the central axis as shown by Figure 2 4

The two FDC detectors are at each end of this cylinder
The VTX chamber is contained within a cylinder of outer radius 16.2 cm and has
the beam pipe as its central axis. The inner radius of the VTX is 3.7 cm which is just
outs1de the beam pipe. The VTX actually consists of three separate independent layers
concentrlc 1n radial distance, r. Each layer is also divided in equal sections or cells in q)

w1th the innermost layer containing 16 cells and the outer two partitioned into 32 cells

34

 

 

Figure 2.5: The VTX chamber with the layout of its wires.

The VTX chamber was designed to measure the tracks of charged particles. The
cells in the VTX accomplish this by measuring the charge on sense wires resulting from
the passage of charged particles ionizing the gas inside the cell. The ionized gas drifts to
the sense wires due to electric fields shaped by other wires in the cell. Figure 2.5 shows
the layout of the various wires in the VTX.

The Transition Radiation Detector (TRD) lies between the VTX and CDC
detectors. It was designed to help identify whether a particle is an electron. The TRD
uses the principle that highly relativistic particles emit photons when they pass through a
boundary between materials with different dielectric constants. The energy of the
photons is dependent on the mass of the particle and therefore the much lighter electron
can be distinguished from a hadron.

The TRD consists of three concentric layers. Each layer has a radiator and an
X—ray detector. The radiator is a collection of 393 layers of 18 pm thick polypropylene

foil spaced an average 150 um apart. X-rays are detected through the use of a charge

35

CROSS—SECTION OF TRD LAYER 1

OUTER CHAMBER SHELL
70pm GRID WIRE
ALUMINIZED MYLAR
8m

  
  
 

   

15mm .

   
 
 

o
————— —o—.-—____ ._

 

 

 
  

RADIATOR STACK
N 2

C
CONVERSION 0 .
STAGE 0]
30pm ANODE WIRE
/ 100nm POTENTIAL WIRE
23am MYLAR WINDOWS
HELICAL CATHODE STRIPS

Figure 2.6: Cross section of a TRD chamber.

drift chamber. When charged particle pairs from converted X-rays and other charged
particles pass through the chamber, they will produce an avalanche of charge that drifts to
sense wires. The setup is shown in Figure 2.6.

Outside of the TRD is the Central Drift Chamber, or the CDC. Like the VTX it is
also a drift chamber detector and its purpose is also to measure the tracks of charged
particles. The CDC is a 184 cm long cylindrical shell surrounding the TRD with an inner
radius of 49.5 cm and an outer radius of 74.5 cm. The CDC is constructed of four layers
with each layer sectioned into 32 cells with equal (1) coverage.

The (p positioning of the cells between adjacent layers is staggered by a half cell.
This is to insure that a particle will travel through cells on at least two layers. Each of the
. cells contains seven sense wires and two delay lines. The delay lines are placed between
the cell walls and the two outermost sense wires. Charged particles passing through a

cell will ionize the gas with the charge drifting to the seven sense wires.

36

 

 

 

 

 

 

Figure 2.7: Cutaway view of the CDC showing the wire placement in
each cell.

The drift time to the sense wires gives r and (1 information on the track. Accumulated
charge on the sense wire adjacent to a delay line will induce a charge on the delay line.
This charge will travel down the delay line in two pulses traveling to Opposite ends of the
line. The time of arrival of both pulses gives a 2 position for that portion of the track
nearest the delay line. Appendix B details a project to recalibrate this 2 measurement.

Unlike the other central detectors, the Forward Drift Chambers (FDC) are not
cylindrical Shells encompassing the beam line, but are, as Figure 2.4 shows, two roughly
disk shaped detectors placed on both ends of the other central detectors. The purpose of
the FDC is to measure tracks for particles with rapidity outside the range of the CDC and
it does this with the use of drift chambers. Each detector is made up Of three modules.
There are two 9 modules and a CD module sandwiched between them as Figure 2.8
illustrates. AS the figure shows, the O modules are rotated by 45° relative to each other.
As the names suggest, the O modules measure the Booordinate and the (D module

measures the (I) coordinate.

37

 

 

Figure 2.8: The three modules which make up each of the Forward Drift
Chambers.

2.2.2 Calorimeter

The calorimeter for DO is designed to measure the energy of particles from
collisions and also to provide for identification for some of those particles. The
calorimeter is made up of three main sections, the central calorimeter and two end
calorimeters as shown by Figure 2.9. The central calorimeter covers the region lnl _<_ 1,
the end calorimeter south covers 1 S n _<_ 4, and the end calorimeter north, the range
-4 s n s -1.

Each of the three calorimeters is made up of box shaped cells. Figure 2.10 shows
that a cell consists of an absorber and a signal readout board with liquid Argon in the gap
between them. When a particle hits the absorber plate, it produces secondary particles
which then ionize the liquid Argon. The signal readout board is kept at 2 to 2.5 kV

potential relative to the absorber plate so that charge will drift to the signal readout board.

38

 

Dfl LIQUID ARGON CALORIMETEFI

  
     
 
 
 
 
   
    
         
          
  
 
 
    

END CALORIMETER

Outer Hadronic
(Coarse)

Middle Hadronic
(Fine 81 Coarse)

 
 
 
   
   

/ ,

l

 

.. §é¢l
ea.

/ -\ \ .

CENTRAL
CALORIMETER
Electromagnetic
Fine Hadronic
Coarse Hadronic

Inner Hadronic
(Fine 8. Coarse

1r“
Electromagnetic

Figure 2.9: Cutaway view of the calorimeter.

 

K Absorber Plate Pad

Resistive Coat Liquid Argon \
G 10 Insulator J

Gap

/7/ i
.. %/

a

g <———-—- Unit Cell -—>

 

 

\\\\\\\‘

 

 

 

 

 

 

 

 

 

 

It

 

 

 

 

 

Figure 2.10: Representation Of a typical calorimeter cell.

9.)
\D

   

 

 

 

i x.\ “' i 4 . . f . /
.3 . 4 j . .. ’1, 7
__ ___- ‘ , 1
._ . / .
i... _ .w M
I,” ‘

v . _
’ififisvge_.].. » "‘ " V

 

Figure 2.11: A portion of the calorimeter with pseudo-projective towers at
different 71.

The placement and sizes of the cells are such as to make pseudo-projective
towers. That is, the cells are lined up so that if a collision takes place at the center of the
DO Detector, a particle leaving the collision will have a high probability to pass through
a continuous string of cells along a line of constant 17. Figure 2.11 gives a representation
of these pseudo-projective towers.

The central calorimeter is made up of several sections. They are, with increasing
distance from the beam pipe, the electromagnetic (EM), fine hadronic (FH), and coarse
hadronic (CH) sections. There are 32 EM cells in ¢ and 16 FH and CH cells. The EM
section uses 3 mm thick depleted uranium absorbers, and has a total radiation length (X0)
and interaction length (71A) of 20.5 and 0.76 respectively. A 6 mm thick uranium-niobium

alloy is the absorber material for the FH cells. The total radiation length and interaction

40

 

length for the PH section is 96.0 and 3.2 respectively. The CH cells use 46.5 mm thick
copper for their cell’s absorbers. The CH section has 32.9 and 3.2 for its radiation length
and interaction length.

The end calorimeters also have different sections. They are the EM, inner
hadronic (II-I), middle hadronic (MH), and outer hadronic (OH) sections. The IH and
MH are further subdivided into fine and coarse areas.

Between the central and end calorimeters are gaps covering 0.8 S |n| S 1.4. To
reduce the chance of a particle in these regions not being detected, arrays of scintillating
counters were installed in order to lessen the dead regions in the gaps. These
intercryostat detectors (ICD) consist of 384 tiles Of scintillator material. Each tile has
three or four grooves in which are placed wave shifting scintillating fiber. These

scintillating fibers are fed into photomultiplier tubes.

2.2.3 Muon Detector

Encasing the calorimeter is a detector to identify and measure muons. It is made
up of inner proportional drift tube chambers (PDT'S), iron toroids, and outer PDT’s. One
layer of PDT’s lie just inside the iron toroids. The iron toroids generate a magnetic field
which bends muons as they pass through. Two outer layers PDT’s then measure a
muon’s track after it passes through a toroid. By comparing a muon’s track from the
inner PDT’S to the track from the outer PDT’S the extent of the bending of the muon’s
trajectory can be measured. The amount the muon bends provides a measurement of its

momentum .

41

 

 

 

 

Level Level Level
Detector —> —> D —> Tape

0 1 \ 2
Output 450 kHz -200 Hz Level / ~2 Hz

Rate 1.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.12: Schematic view of the different trigger levels.

2.2.4 Triggering System

The goal of the trigger system is to maximize the interesting physics events
recorded. The trigger does this by making preliminary calculations of certain quantities
Of a beam crossing and deciding whether it constitutes an interesting event. This prevents
uninteresting physics from being recorded to tape. Not only does this keep the DO data
sample clean, but it also leaves the detector free to examine succeeding beam crossings
rather than wasting time calculating quantities for an uninteresting beam crossing. The
trigger system consists of 4 levels, named level 0, l, 1.5, and 2. For an event to be
recorded as DO data, it must pass the criteria set by at least three of these levels, with
level 1.5 being required for only a subset of events. Therefore the path an event takes to
tape is to first pass through level 0, then level 1, possibly level 1.5, and finally level 2.
Figure 2.12 illustrates this path.

The central piece in the level 0 trigger is the level 0 detector. This detector
consists of two square arrays of scintillator counters each mounted on the forward face of
one of the end calorimeters and having partial coverage in the 1.9 S lnl S 4.3 range. The
scintillator counters are read out into photomultiplier tubes. This detector basically
determines whether an inelastic collision has taken place, and if so, makes a rough
calculation of the 2 position of the interaction vertex. It determines whether a beam
crossing resulted in an inelastic collision by the criteria that both scintillator arrays must
register a signal within a certain time period of each other. The z coordinate of the vertex

of an inelastic collision is calculated from the time difference between the two signals.

42

With the time between beam crossings at 3.5 us, the level 0 trigger passes events at the
rate of about 150 kHz.

The level 1 trigger uses quick calculations made from the muon detector and
calorimeter level 1 triggers to decide whether a collision is interesting physics. This
decision is made by the level 1 framework.

The level 1 calorimeter trigger makes calculations of the component transverse to
the beam line of several energies such as the total transverse energy, E 7, the hadronic E 7,
the electromagnetic E7, the missing ET, and the E, of the trigger towers. The trigger
towers consist of the pseudo-projective towers in the calorimeter described on page 40.

The trigger framework makes its decisions using 256 bits called AND-OR terms.
Each of these bits are flags indicating whether a certain condition has, or has not been
met by an event. An example of such a condition is testing whether an event had a
minimum amount of hadronic energy. The framework also contains 32 programmable
specific triggers which contain sets of these AND-OR terms. An event passes a specific
trigger only if all of the AND-OR terms match the state required by the specific trigger.
But in order to pass the level 1 trigger an event has to meet the conditions of only one
specific trigger. The level 1 trigger passes events at a rate of about 200 Hz.

Some of the specific triggers are flagged for the level 1.5 trigger. If an event
passes the criteria for one of these specific triggers, then they are passed on to the level
1.5 trigger where several quantities are measured more accurately than was done at
level 1. Appendix A describes a program that monitors Levels 1 and 1.5 operations.

The level 2 trigger is comprised of a network of DEC VAX 4000’s running
filtering software. Here, the filtering software uses quantities calculated more accurately
than at levels 1 and 1.5 which is allowed due to the smaller event input rate. A specific
example of a level 2 filter is the one for obtainingZ events. This filter requires the
presence of two clusters of deposited energy resembling electrons with E, > 20 GeV and

ET > 16 GeV. Events at level 2 are filtered so that events pass only at a rate of 2 Hz.

43

 

Chapter 3
Refinement of DO Z Data

3.1 Introduction

Prior to the publication of the DO Z p, distribution it was felt that some further
studies could possibly refine the already completed analysis [27]. Some of these
additional studies involved the use Of a Monte Carlo which was used to simulate Z events
in the DO detector. Another project was the development and use of a specialized

program to produce properly smeared Z pT distributions for studies in the high pT region.

3.2 Studies With Fast Monte Carlo

A fast Monte Carlo, named the CMS Monte Carlo, was used in the Z pT analysis
in order to measure the acceptance and as a model for the detector smearing. This Monte
Carlo was first devised for the W mass measurement at DO [28] [29]. This section
describes studies which were performed in order to better understand the behavior of this
Monte Carlo and to learn if there are kinematic regions in which the measured Z Pr

differs greatly from the true pr.

44

 

3.2.1 Examination of Kinematic Variables

The aim of the first study was to learn from the simulator if there is a range for
one or more kinematic quantities in which the smeared, or measured, Z p1. is far off in
value from the unsmeared, or true pr. If such a region was found, then it might have to be
removed through a cut on the data sample. The Z kinematic quantities studied were the
transverse momentum, rapidity, and mass. The 2 of the interaction vertex and some
properties of the most energetic electron were also included in the study. All of the
kinematic variables considered were the smeared values, because these would be the only
values available for measurement in an actual data sample.

In order to examine the effect of these kinematic quantities on the smearing of the
Z p7, profile histograms were made of the absolute value of the resolution as a function of

each of the kinematic quantities. The absolute resolution is defined as
smeared _ Z unsmeared

IZPT T

unsmeared
ZPT

 

. A profile histogram plots the average y value and its 6 in each
bin of x. One advantage of a profile histogram over a two dimensional scatter plot is that
it is easier to notice any trends as a variable changes. Also if the population of events is
heavily concentrated in one region of values in x, then it can appear in a scatter plot that
the Z pr smearing is greater in that region simpler because there are more points to plot in
that region. This problem is avoided with the profile histogram.

It should be noted that with one exception, the requirement, unsmeared Z p, >0.5
GeV, was used in making the profile plots shown in this subsection. This cut was made
because the denominator in the resolution function blows up as p, ——) 0, giving large
resolution values for what are actually insignificant amounts of smearing. The exception
to this cut is the plot of the resolution versus the Z pr, which was kept as is in order to
illustrate this problem with the low p, area. This cut only eliminates 0.85% of the total
sample and still leaves enough low p, events to accurately represent this region in the

plots.

45

Experimentally, the mass of the Z is arrived at by measuring the invariant mass of
the electron pair. From the plot of the resolution versus the Z mass in Figure 3.1, it is
apparent there is no dependence on the mass. The data sample used for the Z p7.
distribution actually only includes events with a measuredZ mass between 75 and 105
GeV in order to minimize the contamination of photon Drell-Yan events. In this mass
range of the profile plot, the average resolution is about 20%.

There also is little dependence of the pT resolution on the Z rapidity as seen from
the plot in the same figure. There is only a 0.5% difference in the resolution between
small absolute y and large absolute y events.

The plot showing the change in p, smearing as a function of pr in Figure 3.2,
shows a steady resolution less than 5% except at the lowest pr. This jump at smallest pT,
as explained earlier, can be attributed to the denominator of the resolution going to zero
in this region. Finally, the plot of the vertex 2 versus the pT resolution shows there is no
systematic dependence on this variable.

The second set of plots in Figure 3.3 and Figure 3.4 shows several kinematic
variables for the highest pT electron in an event. The resolution in Z pT does show a
tendency to get larger as the electron energy becomes smaller, with the peak resolution
near 20%. When theZ p, resolution is plotted against the pseudorapidity of the electron
it can be seen that the resolution is fairly constant at or below 20% over the entire range.
Finally, in examining the effect of the pT of the electron on the resolution, there is a sharp
peak to the resolution of 22.5% at a p, of 40 GeV, though this is within an acceptable
level.

Overall, while the Z [9, resolution does Show a functional dependence on some of
these kinematic variables, the resolution never reaches an unacceptably large level which

would require any additional cutting on the data.

46

x10
10000

8000

6000

4000

2000

16000
14000
12000
10000
8000
6000
4000
2000

 

IIIIIIIIIIIIIIIIIIII

 

IIIIl

(18

(16

0.4

0.2

 

III IIIIIII

 

50

75

100 125 150

mees

 

IIIIIIIIIIIIIIIIIIIIIIIlllllllll

 

 

0.2
0.175
0.15
0.125
0.1
0.075
0.05
0.025

 

IIIIIIIIIIIIIILI O

 

—4

-2

0 2 4

Z rapidity

 

 

,—

b

i—

I.

)_.

'— ~1—
* +
'— _ —+—
>-—++_’_-.—_.__-—_ —o——9—‘+“

L -+— ...
>—
EHIILIIIILJIIIIIJIIII

 

 

 

50 75 100 125 150

res vs. smeared mees

 

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

 

TTII]

—1—

+

IIIIIIIIIIIIIIIIIII

 

 

—4 -2 O 2 4

res vs. smeared Z rapidity

Figure 3.]: Dependence of the absolute resolution on the Z mass and
rapidity.

47

 

 

 

x102
2500

2000
1500
1000

500

IIIleIIIIIIIIIIIIIlIIIIIIII

 

 

O

1 J 1

LIIIIIII

 

 

0

100 200

smeared Z pt

300

 

50000

40000

30000

20000

10000

 

IIIIIIIIIIIIII]ITTIIIIIIIIII

 

 

 

z vertex

0.8

0.6

0.4

0.2

0.5

0.4

0.3

0.2

0.1

 

 

i—IIIIIIIITIIIIIIIIITIIII

I

I‘H‘j—WWA—U 1r-

 

 

i
0

100 200 300

res vs. smeared zeept

 

IIIIIIIIIIIIIIIIIIIIIIITIIIT

p—

 

 

 

 

 

-1

00 0 100

res vs. smeared zvtx

Figure 3.2: Absolute resolution dependence on the Z p, and the z of the
interaction vertex.

48

x10
10000

8000

6000

4000

2000

30000

25000

20000

15000

10000

5000

 

 

 

 

500

 

:

C

h

L

71.1 I l I L I.I 1 l I J I J I I l I I 1
0 100 200 300 400

smeared e1e

E

E

L—

E

 

 

 

smeared eieta

 

0.225
0.2
0.175
0.15
0.125
0.1
0.075
0.05
0.025

0.225
0.2
0.175
0.15
0.125
0.1
0.075
0.05
0.025

 

IIIIIIIIIIIII

i

IIIIIIIIIIIIIIIIIIIIIITIIIII

 

.—

 

l I I I l I l I I II I l I I I l I [_I kfl.1

 

O [Til

100 200 300 400 500

res vs. smeared e1e

 

 

IIIIIIIIIIIIIIIIIII[IIITTTIITTIIIIlll‘lllillll

 

 

res vs. smeared eleta

Figure 3.3: Absolute resolution dependence on the energy and
pseudorapidity of the highest p, electron.

49

x102

 

 

 

 

 

 

 

 

 

 

2000 _— :
5 0.225 :— -
1750 :- E 0
I 0.2 E— -
1500 :— 0.175 E— *
L :
1250 :— 045 :— ’
1000 5- 0.125 :— g
750 E 01 5*
E 0.075 :— +
500 : +
E 0.05 g— +
250 E— E. - -
: 0025 E “*‘Hrfl. ++I+I+++
0 TI I I I I I Igl I l I l I O '—l I I I I I I I I++I I I I-
0 100 200 300 0 100 200 300
'smeared elet res vs. smeared e1 et

Figure 3.4: Absolute resolution dependence on the p, of the highest pT
electron.

3.2.2 Studying Contributions to Smearing Resolution

The purpose of the next study was to discover the contribution to the Z pT absolute
resolution (as defined in the last subsection) from the various detector effects. This was
accomplished by independently turning on smearing for each kinematic quantity one at a
time and analyzing the effects on the resolution. These studies were performed
separately for events which had both electrons in the central calorimeter, “CCCC” events,
and for events with one electron in the CC and the other in the BC, or “CCEC” events.

The detector effects against which the resolution was compared were the EM
calorimeter position resolution, the EM calorimeter energy resolution, the track
resolution, the underlying event contribution to the electron energy, and the resolution in
the z coordinate of the interaction vertex. The EM calorimeter position resolution
quantifies how well the calorimeter is able to determine the trajectory of an electron from
its energy deposition in the calorimeter. The EM calorimeter energy resolution is the
accuracy of the EM calorimeter in measuring the energy of the electron. The track

resolution is the detector’s accuracy in determining the angle and position of tracks in the

50

CDC and FDC. The underlying event is the product of the noninteracting partons from
the initial proton and antiproton. In real events, these “leftover” partons decay into
particles in which the momentum vectors sum to zero. However, when any of these
particles deposits energy in the same portion of the calorimeter as an electron, this extra
energy will inflate the [Jr of that electron beyond its true value, which will introduce an
error in aZ p, measurement. The resolution on the z coordinate of the interaction vertex
is self explanatory.

The figures of Z pT vs. absolute resolution on the following pages show the
contribution of each of these quantities on the total resolution of the Z pT. The jump in
the resolution at the lowest pT bin for these plots has the same explanation given in the
last subsection and the same cut was made, Z pT >0.5 GeV.

For the EM calorimeter resolution, the Z p7. resolution for CCCC events is a near
constant 2% above pT of 20 GeV. The resolution peaks at 33.5% for the 0—5 pT bin, which
only amounts to a maximum, average smearing of 1.7 GeV. The CCEC events have a
similar distribution, with the resolution peaking at a smaller 24% at the low [9, end. Thus
the EM calorimeter resolution should introduce only small errors to the measured Z pT
distribution. The accuracy of the EM calorimeter position measurement shows itself to
have even a smaller effect on the overall resolution. The Z pT resolution has a value less
than 1% for all p, less than 20 GeV with the CCCC events. Here, the low pT peak is
under 9.5% which gives a maximum average smearing of under 0.5 GeV for that lowest
bin. The smearing of the Z pT is somewhat less for the CCEC events, with the resolution
under 1% for nearly all p,.

The track resolution is a negligible contribution to the Z pT resolution when
compared to the resolutions from the calorimeter measurements. For CCCC events it is
also fairly constant in pr and its value remains below 0.5% for all p, > 20 GeV. A
somewhat improved distribution is seen for the CCEC events, especially for the lower

end of the p, range.

51

The next resolution effect studied, the underlying event contribution, repeats the
pattern of being under 1% for both CCCC and CCEC events for all but the lowest pr.
However unlike in the other cases, the distribution for the CCEC events shows a higher
overall smearing than the CCCC distribution. The 2 vertex resolution has the smallest
contribution to the Z resolution with values less than 105%.

All of these contributions to the Z resolution are small, with each averaging less
than a few percent in the smearing of the Z p, with the largest contribution coming from
the EM calorimeter resolution. Even when the absolute resolution peaks above 30% as it

does in some plots, it amounts to 1.7 GeV in average smearing at most.

52

 

 

0.35
0.3
0.25

0.2

0.1

0.05

0.1

0.08

0.06

0.04

0.02

 

 

 

 

300

 

 

 

 

E i l ”fiu'i'wfh’tflm'l'fllwrih a +4 *1—
O 100 200
emres CCCC
I.
_ 2 .5,“ e’e'rfi’tiw- ‘1" ++ - +
1 1 1 1T 1 I ‘1’ ‘f’TI' Tris-PI]
O 100 200

empos CCCC

300

0.25

0.2

0.1

0.05

0.06

0.05

0.04

0.03

0.02

0.01

0

 

 

 

 

 

 

 

 

 

[:-
-I I I ZIII LI TI 1.7-+14- I+I++II I I 1'1
0 50 100 150 200 250
emreS CCEC
I: "Z~.'. WW-+H+ ++++++ ++I+j>+++|>k
I IJ I I I I I I I I I I ISLI I k-I I I I I
0 50 100 150 200 250

empos CCGC

Figure 3.5: Effect of EM calorimeter energy and position resolutions on
the Z p, absolute resolution. The absolute resolution is plotted as a

function of Z pr.

53

0.05

0.04

0.03

0.02

0.01

0.16
0.14
0.12

0.1
0.08
0.06
0.04
0.02

 

 

 

 

O

 

 

 

 

O

.
E—
C
T
: -~.~.......+..+ .. .1. 1+ _ —
1 1 1 1 I 1+ITI 7*”‘1 +Lw+J+-+J.t
100 200 300
trackres cccc
I
T
t “2...-
I 14T‘I“T*T‘HH.1~*II’+- I_+ 44.41 -1
100 200 300

uevent cccc

0.025

0.02

0.015

0.01

0.005

0.35
0.3
0.25
0.2
0.15

0.05
0

 

 

 

 

 

 

 

 

E
E
L + j-
31.... i 1+ ‘
: +++++++++++++++ +I + ++ ‘l’ +
_1111I1111I111111111I111_1
0 50 100 150 200 250
trackres ccec
E-
1‘:~?W°‘°M~row~‘+ ~41.» L141 1* -1 .1 -
0 50 100 150 200

uevent ccec

Figure 3.6: Z pT absolute resolution dependence on track resolution and
The absolute resolution is plotted as a

the underlying event contribution.
function of Z pT.

54

 

 

-5
x10

 

 

 

 

 

 

 

 

—5
x 10
0225 g 0.25 _—
O.2 :— E
0.175 _— 0.2 :—
0.15 E :
0.125 :— 0-15 :'
0.1 E— :
E 0.1 ~—
0.075 E- C
0.05 E 0.05 :__
0.025 E_—_ ‘ : —‘- ++
: __—~__,_ +4. + + + .. __ ”~W ‘1’ ~ ._
O ”I l I-TT'THTH'VIH'IH‘knfi-ru l O 1111 Iiiimfiffltfih l+IIllll
0 100 200 300 0 50 100 150 200 250
vtxres cccc vtxres ccec

Figure 3.7: Effect of z vertex resolution on Z p7. absolute resolution. The
absolute resolution is plotted as a function of Z p,.

3.3 Development of Fast Smearing Program

One final piece needed for the Z p, analysis was to study the smearing effects at
the high p7. range of the distribution. In order to do this, it was necessary to see the
difference in smeared distributions from different input theoretical distributions with
various parameter values. This study was not feasible with the CMS Monte Carlo used in
the previous studies because it could not produce the necessary statistics in a timely
fashion as this is a region of very small cross section. Therefore, a new specialized
smearing program was developed. The aim of this code was to smear the Z p,
distribution alone and do it much faster than the CMS Monte Carlo, but also to produce a
smeared distribution consistent with the more sophisticated CMS.

It was decided that the best way to produce a faster smearing code consistent with
CMS would be to parameterize the smeared output from CMS. A distribution of

(smeared pT - unsmeared pr) vs. unsmeared pr from CMS was made. This distribution

55

was separated into slices of the unsmeared p7. Then each slice was fit to a double
Gaussian shape. Since a Z p, distribution has high statistics only at low pr, CMS could
never achieve enough high p, statistics needed for fitting with the computing power
available. As an alternative, a fake distribution of pr vs. y, shown in Figure 3.8, was
devised as an input to CMS. It consists of bands spaced over the p, axis, each band
having the same area and shaped to have the correct rapidity distribution for a Z particle.
The bands are spaced 1 GeV apart in the range 0 to 20 GeV pr, and then are spaced 20
GeV apart thereafter. The finer spacing at low p, was used because it was previously
found that the fit parameters change more in that range. Once this distribution was used
as the input distribution to CMS, it was found that the smeared distribution indeed had
enough statistics to be fit to double Gaussians.

After the double Gaussian fits were performed, the fit parameters: normalization,
sigma, and mean were plotted versus pT (Figure 3.9). It was noticed that the mean values
were negligible over all p, values and were subsequently taken to be 0. Plotting also
revealed that one of the Gaussians (labeled “2” in the plots) became insignificant
compared to the other Gaussian above 200 GeV p,. The values for the Gaussian widths
were fit to lines and the relative normalization was fit to a constant.

The fast smearing program, SMEAR_PT, was designed to pick a random p,
value based on an input Z pr distribution, then to choose a random offset and add this to
the p, value to produce a smeared p, value. The random offset was calculated by picking
a random number from a double Gaussian distribution (single Gaussian if p, was greater
than 200 GeV). The parameters for the double or single Gaussian were chosen by using
the fit values to the CMS smeared distribution.

The smeared distribution from SMEAR_PT compared to CMS is shown in
Figure 3.10. Although the peak for the distribution from SMEAR_PT is somewhat

higher, the two histograms match up well at higher pT.

56

   
   

20000

17500

15000

12500

     
  
  

  
       
   

Imum
llHI In
llllllllll

2500 '” ,
O a 1
300

     
   
   

10000

 

 

 

7500

 

 

 

  
   
  
 

      
 
 

 

llllllll‘iilllllllll‘llllIlllllll

    
 
  

J 1

 

5000

 

 

 

lllllllll

 

 

Figure 3.8: Special input distribution to CMS.

57

()2

OJ

 

 

 

 

   

{/1161 0.1801 / 32

P1
P2

 

 

 

   
   

 

 

 

 

 

 

 

 

 

 

Illllll

 

 

 

 

 

 

 

 

 

 

 

 

 

4 .—
: {/ndt 0.1528 / 28
0.3745 — P1 1.236
0.1355E—01 — P2 0.131OE-01
3 _—
2 E
l
[111111111 1111111111l1111l1111|
100 200 0 50 100 150 200
Sigma 1 Fit Sigma 2 Fit
1 .—
0fi75 ,;
(15
(125
lllllllllllllillllL
0 50 100 150 200
Mean 2
{/ndf 0.5539E-01/28
P1 0.6811
P2 -0.6702E-04
OX7
N1
C16

0

 

iii—Lillllllllllllll
50 100 150 200

Relative Normalization Fit

Figure 3.9: Fit parameters plotted versus pr.

58

x103—

1400

1200

1000

800

600

400

200

 

 

888
44832791
36.17
1Ei78

 

 

 

_ ID
Entfies
” Mean
RMS
“'- -:. —— SMEAR_PT DISTRIBUTION
i: t} ———————— CMS DISTRIBUTION
L
;_
L
t

 

lllJJlllllllllllllllllllIgliLllllllllllllllllllli

 

 

20 25 30 35 4O 45 50 55

PT

59

65

Figure 3.10: CMS smeared and SMEAR_PT smeared distributions.

70

 

Chapter 4
N on-Perturbative QCD Fits

4.1 Method of Fitting

In order to perform fits to the non-perturbative parameters a method was needed ‘
to calculate the theoretical cross section distributions by the non-perturbative +
resummed method described in subsection 1.2.4. This calculation was made using the
LEGACY program written at MSU. LEGACY is able to generate triple differential cross

sections in the form for W’s, Z’s, photons, and Higgs particles for both fixed

80
313—18an
target and collider experiments. It can make this calculation for any input [2,, rapidity,
and center of mass energy, ECM. In addition, the calculation can be made with any choice
of PDF and non-perturbative function.

The first step in the fitting process was to produce tables, or grids, of these cross
sections for a range in p, values and g parameter values of the non-perturbative function.
Such grids were made for each experiment to be included in the fitting. After these grids
were produced by LEGACY, a Jacobian transformation would be performed on the grids
if the experimental data was not in the same form as the LEGACY cross section.

Once these grids were made, a fit could be performed for the non-perturbative

parameters. This was done by a x2 minimization with a FORTRAN program linked to

the CERNLIB routine MINU IT [30]. The grids provided the theoretical calculation.

6O

Since MINUIT requires the function it fits to be continuous in the fit parameters,
interpolations were made between grid points using another CERNLIB routine,
DIVDIF [3 1].

Initially, it was important to fit over the largest feasible g parameter space in order
to find the best possible minimum. The limitations on this coverage were due to finite
computational speed and the fact that the non-perturbative function only produces
physically reasonable cross section values in a limited region of the g parameter space.
Producing a set of grids for a global fit could take 6+ hours of running on several VAX
4000 workstations. Therefore the initial fit would be with grids coarsely spaced in g
space. The coarse grids, unless stated otherwise, ranged from 0.03 to 0.93 for gl , 0.05 to
1.0 for g2, and where applicable, -2.0 to 3.0 for g3. In some cases after a minimum was
found with coarse grids, finer spaced grids were produced which were centered around
the minimum found with the coarse grids and the fit was repeated. This was done to
reduce any potential error due to interpolation. In some cases this step was repeated with
yet a third set of grids with finer spacings than the second grid set. The spacings for all

finer spaced grids varied on a fit-by-fit basis.

4.2 Experiments Included in Fits

In order to test that a model is truly universal for all Drell-Yan type processes, it
was important to choose a set of data from a variety of experiments with different values
of t, center of mass energies, and different boson masses. Of the data used in the fits,
three were from collider experiments and two from fixed target experiments. The
collider experiments included the R209 [32], CDF Z [33][34] and the DO Z [27] data.
Fixed target data were used from both the E288 [35] and E605 [36] experiments. The pT
bins used from each experiment were chosen to be in the range that the non-perturbative
functions have an effect on the cross section, p7. < Q, and to provide enough data points

for fitting.

61

 

 

 

 

 

 

CM Boson
Experiment ENERGY 1 Range Mass PROCESS
(GeV) (GeV)
R209 62.0 0007-0031 5—11 p+p—>/.t+u—+X
CDF Z 1800.0 0.003 91.19 p+pbar—>Z
DO Z 1800.0 0.003 91.19 p+pbar—)Z
E288 27.4 0033-0110 5-9 p+Cu—>u+1r‘+X
0033-0054 7-9 _
E605 3” 0073-022 10.5-18 (”CU—”1+“ +X

 

 

 

 

 

 

 

Table 4.1: Experiments included in global fits with some properties for
the data used in the fits.

4.2.1 R209

The R209 experiment was based at the CERN Intersecting Storage Rings, (ISR), a
proton-proton collider. The total «/_ = 62 GeV data set contains 7827 events with an
integrated luminosity of 1.12x1038 cm'z. The R209 data also contains an overall
normalization error of 10%.

Only a subset of R209’s total data sample was actually used in fitting. The
x/E = 62 GeV data, and not the «5 = 44 GeV data, were chosen for inclusion in the fits
because the lower energy data have much lower statistics. Of this data set, the two bins
with photon mass ranges of 5 GeV S Q S 8 GeV and 8 GeV S Q S 11 GeV were
included. Since only the non-perturbative region was of interest, only the data points

with pTS 1.8 GeV were used.

62

To compare the cross sections of LEGACY directly to the published data, the

LEGACY cross sections had to be transformed into the same form as the data, 352—.
T

The steps taken to make the transformation are as follows:

1 Numerically integrate over rapidity to convert Limo ji—

3ptayaQ2 3pT3Q2
80'

812%8Q

3. Separately integrate over the range of each mass bin to remove the dependence on Q2.

2. Divide by 2p, to place the cross section into the form

Each mass bin was integrated by having several identical grids with different

photon masses covering the range of the mass bin. A numerical integration was then
performed for each fimp gl, g2, ...). For example, for the integration over the
5—8 GeV mass bin therTe were identical grids produced using photon mass values of
5.0, 5.5, 6.0, 6.5, 7.0, 7.5, and 8.0 GeV. gig—QTQP g,, g2, ...) values for each of

these masses were then used in a numerical integration of Q2 to obtain

30'
“TOUT: g1, g2, ...) values.
T

4.2.2 CDF

The CDFZ data come from the first collider detector at Fermilab located in the
Tevatron ring. There are actually two data sets used in the fitting. One data set was
taken from the 1988-89 accelerator run, (Run 0). The second data set was from the more
recent 1992-1995 (Run 1) accelerator running. The older data set, with an integrated
luminosity of 4.05 pb", has many fewer events than the more recent results from CDF
and DO. This earlier data set has 290 events with p, S 23 GeV, the p, range used in the
fits. The more recent data set has an integrated luminosity of 110 pb“.

Since the data were published in the form a—o, it was relatively simple to
transform the LEGACY cross section into this form.TThe Q2 dependence disappears

since it is a constant, namely the square of the Z mass. The only necessary step was to

numerically integrate over rapidity.

63

4.2.3 E288

E288 was a fixed target experiment located at Fermilab. The data which were
used from this experiment came from a 400 GeV proton beam impacting a copper target
and producing virtual photons which convert into dimuon pairs. From the published data,
a subset was chosen for inclusion into the fitting. In most cases, the data from 5 to 9 GeV
in photon mass were fit, though in some instances data from the higher mass bins (9-14
GeV) were also included. These higher mass data have much larger fractional errors than
the low mass data and so did not effect the fitting as significantly. In order to confine the
fitting to the non-perturbative regime, only the seven lowest p, bins (pTS 1.4 GeV) were
included. The overall normalization error for the data is listed at 25%.

In order to match the data, the following relation was used to convert the

LEGACY cross sections into invariant cross sections:

.130 1 do

dP3 _ 27WT dPTd)’ .

 

 

(1.17)

With this relation, the following steps were performed to make the transformation:
1. Divide the grid cross section by 211: 1),.

2. Numerically integrate over Q2.

4.2.4 E605

The E605 experiment, like E288 was a fixed target detector located at Fermilab,
though the E605 experiment took its data at a later time. Again like E288, the data
consist of dimuon pairs produced from a proton beam colliding with a copper target. In
this case, however, the proton beam energy was 800 GeV. The data set included in the
fits covered the mass ranges of 7.0-9.0 GeV and additionally 10.5-18.0 GeV in some

cases. In order to stay within the non-perturbative regime, the seven pT bins below 1.4

64

 

 

GeV were used. These data points had an overall normalization uncertainty of 15% and a
point-to-point systematic error of 10% in addition to the statistical errors. The Jacobian
transformation used for E605 is identical to the transformation used for E288 because the

E605 data were also published as invariant cross sections.

4.3 Choice of Non-Perturbative Functions

Because there is as yet little theoretical basis for the form of the non-perturbative
function, fits were made with several functions. One of the functions chosen was the
DWS parameterization (Equation 1.30). Fits using this parameterization will hereafter be
referred to as “2 parameter fits.” The function from the LY parameterization (Equation
1.31) was also used in fitting. Still a third function, given in the equation below, was also

tried in some fits,
Gauss 1: —glb2 — g2!)2 ln[3%;] — glg3b2 ln(100xaxb). (4. 1)

This new parameterization is similar to the LY function, but it yields a pure Gaussian
shape in terms of the b parameter.

Another purely Gaussian parameterization was also tried,

Gauss 2: —glb2 — g2b21n[%)— glg3b2(xa + xb). (4.2)

However this function was fOund not to match the data well in initial fits and so was

discarded.

65

 

 

4.4 Preliminary Fits

The numerous fits which were performed are described in chronological order in
this section and in the following two sections. A table summarizing all of the fits in this
chapter, Table 4.12, is given on pages 87-89. This section describes the results of global
fitting to data predating Run 1 Tevatron Z results. A portion of this work has been

accepted for publication [37] and is reprinted in Appendix C.

4.4.1 Duplication of Original LY Fitting

Before the DO data became available, studies were done to isolate the effects of
such things as the PDF used and normalization uncertainties from other experiments.
The first of these fits, labeled Fit A, was a duplication of the Ladinsky Yuan fit [24], with
the LY parameterization (Equation 1.31). This was done primarily as a double check on
the latest version of LEGACY as well as the fitting code. The fit involved data from the
R209 experiment (5 GeV S Q S 8 GeV), the CDF Z Run 0 data, and the E288 experiment
(6 GeV S Q S 8 GeV). The original LY fit actually consisted of two steps. The first step
was a fit to g2 using the R209 and CDF Z data. Then, with the R209 and E288 data, a fit
was performed for g1 and g3 with g2 fixed at the value from the first step. The three
parameters were not simultaneously fit at the time because of limitations in computing
power and also because it was thought that the higher statistics E288 data would cause
the important data at smaller Ito be neglected. These fits were done using the CTEQ2M
PDF [38] to calculate cross sections. Equation 4.3 lists the fit values from the original

LY fit.

g, = 0113:3334, g2 = 05812;, g3 = 4.51311 (4.3)

66

 

 

 

It was later found that a FORTRAN error in the description of the neutron quark
density in the original LEGACY resulted in an incorrect published value of g1 and g_,.
This error affected only calculations for fixed heavy target experiments so that the g2
value should not have been affected. Equation 4.4 shows the value from refitting. This

refitting resulted in a value of g2 which is consistent with the original LY fit.
Fit A: gI = 0.05, g2 = 0.57, g3 = 0.83 (4.4)

The next fit, Fit B, was identical to the previous fit with the exception that
CTEQ3M [39] was used to calculate the cross sections. Comparing the results of this fit
to that of the previous fit gives a measure of the effect of two particular PDF’s. Equation
4.5 shows the values obtained with CTEQ3M. It can be seen that there is about a one to
two sigma difference in g2 and a significant change in g3. This lends credence to the
assertion that it is only valid to use non-perturbative fit values in a calculation with the
PDF used to obtain those fit values.

In order to more easily compare fits to each other, all subsequent fits were done

using CTEQ3M.
FitB (LY function): gl = 0.08, g2 = 0.47, g3 = 1.6 (4.5)

As a test of the difference between parameterizations, the same fit was done with
the Gauss 1 parameterization and that fit resulted in the values below. As expected, the
g2 value did not differ from the LY parameterization’s value, however there is a
significant difference in the g3 values. The slight change in the g1 values is due to the

coupling between g, and g3.
FitB (Gauss 1): gl = 0.11, g2 = 0.47, g3 = 0.61 (4.6)

67

 

 

 

 

 

 

Parameter Value
gl 0.15
g2 0.40
g3 -0.84
R209
Norm 0.99
E288
Norm 1.47

 

 

 

 

Table 4.2: Fit C values using experimental data from R209, E288, and
CDF Z.

4.4.2 Simultaneous Fitting of All Parameters

The next issue confronted was in the mechanics of the fitting itself. Here, Fit B
was repeated with the exception that all three g parameters were fit simultaneously. In
addition, the normalizations for both the R209 and E288 data set were included as fit
parameters due to the large uncertainty quoted in the published data. The values for Fit C
are given in Table 4.2. Note that the fit to the E288 normalization is 47% away from the
central value, which is well outside the 25% error range.

Data from E605 and additional E288 data were included in Fit D, the next fit.
This was the only change from the Fit C data set. The E605 data included in the fit
ranged in 7.0 to 9.0 GeV in boson mass. The E288 data was extended to include 5.0 to
9.0 GeV in boson mass Fits were performed with the LY, DWS, and Gauss 1 functions.

From an examination of the )8 per degree of freedom, the Gauss 1 fit appears to
provide a better fit than the other parameterizations. The Gauss 1 fit is also the only one
which results in a normalization for E605 which is within the published error range,
though all three fits resulted in normalizations for E288 which were outside of its

published error range.

68

 

 

 

 

 

 

 

 

LY DWS
Parameter Function Function Gauss 1
g, 0.16 0.18 0.21
g2 0.34 0.27 0.47
g3 -0.41 0.00 -0.51
R209
Norm 1.02 1.05 0.93
E288
Norm 1.39 1.29 1.31
E605
Norm 1.28 1.25 1.15
x2 [dof 2.18 2.18 1.65

 

 

 

 

 

 

Table 4.3: Values for Fit D for three parameterizations.

The next fit done used the LY function and added some higher mass E605 data to
the data of the previous fit. Table 4.4 lists the range of each experiment’s data included
in Fit E. It was noticed that again the fit results in a normalization outside the given 25%
error range for E288. Table 4.5 also shows that the fitted E605 normalization also ended
up outside of the 15 % error range of the published value.

Due to the inability to obtain fits which describe the data within the bounds
published by the experiments, it was decided to test the behavior of the fits if E288 was
removed in the next fit, Fit F. E288 was singled out because of the inability to produce a
consistent fit to its normalization in Fits C, D, and E. There were also indications that the
CTEQ collaboration had problems when including E288 in its global PDF fits [40]. Since
E605 provides higher statistics data with the same process as E288 and a similar center of

mass energy, there would be no loss to the diversity of the data set.

69

 

P BOSON

T MASS
Experiment RANGE
(G eV) RANGE

 

 

 

 

(GeV)
R209 0.0-1.8 5.0-8.0
CDF Z 0.0-22.8 81.0
E288 0.0-1.4 5.0-9.0

7.0-9.0,
E605 0.0-1.4 10.5-18.0

 

 

 

 

 

Table 4.4: Data set of fit including extended E605 data set, Fit E.

 

 

 

 

 

 

 

Parameter Value

g] 0.17
82 0.30
g3 -0.20

R209

Norm 1.04

E288

Norm 1.35

E605

Norm 1.30

 

 

 

 

Table 4.5: Fit results for Fit E.

70

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Value Uncertainty

81 0.15 -0.03,+0.04
g, 0.48 -0.05,+0.04
33 -0.58 -0.20,+0.26

R209

Norm 0.96 N/A

E605

Norm 1.14 N/A

 

Table 4.6: Fit for LY parameterization excluding E288, Fit F.

As well as discarding E288, data in the 8-1 1 GeV mass range from R209 was
added. Only the 7-9 GeV mass range from E605 was included in the global data set.
These steps were made with the purpose of balancing the amount of large and small 1'
data. In addition to the LY parameterization, fits were performed using the DWS and
Gauss 1 parameterizations. However, fitting with the Gauss 1 parameterization using
fine grids resulted in multiple minima of the same x2 rather than a single minimum. In
other words, there were many values for the fit parameters which fit the data equally
well. This suggests that the data set used was not adequate to differentiate among theory

curves with different fit parameters and that additional data would serve to pick out a

single minimum.

71

 

 

 

 

 

 

 

Parameter Value Uncertainty

gl 0.24 -0.07,+0.08
g, 0.34 -0.08,+0.07
g3 0.00 N/A

R209

Norm 0.96 N/A

E605

Norm 1.06 N/A

 

 

 

 

 

Table 4.7: Fit for DWS parameterization excluding E288, Fit F.

With this new data set, the normalizations in Fit F now fall within the quoted
experimental error ranges for both the LY and DWS parameterizations. For this fit, an
analysis was-also done in order to calculate the uncertainties on the g values. For the
DWS parameterization, a contour at 1+minimum x2 was plotted versus g1 and g2
(Figure 4.1). The upper and lower uncertainty values for the g, parameter were
determined to be the upper and lower g, extremes on the contour. The LY
parameterization is more complicated because of the correlations among the three g
parameters. It was decided that the uncertainties would be calculated by fixing each g
parameter, one at a time, at its best fit value, and then producing a contour at
1+minimum x2 with the other two g parameters on the x and y axes. The uncertainty for
those two parameters would then be determined exactly as was done with the 2 parameter
fit. These contours are shown in Figure 4.2.

Though the fit did not include E288, a comparison was done to the data using the
g values from Fit F of both parameterizations. In Figure 4.3 it can be seen that the
calculation matches the data well. Also of note is that both parameterizations are

indistinguishable in the plot.

72

2 PARAMETER FIT

92

 

0.425 —
04 —
0.37s —
0.35 —
0325 e

0.3 i—

0275 —

 

 

0.25 ‘Trl111l111l1L11111J141i

l L l L l I l L l I l
0.16 0.18 0.2 0.22 0.24 0.26

l l l
0.28 0.3 0.32

0.34
91

 

Figure 4.1: Contour of x2 +1 for 2 parameter fit, Fit F.

However, comparing pT distributions derived from the LY parameterization to
distributions made from the DWS parameterization for CDF Z reveals something
interesting. Figure 4.3 shows the calculated fit distributions compared to the
experimental data included in the fit. All the plots except the CDF plot show the two
theory curves to be identical. For the CDF plot, there is a noticeable difference at the
peak, suggesting that more precise Tevatron Z data would be able to differentiate between

the two parameterizations.

73

3 PARAMETER FIT

 

0.51 r

0.5 _

 

 

' I '11 I I I I . 11 L4 '1 L J 1 ' I J A ‘ l l A 11 I 111 A 1
0.12 0.1.5 0.14 0.15 0.16 0.1/ 0.18 0.19 0.2 0.21
91

 

g}

-0.¢75 _L
—o.5 l
«0525 _—

-o.575 ’

—0.625 L
—0.65 —

-0.675 :-

 

 

-0.7LJL_._i_iiAAlAAL-4.Llinil.1 .1111
0.‘7 0.1.5 0.14 0.35 0.15 0.17 0.15 0.19

91

 

-0.65 *-

~0.75

 

 

 

1 . ‘1‘ 1 1‘l‘ll'llll‘AlJAJLlllllllltl‘ AA'
0.42 0.4.} 0.44 0.45 046 0.47 0.48 0.49 0.5 0.51
92

‘08 [A111 1: ll

 

Figure 4.2: Contours for determining uncertainties for Fit F with the LY
parameterization. These plots were made with fixed g3, g2, and g1
respectively at their values at the minimum.

74

 

 

0.0.") (pb CeV'z/nucleon)

Ed’o/dp3 at y

(pb/GeV2>

2
T

dOng/do

_-

r23
0

60

40

20

 

 

__“_— 'Y—-' "_'T_ _' Y —'T""T. FTY]

_1 _ _'_T”

91: .24, 92: .34, 93: 0., NORM: 0.92
91: .15. 92: .48. 93: —.58, NORM= 0.79

L J l J l 1 l 1 l l 1 l l l 1 l l l l 1 |

  
  
  
  
 
    
 

2288 data (FRI) 23 (1901) 23)
CTEQ}. s=(27.4 GeV)2

5<0<6 GeV

 
 

6<O<7 GeV

7<O<80eV

 

8<O<9 GeV

 

 

 

 
   

 

 

 

0 0.2 0.4 0.6 0.8 1 1.2 1.4
P1(Gev)
3209 data {PRL ‘7 (1981) 12} CTE03
__ g1: .24, 92: .34, 93: 0., NORM: 1.04
91:15.92: .48,g3= —.58.NORM=1.04
_ 5<0<8 GeV
1
l 1
_ 8<O<110eV
ll111L111lllll!11JiLlllllll‘llllll 1']
O 0.2 0.4 0.6 0.8 1 '.2 1.4 1.6 1.8 2
0, (GeV)

 

—.
I

Ed’a/dp3 ot XF=O.1 (pb GeV‘z/nucleon)
O
I

 

. g1: .24. 92: .34, 93: 0.1101211: 094

3605 data (PHD

13 11991) 2315}
CTEQ}. s=(3a.a GeV)”

91 = .15, 92: .48, gs: —.58, NORM: 0.87

¢ 7<0<8 GeV

o
8<O<9 GeV

  
 

10.5<Q<11.5 CeV

\i+\?

 

 

 

 

 

 

 

 

 

—2
10 f
13.5<Q<1Bcev
1 I l 1 L ' 1 l 1 1 L l 1 I l 1 1 l I I 1 Pl 1 l I
o 0.2 04 0.5 0.8 1 1.2 1.4
Pl(GeV)
, clJ cm 2 data 1 m 67 (1991) 2937 1
700 :. o0 CTEO3. NORM: 1.
F 0
C .1 o 91: 24.92: 54.93: 0
i o 1' 0 91:15.92: .48. 93: -.58
600 3
C °° .
1 °.
A500 — o.
i) O
00
Q o 11
_Q 00
C1400 “ [13 0
v 0
. C - °
(3 » 8
300 ~
>. . 8
0 * 0
b .0 a
D200 —
r. a
I O
100 — + °
. 0
° 1
3
11'11111111]llliLLlllillllll111111111I1L1
o 2.5 5 7.5 10 12.5 15 17.5 20
01(GeV)

Figure 4.3: Plots of LY and DWS parameterization curves superimposed
with data. Note that the NORM values given here are applied to the
theoretical curves and are therefore reciprocals of the values given in

Table 4.6 and Table 4.7.

 

 

 

 

 

 

 

 

 

 

LY DWS
Parameter Function Function Gauss l Gauss 3

gl 0.05 0.03 0.05 0.05

g2 0.72 0.76 0.80 0.80

g3 -2.50 0.00 -2.50 -2.50
R209

Norm 0.93 0.92 0.89 0.89

CDF Run 1

Norm 0.88 0.89 0.89 0.90
D0 Z

Norm 0.99 1.00 1.00 1.00

E605 Norm 1.05 0.88 0.94 0.94

x2 [dof 1.15 1.30 0.88 0.88

 

 

 

 

 

 

 

Table 4.8: Results for global fit including DO and CDF Run 1 data, Fit G.

4.5 Inclusion of Tevatron Run 1 Z Data

At the beginning of year 2000 both the DO and CDF collaborations made
available new distributions of Z pT data taken during Run 1 at Fermilab. Both data sets
have high statistics which would counter balance the high statistics E605 data in any
global fitting. The new CDF and DO data were then added to the global set used in Fit F
and fits were performed for the DWS, LY, and Gauss 1 parameterizations as well as a
new parameterization, Gauss 3, described in subsection 4.5.3. The normalizations for
both of these data sets were fit in all of the parameterizations. The fit values for all the

parameterizations with this data set, Fit G, are given in Table 4.8.

76

 

0.8 —

) atx, =0.l

0.6 _

pb GeV‘2
nucleon

30' (
11;)"

 

 

 

 

 

 

 

 

L0 0.4 ~ —
,
r .
0.2 — —
0 Data
_ normalized LY
----- normalized DSW
. —- -normalized Gauss 1
0 l I I i l l A l l l ' l L 1 1 l l ' L 1 1 A A 1 ' L J
0 0.2 0.4 0.6 0.8 1 1.2 1.4

PT (GeV)
Figure 4.4: Gauss 1 fit distribution for E605.

4.5.1 Gauss 1 Fit

Fit G performed with the Gauss 1 parameterization was found to have a
significantly lower )6 than both the LY parameterization and 2 parameter fits.
Unfortunately this function had problems. First of all, it has the identical insensitivity to
g3 as the LY function fit (see below). Several minima were found which varied in g3
from —2.0 to at least -3.0, the lower boundary of the grids. The minimum given in the
table is for a g3 of -2.5 because it is at the midpoint of the range in which minima were
found. Also, the plot of the fit distribution versus the E605, Figure 4.4, data reveals that
the distribution dips at low pr. Perturbative QCD predicts that for an infinite boson mass,
a 51—;— distribution should flatten out as p, goes to 0 [11]. The judgement was made that

the non-perturbative prediction should give the same behavior for finite mass values.

Because of this theoretical prejudice and the g3 insensitivity, this fit was discarded.

77

4.5.2 LY and DWS Parameterization Fits

Figure 4.5 shows the LY function fit and 2 parameter fit compared to the data.
The plots of both of these fits match each other closely for all the experiments, the most
noticeable exception being a difference in the peaks of the Tevatron Z plots. The gl value
for both fits is very small, at or near the lower edge of the grid in g,. In fact, it was found
that for the 2 parameter fit that a small, negative gl value was preferred by MINUIT.
However, rather than a true minimum, there was a large set of g2 values and negative gl
values which gave equally good fits. In addition it was found that negative gl values
caused a low p7. dip in the prediction curve for E605 like the ones seen for the Gauss 1 fit.

There were also some complications in fitting to the LY function. It was found
that the lowest x2 occurs with g3 values between -3.0 and -2.2. However, the x2 is very
insensitive to the value of g3 in this range. Many minima were found with nearly the

same x2, with g3 values throughout the -3.0 to -2.2 range.

4.5.3 Gauss 3 Fit

After noticing that the fits for all three parameterizations result in a very small
value for gl , it was thought that this could be a reason for the Fit G fits being insensitive
to the g3 value. With a small g1, the third term of the non-perturbative functions also
becomes small. Therefore a large change in g3 would be necessary to effect a change in
the value of the non-perturbative function. It was decided to try a small variation on the
Gauss 1 parameterization, so that gl would be eliminated from the third term:

Gauss 3: —g,b2 — g2};2 4%.?) — g3b21n(100xaxb). ' (4.7)

78

 

J atx, =0.l

 

 

  
   
  

 

  

 

 

 

 

 

2b GeV"
nucleon

(

’a
dp"

 

Ed

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

pb
GeV

 

(

 

 

 

 

 

 

 

 

E605 Data R209 Data
1 ‘r r r ‘ v ' r I v I v I v u v I I v v I T v I 1 v r r 120 1 Y Y fi I 7 v v I I v v v fir W I v v
I 0 Data 1 ' g‘=0.05.gz=0.72.93:-2.5.NORM=1.08 ‘
—normallzed LY . .
----- normalized 08 950.03 92:0.76, 93:0.0. NORM=1.09
' ‘ 100 — -
0.8 . — 0 Data
—normafized LY
; — -normallzed osw
80 P a
h 1
0.6 "' "‘ .D fl>
CA. 0 ..
O
\__/
60 —<
. isle ‘
0.4 1
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_ 91:0.05.gz=0.72,93=~25.NORM=O.95 . 20 r ~
9,=o.oa 92:0.76. 93:00. NORM=1.14 M
O
o 1 l 1 I l ‘ A l l 1 ' 1 l A A I 1 J L 1 l L I 1 ‘L A J. o l J l l l 1 l L L l I l A A l A I A A
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.5 1 1.5 2
P (GeV) P (GeV)
T T
D0 2 Data CDF 2 Hum
700 Y Y 1 V 1 V Y Y Y I Y I T V I V I’ Y Y 800 b V V T T l I V Y Y I V V V I I Y Y T
O 0818 - T 7 0 Da‘a j
I —-—normalized LY . f l 1 —normallzed LY
600 I ..... normaflzad DSW 700 — §i 1 """ normalized DSW ..
eoo ’ -
1
A 500
n > I
C... O
O
\_/ b
400 --
l 0' .5 '
.4 7“ ‘0 .
7 300 ' 3
- 200 5 ~‘
100 *_ 95005. 92:072. 93:25. NORM=1.01 1
100 —
95003 92:076. 93:00, NORM=1.00 - 950-03 92=0-76. 93=0-0. N0RM=1-12 <
1
o - A A A A i A A A A 1 A A A A l A A A A ' 0 A A A A l A A A A l A l A A l A A A A
0 5 1 0 1 5 2 0 0 5 1 0 1 5 20
PT (GeV) PT (GeV)

Figure 4.5: LY function and 2 parameter Fit G compared to each other.
Note that the NORM values given here are applied to the theoretical
curves and are therefore reciprocals of the values given in Table 4.8.

79

Unfortunately Fit G for this parameterization exhibited the same behavior as the
Gauss 1 fit. It too found more than a single minimum between -2.0 and -3.0 in g3. The
minimum included in the table is for a g3 of -2.5 because it lies in the midpoint of the g3
range of found minima. This fit also has the same dip at low p, for the E605 curves

observed for the Gauss 1 fit making this result undesirable.

4.6 Reintroduction of Drell-Yan Data

Eventually it was decided that the data set for Fit G was now overbalanced to the
small T data and that was possibly causing the insensitivity to g3 and the very small g1.
Therefore it was decided to first reintroduce the higher mass E605 data and finally data

from E288 and examine the effects on the fits in steps.

4.6.1 Fitting with Higher Mass E605

The data set in Fit G already included the 7-8 GeV and 8-9 GeV E605 mass bins.
In addition to the data used for Fit G, Fit H included three additional E605 mass bins, the
10.5-11.5, 11.5-13.5, and 13.5-18 GeV bins. Table 4.9 outlines the results for the
different parameterizations.

The fit values for the DWS parameterization essentially do not change from Fit G
to Fit H. Most notable is that this parameterization still retains the very small g]. The
reduced x2 for this particular fit has increased from Fit G also. Examination of the fit for
this function plotted against the data in Figure 4.6 reveals why its reduced x2 is larger.

This function does a poor job of fitting to the E605 data. The fit curves are
outside the error bars for nearly all the data in the three highest mass bins. Also the fit
curve for the second lowest mass bin shows the same type of low p, dip as seen in the

Gauss 1 Fit G result and so makes this fit undesirable for the same reasons.

80

 

 

 

 

 

 

 

 

 

 

 

 

LY DWS
Parameter Function Function Gauss 1
g1 0.03 0.03 0.07
g2 0.70 0.71 0.74
g3 1.00 0.00 -2.00
R209
Norm 0.94 0.93 0.90
CDF Run 1
Norm 0.89 0.89 0.89
D0 Z
Norm 1.00 1.00 1.00
E605 Norm 0.94 0.89 1.11
x2 180 194 101
x2 [dof 1.96 2.09 1.1

 

 

 

 

 

 

Table 4.9: Fit H results.

The fit for the LY function also has nearly identical values to Fit G numbers.
There is however, a significant change for g3. Now, rather than a continuous range of
equally acceptable g3 values, this fit has only one best value for g3. Also the g3 value is
far from the -2.0 to -3.0 range of the previous fit. Like the fit to the DWS function, it also
has a larger reduced x2 than its Fit G counterpart. The similarity between the two
parameterizations continues in the plots. Fit curves for both of these functions are

virtually identical for all the data.

81

 

pb GcV'2
nucleon

) atx, =O.l

i

E .

1

pb
GeV

 

(

do
(1p,-

 

d‘a
dp’

700

600

500

400

300

200

100

E605 Data

 

T 1

YT'I'YTIYYY

    

 

— Normalized LY Function Fit
— - Normalized 2-Parameler Fil
- - - - - Normalized Gauss 1 Fit

 

 

 

 

 

 

 

 

’1..1...1..11..11..11. 11—1.
0 0.2 0.4 0.6 0.8 1 1.2

P (GeV)

T
DO 2 Data

_ T 1 r V I I 7| 1 I ‘ V Y I l ‘i I
C . - Data
, —Normalized LY Function Flt -
~ — -Normalized 2-Parameter Fit -
t ----- Normalized Gauss 1 Fit

 

 

 

 

l A 1 A A I A A A l ‘

 

 

10 15

PT (GeV)

Figure 4.6:

20

pb
GeV 2

 

(

1
T

 

do
(117

 

 

82

120

40

800

700

600

300
2oo

100‘

 

100
80 -

60

20

 

8209 Data

T f l fi‘f 1' r I I

1

 

i
i . Data .
' — Normalized LY Function Fit 1
— - Normalized Z-Parameter Fit 1
----- Normalized Gauss 1 Fit

  
    

 

 

 

 

M
14.11.1111...l.iLi
0.5 1 1.5 2
PT(GeV)

CDF Z Run 1

 

 

 

I l r I '

 

Data
l —- Normalized LY Function Fit 5
— - Normalized 2-Pa1ameler Fil

----- Normalized Gauss 1 Fit

 

 

 

 

20

Plots of Fit H results compared to Data.

 

The fit to the Gauss 1 function also had its g3 value fixed uniquely by the addition
of the higher mass E605 data. However instead of mirroring the LY function values like
it did in the previous fit, its g3 value is very different. The reduced )6 for this function is
the lowest of the three parameterizations also. The reason for this can be seen in
Figure 4.6. This parameterization has the best match to the E605 data, especially the
three higher mass bins. Notice that this fit doesn’t have the low pT dip as do the other two
fits. There is not much of a difference in the Run ] Tevatron Z plots except for some

minor difference in peak height and position.

4.6.2 Bringing E288 Back into Fits

Fit I adds E288 data to the data set of the previous set of fits. Table 4.10 gives the
data set used for this fit. Recall that E288 had to be taken out of earlier fits because of
normalization problems as well as the overabundance of Drell-Yan data at the time. With
the presence of the Run 1 Tevatron data it was felt that it was no longer necessary to fear
Drell-Yan data dominating the fits and that the normalizations would be kept from
floating too far from their nominal values. The results are given in Table 4.11.

The fit to the LY function has several things of interest. First, the 31 value
remains at a very small value like all previous fits which included Run 1 Z data. The g2
value is somewhat smaller than the 0.7 value from the previous two fits and g3 has taken
a big shift in value from Fit H. Also notice that the E288 normalization is now only 3%
outside its error. However, the reduced )6" is quite large indicating the prediction does not
match the data well.

The 2 parameter fit to the DWS function bears similarities to the LY function fit.
It too has the tiny gl value and the same g2. This fit also shows the E288 normalization is

well behaved. Note that it has also the same large reduced )6.

83

 

 

P BOSON

T MASS
Experiment RANGE
( G eV) RANGE

 

 

(GeV)
R209 0.0-1.8 5.0-11.0
CDF Z
(Run 0) 0.0-22.8 91.19

 

CDF Z
(Run 1) 0.0-20.0 91.19

 

D0 Z 0.0-20.0 91.19

 

 

 

 

 

 

E288 0.0-1.4 5.0—9.0
7.0—9.0,
E605 0.0-1.4 10548.0

 

Table 4.10: Data set for Fit 1.

 

 

 

 

 

 

 

 

 

 

 

Parameter Furfgion FIRIVCZiSOIl Gauss 1
gl 0.02 0.016 0.21
g2 0.55 0.54 0.68
g3 -1.50 0.00 -0.60
fig; 1.01 1.02 0.86
(IDES? 1 0.90 0.89 0.89
123%} 1.01 1.01 1.00
E605 Norm 1.07 1.15 1.00
E288 Norm 1.28 1.23 1.190
x2 407 416 176
x2 [dof 3.42 3.47 1.48

 

 

 

 

 

Table 4.11: Fit I results.

84

 

 

 

 

2
J at y = 0.03

(pb GeV'

d’a

 

 

E605 Data R209 Data

 

 

     
     

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 1 T ' ‘ ' l ‘ l ' ' ' l ' ' ' . 140 ' T f ' l l l l l 1
......... L 1
““““““ _ ‘ ’ \ ~ Data 1
’ —Normalized LY Function Fit ‘
' ‘ 120 *' - — -Normalized 2-ParameterF1t ‘
* l nun-Normalized Gausst Fit ‘
r- 6 4
100 ~ 1 ~
° Da‘a ‘ 1 "
-— Normalized LY Function Fit » '
— -Normalized 2-Parameter Flt r: » ~-._
g ----- Nonnaiized Gausst Fil .0 > 30 ~ ‘~ —
o A _ c. o
o 0.1 r . U * ‘
3 _ v F 4
3
q i- - 1
P «1. 60 r- .4
t" ’ g % . 1
% d
. 4o » ~
20 ... —l
i- T ‘
0.01 : j : '-
L A - l A A - l A A A l A A A l A A - l A A A l A I A A 0 . A A A - l A A . A 1 A . A - l A
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.5 1 1.5 2
P (GeV) PT (GeV)
T
DO Z Data CDF Z Run 1
7oop.rw.,,...,....,.... aoo_....,....,....,...
: Data 1 . - Data .
L — Normalized LY Function Fit 1 700 - —N°"“allzed LY WWW F" 1
600 t — -Normalized 2-ParameterFit 1 ’ _ -Normalized 2-ParameterFtt 1
. ...... Normafized Gauss 1 F“ , '° "-Normalized Gauss 1 F"
. 7 600
500 ~ n
1 h 500
> 400 F 1 fig :
j v 400
30° C' 1 £3le .
. 300 .
200 3- — .
1 200 _
100 f ‘ 100 i
o b A A A A l A A A A I A A A A l A A A A ‘ 0 A A A A I A A L A l A A A A i A A A
0 5 1 0 1 5 20 0 5 1 0 1 5 20
P, (GeV) PT (GeV)

Figure 4.7: Fit I Plots compared to Data.

85

 

 

Data
Normalized LY Function Fit
—— - Normalized 2-Parameter Fit

E288 Data ----- Normalized Gauss 1 Fit

I T I l l ‘ I I’ v ' Y 1 V I 1 V l ' T‘fiv

 

 

 

 

l TTT

lLLl

 

 

 

A l J A A i A L L l

O 0.2 0.4 0.6

A l A A l A A A 1 A A

0.8 1 1.2 1.4

A:T (GeV)

 

Figure 4.8: Fit I Plot of E288 compared to fit curves.

The Gauss 1 fit gives the most dramatic results here. For the first time since
including the Run 1Z data, there is a fit with a g1 that is not nearly zero. Also the g2
value is larger than the other two parameterizations for this fit. The big difference from
the other parameterizations can be seen in the x2 and reduced )6". It’s reduced )6 is less
than half the values for the other two functions.

This difference can best be seen from the plot comparisons to the data in Figure 4.7 and
Figure 4.8. For the plots to the Run 1 Z data, both the LY and DWS functions fail to
match the heights and positions of both peaks well. The Gauss 1 parameterization
matches the peaks of the two experiments much better. It actually best matches the two Z
p, distributions over the whole p, range in the fit. Looking at the E288 plot shows that
the Gauss 1 fit does noticeably better than the other two fits. The three fits are
indistinguishable when compared to E605, except for the lowest mass bin where there is
a small difference with the Gauss 1 curve. The R209 plot shows a clear difference
between Gauss 1 and the other two fits for the lowest mass bin. However the )6

contribution from this mass bin is about even for all the functions.

86

 

 

 

 

 

 

 

 

 

 

 

 

 

Fit Function Results Data Set
_ _ _ R209 5<Q<8; CDF Z Run 0;
A LY g1—0.05, g2—0.57, g3—O.83 E288 6<Q<8
_ _ _ R209 5<Q<8; CDF Z Run 0;
B LY g1—0.08, g2-0.47, g3—1.6 E288 6 <Q <8
Gauss 1 g1=0.11, g2=0.47, g3=0.61
C LY g1=0i§2569gfiifioé 535—3034 R209 5<Q<8; CDF 2 Run 0;
E288 Norm = 1.47 E288 6<Q<8
g1=0.18, g2=0.27
D DWS R209 Norm = 1.05 R209 5<Q<8; CDF Z Run 0;
E288 Norm = 1.29 E288 5<Q<9; E605 7<Q<9
E605 Norm = 1.25
g1=0.16, g2=0.34, g3=-O.4l
LY R209 Norm = 1.02
E288 Norm = 1.39
E605 Norm = 1.28
g1=0.21, g2=0.47, g3=-0.51
Gauss 1 R209 Norm = 0.93
E288 Norm = 1.31
E605 Norm =1.15
glzoigdggfi‘fiio; $54020 R209 5<Q<8; CDF 2 Run 0;
E LY E288 Norm ; 1'35 E288 5<Q<9;
E605 Norm ___ 1°30 E605 7<Q<9, 10.5<Q<18.0
g1=0.24, g2=0.34 R209 5<Q<<11;
F DWS R209 Norm = 0.96 CDF Z Run 0;
E605 Norm = 1.06 E605 7<Q<9
g1=0.15, g2=0.48, g3=-0.58
LY R209 Norm = 0.96

 

 

E605 Norm =1.14

 

 

Table 4.12: Summary of fit results.

87

 

 

 

 

DWS

g1=0.03, g2=0.76

R209 Norm = 0.92
CDF Run 1 Norm = 0.89
D0 Z Run 1 Norm =1.00

E605 Norm = 0.88

R209 5<Q<11;
CDF Z Run 0 and Run 1;
D0 Z Run 1; E605 7<Q<9

 

LY

g1=0.05, g2=0.72, g3=-2.50
R209 Norm = 0.93
CDF Run 1 Norm = 0.88
D0 Z Run 1 Norm = 0.99
E605 Norm = 1.05

 

Gauss 1

g1=0.05, g2=0.80, g3=-2.50
R209 Norm = 0.89
CDF Run 1 Norm = 0.89
D0 Z Run 1 Norm =1.00
E605 Norm = 0.94

 

Gauss 3

g1=0.05, g2=0.80, g3=-2.50
R209 Norm = 0.89
CDF Run 1 Norm = 0.90
D0 Z Run 1 Norm =1.00
E605 Norm = 0.94

 

DWS

g1=0.03, g2=0.71

R209 Norm = 0.93
CDF Run 1 Norm = 0.89
D0 Z Run 1 Norm = 1.00

E605 Norm = 0.89

R209 5<Q<11;
CDF Z Run 0 and Run 1;
D0 Z Run 1;
E605 7<Q<9, 10.5<Q<18.0

 

LY

g1=0.03, g2=0.70, g3=1.00
R209 Norm = 0.94
CDF Run 1 Norm = 0.89
D0 Z Run 1 Norm =1.00
E605 Norm = 0.94

 

 

 

Gauss 1

 

g1=0.07, g2=0.74, g3=-2.00
R209 Norm = 0.90
CDF Run 1 Norm = 0.89
DOZRun 1 Norm =1.00
E605 Norm =1.11

 

 

Table 4.12: cont.

88

 

 

 

 

 

DWS

g1=0.02, g2=0.54
R209 Norm = 1.02
CDF Run 1 Norm = 0.89
D0 Z Run 1 Norm = 1.01
E288 Norm = 1.23
E605 Norm =1.15

R209 5<Q<11;
CDF Z Run 0 and Run 1;
DO Z Run 1; E288 5<Q<9;
E605 7<Q<9, 10.5<Q<18.0

 

LY

g1=0.02, g2=0.55, g3=-1.50
R209 Norm = 1.01
CDF Run 1 Norm = 0.90
D0 Z Run 1 Norm = 1.01
E288 Norm = 1.28
E605 Norm = 1.07

 

 

 

Gauss 1

 

g1=0.21, g2=0.68, g3='0.60
R209 Norm = 0.86
CDF Run 1 Norm = 0.89
D0 ZRun 1 Norm =1.00
E288 Norm = 1.19
E605 Norm = 1.00

 

 

Table 4.12: cont.

89

 

i Z

 

Chapter 5

Conclusion

Based on the observations made in the previous chapter, the Fit I fit to the Gauss
1 parameterization is the recommended choice for further theoretical studies as well as

well as for producing distributions to compare to experiments:
Gauss 1: —g,b2 — g2!)2 lnLEgE] — glg3b2 ln(100xaxb), (5.1)
Fit 1, Gauss 1: g, = 0.2133}, g2 = 0.68333, g3 = —0.60:8;3§,. (5.2)

The Fit I series of fits includes the most diverse set of data, as well as the most
balanced set, including high statistics data at both ends of the Trange. The fit to the
Gauss 1 function is singled out because it clearly fits the data best.

The errors in Equation 5.2 were calculated by a slightly different method than the
one described on page 72 for Fit F. First, x2 values were calculated around the minimum
in a range in all three g parameters to obtain, in essence, a three dimensional function,
752(gl,g2,g3). Next, the surface area where this function equaled 1+minimum x2 was
plotted in the three dimensional g parameter space, (Figure 5.1). The extremes in the
coordinates for this surface on each axis was taken as the errors for the respective

parameters.

90

 

 

 

Figure 5.1: Uncertainty contour and two dimensional projections for the
Fit I result with the Gauss 1 parameterization.

Recent work on obtaining errors for PDF parameters, [41], [42], and [43], points
out that using 1+minimum x2 as the confidence limit is idealistic in global fitting where
experiments often make judgements on systematic uncertainties. When a more
satisfactory method of obtaining uncertainties for PDF’s is achieved, perhaps it can be
adopted for use here in the future.

The result in (5.2) certainly doesn’t mark the end of the study of low pT, non-
perturbative QCD . In the future, there will be even more data to include in fits. There
are also more recent PDF’s to use in the theory calculation and it will be interesting to see

the effect of these different PDF’s. Finally, additional theoretical work may give rise to

91

other parameterizations. In fact, it has been suggested [44] that perhaps the non-
perturbative function should have a more explicit dependence on thex values as in the
CSS formulation, and that the Gauss 1 function can be viewed as two terms, with the first

term suggesting a first order expansion in x,

_b2 [g] (1 + g3 ln(100xaxb )) + g2 ln[%):|.

92

Appendix A
The Level 1 Trigger Monitor Program

A.1 Introduction

The level 1 Trigger Monitor Program, TRGMON [45], displays information on
the programming and operational status of the DQ level 1 trigger. It is an important tool
for assuring that the level 1 trigger is operating properly during data taking and that the
trigger framework is programmed as expected. Along with the trigger framework, the
current status is given for the specific triggers, the calorimeter trigger, and the global
thresholds.

It was determined during Run 1a data taking that TRGMON did not provide
enough functionality for those monitoring the level 1 trigger. Some of the data displayed
by the program are average rates and percentages. These averages were computed over
an integration time of approximately five seconds. The problem was that in some cases it
is more important to know these average values over a longer time period, perhaps an
entire run, rather than over the previous five seconds. Therefore, a new version of
TRGMON was written which allowed for a user-chosen integration time rather than a

fixed time.

93

 

A.2 Method of Information Gathering and Computing

The TRGMON program gets its information directly from the Trigger Control
Computer (TCC). As the name implies, the TCC is what actually sets the parameters for
the level 1 trigger through a program called COOR. The TCC at a fixed five second
interval will request a set of data from the trigger framework. This set of data, called the
First level Trigger Data Block, contains all the information on the current state of the
level 1 trigger. How quickly the framework sends this data block is dependent on how
much data the data acquisition system is handling at the time. Therefore, the TCC will
not always have a new data block at precisely five seconds, and so integration times for
rates and percentages computed from these blocks will only be five seconds on average.
Once the TCC receives a new data block from the trigger framework, it will compute the
difference in some scalars from the previous data block it received. These delta values
are saved in a separate data block. After the level 1 data block is received by the TCC
and the delta scalars values are calculated, then it is available to be sent to TRGMON.

In the original version of TRGMON, once the data were received it could
calculate the necessary rates and percentages in the following way. Knowing the
differences and the beam crossing rate, average rates and percentages were calculated by:

delta counts

rate ( in Hz) 2 . , , (A.1)
Beam crossrng per1od - delta beam crossmgs

 

One example calculation is the global level 1 trigger firing rate. To calculate this
rate, the current count of the number of level 1 firings is subtracted from the same count
from the previous five seconds. This difference is then used to get the firing rate through

Equation A. 1.

94

 

 

 

A.3 Implementing User Defined Integration Times

A.3.1 Method Of Calculating Over Larger Integration Times

In order to allow longer integration times it was necessary to make a fundamental
change in the information which was stored by TRGMON and how it was stored. Instead
of having only the quantities for the current beam crossing and for the crossing five
seconds prior to it, the information from the data blocks are stored in ring arrays. When a
new scaler value is received, it is stored in a ring array and it is placed in the location of
the oldest value of that sealer. The “delta counts” value used in Equation A.l is now the

subtraction of the oldest value in the ring array from the newest value.

A.3.2 User Interface

The user is allowed to choose an integration time from a range of five seconds to
two hours. The integration time must be an integer multiple of the refresh rate, although
the user could also choose this interval. These choices determine the sizes of the ring

arrays.

A.3.3 Behavior Under Special Conditions

After a given set of values for the integration and refresh times are chosen, the
displayed rates and percentages will not at first be calculated over the chosen integration
time. There will first be a transitional stage where the ring arrays must become filled.
The delta counts will still be calculated by the difference of the oldest value and the
newest sealer value. Therefore as the ring arrays are in the process of becoming filled,
the integration time will increase with each refresh. This transition state ends when the

user-chosen integration time is reached.

95

 

Another special condition is when the TCC is not receiving new data blocks from
the framework, but continues to receive data on the same beam crossing. In this case,
when TRGMON is given these stale data blocks, the ring buffers will continue to be
filled with the same sealer value. The result will be that the rates and percentages will
degrade to zero with each refresh. TRGMON will however indicate in its display that the

data blocks are stale.

A.4 Testing and Implementation of New TRGMON

Before the revised monitoring program could be used by the DO shift crew, it had
to be tested for accuracy and reliability. The program was first tested on a hardware
simulator of the level 1 trigger located at MSU. After this, it was put into place on the
DO computer to monitor actual data taking runs. The revised TRGMON was being run
in parallel with the older version at this point, with the older version still being used by
DO shift members. After finding the program to be working as designed, the revised

TRGMON was then put in place and used on shifts for the remainder of Run 1.

96

 

Appendix B

The Z Calibration Detector

B.1 Introduction

During the latter part of Run 1a, studies were done [46] in order to compare the
CDC tracks of muons to their tracks in the Muon Detector. It was found that for a muon
the CDC track and the Muon Detector track did not lie along the same line. Specifically,
the Muon Detector track, when extrapolated to the CDC, would be offset in z from the
CDC track. Figure B.l illustrates such a misalignment. Since the accuracy of the z
position hits in the Muon Detector were determined independently [46], it was
determined that there was an inaccuracy in the calculation of the z coordinate for the
CDC hits. Since inaccurate z’s in the CDC hits would result in a track with an incorrect 9
angle, this could have an effect on the accuracy of the determination of the z position of
the interaction vertex. It would also lead to an inaccurate calculation of p2 and p, for the
particle which produced the track. Therefore it was proposed [47] that a simple detector
be constructed to recalibrate the z coordinate for the CDC. The z Calibration Detector,

(ZCD), was constructed and installed during a sh01t off-line period for the DO Detector.

97

 

 

Muon Detector

 

 

 

 

CDC

 

 

 

Figure B]: A Muon Detector track extrapolated to the CDC does not

align with thez positions of the delay line hits.

B.2 Design of Detector

It was necessary to keep the design simple because there was a limited time in

which the DO Detector would be off-line and outside the collision hall. The simple

design consisted of scintillating fibers placed at precisely known positions in z just

outside the CDC. Therefore when an energetic particle passed through a fiber, the

particle’s 2 position from the fiber could be compared with the 2 positions from the delay

line hits. The scintillating fiber detector was made up of twenty separate modules, with

each module containing eight scintillating fibers 2 mm in diameter and 5 in long. Each

scintillating fiber was connected to a clear waveguide fiber which ran into a

photomultiplier tube (PMT). There was one PMT for each module, so that on each

module all eight clear fibers fed into the one PMT. Figure B.2 is a diagram of the layout

of a module.

98

ED ZDErm 3:2 9.6.: 8 .52” \

man 43 oz:
8.; 5 mm a

I I
I I

6.5 in

~33 Eva. mass—3:5 2.5..

wages—.3261

43: 3.. 299.. «:53

dz amoor

 

 

 

Figure B.2: Diagram of a ZCD module and two connecting modules.

99

H ..‘,::<
.wul‘ ’

   
 

outermost CDC layer

ZCD module pair

Figure B.3: A representation of the placement of one of the ZCD modules
over a CDC sector.

Modules were connected in pairs and each module pair was placed to entirely
cover one of the outermost CDC sectors as shown by Figure B.3. Even though each
module pair was placed to cover only one CDC sector in the outermost layer, the CDC
sectors were staggered in d) between layers so that each module pair covered two sectors
in the first and third CDC layer. Figure B.4 shows an edge-on view of the CDC with the

sectors covered by the ten module pairs.

B.3 Acquisition and Selection of Data
Before the calibration of the CDC could take place, the events to be used in the

calibration process needed to be selected. The data were selected to include events which

possessed the necessary characteristics required for calibration. It was also required that

100

 

these quantities could be measured accurately. These were the criteria that were used in

selecting the data to be used in the calibration process:

0 At least one ZCD module must have registered a hit for the event.

0 Only one CDC track was allowed to intersect a ZCD module which registered a hit
fiber. (The reason for this is that if two or more fibers were hit for a given module,
there was no way to tell which fibers were hit because all of the fibers were read by
one PMT.)

0 The ¢ angle of the CDC track must have been within the (1) range covered by the
module which registered the hit. (This requirement weeded out nonradial tracks.)

0 A CDC delay line hit must have been read out from both sides of the delay line.

B.4 Calibration Method

Once the data set was selected, the same calibration process was used for each
event in the data set. The first step in the calibration process was to locate the CDC track
intersecting the module with a hit and extrapolate the track out to the radial distance of
the ZCD fibers. The 2 position for which the extrapolation reached the radial distance of
the fibers was designated Z“, . The fiber which came closest to the extrapolated track
was then the fiber which the particle hit and one could then compare the well-measured
position of the fiber, Zfibe, , to Zm to see how far off the CDC track was. An example of

such an extrapolation is shown by Figure B5.

101

 

 

 

 

 

Figure B.4: Placement of the ZCD module pairs in relation to the CDC
sectors. The numbering of the module pairs is shown as well as the CDC
sectors covered by the ZCD.

102

ZCD fibers
0 O O O O

   
 

 

CDC Zfiber

 

CDC track

 

 

Figure B.5: Extrapolation of CDC track out to the ZCD module to obtain

81'!

For each fiber in the ZCD, a distribution of AZ was plotted,
AZZZflber—Zext' (B1)

Each of these plots were then fit to a Gaussian distribution added to a constant.
Examples of such fits are depicted in Figure B6. The mean AZ of the fitted function,
T2 ,was used as the average offset between Zfibe, and Zen for the fiber. One could then
obtain the average extrapolated position of a CDC track hitting a certain fiber, ZR: , by
adding E to Zflber’
Once A7 and 2&7 were obtained for each fiber, then plots were made of Zfibe, vs.
A7 and Zflbe, vs. Ze—xt for each module and module pair. Each of these distributions were

fl '3 tO a line. The slopes and intercepts of these fitted lines were then used as the

COl‘l‘ections to be applied to CDC hits. Some of these fits are shown in Figure 3.7.

103

 

 

120

100

80

60

4O

20

300

250

200

150

‘IOO

50

DZ BETWEEN DELAY LINE 8 AND ZCD FIBER, DL8.HST(New dr correct)

 

 

   
 

 

  

 

 

 

 

 

lflIIlIIIIITIIIIII]IIIIIIIIIIIII

 

 

 

{/ndf .77
P1
132
P3
P4

_ 10 1011
.. 1976
_. 8.89 / 27
— Pl 91.461: 5.335
- P2 0.1633: 0.3382E-Ol
- PS 05158:}: 0.39405-01
—‘ P4 23.941: 1.396
4‘ 010 “+1 1 1 1 1 1 1 1 1 1 1 1
—5 O 5
DELT Z OF MOD 1 FIBER 1
10 1075
Entries 1 4440
/ 27
237.9: 8.910

0.7080 :1: 0.18836-01
0.5457 :1: 0.2110E—O1
50.96 :I: 1.894

 

 

 

*1 «It 1
«1111. 11111511
1' o . I .
l l 1 L A L J 1 l l 1 l 1_ l l
—5 0 5

DELT Z OF MOD 7 FIBER 5

180
160
140
120
100
80
60
4O
20

250

200

150

100

50

 

 

 

 

10 1012
Entr 2410
{AK 0. 35 / 27
133. 4 :t 5.947

0.2449 :1: 0.263OE-01
O. 6340 :1: 0.2673E—01

 

 

 

 

 

 

 

 

Figure B.6: Typical Fits to AZ for four fibers.

Z. 25. 331 1.410

2’1?“ o“

F 1. +1 1
—5 O 5
DELT Z OF MOD 1 FIBER 2

10 1197

: Entries 2591

_ )8/ 165.83 / 27

_ p1 181.0:t 7.459

— 92 -O..‘5009:l: 0.2049E-01

r- P3 0.55541 0.2385E-01

— NIL 25.46: 1.417

: 1 1

Z 0‘ * “‘0

J 1 l 1 l l l l 1 1 l
—5 0 5

DELT Z OF MOD 19 FIBER 7

 

 

 

 

 

100 Z (CM) OF FIBER vs Z Meos FROMODOLY 8, DL8. HST(New DR correct)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Z/ndf 2596 / 5 ; 2/ndf 8249 / 6 ,
90 P1 0.24231: 0.20545— 01 ____________ 90 P1 —0.2682:t O.—2230E 01 ____________
P2 1-008: 0-3892E—03 ; P2 1.0121 039145—03 g 0
7O 7O
60 60
50 50
4O 4O
30 30
20 20
10 : I . I . iO . . I .
0:1111111111111111111 0:111111111111ll11111
0 2O 4O 60 80 100 O 20 4O 60 80 100
Z FIBER vs Z MEAS MOD.1 Z FIBER vs Z MEAS MOD.2
100 xZ/nd1261.6 / 6 § 100 xZ/ndr 101.8 / 6 g
90 P1 -0.4028:l: 0.2234E-01 ____________ 90 P1 —0.7473E—011 0.1680E-01 ............
P2 1.010: 036905—03 g P2 1.011i 031205—03 §
: z I . 80 2 : :
7O
60
50
4O
3O
20
_ a s s 5 IO
O:lll1lll1lll1lll1lll O Lll1l11411l|1lll1llL
O 20 4O 60 80 1 OO O 20 4O 6O 80 100
Z FIBER vs Z MEAS MODE) Z FIBER vs Z MEAS MOD.4

Figure B.7: Examples of linear fits to Zfiber vs. Za for modules 1-4.

105

 

 

 

 

 

B.5 Results

The values for these corrections are shown in the following tables as originally
listed in [48]. The tables list the values from fits to the data of module pairs rather than
individual modules. Therefore, under column i, data from module 1' and module i+10
were used in the fit. The seven delay lines with “No Data” gave us little or no hit
information and it is suspected that the problem lies with the delay lines themselves

rather than with the ZCD. Also, no data for module pair 10 is listed because it did not

produce enough data for reliable statistics.

The mean value for the slope is 1.01 1:1: .0022 which can result in a difference of
as much as 1.5 cm for the delay line hit positions. This average slope was included with

other CDC recalibration studies which used the Muon Detector to obtain an average

correction to the CDC [49].

106

 

4.x".

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

U5_5v. Zea—:5 ~05:
5.5
H N u a m a q a e
U—L :zmo :ZNN :Zow :55 :55 :53“: :: 5m :zom r93
U—L.U :zmq :: 3 :55 :: 3 :5: 2o U55 :ZB :zam :Zwo
UFN :53 :53 :53 :53 :55 :55 _535 :52 :53
UFNQA :55» :58 :55 :53 :53 2o U58 :55 :55 :5 B
Uru .555 :55 ::5 :53 ::S ::B :95 :zmo :535
U: :55 :53 :58 :53 :55 :E5 :55 :52: :: E
Urm ::um :zmm Zo U555 :58 :55 :: 5 5.23 :: 3 :53:
U—AmA—A :zwm :: _m :: 3 5.35 :55 Zo U55 ::mo 20 U58 :535
Ura 5.535 :55 :58 :55 :53 :53 :55 :53 :Zoo
Ubah. :zom :53 :55 Zo U58 :53 20 U55 :53 :55 :Zom
U—Lq :zfi :ZNm :55: :zom :55 :ZNN #555 :zwo ::m~
U—Am :: _m :: Z :55 :55 :53 :55 55:5 :55 :: B

 

 

 

 

 

 

 

 

 

 

 

ext

Table B. 1: Slopes from linear fits to Zfibe, vs Z—— distributions of module

pairs.

107

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Univ. gems—5 U5:

5.; 5 5 5 a 5 a q 5 5

EL -0055 0.555 0.35 0.005 0.055 0.055 0.555 0.35 0.505

55.5 0.0: -003 -055 0.000 0.555 20 058 0.55 0.15 0.550

:5 0.30 -055 -055 0.050 -003 -0055 -0555 0.505 0.555
25.5 -0055 -055 -0500 -002 0.055 20 058 0.35 -0555 0.550

25 -0005 -05: -0550 0.050 0.555 0.055 0.505 -050... 0.0.5

E... 0.25 -0550 -0505 0.050 -0555 0.25 -0005 0.505 0.555
25 0.505 0.555 20 058 0.550 0.505 -0055 0.35 0.055 0.550

2.5-5 9:: -005 -0555 0.055 0.5: 20 058 0.505 20 058 -0055
2.0 0.555 0.85 -0055 0.35 -0050 -0055 -0500 0.555 0.555

Ea-.. -0055 -0555 -055 20 058 0.505 20 058 0.550 -0555 -005
=5 0.35 -0055 0;: 0.055 0.5.: 0.35 0.005 0.550 0.555

25 0.005 -0550 -0550 0.0: -0005 0.055 0.055 -0555 0.050

 

 

 

 

 

 

 

 

 

 

 

f ext

Table B.2: Intercepts from linear fits to Z lb” vs Z—, distributions of

module pairs.

108

 

 

 

Appendix C

Reprint of Preliminary Fits Paper

F. Landry, R. Brock, G. Ladinsky, and C.-P. Yuan, Phys. Rev. D63 013004 (2001).

109

 

 

 

PHYSICAL REVIEW D, VOLUME 63. 013004

New fits for the non-perturbative parameters in the CSS resummation formalism

F. Landry, R. Brock, G. Ladinsky, and C.-P. Yuan
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824
(Received 25 May 1999; published 5 December 2000)

We update the non-perturbative function of the Collins-Soper-Sterman (CSS) resummation formalism which
resums the large logarithmic terms originating from multiple soft gluon emission in hadron collisions. Two
functional forms in impact parameter (b) space are considered, one with a pure Gaussian form with two
parameters and another with an additional linear term. The results for the two parameter fit are found to be
g, =0.24t3;$ GeV2 and g2=0.34:3;3; GeV’. The results for the three parameter fit are g1=0.15:3j8; GeVz,
g2=0.48:3j& GeVz, and g3= -0.58:3§3 GeV". We also discuss the potential of the full Fermilab Tevatron
run 1 Z boson data for further testing of the universality of the non-perturbative function within the context of

Drell-Yan production,

D01: 10.1 lO3/PhysRevD.63.013004

I. INTRODUCTION

It is a prediction of the theory of quantum Chromodynam-
ics (QCD) that at hadron colliders the production of Drell-
Yan pairs or weak gauge bosons (W3 and Z) will generally
be accompanied by gluon radiation. Therefore, in order to
test QCD theory or the electroweak properties of vector
bosons, it is necessary to include the effects of multiple

gluon emission. At the Fermilab Tevatron (a p13 collider), we
e t about 2x 10‘5 Wt and 6X 105 z bosons produced at

S: 1.8 TeV, per 100 pp" of luminosity. This large sample
of data is useful (i) for QCD studies (with either single or
multiple scales), (ii) as a tool for precision measurements of
the W boson mass and width, and (iii) as a probe for new
physics (e.g., Z’). Achievement of these physics goals re-
quires accurate predictions for the distributions of the rapid-
ity and the transverse momentum of Will bosons and of
their decay products.

Consider the production process h1h2—svx. Denote QT
and Q to be the transverse momentum and the invariant mass
of the vector boson V, respectively. When QT~Q, there is
only one hard scale and a fixed-order perturbation calcula-
tion is reliable. When QT< Q, there are two hard scales and
the convergence of the conventional perturbative expansion
is impaired. Hence, it is necessary to apply the technique of
QCD resummation to combine the singular terms in each
order of perturbative calculation, which yields

d0 1

—~-—— a L+1 +a2 L3+L2)+a3(L5+L4)

dQ; Q;{ S( ) S( S
+a§(L7+L6)+.-.+a§(L+1)+a§(L3+L2)

+a§(L5+L4)+ - - - + a§(L+1)+ a§(L3+L2)

+...}, (1)

where a, is the strong coupling constant, L denotes
ln(Q2/Q%) and the explicit coefficients multiplying the logs
are suppressed.

Resummation of large logarithms yields a Sudakov form
factor [1,2] and cures divergences as QT—tO. This resumma-

0556-282 l/2000/63(1)/013004(10)/$ 15.00

110

63 013004-1

PACS number(s): 12.15Ji, 12.60.Cn, l3.20.—v, 13.35.-r

tion was pioneered by Dokshitzer, D’yakonov and Troyan
(DDT) who performed an analysis in Q1 space which led to
a leading-log resummation formalism [1]. Later, Parisi-
Petronzio showed [2] that for large Q the QT—vO region can
be calculated perturbatively by imposing the condition of
transverse momentum conservation:

" dzb - « " - ~
5(2)( 1; _ * )= iQT-b ikTi-b’ 2
glam 4172¢: .5." ()
in b-space (b is the impact pararheter, which is the Fourier
conjugate of Q7). Their improved formalism also sums some
subleading-logs. They showed that as Q-m, events at Q,-
~0 may be obtained asymptotically by the emission of at
least two gluons whose transverse momenta are not small
and add to approximately zero. The intercept at QT: 0 is
predicted to be [2]

do
«2%

 

A2 770

~ 0‘ o( ETD) . (3)
(Zr-‘0 Q
where 170=Aln[l+l/A] with A=12Cp/(33-2nf), and 170
=06 for n 1:4 and CF=4/3. Collins and Soper extended [3]
this work in b-space and applied the properties of the renor-
malization group invariance to create a formalism that re-
sums all the large log terms to all orders in a,.

Although various formalisms for resumming large
ln(Q2/Q§-) terms have been proposed in the literature [4,5],
we will concentrate in this paper on the formalism given by
Collins, Soper and Sterman (CSS) [6], which has been ap-
plied to studies of the production of single [7—10] and
double [1 1] weak gauge bosons as well as Higgs bosons [12]
at hadron colliders.

II. COLLINS-SOPER-STERMAN RESUMMATION
FORMALISM

In the CSS resummation formalism, the cross section is
written in the form

@2000 The American Physical Society

 

F. LANDRY, R. BROCK, G. LADINSKY, AND C.-P. YUAN

d0'(hlh2-+VX)_ l
szdey (2w)2
+Y(QT!Qv-xlrx2)t (4)

where y is the rapidity of the vector boson V, and the parton
momentum fractions are defined as x1= e’Q/t/IS7 and x2
=e yQ/\/§ with x/E as the center-of-mass (c. m.) energy of
the hadrons h, and h. In Eq. (4), Y is the regular piece
which can be obtained by subtracting the singular terms from
the exact fixed-order result The quantity W satisfies a renor-
malization group equation with the solution of the form

 

Idzb eiérl;W(ons-xl 9x2)

C
W(Q,b,x1,x2)= e 8‘9"” C1'C2)W(-—b,b, xhxz). (5)

Here the Sudakov exponent is defined as

€de _fl'_
cilb’ ii

 

8(Q b C19C2)=A(as(fl) C I)1n

 

 

CiQZ)

+B(as(fl)9C1’C2) v (6)

 

and the x1 and x2 dependence of W factorizes as

use

W C‘ b
Czbr sxlsxz

C
=2ej Cfl.l(_2-lb"c ,b ,x1)th2(Ez%,b,x2). (7)

Here, C”, is a convolution of the parton distribution function
(fa/h) with calculable Wilson coefficient functions (C in),
which are defined through

1d C
material’s-1.3.9)

C
xr.n.( § u= 7,3) (8)

The sum on the index a is over incoming partons, j denotes
the quark flavors with (electroweak) charge e j , and the fac-
torization scale p. is fixed to be C3 lb. A few comments
about this formalism:

(a) The A, B and C functions can be calculated order-by-
order in 01,.

(b) A special choice can be made for the renormalization
constants C, so that the contributions obtained from the
expansion in a, of the CSS resummed calculation agree
with those from the fixed- order calculation. This is the
canonical choice. It has C1: C3: 2e 75=b0 and C2
= C1 lbo=l, where 75 is Buler’ s constant.

(c) b is integrated from O to 0°. For b» l/AQCD , the pertur-
bative calculation is no longer reliable. In order to ac-

PHYSICAL REVIEW D 63 013004

count for non-perturbative contributions from the large b
region this formalism includes an additional multiplica-
tive factor which contains measurable parameters.

We refer the readers to Ref. [9] for a more detailed discus-
sion on how to apply the CSS resummation formalism to the
phenomenology of hadron collider physics.

A. The non-perturbative function

As noted in the previous section, it is necessary to include
an additional factor, usually referred to as the “non-pertur-
bative function,” in the CSS resummation formalism in or-
der to incorporate some long distance physics not accounted
for by the perturbative derivation. Collins and Soper postu-
lated [6]

W;r(b)= ertanfég’lb). (9)

with
b

b =————,
* ./1+(b/b,,,.,,)I

so that b never exceeds bmJr and Wj;(b*) can be reliably
calculated in perturbation theory. (In numerical calculations,
bM, is typically set to be of the order of 1 GeV".) Based
upon a renormalization group analysis, they found that the
non-perturbative function can be generally written as

(10)

Q2
-F1(b)ln E2, —

W%P(b,Q,Qo,xl,x2)=exp Fj/h,(xr ,b)

 

 

 

‘FE/h2(x2.b) . (11)

 

where F1, F 11": and F r”, 2 must be extracted from data with
the constraint that

WfEP(b=0)=l.

Furthermore, the F 1 term only depends on Q, while F ,,, I and
F3”: in general depend on x, or x2, and their values can
depend on the flavor of the initial state partons (j and I: in
this case). Later, in Ref. [13], it was shown that the
F1(b)ln(Q2/Q(2,) dependence is also suggested by infrared
renormalon contributions to the QT distribution.

B. Testing the universality of W”

The CSS resummation formalism suggests that the non-
perturbative function should be universal.l Its role is analo-
gous to that of the parton distribution function (PDF) in any

'Here we mean “universal” within the context of Drellean pro-
cesses, although it may apply in general to other reactions having
the same initial or final state particles as in the Drell-Yan process.

013004-2

111

 

. _ ...-...

 

NEW FITS FOR THE NON-PERTURBATIV E PARAMETERS . . .

fixed order perturbative calculation, as its value must be de-
termined from data. The first attempt to determine such a
universal non-perturbative function was made by Davies,
Webber and Stirling (DWS) [14] in 1985 who fit data avail-
able at that time to the resummed piece (the W1 term) using
the Duke and Owens parton distribution functions [15]. Sub-
sequently, the DWS results were combined with a next-to-
leading order (NLO) calculation [16] by Arnold and Kauff-
man [7] in 1991 to provide the first complete CSS prediction
relevant to hadron collider Drell-Yan data. In 1994, Ladin—
sky and Yuan (LY) [17] observed that the prediction of the
DWS set of WN—P deviates from R209 data (p + p—tp. p.

+x at J§= 62 GeV) using the CTEQZM PDF [18]. In order
to incorporate possible ln(r) dependence, LY postulated a

model for the non-perturbative term, which was different
from that of DWS, as

W”-'(b...=QQox..x2) cxp[— gibz— 1521423)

_8133b1n(100x1x2)]v (12)

where xlx2= r. A “two-stage fit” of the R209, Collider De-
tector at Fermilab (CDF) z (4 ph-l data) and E288 (p
+Cu) data gave [17]

g,=o.11:3;3; GeVz. 5:058:35 GeVz,

83: ”1.5.81 GeV—l,

for Q0: 1.6 GeV and b,,,,,,=0.5 GeV".2 The purpose of the
project described here is twofold: First, we update fits for the
non-perturbative parameters using modern, high-statistics
samples of low energy Drell-Yan data and second we incor-
porate a fitting technique which will track the full error ma-
uix for all fitted parameters. In a subsequent report we will
update these results for the full Tevatron run I datasets [19].
Our results are given in the following sections.

111. FI'I'I'ING PROCEDURE
A. Choice of the parametrization form

At the present time, the non-perturbative functions in the
CSS resummation formalism cannot be derived from QCD
theory, so a variety of functional forms should be studied.

The only necessary condition is that W§EP(b=O) = 1. For
simplicity, we consider only two typical functional forms for
WN-P(b, Q, Q0 ,x1,x2) in b space: (i) 2—parameter pure Gauss-
ian form [DWS form],

CXP[‘81__821“(2(Q2)Ib2 (13)

2A FORTRAN coding error in calculating the parton densities of the
neutron led to an incorrect value for g,.

PHYSICAL REVIEW D 63 013004

TABLE I. Drell-Yan data used in this analysis. Here. 8N is the
published normalization uncertainty for each experiment The CDF
data were from Tevatron collider run 0 of 4 pb".

 

 

 

Experiment Ref. Reaction ./§ GeV (J?) 6N
R209 [23] p+p-tp.+p._+X 62 ~01 10%
5605 [24] p+Cu—+p.+p.'+X 38.8 ~o.2 15%

CDF-Z [25] 104,552+ x 1800 ~o.05 —
15288 [26] p+Cu_.,.+,i-+x 27.4 ~02 25%

 

 

and (ii) a 3-parameter form [LY form],

“pH-erg:ln(2—g—o)]b2-lgi83|n(100xix2)lb .
(14)

with a logarithmic x-dependent third term which is linear in
b. The ln(100x1x2) term is equivalent to ln(7fro) for J};
=0.l.

Both forms assume no flavor dependence for simplicity.
In addition to fitting for the non-pertm'bative parameters, g1,
g2, and g3, the overall normalizations were allowed to float
for some fits. One can also study another pure Gaussian form
with similar x dependence such as

Q
¢XP[ ‘81-82 111(5) ”8183 ln( 100x112) 192-
0

However, we find that current data are not yet precise
enough to clearly separate the g2 and g3 parameters within
this functional form and so it will not be discussed further in
this paper. We also tested a few additional functions which
did not incorporate additional parameters, but did not find
any clear advantage to them when fitting the current Drell-
Yan data. However, as to be shown later, the run 1 W/Z data
at the Tevatron are expected to determine the g2 coefficient
with good accuracy, and these data can be combined with the
low energy Drell-Yan data to further test various scenarios
for x dependence and ultimately, universality.

B. Choice of the data sets

In order to determine the non-perturbative functions dis-
cussed above, we need to choose experimental data sets for
which the contribution to the non-perturbative piece domi-
nates the transverse momentum distributions. This suggests
using low energy fixed target or collider data in which the
transverse momentum (QT) of the Drell-Yan pair is much
less than its invariant mass (Q). Because the CSS resumma-
tion formalism better describes data in which the Drell-Yan
pairs are produced in the central rapidity region (as defined
in the center-of-mass frame of the initial state hadrons) we
shall concentrate on data with those properties. Based upon
the above criteria we chose to consider data shown in Table
I. We have also examined the E772 data [20], from the pro-
cessp+H2—>p. p. +X at fl: 56. 6GeV, and found that it
was not compatible in our fits with the above data, and it is

013004-3

112

 

F. LANDRY, R. BROCK, G. LADINSKY. AND C.-P. YUAN

not included in this study.3 Except where noted, all of the fits
to g1” were done using the CTEQ3M PDF [22] fits.

C. Primary fits

As to be shown later, the E288 data have the smallest
errors, and would be expected to dominate the result of a
global fit. That is indeed the case. When including the E288
data in a global fit, we found that the resulting fit required the
NORM4 to be too large (as compared to the experimental
systematic error) for either the E288 or the E605 data. Fur-
thermore. the shape of the R209 data cannot be well de-
scribed by the theory prediction based on such a fit.

1. FitsAu

As explained above, a straightforward global fit (that is,
one which includes all of the available data) does not give a
satisfactory X2 due to the large systematic uncertainties. We
therefore employed a different strategy for the global fit
based on the statistical quality of the data. We included the
first two mass bins (7<Q<SGeV and 8<Q<9 GeV) of
the E605 data, all but the highest mass bin of the R209
results, and all of the early CDF-Z boson data in an initial
global fit, referred to here as Fit A23 (see Figs. 1, 2, and 3).
In total, 31 data points were considered. We allowed the
normalization of the R209 and E605 data to float within their
overall systematic normalization errors, while fixing the nor-
malization of the CDF-Z data to unity. (The point-to-point
systematic error of 10% for the E605 data has also been
included in the error bars of the data points shown in Fig. 3.)
In addition to the normalization factor for each experiment,
the fitted parameters of our global fit include the coefficients
31 3 and g1” for the 2-parameter and 3-parameter fits, re-
spectively. We found that, for Q0: 1.6 GeV and bum,
= l/(2 GeV), both the 2—parameter and the 3-parameter
forms give good fits, with X2 per degree of freedom about
equal to 1.4.5 The best fitted central values for Fit A2 are:
g1= 0.24 GeVz, 82: 0.34 GeVz. While the central values for
Fit A3 are g.=o.15 GeVz, 32:0.48 GeV’, g3= -—0.58
GeV". The fitted values for the R209 NORM are 1.04 for
both Fits A2 and A3.

2. Uncertainties in the fits

We have also studied the uncertainties of the fitted g pa-
rameters. For the 2-parameter fit, the la error in the x2 plot
(with an approximately elliptical contour) gives —0.07
< 8g1<0.08 GeV2 and —0.08< 6g2<0.07 GeVz. This is
shown in Fig. 4. For the 3-parameter fit, the situation is more

3Using the fitted g values to be given below, the theory prediction
for the E772 experiment is typically a factor of 2 smaller than the
data. Similarly, CTEQ fitting of PDF parameters are not well fit
with these data [21].

Here, and in subsequent discussions, the quantity NORM is the
fitted normalization factor which is applied to the prediction curves
in all that follow: the data are uncorrected.

SWe scan the values of g, and 32 between 0 and l, and g, be-
tween -2 and 3.

PHYSICAL REVIEW D 63 013004

.-

40

 

_ 8209 data rm ‘7 (19.1) 12}
91- .24, 92- .34. 93- 0.. NORM- 1.04
91- .15. 92- .48. 93- -.58. NORM- 1.04

CTEOJ

 

'3
o
I

5<O<8 GeV

§
I

dam/do: (pb/COW)
S

0
O

40

 

2° e<o<r 1 GeV i

VjTII'VI'UYIW'

 

 

 

 

FIG. 1. 12209 data, from p+p-—tp+p._+X at J§=62 GeV,
with an overall systematic normalization error of 10%. The curves
are the results of Fits A2 and A3 and are multiplied by the value of
NORM, as shown in the figure and described in the text.

complicated, as the fitted values of g’s are highly correlated
as shown in Fig. 5. In order to estimate the uncertainties of
the fitted g values, we fix the value (at its best fit value) of
g ’8, one at a time, and examine the uncertainties of the other
two, in a way similar to studying the 2-parameter fit result.
We found that

g, fixed:-0.05< 5g2< + 0.04 GeV’,

 

 

. cor—T data t m_ a? (1991) 2937 l
' ottosmoau- t.

o 1:24. 2- .34. 3:0.
0. 0 31-45. 32- .43. 33- —.5a

Y‘Y‘IVYVI'
n
T
o
.
0

§

 

da(Z)/d°t (Pb/06V)

V—V."T°

 

 

 

FIG. 2. Comparison of 4 pb‘1 CDF-Z data at the Tevatron with
two different theory model predictions. The dots correspond to the
results of Fits A, and A3.

013004-4

113

NEW FITS FOR THE NON-PERTURBATIVE PARAMETERS . . .

 

8605 date PRO 43 (1991) 2015}
”£03,0-“ng

. gl- .24.92- .34.93- 0..NORM- 0.94 _
g1- .15. g2- .48. 93- -.5a. nom- 0.37 ....

E. I 5 7<O<8 GeV
I t .
, t

8<Q<9 GeV

10.5<O<11.5 GOV

u.

Ed’a/dp’ at X,==0.1 (pb GeV"/ nucleon)
OI
T‘

i 1 .5<0<13.5 GOV

10 —- f *
ll 3.$<O<1IB GeV 1

0.2 0.4 0.6 0.0 i 1 .2 1.4

Pt (0W)

 

 

 

FIG. 3. E605 data, from p+Cu—.,i+,i-+x at J§= 38.8 GeV.
with an overall systematic normalization error of 15%. The curves
are the results of Fits A2 and A3 and are multiplied by the value of
NORM, as shown in the figure and described in the text

-0.2o< 6g3<+0.26 GeV“;
g2 fixed:-0.03< 6g l< +0.03 GeVz,
—o.11<sg,<+o.11 GeV";
g3 fixed:-0.02< 631< +0.04 GeVz,

- o.o3< 6g2< + 0.02 GeVz:

constitute a conservative set of uncertainty ranges. With an
appreciation for the complexity discussed above, we charac-
terize the uncertainties of the fitted g’s in the 3-parameter
form conservatively by their maximal deviations among all
of the fixed parameter choices just described. Therefore, the
best fitted g values are:

Fit A2: ,t;,=o.24:‘3;€’,,8 GeVz. gz=o.34:3;3’, ch2;

Fit A3: 5:015:33; GeVz, g2=o.48:3;8; GeVz,

33 = - 0.58:8:7338 GeV' 1;

for the 2 and 3 parameter fits, respectively. In summary, fits
A2 and A3 constitute the main results of this paper. The g1
and g2 parameter spaces of the above two fits do not strongly
overlap because the 3 parameters are found to be highly cor-
related.

3. Cross checks

Given these values of g’s and the fitted normalization
factors for the E605 data, we can calculate the two different
predictions for the other three high mass bins not used in Fits

PHYSICAL REVIEW D 63 013004

2 am I'I'l'

92

 

0.425 '-

0.4 -

0.375 '-

0.325 '-

0.3 *-

O.275 -

 

 

0.25 1111‘IAAAILLLLLIJLJLIH4JLJLIJLLIIl
0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3

IIIAJJ
0.32 0.34

91

 

FIG. 4. The error ellipse on the g, and g2 plane from which the
errors of the 2-parameter fit A; were interpreted.

A23, and the results are also shown in Fig. 3. In order to
compare with the E288 data, we created Fits N23 in which
we fix the g’s to those obtained from Fits A13 and fit for
NORM from the E288 data alone. Figure 6 shows the result—
ing fits are acceptable, with values of NORM close to the
quoted 25%, namely, NORM=O.92 and 0.79 for Fits N13,
respectively. It is encouraging that the quality of the fit for
the E288 results is very similar to that for the E605 data and
that the normalizations are now acceptably within the range
quoted by the experiment. Hence, we conclude that the fitted
values of g ’s reasonably describe the wide-ranging, complete
set of data, as discussed above.

We note that although the CDF-Z data, as shown in Fig.
2, contain only 7 data points with large statistical uncertain-
ties, they prove to be very useful in determining the value of
g2. To test this observation, we performed an additional fit,
L2. Following the method suggested in [17], we set g, to be
zero and fit the g, and 3; parameters using the R209 and
CDF-Z data alone. (Note that for the R209 data, the typical
value of $7 is of the order 0.1, which motivates the choice of
Jr”: 0.1 in the LY form. Effectively, the g3 contribution to
the R209 data can be ignored.) We found that the best fit6
gives g2=0.47 GeVz, which is in good agreement with the
result of the global Fit A3 discussed above. Hence, we con-
clude that the CDF-Z data already play an important role in
constraining the g2 parameter, which can be further im-
proved with large Z data sarnples from run 1 of the Tevatron
collider experiments. We shall defer its discussion to the
next section.

“Also, g, = 0.18 GeVz.

013004-5

114

F. LANDRY, R. BROCK, G. LADINSKY, AND C.-P. YUAN

 

 

 

 

PHYSICAL REVIEW D 63 013004

 

 

 

 

 

 

(a SPWF" 3 PM"!
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14444 11

 

0.12 0.13 0.14

0.16 0.17 0.10 0.19
91

FIG. 5. The error ellipse projections from which the errors of the 3-parameter fit A, were interpreted. (a) g. and g2 plane, (b) g2 and g3

plane, and (c) g, and g, plane.

IV. RUN 1 W AND Z BOSON DATA AT THE TEVATRON

The run 1 W and Z boson data at the Tevatron can be
useful as a test of universality and the x dependence of the
non-perturbative function WyEP(b,Q,Qo,x1,x2). This is
clearly demonstrated in Fig. 2, where we give the predictions
for the two different global fits (2-parameter and 3-pararneter
fits) obtained in the previous section using the CTEQ3M
PDF parametrizations. (The CTEQ4M PDF [27] gives simi-
lar results.) With the large Z boson data sets anticipated from
Tevatron run 1 (la and lb), it should be possible to distin-
guish these two example models.

As shown in Ref. [9], for QT> 10 GeV the non-
perturbative function has little effect on the QT distribution,
although in principle it affects the whole QT range (up to
QT~Q). In order to study the resolving power of the full
Tevatron run 1 Z boson data in determining the non-
perturbative function, we have performed a “toy global fit,"
Fit F 3 as follows. First, we generate a set of fake run 1 Z
boson data (assuming 5,500 reconstructed Z bosons in 24 QT
bins between Qr= 0 and 20 GeV/c) using the original LY fit
results (g,=o.ll GeVz, g2=0.58 GeV2 and g3=- 1.5
GeV' l). Then, we combine these fake-Z boson data with the
R209 and E605 Drell-Yan data as discussed above to per-

013004-6

115

 

NEW FITS FOR THE NON-PERTURBATIVE PARAMETERS . . .

 

 
   

 

 

 

A :20. data (up 23 (1901) 23)
g creos. a-(27.4 GeV)‘
.2
U
3 5<0<6 GeV
\
”5
O
0 1 t-
n P
3 I
t’)
0.
0
ll
>4
u
0
n
a .
‘O
\
.P
'0 _‘ .
“J to — 8<Q<9 GeV
5 o
C 9 \
. 91s .24. 92- .34, 93:- 0., NORM- 0.92 __ o
. 91-.15.92- .48, 93- —ea. NORM- 0.79 ....
I A l 1 A J J L J A I l l 1 I I I I I l l I I J J A
o 0.2 at 0.6 0.3 1 1.2 1.4
Pt(GeV)

FIG. 6. E288 data, from p+Cu-+n+,r +x at JE= 27.4 GeV,
with an overall systematic normalization error of 25%. The curves
are the results of Fits N2 and N3 and are multiplied by the value of
NORM, as shown in the figure and described in the text.

form a global fit. The 3-parameter form results in

Fit F3: ,g,=0.10:‘3;322 GeVz, g,=0.57:3;3; GeVz,

g3=—o.98:8;}3 GeV",

with a X2 per degree of freedom of approximately 1.3.7
These fitted values for the 3’s agree perfectly with those
used to generate the fake-Z data except for the value of 33,
which is smaller by a factor of 2. It is interesting to note that
this result agrees within 20' with that of Fit A3, using only
the current low energy data, although the uncertainties on g ’s
are smaller by a factor of 2.

We have also performed the same fit for the 2-parameter
form and obtained an equally good fit with

Flt F2: g,=0.04:3;3§ GeVz, g2=0.63:3;3; GeVz.
While the new g2 value (0.63 GeVZ) obtained in 1:1th is
very different from that of A2 (0.34 GeVz) given in the pre-
vious section, it is actually in good agreement with the g2
value (0.57 GeVz) obtained from Fit/13. This implies that
the run 1 Z boson data, when combined with the low energy
Drell-Yan data, can be extremely useful in determining the
parameter g2. In Figs. 7—10, we compare the two theory
predictions (derived from F23) with the R209, fake-Z, E605,
and E288 data. As shown, they both agree well with all of
the data.

7This amounts to a shift in the prediction for the mass and the
width of the W boson by about 5 MeV and 10 MeV, respectively
[28].

PHYSICAL REVIEW D 63 013004

140

 

1:209 data (m 47 (1901) 12} crtos
91- 0.04. 92- 0.63, 93- 0.0, Mom- 1.05
gi- 0.10. 92- 0.57. 93- -0.9a. Moau- 1.05

in.
N
0

    
     

5<O<8 GeV

§

1 l V V T I I

donor/d0? (pb/GOV')
8

I V 7 V l I I

00
O

40

I I V I Y T I

8<0<11 GeV
1 t

I ' ‘
o lljlljlllllllllll nlaanmaan

0 0.2 0.4 0.6 0.8 1 1 .2 1.4 1.0 1 .0 2
0. (GeV)

 

 

 

FIG. 7. Comparison of R209 data with two different theory
model predictions obtained from the “toy global fit." The curves
are the results of Fits F2 and F3 and are multiplied by the value of
NORM, as shown in the figure and described in the text.

Before closing this section, we would like to comment on
the result of a single parameter study of the fake Z data, Fit
S1. Given the large sample of run 1 Z data, one can consider
fitting the non-perturbative function with only one,
Q-independent, non-perturbative parameter. With this in
mind, we fitted the fake Z data with the non-perturbative
function

 

Fate 2 data CTE'os

s‘
I

§

0 91- 0.04 92- 0.63 93- 0.00

 

o gi- 0.10 92- 0.57 93- —o.sq

do(Z)/d0r (pb/GeV)

§

§

 

 

IIIELJAIJAJJAILLIAIII

I lllllJlAJlllll'

0 2.5 5 7.5 10 12.6 15 17.5 20

01 (GeV)

 

FIG. 8. Comparison of “fake Z” data (solid curve) with two
different theory model predictions (dots) obtained from the “toy
global fits" F2 and F3.

0 1 3004-7

116

F. LANDRY, R. BROCK, G. LADINSKY, AND C.-P. YUAN

 

3605 6.1:: (PM

3 (1991) 2015)
CIE03.8=(38.8

. 91- 0.04. 92- 0.63, 93- 0.0. nonu- 1.0:—
gl- 0.10. 92- 0.57. 93- —o.sa. nonu- 0a...

A 7<O<8 GeV
l- M
‘ O

‘ i \\
- 8<Q<9 GeV

9

.-
..-

V'l

 

‘
°1
-
I

   
 

'ITY'

10.5<O<11.5 GOV

    

Ed’a/dp" at X,=0.1 (pb CeV"/nucleon)

 

 

 

W..
‘0'“.- u
1 1.s<o<13.s Gav
1 i *
._..___ ~-—~.-....___1
MM...
1°".- 1 ant
’ 135<o<18 GeV
1 I I J I I l L l j l l I I I l I A I I l A I l
0 0.2 0.0 0.5 0.8 1 1 .2 1.4
Pt (cw)

FIG. 9. Comparison of E605 data with two different theory
model predictions obtained from the “toy global fit.” The curves
are the results of Fits F 2 and F 3 and are multiplied by the value of
NORM, as shown in the figure and described in the text

Wii'tbQani.x2)=expi—§.b2]. (is)
and found that
Fit s,:§1=2.1:3;g GeVz,

which gives a good description of the Q1- distribution of the
“fake-Z” data. It is easy to see that this fitted value agrees

 

3200 data (BID 23 (1901) 23}
CTEQ}. a-(27.4 60V)'

 

 

Ed’a/dp’ at y-0.03 (pb GeV"/ nucleon)

    

. t
. 91- 0.04. 92- 0.63. 93- 0.0, NORM- 0.93 _
. g1- 0.10, 92- 0.57. 93- —o.sa. NORM- 0.79 ......

 

 

IIIIAIIIIIIIIIIIJIAIIIII

0 0.2 0.4 0.6 0.8 1 1 .2 1 .4
Pt (GOV)

 

FIG. 10. Comparison of E288 data with two different theory
model predictions obtained from the “toy global fit.” The curves
are the results of Fits F 2 and F3 and are multiplied by the value of
NORM, as shown in the figure and described in the text.

PHYSICAL REVIEW D 63 013004

with that of the 2-parameter fit just by considering the coef-
ficient of b2 in first two terms of Eq. (14). For the results of
F; with the value of the z boson mass, M2: 91.187 GeV/c2,
we obtain 004+ 0.63 ln(MZ/2Q0) = 2.14, which is essentially
the same as the coeficient of b2 in S 1 . One interesting ques-
tion is whether the result of this one-parameter fit alone can
be used to also describe the QT distribution of the W: boson
produced at the Tevatron (at the same energy). A quantita-
tive estimate can be easily obtained by noting again that the
difference between 0.04+0.63 ln(MZ/2Q0)=2.l4 and 0.04
+0.63 ln(MW/2Q0) =2.06 with the W—boson mass M w
= 80.3 GeV/c2, is 0.08, which are essentially the same, given
the uncertainty of 0.09 GeV2 from S 1. We conclude that it is
indeed a good approximation to use the one-parameter fit
result from fitting Z boson data in order to predict the QT
distribution of the W: boson using the CSS resummation
formalism. On the other hand, a single parameter without Q

dependence (i.e. the parameter E, alone) does not give a
reasonable global fit to all of the Drell-Yan data discussed
above. For instance, for the R209 data, the 2—parameter fit
gives 0.04+ 0.63 ln(8/2Q0)=0.6 for the coefficient of Eq.

(13), which is not consistent with the value of g, from S1.
Hence, we conclude that in order to test the universality of
the non-perturbative function of the CSS formalism, one
must consider its functional form with Q (and x) depen-
dence.

V. CONCLUSIONS

The effects of QCD gluon resummation are important in
many precision measurements. In order to make predictions
using the CSS resummation formalism for the QT distribu-
tions of vector bosons at hadron colliders. it is necessary to
include contributions from the phenomenological non-
perturbative functions inherent to the formalism. In this pa-
per, we have extended previous results by making use of
2—parameter and the 3-parameter fits to modern, low energy
Drell-Yan data. We found that both parametrizations result
in good fits. In particular our results are

Fit A2: g,=0.24:3;3§ GeV2, g2=0.34:3;$ GeVz.

Fit A3: g,=0.15:3;3§ GeVz, g2=0.48:3;3§ GeVz,

g3= —0.58:3;§3 GeV".

Each functional form predicts measurably different QT dis-
tributions for Z bosons produced at the Fermilab Tevatron.
We showed that the full Tevatron run 1 Z boson data can
potentially distinguish these two different models. Table II
summarizes all of the fits described in this paper.

In particular, using the results from a toy global fit, we
concluded that the large sample of the run 1 Z data can help
to determine the value of g2, which is the coefficient of the
ln(Q/2Qo) term. Work to incorporate the full run I data is in
progress [19]. Given these new data, one can hope to study
the x dependence of the non-perturbative function in more
detail. We also confirmed that it is reasonable to use a single

non-perturbative parameter E, to fit Z boson data, and use

013004-8

117

NEW FITS FOR THE NON-PERTURBATIVE PARAMETERS . . .

PHYSICAL REVIEW D 63 013004

TABLE II. Summary of the results of all fits described in this paper. The last column denotes the figure

numbers in which the results are displayed.

 

 

 

Flt Contents Label Results Figure
11,, full 2 and A, g, =0.24t3;$ GeV2 1.23.4.5
3 parameter g,=0.34:3;37, Gov2
global fit A 3 g,=0.15:3;3§ ocv2
g,=0.48:8;8; Gev2
g3= —0.58:3:§3 Gev-l
L, g,=0 L, g,=0.18 GeV2
R209 plus CDF g,=0.47 Gev2
N23 fix g’s to Au N, NORM=0.92 6
fit NORM from 13288 N3 NORM=0.79
F,_, g’s from LY F, g, =0.04:3;3§ Gev2 7.89.10
to generate fake Z data g,=0.63:gj8§ GeV2
plus R209 and E605 F, g, =0.10:3;3§ (iev2
35057133; Gev2
g,= —0.98:3;{; GeV"
.51 fake 2 data 51 5:21:33: Gav2

 

 

that result to study the Q,- distribution of the WI boson for
QT<10 GeV. Recently this point has been made in the con-
text of a momentum-space fit [5] using a single parameter.
Such an approach might indeed alleviate the computational
overhead required in order to generate complete y - QT grids
of simulated W bosons necessary for M w analyses. However,
if one is interested in testing the universality property of the
CSS resummation formalism or making predictions about W
and Z boson production at future colliders. such as the CERN

Large Hadron Collider, then one must include the Q (and,
possibly, x) dependent term in the non-perturbative function.

ACKNOWLEDGMENTS

We thank C. Balézs, D. Casey, J. Collins, W.—K. Tung
and the CTEQ Collaboration for useful discussions. This
work was supported in part by National Science Foundation
grants PHY-9514180 and PHY-9802564.

 

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