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This is to certify that the

dissertation entitled

Using Beam Polarization to Study
Ancmalous Electroweak Physics and

Supersymnetry

presented by
Michael C. Wiest

has been accepted towards fulfillment
of the requirements for

 

PhD degree in _Eh¥sics__

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1/98 wagon-mum“

 

USING BEAM POLARIZATION TO STUDY
ANOMALOUS ELECTROWEAK PHYSICS
AND
SUPERSYMMETRY

by
Michael Christian Wiest

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

1997

ABSTRACT
USING BEAM POLARIZATION TO STUDY
ANOMALOUS ELECTROWEAK PHYSICS AND SUPERSYMMETRY
By
Michael Christian Wiest

This thesis considers the phenomenology of two classes of possible fundamental in-
teractions and fields not present in the Standard Model (SM) of high-energy physics.
The topics are associated because each example considers how a collider experiment

with a longtitudinally polarized beam might be used to study the interactions.

It is technically feasible to polarize the proton beam at the Fermilab Tevatron col-
lider. In Part I we model an experiment at a polarized Tevatron, to compare its
capabilities to the existing unpolarized Tevatron. In particular, we estimate the
precision with which the Standard Model (SM) electroweak triple-boson couplings
could be tested; that is, we calculate limits on possible anomalous electroweak triple-
boson couplings, for a polarized and an unpolarized Tevatron. For comparison to
the Tevatron, and as a simpler illustrative calculation, we first calculate the rate of
W+W‘ production at two e+e‘ colliders, LEPII and the proposed N LC. This pro-
cess is sensitive to the triple-boson couplings, so the event rate allows us to estimate
limits which could be placed on anomalous couplings using data from these two col-
liders. We then calculate the rates of W+W’ production and W + '7 production at
the Tevatron pf) collider with and without a longtitudinally polarized proton beam,
to gauge the increase in sensitivity with a polarized beam. Because accurate polar-
ized parton distribution functions (ppdf’s) would be required for precise predictions,
we also consider how measurement of single-W production with a polarized proton

beam could be used to improve the accuracy of the ppdf’s. The physical decays of

the produced W bosons are included in all of these calculations.

In Part II we identify the region of parameter space for which the proposed NLC
would be able to detect a very light (order keV) gravitino, which is required in gauge-
mediated models of supersymmetry-breaking, and allowed in “no-scale” models of
unified supergravity. To map this accessible parameter space, we calculate the
production of gaugino pairs which decay to photons and gravitinos: 3+9." —r XiXi -—>
77GG’. In such an experiment, the polarized electron beam of the NLC would make

it possible to eliminate a background process.

Dedicated with love to my parents,

Toni and Michael

iv

Acknowledgements

I am most grateful to my thesis advisor, Dan Stump, for his invaluable contribution
to my graduate education. My debt to him began to accrue from my first year
at MSU, when he introduced our class to quantum mechanics with his well-known
clarity and authority. Since I became his student, he has been extremely generous
with his time and knowledge, and quite Willing to tailor my research to my interests.
I am very grateful for the opportunity to osmotically improve my understanding of
quantum field theory. I also owe to Dr. Stump an appreciation of what makes a
scientifically interesting problem, and an appreciation for thoroughness of investiga-
tion. When I would begin to drift into the clouds of speculation, Dr. Stump gently
brought me back to earth.

I am grateful to C.—P. Yuan for suggesting the project which became Part I of
my thesis, and for his collaboration in that work. I also thank Dr. Yuan for much
helpful advice, and for teaching me (in class and out) methods of “practical” field
theory, as well as for serving officially on my guidance committee.

If the gift of wisdom excels all others, then the gift of a humourous anecdote
may rank second. Gerald Pollack generously shared both with me. Encountering
Dr. Pollack in the halls of the physics building during my years here greatly reduced
the lab—rat feeling that results from endlessly walking back and forth in a concrete
hall. I also thank Dr. Pollack for serving on my guidance committee.

I thank Chip Brock for serving on my guidance committee and for his course
which introduced me to quantum field theory. Thanks to Aaron Galonsky for serv-
ing on my committee and for encouraging discussions. Thanks to Wayne Repko
for guiding me through a physics “800” project which taught me FORTRAN, and
for radiating a relaxed attitude at me in the hallways. I also gratefully acknowl—

edge Carl Schmidt’s course on supersymmetry and Jeff Kuhn’s gravity course. I

feel lucky for the exposure to Einstein’s wonderful theory and its quantum mechan-
ical sibling. I thank Eric Poppitz, whose talk on gauge-mediated supersymmetry
breaking suggested to me the idea for Part II of this thesis.

In Down and Out in Paris and London, George Orwell defines the idiomatic
French word “debrouillard” to mean something like, “someone who, when asked to
do the impossible, finds a way.” Stephanie Holland of the graduate physics office has
just such a “can—do” attitude. I am grateful to her and to Jules Kovacs for defeating
countless frumious beaurocratic bandersnatches. I also gratefully acknowledge much
similar help from Mary Curtis, Lisa Ruess, Lorie Neuman; and Jeanette Dubendorf,
who helped me with the laborious typing of address labels for post-doc applications.
Thanks to George Perkins for waking me from many computer nightmares. My
teaching assistantships were enhanced by cheerful support from Peter Signell, Gene
Kales, Jack Hetherington, Jerzy Borysowicz, Harry Weerts, Carl Bromberg, and
especially Michelle Li of the CBI administration.

Without my fellow graduate students and post-docs I might well have ended like
the legendary student found naked on a traffic island, crying “Newton was wrong!”
I am grateful for their friendship as well as their help with my studies. Thanks to
Doug Carlson for his collaboration and for sharing his home-brewed beer. Thanks to
Glenn Ladinsky for sharing his expertise and his program for generating polarized
parton distributions. I’m grateful to Joelle Murray, Hung-Liang Lai, Csaba Balazs,
Jim Hughes, Xiaoning Wang, Andre Maul, David Bowser-Chao, Francisco Larios,
Tim Tait, Tom Rockwell, Gian Diloretto, Steve Jerger, and Jim Amundson for all
sorts of technical and moral support. Thank you Kate Frame for invaluable “reality
checks” and your cherished friendship. I also thank Ann Connor for her support
and patience with me during these final months of thesis writing.

Finally I thank my parents, Michael and Toni, and my brother Matthew, and

the rest of my family, who are my foundation. I cannot adequately express my debt

or my gratitude to them.

vii

Contents

List of 'Ihbles xi
List of Figures xv

1 Introduction 1
1.1 The Standard Model ........................... 1
1.1.1 Quantum Chromodynamics ................... 2

1.1.2 The Electroweak Sector ..................... 3

1.1.3 The Higgs Sector ......................... 3

1.1.4 Generations, the CKM Matrix, and CP violation ....... 5

1.1.5 Summary of the Standard Model ................ 6

1.2 Triple-Boson Couplings .......................... 7
1.3 Beam Polarization ............................ 13
1.4 The Organization of This Thesis ..................... 14

I Anomalous Electroweak Triple-Boson Couplings l5

2 Probing Anomalous Triple-Boson Couplings with e‘e’r —+ W' W+ 16

2.1 Introduction ............................... 16
2.1.1 Calculation ............................ 18
2.1.2 Estimation of Sensitivity to Anomalous Couplings ....... 24

viii

2.1.3 Using x2 to Estimate Sensitivities to Anomalous Couplings . . 28
2.1.4 Conclusions ............................ 31

3 Probing Anomalous Triple-Boson Couplings with Polarized Proton—

Antiproton Collisions 35
3.1 Introduction ................................ 35
3.2 W+7 production ............................. 39
3.2.1 Method of Calculation ...................... 39
3.2.2 Results ............................... 42
3.2.3 Limits on An, and A, ...................... 45

3.3 W+ W“ production ............................ 46
3.3.1 Background, Approximations, and Cuts ............. 49
3.3.2 Results ............................... 57
3.3.3 Limits on A5,, A53, «\7, and AZ ................ 62
3.3.4 Interpretation of the Results ................... 64

3.4 Discussion and Conclusions ....................... 65
4 Probing the Polarized Parton Distributions with 125 —> W 75
4.1 Introduction ................................ 75
4.2 W55 production .............................. 77
4.3 Decay of Wi ............................... 86
4.4 Conclusions ................................ 94

II A Light Gravitino and the Supersymmetry Breaking

Mechanism 96
5 Detecting a Light Gravitino at a Future Linear Collider 97
5.1 Introduction to Supersymmetry ..................... 97

ix

5.1.1 Supersymmetry Breaking ..................... 100
5.2 Introduction to the Problem at Hand ................. 102
5.3 Detecting a Light Gravitino at the Next Linear Collider ....... 104
5.3.1 Production of Neutralino Pairs at the NLC ........... 104
5.3.2 Decay of the Neutralino ..................... 106
5.3.3 Background-Free Signal Events ................. 109
5.3.4 Non-Background-Free Signal Events ............... 115
5.4 Discussion and Conclusion ....................... 121
5.5 Additional Comments .......................... 126
The Standard Model Lagrangian 128
Helicity Amplitude Method 131
B.1 Helicity Eigenstates ............................ 131
B.2 Example Calculation ........................... 134

List of Tables

1.1

2.1
2.2
2.3
2.4
2.5

3.1

Current 95% CL. limits on anomalous couplings from direct cross
section measurments. A subscript '7, Z indicates that the photon and
Z anomalous couplings were assumed equal. All couplings but those
being limited were fixed at their SM values. All entries shown were
calculated with A = 1.5 TeV, except the last, from Reference [12], for
which A = 1.0 TeV. ............................

Final state cuts ...............................
LEPII limits on An, from the total cross section. ...........
NLC limits on An, from the total cross section. ............
LEPII limits on Art, from )2“ of cos 0., distribution. ..........

NLC limits on An, from 5&2 of cos 9., distribution. ...........

Cross-section for the process pp —» W+7 with polarized protons, for
different values of the anomalous coupling An,, assuming A, = 0.
Cross-sections are in pb. The branching ratio 2/ 9 for W+ —> e+u¢ or
”+11", and the effect of our cuts, is included. The unpolarized case was
calculated separately using CTEQ2 parton distribution functions, for
comparison. The asymmetry A, defined in Eq. (3.7), is calculated by

fitting the data to a parabola. .....................

10

20
26
26
31
33

3.2

3.3

3.4

3.5

3.6

Cross-section for the process 1013 —» W+7 with polarized protons, for
different values of the anomalous coupling A,, assuming An, = 0.
Cross-sections are in pb. The unpolarized case was calculated sep-
arately using CTEQ2 parton distribution functions, for comparison.

The asymmetry A, defined in Eq. (3.7), is calculated by fitting the

data to a parabola. ...........................

Limits on non-Standard couplings, from pp -) W+7 with polarized
and unpolarized protons. The upper number is for 1 fb’l integrated
luminosity, and the lower number (in parentheses) is for 10 fb'l.

Purely electroweak cross-sections, in pb, for p(R)13 -+ W+ + 2 jets,
with W1" —-> it! where l = e or p; the proton is right-handed. The

cross-section for p(R)13 —> W‘ + 2 jets is the same. The branching

ratio (2/9) x (6/9) = 4/27 and the effect of our cuts are included. . .

Purely electroweak cross-sections, in pb, for p(L)p —> W+ + 2 jets,
with W+ —t iv where l = e or p; the proton is left—handed. The

cross-section for p(L)13 —) W“ + 2 jets is the same. The branching

ratio (2/ 9) x (6/ 9) = 4/27 and the effect of our cuts are included. . .

Purely electroweak cross-sections, in pb, for p13 -—> W+ + 2 jets, with
W+ -» [V where l = e or p; the proton is unpolarized. The cross-
section for p13 —> W‘ + 2 jets is the same. The branching ratio
(2/ 9) x (6/ 9) = 4/27 and the effect of our cuts are included. These
values were calculated independently using the Morfin-Tung ppdf’s;

the cross-section for unpolarized protons is equal to the average of

cross-section for left and right polarized protons. ...........

.69

70

71

3.7

3.8

3.9

3.10

3.11

Electroweak cross-sections, in pb, for W+W‘ production with non-
Standard couplings, with polarized or unpolarized protons, and with
a large-J3 cut, \/3 > 340 GeV. One W decays leptonically, the other
to 2 jets, and the branching ratio 4/27 and the effect of our cuts
are included in the cross-section. Two assumptions on non-Standard

couplings are listed: An, = Aug with A, = A2 = 0, and A, = A2

with An, = Anz = 0. ..........................

QCD background cross-sections for p13 —» Wi + 2 jets, for vari-
ous proton polarizations and kinematic cuts. The unpolarized cases

0(pp —+ W+ 23') = 0(1)}? —> W‘ 2j) were calculated separately using

CTEQ2 parton distribution functions. .................

Limits on anomalous couplings that could be set from pf) —; Wi +
2 jets with polarized or unpolarized protons. The numbers in paren-

theses are for 10 fb‘1 integrated luminosity, and the other numbers

are for 1 fb'1 integrated luminosity. ..................

Limits on anomalous couplings that could be set from pp —> W:t +
2 jets with polarized or unpolarized protons, from events with x/E >

340 GeV. The numbers in parentheses are for 10 fb'l integrated lu-

minosity, and the other numbers are for 1 fb’1 integrated luminosity.

Effect of parton Q scale on the calculated cross-section for background

processes pp —> Wi + 2 jets. These are unpolarized cross-sections,

calculated with CTEQ2 parton distribution functions. ........

74

4.1 Total cross-sections for the processes pi+fi -—§ Wi -> 6* +(l7l, where A
is the proton polarization, L or R. The polarized parton distribution
functions are taken from Refs. [38] and [39]. Also, we consider a
transverse momentum cut 191., > megn. There is no restriction on
lepton rapidity ye; the effect of a cut on 318 may be judged from
Figs. 4.8, 4.9, 4.11, 4.12. ......................... 94

5.1 Production cross-section a and typical decay distance D for the pro-
cess e§e+ —-) xgx? at the NLC, assuming a = 0, for several values
of masses. (Note that the electron beam is right-hand polarized, so

the cross-section is a factor of 2 larger than that for an unpolarized

electron beam.) These numbers are for Mm,” = 106 GeV. ....... 113
B.1 Helicity states in two-component spinor, and bra-ket notation. . . . . 132
B.2 Dirac helicity eigenvectors for massless fermions ............. 133

List of Figures

2.1 Diagrams for e‘e+ —+ W"W+ —> l‘fiqrj’ ................. 19
2.2 Rapidity distributions of the W‘ at LEPII from analytic amplitudes

(smooth) and uncut Monte Carlo calculation (histogram). ...... 19
2.3 Rapidity distributions of the W‘ at NLC from analytic amplitudes

of (smooth) and uncut Monte Carlo calculation (histogram). ..... 20
2.4 (ta/(1yc at LEPII, without cuts, for e+e" —» W+W‘ —> e’fieundd-mp

Solid line is SM, dashed is A5, = —0.5 .................. 22
2.5 (la/(1ye at LEPII, with cuts, for e"’e" —» W+W‘ —+ e-ieuredJI-ed-

Solid line is SM, dashed is A5, = —0.5 .................. 23
2.6 Total cross section at LEPII for e+e‘ —-> W+W‘ —+ l’iiqu’ vs. A5,. . 25
2.7 Total cross section at NLC for e+e‘ —+ W+W’ —» l‘fiqq’ vs. A5,. . . 25
2.8 Total cross section at NLC for e+e" —» W+W‘ vs. A5,. The solid

and dashed curves are for left and right polarized electron beams,

respectively. ................................ 27

2.9 Differential cross sections in cosine of the lepton angle at LEPII and

NLC for two values of A5,. ....................... 30
2.10 323,, at LEPII for e+e“ —) W+W" —> 1‘17qq7 vs. A5, ........... 32
2.11 )2", at NLC for e+e" —» W+W" -—) (“fiqq’ vs. A5,. .......... 32

XV

3.1

3.2
3.3

3.4

3.5

3.6

3.7
3.8

3.9

Polarized parton distribution functions. The curves are zAf(a:) vs a:

for parton types um , dual , um(= dm) , g, which are the most
important partons in our calculations. ................. 38
Feynman diagrams for the process ad —-> e+u7. ............ 40
Total cross-section (with cuts in Eqs.(3.3)-(3.5)) for polarized protons

vs anomalous coupling A5,, assuming A, = 0. The unpolarized cross-
section was calculated separately using CTEQ2 parton distribution
functions. ................................. 42
Total cross-section (with cuts in Eqs.(3.3)-—(3.5)) for polarized protons

vs anomalous coupling A,, assuming A5, = 0. The unpolarized cross-
section was calculated separately using CTEQ2 parton distribution
functions. ................................. 43
Distribution of photon transverse momentum p1,. The solid line is

for A5, = 0, and the dashed line is for A5, = —1; in both cases

A, = 0. .................................. 44
Complete set of electroweak diagrams for n+1] -> d+fi+W+ (—> Z + m) .

47

Example diagrams for QCD production of W+ and two jets. ..... 49
Two-jet invariant mass distribution for the signal process. The solid

line is the result of the complete calculation of the pure electroweak
process p + 13 —+ qq’W+, with W+ —§ in; the dotted line is the result

of the calculation of W+W‘ production, with W+ —t in and W‘ —>

2 jets, with a cut on the two-jet invariant mass (70 < M2,- < 90 GeV). 51
Two-jet invariant mass distribution for the QCD background pro-
cesses p13 -—) W+ + 2 jets. (The cross-section for W” + 2 jets is the

same, for unpolarized scattering.) ................... 52

xvi

3.10

3.11

3.12

3.13

3.14

x/i distribution for the signal process. The solid line is the result of

the complete calculation of the pure electroweak process 1) + 1'5 —i qé’W“ ,
with W+ —> Zia; the dotted line is the result of the calculation of

W+ W“ production, with W+ —> in and W“ —-+ 2 jets, with a cut

on the two-jet invariant mass (70 < M2,- < 90 GeV). ......... 54
Comparison of J; distributions for signal and background processes.

The solid line is the QCD background. The dotted line is the elec-
troweak process, with zero anomalous couplings; the dashed line is

the electroweak process with A5, = A53 2 0.5 ............. 55
Comparison of \/§ distributions for signal and background processes,

for x/S > 340 GeV. The solid line is the QCD background. The dotted

line is the electroweak process, with zero anomalous couplings; the
dashed line is the electroweak process with A5, = A52 = 0.5. . . . . 56
Electroweak cross-section for pip -> W+W“ as a function of anoma-

lous couplings A5, for polarized (L, R) and unpolarized protons. For

each polarization, three cases are shown, corresponding to assump-

tions (X) A5, at 0 and A52 = 0, (0) A5, = 0 and A52 76 0, and

(C1) A5, 2 A52. In all cases A, = A2 = 0. ............... 59
Electroweak cross-section for p313 —> W+W“ as a function of anoma-

lous couplings A, for polarized (L, R) and unpolarized protons. For

each polarization, three cases are shown, corresponding to assump-

tions (x) A, aé 0 and A2 = 0, (O) A, = 0 and A2 at 0, and (D)

A, = A2. In all cases A5, 2 A52 2 0. ................. 60

xvii

3.15 Electroweak cross-section for pAp' —» W+W“ as a function of anoma-

4.1

4.2

4.3
4.4

4.5

4.6

lous couplings, with x/i > 340 GeV. For each polarization, two cases
are shown, corresponding to assumptions (x) A5, = A52 with

A, = A2 = 0, and (O) A, = A2 with A5, = A52 = 0. ......... 61

Polarized parton distribution functions. The curves are :cA f (2:) vs a:
for parton types it and d, which are the most important partons in our
calculations, for two sets of polarized parton distribution functions.
The solid curve is the Nadolsky parametrization [38] and the dashed
curve is the Gehrmann-Stirling parametrization [39]. ......... 78
The function zgf(:c, Q), which has been measured experimentally in
polarized deep-inelastic lepton scattering, calculated using two sets
of polarized parton distribution functions. The solid curve is the
Nadolsky parametrization [38] and the dashed curve is the Gehrmann-
Stirling parametrization [39]. ...................... 79
Diagram for ad —» W+ .......................... 81
Cross-sections do(A)/dyw for p), + p —> W+ in nb, multiplied by
1/9, the branching ratio for the W+ to decay to 6+ + 24,. Proton
polarization A is L or R. The solid and dashed curves are for ppdf’s
from Ref. [38] and Ref. [39], respectively. ............... 82
Cross-sections do(A)/dyw for p,\ + 13 —) W“ in nb, multiplied by
1/ 9, the branching ratio for the W“ to decay to e“ + B... Proton
polarization A is L or R. The solid and dashed curves are for ppdf’s
from Ref. [38] and Ref. [39], respectively. ............... 83
The left-right polarization asymmetry A“; for Wi production in
polarized p - p collisions. The solid and dashed curves are for ppdf’s
from Ref. [38] and Ref. [39], respectively. ............... 84

xviii

4.7

4.8

4.9

4.10

4.11

4.12

Schematic diagram of particle momentum and spin orientations in
uLdR —) W; —+ e314,. Single arrows indicate momentum directions;
double arrows indicate spin orientation. For clarity the (I and e“

momenta are shown at angles to the other particles’, but they should

be imagined as along the same line ....................

Cross-sections do(A)/dy., for p; + 13 —+ W+ -—» e+ + 11.3 in nb, with

proton polarization A = L or R. The solid and dashed curves, labeled

(a) and (b), are for ppdf’s from Ref. [38] and Ref. [39], respectively. .

Cross-sections da(A)/dyc for p; + 13 —-> W“ —» e“ + 17¢ in nb, with

proton polarization A = L or R. The solid and dashed curves labeled

(a) and (b), are for ppdf’s from Ref. [38] and Ref. [39], respectively. .

The left-right polarization asymmetry ALR(8) for W2t production in

polarized p - p collisions, with decay W‘t —’ e": +(17.),. The solid and

dashed curves, labeled (a) and (b), are for ppdf’s from Ref. [38] and

Ref. [39], respectively. ..........................

Cross-sections do(A)/dye for p,\ + 13 —> W+ —> e+ + u, in nb, with a
kinematic cut on the lepton transverse momentum, p1,; > 25 GeV.

The solid and dashed curves, labeled (a) and (b), are for ppdf’s from

Ref. [38] and Ref. [39], respectively. ..................

Cross-sections do(A)/dyc for p; + 1“) —» W“ —+ e“ + 17¢ in nb, with a
kinematic cut on the lepton transverse momentum, p1,, > 25 GeV.

The solid and dashed curves labeled (a) and (b), are for ppdf’s from

Ref. [38] and Ref. [39], respectively. ..................

87

88

88

4.13 The left-right polarization asymmetry ALR(e) for W" production in
polarized p - p collisions, with decay Wi ——» e:h +571, with a kinematic
cut on the lepton transverse momentum, pTe > 25 GeV. The solid
and dashed curves, labeled (a) and (b), are for ppdf’s from Ref. [38]
and Ref. [39], respectively. ........................ 92

5.1 Differential cross-section do/d cos 0 for efie+ —+ x9361), where 0 is the

angle of a neutralino, at \/— = 500 GeV. The mixing angle a is 0, and

the masses are mx? = 100 GeV, me; = 300 GeV (solid curve), and

"hr? = 200 GeV, me; = 600 GeV (dashed curve). ........... 107
5.2 Range of parameters MN," and mx? accessible at the NLC. We allow

Mm,” to vary from 104 to 107 GeV. The shaded region is the range

of MN,” and mac? for which D < 1m and 0' > 0.2 fb, where D is

the typical decay length of x? and a' is the production cross-section

for efie+ —> XiXi at the NLC. The bound 0' > 0.2 fb is equivalent

to observing more than 10 events assuming integrated luminosity 50

fb“1. The cross-shaded region is for 10 cm < D < 1 m, corresponding

to the background-free signal process. Parameter values for this plot

are a = 0 and me; = 300 GeV. ..................... 110
5.3 Differential cross-section in photon energy do/dE, for age“r -—> Xi’Xi —+

77GG at the NLC, assuming a = 0. The mass parameter values are

mx‘f = 100 GeV and me; = 300 GeV. (The normalization of do/dE,

is such that the integral over E, is 2 times the total cross-section,

because the final state has identical photons.) ............. 111
5.4 Same as Fig. 5.2, but with a more precise calculation as described in

Section 5.3.3. ............................... 115

5.5

5.6

5.7

5.8

5.9

Differential cross-section in photon transverse momentum do/dp} for
CBC-i. -—» xg’x? —> 776363 at the NLC, assuming a = 0. The mass
parameter values are mx? = 100 GeV and me; = 300 GeV. (The

normalization of do/dp} is such that the integral over p} is 2 times

the total cross-section, because the final state has identical photons.) 117

Differential cross-section in the sum of photon energies t10'/d(E,1 +
E,,) for efie+ —> XiXi —i 77GG at the NLC, assuming a = 0. The
mass parameter values are mx? = 100 GeV and me; = 300 GeV.

Differential cross-section da/dMgm, in the invariant mass of invisible
particles Mg", for the signal process efie+ —-> xgx? —+ 77GG’, and
for the background process eke+ —» 77Z, with decay Z —-> 1117. The
photon transverse momenta are required to be greater than 20 GeV.

The masses for the signal process are mx? = 200 GeV and mm =

600 GeV. .................................

Total cross-section at the NLC for the process efie+ —» XiXi with a =
0, as a function of mx? for me; = 300 GeV (solid curve), me} = 100
GeV (dashed curve), and me; = 600 GeV (dot dashed curve). (Note

that only mx? < me; is allowed in the models we consider because

x? is the NLSP.) .............................

Total cross-section for the process e§e+ —+ XiXi’ with a = 0, as a
function of VS. The mass parameter values are mx? = 100 GeV,
me; = 300 GeV (solid curve), and mx? = 200 GeV, me; = 600 GeV

(dashed curve). ..............................

. 118

119

120

122

5.10 Range of parameters MN,” and mx? accessible at the NLC with
J? = 500 GeV (solid line), 1TeV (dashed line), and 1.5 TeV (dot-
dash line). The luminosity per year is 50, 200 and 200 fb“1, re-
spectively. Mass parameter values are me; = 300 GeV, 500 GeV and
750 GeV, respectively, and the mixing angle a is 0. The interior of
each triangle is the region where the number of events is greater than
10, and D < 1m. ............................. 123
5.11 Range of parameters MW,” and mx? accessible at the N LC (solid line),
LEP/SLC (dashed line), and LEP-II (dot-dash line). The interior of
each triangle is the region where the number of events is greater than
10, and D < 1m. Parameter values for this plot are a = 0 and
me; = 300 GeV. ............................. 124

B.1 (a) The WW7 vertex. (b) Example diagram for ad —+ W+7 —> e+u7.

Arrows indicate the defined momentum direction. ........... 135

Chapter 1

Introduction

The field denotes
this body, and wise men
call one who knows it

the field-knower.

Know me as the field-knower
in all fields—what I deem
to be knowledge is knowledge

of the field and its knower.

Lord Krishna, Bhagavad Gita

1.1 The Standard Model

The Standard Model (SM) describes the interactions of the most fundamental known
particles in nature. The SM is a quantum field theory [1], which means that the
particles detected in scattering experiments are represented by locally interacting

quantum fields. The Lagrangian of such a theory is specified by a list of its

constituent fields and its symmetries. A symmetry is an invariance of the Lagrangian
under a group of transformations of the fields. The SM Lagrangian is invariant under

“gauge,” or local, transformations corresponding to the group
SU(3)C X 311(2)], X U(1)y,

where S U (N ) can be represented by the group of unitary complex N x N matrices
with determinant one, while a U(1) transformation is equivalent to multiplication

by a complex phase.

1.1.1 Quantum Chromodynamics

The subscript C on the SU(3) group refers to the quantum number associated with
this group, which is called color. Because it governs the interactions of colored
particles, this sector of the Standard model is known as Quantum Chromodynamics
(QCD). The fundamental colored matter particles (fermions) are called quarks. Each
quark field comes in three colors, so the quarks are represented by SU(3) triplets.
There are six of these color-triplet fields, labelled by their “flavor”: up, down,
strange, charm, top, and bottom. The matter fields and the gauge group determine
the gauge bosons. The QCD gauge bosons are called gluons because they hold
together the quarks which make up protons and neutrons. They fall into the adjoint
representation of SU(3), which includes eight independent color states.

QCD has the property that only color-singlet states can be isolated and observed:
quarks or gluons are “confined” within color-singlet bound states, hadrons, which
are the observed strongly interacting particles. The observational consequence of
confinement is that when a scattering process produces quarks or gluons in the final

state, these will “hadronize” to be observed as “jets” of hadrons.

1.1.2 The Electroweak Sector

The S U (2);, x U (1)y sector is called the electroweak theory [2]. These two groups
are lumped together because their product group is broken down to U (1) EM by the
Higgs mechanism, which will be discussed in section 1.1.3. The subscript L means
that only left-handed fermion fields couple to the S U (2) gauge bosons. The Lorentz
character of the interaction vertex is vector minus axial vector, or V — A; this
combination projects out the left-handed component of a fermion.t The left-handed
fermions, including quarks (color triplets) and leptons (color singlets, or “colorless”)
fall into doublets; while the right-handed fields are represented by S U (2) singlets,
which do not interact with the S U (2) bosons. The upper and lower components
of an S U (2);, doublet are distinguished by their flavor name, or mathematically by
the third component of the “weak isospin” quantum number, I3. I3 = i% for the
upper and lower components, respectively. The gauge field is again in the adjoint
representation, so it has three components, called W1", Wf, and W; .

The U (1);» quantum number is called weak hypercharge, denoted by Y. The
U (1)y gauge field is denoted by B“. The values of the weak hypercharge are chosen

to satisfy a relation with Ia and the electromagnetic charge Q (in units of e):
Y —““— 2(Q — I3).

1.1.3 The Higgs Sector

With the inclusion of quarks, leptons, and gauge bosons as outlined above, we
can write down a gauge invariant Lagrangian. However, mass terms for any of
these fields, of the generic form m2<I>l<I>, would not be invariant under a SM gauge

transformation. As the corresponding particles are known experimentally to be

 

lThis statement is strictly true only for massless fermions. Throughout this thesis quarks and

leptons will be approximated as massless. This approximation is good at the collider energies we

will be considering, because E > me“.

massive, the theory described so far is unacceptable. While the dynamics of the
theory, codified in the Lagrangian, respects the SM symmetry, the fields that we
would like to call fundamental are found in Nature to be massive, so they do not
respect the symmetry.

The SM solution to this problem is called the Higgs mechanism [3]. The general
strategy is to associate the gauge non-invariant physics with the ground or vacuum
state, and retain the invariance of the Lagrangian. When the ground state of a
system does not possess the full symmetry of the system’s Lagrangian, the symmetry
is said to be “hidden” or “spontaneously broken.”

We postulate a complex scalar field, called the Higgs field <I>, which is an S U (2)
doublet; and we postulate a Higgs potential that forces the field to have a non-zero
vacuum expectation value (vev), thereby breaking the S U (2);, x U (1)y symmetry.
We may perform a gauge transformation on the Higgs doublet which eliminates
three of its four degrees of freedom, so that the Higgs field can be written in terms
of its vev and one real field. When we transform the Higgs field we must make the
corresponding transformation of the gauge fields. When the Lagrangian is rewritten
in terms of these transformed fields, we find that mass terms have appeared for
three of the gauge bosons, and the mass is related to the vev of the Higgs field. In
the argot, one says that the Nambu-Goldstone boson degrees of freedom have been
“eaten” by the gauge bosons to acquire their mass.

The SM symmetry of the original Lagrangian is “hidden” by the Higgs’ dynam-
ics, but our transformed Lagrangian respects a residual U(1) symmetry, called the
U (1) EM. We require the gauge boson associated with this unbroken symmetry to be
massless, which identifies it as the photon, A”, and fixes the physical combinations
of the fundamental fields, the Wui and Z3. This physical SM Lagrangian is shown
in Appendix A.

1.1.4 Generations, the CKM Matrix, and CP violation

We now have an economical description of massive gauge fields interacting with
fermion matter fields, but Nature is not so simple. There are three copies of each
fermion field which differ only in mass. In addition to the electron field there are
muon and tau fields with the same quantum numbers as the electron, except that
they are orders of magnitude more massive. Similarly, the .S' U (2) L doublet of up and
down quarks is supplemented by the heavier charm-strange doublet and heaviest,
the top-bottom doublet. Each of the three sets of fields is called a “generation,” or
“family,” while a specific field within a generation is identified by its “flavor.”

To complicate matters further, the eigenstates of the interaction Lagrangian for
these various flavors of fields are not equal to the mass eigenstates observed in
scattering experiments. This means that mixing among the generations is present.
The mixing can be parametrized by three angles and a complex phase arranged in
a 3 x 3 matrix in generation space, called the Cabbibo—Kobayashi—Maskawa (CKM)
matrix [4]. It turns out that we can relegate all the minng to affect only the 13 = --%
quarks; the lepton mass eigenstates are equal to their interaction eigenstates. An
important experimentally verified prediction of this formalism is the absence of
flavour-changing neutral currents at “tree level,” the leading order of perturbation
theory. For example, a charm quark very rarely decays to an up quark.

Experiments with the decay of K mesons, containing strange flavor quarks, show
that the weak interactions are not invariant under combined charge-conjugation (C)
and parity (P) transformations. The complex phase in the CKM matrix provides
a convenient parametrization of CP violation, because CP invariance requires that
MEKM = MCKM. (However, it is not yet known whether CP violation in the

Standard Model is solely due to this complex phase in the CKM matrix.)

In this thesis we will neglect the CKM mixing, as the CKM matrix is near enough

to the identity for our calculation.

1.1.5 Summary of the Standard Model

The SM identifies the constituents of matter as a set of interacting boson and fermion
fields. The Lagrangian postulated to describe their dynamics is invariant under a
set of transformations corresponding to the group S U (3)0 x S U (2);, x U (1)y. The
ground state, or vacuum, of this system of fields does not respect the SM symmetry.
The non-zero vev of the Higgs field leads to a set of effective, or physical fields,
including massive vector bosons.

The fermions of the SM are the quarks and leptons. The quarks are “col-
ored,” because they transform as triplets under an S U (3)0 transformation. The
left-handed quarks are also doublets under .5' U (2)1” while the right-handed quark
fields are S U (2);, singlets. The leptons are color singlets and again the left-handed
leptons are S U (2) L doublets, while the right-handed leptons are singlets. There are
three generations of quarks and leptons differing only in mass; the CKM matrix
parametrizes the mixing between generations.

The bosons of the SM are the vector gauge bosons and the scalar Higgs boson.
The Higgs is a colorless S U (2) L doublet. The mediators of the color interaction
are the SU(3)C octet gluons, which are SU (2)1, singlets. The physical electro-
weak bosons are the massive Wi, Z0, and the massless photon A. They are linear
combinations of the S U (2);, X U (1)y gauge bosons, which transform under the
adjoint representation of their respective simple groups.

This set of fields along with the dynamics specified by the Lagrangian has been
adequate thus far to account for the results of high-energy scattering experiments,

although precision tests are under way to test for deviation from the SM. Despite

its phenomenological success, there are reasons to believe that the SM is not a com-
plete theory of elementary particles. For one, the gauge groups and parameters
of the SM appear arbitrary. Also, the equality of the magnitudes of the proton
and electron charges is unexplained unless the SM gauge groups are unified into a
larger simple group such as S U (5) Furthermore, the gravitational interaction is
not included in the theory. It may also be argued that the technical “triviality” of
the Standard Higgs sector implies that the model is only valid below some cut—off
energy scale [5]. In other words, the SM must be considered an effective—as op-
posed to a fundamental—theory. However, like the periodic table of elements, the
Standard Model will continue to provide an accurate and economical description of

the phenomena in its domain.

1.2 Triple-Boson Couplings

Because the generators of the Standard model gauge group do not commute, the
gauge bosons must interact among themselves to preserve gauge invariance. The
form and strength of these interactions are fixed by the gauge invariance, so the
measurement of the triple-boson coupling parameters is an important test of the
Standard Model.

In part I of this work we will focus on the electroweak triple-boson couplings.
In particular, we would like to determine experimentally values of the WW7 and
WWZ couplings. Let Wi“(:c), Z“(a:), and A”(:c) denote the fields of W*, Z0, and

7; then the interaction Lagrangian we consider is
£3 : —ig(W,:,W’“ — WLWH‘XA” sin 0w + Z" cos 6w) (1.1)
—igW:W;(A“"5, sin 0w + Z’“’52 cos 9w)

_X;%,_W“+“WJG(AWA7 sin 0w + Z‘“’A2 cos 9w)

where Aw = 6,,Au —6.,A,,, etc. L3 is CP invariant, and gauge invariant with respect
to U(1)EM electromagnetic gauge transformations.
The parameters 5, and A, are related to the anomalous magnetic moment (FW)

and anomalous electric quadrupole moment (Qw)of the W“, as follows [6]:
— —e—(1 + + A )
”W _ ZMW n7 7 2

—e
Qw = —MW2(5, — A,).

The parameters 52 and A2 are similar WWZ couplings. In the Standard
SU(2)><U(1) gauge theory these coupling parameters have the definite values

5, = 52 = 1 , A, = A2 = 0 (Standard Model).

We define A5 by
A5 = 5 —— 1 .

If A5 or A is significantly different than 0 for either 7 or Z0, then the Standard
SU(2) x U(1) gauge theory is not the complete theory of the electroweak interactions.

We can check that the SM Lagrangian from the appendix reproduces £3 with A5
and A both zero. From Appendix A, the SM electroweak triple-boson interactions

are included in:

1 _ _
£¢11(SM) = — §|Dqu “Dqul2

— -;-[g cos awzij + eAW](W+"W“" — W““W+").

The second line is equal to the second line of £3, because the symmetric part of
W+“W“" contracted with the antisymmetric field tensors vanishes; and because
e = gsin 9w. To expand the first line and show that it includes the first line of £3
above, recall that

D” E 0,, + ig(Z2 cos 0w + A,J sin 0w).

Then the terms cubic in the vector fields are

igWJuKZO'“ cos 9w + A“ sin 0W)W"’ —— (Z°"’ cos 0w + A” sin 6w)W—"]

— igWJu[(Z°’“ cos 9w + A” sin 0w)W+" - (Z0"’ cos 0w + .4” sin 0w)W+"].

Again, since the tensor Wm, is antisymmetric, when it is contracted with any other
tensor only the antisymmetric part will survive; so we may replace the terms in

brackets, giving finally
_ig(W;'uW““ — W,;,W+“)(A" sin 9w + Z” cos 0w),

as claimed.

In the Standard Model, A5 and A can be induced at loop level, but only of
the size of 0(92/16'52) % 0.003 at one-loop level from naive dimensional analysis
[7]. Thus setting stringent experimental limits on the anomalous WW7 or WWZ
couplings is an important test of the Standard Model; actually discovering large
anomalous interactions would be a sign of new physics. We treat L3 as an effective
Lagrangian and only use it for tree-level calculations.

To determine the WWV couplings (where V stands for 7 or Z0) it is necessary
to measure the experimental cross-section, or the distribution of some kinematic
variable, for a scattering process that depends on the WWV coupling, and compare
the measurement to a calculated prediction. In subsequent chapters we will consider
a few processes to estimate their sensitivities to the WWV couplings.

Current (95% confidence) limits on A5 and A from pp —» W7,WW,WZ at
Fermilab, as reported in References [8] — [12] are shown in Table 1.1. The limits
on individual anomalous couplings were calculated assuming the other anomalous
couplings were zero.

Because the anomalous triple-boson amplitudes increase with energy as s /M3y,

to preserve the unitarity of the amplitudes these analyses were performed treating

10

Table 1.1: Current 95% CL. limits on anomalous couplings from direct cross section
measurments. A subscript 7, Z indicates that the photon and Z anomalous couplings
were assumed equal. All couplings but those being limited were fixed at their SM
values. All entries shown were calculated with A = 1.5 TeV, except the last, from
Reference [12], for which A = 1.0 TeV.

 

 

 

Experiment / Luminosity Process pp -—+ Ref. Limits

D0 lu7(l = e,p) [8] -1.6 < A5, < 1.8
13.8pb“1 —0.6 < A, < 0.6
D0 81/ij [9] —0.47 < A5,,2 < 0.63
96 pb‘l —0.36 < A,,2 < 0.39

—1.38 < A5, < 1.70
—1.21 < A, < 1.25
—0.60 < A52 < 0.79
—0.40 < A2 < 0.43

 

 

 

D0 W7, WZ, WW [10] —1.38 < A5,,2 < 1.70
13.8pb“l Combined —0.44 < A,,2 < 0.44
CDF lV7(l = e,p.) [11] —2.3 < A5, < 2.2
19.6pb“1 —0.7 < A, < 0.7
CDF lej [12] —0.89 < A5,; < 1.27

 

 

19.6pb-l —0.81 < A,,2 < 0.84

 

 

 

 

 

11

the anomalous coupling parameters as “dipole” form factors. For example,

A5(s) = US$772? (1.2)

with a similar expression for A. A may be interpreted as the energy scale of non-
Standard gauge boson dynamics. Both Tevatron detector collaborations found that
for sufficiently large A, above about 200 GeV, the limits on non-Standard couplings
are relatively insensitive to A. In this thesis we treat the anomalous couplings as
point couplings, independent of s. That is, we set A5(s) = A5, which is valid
at energy scales much smaller than A. This approximation causes our estimates of
bounds on anomalous couplings to be more stringent than those calculated including
the form factor behavior. Low energy values of anomalous couplings provide a lower
limit on A of order 1 TeV [13]. In the worst case, if A z 1TeV, our NLC limits
(«3 = 500 GeV) would be about 35% smaller than those calculated using form
factor couplings.

The fact that the anomalous vertices violate unitarity at high energies can ac-
tually be used to derive limits on the couplings. The authors of Reference [14] find
for A3 greater than about 1 TeV,

1.86 0.85
[A51] < 7: [A52] < 'A—z,
0.99 9.33

[A7] < 7’ Hz] < (13)

with A in TeV. To compare these unitarity bounds to the bounds obtainable from
direct cross section measurements, we anticipate a result of Chapter 2, namely that
the Next Linear Collider (NLC) could probe anomalous couplings of order 0.01.
Therefore, direct experimental cross section measurements will set stronger limits

on anomalous couplings than these unitarity bounds, unless A is greater than order

 

:The definition of A in Ref. [14] is not identical to Eq. (1.2), but the interpretation as the scale
of new triple-boson dynamics is the same.

12

10 TeV. Inversely, if A is greater than 10 TeV, the unitarity bounds imply that we
could not observe anomalous couplings at the N LC.

One can also derive limits on anomalous couplings by an analysis of precision
(loop-level) electroweak measurements [15]. A DPF study [16] reports that a global
fit of electroweak data, including for example measurements of lepton gyromagnetic

ratios and electric dipole moments, yields the following 10' (68 % C.L.) constraints,

A5, = 0.056:l:0.056

A52 = 4001940044

A, = —0.036i0.034

A2 = 004940.045, (1.4)

assuming that only one anomalous coupling varies at a time. The authors of [16]
consider it unlikely that non-Standard physics will only produce one anomalous
triple-boson vertex. They emphasize that these bounds disappear if the coupings
are allowed to vary simultaneously, due to the possibility of cancellations among the
couplings. In this case one is left with limits of order 1 from precision electroweak
measurements. Furthermore, the authors of [14] caution that bounds such as these,
which make use of loop calculations, ”should be taken with a grain of salt,” because
of ambiguities inherent in calculating loop effects in non-gauge models. They point
out that if one only analyzes unambiguous low-energy quantities, like (9 — 2)”, the
resulting bounds are again only of order 1. (Reference [17] discusses bounds from
(g — 2),, and from the precision measurement of the Z —» b5 branching ratio at
LEPII and BNL.) We conclude that possible bounds on anomalous couplings from
direct experimental cross section measurements are worth investigating; Part I of

this thesis is devoted to calculating these bounds at a few particle colliders.

13
1.3 Beam Polarization

Polarizing one or both incoming beams longtitudinally in a collider experiment can
enhance the precision of studies of parity non-conserving physics, or reduce parity
non-conserving background processes. Because the electroweak interaction violates
parity conservation (e.g. the W couples only to left-handed fields), electroweak
processes are the SM processes most sensitive to polarization.

Experimenters can potentially use this fact in two ways. To study an electroweak
process, a polarized beam would allow the experimenter to choose the polarization
with the higher cross section and greater event rate. On the other hand, to study a
parity-conserving process with an electroweak background, the polarization which
minimizes the background may be chosen. As an example consider single W pro-
duction from e+e“ annihilation, where we may polarize the electron to be right or
left-handed. Because the W coupling to fermions is pure left-handed, we will see
no W’s from a right-handed electron beam. Now, the unpolarized cross section is
the average of the two polarized cross sections. Therefore, if we choose the left
polarization, the polarized cross section will be twice the unpolarized cross section.
This will be true of any process for which the diagrams involve a. W line connected
to a massless initial fermion.

The calculation of polarized cross sections is straightforward in terms of helicity
amplitudes. The helicity eigenstates are definite polarization states of the initial
elementary fermions. At an e+e“ collider this means we simply include the helicity
amplitudes corresponding to the chosen polarization. At a pp collider, we may p0.
larize the initial hadron; however, since we can only calculate amplitudes involving
the elementary fields of the SM, i.e. “partons” within the proton, therefore to relate
the polarization of the beam to the helicity of the interacting constituent “partons”

from the proton, we will need to make use of polarized parton distribution functions

14

(ppdf’s), which give the probability to find a gluon or quark in a polarized proton.
These functions will be introduced in Chapter 3.

1.4 The Organization of This Thesis

Part I of this thesis is concerned with comparing the sensitivities of various collider
experiments to anomalous electroweak triple-boson vertices. In Chapter 2 we will
estimate these sensitivities at two e+e“ colliders. The purpose of Chapter 3 is
to determine to what extent polarizing a p1“) collider, the Fermilab Tevatron, can
enhance its sensitivity to anomalous couplings. In Chapter 3 the importance of the
polarized parton distribution functions will become apparent. Chapter 4 examines
how these might be constrained by studying single W production at a polarized pp
collider.

Part II of this thesis deals with supersymmetry (SUSY). After a brief introduc-
tion to SUSY, we explore the phenomenology of a very light “gravitino” particle,
which arises in certain mechanisms of SUSY-breaking. The role of polarization in

this part will be to eliminate a background process.

Part I '

Anomalous Electroweak
Triple-Boson Couplings

15

Chapter 2

Probing Anomalous Triple-Boson
Couplings with e_e+ —> W—WJr

2.1 Introduction

Processes which produce electroweak bosons offer the most obvious testing ground
for the triple-boson vertices. We will focus here on W pair production, as it is simple
to calculate and has a relatively large cross-section. In part I of this work we are
interested in the potential of beam polarization for improving sensitivities to triple-
boson couplings. The Tevatron, with center of mass energy J3 2 2 TeV, is the
existing collider with sufficient energy to produce a W pair. Before turning to the
p13 collider calculation, we consider production of a pair of W’s from e“e+ collisions
at LEPII and at the proposed Next Linear Collider (NLC). This calculation will be
somewhat simpler than the p13 calculation, so the analysis may be more transparent.

The authors of [18] estimated the sensitivity of LEPII to anomalous triple-boson
couplings through W pair production, without explicitly calculating the decay of
the W’s. We will reproduce their calculation here, including the decays and realistic
cuts, and focusing on one particular anomalous coupling, A5,. They assumed for
LEPII J3 = 190 Gev and a luminosity of 0.5fb“1yr“1, as will we. We will not
consider the partial polarizability of the LEP beam. The proposed NLC would have
J3 = 0.5 TeV, very high luminosity (50 fb“1yr“1), and better than 90% polarization

16

17

of the electron beam [19]. The authors of [20] estimated the sensitivities to anoma-
lous couplings at the NLC, using a x2 analysis of a triple-differential cross-section.
We will instead use a single-differential cross-section, the cosine distribution of the
final electron, which was identified in [18] as the most sensitive variable to the
anomalous couplings. We will thus be able to directly compare our LEPII and NLC
sensitivities. In our NLC calculation we will assume 100% polarization.

In the pp analysis in the next chapter we will use only the total cross-section
to calculate limits on the anomalous couplings, because the uncertainties in the
polarized parton distribution functions (ppdf’s) are incompatible with the precision
we would hope to achieve by a x2 fit. For the e+e“ case at hand, we will also estimate
sensitivities using the x2 statistic to rule out values of anomalous couplings for which
the final state electron cosine distribution, do / dcos9e, deviates significantly from the
standard model distribution. By comparing the sensitivities from the x2 calculation
to those from the total cross-section calculation, we can get an idea of how the
sensitivities in the pi) case could improve when the ppdf’s are better constrained.

A realistic study must include the decay of the W bosons. Following references

[18, 20] we will consider the semi-leptonic decay channel:

e—e+ —) W-W+ —) I‘qu’.
The leptons included are l = e, p, and the hadronic decay channel includes q = u,c.
At the NLC we calculate cross-sections with a left-polarized electron beam, and at
LEPII we consider the unpolarized process. We choose this decay mode because

allowing both W’s to decay leptonically would reduce the cross-section, whereas the

four-jet hadronic mode would have a large QCD background.

18

2.1.1 Calculation

To calculate the cross-sections, we began with the helicity amplitudes corresponding
to the Feynman diagrams in Figure 2.1. The definitions of our helicity eigenstates
are collected in Appendix B.1. Helicity amplitudes are most convenient in a study
involving polarization, as at the NLC, because the helicity amplitudes correspond
to definite polarization states. To calculate a polarized cross section, we include
only the helicity amplitudes corresponding to the chosen polarization, instead of
summing the amplitudes-squared for every helicity.

To calculate a cross-section from the amplitudes, we must integrate the phase
space of the process. We perform this integration using a Monte-Carlo event gen-
erator based on the VEGAS routine [21]. We store all the simulated events in an
HBOOK NTUPLE, from which we can project any differential cross-section.

Figures 2.2 and 2.3 compare the differential cross section in W“ rapidityl of our
Monte Carlo calculation to an analytic calculation based on the amplitudes of [18].
The Monte Carlo distribution is uncut at this stage, but includes the W decay with
a narrow width.

The cuts we imposed on the cross-section calculations are shown in Table 2.1.

 

AR = \/(A¢)2 + (A1,)2 is a measure of the separation between two particles, where
Ad; and A1] are their separations in azimuthal angle and rapidity, respectively. Re-
quiring AR to be greater than 0.7 ensures that after a final parton showers into a
jet it will still be distinguishable experimentally from the other final state particles.
The cuts on AR and on the absolute value of rapidity are imposed on all final state

particles except for the neutrino. The cut on transverse momentum (PT) is imposed

 

lRapidity y is a measure of polar angle defined by y 5 %ln fit, where P, = Pcos 0, 0 being
the angle from the s-axis. The s-axis is defined to point in the direction of the proton beam at
pp colliders, and the electron beam at e+e“ colliders. Zero rapidity corresponds to the “central”
0 = 90 region; large positive or negative rapidity approaches 0 = 0 or 0 = 180 degrees, respectively.
(Throughout this thesis c and A are set equal to l.)

19

(-

 

 

 

47
Figure 2.1: Diagrams for e“e+ —-> W'W+ —* l-l'qu-’

 

 

 

 

 

 

 

 

 

 

hI j W I I I I I I l I I I I I I I I I I Ifi I I I I I I“

- -1

50 — _.

_ J

" “i

40 — —

.8. _ .l

v - 4

30 — '—

4. - / -

CU - q

\ L- / -t
b

10 '— —

[— -.

o P‘ ‘ l I I l l l l l L L l l l 1 M J l l l l l l l L l '1]

-0.50 -0.25 0.00 0.25 0.50

Yv

Figure 2.2: Rapidity distributions of the W“ at LEPII from analytic amplitudes
(smooth) and uncut Monte Carlo calculation (histogram).

20

 

 

 

 

 

 

40 I I I j I I I I I I I I I f I I 1 I T II I
~ ' 4
30 — -
[— —4
a t ~
0'. '- -1
v
r- J
20 — —l
-
>s ~ ..
‘U
\ .. ..
b
'6 _ .4
y- u
10 — _
l. -
o J l l l

 

Figure 2.3: Rapidity distributions of the W“ at NLC from analytic amplitudes of
(smooth) and uncut Monte Carlo calculation (histogram).

Table 2.1: Final state cuts.

 

 

 

 

LEPII NLC
Minimum PT 10 GeV 20 GeV
Maximum Rapidity 1.5 1.5
Minimum AR 0.7 0.7

 

 

on all final state particles.

Although these cuts reduce the total number of events, they enhance the depen-
dence of the cross section on anomalous couplings. Figure 2.4 shows the rapidity
distribution of the final electron at LEPII for the SM and for A5, = 0.5, with-
out cuts. Where the cross section is large, at high rapidities, the dependence on
anomalous coupling is small. The largest deviation from the SM occurs at low and
central rapidities, where the cross section is smaller. This means that when we
cut out events with high lycl, we cut out mostly events from the region with weak

dependence on anomalous couplings. Furthermore, since the effect of anomalous

21

couplings increases with energy, the low PT region, which we cut out, is less sen-
sitive than the high pt region. Figure 2.5 shows the electron rapidity distributions
after implementing the cuts of Table 2.1. Still, the cuts we chose are probably not
optimal, so perhaps the limits we find could be improved somewhat.

With these cuts, the total SM cross section is 3.43 picobarns at LEPII, and
0.66 pb at the NLC (with left polarized electron beam). The total cross section as
a function of anomalous photon couplings is shown in Figures 2.6 and 2.7. This
dependence of the cross section on anomalous coupling will allow us to rule out

couplings for which the cross section is significantly different from the SM.

22

 

 

OJ

008

006

004

002

IIIIITIII‘IIITIIWIIITIYIITIIIIII

 

 

 

 

—3 -2 —1 O 1 2 3

 

 

 

Figure 2.4: eta/dye at LEPII, without cuts, for e+e“ ——i W+W“ -> e“17¢u,cdd_,cd.
Solid line is SM, dashed is A5, = —0.5.

23

 

 

0.1 L

(Db)

/dy.

do

0.08 _

0.06 -

0.04 —

0.02 —-

 

 

 

O 111 111 #4 L1 LJIJI l l l l l l 111 l 1 L

—3 -2 -1 O l 2 3

 

 

 

 

Figure 2.5: eta/d3;c at LEPII, with cuts, for e+e“ -—> W+W“ —> Figural“. Solid
line is SM, dashed is A5, = —0.5.

24

2.1.2 Estimation of Sensitivity to Anomalous Couplings

Here follows an explanation of how we estimated the sensitivity of W pair production
to anomalous triple-boson couplings. We assume the anomalous couplings are in
fact zero; the limits we place on their allowed values from hypothetical experiments
at LEPII and N LC indicate the sensitivity of these experiments to the anomalous
couplings.

For simplicity, we focus on the A5, coupling, assuming that A, 2 A52 = A2 = 0.
To estimate the experimental limit that could be placed on an anomalous WW7
coupling, we must estimate the uncertainty in an experimental measurement of the
cross section a’. For this analysis we simply assume that the standard deviation in
the number of events N is EN = \fN, i.e. that N obeys Poisson statistics. (For
our purposes we do not include the efficiency of the detector.) The measured cross
section would be a' = N / L where L is the integrated luminosity. The 3-sigma upper
limit on 0' (i.e. 99.7% confidence level) expected from Poisson statistics would be
(N + 3\/N ) / L. Thus the measurement would rule out a cross-section larger than

503 = 3% . (2.1)

Similarly, for a 1-sigma (68.3% C.L.) or 2-sigma (95.5% C.L.) limit, we replace the

0' + 603, where

3 in this equation with a 1 or a 2, respectively.

To estimate the limit that could be placed on A5, or A,, assuming the anomalous
couplings are in fact zero, we compare the uncertainty 603 to the variation of the
calculated a as a function of the anomalous coupling. At the 3-sigma confidence

level A5, would be in the range with
|a'(A5,) — a'(A5, = 0)] < 503 . (2.2)

We obtain the results in Tables 2.2 and 2.3, for the experimental limits that

25

 

 

 

 

_ I I I I I I 1% I I I I I I I I l I ']

4.0 —- _

3.6 —- —

fa _ ..
3

E '- -4

3.6 — .—

“ 'l

- .J

— J

" l l 1 1 1 1 l 1 1 1 1 1 1 1 1 l l d

—1 0 1 2
Ax?

Figure 2.6: Total cross section at LEPII for e+e“ —§ W+W“ —+ l“i7q§’ vs. A5,.

 

-t
-(
—t
.4
q
.4
'i
.4
—1
.4
fi
.1

0.675

0.670

0.665

0.660

9m (Pb)

0.655

0.650

WTIIIIIIIWIIITIIIIIIIIIIIIII

 

 

1 MA I L J l 1 1 l l 1 l 1 1 I 1 l l M l 1

-0.010 —0.005 0.000 0.005 0.010

D-
#-

 

Figure 2.7: Total cross section at NLC for e+e“ —> W+W“ —> l“i7q§’ vs. A5,.

26

Table 2.2: LEPII limits on A5, from the total cross section.

 

 

LEPII Total Cross Section Limits
1 Sigma 2 Sigma 3 Sigma
A5, +1.35 +1.52 +1.68
—0.23 -0.41 —0.56

 

 

 

Table 2.3: NLC limits on A5, from the total cross section.

 

 

N LC Total Cross Section Limits
1 Sigma 2 Sigma 3 Sigma
A5, +0.0031 +0.0062 +0.0095
~0.0029 —0.0058 -0.0085

 

 

 

 

could be set on the anomalous couplings of the photon. We emphasize that the
numbers in Tables 2.2 and 2.3 do not include possible experimental uncertainties.

This procedure gives us limits on anomalous couplings, assuming the anomalous
couplings are in fact zero. In an experiment we would want to find the most probable
value of the couplings based on our measurement. A glance at Figures 2.6 and 2.7
shows that for any measured value of the total cross section there correspond two
values of anomalous coupling. This ambiguity possibly can be avoided with the
polarization capability of the N LC.

Figure 2.8 shows the total cross section as a function of A5, for each polarization,
for the process e+e“ —+ W+W“ at the NLC. We calculated these cross sections from
analytic amplitudes given in [18]. To see how these plots may resolve an ambiguity
in A5,, imagine we perform the experiment with a left-handed electron beam and
find a cross section of 15 pb. This corresponds to A5, of about -0.1 or 0.26. If we
then measure the right-handed cross section, a value of 1 pb means A5, = -0.1;
while 2 pb means A5, = 0.26. In other words, after finding two possible values of
A5, from the left-handed run, the right-handed run will distinguish between them

 

 

 

 

 

25 I
20 ~ _
L I
l- ..
._. 15 —- -—
.D - .1
3 p- «I
f" r- .1

9 _
D '1
10 — \ _
_ \ d
- \ .
t- \ / ‘
r / ‘
\

5 —- \ / —
b- \ / d
- \ / ‘
_ \ / ‘
_ \ / .

l l 1 l l l l l l l\ N l [Al/l l l l 1 1 l I l

-0.50 -0.25 0.00 0.25 0.50

At.

Figure 2.8: Total cross section at NLC for e+e“ —> W+W“ vs. A5,. The solid and
dashed curves are for left and right polarized electron beams, respectively.

if the right-handed cross section is significantly different at the two values of A5,.

Figure 2.8 also shows why we chose to study the left polarized process at NLC.
The right-handed cross section is much smaller, without showing any greater de-
pendence on the anomalous couplings. The right-handed cross section is smaller

because the t-channel diagram in Figure 2.1 is dominant, and vanishes in the right-

polarized case due to the pure left-handed coupling of the W boson.

28

2.1.3 Using x2 to Estimate Sensitivities to Anomalous Cou-
plings
We also found limits on anomalous couplings by a x2 analysis. The x2 statistic
provides a measure of the deviation of a difierential cross section from the SM
distribution. The strategy is similar to the above. Instead of limiting the deviation
of the total cross section from the SM, we limit the x2.

The reduced chi squared is defined by [22]

(0k - Etc)2
X3111“ _ (112——

El:
where d is the number of degrees of freedom. For our purposes, (I will be the

(2.3)

number of bins into which we histogram a differential cross section. 0,, is the
number of events observed, and E1, is the number of events expected, in the le‘”
bin. The subscript cap is a reminder that this quantity is defined in terms of an
experiment. In our theoretical study we are not making an actual observation, but
merely simulating a possible experiment. N = Lo relates the number of observed
events to the luminosity and the theoretical cross section. Using this formula to
relate the numbers of events in Eq. (2.3) to cross sections, we define a theoretical

reduced “chi squared” in terms of a binned differential cross section:

 

)Ith(A"7): £§1(UAM’U;M:SMk)a (2'4)

which is a function of the anomalous coupling. a". is the integrated cross section
in the k‘“ bin. The quotes around “chi squared” indicate that it is a theoretical
quantity, not distributed according to the chi-squared distribution— a distribution
which refers to the results of experiments. We will nevertheless treat the theoretical
chi squared as a deviation from the SM that could be measured if the anomalous
couplings were not zero. Then, assuming the anomalous couplings are zero, we rule
out those anomalous couplings that correspond to “unlikely” chi squareds (at some

confidence level).

29

We will choose a distribution, calculate the chi squared as a function of anoma-
lous coupling, and rule out large chi squareds. The authors of [18] found that the
angular distribution of the final electron is most sensitive to the couplings we are in-
terested in, so we will use the cosine distribution, do/d cos Be, as they did. Figure 2.9
shows this differential cross section at each collider for two values of A5,.

For calculational convenience and to remove any bias in )2,”,,(A5,) due to Monte
Carlo fluctuations, we fitted our Monte Carlo data to a smooth function. The func-
tional form we choose is cubic times Gaussian in the electron cosine and quadratic
in anomalous coupling. We rebin the smoothed distribution to use Eq. (2.4).

In principle we can bin our differential cross section into an arbitrary number of
bins. The sensitivity to anomalous couplings of a distribution is dependent on how
we bin that distribution. One restriction on our binning is that the bin widths must
be large enough so that the number of events corresponding to the cross section in
a particular bin is statistically significant. This means the number of bins cannot
be too large. This turns out not to be a limiting factor for the process we are
considering. We calculated sensitivities for various numbers of bins, and found that
dividing do/d cos 9, into 2 or 3 bins is most advantageous. This is initially suprising,
as one might imagine that dividing the distribution into more bins increases our
sensitivity to detailed features of the distribution. However, if there are no “features”
to be distinguished in the anomalous distribution, increasing the number of bins is
actually a disadvantage. This conclusion can be made plausible by considering the
chi-squared for an experiment whose observed and predicted distributions differ only
by a constant. In this case the highest value of 52:” will be obtained for only one bin,

which is equivalent to calculating the squared deviation of the total cross section.

30

 

 

  
 
    
 

    

 

 

 

 

 

 

 

s s:~
g 4 :_ LEPII Unpol.
{ :
e : ,
P 3 5 Sohd Axe/=0
E 1335th AK,=-1
2 r
1 :__ ______________ ‘r—
o I. l l l L l l l l l l l l l l l l l l l l l l l I. l 1 L J J. I l l 4 l l
-0.8 -O.6 -o.4 -0.2 0 0.2 0.4 0.6 0.8
cost
A 1.6
n :
3 14 '—
§ ,2 =_ NLC Left Pol.
8 ' :
S 1 3—
e : .
0.8 _"_'_ Solid M1=O
0.5 E. Dashed AK,=0.0I
I—
0.4 :—
0.2 :—
0 .- L 1 1 J 1 LJ 1 L JA 1 l L l l L l I l 1 l l l 1 L L l l 1 l l l
—o.s -0.6 -0.4 -o.2 o 0.2 0.4 0.6 0.8
cord

 

 

 

Figure 2.9: Differential cross sections in cosine of the lepton angle at LEPII and
NLC for two values of A5,.

31

Table 2.4: LEPII limits on A5, from )22 of cos 9., distribution.

 

 

LEPII 722 Limits on A5,

(1 1 Sigma 2 Sigma 3 Sigma

1 +1.35 +1.52 +1.68
—0.23 —0.41 —0.56

2 +0.35 +1.57 +1.73
-0.23 -0.36 -0.46

3 +0.42 +1.60 +1.75
—0.26 —0.37 —0.46

 

 

 

 

 

 

The highest value of the reduced chi-squared, however, does not necessarily cor-
respond to the greatest sensitivity to anomalous coupling. We rule out a particular
value of A5, if 2,2,,(A5,) is greater than some cut-off 522(d, CL), which is given by
the chi-squared distribution and depends on the number of bins d and the desired
confidence level. The question of the optimal number of bins to use is therefore
answered by the detailed calculation.

Plots of 523,, versus anomalous coupling are shown in Figures 2.10 and 2.11, for
2 bins. We report estimated limits on A5, from LEPII and the NLC in Tables 2.4
and 2.5; for 1, 2, and 3 bins. We note that higher numbers of bins than 3 give poorer
limits on A5,. Also note that the limit calculation for a chi-squared with one bin
is equivalent to the calculation from deviations in the total cross section discussed

above.

2.1.4 Conclusions

We estimated sensitivities to A5, at LEPII and the proposed NLC. W pair
production is sensitive, at the 3 sigma level, to anomalous couplings of order 1 at

LEPII, and less than order 0.01 at the N LC. We conclude that the NLC would be

able to probe the loop-level corrections to the triple-boson couplings.

32

 

 

 

 

 

c-4
.1
——l
.1

 

y.—
p-

l l l l l l l

lLLllllllllllllJllllllll

 

 

—0.010 0.010

 

Figure 2.11: 2%,, at NLC for e+e“ —> W+W“ —4 l“z7q§’ vs. A5,.

 

33

Table 2.5: NLC limits on A5, from 522 of cos He distribution.

 

 

NLC 5&2 Limits on A5,

d 1 Sigma 2 Sigma 3 Sigma

1 +0.0031 +0.0062 +0.0095
-0.0029 -0.0058 -0.0085

2 +0.0038 +0.0055 +0.0077
-0.0036 -0.0052 -0.0071

3 +0.0041 ' +0.0062 +0.0083
-0.0039 -0.0058 -0.0076

 

 

 

 

 

 

The LEPII sensitivities we find are worse than the sensitivities estimated in [18]
to the anomalous Z couplings. This disparity is not surprising, since our choice
of one W decay channel reduces the total number of events. Those authors ex-
pected about 4000 events at LEPII, whereas after cuts we find about 1600 events.
Assuming similar sensitivity to photon and Z couplings, the reduction in statistics
corresponds to a reduction in estimated sensitivity. In fact, a W pair production
experiment cannot distinguish an anomalous Z coupling from an anomalous photon
coupling from the total cross section alone. In the event of a non-standard exper-
imental result, one may resolve this ambiguity with complementary data from an
unambiguous process, such as W plus photon production, which only depends on
the photon couplings. (In the next chapter we will estimate sensitivities to W plus
photon production at a pp collider.)

We find that using a x2 analysis generally improves our sensitivity only marginally,
by about 15% or less. It turns out that the optimal number of bins for a chi-squared
analysis to limit A5, is only two, because for the very small non-standard cou-
plings we are simulating, there is little difference in shape between the Standard
and non-standard differential cross sections.

In the next chapter we estimate sensitivities to anomalous couplings at the

34

Tevatron, and at a proposed Tevatron with a polarized proton beam. The fact
that the protons and anti-protons are not elementary particles will add a level of

complication to the calculation.

Chapter 3

Probing Anomalous Triple-Boson
Couplings with Polarized
Proton—Antiproton Collisions

3.1 Introduction

In the last few years the SPIN collaboration has shown in various technical notes
that it is feasible to polarize the proton beam, longitudinally or transversely, during
the colliding mode of the Tevatron [23]. Taking proton polarization as a possibility,
we examine one of the possible physics topics that could be pursued with such a
beam configuration - to study tri-boson couplings of the weak gauge bosons. (Other
interesting physics topics involving polarization at the Tevatron collider can be found
in Ref. [24].) I

In this chapter we consider two processes in proton-antiproton collisions

19+13-->W+ (+1“ Vz)+7,

p+f)—>Wi (a£i(§})+WI(—+2jets) .

 

'After this work was completed, we learned that a polarized colliding beam is less likely to
be built in the near future at the Tevatron. However, since a polarised collider is technically
feasible, we believe it is still interesting, and potentially useful, to explore theoretically the possible
experiments with such a machine.

35

36

For t we include both e and p. The unpolarized cross-section for pp —t W“7 is
equal to that for pp -+ W+7 in a CP invariant theory such as £3, the tri-boson La-
grangian defined in Section 1.2. However, we are interested in scattering of polarized
protons, for which W+7 is more interesting than W“7, as we will see below.

The purpose of this chapter is to explore the experimental search for anomalous
couplings in proton-antiproton collisions, assuming the protons are longitudinally
polarized. The antiprotons are assumed to be unpolarized. With the Tevatron
collider in mind [23, 24], we consider the center-of-mass energy equal to 2 TeV.

The reaction cross-section for a process involving the WW7 or WWZ coupling
depends on the longitudinal polarization of the proton through spin-dependent par-
ton distribution functions. The coupling of W5 to quarks (ud or other flavor com-
binations) is a V—A (vector—axial vector coupling) interaction, so the parton-level
cross-section depends strongly on the helicities of the quarks: for massless quarks a
Wi couples only to left-handed (L) quarks and right-handed (R) antiquarks. Thus
the parton-level process depends strongly on helicity. The question is whether the
proton process depends strongly on proton helicity. If a polarized proton contained
equal parton densities of left-handed and right-handed quarks, then the proton
cross-section would not depend on the proton helicity. However, we know that the
densities of L and R quarks are not equal for polarized protons. Therefore, the pp
cross-section will be different for left-polarized and right-polarized protons.

Our calculations of the polarized-proton cross-sections depend on polarized par-
ton distribution functions (hereafter abbreviated ppdf’s), and these are only known

with limited accuracy. The ppdf’s are defined as follows: For any parton type f, we
define

5(2) = gush/W»

= density of L (or R) partons in a L (or R)‘proton,

37

12(2) = gas—Are»

= density of L (or R) partons in a R (or L) proton,

where :c is the fraction of the proton momentum carried by the parton. There are

nine different parton types

f=uvalsdualsusea=aseasdsea= seaagss=§sc=asb2bstzt°

In the ppdf’s we used, the u and d sea distributions are equal, but different than

the s and c distributions, and the b and t distributions are zero
u,ea(:c) = dm(2:) ; b(z) = 0 ; t(:s) = 0 .

Figure 3.1 shows the polarization dependence of the ppdf’s we used, by plotting
a: A f(:1:) for several parton species. The ppdf’s depend on momentum scale Q; i.e.
f5: = fi(a:,Q2). Fig. 3.1 corresponds to Q=80 GeV. (These ppdf’s are calculated
from a program based on Morfin-Tung parton distribution functions [25, 26].) The
ppdf’s have been measured, to some limited precision, from polarized deep-inelastic
lepton scattering [27]. For example, recent data from the Spin Muon Collaboration
(SMC) at CERN provide a measurement of the polarization difference, integrated
over :1: and weighted by eg/e2 [28]:

r1 62 1 -
I E .1. 2’) j,- [Af(=v,Q’) + 23:32.00]

' 0 f=u,d,s,c

0.142 :l: 0.008 i 0.011,

where the momentum scale is Q2 = 10 GeV’. The ppdf’s used in our calculations

have

I = 0.138 for Q2 = 10 GeV2 ,

I = 0.163 for Q2 = (80 GeV)2 ,

38

1.00

 

0.75

 

 

 

 

 

0.50
goes
<
>4
0.00
dal
bllllllLiJlivllll[IAIJJ—thllLle
0.0 0.1 0.2 0.3 0.4 0.5 0.6
X

Figure 3.1: Polarized parton distribution functions. The curves are zAf(:r) us :1: for

parton types um; , dual , u,ea(= duo) , g, which are the most important partons in
our calculations.

where the Qz-dependence is determined by renormalization group equations. For
Wi production the relevant momentum scale is of order Mw. The spin-dependence
of the quark densities is rather small, as indicated by the small value of I ,l (30
one question that motivates our study is whether the cross-sections for these Wi-
production processes depend significantly on the proton helicity.

In Section 3.2 we calculate the cross-section for the process pip —» W+7, where
A = L or R denotes a left-handed or right-handed proton. This process is sensitive
to the WW7 anomalous couplings. In Section 3.3 we consider the process mp —§
Wi WI, which is sensitive to both WW7 and WWZ anomalous couplings. The
purpose of these calculations is to explore whether polarization of the protons can

increase the sensitivity of measurement of anomalous couplings.

 

II = .138 is small compared to the prediction I = .17, based on the Ellis-Jafi'e sum rule, which
assumes vanishing strange sea polarisation [29].

39

3.2 W+7 production

The Feynman diagrams for the process p + p —> W+ + 7 are shown in Figure 3.2.
One diagram has a WW7 vertex, so the cross-section depends on the anomalous
photon coupling parameters A5, and A,. This process can be used to place lim-
its on the anomalous couplings. Calculations of the unpolarized cross-section with
anomalous couplings were described in References [30] and [31]. For polarized scat-
tering we expect the cross-section for left-polarized protons to be larger than for
right-polarized protons, because the produced W+ line is always connected to a
left-handed quark line. We only consider the process pip -’ W+7, and not W“7,
because the former is more sensitive to the proton helicity. A W+ comes from a u
quark, whereas a W“ comes from a d quark. The helicity dependence is stronger
for 11. than d in a proton, as seen in Figure 3.1. To investigate whether polarizing
the proton beam would yield better limits on A5, and A,, we have calculated the
polarized and unpolarized cross-sections. The results of this study are reported in

this section.

3.2.1 Method of Calculation

The cross-section for p; + p —) W+ + 7, where A is L or R for left-handed or

right-handed protons, and with subsequent decay W+ —-> I“ + 11;, is expressed as

fl

a(,\) = i... dzdz’[&LR(2P1,2:'P2)ui(a:)d(z') (3.1)

+5'LR(€D’P2: 3P1)J=F(3)“(z’)l

where the notation is as follows: The parton cross-section &LR(p1,p2) is for the
process uL(p1)+ dR(p2) —> W“ +7. The upper sign on ui(:c) and d;(:c) is for A = L

and the lower sign is for A = R. The factor of % averages over the spins of the

40

 

 

 

 

 

Figure 3.2: Feynman diagrams for the process ud —+ e+u7.

unpolarized p. The first line in Eq. (3.1) corresponds to u,d coming from p,p', and
the second line corresponds to d,u coming from p, 13', respectively. a: and z’ are the
parton momentum fractions in the proton and antiproton respectively. The parton

distribution functions are, for example,

ui(:c) = u quark with same/opposite helicity as p
d(:c') = d quark in unpolarized p
d;(:c) = d quark with opposite/ same helicity as p
11(3') = u quark in unpolarized 5.
We also add the contribution for the parton process c + 3 -—» W+ + 7, which is,
however, small. (We ignore Cabibbo-Kobayashi-Maskawa mixing in this work.)

Finally, we add the cross-sections for two lepton decay modes of the W+; that is, t

can be either e or p.

41

The parton cross-section is calculated from helicity amplitudes for the reaction.
In this reaction there is only one nonzero helicity amplitude, with Au = L and
Ag = R, because we approximate the quark masses as 0. We calculate the helicity

amplitude M [,3 analytically. Then the parton cross-section is

r
a“. = d<I>|MLR|2 (3.2)

where f d<I> indicates a phase space integral.

The phase space and :c, z’ integrations are performed by a Monte-Carlo program,
based on the Vegas Monte-Carlo integration routine [21]. The style of the full Monte-
Carlo program is the same as the program PAPAGENO [32].

The kinematic cuts we impose on the final (1' and 7 are:

 

rapidity I’ll] < 3 9 [777] < 3 : (3'3)
transverse momentum pm > 20 GeV , p1, > 20 GeV , (3.4)
AR 2 \/(A1,)2 + (A03)? > 0.7 , (3.5)

where AR is the separation of the PL and 7 in 1] — 96 space. The only cut on the

neutrino is a transverse momentum cut p1,, > 20 GeV; that is, we require
ET > 20 GeV . (3.6)

At the parton level, there is no background to this process from other interac-
tions, as long as we require the W+ to decay to leptons. There is an experimental
backround due to confusion between jets and photons in the detector [31]. How-
ever, we do not consider the experimental background here, because our interest is

to examine the effect of proton polarization, compared with the unpolarized case.

 

2.0 I I I I I I I I I I IN I I I I I I I

_ C] p(L) _
- <> p(R) ‘
= X unpolarized (CTEQ2) “

V \P”!

 

 

 

 

Figure 3.3: Total cross-section (with cuts in Eqs.(3.3)—(3.5)) for polarized protons
us anomalous coupling A5,, assuming A, = 0. The unpolarized cross-section was
calculated separately using CTEQ2 parton distribution functions.

3.2.2 Results

Tables 3.1 and 3.2 show the results of our calculations: the W+7 production cross-
section for polarized and unpolarized protons with different values of anomalous
couplings A5, and A,. The cross-section includes the branching ratio 2 / 9 for W+ —+
(+114, where the decay modes I = e and l = p. are added. (The branching ratio factor
2 / 9 is included in all cross-sections reported hereafter.) The cross-section is smallest
for the Standard Model values A5, = 0 and A, = 0. The cross-section depends more
strongly on A, than on A5,. Figures 3.3 and 3.4 show plots of the cross-section as
A5, assuming A, = 0, and us A, assuming A5, = 0. As expected, the cross-section
is larger for left-handed protons; 0'(L) is roughly 3 times U(R), and so roughly 1.5

times the unpolarized cross-section. The unpolarized cross-section is by definition

equal to %(o(L) + o(R)).

43

 

 

 

 

8 .- I I I I I I— I I I I I I I I I I I T I
II
c [31304)
°p(R)
6 Xunpolarized (CTEQZ)
3 l.
3' 4’ "
b l'
l.
2 in—
L
o _ I l I I I I I I l l l l I I I I I I
-2 -1 0 1 2

Figure 3.4: Total cross-section (with cuts in Eqs.(3.3)—(3.5)) for polarized protons
vs anomalous coupling A,, assuming A5, = 0. The unpolarized cross-section was
calculated separately using CTEQ2 parton distribution functions.

Figures 3.3 and 3.4 show only the total cross-section, for the cuts specified in
Eqs. (3.3)-(3.5). Analysis of differential cross-sections with respect to relevant kine-
matic variables—such as the x2 analysis of the previous chapter—may provide more
precise tests of anomalous couplings [31]. For example, Figure 3.5 shows the dis-
tribution of p7, for polarized proton scattering, with A5, = 0 (solid line) and
A5, = —1 (dashed line). The shapes of the distributions are similar for left- or

right—handed protons, but there is an overall difference of magnitude.

44

 

 

 

 

 

 

 

 

 

 

 

E
- r-
S
3 -1 -1
'o L’ E
B ' t
'0 .
- -2
10 :- 10 r
i t
l- n-
-3 .3
10 r 10 .—
_ ;
l I I I l I I 1 I l l I 7
0 50 100 150 0
pn(GeV)

 

Figure 3.5: Distribution of photon transverse momentum p1,. The solid line is for

A5, = 0, and the dashed line is for A5, = —1; in both cases A, = 0.

45

3.2.3 Limits on A5, and A,

Tables 3.1 and 3.2 are results from the Monte Carlo calculations of cross-section
vs anomalous couplings. There is a fairly large effect of proton polarization: 0'(L)
is generally about a factor of 3 larger than 0'(R). But to see whether an experi-
ment with polarized protons would yield a significantly better measurement of the
anomalous couplings, we must estimate the experimental limit that could be set on
A5, or A,, for a given integrated luminosity. The sensitivities to these anomalous
couplings are estimated using the same procedure as in Chapter 2. That is, we as-
sume the number of events N detected is predicted by the SM, and an uncertainty
in this number given by W. (This is an underestimate of the experimental un-
certainty; for instance, it does not take into account the experimental background
of jets misidentified as photons.) The anomalous couplings which correspond to a
three-sigma fluctuation of the number of events are the largest anomalous couplings
consistent with a SM result, at the 99.7% confidence level; in other words these
values of the anomalous couplings are the 30' limits on the couplings.

The left-polarized proton provides a better limit on A5, or A,, because it has
a larger cross-section. The improvement in precision from left-polarized protons,
compared to unpolarized, is not very great, because 0'(L) is only about 1.5 times
larger than 0(unpolarized), and because the precision on 0' is only proportional to
01/2. The result is that the total cross-section for left-polarized proton scattering
can provide a better limit on A5, or A, than that for unpolarized scattering, better
by about 10 to 20 %.

Another way to consider the effect of proton helicity is to calculate the left-right

asymmetry, defined by

_ a(Ll—"(12)
A- 0'(L) + 0'(R) .

An asymmetry measurement may be accurate experimentally because systematic

 

(3.7)

46

errors cancel in the ratio. The possible range of A is —1 S A S 1, and A is equal to 0
in a left-right symmetric world. For the parton process qyj’ —-> W+7, we have A = 1
because only left-handed quarks contribute. For the proton process, the asymmetry
is reported in Tables 3.1 and 3.2; we find A z 0.5. However, A depends only weakly
on A5, and A,: A varies by 0(10%) over the range of anomalous couplings consid-

ered. Thus a measurement of A to determine A5, or A, would require high statistics.

3.3 W+W“ production

The process pp—)W+W“ provides a way to test the Standard Model WWV vertices
for both V = 7 and V = Z0, although we may not be able to distinguish the two
couplings from each other unless we can compare to results from the W7 production

process discussed above. We consider the electroweak process
p(A)+p—>W+(—>Z+u¢)+W“(—+d+1“1), (3.3)

and also the process in which the W“ decays leptonically while the W+ decays to 2
jets. The diagrams for the production of two W’s include two electroweak vertices,
in contrast to the background diagrams (discussed below), which only contain one
electroweak vertex. The complete set of Feynman diagrams with the final state Zydfi
includes diagrams that do not have the form of W+W“ production. Figure 3.6 shows

the complete set of Feynman diagrams for the parton process
u+fi-—>W++d+fi, WithW+—+Z+V(; (3.9)
there is similarly a set of diagrams for the process
d+J—>W++d+0, withW+—»Z+u¢. (3.10)

Also, there are similar diagrams for production of W“ +u+d, with W“ —» (+175.

(The complete set of diagrams for the final state fade], or lfiud, includes additional

7.Z ’7 P

u (I

 

     

If It

(I
(l
1.1 —4— _
II
+ ,'+
II “M
u

u—.— C
M d 11

Figure 3.6: Complete set of electrdweak diagrams for u+fi —) d+i1+W+ (—) Z + V1) .

a-1

 

diagrams in which the leptons are not decay products of a single narrow Wi.)
These parton processes involve the WWi7 and WWZ vertices in some Feynman
diagrams, so the cross-sections depend on anomalous WWV couplings, i. e. the
parameters A5,, A5 2, A,, A2. In this section we study the cross-section as a function
of these non-Standard parameters, for polarized protons. The purpose is again to
see whether an experiment with a polarized proton beam would yield a stronger test
of the electroweak triple-boson vertices.

Some, though not all, of the Feynman diagrams in Fig. 3.6 have the form of
W+W“ production, followed by decays of the W’s, one leptonically and the other
into two jets. At 2 TeV center-of-mass energy, the cross-section is dominated by
the W+W“ production. Therefore, as explained further below, we approximate the
cross-section by W+W“ production. The cross-sections for (Bud and fluid final
states are equal in this approximation, because either W is equally likely to decay

leptonically.

48
A related process is W35 Z0 production, where the Z0 decays to 2 jets, e. g. ,
u+d—>W+(—>l++iq)+ZO(—>q+rj) . (3.11)

In our calculations we ignore the W5: Z 0 production. In a theoretical calculation
the W+ W“ production is distinguishable from the W520 production. However,
in an experiment these two processes are tangled together, because they are both
observed as W5 + 2 jets. In our W+W“ calculation we impose a kinematic cut on
the invariant mass M2,- of the 2 jets, making M2, approximately equal to the W5:
mass; specifically we take M2,- between 70 and 90 GeV. (This cut reduces the QCD
background of W“: + 2 jets, as discussed in the next subsection.) But even with this
cut on M2,- there would still be an overlap between W+W“ production and WIZ0
production. The purpose of our calculation is a theoretical study of the effect of
proton polarization on the cross-section. A complete analysis of experimental data
would need to include both the W+W“ and W‘IZ0 processes together.

The method of calculation is similar to that of W7 production in Section 3.1,
but with some differences. In Section 3.2 only one parton helicity combination
contributes, u(L)d(R). Here two helicity combinations contribute, for example
u(L)fi(R) and u(R)1“1.(L), and we include both. In fact, for Standard couplings
the contribution from u(R)fi(L) is very small compared to u(L)i(R), because of
interference between Feynman diagrams, so the parton-level process still depends
strongly on quark helicities. A more important difference is that in Section 3.2 there
was no parton-level background process, whereas here we have a large background

from electroweak+QCD processes.

49

7.2

“.1

Figure 3.7: Example diagrams for QCD production of W“ and two jets.

3.3.1 Background, Approximations, and Cuts

The parton-level background to this process is the production of Wi +2 jets by pro-
cesses with one electroweak vertex and one QCD vertex. Two example background
diagrams are shown in Figure 3.7. All combinations of quarks and gluons in the
initial and final states are included in the complete set. These processes do not in-
terfere quantum mechanically with our doubly electroweak signal process, because
they have a different color structure. For example, the qfig vertex is color octet,
where 9 denotes the gluon, whereas q§7 or qqZ" vertices are color singlet. How-
ever, the final states are indistinguishable experimentally, so the doubly electroweak
reaction is hidden in a background of electroweak-QCD reactions.

We have calculated the background cross-section from the helicity amplitudes for
a complete set of W2t + 2 jets processes [33], with the polarized parton distribution
functions described in Section 3.1. The background reactions should depend strongly

on the helicity of the proton: The produced W‘t must connect to a left-handed quark

50

or right-handed antiquark for massless quarks, by the V-A coupling; the density
of quarks of given helicity depends on the helicity of the proton. By contrast,
for the signal reaction the diagrams with WW7 or WWZ vertices do not require
any specific helicities of the incoming quarks. Thus the signal reaction will have
a different dependence on proton helicity than the background. The important
question is whether the signal-to-background ratio in the measurement of anomalous
WWV couplings is better for left- or right-polarized protons.

To reduce the background we impose a cut on the invariant mass M2,- of the 2
jets, putting ng approximately equal to MW. The solid curve in Figure 3.8 shows
the distribution of the invariant mass of the two jets produced by the complete
unpolarized doubly electroweak process with final state qé’ W“, with W“ —-> in, for
J3 = 2 TeV. (Figures 3.8 through 3.12 are 30 bin histograms; the vertical axes are
labelled in picobarns per bin.) The qq’ invariant mass is peaked at the W“ mass.
Because the cross-section is dominated by the W“ resonance, we approadmate the
calculation by keeping only the Feynman diagrams that produce a W“W“ pair,
which is an accurate simplifying approximation. Furthermore, we require the qrj’
invariant mass to be approximately equal to the W“ mass: we calculate the cross-
section only for events with MW: between 70 and 90 GeV. The dotted curve in
Figure 3.8 shows the 2-jet mass distribution when we neglect all but the W“W“ pair
production diagrams and also require the 2-jet mass to be between 70 and 90 GeV.
With this M2,- cut, the complete doubly electroweak process is practically the same
as production of W“W“ followed by leptonic decay of Wi and quark-antiquark
decay of WI. For comparison, Fig. 3.9 shows the 2-jet mass distribution of the
background processes with the kinematic cuts listed in Eqs. (3.12)—(3.14) below. The
cut 70 GeV < qul < 90 GeV reduces the total background significantly, because

there is no resonant effect in the background processes.

51

 

lbin)
O
o s
a 8.

do/dMZE (pb
O as
U.) U!

0.25

.O
N

 

0.15

 

0. l
0.05

 

 

 

O

o IIIIITIII'IIIIIIIIIIIIIIIIIIIIIIIIIIITIIITIITII

25 50 75 100 125 150 175 200 225
M2,. (GeV)

Figure 3.8: Two-jet invariant mass distribution for the signal process. The solid line
is the result of the complete calculation of the pure electroweak process p + p —>
qi’ W“, with W“ -+ fill; the dotted line is the result of the calculation of W“ W“
production, with W“ —> Z!" and W“ —> 2 jets, with a cut on the two-jet invariant

mass (70 < ng < 90 GeV).

52

 

’E 1.
g 1-
as-
V b
5? .
\4—
b ..
'0 .
31—
2.—
P

 

 

-
0 l IIIIIIIIJJILIIIIIIIIIIIIIIIIIIllIllIl Ill

0 25 50 75 100 125 150 175 200 225
M2,(GeV)

 

Figure 3.9: Two-jet invariant mass distribution for the QCD background processes
pf) -+ W“ + 2 jets. (The cross-section for W“ + 2 jets is the same, for unpolarized
scattering.)

53

In addition to the ng-cut just described, we impose the following kinematic

cuts on all the final-state particles except the neutrino:

 

rapidity [17] < 3, (3.12)
transverse momentum pr > 15 GeV, (3.13)
AR 2 J(An)2 + (A41)2 > 0.7 , (3.14)

where AR is the separation of any pair of final particles not including the neutrino.
In the case of the neutrino, we impose only a transverse momentum cut, p7,, > 15
GeV. That is,

ET > 15 GeV. (3.15)

Figure 3.10 is another comparison between the complete doubly electroweak calcula-
tion, shown as the solid line, and the simplified approximation (W“W“ production
with subsequent W:h decays), shown as the dashed line. Figure 3.10 compares the
J5 distributions, where J5 is the center of mass energy of the parton-level process.
Again, the two calculations are practically equal.

We also consider, separately, a cut requiring large J5, specifically J5 > 340 GeV,
The variable J5 is important because the effect of anomalous coupling increases with
J5. Figure 3.11 compares signal and background as a function of J5. This figure
shows why large J5 is interesting: The dependence on A5 and A is stronger, and the
signal-to—background ratio is larger, for large J5. Figure 3.12 compares signal and
background for J5 > 340 GeV. On the other hand, this energy cut reduces drasti-
cally the total number of events, so it becomes a question of detailed calculation to
see whether it is a real advantage, given the available luminosity. Our calculation
is for J? = 2 TeV. A pp collider with higher center-of-mass energy would produce
more events in the interesting region of phase space with large J5, and provide

stronger tests of the anomalous WWV couplings.

54

 

do/d‘ls (pb/bin)
O

 

O
O
00
I I I I I I I I I I I I I41 I41 I T41 I I I I

 

 

 

 

 

0.06
0.04
0.02
0 llllIllllIlI llllIllllIIlllIllllIlIIlIIIIl I l
O 50 100 150 200 250 300 350 400 450
xls (GeV)

Figure 3.10: J5 distribution for the signal process. The solid line is the result
of the complete calculation of the pure electroweak process p + p —-> qq’ W“, with
W“ —-» In; the dotted line is the result of the calculation of W“W“ production,
with W“ Hill; and W“ —> 2 jets, with a cut on the two-jet invariant mass

(70 < M2,- < 90 GeV).

55

 

1.4

1.2

do/dVS (pb/bin)

0.8

 

0.6

0.4

IIIIIIIIIIIW—IrIIIIIIIlIIII

0.2

 

l -"”"s:.
‘4‘. "olo
111111111111 111111111l111‘1‘I-1-1""'

O
O 50 100 150 200 250 300 350 400 450
«Is (GeV)

 

 

 

 

Figure 3.11: Comparison of J5 distributions for signal and background processes.
The solid line is the QCD background. The dotted line is the electroweak process,
with zero anomalous couplings; the dashed line is the electroweak process with

A5, = A52 = 0.5.

[l

56

 

 

 

 

 

 

g 0.05 —
6 :
(VJ .—
E 0.04 :-
b .-
‘D ..
0.03 -
0.02 :-
0.01 :-
0 I“1 1 1 I1 1 1 111 1 1 1 l L1..1.I-r.1-I'r‘l‘I'-1-s-1-o
300 350 400 450 500 550 600
43 (GeV)

Figure 3.12: Comparison of J5 distributions for signal and background processes,
for J5 > 340 GeV. The solid line is the QCD background. The dotted line is the
electroweak process, with zero anomalous couplings; the dashed line is the elec-
troweak process with A5, = A52 = 0.5.

57

It is difficult to determine J5 accurately in an experimental event, because
that requires measurement of jet momenta and missing neutrino momentum. The
z-component of the neutrino momentum p,,z can be obtained up to a two-fold am-
biguity by solving the mass contraint for the W-boson, M3,, = (pg + p..)2. One way
to choose p”z is to select the solution with smaller absolute value, because a hard
scattering process tends to produce final products in the central rapidity region with
large transverse momenta. Given pa, it is straightforward to calculate J5 from the
4-momenta of l, u, and the 2 jets. An alternative way to select large J5 is to require
large transverse mass of the final state of the hard scattering process [34]. However,

we have not pursued these approaches in this work.

3.3.2 Results

For the cross-section calculations that follow, we set Q = J5, where Q is the mo-
mentum scale used in the parton distribution functions . The uncertainties we quote
are only the statistical Monte-Carlo uncertainties. The theoretical uncertainty due
to the choice of Q scale is discussed briefly later. The cross-section includes the
branching ratio 2 / 9 for leptonic decay of one W into either electron or muon plus
neutrino. The cross-section also includes the branching ratio 6/9 for 2-jet decay of
the other W, W“ —+ ud and W“ —-) c5. (Again, we ignore CKM mixing.) The
overall branching ratio (2/9) x (6/9) = 4/27, as well as the effect of the cuts of
Eqs. (3.12)-(3.14), are always included in the cross-sections reported hereafter for
W“W“ production. Note that we consider separately the rates for l“ + jets and
l“ + jets.

Figures 3.13 to 3.15 show the results as plots of signal cross-section vs anomalous
coupling parameters. Since our signal process involves the creation of a W“ W“

pair with subsequent decays of the W“ and W“, the signal cross-section is the same

58

regardless of which W decays to leptons, in the narrow-width approximation. This
is not true of the background cross-section, when the proton beam is polarized.
(For unpolarized scattering the production rates of W“ and W“ are equal by CP
invariance; but for scattering of polarized protons on unpolarized antiprotons, CP
invariance does not apply.) For instance, for a left-handed proton the background
process p(L) + p —) W“ + 2 jets has a larger rate than p(L) + p -> W“ + 2 jets ,
because the probability of finding u(L) (which produces W“) inside p(L) is larger
than d(L) (which produces W“), as implied by Fig. 3.1. (Remember that u(R)
and d(R) do not contribute to the constituent cross-section of the W“ + 2 jets
background process because the weak charged current is left-handed.) Thus we may
calculate the W“ + 2 jets and W“ + 2 jets possible backgrounds, and then choose
the W charge for which the background cross-section is smaller. Since the signal
process is symmetric with respect to the charge of the W that decays leptonically, the
process with the smaller background has a better signal-to-background ratio. In light
of the cross-section inequalities mentioned above, it is advantageous experimentally
to observe the W“(—> £17) + 2 jets mode for a left-polarized proton beam, and
similarly the W“(—) Lu) + 2 jets for a right-polarized proton beam. Hereafter we
apply this strategy when comparing signal and background rates in the tables.
The program we have used to calculate the electroweak+QCD background, which
comes from many processes [33], is set up to calculate the cross-section for produc-
tion of W“ + 2 jets. To find the cross-section for production of W“ + 2 jets, we
calculate the rate for W“ production with polarized anti-protons, which is equal to

the rate we want by the approximate CP invariance of the SM:
0(p + p(L) —+ W- + 2 jets) = 0'(p(R) + p —-> W“ + 2 jets) (3.16)

0(1) + p(R) —+ W— + 2 jets) = 0'(p(L) + p —) W+ + 2 jets) (3.17)

Tables 3.4 to 3.8 give the calculated cross-sections. Table 3.8 lists the background

59

 

2.0 IIIIIIzIIIrIkIIIIIIWHIII IIIIIIJ'III'IIIIIIIIIII IIIIIIJIIIIIIIIIIIIIIII

l a) p L) b) p(R) ' e) unpol. ‘

I me, ~- m, a
’- L' “2 7 ' m: d
1 - nix,=Ax, -~ Q57=A5zJ

   
 

I l I I I I I l I I L I I l I

 

 

 

 

- -L ..
0.0 ILILLLLIIIIIIIIIILLIII lIllIlIlIIIIIlIIIlIIIII lIIIlIIIJIIIIlIIIlllIII
-1.0-0.5 0.0 0.5 1.0-1.0-0.5 0.0 0.5 1.0-1.0-0.5 0.0 0.5 1.0
A5 A5 A5

 

Figure 3.13: Electroweak cross-section for my —> W“ W“ as a function of anomalous
couplings A5, for polarized (L, R) and unpolarized protons. For each polarization,

three cases are shown, corresponding to assumptions (x) A5, 76 0 and A52 = 0,
(0) A5, = 0 and A52 79 0, and (C1) A5, = A52. In all cases A, = A2 = 0.

60

 

 

 

    

2.0 IIIIIIIIIIIIIIIIII IIIIIIIIIIII IIII IIIII III IIIII IIIII IIIIIII
- ta) p(L) .. 1 p115 7.) unpol 1
- n A, -- A, .
- ~~ <3. -- a. .
' “[- Q7=xz "" Q7=A3 "‘
16— —— —— —
1- -p— d- -I
- -u- -- -1
75 ' ‘7 " '
310— ~— ——
b - -(r— .11. .1
- -1- ul- -1
.— -]1- dp -l
- .11— db -]
0.5— + _
- + 1. ..
.- -11— c1- -1
o 1L1L1l111111111l1111l1 1|111111111I1111l1111l1 1I1111I1|11I1111l111111

 

 

 

 

 

0.
-1.0-0.5 0.0 0.5 1.0-1.0-0.5 0.0 0.5 1.0-1.0-0.5 0.0 0.5 1.0
A A A

Figure 3.14: Electroweak cross-section for p Ap —» W“W“ as a function of anomalous
couplings A, for polarized (L, R) and unpolarized protons. For each polarization,
three cases are shown, corresponding to assumptions (x) A, at 0 and A2 = 0, (O)
A,=0andA27é0,and(C|) A,=A2. InallcasesA5,=A52=0.

61

 

     

 

 

 

 

 

m-IIIIIIIIIIIIIIIIIIIIIII IIIfiIIIIIIIIIIIIITIIII
' (b) p(R) ‘
'- m7=AICf
% _ Q7=Az J
U
0 qh ‘
V 4»- -
(‘3
(N. «L— a-1
)9) —4— ——4
Lo .0— .1
.9.
3
3'
b
r- 4L d
0. I111jl1111l111ll111111 11111111111 1111i1111|1 lllllllllllllllllllllll
-1.0-0.5 0.0 0.5 1.0-1.0-0.5 0.0 0.5 1.0-1.0-0.5 0.0 0.5 1.0
AIC.A AIC,A AKA

Figure 3.15: Electroweak cross-section for pgp' -+ W+W' as a function of anoma-
lous couplings, with J} > 340 GeV. For each polarization, two cases are shown,
corresponding to assumptions (x) An, = Aug with A, = A2 = 0, and (O) A, = A2
with An, = Anz = 0.

62

cross-sections for different proton helicities. As expected, 0’(L) is larger than U(R)
for W+ production: the W+ must come from a u(L), and the density of u(L) is
larger in p(L) than in p(R). On the other hand, 0'(L) is smaller than U(R) for W“
production: the W‘ must come from a d(L), and the density of d(L) is smaller in
p(L) than in p(R), because Ad(:c) is mostly negative, as shown in Fig. 3.1.

Table 3.8 lists cross-sections calculated using the polarized parton distribution
functions of Section 3.1, and also, for comparison, unpolarized cross-sections calcu-
lated using CTEQ2 parton distribution functions [35]. We have used the leading-
order set CTEQ2L for our studies. The CTEQ2 results are consistent with the
average of the two proton helicities, within the uncertainty of the Monte Carlo cal-

culations.

3.3.3 Limits on An.“ Anz, A,, and AZ

We estimate limits on anomalous couplings that could be set by experiments at
«3 = 2 TeV. Since there is a large background for this process, the limits depend
on the background cross-section 03. Again, as in our previous analyses, we assume
that in an experiment with N events the standard deviation of N is EN = x/IV. At

the three-sigma confidence level, the uncertainty of a’ is

0’
603 = 3\/% , (3.18)

which is the same as in Chapter 2, except that here a' is the sum of signal and
background cross-sections. We consider integrated luminosity L = l fb'1 and L =
10 fb'l.

To calculate the limit that could be set on an anomalous coupling parameter, e.g.
A5,, assuming the actual value of the parameter is zero, we compare the statistical

uncertainty 603 to the variation of the calculated cross-section as a function of A5,.

63
At the three-sigma confidence level, Art, is in the range with
|0'(An,) — U(An, = 0)| < 603 . (3.19)

For example, Figure 3.13c shows 0’ vs A1: for unpolarized scattering. Three possi-
bilities are shown: (1') the effect of An, with Anz = 0, (ii) the effect of Aug with
An, = 0, and (iii) the effect if An, = A52. The background cross-section is given
in Table 3.8. Then Tables 3.9 and 3.10 list the limits that could be set on An (and
also A) for these three cases. The estimated limits in Table 3.9 are from events with
arbitrary fl, and the limits in Table 3.10 are from events with \/§ > 340 GeV.

Our purpose is to determine whether measurment of the cross-section with polar-
ized protons leads to stronger limits on anomalous couplings than with unpolarized
protons. Tables 3.9 and 3.10 show that the limit is stronger with polarized protons,
assuming the same integrated luminosity as for unpolarized scattering. The con-
straints on either An, or A, alone from the W+ W“ channel are not as strong as
those obtained from studying the W+7 channel. Though the total W7 cross section
is much larger than the W+W" cross section, after including the W decays and cuts
the signal cross sections are comparable (unpolarized cross sections 0.47 and 0.7pb,
for W7 and W+W", respectively). Therefore the superior sensitivity of the W7
channel in our study is not explained by a large cross section for that channel. The
W7 channel is more sensitive to the photon couplings simply because no Z couplings
contribute to the event rate. In other words, some of the WW events are due to Z
interactions, which are independent of the photon couplings, so a given anomalous
photon coupling will result in a smaller fractional deviation from the SM event rate.
The W7 and WW channels are therefore complementary, since the WW channel
probes both W and photon couplings, without distinguishing between them, while
the W7 channel limits only the photon couplings.

The constraints on A52 and A2 are about a factor of 2 better than those on

64

An, and A, from the W+W’ channel. For models with a linear SU(2) symmetry,
such that n, = 132 and A, = A2, the constraint on An, is about a factor of 2 better
than from the W+7 channel, while the constraint on A, is about the same as from
the W+7 channel, after selecting the large \/§ region. In general, selecting large J}
(> 340 GeV) improves the significance of signal to background by about a factor of 2.

3.3.4 Interpretation of the Results

The limits described above show how this process, W+W' production with polar-
ized protons, tests the WWV vertex. The estimated limits on anomalous couplings
include only simple statistical uncertainty, based on Poisson statistics (EN = x/IV);
they do not include experimental uncertainty due to detector inefficiency or theo-
retical uncertainty of parton distribution functions or Q scale.

For example, the background calculation depends on the choice of parton mo.
mentum scale Q. Table 3.11 shows how the background cross-section, for unpolar-
ized scattering, depends on the choice of Q. We used Q = J; in our calculations.
Q = 2Mw or Q = fl / 2 would also be reasonable choices. The cross-sections for the
three choices differ by about 20%. This theoretical uncertainty due to dependence
on scale choice is larger than the statistical uncertainty estimated above. It is thus
difficult to extract the signal cross-section unless the uncertainty in the background
cross-section is reduced, by including higher-order QCD effects to reduce the de-
pendence on Q scale. On the other hand, given large luminosity the details of the
QCD background processes, e.g., the shapes of \/§ distributions, can be measured
directly from data to discriminate between different theoretical predictions from dif-
ferent scale choices. Therefore we anticipate that background rates will eventually
be better known, and allow us to extract signal rates.

Our purpose in this work is to study the effect of proton polarization. For this

65

study we have analysed only the total cross-section as a function of An and A.
Stronger constraints on An and A may be set by analysing, as in Chapter 2, dif-
ferential cross-sections with respect to variables for which the distribution of events
is sensitive to Art or A [30, 31]. It would be interesting to find kinematic variables
dependent on proton helicity for which the distribution of events is sensitive to A1:

or A, to further test the triple-boson couplings with polarized proton scattering.

3.4 Discussion and Conclusions

We have studied the potential of a polarized proton beam at the Tevatron collider for
measuring the tri-boson couplings WW7 and WWZ. Because the polarized parton
distribution functions in the relevant kinematic region (i.e., :e-values) are not yet
precise enough to give definite detailed predictions about the rates of the signal and
the backgrounds, we have concentrated on the comparisons between the total event
rates from a polarized- and an unpolarized- proton beam. As summarized in Tables
3.3, 3.9 and 3.10, we found that with a polarized proton beam the limits on non-
standard parameters An,, A,, Anz and A2 are somewhat improved compared to
those obtained from an unpolarized proton beam. We anticipate that these results
can be further improved by studying detailed distributions of relevant kinematic
variables. Generally a factor of 2 improvement in measuring these non-standard
parameters is expected, after selecting kinematic regions where the signal becomes
more important, as illustrated in Tables 3.9 and 3.10.

One interesting feature that we found about the polarized collider program is
that it is possible to select the polarization state of the proton beam to enhance the
ratio of signal to background for a specific charge mode of the final state. This was

demonstrated in studying the process pp —) W+W" —+ (£17); + 2 jets. For the

66

final state with a positive charged lepton (1"), one can select the polarization of the
proton beam to be right-handed to improve the signal-to-background ratio because
the QCD background process p(R)p —) W+(—+ (+14) + 2 jets has a smaller rate
than p(L)p' —» W+(-—> (+14) + 2 jets. The signal rate, in contrast, is independent
of the charge mode of the isolated lepton from the W-boson decay because either
W+ or W' can decay into the charged lepton. We expect that similar tricks can be
applied to other polarized physics measurements, such as the lepton + jet mode of
the it pair production from qt} and 99 fusion processes.

We are now in a position to compare the sensitivity of a Tevatron experiment
with a polarized beam (“polarized Tevatron”) to those of the NLC and LEPII.
Tables 2.2 and 2.3 show limits on anomalous photon couplings from LEPII and
the N LC, calculated from the variation of the total cross section; and Tables 3.9
and 3.10 show limits from a polarized Tevatron. Comparing 3-sigma limits on An,
and A,, it is immediately evident that the N LC is superior to the other colliders
by about two orders of magnitude. The polarized Tevatron gives somewhat better
limits than LEPII, especially when we capitalize on high energy events with greater
dependence on anomalous couplings by cutting events with x/g < 340 Gev. Note,
however, that a complete analysis of an actual Tevatron experiment would have
to include the contribution from WZ production, which depends on anomalous Z
couplings. In addition, to extract such precise limits from the Tevatron one would
have to reduce the momentum-scale choice uncertainty by calculating the next-to-
leading-order corrections to the cross section. Due to the relatively wide range of
center-of-mass energies for the Tevatron hard-scattering events, it would also be
important to include the energy dependence of the anomalous couplings, especially

if the scale of anomalous physics A (discussed in Chapter 1) were near its lower

bound.

67

The Tevatron limits should improve mildly, as did those from the e+e‘ collid-
ers, were we to use differential cross section information to calculate them, rather
than merely the total cross section. However, reliable predictions of distributions of
kinematic variables will require more precise polarized parton distribution functions.
Measurements of polarized proton-antiproton reactions will yield new information
on spin-dependent parton distributions. Thus the use of polarized scattering to test
fundamental physics, and the determination of the spin structure of the proton,
would proceed together. In the next chapter we investigate the potential improve-
ment of ppdf accuracy from studying single-Wt production with a polarized proton

beam.

68

Chapter 3 Tables

Table 3.1: Cross-section for the process pp —> W+7 with polarized protons, for
different values of the anomalous coupling A5,, assuming A, = 0. Cross-sections are
in pb. The branching ratio 2/ 9 for W+ —+ e+uc or [fix/u, and the effect of our cuts,
is included. The unpolarized case was calculated separately using CTEQ2 parton
distribution functions, for comparison. The asymmetry A, defined in Eq. (3.7), is
calculated by fitting the data to a parabola.

p113 —> W+7 cross-sections in pb

 

Art,
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0

p(R)

0.49 :1 0.01
0.39 i 0.01
0.29 :t 0.01
0.25 :1: 0.01
0.25 :1 0.01
0.27 a: 0.01
0.32 :1: 0.01
0.41 :I: 0.01
0.53 :I: 0.02

p(L)

1.44 d: 0.01
1.08 :1: 0.01
0.86 a: 0.01
0.72 d: 0.01
0.69 :1: 0.01
0.75 :l: 0.01
0.94 :1: 0.01
1.23 i 0.01
1.57 :1: 0.02

Unpol

0.98 i 0.01
0.75 :l: 0.01
0.59 :l: 0.01
0.51 :l: 0.01
0.47 :l: 0.01
0.54 :l: 0.01
0.63 i 0.01
0.83 :l: 0.01
1.09 :l: 0.02

A

0.514
0.504
0.491
0.480
0.475
0.480
0.491
0.504
0.514

 

Table 3.2: Cross-section for the process pf) —-> W+7 with polarized protons, for
different values of the anomalous coupling A,, assuming An, = 0. Cross-sections
are in pb. The unpolarized case was calculated separately using CTEQ2 parton
distribution functions, for comparison. The asymmetry A, defined in Eq. (3.7), is
calculated by fitting the data to a parabola.

 

 

PX? -> W+7 cross-sections in pb.

 

A7

p(R)

p(L)

Unpol

A

 

2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
—2.0

1.73 :l: 0.03
1.11 :l: 0.02
0.62 :l: 0.01
0.35 :l: 0.03
0.24 :t 0.01
0.34 :l: 0.01
0.57 :l: 0.01
0.99 :l: 0.03
1.64 i 0.03

7.47 :l: 0.03
4.47 :l: 0.02
2.32 :I: 0.01
1.14 :l: 0.03
0.68 :l: 0.01
1.05 :l: 0.01
2.19 :1: 0.01
4.13 :l: 0.03
6.87 :l: 0.03

4.55 :l: 0.03
2.72 :t 0.02
1.50 :l: 0.01
0.75 :l: 0.03
0.49 i 0.01
0.71 :l: 0.01
1.39 :l: 0.01
2.59 :t 0.03
4.16 i 0.03

0.613
0.602
0.580
0.530
0.478
0.530
0.580
0.602
0.613

 

69

Table 3.3: Limits on non-Standard couplings, from pp —2 W+7 with polarized and

unpolarized protons. The upper number is for 1 fb‘1 integrated luminosity, and the
lower number (in parentheses) is for 10 fb‘l.

 

 

 

 

Parameter Limits
p(R) p(L) Unpol
An, i0.89 :l:0.62 :l:0.70
(:l:0.50) (:l:0.35) (:l:0.39)
A, :l:0.36 $0.22 :l:0.26

(i020) (10.13) (40.14)

70

Table 3.4: Purely electroweak cross-sections, in pb, for p(R)p’ —> W+ + 2 jets, with
W+ —> it! where l = e or p; the proton is right-handed. The cross-section for
p(R)p —-> W” + 2 jets is the same. The branching ratio (2/9) x (6/9) = 4/27 and
the effect of our cuts are included.

 

 

Electroweak cross-sections in pb;

right-polarized proton.

 

 

 

 

 

Value Varied parameter

An, Anz An, = Artz
1.00 0.604i.012 1.120:l:.020 1.064:l:.020
0.75 0.534i.012 0.774:l:.016 0.816:l:.016
0.50 0.496:l:.012 0.600:l:.012 0.588:l:.012
0.25 0.456i.012 0.484:l:.012 0.484i.012
0.00 0.456:l:.012 0.456:l:.012 0.456:l:.012
-0.25 0.484:l:.012 0.508:l:.012 0.516:l:.012
-0.50 0.540:l:.012 0.628:l:.012 0.676zlz.012
-0.75 0.6202t.012 0.822:|:.014 0.908zl:.016
-l.00 0.716i.016 1.172i.024 1.356:l:.028

A, A2 A, = A2
1.00 0.740:l:.020 1.344:l:.028 1.444:t.032
0.75 0.602i.014 0.952:l:.018 1.004i.020
0.50 0.524:l:.012 0.688:l:.016 0.712:l:.016
0.25 0.472i.012 0.520:l:.012 0.520:l:.012
0.00 0.456:l;.012 0.456:l:.012 0.456:l:.012
-0.25 0.480:l:.012 0.504:l:.012 0.516:l:.012
-0.50 0.548:l:.012 0.664:l:.016 0.708:l:.016
-0.75 0.642:l:.016 0.920i.018 l.000:l:.020
-1.00 0.768:l:.016 1.2882t.024 1.444:l:.028

 

 

71

Table 3.5: Purely electroweak cross-sections, in pb, for p(L)}? —I W+ + 2 jets,
with W+ —+ [V where l = e or p; the proton is left-handed. The cross-section for
p(L)p’ —> W" + 2 jets is the same. The branching ratio (2/9) x (6/9) = 4/27 and
the effect of our cuts are included.

 

 

Electroweak cross-sections in pb; left-polarized proton.

 

 

 

 

Value Varied quantity

An, Alcz An, = Anz A, A2 A, = A2
1.00 1.08:l:.02 1.88:l:.04 2.68:t.08 1.26:l:.04 2.60:l:.04 3.84:l:.08
0.75 1.02i.02 1.48:l:.04 1.92:l:.04 1.10:l:.02 1.84:l:.04 2.60:l:.08
0.50 098212.02 1.16:l:.02 1.34:l:.04 1.04:l:.02 1.38:l:.04 1.72:l:.08
0.25 0.98:l:.02 1.00i.02 1.02i.02 0.98:l:.02 1.06:l:.02 1.16:l:.02
0.00 0.96:l:.02 0.96:l:.02 0.96i.02 0.94:l:.02 0.94zl:.02 0.98:l:.02
-0.25 0.98:l:.02 1.08i.02 l.16:l:.04 1.00:l:.02 1.06:l:.02 1.16:l:.04
-0.50 1.04:l:.02 1.32:l:.02 1.58:l:.04 1.04:l:.02 1.40i.04 l.72:l:.04
-0.75 1.14:l:.02 1.72:l:.04 2.28:l:.04 1.14:l:.02 1.86:l:.04 2.60:l:.08
-1.00 1.242t.04 2.24:l:.04 3.20i.08 1.26:l:.04 2.56i.04 3.88:l:.08

 

 

Table 3.6: Purely electroweak cross-sections, in pb, for p}? —* W+ + 2 jets, with
W+ —> iv where l = e or p; the proton is unpolarized. The cross-section for
p1“) —> W‘ + 2 jets is the same. The branching ratio (2/9) X (6/9) = 4/27 and the
effect of our cuts are included. These values were calculated independently using
the Morfin-Tung ppdf’s; the cross-section for unpolarized protons is equal to the

average of cross-section for left and right polarized protons.

 

Electroweak cross-sections in pb; unpolarized proton

 

 

 

 

Value Varied quantity

AIS, Alcz An, = Alcz A, A2 A, = A2
1.00 0.84:l:.02 1.50:l:.04 l.88:l:.06 1.00:l:.02 1.98i.06 2.64i.06
0.75 0.78:L-.02 1.14:l:.02 1.36:l:.02 0.86:l:.02 l.40:l:.04 1.80:l:.06
0.50 0.74:l:.02 0.88:l:.02 0.96:l:.02 0.78:l:.02 1.04:l:.02 l.22:l:.02
0.25 0.72:l:.02 0.74:l:.02 0.76:l:.02 0.72:l:.02 0.80:l:.02 0.84:l:.02
0.00 0.72:l:.02 0.72:l:.02 0.72i.02 0.70:t.02 0.70:l:.02 0.70i.02
-0.25 0.72:l:.02 0.80:l:.02 0.84:l:.02 0.74:l:.02 0.86:l:.02 0.84:l:.02
-0.50 0.78:|:.02 0.9821302 1.14:l:.02 0.80i.02 1.04:l:.02 1.22i.02
-0.75 0.88:l:.02 l.28:l:.02 1.60:l:.02 0.88:l:.02 1.38:l:.04 1.80:l:.06
-l.00 0.98:l:.02 1.70:l:.04 2.28:l:.06 1.02:l:.02 1.92;l:.04 2.66:l:.06

 

 

72

Table 3.7: Electroweak cross-sections, in pb, for W+W’ production with non-
Standard couplings, with polarized or unpolarized protons, and with a large-J's;
cut, «3 > 340 GeV. One W decays leptonically, the other to 2 jets, and the branch-
ing ratio 4 / 27 and the effect of our cuts are included in the cross-section. Two
assumptions on non-Standard couplings are listed: An, = Anz with A, = A2 = 0,

and A, = A2 With An, = ANZ =0.

 

Electroweak cross-sections

in pb, with 4/3 > 340 GeV

 

 

 

 

 

 

 

Value Varied parameter
An, = Anz

p(L) p(R) Unpol
0.75 0.720:l2.016 0.2082t.004 0.456:}2.012
0.50 0.3802t.008 0.110:l:.002 0.236:l2.004
0.25 0.180:l:.004 0.052:l:.002 0.112:t.002
0.00 0.132:l2.004 0.038:l:.002 0.082:l:.002
-0.25 0.2282t.004 0.066:l2.002 0.142:l2.004
-0.50 0.4842l:.012 0.138:l:.002 0.304i.008
-0.75 0.876:l2.016 0.244:l:.004 055632.012

A, = A2

p(L) p(R) Unpol
0.75 1.196i.028 0.3162l:.008 0.756:l:.012
0.50 0.592:l:.012 0.162:l:.004 0.368:t.008
0.25 0.248:l:.008 0.070:l:.002 0.152:l:.004
0.00 0.132:|:.004 0.0382L-.000 0.082:l:.002
-0.25 0.2441008 0.070:l2.002 0.156:l:.004
-0.50 0.604:l:.016 0.166:l:.004 0.376:l2.008
-0.75 1.160:l:.028 0.316:t.008 0.732212.016

 

 

 

73

Table 3.8: QCD background cross-sections for pp ——> Wi + 2 jets, for various proton
polarizations and kinematic cuts. The unpolarized cases U(pp -—» W+ 2j) = U(pp —I
W" 2j) were calculated separately using CTEQ2 parton distribution functions.

 

W+2 jet Background

 

process 0' (pb)

 

without 2 jet mass cut:

p(R)p —) W" 23' 50.56$.58
p(L)p —» W‘ 2j 41.36$.52
p(L)p —> W+ 2j 65.46$.78
p(R)p --) W+ 2j 26.84$.28
pf; —> W+ 2j 45.30$.50

 

with 2 jet mass cut (70-90 GeV):
p(R)p —> W‘ 2j 7.324$.070
p(L)p _. w- 23' 531022.056
p(L)}? —+ W+ 2j 9.4041092
p(R)p —> W+ 2j 3.874$.034

p5—+W*‘2j

6.514$.060

 

with 2 jet mass cut
and \/§ > 340 GeV:
MR)? -’ W‘ 21'
p(L)i5 -* W’ 23'
p(L)? -—> W’“ 21'
p(R)I3 —+ W” 23'
7213 -> W+ 22'

0.348$.004
0.308$.004
0.520$.004
0.144$.004
0.358$.002

 

Table 3.9: Limits on anomalous couplings that could be set from p'p -» W* + 2 jets
with polarized or unpolarized protons. The numbers in parentheses are for 10 fb-l
integrated luminosity, and the other numbers are for 1 fb‘l integrated luminosity.

 

 

 

Parameter Limits

p(R) p(L) Unpol
An, $0.89 $1.10 $1.06
Anz $0.56 $0.47 $0.54
A16,=Anz $0.53 $0.35 $0.44

($0.30) ($0.20) ($0.24)
A, $0.79 $0.77 $0.83
Az $0.47 $0.37 $0.44
A,=Az $0.44 $0.29 $0.36

($0.25) ($0.16) ($0.20)

 

 

74

Table 3.10: Limits on anomalous couplings that could be set from p13 —5 W“: +
2 jets with polarized or unpolarized protons, from events with «5 > 340 GeV. The

numbers in parentheses are for 10 fb"1 integrated luminosity, and the other numbers
are for 1 fb‘1 integrated luminosity.

 

Parameter Limits
p(R) P(L) Unpol
An,=Anz $0.34 $0.23 $0.28
($0.19) ($0.13) ($0.16)
A,=Az $0.28 $0.18 $0.23
($0.16) ($0.10) ($0.13)

 

Table 3.11: Effect of parton Q scale on the calculated cross-section for background
processes pp' —+ Wi + 2 jets. These are unpolarized cross-sections, calculated with
CTEQ2 parton distribution functions.

 

Q scale 0 (pb)
.5 6.420 4 .056

75/2 8.5384074
2Mw 7.3984068

 

Chapter 4

Probing the Polarized Parton
Distributions with p13 —> W

4.1 Introduction

In the preceding chapter we examined the potential of a Tevatron with a polarized
proton beam for studying anomalous electroweak physics. Other possible physics
topics to explore with a polarized beam are outlined in Reference [24]. However, in
order to use polarized p—p scattering as a probe of interesting physics, it would be
necessary to know accurately the parton distribution functions for polarized protons.

Polarized parton distribution functions (hereafter abbreviated ppdf’s) have been
measured to some accuracy in deep-inelastic polarized lepton scattering [36, 27, 28].
These experiments have revealed the spin structure of the proton, which is inter-
esting in its own right. Several parametrizations of ppdf’s have been published
[37, 26, 38, 39] based on this data. The purpose of this chapter is to examine
whether single-W3: production in polarized p—p collisions, with longitudinally po-
larized protons, could be used as another, complementary, method to constrain the
ppdf’s.

With the Tevatron collider in mind [23], we consider the center-of-mass energy

equal to 2 TeV.l

 

lAfter this work was completed, we learned that a polarized colliding beam is less likely to

75

76

The idea to study ppdf’s from the effect of proton polarization on the process p+
p -+ W‘h has been considered previously, in Ref. [40], in which the cross-section for
producing W* was calculated, treating the W3: as a stable particle. Here we will take
into account also the leptonic decay W* —> ei +972, including a possible kinematic
cut on p1 of the cf; i.e., we are concerned with the experimentally measurable
processes p + p —> (Wi —> e* +971) + X.

The cross-section for W": production depends strongly on the helicities of the
quarks. For massless quarks a W* couples only to left-handed (L) quarks and right-
handed (R) antiquarks, because of the V—A form of the weak coupling. We know
that the densities of L and R quarks in a proton depend on the proton helicity.
Therefore, the cross-section for p,\ + p —) Wi, where A = L or R indicates the
helicity of the proton, depends on A. Measurement of the polarization dependence
would provide some information on the ppdf’s. The question which motivates our
study is whether such data could determine the ppdf’s accurately.

The ppdf’s are defined as follows: For any parton type f, we define

Me) = -:;(f(z)+Af(=v))

2 density of L (or R) partons in a L (or R)‘proton,

12(2) = ass—Am»

2 density of L (or R) partons in a R (or L) proton,

where a: is the momentum fraction of the parton. Thus A f(:c) is the difference
between partons with the same helicity as the proton and partons with opposite

helicity. There are 13 different parton types

f=mm¢aqammiaaafi

 

be built in the near future at the Tevatron. However, since a polarised collider is technically
feasible, we believe it is still interesting, and potentially useful, to explore theoretically the possible
experiments with such a machine.

77

In our calculations we neglect the 5,5 and t,t distributions. Also, the gluon distri-
bution does not contribute in our calculation, which is only a tree-level calculation.

We have used two recently published parametrizations of ppdf’s [38, 39]. (Specif-
ically, we use set 2 from Ref. [38] and set B from Ref. [39].) We calculated these
ppdf’s with a computer program written by Dr. Glenn Ladinsky. Figure 4.1 shows
zAf(:c) vs a: for f = u and d. The ppdf’s depend on momentum scale Q; i.e.,
ft = fi(:v, Q). Figure 4.1 is for Q=5 GeV. These distributions were constructed by

fitting to data for the polarized structure function gf(:e, Q); in the parton model

511(20): $232: [Amen + A§(z,Q)l . (4.1)

Figure 4.2 shows plots of zgf(z, Q) vs a: calculated from the two sets of ppdf’s we
used, again for Q = 5 GeV.

The difference between the two parametrizations of ppdf’s, illustrated in Figs. 4.1
and 4.2, indicates the uncertainty of our current knowledge of the ppdf’s. Either
parametrization is a reasonable fit to existing data from polarized deep-inelastic
lepton scattering, within the uncertainty of the data. We will use the difference
between these two sets of ppdf’s to gauge the current uncertainty of the ppdf’s. We
will consider whether measurements of W1t production in p113 collisions could be

used to reduce this uncertainty.

4.2 Wi production

In the parton model, the cross-section for p,\ + p —) W+ + X, where A is L or R
for left-handed or right-handed protons, with subsequent decay W+ —1 e+ + 11.3, is
expressed as

5(1) = :1 3535' [&LR(2P,2'P')ui(z)d(z') (4.2)

NIH

+&LR(2’P’, 5p)3‘,(5)a(5')]

78

 

_Illl| I I'll—fill I IIIIIIL

x Af(x,Q)

 

d
1111' l llllllll I lllllJl

-0.2
0.005 0.010 0.020 0.050 0.100 0.200 0.500 1.000
X

L.

 

 

 

Figure 4.1: Polarized parton distribution functions. The curves are :cA f (2:) vs a: for
parton types 11. and d, which are the most important partons in our calculations, for
two sets of polarized parton distribution functions. The solid curve is the Nadolsky
parametrization [38] and the dashed curve is the Gehrmann-Stirling parametrization

[39].

79

 

 

0-10 [III I l IIIIIII I I T [Ill
- solid: Ned. j

- dashuzc-S
0.08[— _
C I
8 0.06-— __
x" p— .-
V I— q
A" _ _
no _ ._
)4 0.04— —
[- _
0.02— .—
Z / I

/ /

.. \ _
[L111 1 Lllllll l 1 1l11‘

 

 

 

0.00
0.005 0.010 0.020 0.050 0.100 0.200 0.500 1.000
x

Figure 4.2: The function zyflz, Q), which has been measured experimentally in
polarized deep-inelastic lepton scattering, calculated using two sets of polarized
parton distribution functions. The solid curve is the Nadolsky parametrization [38]
and the dashed curve is the Gehrmann-Stirling parametrization [39].

80

where the notation is as follows: The parton cross-section &LR(p1, p2) is for the pro-
cess uL(p1) + dR(p2) -—> W+ —> e+ + Vc; note that u must be left-handed and d
right-handed because of the V—A coupling. The upper sign on ui(z) and d4(a:)
is for A = L and the lower sign is for A = R. The first line in Eq. (4.2) corre-
sponds to u,d coming from p,p, and the second line corresponds to d,u coming
from 12,13, respectively. a: and :e’ are the parton momentum fractions in the proton

and antiproton respectively. The parton distribution functions are, for example,

ui(a:) = u quark with same/ opposite helicity as p
d(z') = d quark in unpolarized p
d4(z:) = d quark with opposite/ same helicity as p

11(22’) 2 u quark in unpolarized 13.

(We evaluate the parton distribution functions at momentum scale Q = 80 GeV
for W15 production.) The factor of 1 / 2 is from the average over 1') helicities. We
also add the contribution for the parton process c + 3 —) W+ —2 6+ + 14,. (We
ignore Cabibbo-Kobayashi-Maskawa mixing in this work.) The cross-section for
p; + p —2 W" —> e" + 176. is given by a similar expression, but with parton densities
di(z)u(z’) and fi;(z)d(z’).

We have calculated U(A) from Eq. (4.2) by a Monte Carlo calculation based on
the program PAPAGENO [32], modified to include the two sets of polarized parton
distribution functions.

As a preliminary calculation, in this Section we consider the processes p; + 13 —->
Wi treating Wi as a stable particle. Figure 4.3 shows the Feynman diagram for

the hard production process to produce a W+. (In Section 4.3 we include the Wi

decay.) Integrating the squared amplitude over the W phase-space yields the

81

W+_

E:

Figure 4.3: Diagram for ud —+ W+
color-averaged parton-level cross-section, summed over W polarizations:
. I “'92 I 2
oLR(:e,z) = -3—-6(zz s — MW) . (4.3)

The 5-function sets the invariant mass of the incoming partons equal to the invariant
mass of the W.

In this case the cross-section U(A) may be calculated analytically, i.e., without
a Monte Carlo program, by inserting this 6'“;— which treats the Wi as a stable
particle of mass M w—into Eq. (4.2), instead of the hard cross-section which includes
the W decay. To find the differential cross-section in rapidity, include in Eq. (4.2)
a factor

1 EW + PzW

which by going to the pp center-of-momentum frame can be shown to be equal to

1 :c

82

 

 

 

 

 

P). 'l' P " WI
0.4 .__I r l l l 7 r T r l r l I I l l I r :—
r-solld: Ned. A: d
Tduhu: G-B 2
L. ..
A 0.3 _ —
,g _ _
‘2 I j
>~ _ _
3

b 0.2 _— —
"U _
O: .. ..
\ - ..
v-t _ ..
0.1 — A
L _

0.0 l l l
—4 -—2 0 2 4

Figure 4.4: Cross-sections do(A)/dyw for p; + p —+ W+ in nb, multiplied by 1/9,
the branching ratio for the W+ to decay to e+ + u... Proton polarization A is L or R.
The solid and dashed curves are for ppdf’s from Ref. [38] and Ref. [39], respectively.
This 6—function, together with the 6-function from Eq. (4.3), relates the rapidity yw

of the W* to the parton momentum fractions 2: and z’:

= _ Uw = __ -uw .
:1: fl e , a: \/3 e , (4 4)
and yields the differential cross-section
do A ‘Irg2 — _
713%.?) = E‘ [u1(4)d(5') + d4(z)u(z')] . (4.5)

For W“ production interchange v. and d. The result is shown in Figures 4.4 and 4.5.
Figure 4.4, which is for W“L production, has graphs of édo/dyw vs yw, where yw
is the rapidity of the W+. Figure 4.5 is the same for W‘ production. The cross-
sections have been plotted in Figs. 4.4 and 4.5 multiplied by 1/9, which is the
branching ratio for W53 to decay to ei +52, because in an experiment the Wi

would be observed from its decay to leptons.

83

 

 

 

 

 

P11 + P " W-
0.4 IIIIIIFFIIFIIIIIIfi;
“solid: Ned. '1
~duhu: G—S -
A 0.3 _ ‘—
.D _ q
5 _ .
g >— _.
'U f “
E 0.2 — —
L ..
'U

o: F “
\ - _
fl D-I -
0.1 — '—
1- -1
r- \ -1

0 0 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1
-4 -2 0 2 4

Y1

Figure 4.5: Cross-sections da'(A)/dyw for p; + 13 —1 W“ in nb, multiplied by 1 / 9,
the branching ratio for the W“ to decay to e“ + 17... Proton polarization A is L or R.
The solid and dashed curves are for ppdf’s from Ref. [38] and Ref. [39], respectively.

The cross-section for 1);, + p ——+ WJr is greater than that for p3 + p —» W+,
because W+ production is dominated by the reaction of 11.1, from the p and d3 from
the p". Since Au(:c) is positive, as we see from Fig. 4.1, 11.1, has greater density in pL
than in pH, making the cross-section larger for p1,. Similarly, 0(pR + ‘p -+ W“) is
greater than U(pL + p —> W“) because W“ production is dominated by dL from the
p. Figure 4.1 shows that Ad(z) is negative, so dL has greater density in p}; than in
PL-

We can display the spin dependence by plotting the left-right polarization asym-
metry ALR(W), defined by

d0(L)/dyw - 40(R)/dyw

do(L)/dyw + da'(R)/dyw . (4.6)

ALR(W) =

 

For w+ production Eq. (4.6) shows that ALR(W) is simply related to Au(z) and

84

Px‘l'P ”Wt

 

IIIIIIIIIITIIIIVII

r1
gm

1 . 0
solid: Ned.
dashes: G-S

Ij

0.5

0.0

Am(wl

 

—0.5

lJlJLllllllllLlllllll

IIITTITIFIIIIIIIFIT

-1.0

 

 

L14 1 l l I 1 [LI 1 l 11 l_111
—4 -2 0 2 4
Y1

 

Figure 4.6: The left-right polarization asymmetry ALR for W3: production in polar-
ized p — p collisions. The solid and dashed curves are for ppdf’s from Ref. [38] and
Ref. [39], respectively.

Ad(:c); we have

Au(a:)d(:e’) — Ad(z)1“1.(z’)
u(z)d(z’) + d(z)1“1(2:’)

The polarization dependence of the sea quarks is small, so ALR(W) is approximately

Au(z)/u(:c). Similarly, for W“ production ALR(W) is approximately Ad(z)/d(:e).

 

ALR(W) = (4.7)

The left-right polarization asymmetry ALR(W) is shown in Figure 4.6, as a
function of yw, for both W+ and W“ production. The asymmetry is greatest for
yw > 0, i.e., at large 2:, where the asymmetry may be as large as 50 percent.
For yw < 0, the asymmetry is small because yw < 0 corresponds to small 2:, where
A f (z) is small. This result is in accord with the intuition that the helicity of partons
with small a: is not much correlated with proton helicity.

In the case of W+ production the two sets of ppdf’s give fairly similar results
for ALR(W). The W+ asymmetry depends on Au(z), which is relatively well deter-

mined by deep-inelastic polarized lepton scattering. In the case of W“ production

85

the two sets of ppdf’s give dissimilar results for ALR( W) The W“ asymmetry de-
pends on Ad(:e), which is less well determined by measurement of gf(:1:, Q) because
Aq(z) is weighted by e: which is 1/9 for d compared to 4/ 9 for 11.. Thus the process
p + I“) —1 W“ would be the most likely to give accurate new information on the
ppdf’s, specifically on Ad(:c), as has been pointed out by Nadolsky [40].

A significant difference between the two sets of ppdf’s occurs at very large yw, say
yw > 2, which corresponds to the limit 2: —) 1 by Eq. (4.4). The two parametriza-
tions of ppdf’s treat the limit a: —> 1 differently. Nadolsky [38] imposed the require-
ment that u_.(:c) —1 0 and d_(:1:) —> 0 as a: —> 1; equivalently, Au(z) —> 11(2) and
Ad(:e) —-> d(z). This requirement, first derived in Ref. [41] by an argument based on
first-order perturbation theory, means that if a parton carries all the momentum of
the proton (a: = 1) then it has the same helicity as the proton. With this require-
ment the asymmetry ALR(W) must approach +1 as :1: —> 1, i.e., as yw approaches
its maximum value. (Ad(z) and d(z) are negligible at z = 1.) Indeed ALR(W) for
the Nadolsky ppdf’s is increasing toward 1 as yw increases. Testing experimentally
this interesting helicity behavior at large a: is emphasized in Ref. [40]. Gehrmann
and Stirling [39] did not impose the requirement A f(z) —+ f(:1:) as a: —) 1, but
rather fitted Au(z) and Ad(:e) only in the limited range of a: where data exists on
gf(z,Q), extending out to a: z 0.6; therefore their ppdf’s should only be applied
for yw < 2.7. Of course, we exaggerate the difference between 0’(L) and U(R) at
very large yw by plotting the left-right asymmetry ALR(W), which is a ratio; both
da'(L)/dyw and do(R)/dyw approach 0 for large yw, so ALR(W) is a ratio of very
small cross-sections there. In any case we are interested in determining the ppdf’s
for all :13, not just in the limit :1: —> 1.

The results shown in Figs. 4.4 — 4.6 give a good indication of the effect of proton

polarization, but are not directly measurable because the W‘t is unstable.

86

4.3 Decay of Wi

Next we include the leptonic decay of the Wi to consider the processes p,\ + p -)
(W53 -> ei +573.) + X. It is this process that would actually be observed in an
experiment. Of course only the e* would be detected. The question addressed here
is whether the Wi decay would markedly reduce the sensitivity of the polarization
asymmetry ALR to the ppdf’s.

Figure 4.8 shows do(A)/dy¢ for the process p,\ + p -—> (W+ —1 e+ + 14,) + X, and
Figure 4.9 shows the same for the process p; + p —-> (W“ —+ e“ + 17,) + X, where
3,.: is the rapidity of the e*. These cross-sections were calculated by a Monte Carlo
calculation from the tree-level formula.

We note that the distributions of y.3 are spread out and shifted relative to the
distributions of yw; the shift is toward negative rapidity in the case of e+ and
W+, and toward positive rapidity in the case of e“ and W“. These shifts can be
understood as a consequence of the V-A coupling between W* and eflfl-l, together
with angular momentum conservation. Consider, for example, pm? -—1 W11 The
partons involved must be a left-handed u colliding with a right-handed d, as shown
schematically in Figure 4.7. Because a 111, in a pL tends to have relatively large
a: (see Fig. 4.1) compared to the d); from the 1“), the W+ comes out preferentially
in the direction of the 11;, (see Fig. 4.4), which is the case depicted in Figure 4.7.
Angular momentum conservation—conservation of the double-arrow directions in
Figure 4.7—along with the requirement that the e+ be right-handed, by the V—A
coupling, force the positron direction to be against the W+ direction. In the lab
frame this corresponds to a negative shift in rapidity, because a Lorentz boost from
the W+ rest frame to the lab frame simply adds the W+ rapidity (zero in its rest
frame) to the positron rapidity.

Figure 4.10 shows the left-right polarization asymmetry ALR(e) for both e+ and

87

 

> W 2 ”L

 

Figure 4.7: Schematic diagram of particle momentum and spin orientations in
uLdR —-» W3 —) efiuL. Single arrows indicate momentum directions; double ar-
rows indicate spin orientation. For clarity the d and e+ momenta are shown at

angles to the other particles’, but they should be imagined as along the same line.

0.4
0.3

TE

5

>3

1, 0.2

\

b

'6
0.1
0.0

88

pA+p +W+ae++u.

 

 

 

IIIITIIIIIIIIIIIII

llllllJllllllllllll

 

 

_'
C (011.11 "=L
_ (b) H
1
.—
l I l I J_L-
-4 -2 0 2

Figure 4.8: Cross-sections do(A)/dy¢ for p; +13 —1 W+ —-1 13* + 116 in nb, with proton
polarization A = L or R. The solid and dashed curves, labeled (a) and (b), are for

ppdf’s from Ref. [38] and Ref. [39], respectively.

0.4
0.3

’3

5

3; 0.2

\

b

'6
0.1
0.0

IDA-+5 4W--’e-+l7.

 

 

a) Nod.
) e-s

 

IIIIIIIIIIIII llIl

 

lllllllllllllllllll

 

 

-4 —2 0 2

Figure 4.9: Cross-sections do(A)/dye for p,\ +15 -—> W“ —1 e“ + 173 in nb, with proton
polarization A = L or R. The solid and dashed curves labeled (a) and (b), are for

Ye

ppdf’s from Ref. [38] and Ref. [39], respectively.

 

89

1321+? »W*->e*+v

 

l

IIIIITITIIIfiIIIIIII

1.0 —'

(a) Nod.
00 0’3

jrl

0.5

IIIII

   
 

 

 

 

’6
‘g 0.0
< I
: e . ‘45
—o.5 —— J
-1.0 L- —
L l l l I l l l l l l l I l l l l l l d
—4 -2 0 2 4

Yo

Figure 4.10: The left-right polarization asymmetry ALR(e) for W“: production in

polarized p — p collisions, with decay W33 —> 13i +971. The solid and dashed curves,
labeled (a) and (b), are for ppdf’s from Ref. [38] and Ref. [39], respectively.

e“ production,
= da(L)/d3/e “’ d”(Bl/dye:
d0(L)/dye + 40(R)/dye '

As before, ALR(e) depends mainly on Au(:1:) for 6*, and on Ad(:e) for e“. However,

 

A1010?) (4.8)

a: is not simply related to ye. Therefore a real experiment, where only 31.3 can
be measured, would not determine Av.(:1:) or Ad(:c) directly; there is no direct
relation analogous to Eq. (4.7) for ALR(e). Instead, a calculation of da'(A)/dy,,
computed from parametrized ppdf’s, would be fitted to the data. The asymmetry
for ei production is somewhat less than the asymmetry found in Section 4.2 for
W* production, but is still significant. We will quantify this statement below by
estimating the luminosity which would be required to distinguish the two ppdf’s.
In an experiment, the charged leptons would be observed as a function of their
transverse momentum p1»... It is interesting to consider events with large p1,. There-

fore we plot in Figures 4.11 and 4.12 the cross-sections for W+ and W“ production,

90

pA+§+W++e++m

 

  

 

 

  

 

I I I I I Ij I I I 1 I I I I I

A : (a) 11.11 :
.o _ (b) 0-8 —1

Fl
v - -1
% 0.3 L— -—
[— _

(D
m _ .1
N v- .-
A. .. ..
I" ,_ —_

Q 0.2 _

1...
o - ..
“I! 1— -1

0
'55 - ..
b - _
'U _ -
2 a

0.0

-4 4

Figure 4.11: Cross-sections alo(A)/dy.3 for p,\ + p —-> W+ —-> e+ + we in nb, with
a kinematic cut on the lepton transverse momentum, p1, > 25 GeV. The solid
and dashed curves, labeled (a) and (b), are for ppdf’s from Ref. [38] and Ref. [39],
respectively.

with decay Wi —> ei +9723, for events with pTe > 25 GeV; Figure 4.13 shows
the corresponding polarization asymmetry. With this high-p1,, cut there is again a
significant asymmetry.

The left-right polarization asymmetry of do(A)/dy., with p1,, > 25 GeV exhibits
large differences between the two sets of ppdf’s that we have used in our calculations,
which implies that this process is sensitive to the ppdf’s. The region of large rapidity
is especially interesting. Indeed, ALR(e) with p1, > 25 GeV resembles ALR(W) of
the underlying W:k production, for large rapidity. Specifically, for the Nadolsky
ppdf’s [38] ALR(e) increases at large 3],, which reflects the increase of ALR(W) at
large yw. Events with large yw are interesting because they come from partons with
large a: in the proton, and we would like to verify the helicity behavior of partons

at large 2:, that is, that f_(:c) —> 0 as a: —> 1. However there is an experimental

91

p,+p +W--+e'+17.

 

 

 

 

0.4 __I_ I I I I I T I I I I W I I I I I 1 —
: (a) Ned. :
’3 _ (b) 6-8 _
e L _
3; 0.3 — —
o - _
1:: _ _
cu - 2
A _ _
13 2.. ._
C, 0.2 J
1. _
O *- q
H L— —
:5 r 4
b l 2
v _
0.0 '
-4 4

 

Figure 4.12: Cross-sections do'(A)/dy.3 for p,\ + I“) —+ W“ —» e“ + 17, in nb, with
a kinematic cut on the lepton transverse momentum, pTe > 25 GeV. The solid
and dashed curves labeled (a) and (b), are for ppdf’s from Ref. [38] and Ref. [39],

respectively.

92

px+1“)-1W*->e*+v

 

    

 

 

 

I—I I I I T I I j Ij I I I I I I I I
1.0 "'— '—
: (a) Ned. -
_ (b) 0-8
’13
u 0.5
3 4
A -1
13 s
9‘ 0.0 "‘
In "' -
3 t e 1
’0? "‘ O {1'11“ ‘
V —0.5 —
.9 - "51
~ :1
-1.0 — —‘
- l l l l l l l l l I l l l I I l l l l d
—4 -2 0 2 4

Yo

Figure 4.13: The left-right polarization asymmetry ALR(e) for Wit production in
polarized p — p collisions, with decay Wi —» ei +573” with a kinematic cut on the
lepton transverse momentum, me > 25 GeV. The solid and dashed curves, labeled
(a) and (b), are for ppdf’s from Ref. [38] and Ref. [39], respectively.

93

difficulty here: It is difficult for a collider detector to detect charged leptons well
with 3],, > 2.5, and the range of y., where ALR(e) increases strongly with y,, is in
the range of '17c > 2.5. Thus we find that a precise test of the helicity behavior of
partons at large 3 would require a detector with large coverage in rapidity, extending
to y., > 2.5. This result shows the importance of the Wi decay. In terms of the
W*, the large-a: helicity behavior is already tested to some degree at yw < 2.5; but
in terms of the ei, the large-:1: helicity behavior is not really tested until 378 > 2.5.
Finally, we estimate the integrated luminosity (f 1:) that would be needed to
measure the ppdf’s accurately. Over a large range of y, with 37.3 > 0, the asymmetry
ALR(e) is of order 0.3, and the diflerence between the two sets of ppdf’s, which
we are using to gauge the current uncertainty of the ppdf’s, is of order 0.1, i.e.,
about 30 percent of the asymmetry. An experiment to distinguish between the two
sets of ppdf’s, and thus provide a more accurate measurement of the ppdf’s than
deep-inelastic scattering, would need to measure the polarized cross-sections to an
accuracy of order 1 percent. The total polarized cross-sections, with and without
the p76 cut, are listed in Table 4.3. (There is no kinematic cut on the rapidity
ye for the cross-sections in Table 4.3; the reduction of 0'(A) for an experimentally
accessible range of ye, say [ye] < 2.5, can be judged from Figs. 4.8, 4.9 and 4.11,
4.12.) By Poisson statistics, a measurement of the cross-section to an accuracy of
1 percent would require «Io/0' = 1/\/N = 0.01, which implies a number of events
N of order 104 events. For the cross-sections in Table 4.3, which are of order 1
nb, the required f L: is of order 10 pb“1. But those are total cross-sections. Larger
integrated luminosity would be needed to measure difl'erential cross-sections to 1
percent accuracy. To have 10“ events in a rapidity bin Ay requires f £ = 104(do/dy-
Ay)“1, which is of order 100 pb“1/Ay. To measure the polarization asymmetry
would require this large I 1C for each proton polarization (L and R). These are only

94

Table 4.1: Total cross-sections for the processes p; + 1“) —+ W:h -—> ei +9733, where
A is the proton polarization, L or R. The polarized parton distribution functions
are taken from Refs. [38] and [39]. Also, we consider a transverse momentum cut
pr... > megn. There is no restriction on lepton rapidity y,; the effect of a cut on ye
may be judged from Figs. 4.8, 4.9, 4.11, 4.12.

 

 

 

 

charge A ppm-n 0’(A)(nb) ppdf
W+ L 0 1.298 Ref. [38]
1.211 Ref. [39]
WJr R 0 0.638 [38]
0.668 [39]
w- L 0 0.817 [38]
0.829 [39]
W“ R 0 1.117 [38]
1.048 [39]
W+ L 25 (GeV) 0.901 [38]
0.841 [39]
W+ R 25 (GeV) 0.442 [38]
0.464 [39]
W“ L 25 (GeV) 0.566 [38]
0.576 [39]
W“ R 25 (GeV) 0.775 [38]
0.728 [39]

 

order of magnitude estimates, but they demonstate that high luminosity, e.g., f I.
as large as 1 lb“, would be necessary to measure the ppdf’s at an interesting level
of accuracy. For regions of ye and PT: where the cross-section is very small, such as
the region of large ye, the required integrated luminosity would be correspondingly

larger.

4.4 Conclusions

We have calculated the cross-sections for Wi production, with subsequent decay to
65573,, in polarized proton - unpolarized antiproton collisions at center-of-mass en-
ergy 2 TeV. The left-right polarization asymmetry is sensitive to the spin-dependent

parton distributions Au(:v) in the case of W1“, and Ad(z) in the case of W“. We

95

used two recently published parametrizations of polarized parton distribution func-
tions to gauge the current uncertainty of Au(z) and Ad(:e), as measured from deep-
inelastic scattering. Wi production can provide a second method of measuring
these important distributions.

The accurate determination of the ppdf’s would be a necessary step in a program
to use polarized proton scattering as a probe of interesting new physics. If a polarized
proton beam were available at the Tevatron collider, the W3% production processes
would be used to obtain new data on ppdf’s. We find that with large luminosity,
i.e., f .C of order 1 fb“1 for each proton polarization, the functions Au(z) and Ad(z)
can be measured to a few percent accuracy over a large range of z. The interesting
large-:1: region can be well measured only if the rapidity coverage of the detector
extends to quite large 3],, at least beyond yc = 2.5.

Experimental studies of spin physics and the ppdf’s will soon be possible at
the BNL Relativistic Heavy Ion Collider (RHIC). A recent preprint [42] examines
the potential of various processes, including single W production, for constraining
the ppdf’s. Reference [43] focuses on W and Z0 production at RHIC, while other

processes at RHIC are discussed in References [44, 45].

Part II

A Light Gravitino and the
Supersymmetry Breaking
Mechanism

96

Chapter 5

Detecting a Light Gravitino at a
Future Linear Collider

5.1 Introduction to Supersymmetry

A too rapid unification, and an excessive appliance to parts and particulars, are the

twin dangers of speculation.

—Ralph Waldo Emerson

Many physicists tacitly live according to a faith that the fundamental interac-
tions, indeed all Natural phenomena, will eventually allow some unified descrip-
tion. Even if this is a mystical delusion, at very high energies, such as presumably
were common in the early universe, quantum gravity effects are important—but
are neglected in the Standard Model (SM). In this context, a puzzling feature
of the SM becomes apparent. The natural scale of gravity is the Planck mass
Mp = m z 1019 GeV, while the scale of the electroweak interactions is about
100 GeV. The tiny ratio between these scales 00“”) is unexplained at present. This
mystery is called the “hierarchy problem.”

Moreover, even should we accept this hierarchy as an a priori constraint, loop
corrections to SM masses will include Planck-scale fields. To end up with physical

masses of order 100 GeV, we would therefore have to multiply each such loop by a

97

98

constant of order 10“”. Unless such miniscule constants are explained at a deeper
level we feel that this is a “naturalness” problem. This “fine-tuning” will have to
be repeated at each order of perturbation theory.

Supersymmetry (SUSY) is a symmetry which solves the fine-tuning problem.
That is, if we define a SUSY theory with a hierarchy of mass scales, the hierarchy will
not be spoiled by loop corrections. In terms of Feynman diagrams, this is because
every boson loop will be cancelled by a matching fermion loop including a factor of
—1, according to the usual Feynman rule for fermion loops. This cancellation, which
preserves a hierarchy against higher-order corrections, is the practical motivation for
SUSY.

But SUSY is also aesthetically attractive, because the symmetry relates fermions
to bosons, the two fundamental classes of field in relativistic quantum field theory.
From the point of view which asks for unification, nothing could be more natural.
In order to accomplish this relation between bosons and fermions, it turns out that
one must involve the generators of the Poincaré group. Specifically, the generators

of SUSY are fermionic operators Q and Q which obey the anti-commutative algebra:

{Q01 Q0} = 2P“(7u)037 (5'1)

where P” is the four-momentum operator, and a and B are spinor indices. It is
satisfying that the generators of SUSY and of space-time translations are woven
together inevitably.

This connection with the Poincaré group is a clue to another attractive feature

of SUSY. In addition to Eq. (5.1), the SUSY generator also commutes with the

generator of space-time translations:

[Qm P“] = 0.

That is, SUSY is a global symmetry, independent of space-time coordinates. SUSY

can be made local, or gauged. The resulting theory includes a spin-2 field and

99

its spin-3/ 2 partner, identifiable as the graviton and the gravitino. In other words,
local SUSY is a (non-renormalizable) theory of quantum gravity, called supergravity
(SUGRA). Part II of this thesis deals with the phenomenology of a light (order keV)
gravitino.

This symmetry between bosons and fermions implies that every fermion has
a bosonic partner with the same SM quantum numbers. Reflection on the SM
particles reveals that none of them can be SUSY partners of each other. For example,
the neutrino cannot be the fermionic partner of the photon, because the neutrino
transforms differently under S U (2)L. Therefore if SUSY actually is a symmetry of
nature, the SUSY partners (“sparticles”) have not been observed. The hypothetical
sparticles are named according to their SM partners. The scalar partners of fermions
are known as “sfermions,” whose names are derived by adding an “s” prefix to the
name of their SM partner to get “squarks” and “sleptons.” The fermionic partners of
the bosons are named with an “ino” suffix: the “gauginos” are “gluinos,” “winos,”
“zinos,” “photinos,” and “Higgsinos.” However, the mass eigenstates of the fermionic
sparticles need not be the same linear combination of unbroken SM fields as the SM
gauge bosons. That is, in general we must consider “neutralinos” and “charginos,”
not specific combinations like the photino. Sparticles are denoted by a tilde over
their SM counterpart. For example, W is the wino field.

Only a few of these new fields will enter into our calculation in this chapter.
We will be interested in the production of gravitinos, the fermionic partner of the
graviton in SUGRA. The diagrams for the production of gravitinos from e+e“ an-
nihilation involve an exchange of the scalar partner to the electron field, called the
selectron. Because right-handed and left-handed electrons are independent fields,
they each have a scalar partner. These are named the right and left selectrons al-

though, again, they are scalars. The exchange of a selectron produces neutralinos

100

which will decay to gravitinos if the gravitino is light enough. The neutralino field
is an unknown combination of the neutral gauginos and Higgsinos.

The SM fields are conveniently distinguished from their superpartners by intro-
ducing a global symmetry called R-parity. The SM fields are R—even, while their
superpartners are R—odd. If R-parity is assumed to be conserved, then lepton and
baryon number are conserved l, and R-odd particles can only be produced in pairs

[47]. We assume R-parity conservation throughout this work.

5.1.1 Supersymmetry Breaking

If supersymmetry is unbroken, then particles must be degenerate in mass with their
superpartners. As we noted above, no superpartners have been observed. Therefore,
in our low-energy world, SUSY must be broken to be a viable theory of particles. As
long as the breaking mechanism does not generate any new quadratic divergences,
which will contaminate weak-scale physics with Planck-scale masses, then our theory
will still stabilize the hierarchy as we wish. Such SUSY breaking is called “soft”; and
the (relatively few) Lagrangian terms which satisfy this constraint are called “soft
SUSY-breaking terms.” These may be determined experimentally, but ultimately
we would like to explain their origin. In fact, Haber said recently [48], “The origin
of supersymmetry breaking is perhaps the most pressing theoretical problem in
fundamental theories of supersymmetry.”

We noted above that SUSY solves the fine-tuning problem by stabilizing the
hierarchy of mass scales against radiative corrections. It has been suggested that
models of dynamically broken SUSY may explain the hierarchy itself [49]. It is also

encouraging to note that in many supersymmetric models electroweak symmetry

 

lR—parity is imposed by hand in the minimal supersymmetric SM. Some theories, such as the
minimal left-right symmetric SUSY SM, automatically guarantee R-parity conservation at the
weak scale [46].

101

is broken dynamically by the renormalization group running of one of the Higgs
masses [50].

Since the origin of SUSY breaking is an unsolved problem, the subject is still
rather technical. Because we will be examining the phenomenology of a light grav-
itino, we will only consider models of SUSY breaking derived from more fundamental
models of broken SUGRA. Here we will merely distinguish two broad classes of mod-
els. In both types of model one posits a “hidden” sector of heavy fields in addition
to the SM “visible” sector. The hidden sector gauge symmetry is assumed to break
itself dynamically, for example by instantons [49] or by condensation of a gaugino
field [51]. This breaking is transmitted to the visible sector via some “messenger”
interaction. The messenger interaction distinguishes our two classes of models. In
older models, the visible and hidden sectors interact only gravitationally; these are
“gravity—mediated” models of SUSY breaking. More recently [52], there has been
a renewal of interest in “gauge-mediated” models of SUSY breaking, in which the
breaking of the hidden sector symmetry is transmitted to the visible sector by some
additional gauge interaction.

The gauge—mediated models discussed in Ref. [52] have been found to require
a very small mass for the gravitino [52]. Therefore the existence of a light grav-
itino is a definite prediction of the whole class of gauge-mediated SUSY breaking
models. Ruling out a light gravitino experimentally would mean ruling out the
gauge—mediated SUSY breaking mechanism. Discovering a light gravitino, on the
other hand, does not rule out all gravity—mediated models; for example, certain
“no-scale” unified SUGRA models [53] allow a very light gravitino. Should a light
gravitino be found, these models would have to be distinguished from each other by

the mass spectrum of the remaining sparticles.

102

5.2 Introduction to the Problem at Hand

Recently, Dine, Nelson and Shirman [52] have revisited a class of models in which
SUSY is dynamically broken at a low scale, within a few orders of magnitude of
the weak scale. Certain technical problems with these models have been solved, so
that they are more attractive as realistic fundamental theories. They noted that
in such models the lightest supersymmetric partner (LSP) is the gravitino (C) and
the next to lightest supersymmetric partner (NLSP) is a neutralino x3), which is a
mixture of neutral gauginos and Higgsinos.l Ellis, Enqvist, and Nanopoulos had also
noted as early as 1984 that “no-scale” models of dynamically broken SUGRA allow
a very light gravitino [53]. These models may have a bearing on the cosmological
constant problem and the strong CP problem,1 as well as dynamically creating a
mass hierarchy. Motivated by these theoretical possibilities, we consider in this
chapter the production of neutralinos and their subsequent decay into gravitinos.
The decay channel of the neutralino which we focus on is the decay into a photon
and a light gravitino.

In the gauge-mediated models the mass of the gravitino m3/2 is in the range
1eV < m3/2 < 10 keV. As in the spontaneously broken electroweak theory, where
the W mass is proportional to the vacuum expectation value of the Higgs field and
the weak coupling, so here the gravitino mass is related to the SUSY breaking scale

and the gravitational coupling [47]. We define a mass parameter

 

Msusy = \/m3/2MPlanck1 (5'2)

where the Planck mass is M plum], = 1/\/CN ~ 1019 GeV. In the gauge-mediated
models MN,” is in the range 105 — 107 GeV. This mass parameter may be the scale of

SUSY breaking, but in general we must only consider Mm,” to be a parameter which

 

lIn N=2 supersymmetric models of this type it is likely that the NLSP will be the right slep-
ton [54]. We do not treat this possibility here.

:The strong CP problem is described for example in Kaku’s book [1].

103

determines the gravitino mass and its effective coupling strength. In particular, in
the “no-scale” unified SUGRA models of Reference [53], SUSY may be broken by
gaugino condensation at scales above 107 GeV, yet the gravitino can be even lighter
than 1 eV.

In most other models of SUSY breaking, such as the minimal supersymmetric
standard model (MSSM), the gravitino mass is of order the weak scale and the
supersymmetry-breaking occurs at a scale in the range 1010 — 1011 GeV. Usually, in
the MSSM, the lightest supersymmetric partner is a neutralino, which is then stable
assuming R-parity conservation. Thus the decay of the neutralino into a photon
and a light gravitino probably signals “low-energy” SUSY breaking; in any case it
narrows the field of possible breaking scenarios.

In this chapter we study the possibility of detecting a light gravitino from the
decay of massive neutralinos (the N LSP, with mass mac?) which are produced in
pairs from e“e+ annihilation at the Next Linear Collider (NLC, a proposed collider
introduced in Chapter 2) with a right-hand polarized electron beam. (Use of a
right-hand polarized beam is to suppress background contributions, as discussed in
Section 5.3.4.) For this study, we assume that the center-of-mass energy \/S of the
NLC leptons is 500 GeV and the N LC luminosity is 50 fb“1 per year. Accordingly,
we ignore the mass of the electron and the mass of the gravitino. The experimental
signature of the signal event of interest is two photons with large missing energy.
The missing energy belongs to the two gravitinos, which escape undetected, because
their interactions are so weak. By calculating the cross section for this process as a
function of MN,” and mx?’ we will determine what region of this parameter space
would be accessible at the N LC. The accessible parameter range for current electron
(LEP/SLC) and hadron (Tevatron) colliders is also briefly discussed; detection of
a light gravitino at these colliders is more thoroughly analysed in Refs. [55] and [53].

104

5.3 Detecting a Light Gravitino at the Next Lin-
ear Collider

5.3.1 Production of Neutralino Pairs at the NLC

A pair of neutralinos, the NLSP’s, can be produced at the NLC via the tree-level
process e“e+ —-> x9361). At tree level, the particle exchanged in the t- and u-channel
Feynman diagrams are either left or right selectrons, and that of the s-channel
diagram is the Z-boson. The t- and u-channel diagrams vanish if x? is Higgsino-like
because the Higgsino-electron-selectron coupling is zero for a massless electron. The
s-channel diagram vanishes if x‘,’ is gaugino-like because there is no tree-level Z-R-
B or Z-WP-IIE interaction, because the Z is a combination of W3 and B, which
interact neither with each other nor themselves. (R is the supersymmetric partner
of the U (1)y gauge boson B, and W'-3 is the supersymmetric partner of the third
component of the S U (2);, gauge boson W3.)

In general the neutralino is a mixture of the neutral gauginos (B and I75) and
neutral Higgsinos. Determining its gaugino and Higgsino components will be im-
portant for distinguishing different models of SUSY breaking. For example, in the
constrained MSSM (CMSSM), which narrows the MSSM’s parameter space with
some cosmological and unification assumptions, the lightest neutralino x? is most
often primarily R, or “bino-like” (see [56] and [57]).

The NLC will provide a powerful tool for probing the content of the neutralino.
Calculation of 3+6“ cross sections is simpler than for hadron collisions, and more
than 90% polarization of the NLC electron beam is expected [19]. For simplicity, in
this paper, we shall assume that the electron beam at the NLC can be 100% right-

hand polarized. The assumption of 100% polarization will not significantly affect our

105

conclusions as compared to a 90% polarized beam, but the polarization capability
will simplify calculation and eliminate a background. For a right-polarized electron
beam, the WP component of the neutralino does not contribute to neutralino pair
production in e§e+ collisions because the I73 coupling to fermions is pure left-
handed, due to the .S' U (2);, gauge symmetry. As for the bino component of the
neutralino, recall that B may be written as a linear combination of photino 7 and
I75; specifically,

R = 7/ cos 9., — Wtanflw , (5.3)

where 9,, is the weak mixing angle (sin2 9., = 0.23). We decompose the bino this way
so that we may compare the cross section for production of mass-eigenstate photinos
to production of general neutralino mass-eigenstates. If x‘,’ is photino-like, then the
production of xgx? from a right-hand polarized electron beam occurs only through
the 7 component in Eq. (5.3). Similarly, for a general gaugino-like neutralino, only
the mass-eigenstate component of the bino will contribute to Xi’Xi’ production, while
the I75 remainder will not contribute when the electron is right-polarized. For the
rest of this thesis we assume that the neutralino x‘,’ is gaugino-like. The case that
x? is Higgsino-er is discussed in the paper upon which this chapter is based [58].

For a gaugino-like neutralino we define a mixing angle a by
x3): Bcosa+fi738ina . (5.4)

Then the cross section for xfi’x? production from a right-handed electron is, by Eqs.

(5.4) and (5.3),

+

 

(5.5)

_ cos a 4
U(eRe

0 0 _ — + ~~

—> XIXl) ”(€126 -* 77) (co8 9...
in terms of the cross section for photino-pair production. In the numerical calcula-
tions below we always report results for a = 0, corresponding to x? being R-like; it

is straightforward to determine the numbers for other values of a.

106

The cross section for efie+ —1 77 has been published [59, 60]. There are two
Feynman diagrams remaining for this process after our assumption of a gaugino-
like neutralino and a right-handed electron beam, representing the t-channel and
u-channel exchange of a right selectron 3;}. The differential cross section do/ d cos 0
for efi(p1)+ e+(p2) —1 Xi)(P3) + x?(p4), where 9 is the angle of an outgoing neutralino

x? relative to the c“ direction, is

do e4 1 mi, “mi? 2 "—mie 2
dc089—161rS — S mg-k— + mg—u

”mi: “212)“ (5.6)

_ 2 2
(me-h — t)(mc-;z — u) cos 0,”

 

 

 

 

where t = (p1 — p3)2 and u = (p1 — p4)“. (Because the final state x9361) contains
identical particles we must specify the normalization of the differential cross section.
The normalization of do/d c080 is such that the integral over c080 from —1 to
1 is 2 times the total cross section.) This cross section is shown in Fig. 5.1 for
x/S = 500 GeV, for a = 0 and two cases of the mass parameters: mx‘,’ 2 100 GeV,
me; = 300 GeV; and mx? = 200 GeV, me; = 600 GeV.

5.3.2 Decay of the Neutralino

Since the photino component of the neutralino x? would mainly decay into a photon
and a gravitino for models ( [52], [53]) in which the gravitino is extremely light, the
experimental signature of the signal event we consider here is two photons and
missing energy. The missing energy belongs to the two gravitinos. Other signals
could be considered that involve the zinc 2 component of x?, which would decay
to a Z-boson and a gravitino; these signatures are 7Z with missing energy and Z Z
with missing energy.

The rate of the neutralino decay x? -—1 76' can be related to the rate of the

photino decay 7 ——7 7C from the fact that a gaugino-like neutralino is a mixture of

da/d cosQ (fb)

300

250

200

150

100

50

107

 

IIIIIIIIWI

 

  

l I

lllllllllllllllllL lLLLllllll

 

 

1.0

Figure 5.1: Differential cross-section do/ d mad for esze+ —1 xfi’xg’, where 0 is the
angle of a neutralino, at J3 = 500 GeV. The mixing angle a is 0, and the masses are
mxo = 100 GeV, me; = 300 GeV (solid curve), and mx? = 200 GeV, me; = 600 GeV

(dashed curve).

108

7 and Z
x? = 7 cos(a — 9",) + Zsin(a —— 0w) , (5.7)

and the fact that at tree level only the 7 component decays to 76, with nearly 100%

branching ratio. Therefore we have
I‘(x(,) —+ 7C) = I‘(7 -——1 7C) cos2(a — 0w) . (5.8)

The photino decay rate has been calculated [61, 59] to be

m5

 

(5.9)

where m is the photino mass, assuming the photino is a mass eigenstate. In general,
however, 7 is not a mass eigenstate, and in Eq. (5.8) we use the neutralino mass

mxc]; that is,
5

.. m 0
I‘(x(1’ —> 70) = 87% cosz(a — 9w) . (5.10)

The parameter (1 is related to our M,,,,,, by [47]

\/§ \/§

4: M 0,, =—M2 . 5.11
{—41rm3/2 Pl cl: “—417 susy ( )

 

The decay rate in Eq. (5.9) is derived by noting that at energies large compared to
the gravitino mass, only the spin-1/2 component of the spin-3 / 2 gravitino interacts.
The spin-1/2 component of the gravitino is called the Goldstino, a fermionic ana-
logue of a Goldstone boson, because it is the massless excitation arising from the
breaking of SUSY. When the local SUGRA symmetry is broken, the two compo—
nents of the Goldstino combine with the two components of the massless gravitino,
to make a four-component massive gravitino. Then 7-7-6 vertex factor, including
only this Goldstino part of the gravitino coupling, is W,,7"]1/5/2d [47]. Note that
the decay rate is inversely proportional to the fourth power of the supersymmetry-

breaking scale MW," and is proportional to the fifth power of the neutralino mass.

109

For Mung = 106 GeV and mx? = 100 GeV, which are characteristic values for our

study, the lifetime of the NLSP x9, given by

1 _6M4 1

8118
1': — y

1‘ mi? cos2 (a — 9w) ’

 

(5.12)

is expected to be 5 X 10“10 sec for a = 0. For MW,” in the range 104 to 107 GeV, 1'
can vary from 10“18 sec to 10“6 sec.

The average distance travelled by a neutralino before it decays at the NLC is

D = 73c?" , (5.13)

 

where ['3 is the neutralino velocity 6 = \/1 — 4mig/S and 7 = 1/ W One
must observe the decay photon to make any statement about existence of a light
gravitino, and to detect the photon from x‘,’ —1 76? the decay must occur within the
detector volume. We shall assume that D must be smaller than 1 meter to observe
the decay photon inside the detector. Figure 5.2, which is explained further below,

shows the range of parameters MN,” and mx? for which at least ten signal events

would be observed at the NLC.

5.3.3 Background-Flee Signal Events

In certain circumstances there will be no background to our signal. The signal we
seek is two photons plus missing energy. If the massive neutralinos live long enough
before decaying into photons, then it will be possible to verify that the photons did
not originate at the interaction point (IP) of the collider. In this case there will be
no SM process with the same signature to act as a background. At the NLC the
beam size is expected to be stable with dimensions 0',_X 0,, x 0', = 5 nm x 300 nm x
100 um [62]. This is small enough that we may consider the interaction region to
be a point. Depending on calorimeter technology, the angular resolution for the

trajectory of a few Gev photon is typically in the range 10 to 100 mrad, assuming

110

Msusy (GGV)

 

10 20 50 100 200 500 1000
mxp (GeV)

Figure 5.2: Range of parameters Mm,” and max? accessible at the NLC. We allow
M,,,,,, to vary from 104 to 107 GeV. The shaded region is the range of Mum, and
mx? for which D < 1m and 0' > 0.2 fb, where D is the typical decay length of

x? and o is the production cross-section for efie+ —+ x‘fx? at the NLC. The bound
0' > 0.2 fb is equivalent to observing more than 10 events assuming integrated
luminosity 50 fb“1. The cross-shaded region is for 10cm < D < 1 m, corresponding
to the background-free signal process. Parameter values for this plot are a = 0 and
me; = 300 GeV.

111

 

 

 

 

 

 

2.0 ,- ' I I j I fl I T I—l I I I I I I I r I’Ifi I I I w I I I q
A i I
:99 1 5 _ i
U . ' 1
\ 1-

Q " 1
1.1.1 *- m a
V 1.0 r —
b ..
m r ,
g : :
L— _
b 0.5 _ q
PU " -1
0.0 I. 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1 l4 1_1 1 .

0 50 100 150 200 250 300

E7 (GeV)

Figure 5. 3: Differential cross- -section in photon energy da/dE, for e Re+ —1 X9361

7700 at the NLC, assuming a— — 0. The mass parameter values are 111,, 0 =100 GeV
and m~ = 300 GeV. (The normalization of do/dE, is such that thex integral over
E, is 2 times the total cross-section, because the final state has identical photons.)

the photon originates in the inner portion of the calorimeter. Given a 1 m radius
for the calorimeter, the pointing resolution to the IP would be 1 cm to 10 cm. Note
that the resolution will generally improve as 1/ ‘/E_,-. The typical energy E, of the
photon from the decay of a massive x? is x/S/4; more precisely, the distribution
in photon energy is constant with E, between «S (1 — fl)/4 and \/§(1 + fi)/4, as
shown in Fig. 5.3. To be conservative, we will assume that if a neutralino travels
more than 10 cm before it decays, then the displaced origin of the decay photon can
be well-separated from the IP at the NLC.

If 10 cm 5 D S 1 m, then the typical signal event occurs within the detector but
away from the IP, and the event is background-free. The range of Mm,” and mx?
in which this condition on D is satisfied is the cross-shaded region in Fig. 5.2. We

allow Mm,” to vary between 104 and 107 GeV in the figures; this range is within a

112

few orders of magnitude of the weak scale, as required in the models of Ref. [52].
This range corresponds to gravitino masses allowed in the SUGRA models of [53],
although in these models SUSY is actually broken at a higher scale than M,.,,,,. We
see from figure 5.2 that for mx? = 100 GeV, the background-free signal occurs for
Mm,” around 106 GeV. Conversely, for Mm", = 106 GeV, the background-free range
of mat? is from 85 GeV up to 122 GeV. (We note that for fixed Mm," and s/S', D
is proportional to fl/mgg.) However, for the lowest Mm.” considered, 104 GeV, the
background-free range of mx? would be below 10 GeV. Since we are considering a
signal with no background, observing just one event would be in principle sufficient
to declare the discovery of the signal. The integrated luminosity of the NLC is
50 fb“l per year, so the N LC experiment is sensitive to a signal production cross
section larger than 0.02 fb.

We find that over a reasonable range of the mass parameters involved in the
Feynman diagrams, the cross section is larger than 1 fb. Because we chose to polarize
the electron beam right-handedly, the production of XiXi) occurs by exchange of a
right selectron £72, and the cross section depends on the mass of £73. (In this work,
we shall ignore the possible small mixing between 3:71; and ET, so that 31'; is the
mass eigenstate with mass mat.) We are assuming, in accord with the models of
references [52] and [53], that the LSP is the gravitino and the NLSP is Xi: so 1?];
must be heavier than x9. Also, in both sets of models, we expect me; to be of the
same order as mxcla, because sfermion masses squared ("1:72) depend on a two-loop
diagram while gaugino masses (mx?) depend on a one-loop diagram. (The details
depend on the gauge groups assumed in the model.) In Table 5.1 we give the signal
production cross section 0' for a few choices of mx? and me; at the NLC, assuming
M,.,,,, = 106 GeV. In all cases the cross section is larger than 1 fb. We conclude that
for Mum, = 106 GeV and me; less than 1TeV, the N LC guarantees the discovery of

113

Table 5.1: Production cross-section 0' and typical decay distance D for the process
efie+ —> XiXi at the NLC, assuming a = 0, for several values of masses. (Note
that the electron beam is right-hand polarized, so the cross-section is a factor of 2
larger than that for an unpolarized electron beam.) These numbers are for Mm," =

106 GeV.
mxoL (GeV) m; (GeV) 0' (fb) D (m)

R

 

100 100 670 0.35
100 300 252 0.35
100 600 57.1 0.35
100 1000 11.7 0.35
200 300 84.6 0.0036
200 600 17.6 0.0036
200 1000 3.4 0.0036

 

the background-free signal event if 85 GeV S mx‘.’ 3 122 GeV.

Table 5.1 also shows the typical decay length D in the various cases. In fact,
it is the decay length D, not the production cross section 0', that constrains the
discovery of the background-free signal event, because 0' is larger than 0.02fb for
mxc; < s/S / 2 = 250 GeV and me} < 1TeV except when "‘21? is extremely close to
\/S / 2. Our conclusion does not change even if we require observation of at least 10
signal events to declare discovery of the signal.

The shaded region in Fig. 5.2 shows the values of MN," and mx‘l’ for which
D 3 1m and for which the number of events at the NLC (0' x 50 fb“1) is greater
than 10. The diagonal boundary on the left is the curve D = 1 m, and the vertical
boundary on the right is where the number of events equals 10. The cross section
0' does not depend on Mm,” so the boundary on the right side is a vertical line
near the threshold at "‘71? = x/S/2 = 250 GeV. Figure 5.2 is for me; = 300 GeV,
where the right selectron ET; is the exchanged particle in the process efie+ —+ x‘llxcll.

Increasing me; up to 1 TeV only slightly shifts the vertical boundary on the right

toward the left. The cross-shaded region has 10 cm S D S 1m, corresponding to

114

the background-free signal. The two diagonal boundary curves depend only on the
kinematics, not on the cross section, so they do not depend on the selectron mass
me;I .

This calculation in terms of the average decay length D provides a rough estimate
of the parameters M w,” and "‘11? for which background-free events occur. The cross—
shaded region in Fig. 5.2 is the region of parameter space where the average decay
distance of the neutralino D = fl7cr is between rmgn = 0.1m and rm“ = 1m.
However, the cross section is large enough that there will be events in which the
neutralinos decay in a distance between rmin and rm, even though their average
decay distance is outside that range.

To calculate a more precise parameter range for background-free events, let
P(r)dr be the probability that a neutralino travels a distance between r and r + dr
before decaying

P(r) = %e-'/0 . (5.14)

Then the fraction of events for which both neutralinos decay in a distance between
rmin and rm“, is

I‘Tmax

fBF = ( 130011?)2 . (5.15)

' 7min

If we require that there are more than 10 background-free events, given integrated

luminosity equal to 50 fb“1, then we must have
ofBF > 0.2fb , (5.16)

where 0' is the production cross section. The range of parameters that satisfy this
condition is larger than the simpler estimate in Fig. 5.2, and is shown as the cross
shaded region in Fig. 5.4.

In the rest of the shaded region, i.e., not including the cross-shaded region,
the decays occur within 10 cm of the IP. We assume conservatively it is not possi-

ble to detect a displaced vertex of these decay photons. For these values of M,.,,,,

115

Msusy (GeV)

 

10 20 50 100 200 500 1000
mxp (GeV)

Figure 5.4: Same as Fig. 5.2, but with a more precise calculation as described in
Section 5.3.3.

and mx‘.’ there are intrinsic backgrounds to the signal process, which we discuss next.

5.3.4 Non-Background-Free Signal Events

As shown in Fig. 5.2 and Table 5.1, assuming M,.,,,, = 106 GeV, if mx? 2 120 GeV
then the decay length D is less than 10 cm, which does not satisfy the criterion for
being a background-free signal event. The experimental signature of the signal event
in this case consists of two photons coming out of the interaction region with large
missing energy. Any background will consist of a standard process with a similar
event signature. Before identifying these processes, let us examine the details of the
signal event, so that we may better distinguish them from the features of the various
backgrounds.

To obtain the distributions of the decay photons we evaluated the full correlated

helicity amplitudes including the decays of the two neutralinos. However, we did

116

not find a noticeable difference compared to a simpler calculation in which the
decays of the x‘,”s are treated independently, ignoring the correlation between their
polarizations. Figure 5.3 shows the distribution of the single-photon energy E,, for
mx‘l’ = 100 GeV and me; = 300 GeV. The distribution is approximately constant
with E, between x/S(1 — )6)/4 and \/§(1 + fl)/4, so the order of magnitude of the
photon energies is \/S / 4. Typically, each photon has large transverse momentum
p}. Figure 5.5 shows the p} distribution of either photon, for mx? = 100 GeV
and me; = 300 GeV. If x? is heavy, then the two decay photons are acollinear,
which indicates missing transverse momentum. If x‘,’ is light, then due to the large
momenta of the x9’s the two photons will tend to be more nearly back-to-back,
but the sum of the two photon energies will still be peaked at about s/S / 2, which
indicates missing energy. This latter feature of the signal event is shown in Fig. 5.6,
which is the distribution of the sum of photon energies, for mxc] = 100 GeV and
me; = 300 GeV. (Figures 3—5 are for a = 0.)

Since the signal event consists of two photons, the first obvious background
process to consider is the ordinary QED process efie+ —-> 77. However, the photon
kinematics for this process are much different than for the signal process: The
photons will be approximately back-to-back with combined energy s/S. Initial state
radiation can change the energy by a small amount, but at the NLC it is expected
that the effects of bremsstrahlung will not significantly change the available center-
of-mass energy of the e“e+ system [19]. Hence in events generated from the QED
process efie“ —> 77 , including possible initial state radiation, the sum of the two
photon energies is close to \/S. By demanding that the sum of the two photon
energies is around VS / 2, indicating large missing energy in the event, and that
the two photons are not back-to-back, one can eliminate the large background from

photon pair production.

117

 

 

 

 

 

 

 

'- I l l I I I I I l I I I T “l—I l I I I I I T I I d
. 1
9 1.5 — —
g; I 1
\ I I
59. 1.0 — —
v __ ..
N?" ' ‘
a, . .
pd . .1
\ 0.5 "‘7 '—
b .. ..
"U : ‘
0.0 - l l l l I l l 1 l l 1 1 1 1 I I l l l I l l l d

O 50 100 150 200 250

pl (GeV)

Figure 5.5: Differential cross-section in photon transverse momentum do/dp} for
efie+ —» XIX? -—> 77GG at the NLC, assuming a = 0. The mass parameter values
are mx‘l’ = 100 GeV and me; = 300 GeV. (The normalization of do/dp}- is such
that the integral over p} is 2 times the total cross-section, because the final state
has identical photons.)

118

 

 

 

 

 

g E I I r I I I I I I I I I I I I j I I I I I l I 1 :1
a) . 1
U 1.0 r .2.
\ ~ -
.0 I I
1'; 0.8 :- 1
A“ * g 3
a 0.6 - —
Lil I 2
+ : :
$1 0.4 .- -:
E13 - .
>5 : r
\ 0'2 T. _.
b - .

v 0.0 I. l l_l I l l l l l l I l l l l l l l I l I l

O 100 200 300 400 500

E71+E73 (GeV)

Figure 5.6: Differential cross-section in the sum of photon energies d0'/d(E,l + E,,)
for efie’r ——> x‘fx? -> 770G at the NLC, assuming a = 0. The mass parameter
values are mx? = 100 GeV and me; = 300 GeV.

Another background process at the NLC is efie“ —> 77Z with the Z-boson decay
Z —> 1117. This process has large missing energy carried away by the neutrino pair.
We have calculated the cross section for this background process. The cross section
diverges at low transverse momentum p1 of either photon, so we require 117 to be
larger than 20 GeV for each photon. We find that this background cross section is
14.7fb (including three neutrino flavors from Z-boson decays) which is rather small
(cf. Table 1). For comparison, with the same pT cut on the photons the production
cross section of the signal event efie+ —> XiXi) —> 77GG’ is 210fb for mx? : 100 GeV
and me; = 300 GeV, and it is 16.9fb for mx? = 200 GeV and me; = 600 GeV.
Although the 77Z cross section is comparable to the signal cross section in the limit
of large SUSY-partner masses, this background event can be easily distinguished
from the signal event: The missing energy in the 77Z event is entirely due to the

decay of the Z-boson, so the invariant mass of the invisible particles (the m7 pair)

119

 

0.25

d
‘
fi

I I I I I I I I I I I I I I l

0.20

0.15

0.10

 

0.05

0.00 , .LAL._. m. ln_. .
00 200

1 300 400 500
Minv (GeV)

do/dMinv (fb/GeV)

OIIII IIII IIII IIrI IIII
”—H-LLLHI l I l

 

  

llllllllllllllllllllllll

_I_Ll

h

 

 

Figure 5.7: Differential cross-section da/dMgm, in the invariant mass of invisible
particles M5,", for the signal process efie+ —> XiXi —> 77GG, and for the background
process size” -> 77Z, with decay Z —-» m7. The photon transverse momenta are

required to be greater than 20 GeV. The masses for the signal process are "be? =
200 GeV and me} = 600 GeV.

is peaked at the Z-boson mass. The invariant mass can be determined from the
visible energy, i.e., the energy of the initial state and the two photons. We define
this invariant mass squared to be M3", = (193- +pe+ —p.,l —p.,, )2. The invariant-mass
peak for the 77Z background event is shown in Fig. 5.7, along with the distribution
in M5,", for signal events with mx? 2 200 GeV and me; = 600 GeV. The signal
distribution is broad, so requiring Mim, to be away from the Z-peak to discriminate
against the 77Z events would further suppress this already small background. For
instance, as seen in Fig. 5.7, if we require IMgm, — mzl > 20 GeV, then the ratio of
signal to background becomes 14.6 fb/0.7fb=21, as compared to 16.9 fb / 14.7 fb=1.15
without the invariant-mass cut.

We have noted in the previous section that the search for our signal is limited

by kinematics rather than cross section, so we need not fear such cuts. Figure 5.8

120

 

 

 

 

 

BOO _ LI‘T ‘1 r I I I I I I I F l I I I I r I l J

L \ \ \ .

600 — '-

A :

.0 - .

b 400 — —

b t 1
200 —

O '- I I I I L1 I I I i I I I I i.I~;:-I~l'fi-I—I.
0 50 100 150 200 250

mx? (GeV)

Figure 5. 8: Total cross- s-ection at the N LC for the process e§e+ —» X176? with a— — 0,
as a function of mx° for m~ — —300 GeV (solid curve), men =100 GeV (dashed
curve), and me~ = 1600 GeV (dot dashed curve). (Note that only mx 0 < men is

allowed 1n the models we consider because x1 is the NLSP. )

shows the X976? production cross section a for a few choices of me}, as a function
of mx? at the NLC. The order of magnitude of a is 100 fb. We conclude that it is
possible to observe the non-background-free signal event at the NLC with 50 fb—1
integrated luminosity, for all the shaded region shown in Fig. 5.2 except if mx‘,’ is
extremely close to the threshold for XiXi production, which is \/§ / 2 = 250 GeV for
the NLC.

Before closing this section, we comment on the reason for using a right-hand
polarized e" beam for the proposed experiment. Polarization has little effect on
the backgrounds already discussed. As the QED coupling is pure vector, the cross
section for efie+ —> 77 is independent of electron polarization. Because the vector
part of the e- e-Z coupling is proportional to 1—4 sin2 0",, which is small, this coupling

is approximately pure axial vector. Thus the background which produces a photon

121

pair and a Z boson is also nearly independent of the electron polarization. The signal
process, on the other hand, may differ for the two polarization cases. efie+ —I xgx‘l’
occurs by exchange of a £72, which does not interact with the V73 component of
x?; whereas e; 3+ —-> Xi’Xi’ occurs by exchange of a 31:, which does couple to 1717’.
Furthermore, the masses of ET; and E}: may be different. We might expect the E}: to
be heavier due to extra contributions from the weak interaction [59]. A heavier left
selectron mass would supress the left polarized cross section, leading one to choose
the right polarization.

In any case, running the experiment with a right-polarized electron beam also
eliminates a background we have not yet considered, the process e"e+ —» 14217677-
The complete gauge-invariant set of diagrams for this process includes the diagrams
for e’e+ —> 77Z(—+ 113178), discussed above, but also includes diagrams involving
e-Ve-W interactions. These contributions to e'e+ —» 11317377 arise from Feynman
diagrams in which a t-channel W boson is exchanged between the two fermion lines.
A right-polarized electron beam eliminates this additional background, because the
W coupling to fermions is pure left-handed. Therefore a right-hand polarized elec-
tron beam has a significantly better ratio of signal to backgrounds, compared to
an unpolarized e" beam. Predicting this additional background due to W boson
interactions for the NLC with less than 100% right-hand polarization of the electron
beam, or for current colliders which have weaker polarization capability, is a topic

for a more detailed study.

5.4 Discussion and Conclusion

We have discussed the possibility of detecting a light gravitino at the NLC, a pro-

posed future e‘e+ collider with center-of-mass energy \/§ = 500 GeV and with a

 

300 (- I I I I l I I I I l I I I I l
250 :-
200 '—

150

a (fb)

100 L

50:—

 

 

 

 

0 500 1000 1500
\/S (GeV)

Figure 5. 9. Total cross-section for the process eRe+ —+ X1761 with (1 =0, as a
function of x/_. The mass parameter values are mx 0 =100 GeV, m~ =300 GeV
(solid curve), and mx 0 _ -200 GeV, m~ — —600 GeV (dashed curve).

right-hand polarized e" beam, from the decay x? —-> 763' where the neutralino x?
is produced in pairs in the e'e"’ collisions. The proposed NLC would also operate
at \/§ = lor 1.5 TeV with a luminosity of 200 fb’1 per year. Figure 5.9 shows the
total cross section a'(e,}e+ —> XiXi) for a gaugino-like x, with a- — 0, as a function
of \/§, for two cases of the mass parameters: mx‘,’ = 100 GeV me; = 300 GeV, and
mxg = 200 GeV me; = 600 GeV. If mx? is large then the cross section increases with
\/§; but if mx? = 100 GeV then the cross section actually decreases as \/§ increases
from 500 GeV to 1 TeV.

The ranges of the parameters MW,” and max? that are accessible for the three
modes of the NLC are shown in Fig. 5.10. (For simplicity, we did not separate
the regions of parameters for background-free signal events from those for non-
background-free signal events, as we did in Fig. 5.2.) It is clearly seen that at
500 GeV the NLC will probe a slightly larger region of the parameters MN,” and

 

 

 

 

 

 

107
9
G)
8 106
3:
:1
2M
105
I ' '
104 l I lllJlll l I I lIIlII
10 20 50 100 200 500 1000

mxp (GeV)

Figure 5.10: Range of parameters Mm,” and "‘21? accessible at the N LC with

«3 = 500 GeV (solid line), 1TeV (dashed line), and 1.5 TeV (dot-dash line). The
luminosity per year is 50, 200 and 200 fb‘l, respectively. Mass parameter values
are 111;;z = 300 GeV, 500 GeV and 750 GeV, respectively, and the mixing angle a is

0. The interior of each triangle is the region where the number of events is greater
than 10, and D < 1m.

mxg than a TeV NLC if mx? < 250 GeV. At higher x/g the neutralinos are more
likely to exit the detector before decaying. Therefore, a TeV NLC is needed to detect
a light gravitino through the event signature of 2 photons plus missing energy, only
if the NLSP x? is heavier than about 250 GeV.

We should also consider whether this process can be discovered at a current e‘e+
or zip collider. First consider the case of LEP/SLC, with center-of-mass energy J— =
171.2 = 91 GeV and integrated luminosity 450 pb'1 (which is about the integrated
luminosity at mz when combining all the experiments from LEP and SLC), and
also LEP-II, with center-of-mass energy \/§ = 190 GeV and luminosity 500 pb"1

per year per experiment. Figure 5.11 shows the range of parameters MW," and

mx? accessible by LEP/SLO, LEP-II, and the NLC, superimposed, with the same

124

 

I l ‘
111 1 1 1 1111

1

Msusy (GeV)
01

 

1 1 1111111

I
IlIIIlIl I I IlIIII

10 20 50 100 200 500 1000
mx? (GeV)

 

 

 

 

Figure 5.11: Range of parameters MN,” and mx? accessible at the NLC (solid line),
LEP/SLC (dashed line), and LEP-II (dot-dash line). The interior of each triangle is
the region where the number of events is greater than 10, and D < 1 m. Parameter
values for this plot are a = 0 and me; = 300 GeV.

assumptions as Fig. 5.2.

The accessible region depends mainly the threshold mass 1nch = J? / 2; and on
kinematics, i.e., on the requirement that D < 1 m, where D is the mean distance
traveled by the neutralino before it decays. In the case of a gaugino-like x? at the
N LC we found that the cross section for right-handed electrons is of order 100 fb (cf.
Table 1), large enough to discover the signal even for large me; and mx? near the
threshold. In the case of LEP/SLC or LEP-II with unpolarized e' beam, the cross
section depends on the masses of both selectrons (ET; and 31:), but if we assume
these masses are about equal then we may estimate that the unpolarized cross
section is approximately equal to the right-handed 6' cross section. The boundaries

for LEP/SLC and LEP-II in Fig. 5.11 were calculated with this assumption, with
me; = me; = 300 GeV. For example, the cross section for LEP-II (\/§ = 190 GeV)

125

with mx? = 100 GeV and me; = mg; = 300 GeV is 143 fb. In Fig. 5.11 the triangular
area enclosed by the curve for each collider is the range of parameters such that there
would be 10 events (total for LEP/SLC, or per year per experiment for LEP-II or
the N LC), with average decay length D < 1 m.

Figure 5.11 implies that for Mm,” between 104 GeV and 107 GeV, the range rele-
vant to the model of Ref. [52], only the region of parameter space with mx? 5 30 GeV
is accessible at LEP/SLC; however, an interesting region for Mm,” < 106 GeV will
be accessible at LEP—II. However, because the electron beam polarization at LEP-II
is only of order 50%, there is a background from left-handed electrons: the process
e‘e+ —» lief/C77 in which a W-boson is exchanged between the fermions. This back-
ground may be significant at the level of 10 events. A definitive analysis of a LEP-II
search for the gravitino process must include this background.

Next we turn to the Tevatron pp' collider. We have calculated the cross section
for the process pf) -—1 X936? at fl = 2 TeV, for bino-er x9, using a Monte Carlo
program with CTEQ2 parton distribution functions. We find that the cross section
is 19 fb for mx? = 100 GeV and m; = 100 GeV, where 6 indicates any squark. The
cross section decreases with mg. Because of the small production rate (about a factor
of 35 smaller than a 500 GeV e+e‘ collider) and the additional large backgrounds
in hadron collisions (either from physics processes or from the imperfectness of the
detector), the current total integrated luminosity of the Tevatron is probably too
small to provide a useful search for the light gravitino. With the upgrade of the
Tevatron, at which the luminosity will increase by an order of magnitude (to about
2fb"l per year), one can probably detect the light gravitino for some values of
MW,” and mxcl). This requires a separate study as well. The Tevatron and LEP are
considered in detail in Reference [55].

In conclusion, the proposed NLC could provide a means to search for the decay

126

of a gaugino—like N ext-to-Lightest-Supersymmetric-Particle x? into a gravitino and
photon, Xi —) 761' , for a significant part of the parameter space relevant to models in
which the SUSY-breaking scale is within a few orders of magnitude of the weak scale.
Models with a light gravitino but a higher SUSY-breaking scale would also be probed
by the NLC. The range 105 Gev< Mum, < 107 GeV corresponds to gravitino masses
in the range 1 eV < m3 /2 < 10 keV. This range of parameters could be accessible at
the NLC, depending on the neutralino mass. The neutralino pair production cross
section is large enough, of order 100 fb, that it is not a limiting factor. The limiting

factor is the lifetime of the N LSP, which is proportional to M 4

easy ‘

If Mm,” is too
large, then the NLSP will exit the detector before decaying, and no information on
the gravitino will be obtained. But as shown in Fig. 5.2, if Mm,” is small enough
for a given neutralino mass mxg, then the neutralinos will decay inside the N LC
detector. The decay x? —> 76? can then be used to detect the gravitino by seeing
the two photons with large missing energy from Xi) pair production.

This study was motivated by the class of gauge-mediated dynamical SUSY-
breaking models, which require a light gravitino. We investigated the possibility of
observing the light gravitino at the N LC, and found that for a large region of the
parameter space, the gravitino would be produced copiously and within the detector.
Therefore, non-detection of the gravitino would rule out a region of parameter space
of the gauge-mediated models. Detection would narrow the field of viable SUSY-
breaking models, providing evidence for the gauge-mediated scenario—though not

proof, since other models do exist which allow a light gravitino.

5.5 Additional Comments

After completing this work we received the paper “Experimental Signatures of Low

Energy Gauge Mediated Supersymmetry Breaking”, by S. Dimopoulos, M.Dine,

127

S. Raby and S. Thomas [54], which is similar and complementary to this chapter.

Also, an ee'ry plus missing transverse energy event was reported by CDF at the
Fermilab collider which seems to have no SM interpretation, and may be consistent
with a light gravitino SUSY scenario. This and another SUSY interpretation of the
event are analyzed in Ref. [63]. The authors of Ref. [63] also consider other signals
which can occur at the Fermilab Tevatron or LEPII in the light gravitino scenario.
In Ref. [64], the CDF event is considered in the “no-scale” SUGRA framework.

APPENDICES

Appendix A
The Standard Model Lagrangian

After symmetry-breaking, the physical fields can be written in terms of the S U (3)0 x
SU(2)], X U(l)y fields:
_1_

fi
204‘ = (—B“ sin 0w + WiJ cos 9W)’

W” = (W," 2]: in),

A” = (8‘“ cos 0w + W; sin 0w),

where the Weinberg angle 0w parametrizes the mixing of the electroweak fields. In
the unitary gauge the Higgs field can be written in terms of one real field 1] and a

constant vacuum expectation value v:

‘1’“) = ( [v +111» Nz‘ )

The Lagrangian is then:

LSM : Lboson + Llepton + Lquarlv

Lb..." = -2 mema) — $109 - (1)2
1 _ ..
— 5|D,,W;‘—D.,W,,:‘|2 (AJ)
1

1 1
+ MW2(W;)*W+" — 74-2}:qu + ngzgzow — ZAWAW

128

129

— éflg cos owzf}, + eAW](W+”W’” — W'"W+") (A.2)
+ 71-512mm; — w;w:)2
+ §[92(Iw+12+ IW‘Iz) + (9” + gszszwum + 112)
+ -;-[(¢9"n)(¢9un) - 213112] - Mn" - 2M“.
where Latin indices run from 1 to 8 in color space, and Greek indices run from
1 to 4 in Minkowski space. The terms labelled A.1 and A.2 are discussed in the

Introduction; they are responsible for electroweak gauge tri—boson couplings. In

terms of the physical fields the covariant derivative is
D” E 3,, + ig(Z3 cos 0w + A,, sin 9w),
and the field strength tensors are
Fuu,&(G) = 3,10“ — BuG’us - g.fa,3,aG,.,aGu,a,

Aw, = anAu—BuAu,

23,, = auzg—auzg.

In terms of v and the SU (2);, and U (1)y coupling constants g and g’, the new

parameters are

I

 

sin 0w = ——£——,—,
(9’2 + 92)?
e = g’g
(9” + a”)?
1
MW = avg,
1 1 MW
M = -— ’2 2 ’ = .
Z 2v(g +9 )2 cosOw

n is the real Higgs scalar field.

Finally, the lepton and quark terms are:

Llepton + Lquark :

130

Zfli fi-CQI4“mI‘%fllf

allf

1
+ 2 (K “9:7" 5A0 )‘IGIM
q

9 - _ _
— 75‘ 2: ¢§L’v"[a+w:+a Wuwf“

=eiuirIuiclt

 

_ g " p _ 5 0
2m W .2113,” (v1 017 )pr-

Recall that electroweak doublets have 13 = i; for their upper and lower com-
ponents, respectively. Then the vector and axial-vector couplings of the Z0 to a

fermion are given by
‘vf = I3J—2stin20w,
a, =1 I3’f.

a'i are equal to %(0'1 2]: 1:02), respectively.

Appendix B

Helicity Amplitude Method

B.1 Helicity Eigenstates

In the Weyl basis the gamma matrices are the following 4 x 4 matrices

7‘(10)’ 7‘(a,- 0’ 7‘ 0—1’

where 0',- are the Pauli 2 x 2 spin matrices

01 O—i 1 o
”‘=(10)’ ”22(1' o)’ ”3=(0—1)'

The spin operator for a Dirac particle is % E, where in terms of the Pauli matrices

_. 3’ 0
2“(o 3)‘

We define the helicity operator 11(3) =35 13, with eigenvalues z\ = +1 and —1 corre-

sponding to spin parallel and antiparallel to the momentum 3, respectively.

A four-component Dirac spinor can be written as

Wt),

where 1,1)... and z/1- are two-component spinors. 211 is a helicity eigenstate if for
fermions

1P: = "i = WiAX%a (B.1)

131

132

Table B.1: Helicity states in two-component spinor, and bra-ket notation.

 

u+(.\ = +1) +\/E +|i3||15+ >
MA = —1) +\/E - l 3 ma— >
u_(A = +1) +\/E — | 3 ”13+ >
u-(z\ = —1) +\/E + I P “P— >
MA = +1) +\/E — I 3 Ilfi- >
v+(A = —1) —\/E +|3ll13+ >
MA = +1) —\/E + I 3 ma— >
v_(A = —1) +\/E — | 3 ”15+ >

 

 

 

 

 

and for anti-fermions

1P1 = ”a: = iszpr_%, (B.2)

with wi = l/ E :l: I I; I. Here 70/2 is the eigenvector of the 2 x 2 operator 3 of) with

eigenvalue A = :tl. These two-component eigenvectors XA/g, are

_ cos0/2 __ —e“¢sin0/2
X1/2— e‘¢sin0/2 ’ X"1/2- cosB/2 ’

where 0,96 are the directional polar coordinates of 5. We also introduce a ket

notation for the Weyl two-component eigenvectors

[13+ >5 X1/2, [13" >5 X-1/2-

In this notation, the eigenstates for fermions (vi) and anti-fermions (vi) for each
helicity are listed in Table B.1. The helicity eigenvectors for massless fermions in
four-component Dirac form have either zlq. or 1p- equal to zero; and the non-zero
two-component spinor is proportional to XiA/2- The massless Dirac helicity eigen-
vectors are shown in Table B.2 in terms of Weyl spinors. Because two components

are zero the algebra reduces to two-component spinors acted on by 2 x 2 matrices.

133

Table B.2: Dirac helicity eigenvectors for massless fermions.

 

(W) (a)
(F)

 

Fermion

 

 

 

 

 

Antifermion ID = ( x/Z—ng )
— —1/2

 

The chirality projection operators are defined to be
1
Pi = 5(1 If: ’75).

In the Weyl basis they are diagonal

1 o 0 o
P+=(o 0)’ P‘=(o 1)’

and they project out 30¢, respectively.

Feynman diagram calculations with a fermion propagator include a factor of j) =

07“
u: +
7_(75 0)’

’7’; =(11:F 3)‘

1): 0 p°+;-I;
pO—E-P 0

The Fierz identities are useful for performing contractions. In terms of bra-kets one

my“. In the Weyl basis

where

Thus,

identity is
< ihilj >< kl7=Full >= 2 < ill >< k|j >. (B.3)

Also, for each ket Ifizi: >, we define an associated ket [E > by

IE >= 4413? >.

134
Then it can be shown that
< “Hm-"7111..” >=< II‘Hun-nfim IE > o (BA)
Therefore uniting Eqs.(B.3) and (BA) another Fierz identity is
< i|7§=|j >< kl‘mull >= 2 < ill} >< le >. (B.5)

For a spin-1 field, the right-handed (R), left-handed (L), and longtitudinal (0) po-
larization vectors are

fit

6‘}; = e_‘/§(0, i sin ()5 — cos «)3 cos 0, —i cos ¢ — sin 43 cos 9, sin 9)
-i¢

6’}, = (ET/{(0, i sin 43 + cos ()5 cos 0, —i cos at + sin 43 cos 0, — sin 0)

66‘ = i-(IP I,Esin0cos¢,EsinOsin¢,Ecos€).

0 and d) are spherical coordinates specifying the direction of the boson’s momentum.

The longtitudinal polarization applies only to massive vector bosons.

B.2 Example Calculation

In this section we give an example calculation of a Feynman diagram, using helicity

states. We also briefly discuss the numerical calculation of the cross section.

We calculate the Feynman diagram in Fig. B.1(b), which is one term in the am-
plitude for W7 production (with W decay). This s-channel diagram includes a
triple-boson vertex. For simplicity we assume A, = 0. Then the WW7 vertex
factor, with momentum definitions as shown in Figure B.1(a), is -ig sin OwI‘afl“,

where

1“”’"‘(qv, (7.10) = (q - é)”g°‘" - (p + (1)59” + (p + §)°9“"

+ A~v(p°g"” -p”9“°)- (B-6)

135

(a)

/W‘(q)
I \W+(§)

 

Figure B.1: (a) The WW7 vertex. (b) Example diagram for ad —-> W+7 —-> e+u'y.
Arrows indicate the defined momentum direction.

(The factor for the WWZ vertex would be identical, with sin 9w replaced by cos 9w.)

With the assignment

P = —P3
q = ‘19?
(I = P4

we can write down the contribution to the amplitude from this diagram

-ieg2 1 1
2 (pi - M30013 - MI»)

 

M = [17(2)7aP—U(1)] ‘ [i(6)7aP-v(5)l ° 62(3)I‘“3",
(B.7)

where we have adopted the shorthand u(l) = u(pl), etc. Note that the f i W vertex

has a chirath projection operator P. because of the parity violating V-A coupling.

To simplify the first term in square brackets, we note that

[17(2)7aP—U(1)] = [17(2)P+7aP—u(1)]

136

and that

17(2)P+ = “(2)7013
= (121(2) v1(2))(‘1’ 3)
= (”1(2) 0)

W = (3 00:83)

= («z—in]

From Eqs. (B.1) and (B.2) we deduce that, in order to be non-zero, u_(1) and

(B.8)

121(2) must have A equal to minus and plus one, respectively; that is, the up quark
is left-handed and the d is right-handed. We make a similar simplification of the
other square-bracketed factor in Eq. (B.7). Then we can use Table B.1 to write the

amplitude in terms of bra-kets:

 

 

- 2 "
—Ieg 1 1
M E .
2 (p2 - Mwe- Mm‘/2E‘2E’2E52 6
< 2 — [7+a|1— > - < 6 — |7+e|5- > -e;(3)r“"". (3.9)

Using Eq. (B.6) for Pam, and the Fierz identity, (B.3), we simplify the Lorentz

contractions to get

 

' 2
_ —zeg 1 1
M " 2 (p: — Ms) (1»: — May)‘/2E‘2E’2E52E°
{—2(p4 +p7) - 6*(3) < 2 — |6+ >< 5 +|1— >
+ < 6 — |(167+ 123)+|5— >< 2 — l £1(3)|1— >
+ < 2 - |(164— 163)+|1— >< 6 —| fi:(3)I5-— >

+ A», [< 6 — | p3+|5— >< 2 —| ;;(3)|1— >

 

— < 2 — I 163+I1— >< 6 — I £1(3)|5- >]}, (B-10)

where [1,]: denotes pu'yi.

137

The rest of the cross section calculation is done by computer. Components of
the transition amplitude in the form of Eq. (B.10) are calculated numerically by a
FORTRAN subroutine. To obtain a differential or total cross section, the “hard”
cross section in terms of partons must be convoluted with the parton distribution
functions and integrated over the momentum fractions of the initial partons. The

hard cross section (for massless partons) is

r (2104

&= —.
. 23

|M|2d<I>, (B.11)

where 5 is the partonic center-of-mass energy, M is the matrix element, and d<I> is
the phase-space element appropriate to the number of outgoing particles. The total

hadronic cross section is then
0: 'dzdz'f(z)f'(z')&, (3.12)

where a: and 2’ are the momentum fractions of a parton in the proton and antiproton,
respectively. (This is one term; we also include a term in which the f and f’
originate in the antiproton and proton, with probability f(z’ ) f’ (23)) Our Monte-
Carlo program, based on PAPAGENO [32], evaluates the phase-space and a: integrals
simultaneously by summing the kinematic and matrix element weights of many
events. That is, for each event a set of random 2’s and momenta are generated which
satisfy all kinematic constraints. The 23’s determine the parton distribution weight,
while the momenta determine the matrix element and phase space weights. The
product of these weights, summed over many events, yields the cross section. The
parameters of each event are stored in an HBOOK NTUPLE, so that we may project
out the distribution of events along any kinematic variable, i.e. the differential cross

sections.

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