First Time Measurement of Polarization Observables for the Charged Cascade Hyperon in Photoproduction

First Time Measurement of Polarization Observables for the Charged Cascade Hyperon in Photoproduction
FLORIDA INTERNATIONAL UNIVERSITY
Miami, Florida
FIRST TIME MEASUREMENTS OF POLARIZATION OBSERVABLES FOR
THE CHARGED CASCADE HYPERON IN PHOTOPRODUCTION
A dissertation submitted in partial fulfillment of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
in
PHYSICS
by
Jason Bono
2014
To: Interim Dean Michael R. Heithaus
College of Arts and Sciences
This dissertation, written by Jason Bono, and entitled First Time Measurements of
Polarization Observables for the Charged Cascade Hyperon in Photoproduction, having been approved in respect to style and intellectual content, is referred to you for
judgment.
We have read this dissertation and recommend that it be approved.
Brian A. Raue
Misak M. Sargsian
Cem Karayalcin
Lei Guo, Major Professor
Date of Defense: June 6, 2014
The dissertation of Jason Bono is approved.
Interim Dean Michael R. Heithaus
College of Arts and Sciences
Dean Lakshmi N. Reddi
University Graduate School
Florida International University, 2014
ii
c Copyright 2014 by Jason Bono
All rights reserved.
iii
DEDICATION
This dissertation is dedicated to my entire family, but most significantly to my mother
Sharon. Without her faith and devotion, my early educational struggles might never
have been overcome.
iv
ACKNOWLEDGMENTS
My time spent in fundamental research has been pleasurable and deeply gratifying,
although the trajectory of events preceding this completed work has been somewhat
circuitous. I suppose it’s all part of the fun, though. Unpredictability in research,
numerous successes, failures, dead ends, new ideas, and excitement have been recurrent themes against a background of some of the less energizing aspects of life in
graduate school. Amidst the mixture of light and dark, in the journey which this
work represents, I am grateful for every inch in its broad path. Melodrama aside,
this accomplishment is contingent upon the hard work and devotion of many people
to whom I wish to express my sincere gratitude.
First, my deepest gratitude goes to Lei Guo for his excellent guidance over the past
three years. With a contagious passion for physics, keen intuition and an openness
to new ideas, he has been a great source of inspiration. His attentiveness and clarity
across scales from the big picture to the most subtle aspects of analysis have provided
a rich and open dialog which has been instrumental in my development. Although,
Lei’s uncanny memory for the details of such discussions has worked to my detriment
on an occasion or two. I additionally thank Lei for the answers he did not provide
and for his encouragement to think for myself. His genuine interest in the wellbeing
of his students has provided me solace during important moments. In the midst of
more difficult times, knowing that someone possessed a selfless imperative to help,
was often help enough.
I am extremely gracious for the additional and voluntary guidance which Brian Raue
bestowed. Many key aspects of the analysis directly descended from his expert advise
as a physicist and statistician. His consistent participation through week after week,
of long, and sometimes overly expressive research meetings was paramount to our
results. I have learned so much and consider Brian as my adoptive advisor. Together
v
Brian and Lei have been, in a strict sense, brilliant, but I am especially grateful for the
tremendous amount trust that the two invested in me. Perhaps most importantly, our
research has been a lot of fun. I implore both of them to continue training graduate
students together.
I would like to thank the world class scientists at Jefferson Lab who maintained
smooth operation of the CLAS detector during data collection. Extra thanks goes
to the hardworking members of the g12 collaboration; especially to those who slaved
in data reconstruction and calibrations as graduate students and post-docs, Johann
Goetz, Diane Schott, Mukesh Saini, Craig Bookwalter, Mike Kunkel and Mike Palone.
I thank Kanzo Nakayama, Misak Sargsian and Oren Maxwell as theoreticians with
whom consulting brought fruitful insight. Special thanks goes to Kanzo for this continued correspondence and collaboration. I kindly thank Joerg Reinhold for patiently
teaching me principals of nuclear physics experimentation. I also thank my undergraduate professor James Cresser for encouraging me to further pursue physics. I still
mull over our discussions on quantum decoherence.
Colleagues, horses, from the FIU nuclear physics lab, thank you. Eric Pooser, Hari
Khanal, Dipak Rimal, Rafael Badui, William Phelps, Adam Freese and Marianna
Gabrielyan, you guys were all amazing to work with. Besides being the most creative
and smartest group of people I know, you are also among the most light-hearted and
hilarious, that’s what u r doing. I hope you have benefited from my presence as much
as I have from yours. Special thanks goes to former post-doc Puneet Khetarpal for
sharing his expertise in programing.
Thanks to my siblings Andrew Bono and Kathleen Schecher with whom I delight
in sharing fun challenges and amusing times. From our distinct breed of humor
containing themes and debates such as the spelling of a certain dog’s name, to our
constant search for adventure via land, snow, sea and the otherwise more absurd, the
vi
elements of our friendship are unique and I value them dearly. You both, presumably
without realizing, have kept me on track time and time again.
To my father Jay Bono, thank you for our friendship and for cultivating my curiosity
about the world. I consider myself fortunate having grown up with access to great
ideas and for them having been communicated so engagingly and often. I carry several
details of our early conversations, although the most memorable aspects pertain to
the open-endedness of inquiry and its implied imaginative freedom. I am delighted
that our conversations, which have always been refreshingly fluid, have paradoxically
grown in both sophistication and immaturity.
To my mother and stepfather Sharon and Chuck Carlino, thank you for everything
you have given us, including putting up with the aforementioned adventures. I owe
more to my mother than I may express in this scope. She has held us all together with
her perseverance and seeming omniscient judgment. For her efforts and struggles that
I know, and for those which I remain ignorant, I wish her happiness and awareness
of how great a mother she has been. To Chuck, thanks for taking in our family as
your own, you have been a blessing.
Thanks to my friends who moved half a world away to be in Miami. For those who
expressed that their relocation was friendship driven, I feel special gratitude and
humility.
Finally to Jen McAnney, thank you for being my friend in the relatively frantic time
of writing and for patiently proofreading my work. I’m energized by your spirit and
humbled by your kindness. I hope your attributes are contagious because I would like
to be more like you.
vii
ABSTRACT OF THE DISSERTATION
FIRST TIME MEASUREMENTS OF POLARIZATION OBSERVABLES FOR
THE CHARGED CASCADE HYPERON IN PHOTOPRODUCTION
by
Jason Bono
Florida International University, 2014
Miami, Florida
Professor Lei Guo, Major Professor
The parity violating weak decay of hyperons offers a valuable means of measuring
their polarization, providing insight into the production of strange quarks and the
matter they compose. Jefferson Lab’s CLAS collaboration has utilized this property
of hyperons, publishing the most precise polarization measurements for the Λ and Σ
in both photoproduction and electroproduction to date. In contrast, cascades, which
contain two strange quarks, can only be produced through indirect processes and as
a result, exhibit low cross sections thus remaining experimentally elusive.
At present, there are two aspects in cascade physics where progress has been minimal:
characterizing their production mechanism, which lacks theoretical and experimental
developments, and observation of the numerous excited cascade resonances that are
required to exist by flavor SU (3)F symmetry. However, CLAS data were collected in
2008 with a luminosity of 68 pb−1 using a circularly polarized photon beam with energies up to 5.45 GeV, incident on a liquid hydrogen target. This dataset is, at present,
the world’s largest for meson photoproduction in its energy range and provides a
unique opportunity to study cascade physics with polarization measurements.
The current analysis explores hyperon production through the γp → K + K + Ξ− reaction by providing the first ever determination of spin observables P , Cx and Cz for
the cascade. Three of our primary goals are to test the only cascade photoproduc-
viii
tion model in existence, examine the underlying processes that give rise to hyperon
polarization, and to stimulate future theoretical developments while providing constraints for their parameters. Our research is part of a broader program to understand
the production of strange quarks and hadrons with strangeness. The remainder of
this document discusses the motivation behind such research, the method of data
collection, details of their analysis, and the significance of our results.
ix
TABLE OF CONTENTS
CHAPTER
PAGE
1 Introduction and Motivation
1.1 Overview of the Standard Model of Particle Physics . . . . . . . . . .
1.2 Our Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 The Motivation Behind Our Research . . . . . . . . . . . . . . . . . .
1.3.1 Motivation: Current Status of Cascade Physics . . . . . . . .
1.3.2 Motivation: Testing Predictions from Theory . . . . . . . . . .
1.3.3 Motivation: Recent Analogous Lambda Results . . . . . . . .
1.3.4 Motivation: Vector Meson Dominance . . . . . . . . . . . . .
1.3.5 Motivation: Universal Hyperon Polarization at High Energies
1.3.6 Motivation: Connection Between Polarization Observables and
Production Amplitudes . . . . . . . . . . . . . . . . . . . . . .
1.3.7 Proton Spin Crisis and Baryon Polarization . . . . . . . . . .
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2 The g12 Experiment
2.1 CEBAF Accelerator . . . . . . . . . . . . . . . . .
2.2 Hall B Photon Tagger . . . . . . . . . . . . . . . .
2.3 Hydrogen Target . . . . . . . . . . . . . . . . . . .
2.4 The CLAS spectrometer . . . . . . . . . . . . . . .
2.5 Start Counter . . . . . . . . . . . . . . . . . . . . .
2.6 Drift Chambers . . . . . . . . . . . . . . . . . . . .
2.7 Superconducting Toroidal Magnet . . . . . . . . . .
2.8 Time-of-Flight Detectors . . . . . . . . . . . . . . .
2.9 Cherenkov Counters . . . . . . . . . . . . . . . . .
2.10 Electromagnetic Calorimeters . . . . . . . . . . . .
2.11 Trigger and Data Acquisition System . . . . . . . .
2.12 Event Reconstruction . . . . . . . . . . . . . . . . .
2.13 Cascade Data . . . . . . . . . . . . . . . . . . . . .
2.13.1 Reconstructed Cascade and Lambda Tracks
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3 Formalism and Methodological Framework
3.1 Spin Observables and the Coordinate System . . . . . . . .
3.2 Connection Between Spin Observables and Polarization .
3.3 Self Analyzing Decay . . . . . . . . . . . . . . . . . . . . .
3.4 Calculation of P . . . . . . . . . . . . . . . . . . . . . . .
3.5 Calculation of Double Polarization Observables Cx and Cz
3.6 Acceptance Independence of Cx and Cz . . . . . . . . . . .
3.7 Frame Transformation Effect on Hyperon Polarization . . .
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4 Data Processing and Event Selection
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x
4.1
4.2
4.3
Physics Event Selection . . . . . . . . . . . . . . .
4.1.1 Topology Requirement . . . . . . . . . . .
4.1.2 Missing Mass Selection in the Hypersphere
4.1.3 Vertex Position Selection . . . . . . . . . .
4.1.4 Vertex Timing Selection . . . . . . . . . .
4.1.5 Eliminating Particle Misidentification . . .
4.1.6 The Fiducial Region . . . . . . . . . . . .
4.1.7 Photon-Beam Energy . . . . . . . . . . . .
Kinematic Binning of Data . . . . . . . . . . . . .
Corrections and Calibration . . . . . . . . . . . .
4.3.1 Photon-Beam Polarization Determination
4.3.2 Multiple Photon Events . . . . . . . . . .
4.3.3 CLAS Energy-Loss Corrections . . . . . .
4.3.4 Photon Beam Energy Corrections for g12 .
4.3.5 Final-State Momentum Corrections for g12
4.3.6 Combined Effects of Corrections . . . . . .
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5 Simulation
5.1 Simulation Overview . . . . . . . . . . . . . . . . . . . . .
5.1.1 Generated and Reconstructed Events . . . . . . . .
5.1.2 Detached Vertices . . . . . . . . . . . . . . . . . . .
5.1.3 GSIM . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 GPP . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Tuning Simulation to Data . . . . . . . . . . . . . . . . . .
5.2.1 Simulated Beam Energy Spectrum . . . . . . . . .
5.2.2 Exponential t-slope . . . . . . . . . . . . . . . . . .
5.2.3 Resonance Mass and Width . . . . . . . . . . . . .
5.2.4 Further Comparison of Simulated and Experimental
5.3 Acceptance Functions . . . . . . . . . . . . . . . . . . . . .
5.3.1 Calculation of Acceptance and Uncertainty . . . . .
6 Results
6.1 Induced Polarization P . . . . . . . . . . . . . . . . . .
6.1.1 Comments on Induced Polarization Results . . .
6.1.2 Alternate Methods for Induced Polarization . .
6.1.3 Further Cross Checks for Induced Polarization .
6.2 Transfered Polarization Cx and Cz . . . . . . . . . . . .
6.2.1 Comments on Transfered Polarization Results .
6.2.2 Cross Check for Transfered Polarization Results
6.3 Comparison with Theory . . . . . . . . . . . . . . . . .
6.3.1 Comments On our Comparison with Theory . .
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7 Systematic Uncertainties
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7.1 Uncertainty in Analyzing Power and Beam Polarization . . . . . . . . 126
xi
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.1.1 Analyzing Power . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Beam Polarization . . . . . . . . . . . . . . . . . . . . . .
Binning of Pion Angle in Beam-Helicity Asymmetry . . . . . . . .
Varying the Mass-Hypersphere Radius . . . . . . . . . . . . . . .
Effect of Fiducial Cuts . . . . . . . . . . . . . . . . . . . . . . . .
Non-Cancellation of Acceptance for P . . . . . . . . . . . . . . . .
Study of Non-Cancellation of Acceptance for Cx and Cz . . . . .
Effect of Method for P : A Cross Check . . . . . . . . . . . . . . .
7.7.1 Fitting vs Raw Yield: A Cross Check on Cx and Cz . . . .
Studies on Background: Further Cross Checks . . . . . . . . . . .
7.8.1 Effective Polarization of Background From Lambda Decay
7.8.2 Effective Polarization of Unknown Background . . . . . . .
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8 Conclusions and Outlook
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Bibliography
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VITA
151
xii
LIST OF TABLES
TABLE
1.1
1.2
4.1
4.2
6.1
6.2
6.3
7.1
PAGE
A table of the elementary particles. . . . . . . . . . . . . . . . . . . .
Three-star and above Lambda and Sigma hyperon resonances listed by
the PDG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The degree of longitudinal electron polarization (Pe ) for each Møller
run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A table showing the degree of circular photon polarization (P ) in the
relevant kinematic bins (Eγ ). . . . . . . . . . . . . . . . . . . . . . . .
2
14
78
79
A table summarizing the results for P , with and without acceptance
corrections. There is good agreement between both methods for all six
bins, well within statistical uncertainty. . . . . . . . . . . . . . . . . . 107
A table summary of induced polarization values binned in center-ofmass Ξ− angle and beam energy. . . . . . . . . . . . . . . . . . . . . 109
A table summary of Ξ− polarization values for the nominal binning
scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Systematic uncertainty for Cx, Cz and P arising from various sources.
All sources are detailed in the present chapter. A summary: Binning
refers to width of the binning used, Mass refers to the width of the mass
cuts, Fiducial refers to fiducial cuts, Acceptance refers to acceptance
effects. Total systemic refers to each source, added in quadrature and
Statistical refers to the known statistical uncertainty associated with
the nominal measurement. Finally, Total uncertainty refers to the total
systematic and statistical uncertainties added in quadrature. . . . . . 126
xiii
LIST OF FIGURES
FIGURE
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.1
2.2
2.3
PAGE
The Baryon Octet as organized in the Eightfold Way according to
charge and strangeness. The octet contains spin-1/2 nucleons, sigmas, the lambda, and cascades. The quark composition is additionally
shown. The particle being studied in our analysis, the charged cascade,
is highlighted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Baryon Decuplet as organized in the Eightfold Way according to
charge and strangeness. The decuplet contains spin-3/2 deltas, excited sigmas, excited cascades, and the omega. Quark compositions
are additionally shown. . . . . . . . . . . . . . . . . . . . . . . . . . .
The γp → K + K + Ξ− reaction through an arbitrary mechanism. At
present, the production mechanism which we seek to understand in
our work, is unknown. . . . . . . . . . . . . . . . . . . . . . . . . . .
Cartoon depiction of the cascade’s half-spin arising from its internal
constituents and their dynamics. In general, the overall spin comes
from quantum mechanical addition rules, summing contributions from
valence quarks (ssd for the cascade), virtual quark-pairs (sea quarks)
and gluons; all three types of constituents contain intrinsic spin, and orbital angular momentum. The relative contributions from each source
of spin are unknown, which is in part what the present work examines.
An illustration of cascade photoproduction in the t-channel, arising
through the decay of an intermediate hyperon. . . . . . . . . . . . . .
Possible diagrams contributing to cascade production. . . . . . . . . .
An illustration of Schumacher’s hypothesis. . . . . . . . . . . . . . . .
Cartoon depiction of lambda photoproduction through vector meson
dominance. In this picture, the polarization of the lambda comes from
the strange quark, which is produced through quantum fluctuations of
the photon. Most of the photon polarization transfers to the lambda
in this picture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cartoon depiction of cascade photoproduction through vector meson
dominance. In this picture, the polarization of the cascade comes from
the strange quark, which is produced through quantum fluctuations of
the photon. While the strange quark from the photon may be fully
polarized within the cascade, its effects are diminished since it may
only contribute a fraction to the total cascade spin. Thus, only some
of the photon polarization transfers to the cascade. . . . . . . . . . .
A photograph of an SRF cavity made from superconducting niobium.
The large-scale design of CEBAF and it’s components. . . . . . . . .
An aerial photograph of CEBAF. . . . . . . . . . . . . . . . . . . . .
xiv
6
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2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
3.1
3.2
4.1
4.2
4.3
A profile of the photon tagger. . . . . . . . . . . . . . . . . . . . . . .
A drawing of the target. . . . . . . . . . . . . . . . . . . . . . . . . .
Drawing of CLAS nested within Hall B. . . . . . . . . . . . . . . . .
Schematic of the CLAS. . . . . . . . . . . . . . . . . . . . . . . . . .
A drawing of the CLAS start counter. . . . . . . . . . . . . . . . . . .
A profile drawing of CLAS detecting a two-track event coming from
the target. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Drawing of a cross-sectional view of the DC regions and the toroidal
magnetic field at half current (1930 A). . . . . . . . . . . . . . . . . .
Diagram of the sectors, regions and super layers of the DC. . . . . . .
Drawing of a track going through the five layers of the DC. . . . . . .
Photo of the torus magnet during construction. . . . . . . . . . . . .
A diagram showing the CC response to a single lepton track entering
One of the six segments. . . . . . . . . . . . . . . . . . . . . . . . . .
A diagram of one dissected sector of the EC with the three u-v-w
layering convention illustrated. . . . . . . . . . . . . . . . . . . . . . .
Coincidence trigger criteria. . . . . . . . . . . . . . . . . . . . . . . .
The path of a charged particle going through a magnetic field. . . . .
The γp → K + K + Ξ− reaction through an arbitrary mechanism with
the subsequent Ξ− → Λπ − weak decay. . . . . . . . . . . . . . . . .
The production plane (left plane) is shown; defined in the center-ofmomentum, it contains the incoming photon and recoiling cascade.
The two kaons in general lie above and below the production plane.
The so called decay plane (right plane), is defined in the rest-frame of
the cascade and contains its decay products (pion and lambda). . . .
Cartoon representing Ξ− photoproduction via virtual-meson exchange
in the t-channel and its weak decay. . . . . . . . . . . . . . . . . . . .
Top left: the missing mass spectrum of the K + K + system, showing
the Ξ− peak at 1.32 GeV. Top right: the missing mass spectrum of
the K + K + π − system, showing the Λ peak at 1.11 GeV. Bottom left:
invariant mass spectrum of the Λπ − system, showing the Ξ− peak
at 1.32 GeV. Bottom right: invariant mass spectrum of the Ξ− − π −
system, showing the Λ peak at 1.11 GeV. In all plots a Gaussian is fit to
the signal over a polynomial background. The vertical lines represents
the known lambda or charged cascade masses. The parameter σ of
the Gaussian fit gives CLAS’s “natural” resolution for its associated
quantity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The nominal mass cut r < 1 represents a three sigma cut as shown
in blue. The first
sideband (in red) are the events lying in the hyper√
4
2
while the second sideband (black) lie in the region
shell
1
<
r
<
√
√
4
4
2 < r < 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
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4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
The cut region of spacial vertex distribution. All events lying outside
the plot were excluded. . . . . . . . . . . . . . . . . . . . . . . . . . .
Difference between event-vertex time as calculated by the RF-corrected
tagger and start counter, for events passing all cuts. A one nanosecond
cut was imposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Difference between vertex-time and particle-vertex time as calculated
by the RF-corrected tagger and TOF, for events passing all cuts. A
one nanosecond cut was imposed. Left: fast kaon. Right: slow kaon. .
β vs momentum for the fast kaon overlaid with it’s theoretical curve.
Left: before hypersphere cuts, an extra band is visible comprising
misidentified pions. Right: after hypersphere cuts, the contamination
is eliminated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
β vs momentum for the pion (left) and slow kaon (right) overlaid with
their respective theoretical curves. Both plots show the data after hypersphere cuts and indicate no contamination due to misidentification
or timing inaccuracies . . . . . . . . . . . . . . . . . . . . . . . . . .
Left: shows the π + φ distribution for sector-three in one p and θ bin
along with the upper and lower limits of the fiducial region represented
by the green vertical line. Right: a second-generation plot, fit to a
hyperbola. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Third-generation plots of the fitting parameters from second-generation
fits for sector three. The data are fit to power functions. . . . . . . .
The angular distribution of the proton from exclusive pπ + π − events is
shown. In the top, φ vs θ is plotted, the bottom plots conveys similar
information mapped to mimic the geometry of CLAS. Left: No fiducial
cuts. Right: nominal fiducial cuts on the proton. . . . . . . . . . . . .
The angular distribution of the positive pion from exclusive pπ + π −
events is shown. In the top, φ vs θ is plotted, the bottom plots conveys
similar information mapped to reflect the geometry of CLAS. Left: no
fiducial cuts. Right: nominal fiducial cuts on the positive pion. . . . .
The angular distribution of the negative pion from exclusive pπ + π −
events is shown. In the top, φ vs θ is plotted, the bottom plots conveys
similar information mapped to reflect the geometry of CLAS. Left: no
fiducial cuts. Right: nominal fiducial cuts on the negative pion. . . .
From left to right: φ vs momentum for the proton, negative pion and
positive pion for pπ + π − events. Top: no fiducial cuts. Bottom: nominal fiducial cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
φ vs θ for the fast kaon from K + K + π − (Λ) events are shown. On the
top, simulated Monte Carlo events are plotted, on bottom, the g12
data of our analysis. Left: No fiducial cuts. Right: nominal fiducial
cuts on the fast kaon. . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvi
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73
4.16 φ vs θ for the slow kaon from K + K + π − (Λ) events are shown. On the
top, simulated Monte Carlo events are plotted, on bottom, the g12
data of our analysis. Left: No fiducial cuts. Right: nominal fiducial
cuts on the slow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.17 φ vs θ for the pion from K + K + π − (Λ) events are shown. On the top,
simulated Monte Carlo events are plotted, on bottom, the g12 data of
our analysis. Left: No fiducial cuts. Right: nominal fiducial cuts on
the pion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.18 The red horizontal lines represent the binning in cos θΞcm while the green
horizontal lines represent the binning in Eγ . Left: the g12 data. Right:
Monte Carlo simulated events. . . . . . . . . . . . . . . . . . . . . . .
4.19 An illustration of the angle definitions used in the γp → π + π − p subanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.20 I (φhel
π + ) for our data within the energy range of W = 1.9 − 2.3 GeV. .
hel
4.21 I (φπ+ ) as measured in the previous analysis. . . . . . . . . . . . . .
4.22 On the left, the Ξ− signal in the missing mass spectrum (after all cuts)
of K + K + before ELOSS, photon energy, and momentum corrections is
shown. On the right, the same signal using the corrected four-vectors.
The mass of the Ξ− is around 1.321 GeV. . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
5.5
A cartoon representing our simulated cascade production model. Kf+ast Y ∗
is produced via virtual-meson exchange in the t-channel. The Y ∗ de+
cays to Kslow
Ξ− . The cascade undergoes a subsequent decay to π − Λ.
The decay of Λ to pπ − was additionally simulated. . . . . . . . . . . .
The ffread card used with gsim for our analysis. . . . . . . . . . . . .
The beam energy spectrum for the Ξ− data sample and Monte Carlo
simulation. The Monte Carlo events are in red and are normalized to
the data represented as points with statistical error bars. . . . . . . .
The left shows a diagram of a generic interaction with two particles in
the initial (p1 and p1 )and two particles in the final state (p3 and p4 ) as
referred to in Equation. 5.21. On the right is the analogous model of
the γp → K + K + (Ξ− ) reaction. It’s should be clear from this diagram
and Equation. 5.21 that in this model, t is the momentum transfer
from the photon to the fast kaon. The blue ellipses in both figures
represent arbitrary intermediate processes. . . . . . . . . . . . . . . .
First, events are generated. The initial simulation splits into two sets:
reconstructed events (Rec) that pass through gsim, gpp and a1c before being analyzed, and generated events (Gen) that pass straight to
the analyzer. Acceptance corrections are obtained from the ratio of
reconstructed to generated events. Applying acceptance to data, one
obtains the corrected t-slope spectrum which can in turn be used as
input for the next iteration. . . . . . . . . . . . . . . . . . . . . . . .
xvii
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93
94
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
6.1
6.2
6.3
6.4
6.5
t-slope for the acceptance corrected experimental data (blue), and the
generated Monte Carlo events (red). Agreement within statistical uncertainty is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
An example of the plots that were used to calculate tslope. The top two
plots show the t-spectrum for Monte Carlo generated (left) and reconstructed events (right). The center left shows the calculated acceptance
as a function of t while center right shows the uncorrected t-spectrum of
the data. The bottom right is an overlay fo the uncorrected t-spectrum
of Monte Carlo and Data events. Finally, the bottom left shows the
acceptance corrected t-spectrum of the data with an exponential fit. . 96
An overlay of the data (green) and simulation (red) for the invariant
+
mass m(Ξ− Kslow
) in six distinct beam-energy bins. The data were fit to
Gaussian functions which yields the mass and width of the underlying
hyperon used as in input parameter for simulation. . . . . . . . . . . 97
An overlay of the data (points) and simulation (red solid) for the in+
variant mass m(Ξ− Kslow
) integrated over all bins. . . . . . . . . . . . 98
Shows the measured cascade angle with respect to the z-axis in the
center of mass frame. This quantity depends on the intermediate Y ∗
mass and width, along with the t-slope. Red is simulated events and
the points are the data. . . . . . . . . . . . . . . . . . . . . . . . . . . 99
The invariant mass of the Kf+ast + Ξ− system. Red is simulated events
and the points are the data. . . . . . . . . . . . . . . . . . . . . . . . 99
The magnitude of momentum for all three mesons. Red is simulated
events and the points are the data. . . . . . . . . . . . . . . . . . . . 100
A few typical acceptance functions (A(θπy − )) in various bins of beam
energy and cascade angle. The red lines show the fits to the underlying
acceptance-histograms with their widths representing the uncertainty
as calculated by the covariance matrices of fitting parameters. . . . . 102
The angular distribution of the pion off ŷ in nine bins of energy and
center of mass cascade angle. The forward-backward asymmetry is
used to calculate the induced polarization P . . . . . . . . . . . . . . .
The angular distribution of the pion off ŷ in three bins of energy (top)
and three bins of center of mass cascade angle (bottom). The forwardbackward asymmetry is used to calculate the induced polarization P .
As one cross check for the cancellation of acceptance, the acceptancecorrected angular distribution of the pion off ŷ is shown. . . . . . . .
Angular distribution of the pion off ŷ, integrated over all bins. The
forward-backward asymmetry is used to calculate the induced polarization P . The measured value of P is constant with zero. . . . . . .
Fitted acceptance corrected pion angular distributions and the resulting measurement of P . Agreement with the primary measurements
shown in Fig. 6.2 is evident for all bins within statistical uncertainty.
xviii
104
105
106
107
110
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Monte Carlo events generated with Pgen = 0. The pion angular distributions of the reconstructed events in 9 bins kinematic bins are shown
along with the measured value of Prec . Better than 1σ agreement for
Prec = Pgen is shown in all but one bin, which shows a deviation of less
than 1.5σ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Monte Carlo events generated with Pgen = −0.5. The pion angular
distributions of the reconstructed events in nine bins kinematic bins are
shown along with the measured value of Prec . Close to 1σ agreement
for Prec = Pgen is shown in all bins. . . . . . . . . . . . . . . . . . . .
Beam-helicity asymmetry as a function of pion angle off x̂ (left) and ẑ
(right), integrated over all bins. The linear fit is used to calculate the
transfered polarization Cx and Cz . . . . . . . . . . . . . . . . . . . . .
Beam-helicity asymmetry as a function of pion angle off x̂, in three bins
of energy (top) and three bins of center of mass cascade angle (bottom).
The linear fit is used to calculate the transfered polarization Cx . . .
Beam-helicity asymmetry as a function of pion angle off ẑ, in three bins
of energy (top) and three bins of center of mass cascade angle (bottom).
The linear fit is used to calculate the transfered polarization Cz . . . .
Beam-helicity asymmetry as a function of pion angle off ŷ, integrated
over all bins. The linear fit is used to measure the forbidden transfered
polarization Cy . The measurement is consistent with zero as required.
A comparison of our results with theory for P (top), Cx (middle) and
Cz (bottom) as a function of beam energy. . . . . . . . . . . . . . . .
A comparison of our results with theory for P (top), Cx (middle) and
Cz (bottom) as a function of cascade angle. . . . . . . . . . . . . . . .
A comparison of our results with theory for P in three energy bins. .
Shows the asymmetry as a function of pion angle off the x-axis and
the corresponding Cx measurement for decreasing pion bin width from
left to right. The nominal binning scheme is shown in the middle plot.
Shows the asymmetry as a function of pion angle off the z-axis and the
corresponding Cz measurement for decreasing pion bin width from left
to right. The nominal binning scheme is shown in the middle plot. . .
The signal shown with 2,3 and 4 σ cuts (or radius cuts in the hypersphere). The nominal 3-sigma cut is shown in black. . . . . . . . . . .
The integrated P results for 2,3 and 4 sigma cuts from left to right
respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The integrated Cx results for 2,3 and 4 sigma cuts from left to right
respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The integrated Cz results for 2,3 and 4 sigma cuts from left to right
respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
P . Left: results with no fiducial cuts. Right: results with loose fiducial
cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xix
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128
129
129
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130
131
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17
7.18
7.19
Cx . Left: results with no fiducial cuts. Right: results with loose fiducial
cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Cz . Left: results with no fiducial cuts. Right: results with loose fiducial
cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
The acceptance corrected forward-backward pion asymmetry results
for P in three energy bins and three center-of-mass cascade angle bins. 133
The reconstructed values of Cxrec (left) and Czrec (right) with a generated
value Cxgen = 0.5 and Czgen = 0. . . . . . . . . . . . . . . . . . . . . . 135
The reconstructed values of Cxrec (left) and Czrec (right) with a generated
value Cxgen = 0 and Czgen = 0.5. . . . . . . . . . . . . . . . . . . . . . 135
A typical fit which was integrated to give a background subtracted yield.136
Left: fitting with four pion bins. Right: no fitting with five pion bins.
The binning scheme for the fitting method was defined coarsely to
provide reliable yields. . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Left: fitting with four pion bins. Right: no fitting with five pion bins.
The binning scheme for the fitting method was defined coarsely to
provide reliable yields. . . . . . . . . . . . . . . . . . . . . . . . . . . 137
In the top four plots: events in red are the result of cut on the invariant
mass m(Λ+π − ) and m(Ξ− −π − ) which identifies the pion coming from
the Λ decay. Its presence is shown over the broader spectrum of events.
In the bottom four plots: Events representing cuts in the hypersphere
radius r = 1, 2, 3, 4, 5, 6 are layered over one another. The vertical
lines provide a means of showing how deep within the hypersphere the
“lambda-pion” cuts are. One should note the Λ-pion cuts lie in the
r = 6 hypersphere, far out in the sideband of the primary signal. . . 139
Measurements of the effective Cx and Cz for events in the lambdapion background. This measurement is not meaningful in terms of
polarization observables, but serves as study of background effects. . 140
In the top four plots: events in red are the result of cut on the invariant
mass m(Λ + π − ) and m(Ξ− − π − ) which identifies the “mixed” background. Its presence is shown over the broader spectrum of events.
In the bottom four plots: Events representing cuts in the hypersphere
radius r = 1, 2, 3, 4, 5, 6 are layered over one another. The vertical
lines provide a means of showing how deep within the hypersphere the
mixed-background cuts are. . . . . . . . . . . . . . . . . . . . . . . . 141
Measurements of the effective Cx and Cz for events in the “mixed”
background. This measurement is not meaningful in terms of polarization observables, but serves as study of background effects. . . . . . . 142
xx
Chapter 1
Introduction and Motivation
Experimental nuclear and particle physics provide a glimpse of what is countenanced
by nature. The mantra “All that is permissible is required” hints at the utility
of carefully crafted observation. Specific measurements under controlled conditions
as dictated by theory have provided rich information into the workings of our universe, revealing remarkable order and beauty, as well as peculiarity. Through modern
physics, a strange inner world to which we are blind in daily experience, has begun
to become illuminated. Although, there is still much work to be done.
The present chapter provides an overview of the prevailing theoretical framework
and a few of its shortcomings, including those that encompass our research. The
motivation behind our research is discussed in detail.
1.1 Overview of the Standard Model of Particle Physics
The tradition of atomism dates back to ancient India and Greece, however for brevity,
we will fast forward through our long history of natural philosophy to the dawn of
modern particle physics: the age of quantum theory. In the early 1940s, only a
handful of what were thought of as elementary particles had been identified. Among
them were the electron, positron, muon, photon, neutron and proton [1]. While the
former three particles are still believed to be void of internal structure, the latter
two have been found to be composites of smaller elements. Our current picture
of particle physics is the result of a renaissance-like explosion of experimental, and
theoretical work in the later half of the twentieth century. Such developments came
predominately from cosmic-ray and collider experiments, revealing new particles that
did not fit the traditional atomic scheme of matter in which neutrons and protons
1
were fundamental. Several new classes of subatomic particles including a vast array
of mesons and baryons were identified. Attempts to explain the complexity of the
observed phenomena and growing list of “elementary” particles blossomed into what
is now known as the standard model : an extraordinarily broad and successful theory.
In the current picture, all matter is composed of a few particles categorized as quarks
and leptons, interacting through the exchange of field quanta known as gauge bosons.
All of the aforementioned elementary particles are shown in Table 1.1 along with
the recently observed Higgs boson, the quanta of the Higgs field, responsible for the
breaking of certain symmetries (i.e., electroweak), which endows the leptons and
quarks, along with the Z and W , bosons with mass.
Table 1.1: A table of the elementary particles. The mass, charge, spin and symbol
for each elementary particle is shown. For the quarks and leptons, first, second and
third generations are organized in the first, second and third columns respectively.
Image Source [2]
The exchange of gauge bosons is thought to underpin all dynamical interactions in
the universe [3]. Each fundamental interaction, mediated by different gauge bosons,
is classified into the strong, weak, electromagnetic or gravitational interactions. The
2
gauge boson underlying the latter interaction, referred to as the graviton, is yet to be
observed experimentally; gravity is not included in the standard model but because of
its weakness, is negligible in the subatomic regime. The color, or strong interaction
is mediated by the exchange of gluons. Photons are exchanged in electromagnetic
interactions, while the weak interaction is mediated by the exchange of W and Z
bosons. As it turns out, the weak and electromagnetic interactions are actually differing aspects of the same phenomenon with the unsurprising name of the electroweak
interaction. The phenomenological difference between the electromagnetic and weak
forces at energies below 100 GeV is a result of the non-zero mass of the W ± and Z 0
bosons, which owe their finite mass to the Higgs mechanism [3].
Leptons, given their name because of their small masses [4], interact electromagnetically and weakly but not via the strong force. Intricate collective behavior of
interacting leptons diversifies the characteristics of atomic matter and gives rise to all
phenomena in chemistry and biology, from the distinguishing features of the elements,
to the microscopic processes upon which all life is dependent. Additionally, leptons
are observed to participate in nuclear β-decay among other weak interactions, a process that has a large effect on the composition of the universe but is only observable
through developments in nuclear experimentation.
Quarks, the building blocks of hadronic matter (hadrons refer to all baryons and
mesons), interacting primarily through the strong interaction, exhibit a strange behavior physicists called confinement. Confinement describes the phenomenon that
quarks are never found alone. Although there is no known analytic proof that quantum chromodynamics (QCD), the theory of the strong interaction, should exhibit
confinement, it is generally understood through least-action principals and consideration of the vacuum’s quantum nature. The energy of interaction between quarks
3
does not diminish with distance (r) as shown in the form of the potential,
Vc = −
4 αs
+ κr.
3 r
(1.1)
As bound quarks become spatially separated, a point is reached when it is more
energetically favorable for a quark-antiquark pair to spontaneously appear from the
vacuum, than for Vc to keep increasing according to Equation 1.1. The creation of
pairs reduces the potential energy associated with the binding of the two original
quarks, more than the energy required to produce the masses of the new quarks.
Effectively, an attempt to separate a hadron (a bound state of two or more quarks)
into its constituent quarks by “pulling” it apart results in the production of a “jet”
of hadrons in a process called hadronization.
Besides confinement, quark dynamics have many other peculiarities. To explain how
otherwise identical particles could coexist, bound in the ground state (i.e, the ∆++ ,
a baryon consisting of three u quarks) without violating the Pauli exclusion principle, quarks were postulated to be endowed with an extra degree of freedom called
“color”[5]. The possible colors an individual quark can posses are red, green or
blue, named after the additive primary colors to draw an analogy to color theory in
the visual arts. The anti-quarks can take on complementary color denoted as cyan
(antired), magenta (antigreen) and yellow (antiblue) analogous with the subtractive
primary colors. As a consequence of confinement, all free particles have a neutral
color charge.
The concept that all hadronic matter is composed of quarks was introduced in part
to explain the flavor symmetry present in the enumeration of hadron states. In the
quark model, baryons contain three quarks and mesons contain a quark-antiquark
pair. To be color neutral, baryons have one quark of each color, red, green and blue.
4
In a meson, it’s quark may take on any color with it’s antiquark taking on the corresponding anti-color. The analogy of color charge with actual color only holds to
the extent to which colors add, forming color neutral states (black or white). The
lightest mesons, one example among many being the pion (π), contain various quantum mechanical combinations of first generation (u and d) quarks. Other categories
of mesonic matter contain higher generations quarks such as the kaons (K), which
possess a strange quark, and the heavy mesons comprising charm and bottom quarks.
The lightest baryons contain only first generation quarks. Protons and neutrons, being the lightest and most common baryons, posses quark configurations of uud and
ddu respectively. The least massive baryon with a single strange quark is known as
the lambda (Λ), uds, whereas the lightest baryonic states of multiple strangeness
are the neutral and charged cascades (Ξ) with quark configurations of uss and dss,
respectively.
The masses of the hadrons are only partially a result of the masses of the quarks
they comprise. For example, the proton mass is around 938 MeV while adding up the
masses of it’s constituent quarks gives a mass around 10 MeV, not much more that 1%
of the total. Most hadronic mass actually comes from gluonic binding energy and the
undulating sea of virtual quark-antiquark pairs that appear and disappear on time
scales according to the Heisenberg Uncertainty Principal, ∆E∆t ≥ h̄2 . Virtual quarks
within the hadron are also commonly referred to as current or sea quarks. Indeed,
the simple quark picture is too large a simplification to characterize many aspects of
hadronic physics. The proton spin puzzle refers physicists’ discovery that the nucleon’s
spin-half is not solely the result of parallel-parallel-antiparallel configuration of it’s
constituent spin-half quarks. Uncovering the origin of proton spin is connected with
our research on the doubly-strange charged cascade (Ξ− ) in the present document
and an elaboration on the connection is provided in Section 1.3.
5
The study of baryons containing multiple strange quarks, which is the subject of our
analysis, has played an important role in the development of the quark model. GellMann’s Eightfold Way [6], intentionally alluding to Buddhism’s Noble Eightfold Path,
organizes the mesons and baryons into ordered multiplets. Fig. 1.1 shows the Baryon
Octet, additionally displaying the now know quark makeup of each particle. The
organizational principles of the Eightfold Way applied to spin-3/2 baryons (shown in
Fig. 1.2) required the existence of the Omega minus (Ω− ), which had not yet been
observed. It was predicted that the missing particle would have a strangeness of -3
and electric charge of -1, which was later verified experimentally, winning Gell-Mann
the Nobel Prize in physics. We now know the quark composition the omega to be
sss.
Figure 1.1: The Baryon Octet as organized in the Eightfold Way according to charge
and strangeness. The octet contains spin-1/2 nucleons, sigmas, the lambda, and
cascades. The quark composition is additionally shown. The particle being studied
in our analysis, the charged cascade, is highlighted.
6
Figure 1.2: The Baryon Decuplet as organized in the Eightfold Way according to
charge and strangeness. The decuplet contains spin-3/2 deltas, excited sigmas, excited
cascades, and the omega. Quark compositions are additionally shown.
In connection with the Eightfold Way and also ultimately explained by the quark
model, the degree of freedom “strangeness” was introduced by Gell-Mann to explain
why kaons and hyperons exhibited a suppressed decay. These unusual particles were
created copiously in energetic collisions but decayed slowly, which seemed to defy
the reversibility of particle interactions. Particles with strangeness typically decay
on time scales around ≈ 10−10 s as opposed to the time scale associated with strong
and electromagnetic decay of ≈ 10−23 s. The peculiar signature of such particles was
broadly explained by the postulate that strangeness is a quantity that is conserved by
the underlying processes of hyperon production, but not in the processes behind their
decay. We now know that strong and electromagnetic interactions, those responsible
for the production of strange hadrons, obey the conservation of strangeness while weak
interactions violate the law. Ground state particles that contain a strange quark are
kinematically forbidden to decay through processes in which strangeness is conserved,
and must instead decay weakly.
7
All observed hadronic matter has consisted of two or three bound quarks, however
further hypothetical mesons and baryons are permitted by the quark model. Pentaquarks, hadrons with an extra quark-antiquark pair (totaling 5 quarks), have been
the subjects of rigorous experimental efforts. Although numerous pentaquark observations were reported in the early 2000s [7, 8, 9], all claims have since been discounted
upon further statistical analysis [3] and inability to reproduce the results. Still today, new observations of pentaquark states is a subject of controversy but none have
been accepted by the nuclear physics community. Ruling out previous pentaquark
candidates does not by any means rule out their possible existence. Similarly, mesons
with multiple quark-antiquark pairs may exist such as the tetraquark, which has been
recently observed [10].
In addition to the hadrons such as the pentaquark and tetraquark, which fall nicely
into the quark model classification scheme, several further “exotic” mesons that lie
outside the quark model could hypothetically exist. Candidates of exotic mesons include glueballs, which are excited interacting gluons devoid of quarks; a state only
possible because of the self-coupling nature of the strong interaction. Other possible
states outside of the quark model are the exotic mesons. Exotic or hybrid mesons
are a bound quark-antiquark pair in which the binding gluons are in an excited state,
augmenting the total meson mass and altering its spin and parity quantum numbers. A significant amount of experimental effort has been devoted to the search for
exotic mesons at the Thomas Jefferson National Accelerator Facility (Jefferson Lab,
or Jlab). Jefferson Lab’s GLUEX collaboration has the principal aim of producing
exotic mesons by exciting a proton target with a photon-beam.
The Standard Model’s demonstrable success and broad range of application, along
with rigorous endeavors probing its theoretical self-consistency, places it among the
most successful and sweeping theories in the history of ideas. The Standard Model
8
does however, fall short of being complete. Among the more prominent problems
with the model, many lie with the fact that it not well incorporated with the general
relativistic formulation of gravity, which itself is a well-established theory. As previously mentioned, the standard model does not, at present, extend its quantum field
characterization to gravitational interaction via gauge boson exchange successfully.
Furthermore, there is reason to believe other forms of matter outside of the standard
model exist. Observations of the universe on it’s largest scale show an accelerating expansion indicating the manifestation of dark energy, which may call for the existence
of additional scalar fields. Furthermore, gravitational lensing experiments, galactic
rotational curves and velocity dispersions among other observational evidence calls
for the existence of extra, non-baryonic matter in the universe that interacts gravitationally, dubbed dark matter. Many candidates for dark matter, such as the axion
or dark photon, lie outside of the standard model. Another prominent issue within
the Standard Model is that it does not correctly account for the non-zero mass of
neutrinos or their oscillations.
In addition to a few fundamental problems with the Standard Model, many more of
its unanswered questions reside within the phenomenological aspects of QCD. Deciphering the substructure of hadrons remains one of the great unsolved problems
in physics. Experimental nuclear physics may address theoretical issues including
proton decay, the proton charge radius, the phases of strongly interacting matter,
hadronization, and the role of gluon self-interactions in nucleons.
1.2 Our Research
In this work we provide the first ever determination of three independent spin observables for the charged cascade (Ξ− ) in photoproduction. From the γp → K + K + Ξ−
reaction (shown in Fig. 1.3), we measure the recoiling cascade’s induced polarization,
9
P , along with the degree of polarization transfered from the photon, Cx and Cz .1
Put succinctly, polarization is the projection of a particle’s spin (depicted in Fig. 1.4),
averaged over a sample, with respect to a quantization axis. Extracting the three
polarization observables, P, Cx and Cz constitute measurements of cascade spin projection onto three orthogonal quantization axes. A detailed discussion of the relevant
observables the coordinate system is left to Chapter 3.
Figure 1.3: The γp → K + K + Ξ− reaction through an arbitrary mechanism. At
present, the production mechanism which we seek to understand in our work, is
unknown.
1.3 The Motivation Behind Our Research
The motivation for our research is multifold. One of the most immediate consequences
of measuring cascade polarization will be a better understanding of the underlying
mechanism behind cascade production, and hence the production of strangeness in
general. We intend that the broader influence of our results will help illuminate
questions such as:
• What processes underlie the photoproduction of hyperons, particularly the cascade?
• What are the processes responsible for hyperon polarization?
1
We take on the notational convention commonly used for single meson production. For double
meson production, P , Cx and Cz are often written as Py , Bzx and Bzz respectively.
10
Figure 1.4: Cartoon depiction of the cascade’s half-spin arising from its internal
constituents and their dynamics. In general, the overall spin comes from quantum
mechanical addition rules, summing contributions from valence quarks (ssd for the
cascade), virtual quark-pairs (sea quarks) and gluons; all three types of constituents
contain intrinsic spin, and orbital angular momentum. The relative contributions
from each source of spin are unknown, which is in part what the present work examines.
• Which excited baryons contribute most significantly to cascade photoproduction?
• Is photoproduction a feasible means to study cascade spin?
Elaboration on the motivation behind our research is provided in the following subsections.
1.3.1
Motivation: Current Status of Cascade Physics
Cascade states (Ξ), which contain two strange quarks, are experimentally under explored as compared with the non-strange baryons (N and ∆) and the S = −1 hyperons (Λ and Σ). In particular, the production mechanism for ground state Ξ baryons
remain unknown, also progress is lacking in the observation of the excited Ξ∗ states
that are predicted to exist by flavor SU (3)F symmetry. Flavor symmetry requires a
corresponding cascade resonance for each N ∗ and ∆∗ resonance, the PDG [3] reports
11
there are around 26 N and ∆’s with a status of 3-star and above. Only six such
cascade resonances have been reported with an additional five constituting the resonances with 2- or 1-star status. The abysmal state of global cascade data is largely
because of small production cross sections, cascades, having two strange quarks, can
only be produced via indirect processes. The little experimental progress that has
been made has been insufficient to significantly stimulate theoretical developments,
and aside from Ref. [11], most theoretical efforts have been devoted to pentaquark
production [12].
At present, there is only one existing publication [13] in which differential cross
sections for Ξ photoproduction were measured. The recent results for the γp →
K + K + Ξ− reaction showed a high probability of the cascade recoiling in the direction
of the proton momentum (in the center-of-mass frame); this is usually an indication
of t-channel production. It was suggested by K. Nakayama [11] that the leading
contribution of Ξ photoproduction comes through meson exchange and intermediate
S = −1 hyperon resonances, as depicted (for t-channel) in Fig. 1.5. It is natural to
assume singly strange intermediate hyperons for the reaction, because otherwise, the
it would involve a t-channel exchange of S=2 exotic mesons. The model of Ref. [11]
was the only production model available for comparison, which, matched the experimental cross-section results fairly well. The results of Ref. [13] did not include any
polarization measurements, though it was pointed out that contributions from various hyperons states can be differentiated using the Ξ− spin observables [14]. We
seek to provide the first of much needed Ξ− polarization observables, which will help
determine the mechanisms behind photoproduction of strangeness and the hadronic
processes that give rise to hyperon polarization.
12
Figure 1.5: An illustration of cascade photoproduction in the t-channel, arising
through the decay of an intermediate hyperon.
1.3.2
Motivation: Testing Predictions from Theory
Just as the Λ has been useful in exploring the S = 0 baryon resonance structure [15],
so too can the cascade be used as a sensitive probe for hyperon resonance contributions. The model mentioned in Section 1.3.1 and detailed in Ref. [11] makes specific
predictions for the polarization observables. Such predictions turn out to have a high
sensitivity to the relative contributions from intermediate hyperon resonances. Only
the resonances with sufficient existing information to determine the hadronic and
electromagnetic coupling constants, shown in Table 1.2, were included. The model is
based on a relativistic meson-exchange of hadronic interactions where the production
amplitudes are calculated from the relevant effective Lagrangian in the tree-level approximation. Although the model does not involve loop integration, it is more robust
than a typical tree-level calculation in that it accounts for the final state interaction
effects through the generalized contact terms which keep gauge invariance of the photoproduction amplitudes. Contributing diagrams to production that are considered
in the model are shown in Fig. 1.6.
13
Table 1.2: Three-star and above Lambda and Sigma hyper resonances listed by the
PDG [3]. Source [11]
Figure 1.6: Diagrams contributing to cascade production in the model of [11].
14
More recently, the model of Ref. [11] has been appended in Ref. [14] to include the
Σ(2030), a four-star hyperon resonance with spin-7/2. The motivation for inclusion
of higher spin resonances came from a discrepancy with experimental data reported
by Ref. [13]. In this work, a comparison of experimental and theoretical results for
Ξ− polarization observables will be made to constrain the production model. There
are three variants on the cascade photoproduction model available for comparison, a
summary of the variants is given below:
1. Pure pseudoscalar coupling with intermediate hyperons up to the Λ(1890).
2. Pure pseudovector coupling with intermediate hyperons up to the Λ(1890).
3. Pure pseudovector coupling with intermediate hyperons up to the Σ(2030).
Polarization predictions for all three variants were compared with our results.
1.3.3
Motivation: Recent Analogous Lambda Results
Surprising experimental results have been reported by Bradford and Schumacher in
Ref. [15] regarding the measured Λ polarization in the reaction γp → K + Λ. The spin
observables they measured, Cx and Cz , determined the polarization transfered from
the photon-beam, to the recoiling Λ in the x and z directions, respectively. Together,
Cx and Cz measure the full polarization transfer, and, when taken along with the
induced polarization (P ), they constitute a basis for complete characterization of the
Λ spin orientation. For an elaboration on observables Cx , Cz and P , see Chapter 3.
It was reported that a remarkably large probability of spin transfer was found for all
kaon angles and all beam energies for the experimental energy range 1.7 < W < 2.3
GeV2 . The integrated results showed, when combined with previous measurements of
2
W denotes the center-of-momentum (frame of reference with zero total momentum) energy in
the initial state, which is the amount of energy available to produce the final state particles.
15
induced polarization P , that,
P̄Λ ≡
p
Cx2 + P 2 + Cz2 = 1.01 ± +0.01,
(1.2)
which is consistent with unity to within high precision. Such findings show that
for production with a fully polarized photon-beam, whichever processes may be contributing to production, the Λ would be produced with 100% polarization. Notably,
not a single preexisting model of the production process was in agreement with such
results. As many as six highly developed theoretical models were compared with the
data, all of them failed to describe the observations [16].
In Ref. [16], Schumacher explains his results with an ad-hoc hypothesis invoking vector meson dominance, which postulates the photon has a hadronic component, and
diquark considerations, which, roughly speaking, models baryons as being composed
of a quark bound with a spin-zero double quark. In his explanation, the incoming
photon spontaneously produces a φ (ss̄ quark pair), as shown in Fig. 1.7. The produced ss̄ pair separates and interacts with the proton, undergoing hadronization into
a K + and Λ. In this picture, the polarization of the s quark is preserved after it is
precessed by a spin-orbit interaction. The ud quark pair within the Λ is the main
diquark component, being the only possible spin and isospin zero quark pair. Such
a scenario leaves the polarization of Λ dominated, presumably, from the polarization
of the s quark. One of the issues with Schumacher’s hypothesis however, is that it
contradicts the well known contribution of N ∗ resonances to the production of Λ.
Anisovich pointed out in Ref. [17] that by invoking contributions from the N (1900)
resonance, a quantitative description of Λ data could be obtained through the conventional isobar picture.
16
Figure 1.7: An illustration of Schumacher’s hypothesis. Image source [16]
Although there is still some uncertainty regarding the production mechanism for the
Λ, great progress has been made as a result of the aforementioned spin observable
measurements. Collectively, cross section and polarization measurements [18, 15]
were necessary to identify the various mechanisms for Λ production, including the
identification of numerous intermediate excited nucleon (N ∗ ) resonances. We seek to
make analogous measurements for the Ξ− to explore production and the contributing
S = −1 hyperon spectrum. We expect results for the cascade to differ greatly from
that of the Λ due to fundamentally differing production mechanisms and diquark
configurations.
1.3.4
Motivation: Vector Meson Dominance
[h] The total constituent quark wave function for the Ξ− is [19],
1
ψΞ− = √ [2(s ↑ d ↓ s ↑) + 2(s ↑ s ↑ d ↓) + 2(d ↓ s ↑ s ↑) − (d ↑ s ↓ s ↑) − (s ↑ d ↑ s ↓)
18
− (s ↓ d ↑ s ↑) − (s ↑ s ↓ d ↑) − (d ↑ s ↑ s ↓) − (s ↓ s ↑ d ↑)],
(1.3)
17
while the wave function for the Λ is,
1
ψΛ = √ [(u ↑ d ↓ s ↑) − (u ↓ d ↑ s ↑) − (d ↑ u ↓ s ↑) + (d ↓ u ↑ s ↑) + (s ↑ u ↑ d ↓)
12
− (s ↑ u ↓ d ↑) − (s ↑ d ↑ u ↓) + (s ↑ d ↓ u ↑) − (d ↓ s ↑ u ↑) + (d ↑ s ↑ u ↓)
− (u ↓ s ↑ d ↑) + (u ↑ s ↑ d ↓)].
(1.4)
We see from Equation 1.4 that the constituent quark model predicts that the majority
of the lambda’s spin comes from the s-quark, since the u- and d-quarks are antialigned with one another. We point out that in the aforementioned vector meson
dominance picture of lambda photoproduction, the diquark considerations need not
be invoked to explain its large total polarization, since either way, the s-quark is the
main contributer to the lambda’s spin.
The wave function for the cascade however, shown in 1.3, yields a more intricate
spin structure. With vector meson dominance, one of the two s-quarks within the
cascade has its origin from the photon, so one might expect the cascade to have
similar polarization features as the lambda, but with a diluted overall magnitude,
as can be inferred by comparing Figs. 1.8 and 1.9. The results of our polarization
measurements will be compared with qualitative expectations from the vector meson
dominance polarization mechanism and compared with previous results of the lambda.
As a side note, if diquark effects are manifest, a measurement of Ξ− polarization may
serve as a sensitive probe of it’s d-quark spin contribution. The cascade’s only possible
isospin-zero quark pair is the ss, where the spin-zero configuration dominates over
the spin-one configuration [20]. In this picture, the majority of cascade polarization
comes from the d-quark, which is likely inherited from the initial state proton target
(see Fig. 1.9). In this case, experiments with target polarization would be interesting.
18
For the present work, where the proton is unpolarized, we might expect a small total
cascade polarization, with little kinematic dependence, associated with strong diquark
correlations.
Figure 1.8: Cartoon depiction of lambda photoproduction through vector meson dominance. In this picture, the polarization of the lambda comes from the strange quark,
which is produced through quantum fluctuations of the photon. Most of the photon
polarization transfers to the lambda in this picture.
Figure 1.9: Cartoon depiction of cascade photoproduction through vector meson
dominance. In this picture, the polarization of the cascade comes from the strange
quark, which is produced through quantum fluctuations of the photon. While the
strange quark from the photon may be fully polarized within the cascade, its effects
are diminished since it may only contribute a fraction to the total cascade spin. Thus,
only some of the photon polarization transfers to the cascade.
19
1.3.5
Motivation: Universal Hyperon Polarization at High
Energies
[h] Polarization is a coherent phenomenon that depends on the interference of complex
amplitudes. In reactions with energies well above threshold, there are a large number
of available amplitudes, with different phases. Because of the presumed cacophony
from the addition of the various phases, it was expected that no polarization would
be observed for hyperons from reactions around 10 GeV above their threshold energy.
The 1976 discovery [21] at Fermilab of non-zero induced (or transverse) Λ polarization
in baryon-baryon collision at 300 GeV was highly unexpected. Since the discovery, a
considerable experimental effort has been underway to characterize the polarization of
all the hyperons. As it turns out, Λ, Σ0 , Σ− , Σ+ , Ξ0 and Ξ− are all produced polarized.
The most basic applications of perturbative Quantum Chromodynamics (pQCD),
which assume massless quarks, predicts no polarization at high energies. On the
other hand, recent developments in pQCD do in fact exhibit non-zero hyperon polarization, however the predictive power of such theories are limited. At present,
hyperon polarization is still mysterious and there is no model that can explain the
observations.
Although high energy cascade experiments have been extensively conducted, where
partonic effects dominate, no one has explored cascade polarization near threshold,
in the hadronic regime. The data in the present work provides a unique opportunity
to study cascade polarization in this unexplored region.
20
1.3.6
Motivation: Connection Between Polarization Observables and Production Amplitudes
[H] Single pseudoscalar-meson photoproduction off nucleons is fully described by four
complex amplitudes. The bilinear combinations of these amplitudes enumerate a total
of 16 observables [22, 23]. As it turns out, of these 16 observables, seven are needed
to determine the amplitudes up to discrete ambiguities (and an overall phase), known
as a complete experiment. It is also known that two further measurements (totaling
nine) are sufficient to eliminate all ambiguities (up to an overall phase) [22]. The
set of observables necessary for an amplitude analysis has been shown to include the
differential cross-section,
dσ
,
dt
and three single-polarization observables; namely the
polarized photon-beam asymmetry, Σ, the target-polarization asymmetry, T , and the
induced baryon recoil polarization P . The twelve remaining unique quantities are
double polarization observables that characterize the reactions under different combinations of beam, target, and baryon recoil polarization. The double polarization observables can be categorized into three groups of four, beam-recoil (BR), target-recoil
(TR) and beam-target (BT) observables. The remaining three measurements, which
are necessary and sufficient to uniquely determine the underlying complex amplitudes
up to discrete ambiguities, include any three of the double polarization observables,
so long as the measurements belong to more than one group. For the elimination of
discrete ambiguities, the measurement of two more double polarization observables
suffices provided that no four measurements come from the same group [22, 23, 24].
Similar requirements for complete experiments have been formulated in Ref. [25] for
the case of two pseudoscalar photoproduction off a nucleon. It was shown for double
meson production that one needs at least 15 observables to constitute a complete
experiment. Current theoretical efforts are underway with the intention of charac-
21
terizing complete experiments with a further reduction of observables. For cascade
photoproduction, no polarization observables have been measured to date. The current research, measuring three of the needed polarization observables, P, Cx and Cz , in
the energy range from 2.8 GeV < Eγ < 5.5 GeV. While the realization of a complete
measurement is still infeasible with current nuclear physics instrumentation, there is
much to be learned along the way, and polarization observables provide detailed information on the interference of the production amplitudes. Many Λ and Σ resonances
have been observed and studied, but there are still large gaps to close in the world
database regarding the spectrum of possible hyperon states and their properties.
1.3.7
Proton Spin Crisis and Baryon Polarization
In the naive quark model, the nucleon consists of three bound, spin-half valence
quarks which gives rise to a rather simple spin structure: the one-half overall spin
of the nucleon should arise solely from the alignment of two valence quarks with the
remaining quark in an anti-aligned configuration. In this picture, non-contribution
from orbital angular momentum to the nucleon is assumed since its stability suggests
that it occupies the lowest possible energy state, hence having a total orbital angular
momentum of zero.
An experiment carried out in 1987 at the European Muon Collaboration (EMC) [26]
found that, contrary to widespread expectations from the naive quark model, the
valence quark spin contribution to the total spin of proton was 20 ± 20%, which
was not only far smaller than 100%, but consistent with 0% [26]. The mysterious
result coined the proton spin crisis or proton spin puzzle prompted an ongoing and
vigorous attempt within the nuclear and particle physics community to understand
the spin structure of the proton. Today, a quarter-century after the EMC results were
published, the proton spin puzzle is largely resolved, although still not in its entirety.
22
Physicists realized that in order to decipher the spin structure of the nucleon, a
more complete quantum chromodynamical model must be considered with non-zero
orbital angular momentum. In this model, the nucleon’s spin comes from it being
made up three constitute quarks along with mediating spin-1 gluons in addition to
virtual quark anti-quark pairs that arise from quantum fluctuations. More exactly,
the postulated spin originating from four basic components is given as,
1
1
= Σ + G + Lq + Lg .
2
2
(1.5)
Each term from left to right (on the right-hand-side) represents the spin component of
the quarks, gluons, followed by the angular momentum component of the quarks and
gluons. The quark spin component can be further decomposed into separate quark
contributions,
¯ + (∆s + ∆s̄) + ...
Σ = (∆u + ∆d) + (∆ū + ∆d)
(1.6)
where the unincluded remaining terms in the sum consist of spin components from
the heavier quarks, from which contributions should be negligible due to their large
masses and resulting sparsity in the nucleon.
Current world data, accumulated over a period of 27 years, suggests that the quarkspin contribution term is no longer consistent with zero, having a value of, [3]
1
Σ ≈ 0.33 ± 0.05.
2
(1.7)
The measured value of the quark-spin contribution reduces the nucleon’s proportion
of missing spin to be on the order of 66%. Recent experiments seem to indicate that
most of the quark spin contribution comes from valence quarks, and very little is from
23
the quark sea [27]. Additionally, experiments have also shown that the gluon spin
term G, may account for up to 50% of the missing nucleon spin [27]. The remainder of
the nucleon’s missing spin is believed to lie predominantly in the finite orbital angular
momentum of the valence quarks.
Although significant experimental and theoretical progress has been made in understanding nucleon’s spin structure, the proton spin puzzle continues to be considered as
one of the most important unsolved problems in the field of physics. Some remaining
unanswered questions include:
• What are the underlying factors that determine the relative contributions to
the total spin of the nucleon?
• Why does the quark spin contribute so little?
• Does the missing proton spin arise due to effects from valence quarks, sea quarks
or gluons?
• Is it possible to separate the quark and gluon orbital angular momentum contributions from experiments in a model-independent way?
Further questions can be asked relating the study of hyperons to the spin puzzle which
include,
• Do the heaver quarks, particularly strange quarks, contribute to the spin structure of the nucleon?
• Are the strange quarks themselves polarized within the nucleon?
• Are the spin structures of ground-state hyperons similar to what the naive quark
model suggests?
• Is the spin structure of ground-state hyperons different from that of the nucleon?
24
• Can the individual valence up and down quark contributions be separated
through the photoproduction of hyperons off polarized a nucleon target?
Results of the present analysis when combined with similar future experiments in
CLAS12, involving large amounts of data and target polarization could shed light
on a number of the above questions. Target polarization measurements in particular
could help decipher the separate contributions of the up and down valence quarks in
the nucleon.
25
Chapter 2
The g12 Experiment
The data analyzed in this work, collectively known as g12 (or E04-005), were collected at Thomas Jefferson National Accelerator Facility (Jefferson Lab) using the
CEBAF Large Acceptance Spectrometer (CLAS). The g12 dataset is, at present, the
world’s largest for meson photoproduction in its energy range and provides a unique
opportunity to study the cascade. Physics events were generated in 2008 using a
circularly polarized photon beam incident on a 40-cm unpolarized liquid-hydrogen
target. The photon beam was produced through a bremsstrahlung process using
a 5.71 GeV electron beam provided by Jefferson Lab’s Continuous Electron Beam
Accelerator Facility (CEBAF). The main component of CLAS is a superconducting toroidal magnet coupled with multi-wire drift-chambers, which track the charged
particles emanating from the nuclear reactions of interest and enable momentum reconstruction. The other components of CLAS are mostly utilized to obtain particle
identification and timing information. Since the time of the g12 data collection, CEBAF has undergone an upgrade doubling the electron energy, enhancing or replacing
preexisting spectrometers, and adding a fourth spectrometer (GLUEX). Laboratory
conditions at the time of data collection will be described.
2.1 CEBAF Accelerator
CEBAF was built to provide a platform for scattering experiments to explore the rules
governing the interaction between quarks. The accelerator was engineered to provide
a high quality, continuous-wave electron beam with up to 80% polarization, energies
up to 6 GeV, and currents up to 200 µA. The design makes use of superconducting
niobium radio-frequency (SRF) cavities to provide the acceleration gradient. After
26
construction, CEBAF was the world’s only accelerator to implement superconducting
niobium technology. Earlier accelerators used non-superconducting copper cavities,
which due to resistive heating, would have been economically impractical to power
under CEBAF’s quality specifications. A photograph of a cavity used at CEBAF is
shown in Fig. 2.1
Figure 2.1: A photograph of an SRF cavity made from superconducting niobium. As
electron clusters travel through the cavity, they experience a continuous acceleration
due to a standing electromagnetic wave. The energy resonances can be tuned by
mechanically adjusting the cavity lengths. Image source [28].
The large-scale design of CEBAF comprises two linear accelerators (linacs) connected
by arcs at either far end, reminiscent of a racetrack, as shown in Figs. 2.2 and 2.3.
Electrons are first produced by a laser incident on a photo-cathode and are injected
into the linac. The cavity geometry of each linac results in electron bunches approximately 90 µm in length, separated by 667 picoseconds. After each pass through a
linac, the electrons enter the recirculation arcs made up of bending magnets. The
track that the electron enters within the recirculation arc is determined by it’s momentum. The electron bunches increase in energy by 1200 MeV after every lap around
the track, completing a full lap up to five times before being delivered to experimental
halls. Having a final energy of up to 6 GeV, the beam bunches are injected sequentially into the halls by the beam switchyard, with a separation between bunches of 2
ns [29, 30, 31].
27
Figure 2.2: The large-scale design of CEBAF and it’s components. Image Source [28].
Figure 2.3: An aerial photograph of CEBAF. Image Source [31].
28
2.2 Hall B Photon Tagger
Hall B has a Bremsstrahlung radiator and a photon tagging system know as the
“tagger”, which is shown in Fig 2.4. The tagger enables scattering experiments using a
real photon beam in addition to the electron beam provided by CEBAF. The radiator
typically consists of a thin gold target in the path of the electron beam. A dipole
magnet produces a 1.75 T uniform field that bends scattered electrons where they
are detected by a two-layer set of scintillators. [30, 31, 32]. The position of each
scattered electron in is determined by the first layer of scintillators while the timing
is measured by the second layer.
The Bremsstrahlung process takes place due to the incident electrons decelerating in
the coulomb field of radiator’s nuclei, producing a photon in accordance with energymomentum conservation. The mass of gold nuclei are much greater than the scattered
electron mass, hence a negligible amount of momentum is transferred. Thus one can
write to high precision,
Eγ = Eei − Eef ,
(2.1)
where Eγ is the energy of the bremsstrahlung photon, Eei is the energy of the incident
electron and Eef is the energy of the scattered electron, which is measured by the taggers’s magnetic spectrometer. The scattered electron position along with knowledge
of the magnetic field gives Eef , while the timing is used to identify which electron
bunch created the photon. Tagger timing along with times from particle tracks in
CLAS give the time of creation for the photon responsible for the physics event [32].
2.3 Hydrogen Target
The g12 experiment used an unpolarized cylindrical target that was 40 cm in length
and 4 cm in radius made of liquid hydrogen (H2 ) and housed in Kapton. The H2
target cell is shown in Figure 2.5
29
Figure 2.4: A profile of the photon tagger. Incident electrons are scattered in the
radiator and then steered by a dipole magnet. Energy and timing information of the
scattered electrons are measured by the scintillator array. Image Source [28].
The target was placed 90 cm upstream of CLAS’s geometrical center. The upstream
placement had the advantage that it increased the efficiency of events at small deflection angles by reducing the angular size of the forward hole in the detector. The
main drawback to the upstream placement of the target’s center was that it caused
the drift chamber resolution to suffer as a result of the particle tracks passing on more
oblique angles relative to the detector planes [29, 30, 31, 33].
Figure 2.5: A drawing of the target. Image Source [28].
30
2.4 The CLAS spectrometer
The CLAS detector, which is shown in Figs. 2.6 and 2.7, (acronym for the CEBAF
Large Acceptance Spectrometer) consists of six sectors segmented in φ (azimuthal
to the beamline). Each sector comprises a four scintillator start counter (ST), three
layers of drift chambers (DC), a gas Cherenkov counter (CC), a series of scintillating time-of-flight (TOF) paddles and finally an electromagnetic calorimeter (EC). A
magnetic field produced in the φ direction by a toroidal magnet located in the middle
DC layer, bends charged particles toward or away from the beamline (polar angle θ).
The magnetic field being solely in the φ direction forces the trajectory of each charged
particle to lay in a plane, allowing for a simplified reconstruction algorithm. Whether
inward or outward bending of trajectory takes place depends on the polarity of the
toroidal magnet and sign of the particle charge [29, 30, 31, 33].
Figure 2.6: Drawing of CLAS nested within Hall B. The electron beam enters the
tagger on the right and produces a photon beam. The photon beam continues through
the target near the center of CLAS where physics events take place. The photons that
do not interact with the target pass through the detector to the main beam dump on
the left. Image Source [30]
31
Figure 2.7: Schematic of the CLAS. CLAS is approximately 4 meters in radius. Image
Source [28].
2.5 Start Counter
The start counter (ST) shown in Fig. 2.8 surrounds the target is the inner-most
detector system in CLAS, and is only used in photon-beam experiments. The purpose
of the ST is to collect timing information for the outgoing particles emanating from
the target. The basic design is a hexagonal array of 24 scintillator paddles. For
coverage at smaller angles in θ. The downstream ends of the scintillators are bent
inwards converging into a six-sided cone [29].
The start counter provides timing measurements close to the event vertex. It was
used in the trigger logic in coincidence with the tagger readout and the time-offlight readout. Additionally, the start counter helps select the appropriate RF-beambucket, which leads to precise measurements of the photon-time, and hence vertextime. Events with a large number of tracks give better timing estimates since they
provide repeated measurements. The hexagonal segmentation of the ST provides
flexibility for trigger logic as individual segments can be excluded [29].
32
Figure 2.8: A drawing of the CLAS start counter. Image source [28]
2.6 Drift Chambers
Each of the six sectors of CLAS contain three large multi-layer drift chambers for
tracking charged particles emanating from the event vertex within the target [34].
The geometry of the detector, as shown in Fig. 2.9, provides good tracking resolution,
efficiency, and large acceptance [34]. The entire detector contains 35,148 individually
instrumented hexagonal drift cells [34]. Fig. 2.10 shows the toroidal magnetic field in
relation to the regions of the DC [29].
All regions of the DC contain two superlayers as shown in Fig. 2.11 which comprise
six layers strung with 20 µm diameter gold-plated tungsten sense wires and 140 µm
diameter gold-plated aluminum field wires [34]. The field wires are arranged to form
a hexagon drift cell with a single sense wire in the middle as shown in Fig. 2.12.
The sense wires receive a current when a charged particle of sufficient energy passes
through the DC, ionizing a non-flammable 90% argon 10% carbon-dioxide gas mixture. The ionized electrons are accelerated towards their nearest sense wires by the
electric gradient, which is produced by the surrounding field wires [29]. Further details
on the CLAS drift chamber can be found in Refs. [29], [34] and [35].
33
Figure 2.9: A profile drawing of CLAS detecting a two-track event coming from the
target. Image Source [31].
Figure 2.10: Drawing of a cross-sectional view of the DC regions and the toroidal
magnetic field at half current (1930 A). Image Source [30]
34
Figure 2.11: Diagram of the sectors, regions and super layers of the DC. Image Source
[36]
Figure 2.12: Drawing of a track going through the five layers of the DC. Image Source
[28]
35
2.7 Superconducting Toroidal Magnet
The toroidal magnetic field in CLAS is produced by six kidney-shaped superconducting coils [29] in between each of the six sectors of the drift-chamber (DC), which can
be seen in Fig. 2.7. Each coil consists of four layers of 54 windings of superconducting
niobium (NbTi/Cu), stabilized by aluminum [29].
During the g12 production runs, the magnet operated at a current of 1930A (corresponding to half capacity) with a half-maximum field of 1.75 T [18] and a polarity
such that negatively charged particles are bent inward towards the beam line. The
benefit of running at half field-strength was that it increased the efficiency of detecting negatively-charged particles. Higher field strength tends to bend the inward-going
negative-tracks more closely to the “hole” in the forward region where they cannot be
detected. The main drawback of the half field-strength operation is that it decreases
momentum resolution [31, 29].
Figure 2.13: Photo of the torus magnet during construction. Image Source [28]
2.8 Time-of-Flight Detectors
Accurate time-of-flight information is essential to particle identification. Along with
the vertex time, time-of-flight provides a calculation of velocity for each particle.
The velocity, in conjunction with momentum from the drift chambers, yields particle
36
mass, which is used to identify the particle. The CLAS time-of-flight (TOF) detector
is a scintillating shell surrounding the Cherenkov counter region about 5 m from the
center of the target. Each of the six identical sectors are made up of 57 scintillator
paddles of 5.08 cm thickness and various lengths. Each paddle has a PMT attached
to both ends. The detector system with nearly 100% efficiency for minimum ionizing
particles provides timing measurements with a resolution between 150-200 ps. Timing
information from the TOF is integrated with the start counter in the trigger logic [29,
37].
The 57 scintillator paddles are bundled into three groups in lab-frame polar angle (θ):
• 8.6◦ < θ < 49.9◦ : scintillator 1-23: width = 15 cm
• 49.9◦ < θ < 131.4◦ : scintillator 24-53: width = 22 cm
• 134.2◦ < θ < 141.0◦ : scintillator 54-57: width = 15 cm
Further information on the time of flight detectors can be found in Ref. [37].
2.9 Cherenkov Counters
The CLAS Cherenkov counter (CC), shown in Fig. 2.14 surrounds the region-three
drift chamber and is used exclusively in particle identification for lepton-pion discrimination. The CC is capable of detecting leptons within 8◦ < θ < 45◦ . As with all
CLAS subsystems, the CC is divided into six identical sectors in φ [29].
Each fixture is filled with a perfluorobutane (C4F10) gas medium with an index of
refraction of 1.00153. The velocity of any hadron track with energy less than 2.5 GeV
is less than the speed of light in perfluorobutane and thus will pass through the CC
without emitting Cherenkov radiation. Leptons, being lighter however, will mostly
be traveling above the threshold velocity and thus will emit light and be detected,
which provides a means to distinguish between leptons and pion.
37
Figure 2.14: A diagram showing the CC response to a single lepton track entering
One of the six segments. Lepton tracks emit Cerenkov radiation as they pass through
the perfluorobutane (n = 1.00153) gas medium. The Cerenkov light first travels to
and is focused by the elliptical outer-surface mirror onto the hyperbolic inner-surface
mirror where it is finally directed into one of the photomultiplier tubes at either end
of the fixture. Image Source [28]
The Cherenkov detector was not used in the present work. More information on the
design and construction of the CC can be found in Ref. [29] and [38].
2.10 Electromagnetic Calorimeters
In the very outer layer of CLAS lies the electromagnetic calorimeter (EC) as shown
in Fig 2.15. The EC is a 102 ton, six-sector, triangular subsystem composed of 39
alternating layers of lead sheets and 36, 10 mm-wide scintillating bars. Each layer is
sequentially rotated by 120◦ repeating after three layers. The naming scheme for the
layers is u-v-w (see Fig. 2.15). The convention provides the ability to make position
measurements with a resolution on the order of the scintillator thickness. Since the
EC is designed to absorb all of the kinetic energy associated with each particle track,
the readout can be related to the total energy of the track. The utility of the EC
is multifold: In electron beam experiments, the EC is useful to detect scattered
electrons originating from the beam. The EC also provides reliable reconstruction
of high energy neutral particles such as photons and neutrons. Furthermore, due
to the EC’s fast response time, it is useful as input for the trigger logic, which was
38
it’s primary use in g12 [30, 33]. More information on the design, construction, and
performance of the EC can be found in Ref. [39].
Figure 2.15: A diagram of one dissected sector of the EC with the three u-v-w layering
convention illustrated. Image Source [28].
2.11 Trigger and Data Acquisition System
The data acquisition (DAQ) used by CLAS is a multi-layer suite of electronics for
recording signals in real time. The experimental trigger is a user-defined logical set of
coupled subsystem signal-criteria, used to distinguish desired events from background.
The primary trigger required a 100 ns coincidence between the tagger, ST and TOF,
and also requiring two-track events have high incident photon energy [30, 31, 33].
Fig. 2.16 shows an example of the basic trigger logic used in data collection for a
single sector. A detailed discussion of the trigger configurations used in g12 is found
in Ref. [30].
The g12 experiment recorded events faster than any previous CLAS experiment, with
a trigger rate of 8 KHz. As a consequence of the high trigger rate, g12 amounted to
the world’s largest dataset for the photoproduction of mesons in this energy range;
26B events were recorded, totaling 121 TB of data and an experimental luminosity
of 68 pb−1 [30, 31, 33].
39
Signals that pass the trigger are recorded in an event-based data format where an event
occupies a set time interval. At the end of each event’s time interval, the signals from
every CLAS subsystem are read out and recorded. After going through a type of
high-pass voltage-filter known as a discriminator the subsystem signals are digitized
by two electronic components: Analog-to-digital converters (ADC), which report the
time integral of the signal voltage and time-to-digital converters (TDC), which report
the time at which a signal arrives. The digitized data stream is then integrated and
written to disk in the event-based data format while being monitored by the DAQ
system before being written to a tape-silo for long-term storage [33, 30, 31].
Figure 2.16: Illustrated are three distinct coincidence trigger criteria. ECPinner and
ECPtotal represent a photon signal above threshold for the inner layer of the EC and
entire EC respectively. ECEinner and ECEtotal represent an electron signal above
threshold for the inner layer of the EC and entire EC respectively. ST0,1,2,3 are the
four TDC signals for the start counter. Image Source [30]
2.12 Event Reconstruction
The toroidal magnetic field geometry allows event reconstruction to be performed in
each sector independently since, in general, the particle tracks will not transverse from
one sector to the other. Reconstruction is done via hit-based tracking and time-based
tracking. Hit-based tracking within the DC is done by finding tracks that produce
40
aligned clusters of superlayer hits within six layers of the DC in any given region.
The segments that align to a physically permissible curve in all three layers are then
selected to construct a full track [30, 31, 33]. The curvature of this track gives the
magnitude of it’s momentum by,
p=
l2 qB
,
8s
(2.2)
where p is the track momentum, l is the chord length, q is the charge of the particle, B
is the field strength and s is the sagitta length. The chord and sagitta are diagrammed
in Fig. 2.17. To further eliminate reconstructed tracks that are not associated with
a true particle track, each track is extrapolated back to the TOF detector where
only those that have an aligned hit are selected. The time-based tracking refines the
momentum and track-path resolution; it limits DC hits, according to its drift time,
to those falling within a specified time window [30, 33]. All good tracks are further
Figure 2.17: The path of a charged particle going through a magnetic field. When the
radius of curvature is large when compared with the sagitta, the radius of curvature
l2
can be approximated by r ≈ 2s
where l is the chord length and s is the sagitta length.
Image source [30]
extrapolated to the remaining components of CLAS, if hits show up in all components
within the appropriate volume, the track is kept.
In the final reconstruction, a known hadronic state is assigned to the track based
41
on its charge and mass calculation. The measurement of the mass is made through
momentum and timing measurements,
m2 =
p2
(1 − β 2 )
β2
(2.3)
with,
β=
t0 − t
.
cl
(2.4)
Above, p is the measured momentum, t0 is the time of the TOF hit, t is the calculated
vertex time and l is the track length from the vertex to the hit TOF paddle. The
initial particle identification (PID) is assigned according to the following threshold
criteria for the measured mass (m),



π
m < 0.3





 K 0.35 < m < 0.65
P ID =


p
0.8 < m < 1.2





 d
1.5 < m < 2.2
GeV
GeV
(2.5)
GeV
GeV
Tracks with calculated masses outside of the above ranges are not identified[30, 31, 33].
2.13 Cascade Data
The g12 data had a total luminosity of 68 pb−1 . The g12 dataset, is at present the
world’s largest for photoproduction near its energy range and provides a first time
opportunity to study the polarization of the Ξ− in photoproduction. By detecting
K + K + π − (Λ) in the final state, the full kinematics for the γp → K + K + Ξ− reaction
were reconstructed. After all event selection criteria, we detected a nearly background
free signal of approximately 5000 events in which the cascade undergoes the weak
decay Ξ− → Λπ − , a globally unprecedented yield. A higher yield for cascade events
is possible through the exclusion of pion detection requirements, however the angular
42
distribution of one of the cascade decay products is necessary to infer its polarization.
Although the detection of the pion results in a decreased signal yield because of the
decrease in overall efficiency, the pion detection nearly eliminates the background by
providing additional kinematic requirements. Extensive details on the method of data
analysis are provided in the ensuing chapters.
2.13.1
Reconstructed Cascade and Lambda Tracks
The four-vectors of the missing Λ and intermediate Ξ− are reconstructed through
energy and momentum conservation. Momentum of the Ξ− is identified from the
missing momentum of the K + K + system while the momentum of the Λ is identified
with the missing momentum of the K + K + π − system,
pΞ− = pmissKK ,
(2.6)
pΛ = pmissKKπ .
(2.7)
and
The cascade four momentum is found by,
pΞ −
 q
m2Ξ− + p2Ξ−



px

=

py


pz
43





,



(2.8)
where mΞ− is the known mass of the Ξ− , pΞ− represents it’s three momentum and
px , py , pz , the Cartesian components thereof. Similarly for the Λ,
 p
m2Λ + p2Λ



px

pΛ = 

py


pz





.



(2.9)
Above, mΞ− = 1.32131 ± 0.00013 GeV is the known Ξ− mass, and similarly mΛ =
1.115683 ± 0.000006 GeV [3]. With knowledge of each four-vector for a sample of
γp → K + K + π − (Λ) events, the polarization of the Ξ− can be determined.
44
Chapter 3
Formalism and Methodological Framework
The present work seeks to make a first-time determination of the recoil-polarization
P , along with the double polarization observables Cx and Cz for the Ξ− hyperon in
the γp → K + K + Ξ− reaction 1 . Loosely speaking, Cx and Cz measure the degree of
polarization transfer from the circularly polarized photon-beam to the Ξ− while P is
a measure of it’s induced polarization.
The weak decay of the cascade, as depicted in Fig. 3.1, provides the physical basis
for the methods used in extracting our results, which are detailed in the remainder
of the current chapter.
Figure 3.1: The γp → K + K + Ξ− reaction through an arbitrary mechanism with the
subsequent Ξ− → Λπ − weak decay.
3.1 Spin Observables and the Coordinate System
As a consequence of the fact that cascade production occurs via electromagnetic and
strong interactions, parity is conserved in the transition from the initial γp state to the
final K + K + Ξ− state2 . The parity conserving nature of cascade photoproduction has
1
For double meson production, P , Cx and Cz are often written as Py , Bzx and Bzz respectively.
2
Note that final state may refer to either K + K + Ξ− or K + K + π − (Λ) depending on the context.
45
an important implication; induced polarization is strictly prohibited along vectorialaxes while transfered polarization is prohibited along pseudo-vectorial-axes. The last
point follows from the respective odd and even parity of vectors and pseudo-vectors
and the fact that polarization observables take on the parity of their quantization
axes. We will choose our coordinates in a way where it is natural to identify the
induced and transfered polarizations separately.
For the sake of clarity in defining our coordinates, the reaction may be written in
terms of its four-vectors, chosen in the center-of-momentum frame,
γ(q1 ) + p(q2 ) → K + (k1 ) + K + (k2 ) + Ξ− (p).
(3.1)
Above, q1 , q2 , k1 , k2 , p represent the four-momenta of each respective particle while
their associated three-momenta will be denoted by, q~1 , q~2 , k~1 , k~2 , p~. For convenience,
we will call the four-vector representing our two meson system k ≡ k1 + k2 . The
reaction or production plane contains the proton and the recoiling Ξ− as shown in
Fig. 3.2.
The production plane is defined by the unit vector,
ŷ ≡
q~2 × p~
,
| q~2 × p~ |
(3.2)
q~1 × ~k
,
| q~1 × ~k |
(3.3)
or, the more conventional equivalent,
ŷ ≡
which is the cross production of the beam and two-kaon momentum. The ẑ axis is
46
Figure 3.2: The production plane (left plane) is shown; defined in the center-ofmomentum, it contains the incoming photon and recoiling cascade. The two kaons in
general lie above and below the production plane. The so called decay plane (right
plane), is defined in the rest-frame of the cascade and contains its decay products
(pion and lambda).
defined to be along the photon momentum,
ẑ ≡
q~1
,
| q~1 |
(3.4)
while x̂ is defined to give a right-handed coordinate system,
x̂ ≡ ŷ × ẑ.
(3.5)
Note that in the above definition of coordinates, x̂ and ẑ are vectors while ŷ is a
pseudo-vector. As mentioned, pseudo-vectors remain invariant under parity transformations P. The beam and two-meson momenta have negative parity,
Pq1 = −q1
Pk = −k.
47
(3.6)
Thus referring to the above equation and the coordinate definitions we find,
P x̂ = x̂
P ŷ = −ŷ
(3.7)
P ẑ = ẑ
The respective parity of each coordinate implies induced cascade polarization is constrained along the pseudovector ŷ, normal to the production plane. Similarly, polarization transfer to the hyperon is constrained within the production plane, along the
vectors x̂ and ẑ. Our polarization results will be expressed in terms of the center-ofmomentum coordinate system.
Because of differences in the literature regarding polarization observables, it is important to be clear regarding one’s definitions so that meaningful comparisons can be
drawn. We follow the convention used by K. Nakayama in [40]. The spin observables,
u
) are
expressed in terms of the reaction’s differential cross sections ( dσ
dΩ
dσu
dσu
1
P ≡
Py = T r[MM†~σ · ŷ],
dΩ
dΩ
6
(3.8)
dσu
1
dσu
Cx ≡
Bzx = T r[M~σ · n̂z M†~σ · x̂],
dΩ
dΩ
6
(3.9)
dσu
dσu
1
Cz ≡
Bzz = T r[M~σ · n̂z M†~σ · ẑ],
dΩ
dΩ
6
(3.10)
where M represents the
0
spin-matrices, σx = 
1
amplitude
for our reaction 
and σx,y,z


 are the usual Pauli
1
0 −i
1 0 
 , σy = 
, σ z = 
. It follows that the
0
i 0
0 −1
48
spin observables are expressed as,
P =
dσ
(+)
dΩ
dσ
(+)
dΩ
−
+
dσ
(−)
dΩ
,
dσ
(−)
dΩ
(3.11)
where ± indicates parallel or anti-parallel spin of the recoil Ξ− relative to the ŷ-axis.
The beam-recoil asymmetries Ci (i = x, z) are similarly expressed,
Ci =
dσ
(+, +) +
[ dΩ
dσ
[ dΩ (+, +) +
dσ
(−, −)]
dΩ
dσ
(−, −)]
dΩ
dσ
− [ dΩ
(+, −) +
dσ
+ [ dΩ (+, −) +
dσ
(−, +)]
dΩ
dσ
(−, +)]
dΩ
(3.12)
where (±, ±) denotes the (positive/negative) photon helicity and the (positive/negative)
helicity of the recoil Ξ− across the î-axis respectively.
3.2 Connection Between Spin Observables and Polarization
The spin-dependent cross section for the γp → K + K + Ξ− reaction can be written [15,
16] in terms of the polarization observables as,
ρΞ
dσ
dσ
=
|unpol {1 + σy P + P (Cx σx + Cz σz )}.
dΩ
dΩ
(3.13)
P ∈ [−1, 1] denotes the degree of circular photon-beam polarization, σx,y,z are the
Pauli spin-matrices and ρΞ is two times the density matrix for an ensemble of Ξ
hyperons given explicitly as
ρΞ = (1 + ~σ · P~Ξ ).
(3.14)
The Ξ− recoil polarization and spin observables can be related by equating the expectation value of the Pauli-spin operators acting on the density matrix, i.e. the trace
49
P~Ξ = T r(ρΞ~σ ) [15, 16] leading to,
PΞ x = P C x ,
PΞy = P,
(3.15)
PΞ z = P C z .
Thus, the induced or ŷ-component of hyperon polarization PΞy is equivalent to the
spin observable P . The transfered, or x̂- and ẑ-components of the hyperon polarization are equal to Cx and Cz multiplied by the photon-beam polarization P .
3.3 Self Analyzing Decay
The connection between the hyperon recoil polarization and the spin observables in
Equation 3.15 indicates that knowledge of the beam polarization P , along with a
measurement of P~Ξ , yields Cx , P and Cz .
As previously mentioned, there is some confusion in the literature regarding the sign
convention for the polarization observables. Part of the confusion involves instances
of incorrect quotations for decay angular distributions of hyperons. Consequentially
we found it important to explicitly define what is being measured, thus we provide a
derivation of the decay angular distribution of the Ξ− .
The Ξ− recoil-polarization (P~Ξ ) can be measured by exploiting the self analyzing
nature of it’s decay Ξ− → π − Λ. The relationship between the angular distribution of
the decay products and the polarization can most easily be seen by first considering
a single Ξ− prepared in the spin-up state across an arbitrary î-axis. Consider our
initial spin-half particle (Ξ− ) prepared at rest, subsequently decaying to a final state
consisting of a pseudo-scalar (π − ) and a spin-half (Λ) particle. By conservation of
total angular momentum, the final state may only have orbital angular momentum
l = 0 or l = 1. For the l = 0 (s-wave) final state, the spin of the Λ must be spin up
50
(X + ) and the contribution to its full angular wave function can be written as,
Ψs = as X + Y00
(3.16)
where as is the relative amplitude for the s-wave. For the l = 1 (p-wave) state, the
spin of the Λ may be up (X + ) or down (X − ). The p-wave contribution to the total
wave function can be written as,
1
11
2
2
1
11
2 2
Ψp = ap (X + Y10 C 120 12 + X − Y11 C −1
),
11
2
(3.17)
2
where ap is the relative amplitude for the p-wave, and the C variables are the appropriate Clebsch-Gordan coefficients. The total wave function is then the sum of
contributions from s and p waves.
1
11
2
2
1
11
2 2
Ψ↑ = Ψs + Ψp = as X + Y00 + ap (X + Y10 C 120 12 + X − Y11 C −1
),
11
2
(3.18)
2
or,
1
11
2
2
1
11
2 2
Ψ↑ = X + (ap Y10 C 120 12 + as Y00 ) + X − ap Y11 C −1
.
11
2
(3.19)
2
The Clebsch-Gordan coefficients and the spherical harmonic functions are given by,
1
11
C 120 12 = −
2
2
Y00 = 1, Y10 =
p
p
1 1
1
2 2
1/3, C −1
2/3,
1 = −
1
√
2
(3.20)
2
p
3 cos θ, Y11 = − 3/2eiφ sin θ.
(3.21)
We may write,
Ψ↑ = X + (−ap cos θ + as ) + X − ap eiφ sin θ.
51
(3.22)
By orthonormality of the spin wave functions we have,
Ψ2↑ = |as |2 + |ap |2 + (a∗p as + ap a∗s ) cos θ,
(3.23)
Ψ2↑ = |as |2 + |ap |2 + 2Re(a∗s ap ) cos θ.
(3.24)
or,
By letting,
α=
2Re(a∗s ap )
,
|as |2 + |ap |2
(3.25)
where α is known as the analyzing power or decay asymmetry with value of −0.458 ±
0.012 [3], we finally arrive at, after normalization,
1
|Ψ↑ (θ)|2 = (1 + α cos θ).
2
(3.26)
One can immediately write the final state wave function corresponding to the case
where the initial Ξ− is prepared in the spin down state by rotating θ by 180◦ ,
1
|Ψ↓ (θ)|2 = (1 − α cos θ).
2
(3.27)
For an ensemble of N cascades with n↑ prepared with the spin-up state and n↓ prepared in the spin-down state, the total angular distribution of the decay-lambda is
given by the weighted sum,
n(θ) = n↑ |Ψ↑ (θ)|2 + n↓ |Ψ↓ (θ)|2 .
(3.28)
By identifying the hyperon recoil-polarization (PΞn̂ ) across the n̂-axis as,
PΞn̂ =
n↑ − n↓
,
n↑ + n↓
52
(3.29)
and making explicit that θ = θΛn , we arrive at,
n(θΛn ) =
N
(1 + PΞn̂ α cos θΛn̂ ).
2
(3.30)
Equation 3.30 describes the polar-angular distribution of the Λ off the n̂-axis in the
Ξ− rest frame and it’s relation to the Ξ− polarization. In order to obtain a similar
distribution for the pion, one should recognize that in the Ξ− rest frame, the pion
and the Λ are produced back-to-back. This equates to a 180◦ rotation giving,
n(θπn ) =
N
(1 − PΞn α cos θπn ).
2
(3.31)
The distribution in terms of the negatively charged pion as opposed to the neutral
lambda is more convenient from an experimental stand point since it is generally more
easy to detect charged particles. What has been shown is that from the rest frame of
the cascade, it’s decay angular distribution measured against the previously defined
center-of-momentum coordinate-system yields its polarization.
3.4 Calculation of P
The equivalence of the recoil polarization’s y-component, PΞy , and the spin-observable
P , along with Equation 3.31, allows for a direct measurement of P . This can be done
multiple ways, one by fitting the acceptance corrected number of events as a function
of cos θπŷ to a first-degree polynomial,
y = p0 + p1 cos θπŷ .
(3.32)
The fitting parameters (p0 , p1 ) relate to the recoil asymmetry (P ) by,
p1 =
−N P α
2
53
(3.33)
or,
−p1
.
αp0
P =
(3.34)
The above method requires extensive Monte Carlo simulation, which was carried out
to correct for acceptance effects of the detector (see Chapter 5).
Similar to the first mentioned method of measuring P , a relation between the Ξ− polarization and the forward-backward-asymmetry (Ay ) of the pion angular distribution
can be derived. The forward-backward asymmetry is defined as,
Ay ≡
N y+ − N y−
,
N y+ + N y−
(3.35)
where N y+ and N y− represent the number of events with cos θπ as positive and negative respectively. The relation between induced polarization P and Ay can be seen
by considering,
N y− =
Z0
n(θπ ) d cos θπ =
αP
N
(1 +
)
2
2
(3.36)
n(θπ ) d cos θπ =
αP
N
(1 −
).
2
2
(3.37)
−1
and
N
y+
Z1
=
0
Thus the asymmetry can be expressed as,
P =
−2Ay
.
α
(3.38)
The advantage of using a forward-backward-asymmetry method for calculating P
is that, as will be demonstrated with a Monte Caro simulation in Chapter 7, the
detector acceptance mostly cancels. The cancellation follows from the polarization
axis ŷ pointing isotropically in the lab-frame-defined azimuthal angle φlab and the
geometry of CLAS.
54
The statistical uncertainty in the asymmetry measurement of P is related to the
Poissonian uncertainty in N y+ and N y− , where for typographical simplicity we momentarily denote as N+ and N− respectively. Taking the partial derivative,
∂A
±2N∓
=
,
∂N±
(N+ + N− )2
(3.39)
we find total statistical uncertainty, through quadrature, is given as,
δAy =
q
2
2
2
,
+ N+2 δN
N−2 δN
−
+
(N+ + N− )2
(3.40)
δAy =
2
(N+ + N− )2
(3.41)
or,
q
N−2 N+ + N+2 N− .
3.5 Calculation of Double Polarization Observables Cx and Cz
The double polarization observables Ci (i = x, y) characterize the transfered photon helicity to the Ξ− in the x- and z-directions (represented as PΞx and PΞz ).
An important experimental aspect is that the photon-beam helicity P was flipped
(P → −P ) at a rate of 30 Hz. Assuming equal photon intensities for positive and
negative helicity states, there is a net beam polarization of P̄ = 0.
There are two ways to measure the transfered polarization. One way is to bin in
forward and backward photon helicity bins and fit the acceptance-corrected pion
distributions, similarly to how P was measured. However, the need for helicity binning
in the mentioned method increases the statistical uncertainty, approximately by a
√
factor of 2. Furthermore, the acceptance ((θŷπ )) as a function of pion angle with
ŷ is much more well behaved than with x̂ and ẑ: while (θŷπ ) is even about θŷπ = 0,
relatively flat and non-zero for all angles, (θx̂π ) and (θẑπ ) are highly asymmetric,
non-zero for only a small subset of angles, and contain regions of high gradients. In
55
fact, measurements of Cx and Cz based on fitting acceptance-corrected pion angular
distributions were attempted and proved to be reliable only in the small sub range of
pion angles where the acceptance is “well behaved”.
The most straightforward and reliable way to extract Cx and Cz employs the photonhelicity asymmetry,
A=
N+ − N−
,
N+ + N−
(3.42)
where N + and N − are the number of events associated with positive and negative
photon-beam helicity states respectively. The form of asymmetry as a function of
pion angle can be seen by associating N ± in Equation 3.42 with PΞi = ±|P |Ci in
Equation 3.31 yielding,
A(cos θπî ) = −|P |αCi cos(θπî )
(3.43)
Equation 3.43 states that the slope of asymmetry with respect to cos(θπî ) is directly
proportional to the transfer polarization observable Ci . Ci is thus obtained by determining the slope of a two-parameter linear fit to A(cos θπî ) in each relevant kinematic
bin. The photon-helicity asymmetry method has the benefit that to first approximation, the detector acceptance and various other systematic effects which do not have
a photon-helicity dependence cancel.
Monte Caro simulations matching the data showed that the non-cancellation of acceptance had a larger effect for the photon-helicity asymmetry than for the forwardbackward asymmetry used to calculate P . In fact, acceptance effects turn out to be
the largest source of systematic error for the measurement of Cx and Cz . For a more
detailed discussion on the effect of detector acceptance see Section 3.6 or Chapter 7.
Note that if the photon-beam intensity was not equal for positive and negative helicity
phases, there would be a non-zero net beam-polarization. Since this quantity is not
56
covariant with the pion angle, it does not effect the slope of asymmetry in Equation
3.42, and therefore does not affect the measured values of Ci . Similarly, the reaction’s
intrinsic beam asymmetry Σ, also being independent of pion angle will not affect the
measured values of Ci .
3.6 Acceptance Independence of Cx and Cz
To examine the extent to which the acceptance cancels in the photon-helicity asymmetry, consider,
N+ − N−
=
A= +
N + N−
n+
+
n+
+
−
+
n−
−
n−
−
(3.44)
If + 6= − , acceptance does not cancel in the asymmetry. Let the phase-space distribution of K1+ , K2+ , π − be represented by the random variables Y1 , Y2 , X respectively.
We assume the total acceptance for detecting the final state K + K + π − can be written
as the product of the acceptance associated with each final state particle.
= C(Y1 )(Y2 )(X),
(3.45)
where C is an overall factor representing all other acceptance effects that are not
helicity dependent. Strictly speaking, the incoming photon’s helicity does not change
the form of each acceptance function, it rather changes the underlying distributions
Y1 , Y2 and X. Let the distributions associated with positive and negative helicity be
Y1± , Y2± , X ± . The total acceptance for positive and negative photon helicity (± ) is
given by
± = C(Y1± )(Y2± )(X ± )
(3.46)
Consider that all dominant underlying mechanisms in the γp → K + K + Ξ− reaction
has the kaons coming from parity-conserving strong or electromagnetic processes. It
follows that all the kinematic distributions for both kaons are independent of the
photon-beam helicity. The only helicity dependent parts of the reaction are those
57
following a weak decay, which strictly is limited to the decay of ground-state hyperons.
The two ground state hyperon decays in the reaction are Ξ− → Λπ − and Λ → pπ − .
Only the Ξ− decay contributes to the acceptance, since the signal is mostly free of
background from the Λ decay. Furthermore the acceptance can only be dependent
on the projection of the π − angular distribution on the x̂- and ẑ- axis since the
polarization cannot be transfered to the Ξ− in the y-axis direction. Thus,
± = C(X ± ),
(3.47)
where the non-helicity-dependent factors of the acceptance have been absorbed in C.
Expanding X ± ,
X ± = {x± , y, z ± },
(3.48)
where x, y, z respectively represent the polar angle distribution of the pion off x̂−, ŷ−
and ẑ for positive (negative) photon helicity states. Finally the acceptance takes on
the form,
± = C(x± )(z ± ).
(3.49)
Out of the two terms above, one, which we shall call the finite binning effect, should
be (shown in Chapter 7) considerably smaller than the other. The finite binning term
is (x± ) when binning the asymmetry in cos θπx (for the calculation of Cx ) and (z ± )
when binning in cos θπz (for the calculation of Cz ). The insignificance of the finite
binning effect can be understood by considering that the distribution surrounding
a point in cos θpx or cos θpz , is limited by the bin width. The distribution, call it xi ,
−
approaches invariance under helicity flipping x+
i = xi as the bin width becomes
arbitrarily fine. Of course the bin width must be finite and as a result there will be a
an effect, albeit a small one. The systematic uncertainty from the two leading-order,
effects were estimated by simulating events as discussed in Chapter 7.
58
3.7 Frame Transformation Effect on Hyperon Polarization
Because of the precession of spin under Lorentz transformations, baryon spin projections may possess differing values in different reference frames. The current analysis
measures the cascade spin projections in it’s rest frame, here we will show that such
measurements remain invariant under a Lorentz-boost to the center-of-momentumframe. The mentioned effect, commonly referred to as Wigner-Thomas precession,
comes about from the non-commutativity of boosts and rotations.
Here we adopt an argument similar to one presented in Ref. [15]. Consider a particle
p in a frame S with velocity β~ and polar angle θ with respect to a boost to a frame
S̃. The velocity and polar angle of p as viewed from S̃ with respect to the boost
˜
direction is denoted β~ and θ̃. Additionally, let the corresponding Lorentz factor for
the particle in S and S̃ be γ and γ̃, respectively, while letting the Lorentz factor for
the boost from S to S̃ be denoted by Γ. The Wigner-Thomas precession angle αW
about the y-axis for an arbitrary boost within the x, z plane has been shown to be,
sin αW =
1+Γ
sin(θ − θ̃)
γ + γ̃
(3.50)
[15, 41, 42]. For this analysis, the boost from the hyperon rest-frame S̃ to the labframe S is by definition, in the direction of the hyperon momentum, meaning zero
polar angle between the frame boost and and the particle momentum (hence θ̃ =
θ = 0 ). It follows that spin precession angle, αW , is zero meaning that the hyperon
polarization is left unchanged when boosting from the hyperon rest-frame to the
center-of-momentum-frame [15, 16].
59
Chapter 4
Data Processing and Event Selection
This chapter details the method of extracting the Ξ− → Λπ − events from the 121
Terra bites of raw data. We derived various algorithms intended to maximize our
sample, whilst minimizing background contamination from other reactions. As a
consequence of the low cross sections for cascade production, the vast majority of
physics events composing the g12 dataset are extraneous to our analysis. Out of the
some 26 billion reconstructed events, only 5143 survived the final selection criteria.
Additionally, we derived important corrections and calibrations that were applied to
the data, which are discussed in Section 4.3.
4.1 Physics Event Selection
4.1.1
Topology Requirement
Figure 4.1: Cartoon representing Ξ− photoproduction via virtual-meson exchange in
the t-channel and its weak decay.
Figure 4.1 shows a toy model of our reaction. Events with all three mesons, K + K + π − ,
were required in the final state. Specifically, the topology requirements implemented
were,
• Number of K + = 2,
60
• Number of π − = 1,
• Number of p = 0 or 1,
• Number of K − = 0,
• Number of π + = 0.
The number of protons was permitted to float since a large amount of data were
expected to contain them due to the Λ → π − p decay. Our dominant source of
background comes from pions masquerading as kaons in the particle identification
logic. Meson misidentification is further discussed in latter sections of the chapter.
As will be shown, events with the decay Ξ− → π − Λ can be selected from the data by
imposing kinematic requirements on the detected mesons.
4.1.2
Missing Mass Selection in the Hypersphere
In our topology there are four useful kinematic constraints that comprise two missing
mass 1 and two invariant mass requirements. Rather than cutting on all four quantities individually, they are taken together in a single cut as orthogonal displacements
within a hypersphere. All four kinematic quantities are divided by their associated
detected widths (3σ) and taken as coordinates in the 4d space. The four kinematic
constraints for our reaction are evident in the diagram in Fig. 4.1. The cuts are given
separately as,
1. Require the missing mass of the (γ + p) − (K + + K + ) system be the known
mass of the Ξ− .
1
Missing mass is the conventional name attributed to invariant mass of an initial state minus a
final state.
61
2. Require the missing mass of the (γ + p) − (K + + K + + π − ) be equal to the
known mass of the Λ
3. Require the invariant mass of the π − + Λ system be equal to the known mass
of Ξ−
4. Require the effective mass of the π − − Ξ− system be equal to the known mass
of Ξ−
The amount by which the constraints were allowed to float was determined by the
resolution of CLAS. The resolution of all four quantities were measured by fitting
each corresponding distribution to a Gaussian function. Each fit and their associated
widths are displayed in Fig. 4.2.
For typographical ease, a short hand notation for the missing mass, M M (K + + K + )
and M M (K + +K + π − ) will be used. Similarly, M (π − +Λ) and M (π − −Ξ− ) denote the
relevant invariant and effective mass quantities. The “hypersphere” has coordinates
defined as,
x1 = (M M (K + + K + ) − Ξ−
mass )/(3σ1 ),
(4.1)
x2 = (M M (K + + K + + π − ) − Λmass )/(3σ2 ),
(4.2)
x3 = (M (Λ + π − ) − Ξ−
mass )/(3σ3 ),
(4.3)
x4 = (M (Ξ− − π − ) − Λmass )/(3σ4 ),
(4.4)
r=
q
x21 + x22 + x23 + x24 ,
62
(4.5)
Figure 4.2: Top left: the missing mass spectrum of the K + K + system, showing the
Ξ− peak at 1.32 GeV. Top right: the missing mass spectrum of the K + K + π − system,
showing the Λ peak at 1.11 GeV. Bottom left: invariant mass spectrum of the Λπ −
system, showing the Ξ− peak at 1.32 GeV. Bottom right: invariant mass spectrum
of the Ξ− − π − system, showing the Λ peak at 1.11 GeV. In all plots a Gaussian
is fit to the signal over a polynomial background. The vertical lines represents the
known lambda or charged cascade masses. The parameter σ of the Gaussian fit gives
CLAS’s “natural” resolution for its associated quantity.
63
where σn denotes the Gaussian width of the associated quantity as measured in
Fig. 4.2. Loosely speaking, a cut on the hypersphere radius r represents a cut on
“sigma”; a “three sigma” cuts equates taking events within the hypervolume defined
by r < 1. Fig. 4.3 shows the mass spectra after various cuts on the hypersphere radius.
The sample used in our analysis is the result of a r < 1 cut (shown in blue). Overlaying the signal in Fig. 4.3, are two sidebands taken from the two equal-hypervolume
√
√
√
outer layers, 1 < r < 4 2 and 4 2 < r < 4 3. A nearly background-free sample of
Ξ → π − Λ events is evident.
Figure 4.3: The nominal mass cut r < 1 represents a three sigma cut as shown√in
blue. The first sideband (in red) are the events lying
in the hyper-shell
1<r< 42
√
√
4
4
while the second sideband (black) lie in the region 2 < r < 3.
64
4.1.3
Vertex Position Selection
[h] The event vertex time is taken as the average over all final state particles, required
to lie within or near the physical region of the hydrogen target. Our vertex selection
occupied a cylindrical volume from -120 to -60 cm along the beam, with a radius of 7
cm centered about zero. It is possible to have event vertices outside the target region
due to the long decay time of the cascade, and the resultant detachment of the pion
vertex. Figure 4.4 shows the vertex distribution for the K + K + π − (Λ) events used in
our analysis.
Figure 4.4: The cut region of spacial vertex distribution. All events lying outside the
plot were excluded.
4.1.4
Vertex Timing Selection
The CEBAF Radio-Frequency signal (RF) responsible for the acceleration of electrons
delivered to Hall B provides a precise (≈ 15 ps) means of correcting the tagger time
once the beam bucket is identified. The vertex time can also be calculated as follows:
take the time readout for a particle hit in a region of the detector and subtract the
65
travel time. The calculation requires a knowledge of the particle speed, β. For each
track β is calculated as,
βSTT OF =
βvtxT OF =
lT OF − lST
,
c(tT OF − tST )
c(tT OF
lT OF
,
− tvtx (T AGRF )
(4.6)
(4.7)
where
• lT OF and lST are the path lengths from the vertex ot the time of flight counter
and start counter respectively,
• TT OF and TST are the hit times in the time of flight counter and start counter
respectively,
• tvtx (T AGRF ) is the RF-corrected vertex time.
Agreement of one nanosecond between the event-vertex time as calculated by the
RF-corrected tagger and the start counter was required. The difference between both
methods of calculation are shown in Fig. 4.5. Similarly, the difference between the
RF-corrected tagger event-time and particle-vertex-time as calculated by the TOF
for both kaons is shown in Fig. 4.6.
66
Figure 4.5: Difference between event-vertex time as calculated by the RF-corrected
tagger and start counter, for events passing all cuts. A one nanosecond cut was
imposed.
Figure 4.6: Difference between vertex-time and particle-vertex time as calculated by
the RF-corrected tagger and TOF, for events passing all cuts. A one nanosecond cut
was imposed. Left: fast kaon. Right: slow kaon.
67
4.1.5
Eliminating Particle Misidentification
As mentioned in Section 4.1.1, for CLAS, a dominant source of background in kaon
analyses are pions misidentified as kaons. It turns out however, that our kinematic
hypersphere cuts eliminate the pion background, as can be seen in Fig. 4.7 and 4.8.
Figure 4.7: β vs momentum for the fast kaon overlaid with it’s theoretical curve.
Left: before hypersphere cuts, an extra band is visible comprising misidentified pions.
Right: after hypersphere cuts, the contamination is eliminated.
Figure 4.8: β vs momentum for the pion (left) and slow kaon (right) overlaid with
their respective theoretical curves. Both plots show the data after hypersphere cuts
and indicate no contamination due to misidentification or timing inaccuracies
68
4.1.6
The Fiducial Region
The fiducial regions of the detector are those found to be trustworthy, as the name
suggests. We derived Geometric fiducial cuts for the g12 data, which are cuts based
on the exclusion of events laying outside regions where acceptance is well behaved
and reliably reproduced in simulation. Such regions for all g12 data are expressed as
an upper and lower limit of the difference in azimuthal angle between the center of a
given sector, and a particle track. Because of the hyperbolic geometry of CLAS and
the presence of the toroidal magnetic field, the fiducial boundaries on the angle φ are
functions of momentum (p), charge, and polar angle (θ) of each track. The boundaries
were evaluated separately in each sector, nominally defined as the φ values in which
occupancy drops below 50% of that in the respective sector’s “flat” region. The flat
regions were defined as −10◦ < φ < 10◦ . The nominal upper and lower φ limits depend
strongly on particle charge, p and θ, hence the need for functional characterization
and extrapolation.
In order to determine the fiducial limits for charged hadrons, a sample of exclusive
γp → pπ + π − events were sliced into 5x15x6 bins in p, θ, and sector respectively. The
φ distributions for π + and π − were plotted separately in each bin. The upper and
lower φ limits of these first-generation plots were found according to the nominal
fiducial definition of 50% occupancy as illustrated in Fig. 4.9. The results from the
first-generation fits were represented in second-generation plots of φmin and φmax
vs θ as also shown in Fig. 4.9. The data in the second-generation plots were fit
with hyperbolas, chosen since they replicate the projection of the detector. Secondgeneration fitting parameters were then plotted vs p in third-generation plots. These
third generation plots were fit to power functions as shown in Fig. 4.10. Results of the
third-generation fits define the sought after functional form φmin (θ, p) and φmax (θ, p)
69
for each sector. The sector integrated results for positive and negative hadron tracks
compose the nominal fiducial region. Tight cuts and loose cuts were defined as a
contraction and expansion respectively, by 4◦ from the nominal fiducial cuts. The
cuts are shown on the pπ + π − and K + K − π − data in Figs. 4.11-4.17. The effect of the
fiducial cuts on the measured spin-observables are left for discussion in Chapter 7 of
this dissertation.
Figure 4.9: Left: shows the π + φ distribution for sector-three in one p and θ bin
along with the upper and lower limits of the fiducial region represented by the green
vertical line. Right: a second-generation plot, fit to a hyperbola.
Figure 4.10: Third-generation plots of the fitting parameters from second-generation
fits for sector three. The data are fit to power functions.
70
Figure 4.11: The angular distribution of the proton from exclusive pπ + π − events is
shown. In the top, φ vs θ is plotted, the bottom plots conveys similar information
mapped to mimic the geometry of CLAS. Left: No fiducial cuts. Right: nominal
fiducial cuts on the proton.
Figure 4.12: The angular distribution of the positive pion from exclusive pπ + π − events
is shown. In the top, φ vs θ is plotted, the bottom plots conveys similar information
mapped to reflect the geometry of CLAS. Left: no fiducial cuts. Right: nominal
fiducial cuts on the positive pion.
71
Figure 4.13: The angular distribution of the negative pion from exclusive pπ + π −
events is shown. In the top, φ vs θ is plotted, the bottom plots conveys similar
information mapped to reflect the geometry of CLAS. Left: no fiducial cuts. Right:
nominal fiducial cuts on the negative pion.
Figure 4.14: From left to right: φ vs momentum for the proton, negative pion and
positive pion for pπ + π − events. Top: no fiducial cuts. Bottom: nominal fiducial cuts.
72
Figure 4.15: φ vs θ for the fast kaon from K + K + π − (Λ) events are shown. On the top,
simulated Monte Carlo events are plotted, on bottom, the g12 data of our analysis.
Left: No fiducial cuts. Right: nominal fiducial cuts on the fast kaon.
Figure 4.16: φ vs θ for the slow kaon from K + K + π − (Λ) events are shown. On the top,
simulated Monte Carlo events are plotted, on bottom, the g12 data of our analysis.
Left: No fiducial cuts. Right: nominal fiducial cuts on the slow.
73
Figure 4.17: φ vs θ for the pion from K + K + π − (Λ) events are shown. On the top,
simulated Monte Carlo events are plotted, on bottom, the g12 data of our analysis.
Left: No fiducial cuts. Right: nominal fiducial cuts on the pion.
4.1.7
Photon-Beam Energy
The threshold center-of-momentum energy Wmin for production of the final state
K + K + Ξ− is simply the sum of each particle’s mass-energy. The threshold photon
energy (Eγmin ) for producing K + K + Ξ− off a fixed proton target can be computed by
considering the conservation of mass-energy.
m2i = m2f =⇒ (pγ + pp )2 = (pK1+ + pK2+ + pΞ− )2 ,
(4.8)
where mi and mf represent the invariant mass of the initial and final state respectively,
and px represents the energy-momentum four-vector for particle x in the reaction.
Carrying out the operation on the left hand side gives,
pγ pγ + pp pp + pγ pp + pp pγ = (pK1+ + pK2+ + pΞ− )2 .
74
(4.9)
Considering the proton has zero momentum in the lab-frame and that for the lowest
energy final state, pK1+ , pK2+ and pΞ− will have zero momentum, one obtains,
m2p + 2mp Eγmin = (mK1+ + mK2+ + mΞ+ )2 .
(4.10)
Which, using the known values of the particle masses gives a value of
Eγmin =
(mK1+ + mK2+ + mΞ+ )2 − m2p
2mp
≈ 2.37 GeV
(4.11)
Where the known masses used in this calculation are,
• mK + = 0.49367 GeV
• mp = 0.93827 GeV
• mΞ− = 1.32171 GeV
Although the threshold is around 2.37 GeV, events created near this energy are not
detected since CLAS is inefficient at reconstructing low speed tracks and also because of low cross sections below 3 GeV [13]. The photon-energy range for cascade
production in the data is 2.8-5.5 GeV.
4.2 Kinematic Binning of Data
For our analysis, the polarization of the cascade was measured in bins of photonbeam energy, Eγ , and center-of-momentum cascade polar angle (see Fig. 3.2), θΞcm .
For convenience the cascade angle is expressed in terms of it’s cosine, that is, cos θΞcm .
Various binning schemes were considered. Most of our results were measured with
data separated in three bins with an equal number of events in each. Figure 4.18
shows the primary binning scheme.
75
Figure 4.18: The red horizontal lines represent the binning in cos θΞcm while the green
horizontal lines represent the binning in Eγ . Left: the g12 data. Right: Monte Carlo
simulated events.
4.3 Corrections and Calibration
4.3.1
Photon-Beam Polarization Determination
The electron beam produced with a polarized laser incident on gallium arsenide
allows for longitudinal polarization of the electrons [43] and in turn, due to the
bremsstrahlung process, circular polarization of the photon beam. The accurate
measurement of the polarization transfered from the photon beam to the produced
hyperons (Cx and Cz ) requires knowledge of the beam polarization. Such knowledge
is ascertained by knowing the magnitude of incident electron-beam polarization, and
the helicity orientation of the electron beam bunch responsible for the event (in the
lab-frame). The Maximon-Olsen formula relating incident electron beam polarization,
76
with the photon polarization is given by [44],
P (Eγ ) =
x(4 − x)
Pelec
4 − 4x + 3x2
(4.12)
where x = Eγ /Eelec is the ratio of photon energy, Eγ , to beam energy, Eelec . The
g12 experiment ran with a constant electron energy of Eelec = 5.715 GeV. The polarization of the electron beam was measured regularly using the a Møller polarimeter.
The polarimeter measures electron polarization by making use of the helicity dependent nature of Møller scattering [29, 35]. The Møller measurements, summarized in
Table 4.1, were performed regularly (every few days) during g12. We wrote a simple
calculator for the collaboration that computes the run-integrated flux-weighted average of the photon-beam circular polarization in a user-specified energy range, the
output of which is given in Table 4.2.
An important experimental aspect of g12 is that the electron-beam helicity was flipped
at a rate around 30 Hz. While the helicity information was recorded and stored in
the HEVT bank for each event, the convention for bit encoding has been known to
change from real-time to delayed-time recording. Further considerations, such as the
half-wave plate orientation also had to be accounted for.
The only sure way to pin down the absolute beam helicity orientation for our data
was to analyze a well known helicity-dependent reaction. Well-established results in
+ −
Ref. [45] in the beam-helicity asymmetry, I (φhel
π ), for the reaction, γp → pπ π ,
were reproduced. The mentioned reaction was shown to have a specific helicity-frame
defined φhel
π + -dependent structure. If the helicity convention was reversed then we
hel
would observe I (φhel
π + ) → −I (φπ + ).
The sub-analysis we performed required exclusive pπ + π − events in the final state. To
ensure exclusivity we required zero missing energy and momentum within detector
77
Table 4.1: The degree of longitudinal electron polarization (Pe ) for each Møller run.
resolution. The helicity frame (shown in Fig. 4.19) was defined to be the rest frame
of the hypothetical parent meson of the two pion system, with ẑ aligned along its
center-of-mass defined momentum. φhel
π + is the angle between the proton-production
plane, and the plane containing both pions. The beam helicity asymmetry is given
by,
I =
N+ − N−
,
N+ + N−
(4.13)
where N ± indicates the number of events with positive (negative) photon helicity.
Figs. 4.20-4.21 show the I (φhel
π + ) for g12 and in the analysis of [45]. The lab-frame
electron helicity readout was taken from the HEVT bank (HEVT→hevt[0].TGRPRS).
78
Table 4.2: A table showing the degree of circular photon polarization (P ) in the
relevant kinematic bins (Eγ ).
The results of our I (φhel
π + ) analysis and the previously published results showed a
positive (negative) HEVT readout indicates positive (negative) photon helicity. The
reproduction of beam-helicity asymmetry for double charged pion production also
served as a way to test the accuracy of the calculated photon polarization magnitude:
both results’ wave amplitudes were in good agreement.
Figure 4.19: An illustration of the angle definitions used in the γp → π + π − p subanalysis. θcm is defined in the center-of-mass frame. θ and φ are defined in the rest
frame of the π + π − system as the polar and azimuthal angles. The z direction is
along the total momentum of the π + π − system. Image source [45]
79
Figure 4.20: I (φhel
π + ) for our data within the energy range of W = 1.9 − 2.3 GeV.
Figure 4.21: I (φhel
π + ) as measured in the analysis [45]. The results of are shown in
bins of W from 1.9 to 2.3 GeV.
80
4.3.2
Multiple Photon Events
The trigger duration and limited timing resolution of the detectors results in multiple
photons being read out by the tagger for each event. Multiple-photon events compose
around 10% of the data; three photon selection algorithms were considered,
• Choose photon at random,
• Choose the highest energy photon,
• Reject multiplicity events.
The method contributing most constructively to the signal-to-background ratio was
to choose the higher energy photon when multiplicity occurred. Photons with higher
energy have a greater probability of producing cascade events although a background
is introduced. The mass-hypersphere cuts however eliminate most of the background
from photons which were chosen incorrectly.
4.3.3
CLAS Energy-Loss Corrections
The standard CLAS energy loss correction software (ELOSS) developed by Eugene
Pasyuk was implemented to account for the energy dissipation experienced by particles as they pass through the material between the target and region-one drift chambers. Using the Beth-Bloch equation, the software calculates each track’s energy deposition based on the measured particle track-momentum, it’s path through CLAS,
and a GEANT model mimicking the materials and design of CLAS. The initial energy
of a non-decaying particle at the event vertex is simply,
Einitial = Emeasured + Eloss .
81
(4.14)
Above, Einitial is the track energy at the event vertex, Emeasured is the track energy,
and Eloss is the dissipated energy of the track [46].
4.3.4
Photon Beam Energy Corrections for g12
Inaccuracies in the photon energy as measured by the tagger arise due to geometric
distortions in the taggers’s focal plane. These distortions cause a shift in the recoil
electrons’ measured locations, resulting in an incorrect energy readout. The effect
was studied for each event by a comparison of the measured photon energy and the
energy required for the inclusive γp → pπ + π − reaction. A systematic correction
(ECOR) was derived by Mike Kunkel and Johann Gotez for the g12 dataset and was
implemented in the present analysis. The effect that ECOR had on our sample is
discussed later in this chapter [33].
4.3.5
Final-State Momentum Corrections for g12
Distortions in the toroidal magnetic field and small spatial misalignments of the drift
chamber during data collection result in a systematic shift in the measured particle
track momenta. In order to correct for the shift in momentum, we implemented a
set of corrections (PCOR) that were derived by Johann Goetz through analysis of
pπ + π − events. Further details on the momentum corrections can be found in [33].
4.3.6
Combined Effects of Corrections
Fig. 4.22 shows the combined effect of ELOSS, photon energy, and momentum corrections on the cascade signal in our data. An increase signal quality is evident from
the 10% narrowing of the mass peak.
82
Figure 4.22: On the left, the Ξ− signal in the missing mass spectrum (after all cuts)
of K + K + before ELOSS, photon energy, and momentum corrections is shown. On
the right, the same signal using the corrected four-vectors. The mass of the Ξ− is
around 1.321 GeV.
83
Chapter 5
Simulation
Generally speaking, the measurement of a physical observable is sensitive to the
apparatus with which the measurement is performed. In the present analysis, the
polarization observables are accessed through an angular distribution. While the
underlying distributions themselves are highly sensitive to the detector’s acceptance,
such effects, to first order, cancel in all but one of our measurement techniques.
Discussions on the cancellation of acceptance can be found in Chapters 3 and 7.
Simulated events were used for the calculation of acceptance corrections, estimation
of certain sources of systematic uncertainty, and for cross checks of our results. The
simulation used in the present analysis is shown to match to the data. Details of the
simulation process, tuning, and characterization of acceptance effects are outlined in
this chapter.
5.1 Simulation Overview
For CLAS, the acceptance is a function that describes the likelihood of detecting
and reconstructing an event. The acceptance for a given particle depends mostly
on its momentum, charge and scattering angle. Characterization of the detector’s
response for an event is a formidable challenge. A class of techniques commonly used
in the physical sciences know as Monte Carlo methods are required. Such methods
are utilized for problems with a high degree of complexity as a means to bypass the
need to obtain closed-form solutions or deterministic algorithms.
The Monte Carlo methods are computational algorithms that obtain numerical results
by employing repeated sampling of a random number generated according to a weighting distribution. In order to characterize the γp → K + K + Ξ− → K + K + π − (Λ) →
84
K + K + π − (pπ − ) reaction-chain, and associated response of CLAS, pseudo-randomly
generated four-vectors corresponding to the appropriate particle masses were obtained. A t-channel model was implicit in the probabilistic weighting factors of the
four-vector generator. Our generated sample of events were then run through a computational model of CLAS. The ratio of events that survive the simulated reconstruction process to the number generated, is the acceptance. Finley tuning the underlying
probability distribution for the random four-vector generation to match the physics
data is a challenge in itself. In general, tuning must be performed iteratively.
The CLAS software GENR8 was used to generate the t-channel phase space events.
The production model of simulation is shown in Fig. 5.1. After generation, the events
were run through a program gamp2part, which we modified so that its functionality
included the ability to mimic the detachment of the π − vertex from the event vertex.
After the generated event’s vertex-positions were modified, they were sequentially
run through the standard CLAS processing software, gsim [47], which simulates the
physical effects of the detector, gpp which smears the simulated signals, and a1c for
reconstruction.
Figure 5.1: A cartoon representing our simulated cascade production model. Kf+ast Y ∗
+
is produced via virtual-meson exchange in the t-channel. The Y ∗ decays to Kslow
Ξ− .
−
−
The cascade undergoes a subsequent decay to π Λ. The decay of Λ to pπ was
additionally simulated.
85
5.1.1
Generated and Reconstructed Events
GENR8 was used to produce a simulation of the γp → K + K + Ξ− → K + K + π − (Λ) →
K + K + π − (pπ − ) reaction. The angular distribution of the final state particles were
controlled by mimicking the t-channel mode of production using t-slope as an input
parameter. The simulated events produced by GENR8 before being processed are
referred to as generated events. The generated events are then passed through further
processing and similar event selection criteria as the data. Surviving events constitute
what are referred to as the reconstructed events.
5.1.2
Detached Vertices
The detached vertex of the pion is a consideration that affects acceptance and had to
be accounted for. Since GENR8 is not equipped for this functionality it was achieved
by modifying the preexisting code of gamp2part. The π − in this analysis comes
from the decay of Ξ− , hence, each event’s Ξ− velocity along with mean lifetime
(τΞ− = 0.1639±0.0015 [3]) can be used to calculate the average π − vertex detachment
from the reaction vertex. The Ξ− vertex is only negligibly distant from the event
vertex following from the fact that excited baryons decay strongly.
For the calculation of vertex detachment, random numbers were generated with an
exponential probability distribution representing the Ξ− lifetime in its rest frame,
which is a known parameter:
f (t) = f rest (t) = αe−αt
86
(5.1)
where the decay constant α is related to the mean lifetime τ by
τ = 1/α.
(5.2)
f (t) was calculated starting from a pseudo random number generator with a uniform
probability distribution p(x) ∈ [0, 1] given by,


 dx : 0 < x < 1
p(x)dx =

 0 : otherwise
(5.3)
The fundamental transformation law of probabilities can be invoked to determine the
desired form. This law is stated as,
|f (t)dt| = |p(x)dx|
(5.4)
or,
f (t) = p(x)|
dx
|,
dt
(5.5)
which enables one to relate the random variable x with the random variable t. Since
p(x) is a constant we have,
f (t) =
dx
.
dt
(5.6)
x can be solved for by taking the definite integral,
Z
x=
t
f (z) dz.
(5.7)
0
Above is the relationship between the source random variable and the known target random variable. In order to evaluate the probability distribution of the source
87
random variable, the above relationship must be inverted. Time, t, is of the form,
t = G(x)
(5.8)
and using the form of f (t) we get,
Z
t
αe−αz dz,
(5.9)
1 −αt 1
+ ) = 1 − e−αt .
e
−α
α
(5.10)
x=
0
or,
x = α(
Inverting the above equation,
t = G(x) =
ln(1 − x)
.
−α
(5.11)
Finally yielding,
t = −τ ln(1 − x).
(5.12)
The random number t represents the Ξ− lifetime at rest. The lab-frame and rest-frame
lifetimes are related by time dilation,
t = trest =
tlab
,
γ
(5.13)
where the Lorentz factor (γ) is given by,
1
1
.
γ=p
=p
1 − v 2 /c2
1 − β2
(5.14)
Above, c is the speed of light. The relative velocity between inertial reference frames
(v) is simply the measured Ξ− velocity, which can be found through its momentum
88
by virtue of the final state kaons.
β=
v
p
=p
.
c
p2 + m2 c2
(5.15)
In natural units the Lorentz factor is
1
,
γ=q
p2
1 − p2 +m2
(5.16)
trest
tlab = q
.
p2
1 − p2 +m
2
(5.17)
so,
Hence the Ξ− flies a distance between its creation and decay given by,
d = vtlab = cβtlab =
cγptrest
.
E
(5.18)
Note that although natural units are used to relate energy mass and momentum, the
speed of light above is in cm/ns to maintain consistency between the units of velocity
with units of time and length in the analysis software.
5.1.3
GSIM
A GEANT based model of the detector known as GSIM was implemented to estimate
acceptance. GSIM takes the kinematics for each particle track and simulates corresponding detector responses for each subsystem of CLAS. The particle kinematics
are integrated with a model of the toroidal magnetic field and the CLAS detector
to simulate a signal. For the familiar reader, values of parameters used in the GSIM
ffread card are provided in Fig. 5.2. More information on GSIM can be found in
Ref. [47].
89
Figure 5.2: The ffread card used with gsim for our analysis.
5.1.4
GPP
GPP uses the output of GSIM and smears the timing signals by randomized displacements, the extent of which is specific to user input and the characteristics of each
subsystem. The necessity of this program comes from the fact that GSIM tends to
output better resolution than real signals from CLAS. Some of GPP’s flags and parameters are found in Fig. 5.2. The broadened signals include the readout from the
drift chambers and time of flight scintillators. Broadening of scintillator signals are
done according to their dimensions, i.e. a longer scintillator requires a wider smearing function. After the simulated signals have been processed, they go through the
reconstruction software a1c, which is described in Chapter 4, and processed with the
same analysis code as the real data.
5.2 Tuning Simulation to Data
In order to ensure the simulated events are representative of the data, a comparison
was made between the accepted events and the data for a number of quantities. The
Monte Carlo data were generated iteratively with tuning of parameters including:
90
• Beam energy spectrum
• Exponential t-slope, which is a quantity related to Mandelstam variable t and
described in Section 5.2.2
• Production-resonance mass and width
• Input Ξ− polarization
This is not an exhaustive list; there are many other kinematic quantities to which the
acceptance may be sensitive. It will be shown in this section however, that the four
quantities listed above provide a sufficient basis for matching Monte Carlo events to
the data and determining the acceptance.
5.2.1
Simulated Beam Energy Spectrum
Tuning the simulated photon-beam energy spectrum to the data’s began with generating events according to a reasonable but arbitrary spectrum. Matching was done
by breaking our data and the first iteration’s reconstructed events into 27 photonbeam energy bins from 2.8 Gev to 5.5 GeV (100 MeV wide). For each energy bin, a
weighting factor Wi was calculated by,
Wi = Nidata /Nirec1 .
(5.19)
Where Ni is the number of reconstructed events in each energy bin for data or simulation. The next iteration of Monte Carlo events had an energy spectrum obtained
by multiplying the first iteration’s number of generated events with the respective
weighting factor, then multiplied by an overall normalization.
Nigen2
=
P rec1
N
gen1
Wi Ni P idata ,
Ni
91
(5.20)
where Nigen2 is the number of Monte Carlo events generated in the second iteration.
In the absence of other changes to kinematics of the simulation, the iterative process
would only need to be repeated twice. However since other kinematic quantities were
being tuned simultaneously, beam tuning was repeated multiple times. An overlay of
the final beam energy spectrum for data and simulated events is shown in Fig. 5.3.
Figure 5.3: The beam energy spectrum for the Ξ− data sample and Monte Carlo simulation. The Monte Carlo events are in red and are normalized to the data represented
as points with statistical error bars.
5.2.2
Exponential t-slope
The t-slope is a parameter in the relationship between differential cross section and
the Mandelstam-variable t for a two-body nuclear interaction, t is given by,
t = (p1 − p3 )2 = (p2 − p4 )2 ,
92
(5.21)
where pi represents the ith particle’s four-vector. Refer to Fig. 5.4 for further clarification.
t measures momentum transfer from the photon to the Ξ− . The t-slope
Figure 5.4: The left shows a diagram of a generic interaction with two particles in
the initial (p1 and p1 )and two particles in the final state (p3 and p4 ) as referred to
in Equation. 5.21. On the right is the analogous model of the γp → K + K + (Ξ− )
reaction. It’s should be clear from this diagram and Equation. 5.21 that in this
model, t is the momentum transfer from the photon to the fast kaon. The blue
ellipses in both figures represent arbitrary intermediate processes.
is denoted as α and is defined according to an assumption regarding the form of the
cross section,
−t
dσ
= Ae α .
dt
Above,
dσ
dt
(5.22)
is the differential cross section with respect to t and A is a constant.
The t-slope of our data was determined by measuring the acceptance corrected yield
in bins of t, then fitting the plot of yield vs t with an exponential function. The fitting
procedure was done in 8 separate photon-energy bins. Tuning the t-slope spectrum
began with a reasonable guess. The first iteration of Monte Carlo events provided
rudimentary acceptance corrections that were applied to the data, in turn providing
a first estimate on its t-slope spectrum. The resultant acceptance corrected t-slope
spectrum was then used as an input for the second iteration of Monte Carlo event
generation. The process was repeated until convergent and stable t-slope spectra were
93
obtained. The flow chart displayed in Fig. 5.5 depicts the iterative process. Fig. 5.6
shows the t-slope spectra for simulated and experimental data, and Fig. 5.7 shows a
number of quantities used for matching.
Generate
Events
Analyzer
Simulate
detector
acceptance
Analyzer
tslope input
Gen
Acceptance
corrected
t-slope
Rec
Compute
acceptance
Data
Polarization
Figure 5.5: First, events are generated. The initial simulation splits into two sets:
reconstructed events (Rec) that pass through gsim, gpp and a1c before being analyzed,
and generated events (Gen) that pass straight to the analyzer. Acceptance corrections
are obtained from the ratio of reconstructed to generated events. Applying acceptance
to data, one obtains the corrected t-slope spectrum which can in turn be used as input
for the next iteration.
94
Figure 5.6: t-slope for the acceptance corrected experimental data (blue), and the generated Monte Carlo events (red). Agreement within statistical uncertainty is shown.
95
Figure 5.7: An example of the plots that were used to calculate tslope. The top two
plots show the t-spectrum for Monte Carlo generated (left) and reconstructed events
(right). The center left shows the calculated acceptance as a function of t while center
right shows the uncorrected t-spectrum of the data. The bottom right is an overlay
fo the uncorrected t-spectrum of Monte Carlo and Data events. Finally, the bottom
left shows the acceptance corrected t-spectrum of the data with an exponential fit.
96
5.2.3
Resonance Mass and Width
Our simulated production model assumes contributions from various intermediate resonant hyperon decays. In the case of Ξ− photoproduction the primary resonances are
believed, due to strangeness conservation in the strong and electromagnetic interactions, to be excited states of the Λ and Σ (i.e. Y ∗ ) having decay modes Y ∗ → K + Ξ− .
The coupling amplitudes across the Y ∗ spectrum for Ξ− production are unknown,1
but the effective resonance mass spectrum for the simulation was matched to the
data in various energy bins. An overlay of Y ∗ mass spectra for data and simulation
is plotted in Fig. 5.8 and 5.9.
Figure 5.8: An overlay of the data (green) and simulation (red) for the invariant
+
mass m(Ξ− Kslow
) in six distinct beam-energy bins. The data were fit to Gaussian
functions which yields the mass and width of the underlying hyperon used as in input
parameter for simulation.
1
Furthering knowledge on cascade production amplitudes is one of the goals of our analysis.
97
Figure 5.9: An overlay of the data (points) and simulation (red solid) for the invariant
+
mass m(Ξ− Kslow
) integrated over all bins.
5.2.4
Further Comparison of Simulated and Experimental
Data
There are many kinematic degrees of freedom in the simulation: vertex position and
momentum for all three final state particles, along with beam energy, to name a few.
Various combinations of the above mentioned variables compose observables that
were tuned to the data. A number of additional quantities that were not directly
tuned may be compared to ensure the simulation is representative of the data. Good
agreement between data and simulation are shown in Figs. 5.10, 5.11 and 5.12.
98
Figure 5.10: Shows the measured cascade angle with respect to the z-axis in the
center of mass frame. This quantity depends on the intermediate Y ∗ mass and width,
along with the t-slope. Red is simulated events and the points are the data.
Figure 5.11: The invariant mass of the Kf+ast + Ξ− system. Red is simulated events
and the points are the data.
99
Figure 5.12: The magnitude of momentum for all three mesons. Red is simulated
events and the points are the data.
5.3 Acceptance Functions
Histograms of acceptance vs. pion angle across x̂, ŷ and ẑ (axes defined in Chapter 3)
were plotted in several hundred bins of beam-energy and center-of-mass cascade angle.
The plots were fit to ten-degree polynomials in pion angle. Our final integrated
acceptance function was defined by fits in all bins and was applied to each event in
the data.
5.3.1
Calculation of Acceptance and Uncertainty
The acceptance () in a given kinematic bin is determined by the ratio of reconstructed
events (r) to generated events (g).
r
= .
g
100
(5.23)
Since r and g are correlated, the uncertainty of the acceptance can not be calculated
using the method of quadratures. However the expression for the acceptance can be
de-correlated by,
=
r
,
r+n
(5.24)
where n is the number of events which failed to be reconstructed. The uncertainty
can thus be determined through quadrature,
δ2 = (
∂ 2
∂
δr ) + ( δn )2 .
∂r
∂n
(5.25)
Which leads to the uncertainty in the efficiency,
δ =
1p
r(1 − ).
g
(5.26)
The efficiency was plotted with cos θπi − where ı = x, y, z (for a discussion on the
coordinate system see Chapter 3). Ten-degree polynomials sufficed as fitting functions
to reflect the structure of acceptance. The three acceptance functions,
A(θπx,y,z
− )
=
10
X
cx,y,z
xn ,
n
(5.27)
n=0
represent the acceptance of events in kinematic bins of photon energy (Eγ ) and center
of mass Ξ− angle with the z axis
2
(cos θΞ− ). Fig. 5.13 shows a number of typical
acceptance functions in various bins. For simpler notation, Aj will be used to represent
a acceptance for an event j.
If N is the detected-number of events, then applying the acceptance on each event
2
As a reminder, the z-axis is parallel with the photon momentum in the center of mass frame.
101
gives the corrected-number of events N 0 as,
N
X
1
N =
.
Aj
j=1
0
(5.28)
To approximate the uncertainty in N 0 , consider that the acceptance Ai varies little
from the average acceptance A within each kinematic bin. One can write,
N0 ≈
N
.
A
(5.29)
The square of the relative uncertainty is then,
(
δN 0 2
δN 2
δA
) =(
) + ( )2 .
0
N
N
A
(5.30)
So,
r
δN 0 = N 0
(
δN 2
δA
) + ( )2 .
N
A
(5.31)
1
δA
) + ( )2 .
N
A
(5.32)
Or,
r
0
δN = N
0
(
Figure 5.13: A few typical acceptance functions (A(θπy − )) in various bins of beam
energy and cascade angle. The red lines show the fits to the underlying acceptancehistograms with their widths representing the uncertainty as calculated by the covariance matrices of fitting parameters.
102
Chapter 6
Results
This chapter provides the outcome of our cascade polarization measurements in the
reaction γp → K + K + Ξ− . Summary tables and discussions for the induced and transfered polarization are provided in Sections 6.1.1 and 6.2.1 respectively. A comparison
between results and predictions from the only known theoretical model of cascade
photoproduction is also provided in Section 6.3, with a discussion on the comparison
in Section 6.3.1.
6.1 Induced Polarization P
The observable P was measured in nine kinematic bins as shown in Fig. 6.1. We
found P to be consistent with zero in all but one bin, although the results also match
the predictions of Ref. [11]. The single non-zero bin yielded a value P = 0.233 ± 1.84,
which constitutes a 1.26σ deviation from zero, which is likely to be a statistical
fluctuation. Rebinning of the results according to various schemes consistently yielded
P = 0 within statistical uncertainty. Integrating over all bins, we found P = 0.027 ±
0.061. Higher precision measurements in three energy bins and three cascade angle
bins are shown in Figs. 6.2 and 6.4. Additionally, we applied acceptance corrections
based on data-tuned Monte Caro events as described in Chapter 5. We found that
acceptance corrections had virtually no effect on P when measured by the forwardbackward asymmetry method, as is demonstrated in Fig. 6.3 and Table 6.1. Thus,
the measurements without acceptance corrections were taken as the nominal results.
The systematic uncertainty introduced to P by acceptance is quantified in Chapter 7.
103
Figure 6.1: The angular distribution of the pion off ŷ in nine bins of energy and
center of mass cascade angle. The forward-backward asymmetry is used to calculate
the induced polarization P .
104
Figure 6.2: The angular distribution of the pion off ŷ in three bins of energy (top)
and three bins of center of mass cascade angle (bottom). The forward-backward
asymmetry is used to calculate the induced polarization P .
105
Figure 6.3: As one cross check for the cancellation of acceptance, the acceptancecorrected angular distribution of the pion off ŷ is shown.
106
Table 6.1: A table summarizing the results for P , with and without acceptance
corrections. There is good agreement between both methods for all six bins, well
within statistical uncertainty.
Figure 6.4: Angular distribution of the pion off ŷ, integrated over all bins. The
forward-backward asymmetry is used to calculate the induced polarization P . The
measured value of P is constant with zero.
107
6.1.1
Comments on Induced Polarization Results
Our results for P , summarized in Table 6.2, which hold independent of beam and
target polarization, show small or zero induced-polarization for the cascade within our
experimental kinematic range. The near-zero integrated value of P differs significantly
from what was reported for Λ production under similar conditions in Ref. [48], which
showed prominent structure for γp → K + Λ for P , with values as high as ±0.5 for
many bins. However, finer binning and statistical precision was achievable for the Λ
analysis due to higher production cross sections. Probing for narrow structures in P
for the cascade must wait until enough data are available to bin more finely. Although
not directly reported, the bin-integrated results of [48] “cancel” among bins to some
degree, in particular for integrating forward and backward angles, which makes better
correspondence with our results. In any case, our P results suggest the polarization
mechanism for the cascade and Λ differ to some degree, although both mechanisms
still remain speculative. There is no reason to assume, that induced-polarization
features for doubly and singly strange baryons should resemble one another, especially
when comparing single-meson and double-meson production. On the other hand,
possible explanations for the differing features of induced polarization could pertain
to different diquark configurations of the Λ and Ξ− .
Interestingly, our observation of a small induced polarization for the cascade corresponds well with expectations derived from diquark considerations. In the diquark
picture, the cascade inherits its spin from the down-quark which in turn, is likely
inherited from the unpolarized proton target. Future studies on target polarization
transfer could help further elucidate the role of diquark effects and down-quark contributions to the cascade polarization mechanism.
108
Table 6.2: A table summary of induced polarization values binned in center-of-mass
Ξ− angle and beam energy.
Induced polarization, which relates to interference between real and imaginary parts
of its production amplitudes, provides a sensitive test for possible production mechanisms. Our results are compared with the theoretical models of Refs. [11, 14] and
are discussed in Section 6.3.
6.1.2
Alternate Methods for Induced Polarization
For an alternative way to measure P , the acceptance corrected angular distribution
of the pion off ŷ was fit to a first degree polynomial, as described in Chapter 3. This
method showed good agreement with the primary algorithm used to extract P in
Figs. 6.1 and 6.2. The acceptance was characterized as a function of beam energy,
cascade angle, and pion angle then applied to the data in Fig. 6.5.
6.1.3
Further Cross Checks for Induced Polarization
[h] Additional cross checks for acceptance cancellation were performed by simulating
events with various effective Ξ− induced polarization values, Pgen . The measured
value for the reconstructed events, Prec , were consistently in agreement with what was
109
Figure 6.5: Fitted acceptance corrected pion angular distributions and the resulting
measurement of P . Agreement with the primary measurements shown in Fig. 6.2 is
evident for all bins within statistical uncertainty.
110
generated. The agreement Pgen = Prec demonstrates the cancellation of acceptance
for the forward-backward asymmetry method. Fig. 6.6 shows the measured value
Prec = 0 deviating less than 1σ from P = 0 for reconstructed Monte Caro events
corresponding to a generated value of Pgen = 0. We found MC events generated
with higher input polarization Pgen deviated more than those generated with low or
zero polarization. Fig. 6.7 shows consistency with the measured value Prec = −0.5
deviating less than 1σ in all but one bin for a generated value of Pgen = −0.5.
Figure 6.6: Monte Carlo events generated with Pgen = 0. The pion angular distributions of the reconstructed events in 9 bins kinematic bins are shown along with the
measured value of Prec . Better than 1σ agreement for Prec = Pgen is shown in all but
one bin, which shows a deviation of less than 1.5σ.
111
Figure 6.7: Monte Carlo events generated with Pgen = −0.5. The pion angular
distributions of the reconstructed events in nine bins kinematic bins are shown along
with the measured value of Prec . Close to 1σ agreement for Prec = Pgen is shown in
all bins.
112
6.2 Transfered Polarization Cx and Cz
Integrated over all kinematic bins, we found Cx = 0.017 ± 0.177, a results consistent
with zero. A negative value, Cz = 0.301 ± 0.145 was additionally observed as shown
in Fig. 6.8. The finest possible binning scheme with reasonable precision comprised
three energy bins integrated over all cascade angles, and three cascade angle bins,
integrated over all energy. Our binned results in Fig. 6.9 showed Cx to be consistent
with zero in all but one bin. Fig. 6.10 shows Cz to have statistically significant
departures from zero, with the highest measured value of Cz = −0.589 ± 0.253 and
the lowest Cz = −0.001 ± 0.217.
Figure 6.8: Beam-helicity asymmetry as a function of pion angle off x̂ (left) and ẑ
(right), integrated over all bins. The linear fit is used to calculate the transfered
polarization Cx and Cz .
113
Figure 6.9: Beam-helicity asymmetry as a function of pion angle off x̂, in three bins
of energy (top) and three bins of center of mass cascade angle (bottom). The linear
fit is used to calculate the transfered polarization Cx .
114
Figure 6.10: Beam-helicity asymmetry as a function of pion angle off ẑ, in three bins
of energy (top) and three bins of center of mass cascade angle (bottom). The linear
fit is used to calculate the transfered polarization Cz .
115
6.2.1
Comments on Transfered Polarization Results
Our results, summarized in Table 6.3, show a small to intermediate degree of polarization transfer from the photon-beam to the recoil-hyperon. The bin-integrated
values of Cx = 0.017 ± 0.177 and Cz = 0.301 ± 0.145 depart greatly from the corresponding observables reported by Ref. [15] for Λ photoproduction. In Ref. [15],
Cz was large in most kinematic bins often with a value near unity, additionally an
empirical constraint Cz ≈ Cx + 1 was observed, which we do not see for the cascade.
Most significantly for Λ, maximal transfer of polarization in all observed bins occurs;
in the case of total circular beam polarization, the Λ recoils with 100% polarization
p
ie RΛ ≡ P 2 + Cx2 + Cz2 ≈ 1. In contrast, we have shown that the integrated total
polarization for the cascade is R̄Ξ ≈ 0.3. Aside from the difference in magnitude we
observed, we see a similarity between the cascade and the lambda: the total polarization almost entirely comes from Cz (or R̄ ≈ Cz ). We qualitatively understand the
polarization results of the photoproduced cascade and lambda using the vector meson
dominance picture as described in Chapter 1. Our Cx and Cz results suggest that
cascade polarization mechanism is similar to that of the lambda.
We did find, within the limited binning scheme, statistically significant kinematic
dependence in Cz but not in Cx . We measured the highest values of Cx = 0.322±0.304
and Cz ≈ 0.589±0.253 in their respective maximal bins. This is not contrary with the
diquark picture, as a measurement of large induced polarization P would have been.
It is conceivable that the polarized photon could have a preference to produce cascades
from protons who’s down-quarks are aligned (or anti-aligned) with the photon spin.
In this indirect way, the photon spin could be found to be correlated with the recoil
cascade spin, manifestly as Cx and Cz , with the spin transfered from the target. If the
photon directly interacts with a valance quark within the proton, the struck quark
116
Table 6.3: A table summary of Ξ− polarization values for the nominal binning scheme.
must have its initial spin anti-aligned with that of the photon in order to have a
spin-half final state. In this scenario, the final s-quark, must have spin parallel with
the initial photon. Subsequent interactions and spin-orbit coupling could precess its
polarization but conserve its magnitude. This would manifest as correlations between
the photon helicity, and the recoil spin, and be seen in transfered polarization. What
we can say regarding the diquark picture at this point is limited however. Experiments
involving target polarization may be of great use to decipher the spin structure of the
cascade.
Besides probing spin structure and polarization mechanisms, transfered polarization
observables tell us about the interference between the complex production amplitudes
and can provide tests for possible cascade production mechanisms. Our results are
compared with three distinct parameterizations of the theoretical model of Ref. [11]
in Section 6.3. Cross checks for our measurement of Cx and Cz are subsequently
discussed in Section 6.2.2.
117
6.2.2
Cross Check for Transfered Polarization Results
As shown in Chapter 3, parity conservation of the weak interaction forbids polarization transfer in the ŷ direction, i.e. Cy = 0. As a cross check for our method,
we measured Cy to ensure this physical constraint is observed in the data. Fig. 6.11
shows the measured value of Cy integrated over all energy and cascade angle. As
required, we measured a value consistent with zero.
Figure 6.11: Beam-helicity asymmetry as a function of pion angle off ŷ, integrated
over all bins. The linear fit is used to measure the forbidden transfered polarization
Cy . The measurement is consistent with zero as required.
6.3 Comparison with Theory
Our final results were compared with theory as displayed in Figs. 6.12, 6.13, and 6.14.
The theoretical predictions of Ref. [11] are based off relativistic meson-exchange of
cascade production with intermediate hyperon resonances, as described in Chapter 1.
The model has three variants, the first two (A and B) include all hyperon resonances
118
up to Λ(1890); the first uses pure pseudoscalar coupling, while the second uses pure
pseudovector coupling. The third variant of the model (C) is described in Ref. [14],
which appends the hyperon resonance spectrum of Ref. [11] to include higher spin
intermediate states, namely the Σ(2030). Overall, the predicted magnitudes of polarization for all three models are in agreement with our results. A more detailed
discussion is left for the following section.
119
Figure 6.12: A comparison of our results with theory for P (top), Cx (middle) and Cz
(bottom) as a function of beam energy. The theoretical predictions of Ref. [11] are
based on pseudoscalar (solid red) and pseudovector (striped blue) meson-exchange as
described in Chapter 1. Additional high spin hyperon resonances were introduced in
Ref. [14] (dotted green). Displayed results are integrated over all energy bins.
120
Figure 6.13: A comparison of our results with theory for P (top), Cx (middle) and
Cz (bottom) as a function of cascade angle. The theoretical predictions of [11] are
based off pseudoscalar (solid red) and pseudovector (striped blue) meson-exchange
in the t-channel as described in Chapter 1. Additional high spin hyperon resonances
were introduced in Ref. [14] (dotted green). Displayed results are integrated over all
cascade angles.
121
Figure 6.14: A comparison of our results with theory for P in three energy bins.
The theoretical predictions of [11] are based off pseudoscalar (red) and pseudovector
(blue) meson-exchange in the t-channel as described in Chapter 1. Additional high
spin hyperon resonances were introduced in Ref. [14] (green).
122
6.3.1
Comments On our Comparison with Theory
Overall, our results for all three polarization observables are in agreement with the
predictions from theory. Our measured values of P indicate a preference for pseudoscalar coupling (in variant A) over pseudovector coupling (in variant B), although
most of the data lie somewhere between variants A and C. Additionally, Fig. 6.14
indicates the kinematic dependence of P to differ from the predictions, however this
may be attributed to statistical fluctuations.
Similar to P , the results of Cx are in good agreement with the theory, although
no single variant fits the data significantly better than the others. The furtherest
departure from the predictions for Cx is in the energy-integrated results which is
shown in Fig. 6.13, again, differences are within statistical precision.
Finally, the measurement of Cz is also in agreement with the theory and there is no
discernible preference for any given variant. While the magnitudes are in agreement,
the energy-integrated Cz results as a function of cascade angle appear to depart from
theory. We do not show monotonically increasing Cz as cascade angles become more
forward in the center-of-mass frame as predicted, however as previously mentioned,
the data sample suffered from a lack of events in the forward regions. The lack of
forward cascade events is mostly because of differential cross section effects. Unlike
in single meson production, Cz is not necessarily required by conservation of angular
momentum to go to unity at extreme forward or backward cascade angles since the
double meson system can carry orbital angular momentum.
The discrepancies between our results and the theory could suggest additional intermediate hyperon resonances may be present in production mechanism of our reaction.
There are a number of known, high-spin hyperon resonances, the Λ(2100) for example,
that were not included in the model calculations. The overall qualitative agreement
123
of our results and the theory does however corroborate the small body of preexisting evidence [11, 13] for cascade production through the excitation of intermediate
hyperons via relativistic meson exchange in the t-channel.
124
Chapter 7
Systematic Uncertainties
The systematic uncertainty of a measurement M can be estimated by,
δsys (M ) = |M − Malt |,
(7.1)
where Malt is the outcome after an aspect of the measurement has been varied appropriately. We would however expect, even in the absence of systematic uncertainty, to
have statistical fluctuations associated with repeated measurements. Care should be
taken to separate the statistical fluctuations from the systematic uncertainty, as to
avoid overestimating the latter.
However, our data do not have the statistical precision needed to accurately estimate
the systematic uncertainty; all observed fluctuations were well within the statistical
uncertainty. We instead chose to “over estimate” the systematic uncertainty by not
performing a separation.
In general, the total systematic uncertainty of a measurement comes from many
sources. Let the uncertainty arising from a single source be denoted by δs (M ). If
the true correlation between each source of uncertainty is not known, as it is often
the case, it is safest to over estimate by assuming zero-correlation and adding in
quadrature,
δsys (M ) =
s
X
(δs (M ))2 .
(7.2)
s
The sources of systematic uncertainty were individually studied and summarized in
Table. 7.1.
125
δ(P )
δ(Cx )
δ(Cz )
α
negligible negligible 0.009
P
N/A
negligible 0.008
Binning
negligible
0.105
0.037
Mass
0.021
0.067
0.019
Fiducial
0.008
0.005
0.048
Acceptance
0.018
0.005
0.020
Total systematic
0.029
0.124
0.067
Statistical
0.061
0.177
0.145
Total uncertainty
0.067
0.216
0.159
Table 7.1: Systematic uncertainty for Cx, Cz and P arising from various sources.
All sources are detailed in the present chapter. A summary: Binning refers to width
of the binning used, Mass refers to the width of the mass cuts, Fiducial refers to
fiducial cuts, Acceptance refers to acceptance effects. Total systemic refers to each
source, added in quadrature and Statistical refers to the known statistical uncertainty
associated with the nominal measurement. Finally, Total uncertainty refers to the
total systematic and statistical uncertainties added in quadrature.
We found that the statistical uncertainty far outweighed the systematic uncertainty,
which, due to our low statistics was anticipated. Note that if separation of statistical
uncertainty was performed, the systematic uncertainties would contribute virtually
nothing and would appear to be inconsequential to our measured spin observables.
The remainder of the present chapter shows and discusses how the individual sources
of uncertainty were computed.
7.1 Uncertainty in Analyzing Power and Beam Polarization
We calculated that the limited precision of the analyzing power contributed an uncertainty of 2.6% for our results, giving δα (P ) = 0.0007 ≈ 0, δα (Cx ) = 0.0005 ≈ 0, and
δα (Cz ) = 0.0081, for the integrated results. The limited photon-beam polarization
precision gave 3% error, yielding δP (Cx ) = 0.0005 ≈ 0 and δP (Cz ) = 0.009. The
calculations of the relative uncertainties are outlined below.
126
7.1.1
Analyzing Power
From Equation 3.43, we find the uncertainty in Cx and Cz due to the uncertainty in
δα = 0.012 [3] to be,
∂ A(cos θπî )
δα
δα (Ci ) = |δα
| = |Ci | = |Ci |0.026,
î
∂α |P |α cos(θπ )
α
(7.3)
which is 2.6% relative error. Similarly for P , Equation 3.38 gives,
δα (P ) = |P
7.1.2
δα
| = |P |0.026.
α
(7.4)
Beam Polarization
The uncertainty in the electron beam polarization largely came from the Moller measurements which were recorded with a precision close to 1.5%. To additionally account
for uncertainty in the photon-beam energy spectrum, we take the uncertainty in the
photon-beam polarization to be 3%. This translates into,
δP (Ci ) = |Ci |0.03.
(7.5)
7.2 Binning of Pion Angle in Beam-Helicity Asymmetry
The observables were found to vary when the binning scheme of the pion angle was
altered. Figures 7.1 and 7.2 show the measured values of Cx and Cz respectively,
with decreasing bin width from left to right. The average deviation from the nominal
measurements gives δbin (Cx ) = 0.105, δbin (Cz ) = 0.037.
127
Figure 7.1: Shows the asymmetry as a function of pion angle off the x-axis and the
corresponding Cx measurement for decreasing pion bin width from left to right. The
nominal binning scheme is shown in the middle plot.
Figure 7.2: Shows the asymmetry as a function of pion angle off the z-axis and the
corresponding Cz measurement for decreasing pion bin width from left to right. The
nominal binning scheme is shown in the middle plot.
7.3 Varying the Mass-Hypersphere Radius
One way to estimate the systematics introduced by possible background contamination is to vary the radius for the hypersphere cut. By increasing or decreasing
the radius, we let in more or less events from the sideband. Fig. 7.3 shows the cascade signal with multiple layers of radius cuts in the hypersphere, while Figs. 7.4,
7.5 and 7.6 show the effects on observables for differing cuts. The average deviation from the nominal measurements gives δmass (P ) = 0.021 , δmass (Cx ) = 0.067 and
δmass (Cz ) = 0.019.
128
Figure 7.3: The signal shown with 2,3 and 4 σ cuts (or radius cuts in the hypersphere).
The nominal 3-sigma cut is shown in black.
Figure 7.4: The integrated P results for 2,3 and 4 sigma cuts from left to right
respectively
129
Figure 7.5: The integrated Cx results for 2,3 and 4 sigma cuts from left to right
respectively
Figure 7.6: The integrated Cz results for 2,3 and 4 sigma cuts from left to right
respectively
7.4 Effect of Fiducial Cuts
The effects of the fiducial cuts which are described in Chapter 4 were observed to
have a small effect on our results. Figures 7.7, 7.8 and 7.9 show a comparison of
results without fiducial cuts and with “loose” fiducial cuts. The deviation from two
measurements gives δf id (P ) = 0.008 δf id (Cx ) = 0.005 and δf id (Cz ) = 0.048.
130
Figure 7.7: P . Left: results with no fiducial cuts. Right: results with loose fiducial
cuts.
Figure 7.8: Cx . Left: results with no fiducial cuts. Right: results with loose fiducial
cuts.
131
Figure 7.9: Cz . Left: results with no fiducial cuts. Right: results with loose fiducial
cuts.
7.5 Non-Cancellation of Acceptance for P
The bin averaged deviation of P as calculated from the uncorrected forward-backward
pion asymmetry and the acceptance-corrected forward-backward pion asymmetry
gives, δac (P ) = 0.018. Each deviation was taken as the difference of values between
what is shown in Figs. 6.2 and 6.3 for each respective bin, then averaged. Note that
averaging δac (P ) over multiple bins inflates statistical fluctuation and thus overestimates the associated uncertainty. We chose to overestimate, to demonstrate the
slightness of acceptance’s effect on our results.
7.6 Study of Non-Cancellation of Acceptance for Cx and Cz
[H] The calculation of Cx and Cz was based on the fact that, to first order, the acceptance of the detector cancels in the asymmetry. All second order non-cancellation
comes from helicity dependent kinematic distributions of the detected particles.
In the γp → K + K + Ξ− reaction, both kaons come either from the decay of excited
baryons (Y ∗ ) or through other parity-conserving strong or electromagnetic interactions. Thus all kinematic distributions for the kaons are independent of the photon-
132
Figure 7.10: The acceptance corrected forward-backward pion asymmetry results for
P in three energy bins and three center-of-mass cascade angle bins.
beam helicity. The only helicity dependent parts of the reaction are those following a
weak decay of ground-state hyperons. The pion, in the decay Ξ− → Λπ − , contributes
to non-cancellation of acceptance. However, only the projections of the pion angular
distribution off x̂ and ẑ are photon-helicity-dependent since the photon-helicity can
not be transfered to the Ξ− in the y-direction.
The measurement of the spin observables from Monte Carlo events were found to
deviate linearly according to
δacc (Ci ) =
z
X
Cj Eij ,
(7.6)
j=x
where Eij is an error matrix that was computed using Monte Carlo events and i takes
on values of x or z. The elements of the 2x2 matrix Eij were estimated by simulating
events with an effective cascade polarization in the x̂ and ẑ directions (corresponding
133
to Cxgen and Czgen ) then determining the deviation between generated and measured
(Cxrec ,Czrec ) values after reconstruction.
As shown in Fig. 7.11, the values Cxgen = 0.5, Czgen = 0 were generated corresponding
to measured values of Cxrec = 0.483, Czrec = 0.098 yielding Exx = 2(0.017) = 0.34
and Exz = 2(−0.002) = −0.004. Similarly, as shown in Fig. 7.12, the values Cxgen =
0, Czgen = 0.5 were generated corresponding to measured values of Cxrec = 0.116, Czrec =
0.461 yielding Ezx = 2(−0.116) = −0.232 and Ezz = 2(0.039) = 0.079. The error
matrix is given by,


 0.34 −0.004
Eij = 

−0.232 0.079
(7.7)
Applying the error matrix on our measured spin observables from the data Cx = 0.017
and Cz = 0.301 gives,
δacc (Cx ) = 0.017Exx + 0.301Exz = 0.0057 − 0.0012 = 0.0045 ≈ 0.005,
(7.8)
δacc (Cz ) = 0.017Ezx + 0.301Ezz = −0.0039 + 0.0238 = 0.0199 ≈ 0.02.
(7.9)
Although we found the second order acceptance effects to be asymmetrical, they were
taken in as typical symmetric systematic uncertainties.
7.7 Effect of Method for P : A Cross Check
The bin averaged deviation of P as calculated from the fitted acceptance-corrected
and uncorrected forward-backward pion asymmetries gives, δmethod (P ) = 0.068. Each
deviation was taken as the difference between values in Figs. 6.2 and 6.5, for each
respective bin. Fitting the acceptance corrected angular distribution is inherently
less reliable than the forward-backward asymmetry method. A reliable measure of
systematic uncertainty introduced by acceptance effects is calculated in Section 7.5.
134
Figure 7.11: The reconstructed values of Cxrec (left) and Czrec (right) with a generated
value Cxgen = 0.5 and Czgen = 0.
Figure 7.12: The reconstructed values of Cxrec (left) and Czrec (right) with a generated
value Cxgen = 0 and Czgen = 0.5.
135
7.7.1
Fitting vs Raw Yield: A Cross Check on Cx and Cz
Another way to quantify the effects of background contamination is to compare results which required no fitting method due to the absence of background, with results
derived from a background-subtracted fitting algorithm. The background was heavily
reintroduced by relaxing two of our four mass constraints, analogously going from a
cut in the hypersphere to a cut in the circle. Fig. 7.13 is an example of a typical
fit which was integrated to give a background-subtracted yield. The results derived
from the fitting algorithm were compared with “background free” results in Figs. 7.14
and 7.15. The average deviation of the two measurements give δyeild (Cx ) = 0.046 and
δyeild (Cz ) = 0.009, both within statistical uncertainty. The comparison between fitting and raw yield serves only as a cross check. A more reliable measure of systematic
uncertainty introduced by background is calculated in Section 7.3.
Figure 7.13: A typical fit which was integrated to give a background subtracted yield.
136
Figure 7.14: Left: fitting with four pion bins. Right: no fitting with five pion bins.
The binning scheme for the fitting method was defined coarsely to provide reliable
yields.
Figure 7.15: Left: fitting with four pion bins. Right: no fitting with five pion bins.
The binning scheme for the fitting method was defined coarsely to provide reliable
yields.
137
7.8 Studies on Background: Further Cross Checks
7.8.1
Effective Polarization of Background From Lambda Decay
Although the data were found to be mostly background free, the primary source would
likely come from contamination as a result of the Λ decay Λ → pπ − . In this scenario,
the pion from the Λ could be misidentified as coming from the cascade decay. This
“Λ background” is shown Fig. 7.16 and was estimated at most to constitute a few
percent of the signal. For the background to contribute significantly to the systematic
uncertainty, it’s measured effective polarization would have to be well over 100%
(possible since the effective values do not correspond to physical observables) due to
its minute presence. The effective polarization of the Λ background was measured
for the sum of all kinematic bins and shown in Fig. 7.17. We found a statistically
significant “effective” polarization of Cz = −0.625, which would contaminate our data
sample by less than 1%, thus, δΛ (Cz ) ≈ −0.006. In any case, the varying hypersphere
cut includes this number so δΛ (Cz ) was not taken as a separate source of systematic
uncertainty. The analysis on the Λ-pion background serves merely as a cross check of
previous systematics.
138
Figure 7.16: In the top four plots: events in red are the result of cut on the invariant
mass m(Λ + π − ) and m(Ξ− − π − ) which identifies the pion coming from the Λ decay.
Its presence is shown over the broader spectrum of events. In the bottom four plots:
Events representing cuts in the hypersphere radius r = 1, 2, 3, 4, 5, 6 are layered over
one another. The vertical lines provide a means of showing how deep within the
hypersphere the “lambda-pion” cuts are. One should note the Λ-pion cuts lie in the
r = 6 hypersphere, far out in the sideband of the primary signal.
139
Figure 7.17: Measurements of the effective Cx and Cz for events in the lambda-pion
background. This measurement is not meaningful in terms of polarization observables,
but serves as study of background effects.
7.8.2
Effective Polarization of Unknown Background
Just as the Λ-pion contamination was examined, a similar study can be done for the
background that comprises a mix of unknown sources. This “unknown” or “mixed”
background is mostly from particle misidentification and is shown Fig. 7.18. The
mixed background was found to have a zero effective polarization. If there was a
large presence of mixed background in our signal, it would dilute the measured polarization. The varying of the hypersphere cuts include the small dilution effect so a
separate systematic uncertainty need not be introduced. The systematic uncertainty
associated with the discrepancy of results deriving from background-subtraction and
non-background subtraction yield method also accounts for this the mixed background.
140
Figure 7.18: In the top four plots: events in red are the result of cut on the invariant
mass m(Λ + π − ) and m(Ξ− − π − ) which identifies the “mixed” background. Its
presence is shown over the broader spectrum of events. In the bottom four plots:
Events representing cuts in the hypersphere radius r = 1, 2, 3, 4, 5, 6 are layered over
one another. The vertical lines provide a means of showing how deep within the
hypersphere the mixed-background cuts are.
141
Figure 7.19: Measurements of the effective Cx and Cz for events in the “mixed”
background. This measurement is not meaningful in terms of polarization observables,
but serves as study of background effects.
142
Chapter 8
Conclusions and Outlook
This work presents the first ever determination of polarization observables for the
charged cascade hyperon in the reaction γp → K + K + Ξ− . We utilized the parityviolating weak decay of the cascade as a means to measure its polarization through
the angular distribution of its decay products. A background-free sample of over
5000 Ξ− → Λπ − events composed the data, a globally unprecedented yield for photoproduction. We measured the induced polarization P , along with the degree of
polarization transfer from a circularly polarized photon beam, Cx and Cz .
Overall the induced polarization P , along with transferred polarization Cx and Cz ,
were found to agree with predictions from the only known model of cascade photoproduction. The theory models hadronic interactions based on relativistic meson
exchange, with cascade production arising due to the excitation and decay of resonant,
singly strange hyperons. Among the included diagrams, the ones involving t-channel
kaon exchange dominate the overall contributions to production, which reproduces
the preexisting data on the cascade cross sections. A model including singly strange
intermediate hyperons for the γp → K + K + Ξ− reaction is a natural choice, because
otherwise, it would involve a t-channel exchange of exotic mesons with S=2.
Our results for P , which due to the nature of our measurement technique allowed
for finer binning than for Cx and Cz , showed the closest agreement with the theory.
We found that the best match for P came from the model parameterization that
only took pseudoscalar mesons as candidates for exchange particles. However within
uncertainty, our findings are also consistent with zero, not far from agreement with
the other two variants of the model. More data at lower energies, especially closer to
143
threshold, will help paint a descriptive picture and provide more stringent constraints.
There were no events detected below 2.8 GeV while threshold is around 2.37 GeV.
The lack of data below 3 GeV is mostly due to small cross sections, as was anticipated
from the known form of the differential cross section. The acceptance corrected beam
energy profile in our data also indicates the cross section drops off rapidly below 3
GeV. Additionally, the detector efficiency falls off at energies approaching threshold.
Although our results were broadly in agreement with theory, there were also some
interesting discrepancies. The most prominent disagreement was in Cz as a function
of center-of-mass cascade angle; the general trend of our results appears to diverge
from theory. Before a definitive conclusion can be drawn on this front, more data need
to be obtained in the forward region. The non-rising Cz for forward cascade angles
is a feature of our data that is permissible by angular momentum conservation for
two-meson production, in contrast with the case of single meson production. We are
eager to compare this feature of our results with future high statistics measurements
made by CLAS12.
As expected, we found the total polarization magnitude for the cascade was much
lower than what has been reported for the Λ. The integrated results show R̄Ξ ≈ 30%,
while for the lambda, R̄Λ ≈ 100%. An interesting similarity between the lambda and
cascade is that most of their total polarization comes from Cz , i.e, R ≈ Cz in most
bins. We qualitatively explain both features of our results invoking a vector meson
dominance picture in which one of the strange quarks in the final state cascade comes
from the photon, and the other, from the decay of an intermediate hyperon resonance.
A similar picture has been suggested to explain the previous lambda polarization data
for photoproduction.
If the naive diquark model is an accurate picture for the internal dynamics of the
cascade, then its total spin would largely be due to the down-quark. In this picture,
144
photoproduction of the cascade off a proton target should exhibit large polarization
transfer from the target, from which the polarizing down-quark is inherited, but one
would see little transfer from the beam. The non-zero values we observed for Cz are
thus in disagreement with this picture. We point out that it is perhaps unnatural
to suppose that the two s quarks strongly correlate as a diquark since this state is
not overall antisymmetric in spin, flavor, and color. Regardless, future experiments
with target polarization could prove invaluable for probing the spin structure of the
cascade.
In addition to constraining current and future theoretical production models, our
results taken in conjunction with subsequent polarized target experiments will facilitate progress towards a complete measurement for the γp → K + K + Ξ− reaction.
Complete measurements for cascade photoproduction would allow for unambiguous
calculation of amplitudes i.e. a model independent identification of any missing hyperon resonances. While the realization of a complete measurement is still infeasible
with current nuclear physics instrumentation, there is much to be learned along the
way, and polarization observables provide detailed information on the interference
of the production amplitudes. Many Λ and Σ resonances have been observed and
studied, but there are still large gaps to close in the world database regarding the
spectrum of possible hyperon states and their properties.
For the near future, it should be possible to make a first time measurement of the beam
asymmetry Σ using our data, provided the experimental photon helicity asymmetry
can be accurately accounted for. Additionally, there are other cascade polarization
observables, specific to two meson production that we plan on extracting. Within the
next few years, CLAS12 may be able to produce and reconstruct around a million
Ξ− → Λπ − events.
145
To conclude, this work has produced the only three standing measurements for the
cascade polarization observables in photoproduction. We have a qualitative understanding of our results from the vector meson dominance picture of the photon. Additionally, our data have corroborated the only theoretical model for cascade photoproduction and provided constraints for its parameters, and for the parameters of future
models. We have taken a first step towards a characterization of the cascade’s underlying complex amplitudes of production. Such amplitudes, which can in principal
be extracted from cross section and polarization and observables such as the three
we have provided, will help elucidate the spectrum of excited hyperon resonances.
Finally, our work has established the feasibility of cascade studies near threshold energies. There is still much to be learned from further studies of this data set and in
cascade data sets to come, in particular with CLAS12. We hope our results will stimulate the otherwise slow progress in experimental and theoretical cascade physics, and
in turn help bring about a phenomenological understanding of the photoproduction
of strangeness.
146
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150
VITA
JASON STEPHAN WILLIAM BONO
2005-2006
B.Sc., Mathematical Physics
Macquarie University
Sydney, Australia
2007-2009
M.S., Physics
University of Miami
Miami, Florida
2009-2014
Ph.D., Physics
Florida International University
Miami, Florida
SELECTED PUBLICATIONS AND PRESENTATIONS
1. Jason Bono, Newport News, Virgina (2014). First Time Measurements of
Polarization Observables for the Charged Cascade in Photoproduction.
2. Jason Bono, Newport News, Virgina (2013). Spin Observables for the Ξ− in
Photoproduction, Fall Meeting of the American Physical Society, Division of
Nuclear Physics.
3. Jason Bono (2014). Polarization of the Cascade Hyperon in Photoproduction,
Proceedings of Science.
4. Jason Bono, Montevideo, Uruguay (2013). Polarization of the Cascade
Hyperon in Photoproduction, 10th Annual Latin American Symposium on
Nuclear Physics and Applications.
5. Jason Bono, Plymouth, New Hampshire (2012). A First Time Polarization
Measurement of the Ξ Baryon in Photoproduction, Gordon Research
Conference on Photonuclear Reactions.
6. Jason Bono, St. Petersburg, Florida (2012). Search for a New Ξ Resonance,
Conference on Intersections of Particle and Nuclear Physics.
151
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