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'I‘ "i TMI"Ia-~‘.II°II . , u. 0'" 'H' "' ‘.4I|1'4;X‘I‘IL" ’3 VIM}: 'II‘ ‘15; II'UI M ' ' 2:1‘ 3: ‘11“ Juli" (Lt; “I! 1I" :’ lu‘ VI - “if!" l— IIIIIIIIIIIIIIEIIIIIIISIII This is to certify that the dissertation entitled Nucleon Structure Functions from Deep Inelastic Charged Current Neutrino Scattering presented by William Gilbert Cobau has been accepted towards fulfillment of the requirements for Ph.D. degree in Physics iétfltmufl gfl/L/ Major professor Date July 20, 1992 MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 —-——————.—— ——————— —~— — 7 —‘ — —— —— 7 7v ,7 VA 7 i 7 7 7 I ' LIBRARY I Michigan State Universlty PLACE iN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE i I____I -____I“: VIII ILJEZ’LJ [3:153 ”1-K"? I—‘i I MSU Is An Affirmative ActiorVEquel Opportunity Institution cmmt L___ NUCLEON STRUCTURE FUNCTIONS FROM DEEP INELASTIC CHARGED CURRENT NEUTRINO SCATTERING By William Gilbert Cobau A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR or PHILOSOPHY Department of Physics and Astronomy 1992 ‘ /_j,") 5 .' r //4 ABSTRACT Nucleon Structure Functions from Deep Inelastic Charged Current Neutrino Scattering By William Gilbert Cobau From a large data set of charged current neutrino—nucleon interactions, double differential cross sections and nucleon structure functions have been measured. In addition, the Quantum Chromodynamics parameter AQCD has been measured from the Q2 evolution of the extracted structure functions. For Marcia, who fills my life with love and energy. iii ACKNOWLEDGEMENTS This thesis would not have been possible without the effort of many people besides myself. First, I would like to thank my wife Marcia Maria Campos Torres for putting up with me and encouraging me to finish this thesis, as quickly as possible. Without the efforts of my fellow E733 graduate students, this work would not have been possible. Special thanks go to my fellow hunchbacks, George Perkins and Robert Hatcher with whom I spent many 12 hour owl shifts. E733 was not an example of “Too many Scientists and not Enough Hunchbacks.” Three Hunchbacks were sufficient. Elizabeth Gallas’ work on the flux file has been essential to this work. My Advisor Raymond Brock deserves special credit for believing thatI could earn my PhD and then giving me the freedom to explore my own ideas and for telling me when I was too far-a-field. While spending time at Fermilab, Stuart Fuess has . served as my Pseudo-Advisor. Both Chip and Stu have spent large amounts of time discussing all parts of this analysis and giving me critical feedback on both the analysis and the actual thesis. I would also like to thank ”Supertech” Ron Olsen for much help in running the experiment. Ron was the Lab C technician during E733. There were few problems that Ron could not deal with quickly and efficiently. Finally, I would like to thank my parents, John and Arlene Cobau, for their love and support. Special thanks go to my father for helping with the proof reading of this thesis-not an easy task for a non-scientist. iv Table of Contents List of Tables ................................................................................................ ix List of Figures ................................................................................................ x Introduction and Theory .............................................................................. 1 1.1 Introduction ................................................................................................................ 1 1.1.1 Why Neutrino Scattering? ................................................................................. 2 1.1.2 This Chapter ....................................................................................................... 4 1.2 Theory ......................................................................................................................... 4 1.2.1 The Theory of Neutrino-Nucleon Scattering ................................................... 5 1.2.2 Quark—Parton Model ........................................................................................ 7 1.2.3 Quantum Chromodynarnics ............................................................................. 9 1.2.4 Neutrino—Nucleon Scattering From a Quark—Paton Model Perspective. 11 1.2.5 Structure Function Evolution .......................................................................... 16 13 This Thesis ................................................................................................................. 18 The Experiment ........................................................................................... 20 2.1 Introduction .............................................................................................................. 20 2.2 The Exposures ........................................................................................................... 20 2.2.1 1982, Experiment 594. ...................................................................................... 21 2.2.2 1985, Experiment 733, Part 1. .......................................................................... 21 2.23 1987, Experiment 733, Part 2. .......................................................................... 25 23 Beams ........................................................................................................................ 25 23.1 DiChromatic Narrow Band Beam .................................................................. 26 2.3.2 Quad-Triplet Wide Band Beam ...................................................................... 26 2.4 Detector ..................................................................................................................... 30 2.4.1 Front Veto Configuration ................................................................................ 30 2.4.2 Target-Calorimeter .......................................................................................... 33 2.4.2.1 Flash Chambers ...................................................................................... 34 2.4.2.2 Proportional Planes ................................................................................ 38 2.43 Spectrometer .................................................................................................... 39 2.43.1 Charge Division ...................................................................................... 40 2.43.2 Drift ......................................................................................................... 40 2.4.4 Event Display ................................................................................................... 41 2.5 Event Reconstruction and Measurement ................................................................ 41 V 2.5.1 Vertex Finding. ................................................................................................ 46 25.2 Muon Finding and Fitting ............................................................................... 46 2.53 9 ....................................................................................................................... 47 2.5.4 5’; ....................................................................................................................... 47 25.5 v ........................................................................................................................ 48 2.6 Final Event Sample ................................................................................................... 52 2.6.1 Cuts ................................................................................................................... 52 2.6.1.1 Acceptance Cuts ..................................................................................... 52 2.6.1.2 Measurement Quality Cuts .................................................................... 55 2.6.1.3 Physics Cuts ............................................................................................ 55 2.6.2 Sample .............................................................................................................. 56 Monte Carlo Simulation ............................................................................. 58 3.1 Introduction .............................................................................................................. 58 3.2 Beam Simulation ....................................................................................................... 58 33 Interaction Simulation .............................................................................................. 59 33.1 Fermi-Motion Simulation ................................................................................ 59 33.2 Neutrino—Nucleon Interaction ........................................................................ 62 3.4 Event Simulation ...................................................................................................... 62 3.4.1 Muon Simulation ............................................................................................. 63 3.4.1.1 Calorimeter Tracking .............................................................................. 65 3.4.1.2 Spectrometer Simulation ........................................................................ 68 3.5 Analysis Of Monte Carlo Events .............................................................................. 69 3.5.1 Event Classification and Muon Angle ............................................................ 69 3.5.2 Muon Fitting .................................................................................................... 69 3.53 Hadron Energy ................................................................................................ 71 3.5.4 Comparison Of Accepted Integral Distributions of Data and Monte Carlo .71 Structure Function Extraction ..................................................................... 78 4.1 Introduction .............................................................................................................. 78 4.2 Differential Cross Section Extraction ....................................................................... 78 4.2.1 Total Cross Section .......................................................................................... 79 4.2.2 Correction for Target Neutron Excess ............................................................ 79 43 Structure Function Measurement ............................................................................ 81 4.4 Systematic Error Analysis ........................................................................................ 82 4.4.1 Acceptance ....................................................................................................... 82 4.4.2 Measurement Biases ........................................................................................ 83 4.5 Scale Errors ................................................................................................................ 85 4.6 Results ....................................................................................................................... 85 4.6.1 Binning ............................................................................................................. 87 4.6.2 Final Cross Section Results .............................................................................. 87 4.63 Final Structure Function Results ..................................................................... 88 Results and Conclusions ........................................................................... 117 5.1 Introduction ............................................................................................................ 117 vi 5.2 Structure Functions Comparisons ......................................................................... 117 52.1 Q1 Evolution ................................................................................................... 117 5.2.2 CDHSW .......................................................................................................... 118 5.23 CCFR .............................................................................................................. 124 53 Parton Distribution Function Fitting ..................................................................... 128 5.4 A00) Fitting .............................................................................................................. 129 5.5 Conclusions ............................................................................................................. 134 Appendices The FMMF Collaboration ........................................................................ 139 Hadron Calibration ................................................................................... 140 8.1 Introduction ............................................................................................................ 140 8.2 Calorimetry ............................................................................................................. 141 8.2.1 Flash Chamber ............................................................................................... 142 8.2.1.1 Properties of Flash Chambers .............................................................. 142 8.2.1.1.1 Saturation .................................................................................... 142 8.2.1.1.2 Detector and Environmental Effects ......................................... 143 8.2.1.1.3 Residual Ionization ..................................................................... 144 8.2.1.2 Flash Chamber Calorimetry Algorithms ............................................. 145 8.2.1.2.1 SHOWER .................................................................................... 146 8.2.1.2.2 EHFC ........................................................................................... 148 8.2.2 Proportional Planes ........................................................................................ 155 8.2.2.1 Monitoring ............................................................................................ 158 8.2.2.2 Calorimetry ........................................................................................... 159 8.23 Corrections for Muon Energy Loss in Calorimeter ...................................... 159 8.23.1 Explicit Elimination .............................................................................. 159 8.2.3.1 Statistical Subtraction ............................................................................ 159 83 Scale Determination ................................................................................................ 160 83.1 Test Beam ....................................................................................................... 160 83.1.1 Method .................................................................................................. 161 83.1.2 Results ................................................................................................... 162 83.2 Neutrino Data ................................................................................................ 167 83.2.1 Method .................................................................................................. 170 8.3.2.2 Results ................................................................................................... 171 83.3 The Self Consistency of the Determined Scales ............................................ 175 8.3.3.1 Proportional Plane Results ................................................................... 175 833.2 Re-Analysis of Test Beam Flash Chamber Results .............................. 180 8.33.2.1 Is There Any Reason to Believe the Test Beam Data? .............. 186 8.4 Conclusions ............................................................................................................. 187 The Physical Cross Section ....................................................................... 188 CI Introduction ............................................................................................................ 188 C2 Radiative Corrections ............................................................................................. 188 vii C.2.1 Box Diagram .................................................................................................. 190 C.2.2 Final State Radiation ...................................................................................... 191 C3 Slow Rescaling Model ............................................................................................ 194 CA Application .............................................................................................................. 195 C5 Conclusions ............................................................................................................. 196 Systematic Error Analysis ......................................................................... 197 D.1 Introduction ............................................................................................................ 197 D2 Procedure ................................................................................................................ 197 D.2.1 Calculation of Systematic Errors from Measurement Biases ...................... 197 D.2.1.1 The Method in Detail ........................................................................... 198 D.2.1.2 Scale Errors ............................................................................................ 200 D.22.2.1 9 .................................................................................................. 200 0.22.2.2 E’; ................................................................................................. 201 D.2.2.23 v ................................................................................................... 202 D.2.13 The Final Ensemble ............................................................................... 202 D.2.2 Acceptance Uncertainties .............................................................................. 2(B D3 Results ..................................................................................................................... 2(8 Alternative Structure Functions ............................................................... 206 El Introduction ............................................................................................................ 206 E2 Structure Functions with Non-Linear Cross Section ............................................ 206 E3 Structure Functions without Slow Rescaling Correction ...................................... 207 List of References ...................................................................................... 216 viii List of Tables Table 2.1. Average Properties of Target and Detector Sampling ................................. 35 Table 2.2. Event Statistics ................................................................................................ 57 Table 4.1. Final Event Sample ........................................................................................ 86 Table 4.2. Binning limits for Structure Function Extraction ........................................ 89 Table 4.3. Anti-Neutrino Differential Cross Sections .................................................... 90 Table 4.4. Neutrino Differential Cross Sections ........................................................... 101 Table 4.5. FMIVIF Structure Functions .......................................................................... 113 Table 5.1. Results of Parton Distribution Fitting .......................................................... 130 Table 5.2 Results of Non-Singlet AQCD Fitting ............................................................. 135 Table 8.1. Scale Refitting Results .................................................................................. 172 Table 8.2. Proportional Plane Refitting Results for Pulse Height ............................... 185 Table D.1. Variations in Calibration Constants ............................................................ 204 Table E.1. Structure Functions Using CCFR Cross Section ......................................... 208 Table 8.2. Structure Functions Without Slow Rescaling Correction .......................... 212 ix Figure 1.1. Figure 1.2. Figure 13. Figure 1.4. Figure 2.1. Figure 2.2. Figure 2.3. Figure 2.4. Figure 2.5. Figure 2.6. Figure 2.7. Figure 2.8. Figure 2.9. Figure 2.10. Figure 2.11. Figure 2.12. Figure 2.13. Figure 3.1. Figure 3.2. Figure 3.3. Figure 3.4. Figure 3.5. Figure 3.6. Figure 3.7. List of Figures eN and vN Scattering Feynman Diagrams .................................................. 3 Primitive Vertices for QED and QCD ........................................................ 10 Feynman Diagrams For vne Scattering ....................................................... 12 Feynman Diagrams For vuq Scattering ...................................................... 14 Neutrino Gate Structure ............................................................................. 23 HiE Trigger Turnon .................................................................................... 24 DiChromatic Beam Schematic .................................................................... 27 Neutrino Energy Versus Beam Radius for DiChromatic Beam Schematic ............................................................................................... 28 Quadrupole Triplet Beam Schematic ......................................................... 29 Neutrino Energy Versus Beam Radius for Quadrupole-Triplet Beam ....31 FMMF Detector Schematic ......................................................................... 32 Flash Chamber Schematic ........................................................................... 36 Ieft/ Right Ambiguity ................................................................................ 42 Event Display .............................................................................................. 43 Event Analysis Diagram ............................................................................. 45 Calorimeter Energy Loss Correction to Fit Muon Energy ........................ 49 Fractional v Resolution as Function of v .................................................... 53 Fermi Momentum Distribution .................................................................. 60 Effects of Fermi-Motion on Perceived x Distribution ................................ 61 Differential Cross Sections for 8 Rays, Bremsstrahlung, Pair Production and Inelastic Nucleon Scattering for 100 GeV Muon Traversing Iron ........................................................................... 64 Spectrum of Energy Lost by 100 GeV Muon Traversing 20 cm of Iron ...66 Monte Carlo Muon Momentum Resolution .............................................. 67 Comparison of Data and Monte Carlo Drift Hit Distributions ................ 70 Comparison of Data and Monte Carlo v Distributions ............................. 73 Figure 3.8. Comparison of Data and Monte Carlo E” Distributions ........................... 74 Figure 3.9. Comparison of Data and Monte Carlo 6” Distributions ............................ 75 Figure 3.10. Comparison of Data and Monte Carlo Ev Distributions ........................... 76 Figure 3.11. Comparison of Data and Monte Carlo Iron Distributions ........................ 77 Figure 5.1. Effects oiQ’Evoiution .............................................................................. 119 Figure 5.2. The FMMF Structure Functions ............................................................... 120 Figure 5.3. Comparison of FMMF and CDHSW Structure Functions ...................... 121 Figure 5.4. Comparison of FMMF and CCFR Structure Functions .......................... 125 Figure 5.5. FMMFParton Distribution Functions at Q2: 5.0 GeV2 ........................... 131 Figure 5.6 FMMFParton Distribution Functions at Q2=50GeV2 ............................ 132 Figure 5.7. Result of Non-Singlet Fit Using FNS=xF3 for 0.0 < x < 0.7 ........................ 136 Figure 5.8. Result of Non-Singlet Fit Using FNS=xF3 for 0.0 < x < 0.3 and FNS=F2 for 0.3 < x < 0.7 .................................................................. 137 Figure 5.9. Result of Non-Singlet Fit Using FN5=F2 for 0.3 < x < 0.7 .......................... 138 Figure 8.1. Variation of Flash Chamber Response ..................................................... 147 Figure 8.2. Multiplicity Correction .............................................................................. 149 Figure 83. Efficiency Correction ................................................................................. 150 Figure 8.4. Construction of Module Transition Curves ............................................. 152 Figure 8.5. Module Transition Curves ........................................................................ 153 Figure 8.6. Variations in Module Response as Measured by Transition Curves ..... 154 Figure 8.7. Spatial Variations in Module Response ................................................... 156 Figure 8.8. Effects of Response Curve Corrections .................................................... 157 Figure 8.9 Raw Test Beam Distributions ................................................................... 163 Figure 8.10. Corrected Pulse Height vs. Test Beam Momentum ................................ 164 Figure 8.11. Response Corrected Raw Hits vs. Test Beam Momentum ..................... 165 Figure 8.12 Calibrated Test Beam Distributions ......................................................... 166 Figure 8.13. Accepted v Distribution—EHFC Scale ...................................................... 168 Figure 8.14. Accepted y Distribution-EHFC Scale ...................................................... 169 Figure 8.15. Accepted v Distribution—Rescaled EHFC Scale ....................................... 173 Figure 8.16. Accepted 3; Distribution—Rescaled EHFC Scale ....................................... 174 Figure 8.17. Accepted v Distribution-EHPR Scale ...................................................... 176 Figure 8.18. Accepted y Distribution-EHPR Scale ...................................................... 177 Figure 8.19. Accepted v Distribution-Rescaled EHPR Scale ....................................... 178 Figure 8.20. Accepted y Distribution-Rescaled EHPR Scale ....................................... 179 Figure 8.21. Accepted v Distribution—1985 EHPR Scale .............................................. 181 Figure 8.22 Accepted y Distribution-1985 EHPR Scale .............................................. 182 xi Figure 8.23. Accepted v Distribution—Rescaled 1985 EHPR Scale .............................. 183 Figure 8.24. Accepted 3] Distribution-Rescaled 1985 EHPR Scale .............................. 184 Figure C.1. Feynman Diagrams for Radiative Correctiom . ...................................... 189 Figure C.2. Box Diagram Corrections to the leading Order Neutrino—Quark Cross Sections ...................................................................................... 192 Figure D.1. Division of a Gaussian for Systematic Error Measurement .................... 199 Figure D.2. Ensembles of Structure Functions ............................................................ 205 xii Chapter 1 Introduction and Theory 1.1 Introduction The structure of the world around us is a recurring theme in mankind's seardi for knowledge. The ancient Greeks first gave man the concepts of elements and atoms as the building blocks of matter. In the past century, our understanding of the building blocks of matter has increased greatly. Atoms have gone from theoretical constructs to real physical objects that we can manipulate. We also know that atoms have a structure that we can describe and explain using Quantum Mechanics. At the center of the atom is the nucleus, which consists of nucleons-protons and neutrons. The structure of these nucleons is the topic of this thesis. In the past thirty years, elementary particle physicists have come to believe that the nucleons are not elementa- ry particles but are composites, like the nucleus and atom. The constituents of the nucleons have been named quarks and gluons or, in general, partons. Quarks are strongly interacting particles which carry a charge of K or 36 the electron charge. Glu- ons are the carriers of the strong force and are uncharged. The theory of Quantum Chromodynamics (QCD) describes the interactions of strongly interacting particles such as quarks and gluons. This thesis presents measurements of the structure of the nucleon and of a fun- damental constant of QCD, A00). Measurements of the nucleon structure are done in scattering experiments. The nucleon is probed by the scattering of high energy particles such as the electron, the neutrino, or the pion. By examining the results of the high energy collisions, one can 2 gather information about the internal structure of the struck object. The energy of the probe being used determines the size of the structures that can be Observed, the higher theenergyoftheprobe, the finerthestructure thatcanbeobserved. Scattering experiments, sudi as the one presented in this thesis, have discovered the structure of the atom, then the structure of the nucleus, and now are probing the structure of the nucleon. In the future, it is possible that scattering experiments will find that the partons that we now believe are fundamental are themselves composed of some new set of particles. This thesis presents an analysis of charged current neutrino—nucleon scattering. Neutrino—nucleon scattering experiments belong to a larger group of scattering experi- ments termed Deep Inelastic Scattering (Dis) experiments. These experiments include Electron—Nucleon, Muon—Nucleon and Neutrino—Nucleon scattering experiments. The Feynman Diagram for these processes are shown in Figure 1.1. The experiments are termed Deep Inelastic Scattering because one looks for inelastic events with large momentum transfers to the nucleon system. The large momentum transfers probe deeply into the internal structure of the struck nucleon. 1.1.1 Why Neutrino Scattering? Neutrino experiments are inherently difficult because neutrino beams are diffi- cult to produce and the cross sections for neutrino interactions are very small compared to that for electron or muon interactions. Then the reader may wonder: ”Why bother? Why not use electrons and muons exclusively to probe the structure of the nucleon?” The an- swer to this question is that neutrino-nucleon scattering provides additional and complementary information. Electron—nucleon or muon—nucleon scattering is predom- inantly an electromagnetic process using the virtual photon as the probe of the nucleon Electromagnetic interactions conserve parity. In contrast, neutrino—nucleon scattering is strictly a weak process using the virtual W (or Z) to probe the nucleon. Parity is not Figure 1.1. eN and vN Scattering Feynman Diagrams. Shown are the Feynman Diagrams for electron-nucleon scattering and charged current neutrino— nucleon scattering. Diagram (a) describes the electron-nucleon scattering. The incoming electron, e, emits a virtual photon, 7, resulting in the outgoing e’. The virtual photon strikes the nucleon, N, destroying the nucleon. The resulting debris is denoted by X This same diagram can be used to describe uN scattering. Diagram (b) describes neutrino—nucleon scattering. This diagram is very similar to the diagram above. The incoming neutrino, v, emits a virtual W and the outgoing lepton is a muon, u. The W strikes the nucleon, N, and the destroyed nucleon continues on as the hadron system, X. 4 oomerved in weak interactions. Thus neutrino-nucleon interactions are sensitive to the parity violating parts of the nucleon structure which are not available in charged parti- cle scattering. As we will see, this makes neutrino scattering sensitive to both the valence and quantum sea parts of the nucleon and allows one to separate the valence part of the nucleon from the quantum sea. 1.1.2 This Chapter This Chapter will present an overview of the theory of Neutrino—Nucleon scat- tering and the Quark-Parton Model. There will then be some preliminary discussion of the data and analysis presented in this thesis. Finally, the contents of this thesis will be outlined. 1.2 Theory The theory of DIS, in general, and neutrino—nucleon scattering, specifically, is well understood. This thesis does not pretend to give a comprehensive review of the theory of DIS. There are many excellent text books (such as, Halzen and Martin 1984; leader and Predazzi 1983; Quigg 1983; and Cheng and Li 1984) with large sections de- voted to D15 and the Quark—Parton Model. This section will provide an overview of neutrino—nudeon scattering and try to look at weak interactions in the context of the Quark-Panon Model. Using Lorentz Invariance and the known structure of Weak interactions, one can derive the form of the differential neutrino—nucleon cross section, This derivation is model independent. It is in this context that structure functions have a profound impor- tance because they will allow a description of the structure of the nucleon without resorting to complex theoretical interpretations. A discussion of the Quark—Patton Model follows. The Quark—Parton Model provides a foundation in which to interpret the structure function results that were ob- tained previously. The Quark—Patton Model provides a context in which we will 5 examine the neutrino—quark scattering. The theory of Weak interactions allows one to calculate the cross section for neutrino—free quark scattering in the same way one might calculate neutrino—electron scattering. From the results of neutrino—quark scattering, one will construct the differential cross sections for neutrino—nucleon scattering and determine, using the derived cross sections, the correspondence between the structure functions and the quark distributions. 1.2.1 The Theory of Neutrino—Nucleon Scattering One can use the methods of Lorentz Invariance and the structure of weak inter- actions (see Cheng and Li 1984 or Halzen and Martin 1984) to calculate the cross section for the inclusive process, v(k)+N(p) —> l(k’)+X(P’). (1.1) This approach has the advantage that it is somewhat model independent. One begins with the Feynman diagram shown in Figure 1.1b and defines the standard Lorentz In- variants, q = k — k’ (1.2) V = p - q/M. (1.3) The effective Lagrangian for Figure 1.1 can be written as, L.“ = -%Ill‘ +h-c. (1.4) where, 1* =1? +13. (15) In Equation 1.5, we break the charge—diarge current into lepton and hadron pieces. The lepton piece I f shows the standard V-A behavior of the Weak Interaction with the form, I," =V,y‘(1—75)e+Vuy‘(1-ys)p+... (1.6) One can then calculate the amplitude of the interaction as, 6 1:9)=§§E(k'>r.(1-tweets). (1.7) The differential cross section can then be calculated from the amplitude, T: V) as, do”) = 1 1 1 dak’ d3P’ " v 2M 213 (2n)32k(, (27:)3210o 1 X52 spins If") 2(2n)‘ 6(k + p — k’ — P’). (1.8) which we can then simplify into the expression, do" GE. , , = lafiw dlqzldv 32 7:152 “’3 (1.9) Now the structure of the interaction is contained in the two matrices, 1w,3 and Wm. The lepton matrix, 104,, is determined by the Weak Interaction and has the form, 1 a, = 8{k,,k;, + k;k,, — k . k’gap + ieamk’7k5}. (1.10) The hadron matrix, Wm3 is much less well defined. There is no theoretical basis to dictate the form of Wu”, so one Chooses the most general form possible: wanii’r‘l) = —W, gap + W2 Paps/M2 ’ iW3 eaprapaqy/Mz +Wiqaqp/M2 +Ws(p..qp +anfi)/M2 (1.11) +i W6 (mp - 4..th )/M2, Where the Wi are Lorentz-invariant functions of V and q2, commonly called structure functions. Now, we can calculate the cross section, do" GHE’ . 9 29 (E+E’) . 26 = —— 2w —+w —+— —w . 1.12 cllqzldv ZnKEI ‘sz 2mg 2 M 5m 2 3 ( ) The W4, W5, and W6 are zero in the limit that ml = 0. In the anti-neutrino case, sign of the W3 term is reversed. To make the conversion to a modern notation based on the concepts of scaling and the parton model, we make the following redefinitions of the structure functions, 7 M W1(V,Q2)E F,(x) (1.13) VW2( v,Q’) a F2(x) (1.14) VW3(V,Q2)5F3(x) (1.15) where, Q2 = -42 (1-16) _ 2 2 x a ‘7 = Q (1.17) p - q 2M v This leads to the recasting of the cross section in the form, dZGv,VN G25 dxdy = 2; [yzxF,+(1—y+;‘g)rzt(y—-;-y2)xr3] (1.18) where, 521:}. 1 19 y p . k Ev ( ° ) We can make a final simplification, by using the Callan—Gross relationship (1969), 2xF,(x) = F2 (x) (1.20) givins dZUV'VN G25 dxdy = 2; [(1—y+-§-y2)F2i(y—%y2)xF3]- (121) In Equation 1.21 we have neglected the term 3? because it is small and decreases as the neutrino energy increases. From this modern form, in the next section, we will be able to make direct associations between these new structure functions and the quarks in the quark—parton model. 1.2.2 Quark-Parton Model To account for the baffling array of ”elementary” particles discovered during the years following World War II, Gell-Mann (1964) and Zweig (1964) suggested that one might account for the properties of the observed baryons and mesons by an under- lying structure Of particles which Gell-Mann named quarks. Quarks were thought to be strongly interacting, spin )6, particles of charges —%e and %e, where e is the magnitude 8 of the charge of the electron. The quarks also carried quantum numbers such as isospin and ”strangeness.” The baryons were thought to be combinations of three quarks (or anti-quarks) and mesons combinations of a quark with an anti—quark This Quark Mod- el was successful in providing a framework for examining hadron spectroscopy and explaining the observed resonances but was initially considered more of an accounting scheme thanamodel ofsome structure underlying the seen spectrumofbaryons and mesons. Starting in 1967, a group of physicists from MIT and SLAC, lead by RE. Taylor, ].I. Friedman, and H.W. Kendall, conducted a series of deep inelastic electron-proton and electron—deuterium scattering experiments at SLAC (See review article by Friedman and Kendall, 1972.) The theoretical expectations in 1967 were that the inelastic spectra would decrease rapidly as a function of increasing four-momentum transfer, Q2. Con- trary to the theoretical expectations, is was found that the inelastic cross section had only a weak dependence on (22 beyond the dependence on Q4 dictated by the photon propagator of Quantum Electrodynamics. This independence of the four-momentum transfer has come to be known as ”scaling.” At the end of the sixties, it was suggested by Feynman (1969) and Bjorken and Paschos (1969) that the unexpected scaling seen in the SLAC inelastic electron—proton scattering data was due to the scattering of the electron Off of partons, constituents of the proton. In the parton picture, the momentum of struck hadrons is distributed be- tween the partons that make up the particular hadron. Each parton carries a fraction of the hadron's momentum, x , where, 2 x E L. (1.22) P ‘ ‘l Summing 1: over all the partons yields, finer an) partons With the 1974 discovery of the charm quark in the form of the I/ w, the quark 9 model suddenly went from an accounting mechanism to the model of the underlying structure of hadrons. Quarks were now considered one of the building blocks of had- tons. 1.2.3 Quantum Chromodynamics The Quark Model described above is successful at describing the known had- ronic resonances but at the cost of the Pauli Exclusion Principle. States such as the A“, which is the combination of three spin-up u quarks, violate the Pauli Exclusion Princi- ple. In an effort to maintain the Exclusion Principle, it was proposed that quarks carried an additional quantum number (dubbed ”color.”) To allow states such as the A”, there must be three colors. Results from the measurement of the cross section for e-eJr —> qq show that the cross section is 3 times greater than that expected for color-less quarks from the measurement of the cross section for e-e+ —) iii-1'. Thus color is real degree of freedom for quarks that does not exist for leptons. Quantum Chromodynamics de- scribes (as its name implies) the interactions of colored particles. Quantum Chromodynamics (QCD) is a theory of strong interactions. There are a number of similarities between Quantum Electrodynamics (QED) and QCD. In QED, the force is mediated by photons. Whereas, in QCD, the strong force is mediated by bosons call gluons. In QED, photons couple only to charged particles. In QCD, the quarks carry color which is similar to charge in that gluons only interact with colored particles. At this point, the two theories, QED and QCD, begin to diverge. Figure 1.2 shows some primitive Feynman vertices from QED and QCD. It is possi- ble to reduce all Feynman diagrams a combination of these primitive vertices. Figure 1.2a shows the basic vertex of QED, a charged particle emits (or absorbs) a photon (either real or virtual) and continues on. From this basic vertex, one constructs all the diagrams of QED. Since the photon is uncharged, the charge of the scattered particle remains the same. figure 1.2b shows a similar diagram for QCD. In this vertex, a colored particle (a electron—photon b >4W quark—gluon c : 3-gluon 4'81”)" Figure 1.2. Primitive Vertices for QED and QCD. Shown are the vertices from Whid‘l all Feynman Diagrams for QED and QCD can be constructed. Diagram (a) shows the only type of vertex in QED, the interaction of a photon with a charged particle such as an electron. Diagram (b) shows a similar vertex in QCD, the interaction of a colored quark with a gluon. Diagrams (C) show vertices unique to QCD. The 3-gluon and 4-gluon vertices are a result of the fact that gluons are colored. ll quark) emits or absorbs a gluon. The difference between this vertex and and the QED vertex is that while the photon is uncharged, the gluon is color ”charged” or colored (or more accurately bicolored). So in Figure 1.2b, for example, a blue quark emits a blue- E gluon and continues on as a red quark. The fact that gluons are bicolored leads to gluon—gluon interactions which have no analog in QED. The gluon—gluon interactions lead to two additional primitive vertices beyond Figure 1.2b which are shown in Figure 1.2c. The self-interaction of the gluons leads to very different properties for QCD when compared with QED. At low energies, the strong coupling constant, as, is of order unity whereas a = K37 so the perturbative techniques used in QED were thought to be unus- able. The self-interaction diagrams of QCD cause a screening effect that makes as shrink as a function of the 4-momentum transfer, Q2. This ”running” of the coupling constant means that at high energies as is small enough to apply perturbative techniques. There is an additional property of QCD called confinement Confinement is the experimental fact that one never observes colored particles. This is presumably because as one tries to separate two quarks, the energy required to separate the two quarks is more than is required to produce a qq pair. While QCD describes how the quarks areheld together inahadron, QCD makesno predictions about I distributions of the constituent partons of a hadron. QCD does make explicit predictions about the how the parton distributions evolve with Q2. The deter- mination of the parton distributions is left to experimenters. 1.2.4 Neutrino-Nucleon Scattering From a Quark-Patton Model Perspective. As a beginning, let us calculate the differential cross section for the process, v“ (k) + e(p) -> 11(k’) + ve(p’). (1.24) The Feynman Diagram for the process is shown in Figure 1.3. The calculation of the cross section for vue scattering is straight forward because all of the particles involved 12 vILL 11- Figure 1.3. Feynman Diagrams For vue Scattering. The figure shows the Feyn- man Diagram for vue scattering. The incoming muon neutrino, v“, emits a virtual W+ becoming the outgoing muon, 11'. The W+ interacts with the electron, e. which becomes an electron neutrino, ve. 13 are point-like. The cross section for this process is simply, 2 §£=§L5 (1.25) dy It This is almost the same diagram as for neutrino—quark scattering. (See Figure 1.4.) For free quarks, the differential cross section would be that shown in Equation 1.25. In the case of neutrino—anti-quark scattering, there is the additional complication of the helic- ity suppression of the interaction. This yields (for free quarks), _ 1.2 dy 7r ( 6) The above cross sections ignore the fact that one never scatters Off of free quarks. The struck quark is always confined within a nucleon We can extend the cross section formulas accounting for the fact that the struck quark does not carry the full momen- tum of the nucleon. The first addition necessary to account for confinement of the quark is that since summfisqmm one must use sqwk. One Can calculate squad“ s...... E (k + 71...... )2 a -2k - 12...... = -2k - mm... = xs......., (1.27) and then substitute squad. (or xs) for the s in Equations 1.25 or 1.26. The probability that a quark carries momentum xp must also be included in the cross section. One can define a parton distribution function (PDF) such that the probability that a quark carries momen- tum xp is q(x)dx. This gives the double differential cross sections in term Of the parton distribution functions: d’O’“1 _ Gfis dxdy — ” xq(x) (1.28) dZO-vfi (3:5 2 _ = 1— . My ” ( y) xq(x) (129) From Equations 1.28 and 1.29 and charge conservation, one can construct the double differential cross section for neutrino—proton (neutron) or anti-neutrino—proton scattering as shown, 14 Figure 1.4. Feynman Diagrams For vuq Scattering. The figure shows the Feyn- man Diagram for vuq scattering. The incoming neutrino, v", emits a virtual W becoming the outgoing muon, u. The W then interacts with the quark, q resulting in the outgoing quark, q’. 15 :12";- 6,?er u dzov” _ Gfis dxdy — 7: x[( where u, d, s, and c are the parton distribution functions for the up, down, strange and 1 - y)2 d(x) + u(x) + (1 - y)2 §(x) + c(x)] (1.31) charm quarks respectively and E, d, '5, and C are the PDFS for the anti-quarks. Equations 1.30 and 1.31 apply equally well for the neutron if one uses the PDFS for the neutron It is customary to treat the proton and neutron as an isospin doublet and to assume that what one writes the PDF d(x), one means dm(x) which is the same as umm(x). The non-valence distributions are assumed to be identical for the proton and the neutron For this thesis, it will be useful to consider the neutrino—nucleon cross section 8y nucleon, one refers to the average of the proton and neutron cross sections. The FMMF target is almost isoscalar, i.e. has equal number of protons and neutron, so the target approximates a nucleon target. Using the above convention of the PDFS, one finds the differential neutrino—nucleon cross section is, dzo'" 1 {£1de + dzow} dxdy 2 dxdy dxdy =%§x{u(x) + d(x) + 2(1 — y)2 fi(x) + 25(x) + 2(1 - y)2 C(x)}, (132) = %;Lsx{u(x) + d(x) + 5( x) + §(x) + (1 - y)2 [Ii(x) + d(x) + C(x) + E(x)]} Where we have used the relationships that fi(x) = d(x), s(x) = 's'(x), and C(x) = “d(x). Next, we regroup the PDFS, giving, ‘52:: z9534(1—y+iy2)[u(x)+d(x)+s(x)+c(x)+fi(x>+3(x)+§(x>+fi(x>l — %y2)[u(x) + d(x) + 25(x) — not) — an) — 2c(x)]} (1.33) Comparing Equation 1.33 with Equation 1.21, one can make the immediate correspon- dence between the quark distributions and the structure functions F; and xF3. For F2, one finds, 16 F,“ = xu+ xd+ xs+ xc+ xTi+ xd+ x§+ x6 = xq+ xq = F27”, (1.34) where q(x) and §(x) are now the sums of the quark and anti-quark distributions. Equa- tion 134 shows that F; is the same for neutrinos and anti-neutrinos and that F2 can be regarded as the sum of the quark and anti-quark distributions. For xFa, the correspondence between the quark distributions and xF3 differ be- tween the cases of vN and VN scattering. Comparison between Equations 121 and 1.33 gives, xF3"N = xu+ xd+ 2xs- xfi— xd— 2x6 z xq— xq. (1.35) for the case of vN scattering. The same analysis for VN scattering shows, foN = xu+ xd+ 2xc- xfi— xd— 2x5 z xq— xq. (1.36) To first-order, the structure functions are the same but when one includes the strange and charm sea in the calculation, one finds that, xF3"N at xFSW . (1.37) It will be necessary to correct for the differences in xF3, when one calculates F2. 1.2.5 Structure Function Evolution In the parton model, the parton distribution and structure function are constants as a function of Q2. QCD makes no prediction about the form of the quark or parton distributions but it does predict that the parton distributions are not just functions of x but are functions of Q2. The prediction of QCD goes beyond simply saying that the parton distributions are functions of Q2, it quantitatively predicts the dependence of the parton distributions on Q2. From the evolution of the the parton distributions, one can measure the parameter, AQCD. ' The vertices shown in Figure 1.2 result in the quark and gluon distributions changing as a function of Q2. The higher the Q2, the shorter the distances probed and the higher the probability that one will separately resolve the quark and the soft gluon it 17 has emitted. Thus as Q2 grows, the average 1: of the struck parton decreases and the parton distributions and structure functions evolve with Q2. QCD makes a quantitative prediction about the Q2 evolution of the structure functions and parton distributions. The evolution of the structure functions is described by the Altarelli—Parisi equations (Altarelli and Parisi 1977): dFNS angzz l d I. 2 2 d1°8Q2= 21: £72quth )F~s(Z'Q )1 (Laser) (“:5 a. Q2 le 2 2 2 2 dlogQa = 2(7r ) £7[qu(§,(2 )Fs(z,Q )+ch(§,Q )G(z,Q )] (1.38b) dG a, Q2 1d 2 2 2 2 dlogQ2 = 2(,r )I‘flpcctz‘rQ )G(z,Q )+Peq(%,Q )Fs(z,Q )] (1.38c) where FNS is the non-singlet structure function, F5 is the singlet structure function and G is the gluon structure function. FNS corresponds to the valence quark distributions which is simply xF3. F5 corresponds to the valence and sea quark distributions which is F2. PW ch, and Pa; are ”splitting functions” and have the form. The splitting functions describe the two contributions to the Q2 evolution of the structure function (or parton distributions) at a given x0: 0 A higher 3: quark emits a gluon and now carries momentum xop. 0 A quark carrying momentum xop emits a gluon and now is a lower x quark These contributions result in the number of low 1: partons growing as a function of (22 while the number of high x partons shrinks. As we have seen the structure functions are simply the sum and differences of parton distributions and thus exhibit the same (22 behavior as the parton distributions. To examine the evolution of FNs, one must first regularize the Altarelli-Parisi Equation 1.38a. The regularization yields: 18 31: 813N_S(x,t)= a,(t) at [3 + 4log(1 - x)]FNS(x,t) + 21:}ng - z2 )FNS(x/z,t) - 2FN5(x,t)], (1.38) where t = log(Q2 / A2 ). Now the evolution of the non-singlet structure function is a direct function of t. Using Equation 1.38, it is possible to measure AQCD. 1.3 W The thesis reports on the measurement of Neutrino—Nucleon Structure Func- tions by a collaboration of Michigan State University, Fermi National Accelerator Laboratory (Fermilab), Massachusetts Institute of Technology and the University of Florida (the FMMF collaboration) in a series of experiments conducted at Fermilab during the 19805. The FMMF collaboration is a group of approximately 40 physicists. The collabora- tors are listed in Appendix A. Fermilab is a United States Govemment laboratory located in Batavia, Illinois, approximately 45 kilometers west of Chicago. Fermilab was constructed in the late 19605. The main purpose of Fermilab is the operation of a Proton-Synchrotron which provides high energy particle beams for use by elementary particle physics experi- ments. By the beginning of the 19805, the Fermilab Main Ring provided 400 GeV protons to experimental areas. Between 1982 and 1984, the accelerator complex at Ferm- ilab was upgraded by the installation of a new Proton-Synchrotron constructed of high field superconducting magnets. The new accelerator (dubbed the TeVatron) is located in the same tunnel as the older Main Ring and is capable of delivering 800 GeV protons to the Fixed Target Areas. In the late 19705, the FMMF collaboration constructed a detector at Fermilab which consisted of an extremely fine grained target—calorimeter and a muon spectrometer. From 1980 to 1988 the detector was exposed to neutrino beams created using protons 19 extracted from Fermilab’s Main Ring and Tevatron. The series of experiments are de- scribed in detail in Chapter 2 Chapter 2 continues with a description of how individual evarts are measured. Chapter 3 describes the Monte Carlo used in the correction of the data for acceptance and smearing. Chapter 4 discusses the extraction of the double dif- 2 ferential cross sections, -d—O;, and the structure functions, F; and xF3, and presents the measured cross sectionsdzfrzl structure functions. Chapter 5 compares the FMMF struc- ture functions presented in Chapter 4 with those measured by other DIS experiments. In Chapter 5, a measurement of A00, will also be discussed. A number of Appendices are also included at the end of the thesis to provide more detailed explanation of the analy- sis used in this thesis. Chapter 2 The Experiment 2.1 Intmdrrstinn The FMMF detector was located in Lab C at Fermi National Accelerator laborato- ry. [ab C is located at the end of the Neutrino Center beam line. Neutrino Center serviced experimental areas for up to 4 experiments, all of which could be run simultaneously. The FMMF detector was exposed to neutrino beams during 3 separate periods, in 1982, 1985 and 1987-88 (referred to as 1987). The data from each of these runs is included in this thesis. This chapter initially describes the individual exposures, briefly describing the neutrino beam characteristics, the trigger (or triggers) used and any differences in the detector instrumentation. The neutrino beams and the FMMF detector are then dis- cussed in detail. A discussion of event measurement follows. The chapter concludes with a discussion of the final data sample. This thesis covers data taken over six years and an experimental effort (including construction, data taking and analysis) that required more than ten years. Appendix A gives the names and affiliation of all of the scientists involved in the building, the oper- ation and / or the analysis of the data taken using the FMMF detector. 2.2 W This section summarizes the three exposures, including the neutrino beams and differences in the instrumentation of the FMMF detector. 20 21 2.2.1 1982, Experiment 594. During 1982, the FMMF detector was exposed to a narrow band DiChromatic beam. The DiChromatic beam is described in detail below. For the DiChromatic beam, 400 GeV protom were used to create a secondary beam of pions (1r’ 5) and kaons (K’s) which were then sign and momentum selected and allowed to decay into a beam of neutrinos that passed through the FMMF detector. The momentum selection resulted in a strong correlation between neutrino energy and beam radius. For this thesis, data from 4 sec- ondary momentum settings were used, —165 GeV/c, +165 GeV/c, +200 GeV/c and +250 GeV/c. When positive (negative) secondaries were selected, a beam of neutrinos (anti-neutrinos) was produced. The standard trigger used in 1982 for the FMMF detector is based on energy depos- ited in the target-calorimeter. This trigger was 50% efficient for a 5 GeV energy deposition and fully efficient for 10 GeV. This standard trigger is referred to as the P111 trigger. During the 1982 running, the muon spectrometer was instrumented using a charge division read-out scheme. 2.2.2 1985, Experiment 733, Part 1. In 1985, the FMMF detector was exposed to a wide band beam produced with 800 GeV protons from the Tevatron. The beam is described as the Quad-Triplet beam (am) because of its optics and it is also described below. The primary feature of this beam is its high energy (due to the use of 800 GeV incident protons) and the lack of secondary momentum selection, which results in a large neutrino flux and a beam of neutrinos and anti-neutrinos. In 1985, a much more complicated trigger scheme was used. The goal of the new triggering scheme was to maximize the number of dimuon and the high Q2 events observed while attempting to maintain as large a minimum bias set as possible. To supplement the PIT-l trigger, two special triggers designed to enhance the number of 22 dimuon and high (22 events were built. To accommodate competition among the rare process triggers while ensuring at least one trigger per spill, the neutrino spill was divid- ed into two parts, a ”beam gate" and "tail gate” (See Figure 21.) The beam gate was the first part of the neutrino spill and was devoted to the triggers designed to isolate special physics signals. The tail gate followed the beam gate was devoted to the minimal trigger with the purpose of maximizing the amount of data taken over the course of the run. The first of these special triggers was the HiE trigger. The HiE trigger was designed to enhance the sample of high (22 events. The HiE trigger was simply the P'I'H trigger with a higher threshold. The HiE trigger was fully efficient for 100 GeV energy depositions. Figure 2.2 shows the measured trigger threshold of the HiE trigger. The second of these special triggers was the 2MU trigger. It was designed to err- hance the sample of events with two (or more) muons. The 2MU trigger was also based on the PIT-I trigger. When a PIT-I trigger was observed, the spectrometer planes were inter- rogated as to how many hits were recorded. It was required that 3 out of the 8 spectrometer planes have more than two hits. Because of the time required to interro- gate the spectrometer planes, the 2MU trigger was delayed relative to the PIT-I and HiE triggers. The 2MU trigger events were not used in this thesis because the acceptance of the 2MU trigger was not well understood. In addition to the previously mentioned triggers, there was a Quasi-Elastic trig- ger. The Quasi-Elastic trigger events were not used in this thesis. The beam gate was designed for use with the special triggers that enhanced the dimuon and high Q2 signals. To this end, during the beam gate, any HiE or 2MU trigger was taken (assuming that another trigger had not been honored previously). The PIT-i trigger was prescaled by a factor of ten or eleven (i.e. only every tenth or every eleventh PTH trigger was taken.) The Quasi-Elastic trigger was also prescaled by three. This trig- gering scheme resulted in approximately equal live-times for all four triggers during the beam gate. 23 mtenszty J. Neutrino Spill 0}? Beam Gate on Ofi‘ Tail Gate on Figure 2.]. Neutrino Gate Structure. Shown is a schematic of the neutrino beam spill and the neutrino gates. The top graph shows the intensity of neutrinos as a function of time. The middle graph shows the ”beam gate.” The bottom graph shows the ”tail gate.” The width of the neutrino spill was 2—3 millisec- onds. The beam gate covers the first part of the neutrino spill while the tail gate involves the last part of the neutrino spill. During the beam gate, the triggers for rare processes were given preference. If no trigger was taken dur- ing the beam gate, any available trigger was taken during the tail gate. This scheme was adopted to maximize the number of rare triggers taken while still maintaining a trigger rate of 1 trigger per neutrino spill. 24 1.0 ‘ Hit t {+++{H _ Ht Hi - t it 1* 1 0.8; l : i 8‘ 0.6— r s i h be 0.4- g i 0.2; 0.0. r+lrtrrlrrrrl_rttrlrrrrlt O 40 80 120 160 200 240 Energy Deposited in Calorimeter (GeV) Figure 2.2. HiE Trigger Turnon The threshold of the HiE trigger as a function of energy deposited in the calorimeter is shown as measured in the data. A fit to the turnon is also shown. 25 The tail gate was designed to maximize the size of the complete data set by en- suring that at least one trigger was taken during every neutrino spill. During the tail gate, the PIT-I trigger was no longer prescaled, which allowed any minimum bias event that occurredinthe tail gate tobe taken The experimenters adjusted the relative widths of the beam and tail gates, based on neutrino beam intensity, to maximize the number of rare triggers while maintaining a high event to spill ratio. Between the 1982 and 1985 exposures, the muon spectrometer was upgraded from the previous charge division read-out to a drift read-out. This change substantially improved the position resolution of the spectrometer. For the higher energy muons produced by the Tevatron Quad-Triplet beam, the improved position resolution was essential. 2.2.3 1987, Experiment 73, Part 2. From April 1987 through the end of January 1988, the FMMF detector was again exposed to neutrinos from the Quad-Triplet beam. During this exposure, the triggering scheme was much simpler than in 1985. As in 1982, the PIT-l trigger became the primary trigger. The prescaling of the P'I'H trigger was eliminated along with the 2MU and HiE triggers. A new second trigger, the cc trigger, was added. The CC trigger was designed to capture charged current events. The CC trigger required that a muon penetrate into the spectrometer, traverse 205 cm of iron and hit the second timing counter located in the fourth gap of the spectrometer. (See Figure 2.7) Approximately half of the data for this thesis came from this exposure. 2.3 Beams The two separate neutrino beams to which the FMMF detector was exposed dif- fered dramatically. The DiChromatic beam from the 400 GeV era at Fermilab was a low flux, relatively low energy neutrino beam with a very well defined neutrino energy (E V) 26 versus radius (R) relationship. The Quad-Triplet beam was a high energy, high flux neutrino beam with much less correlation between E, and R. 2.3.1 DiChromatic Narrow Band Beam The DiChromatic beam is described in great detail elsewhere (Edwards and Sci- ulli 1976). 400 GeV protons were incident on a one interaction length beryllium-oxide target. The secondary it’s and K’s were momentum-selected by the helical DiChromatic magnet train. Finally, the secondaries were allowed to decay in a 300 m evacuated decay space. To reduce the background of wide band neutrinos, the beam is designed so that the momentum vector of the secondaries never pointed in the direction of the detector until the secondaries reached the decay space. A schematic of the beam is shown in Figure 2.3. The momentum-selection of the DiChromatic train results in a secondary mo- mentum spread of approximately 10%. This small momentum bite and the kinematics of the dominant two-body decays of the secondary mesons gives the beam its well de- fined E., versus R relation. Figure 2.4 shows a scatter plot of Ev versus R for accepted charged current events at one of the neutrino settings. One can see the two well defined narrow bands of neutrino energy versus radius corresponding to decays from 1t’s and K’s. This two parent structure is typical of neutrino beams. What is not typical of other wide band neutrino beams is how strongly correlated E V and R are in the DiChromatic beam The penalty one pays for this excellent Ev versus R relationship is an order of magnitude reduction in neutrino flux. 2.3.2 Quad-Triplet Wide Band Beam The Quad-Triplet beam contrasts markedly from the DiChromatic beam. The Tevatron qu has a higher mean energy, much higher maximum energy and much high- er instantaneous flux. There is no attempt to select the sign or momentum of the secondaries. The lack of sign selection results in a beam that contains both neutrinos and 27 .638 2 5: mm 9:380 .88me E St E 6825 85 29? 83.8668 8m 8:532 m3 mafia—m8 on... mo 38 < $88 9 332? can €3.36 .8 rug—om 835808 can :wfi 55 8a 533 MM 98 we: warmed «owns s so .835 8c 898 >eo 8m .3933 acme 656865 .3 25mm 888% 9W0; "~25 _ ,, 5c 5.3%.me fight». 28 § VIIIIUII‘TU 280 240 , Accepted Events N 8 LIL llll JLL 1 00 200 3CIO O O 1.,UiiilroiTrU]IIIU'TYTI'I‘IIITT A . . > 200 . Neutrmo Energy (GeV) 6 '- t - s“. ..o' : ‘3 n - - . ' \r _ .~ I. e .-; ~ '.~ - 2;; :: ‘. . 1 . 160 ‘ -- -:-:< :2;- i .-..='r-'; ."-'{ cu“";\"=- s-“-'33.‘r"‘.= . ’1 1 - .v - I‘. "' 2.; I: .2. u ‘ ::"§_-‘ ' : .I "' I 3.x . 1"." 'i . "I: s z." m ‘ ’ 1‘ :‘i? ' it? ‘31". J ‘2‘ ‘3 ..s ‘ -". ‘. " x t... ":: .- . V. :‘ s; ”‘5 .3. I. .- g . _. ‘ 3s; :2 ‘n. .a; '1: - 120 s - . -' =. a ' .r '.‘_ .'.' 2 ,.:.'.-;:.-. :.=r:.:-. .: 80 :.?‘-’-.-;~.; --.=.,'..t;;-~ v-fl-r .e-s-c - .;;¢;&~;rnpngc J . -:.. 's‘grt..€l\}'§ $‘$§t’- . O.-.f‘. . ’I-"z. .k 1' \ J it . . A. .l'l'" t w ..r .s,‘ “-‘r '0’ If ’ a‘ u- \ . ‘0‘. 5 y -\ fl. 40 "" *’ '~ ; l“ - . ‘ p l . .'~ n I .’ I U I I 1 Y I 0 llllllllllljlllllllllllLllLLLLlLLJlllll 25 50 75 1 00 125 150 175 200 Beam Radius (cm) 0 Figure 2.4. Neutrino Energy Versus Beam Radius for DiChromatic Beam. Shown is a scatter plot of reconstructed neutrino energy versus beam radius for ac- cepted events from the data for the +165 setting. One can note the two bands from pion and kaon decay. The inset in the upper right-hand corner shows the projection of the neutrino energy distribution. 29 WE are a so: a 9:55 ceases "SE 9.: S 6835 55 83a Cancun 8m Edema @338." m5 mo 26* < $83 9 3.32? can U838 :65 was 523 MM can we 95me “swam“ a so can? as 228a >8 8m geranium :88 535. 28385 .3 25mm HOHUOuO—u >90 cow 988nm \\\\ \\\\ mummm >88 «\\\\‘\V tx\\\\-~ mm 30 anti-neutrinos, with integral anti-neutrino flux being approximately 30% of the total flux. The wide range in momenta of the secondaries broadens the E Vversus R bands for each of the parents. Details on this beam are available elsewhere (Stutte 1985). As one can see in Figure 2.5, the beam is extremely simple. 800 GeV protons were pointed at the neutrino detectors and then struck a one interaction length beryllium-oxide target. The secondaries are then focused in a point-to-parallel fashion by 4 sets of 2" x 2" quadru- poles. The focusing is optimized for a secondary momentum of 300 GeV/c. An example of the EV versus R relationship for the Quad-Triplet beam is showed in Figure 2.6. 2.4 Detector The FMMF detector is described in great detail elsewhere (Bogert et al. 1982; Tart- aglia 1984; Brock et a1. 1992; Strongin 1988). A schematic of the detector is shown in Figure 17. The detector was composed of a target-calorimeter followed by a muon spec- trometer. A veto was used at the upstream end of the detector to eliminate non—neutrino background in triggering. This section will briefly describe the parts of the FMMF detector relevant to this thesis. Differences between the detector configuration during different exposures will also be noted. 2.4.1 Front Veto Configuration During all three exposures, a veto system was used at the front of the detector to reject events caused by incoming charged particles. The most common cause of these charged particles was the interaction of neutrinos in the material (or other detectors) upstream of the PM detector. The veto was then made part of the trigger (i.e. TRIGGER = TRIGGER CONDITION . VIE—E). During the 1982 run, a set of liquid scintillator tanks with photo-multiplier tubes (mr’s) for read-out was used as a veto wall. Due to the low instantaneous rate of the DiChromatic beam, the activity in this veto configuration was minimal. For the 1985 run, a single veto wall of 8 5'x8'x1" acrylic scintillator counters with 31 500 ' ?§ I ~ 200. E ~ Loo L Accepted Events L. L .L L L._l j l 1 . , . 200 400 Neutrino Energy (GeV) 1 I 1 Co." Q int O :> ,. . :- '_'- . . . . -. ‘. ~ ~’ . ' ' . ’- \a 300 — - . a L ' a. .'_ .2 ‘. . ~ . \ .. _ . .- . r . ‘2'. D. -. .1. '. .’ v. .. I, . . : ‘ ‘l ; IL: .- 0 . b s , . - :;z.- ',. 'f. f t. . 3‘; .‘ V. ~' . L“. .- '- s ‘ '*. v .' '1‘. x " - P . ° ' .' v ) ‘ ' P . . ' " I . ' .' n u o . ' - ~_ '- - I I- .- l. .' - .. 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I- .u. c' . - .._I'- "'3"; " .f'. -'_-.-' . * -:’--".?;’J.‘~‘o"~ *st..,-.--. ~.-.-:~..- s "'3 .c. ‘ 0 " '. e ‘3 I'D. -. "'f. ‘.‘ -'- ,- . ~— .- I: ~ ..--- a err-"91; ,2: .. .. _ - q .. .-,_-:'-_r. v. ,u, ~ . ' .a--T,.'.\" V. .-"-.; :r - 7'. p,.'. .4 ."..$-‘ .. ”o - .‘ ' . . - . ’. 0" .. ' , Q' - 7. c' ' - " r s ' ' - 0 llllllllllllllllllllJllllllll'llllllLlL O 25 50 75 100 125 150 175 200 Beam Radius (cm) Figure 2.6. Neutrino Energy Versus Beam Radius for Quadrupole-Triplet Beam. Shown is a scatter plot of reconstructed neutrino energy versus beam radius for accepted events from a portion of the 1987 data. Note that the two bands from pion and kaon decay are still present but the bands are much less dis- tinct when compared to the DiChromatic beam. The inset in the upper ri t-hand corner shows the projection of the neutrino energy distribution. 32 Spectrometer a E E Target Calorimeter 5 : a; : I r I E ZOmeters 3 Z: : Z I ' E I E = l ' :‘S I :i' t = I ' C: I :.' = I ' L I L; = l : .2: I :. = I —- m = I ' J l E“ = l ' T‘ I .f: = ' ' ml l ”I = I ' 1:. I 11 "-" I ' I i: i = ' ' l i - + F3 i 3i Calorimeter E; 1 11 Drift % I Stations Je r Mule of Target Calorimeter \ 24 Foot q III F H F W Tomidal _. Ma ets 8 “ . . -. 3 :5; i T . . 4: E U I. E 1 I U "' *1 r I S '" g L E I I; : g I: j d l l g g '43 592 Flash Chambers a I L . H L E 37 Proportional Planes E I a r l J 1 H E E 340 Tons of Target Material —l H I H J n i J I H i1 a I. L i 8 - 24’x24' Drift Planes 8 - 12’x12’ Drift Planes c; >4 '< N 1 HI IIJ XUXYXUXYX D Flash Chamber \V Sand Absorber Plane Steel Shot Aborber Plane Figure 2.7. FMMF Detector Schematic. The construction of the FMMF detector is shownThe upstream end of the detector consists of the target—calorimeter with its two types of detectors: proportional planes, and flash chambers. Downstream of the target-calorimeter is the muon spectrometer with its toroi- dal magnets, and its position measurement stations. This figure shows the detector in its 1987 configuration. 33 wave-shifter to PMT read-out was built. Within the first few minutes of running, it was discovered that the single veto wall was insufficient. The on: created a ”sky-shine” of soft neutrons that caused an extremely high incoherent singles rate resulting in a 100% dead-time. To resolve this problem, a new veto was quickly assembled using the first two calorimeter proportional planes and an existing set of liquid scintillator tanks that formed a plane located between bays 1 and 2. The proportional planes and liquid scintil- lator plane were run in coincidence to form an effective veto while avoiding the dead time problems of the single wall. Any interaction in Bay 1 was vetoed, eliminating Bay 1 from the fiducial volume of the detector. For the 1987 run, a set of double acrylic scintillator walls was constructed at Mich- igan State University. These walls were constructed in a modular fashion and hung in the front of the detector on a structure of UNIsrRU'rW. The two separate walls were then run in coincidence to provide an effective low rate-veto. The new double veto wall pre- formed flawlessly. 2.4.2 Target-Calorimeter The target-calorimeter served three purposes: 0 Target material for neutrino interactions. Tracker for measuring the paths of outgoing muons. 0 Calorimeter for measuring the energy transferred to the nucleon sys- tem by the neutrino interactions. The target-calorimeter was built from 2 types of detectors and 2 different target-absorb- er planes. The detectors used in the calorimeter were 592 (608, in 1982) roughly 4.5 m x 4.5 m flash chambers in 3 views and 37 12' x 12' proportional tube planes in 2 views. The flash chambers are binary devices made of corrugated plastic which provided extreme- ly fine position resolution for single particle tracking and calorimetry information. The proportional planes were standard gas-wire pr0portional devices used for both the trig- gering of the detector and in calorimetry. The target-absorber planes were constructed 34 of Plexiglass tubes filled with either sand or steel shot. The target-calorimeter was con- structed in 37 sub-units called modules. The standard module (see Figure 2.7) consisted of 4 ”beams” of flash chamber detectors and absorber followed by a proportional plane. A beam consisted of a flash chamber followed by a sand absorber plane followed by a second flash chamber followed by a steel shot absorber plane. The pattern was repeated so that the beam consisted of 4 flash chambers and 4 absorber planes. The complete module contained 16 flash chambers, 8 sand absorber planes, 8 steel shot absorber planes and 1 proportional tube plane. The total target-calorimeter had a mass of approximately 300 metric tons with about 100 metric tons in the fiducial region The density of the calorimeter was 1.35 g/ 0113. Table 2.1 gives a complete list of the target-calorimeter properties. 2.4.2.1 Flash Chambers The flash chambers provide the extremely fine sampling and segmentation of the FMMF detector. There were 592 (608 in 1982) flash chambers sampling every 0.2 radiation lengths and 0.04 interaction lengths. Many of the details of the construction and operation of the flash chamber are ignored or only superficially covered in this section. The reader desiring more detail is directed to the references mentioned above or the theses of LA Slate (1985) or (3.]. Perkins (1992). The flash chambers were constructed of approximately 4.5 m long pieces of cor- rugated polypropylene with cells running the length of the corrugated sheet. Figure 2.8 shows a schematic of an individual flash chamber. Flash chamber cells were 5 mm x 5 mm x 4.5 m. The cells were filled with a gas mixture of 90% Helium and 10% Neon. Aluminum foil electrodes covered both sides of each chamber. When an interaction was detected in the detector and a trigger was executed, an approximately 4.5 kV pulse was delivered to one electrode in approximately 60 nanoseconds while the other was main- tained at ground. This was called the ”flash". If there was any ionization left in a cell (by the passage of one or more charged particles), this rapid high voltage pulse resulted in 35 Table 2.1 Calorimeter Properties Detector 1 Flash Proportional Property Average Chamber Plane Sam lin Sam lin Density,p 1.35 g/cm3 4.2 g/cm2 67.5 g/crn2 Radiation Length, X0 14 cm 0.22 X0 3.5 X0 Interaction Length, 3. 85 cm 0.04 A 0.59 A Protons per Nucleus, Z 9.8 — — Neutrons per Nucleus, N 10.4 — — Nucleons per Nucleus, Afi} 20.2 — — Non-isoscalarity, 8 2.97% — - Table 2.1. Average Properties of Target and Detector Sampling. Listed are the average properties of the FMMF target and the sampling fractions of the calo- rimeter detectors. 36 From Hot Plane R1 Bucking Strip Bucking Transformer Copper Pickup Strips (/ 3mmx508mm /' Polypropylene ' Top View - E E E E E 1‘ Ground Bus R2 Magnetostrictive ire _ Magnetostrictive Wand Buck/ing Strip Slde Vlew /10 mils Mylar Ground Bus Polypropylene Spacer J§WDWM , I / :‘ l 508 llllll V: HOYP'ane M S. Wire Phemthed War P°2¥;‘Z?§¥L%".Z§i§23$°' Gas Manifold (pickup region) Figure 2.8. Flash Chamber Schematic. The figure shows the construction of a flash chamber. The pickup region is shown in detail. 37 the formation of a plasma in the cell. The plasma rapidly filled the entire length of the cell. The plasma was detected through the capacitive coupling of the plasma to copper read-out fingers. The entire chamber was read-out using a single magnetostrictive wire which was excited by the current flow in the read-out fingers. The advantage of this type of chamber and read-out system was that large number of cells could be read out (over 700 cells per chamber) using only 2 electronics channels per chamber. The disad- vantage was that one could only detect if ionization was left in a cell or not. There was no way to determine how much ionization was left in a cell (as in an analog device). Be- cause one could detect only whether a cell contained ionization, the flash chamber is often referred to as a digital device or more properly as a binary device. The detection of plasma in a cell is commonly called a ”hit.” The rapid retriggering of the flash chambers was problematic. It was necessary to wait for a few seconds while the residual charge recombined. Any residual ionization left in a cell from the previous trigger would cause that cell to ”re ' ”. The recombina- tion of charge was extremely slow because charge would collect on the walls of cells and the plastic insulator provided no path to ground. During 1982, the flash chambers were not retriggered more often than once every 10 seconds. In 1985 and 1987, it was found that it was possible with the careful control of the electronegative components in the recirculating gas to retrigger as quickly as every 3.5 seconds with minimal refire. A schematic of a module is also shown in Figure 2.7. The chambers were orient- ed in three separate views; X, which provided a vertical sampling, and U and Y, which provided horizontal sampling in two separate views, :10“ from vertical in stereo. The ordering of the flash chambers and absorber planes within a beam was U chamber, sand plane, X chamber, steel shot plane, Y chamber, sand plane, X chamber and steel shot plane. Each module consisted of four beams. In 1982, there were an additional 16 flash chambers which were located at the extreme downstream end of the detector. The first two modules in bay 8 and and the last 38 two in bay 9 were removed between the 1982 and 1985 runs for the installation of a calorimeter drift system which was extremely important in the alignment of the detec- tor, in particular the alignment of the spectrometer. 2.4.2.2 Proportional Planes The 37 proportional planes were positioned between flash chamber modules (16 chambers) and were aligned alternately either horizontally or vertically. The planes were spaced every 3.5 radiation lengths and every 0.59 interaction lengths. The proportional planes were constructed of 1" x 8" x 12' aluminium extrusions. 18 extrusions combined to form a single plane and create an active area of 12' x 12'. Extrusions were divided by thin aluminium webs into approximately 1" square tubes, 12' long. 50 um gold plated wires were strung down the length of the tubes. The tubes were filled with a mixture of 90% Argon and 10% Methane. Positive 1750 V was applied to the wire and the tubes were run in proportional mode. The planes were read out by ganging the signals from 4 tubes together and amplifying the combined signal. This resulted in 36 channels per plane and a lateral sampling of 4". The primary purpose of the proportional planes was for triggering the detector, since the flash chambers are passive devices and require some sort of trigger. The first step in forming a trigger was for the pulse height in any two planes to exceed a very low threshold. Upon observation of this pre-trigger condition, the total pulse heights from all the proportional planes in the fiducial volume were combined and if this pulse height exceeded a pre-determined level, the trigger was satisfied. The standard trigger (known as the PIT-l trigger) discussed above was found by test beam studies to be 50% efficient for 5 GeV of energy deposited in the calorimeter and 100% for 10 GeV. In addition to providing the trigger, in 1985 and 1987, it proved useful to use the proportional planes to augment the flash chamber calorimetry. At the higher energies of the (1113, the flash chambers saturate due to their binary nature, resulting in a constant fractional resolution for higher energies, in contrast the proportional tubes’ fractional 39 resolution improves as the energy deposited increases. Calorirnetrywillbediscussedindetailinalatersectionofthischapterandin great detail in Appendix B. 2.4.3 Spectrometer The muon spectrometer was used to measure the momentum and charge of the muon. The momentum is required to reconstruct the muon energy, and the charge of the muon ”tags” the event as a neutrino or anti-neutrino event. The spectrometer consisted of 7 iron-core toroidal magnets and 4 muon position stations. The magnets were of two types. The spectrometer consisted of the three 24' magnets, followed by the four 12' magnets. The 24' magnets in the upstream part of the spectrometer were 24' in diameter and 60 cm thick while the 12' magnets in the down- stream part of the spectrometer were 12' in diameter and 125 cm thick. The total length in the z (beam) direction of the spectrometer was approximately 1200 cm, of which 680 cm was iron. There were four stations for the measurement of muon position These were located in the gap between the first and second 24' magnets, between the third 24' mag- net and the first 12' magnet, between the second and third 12' magnet and after the last 12' magnet. All the stations were constructed in the same manner as the proportional planes, using an aluminium extrusion similar to that used in proportional planes. The extrusion used contained two layers with cells offset by one half a cell width. Stations consisted of four layers, two measuring horizontal position and two measuring vertical position. The layers with the same orientation in a station were offset from each other by half a wire spacing (0.5 inches). The first two stations were shaped like crosses. These stations were 24' tall and wide but the wings of the crosses were only 12' wide. The last two stations were 12' x 12' planes. As with the proportional planes, 50 pm gold plated wires were strung in the middle of each cell. Positive high voltage was applied to each 40 In 1982, the planes in each station were run in a ”charge division” mode using 90% Argon and 10% Methane, which gave no better than the 1" cell-size spatial resolu- tion. For 1985 and 1987, the planes were run in a drift configuration using 90% Argon and 10% Ethane, which gave a 2 mm resolution. The drift configuration required an additional acrylic scintillator plane for timing of the drift electrons. The two different configurations are described in detail below. 2.4.3.1 Charge Division The charge division scheme is describe in the thesis of Juan Bofill (1984). In the charge division scheme, same ends of 8 (16 for the first station) adjacent wires were resistively connected. The charge was collected by two amplifiers on opposite ends of this resistive network. The more charge collected on a given amplifier, the closer the wire hit is to that amplifier. The hit wire is determined by the ratio Azflr-fh’ q1+42 where q1 and q2 are the charge collected by the two amplifiers. From A, one can deter- mine which wire was hit. This simple scheme results in a single plane resolution of 8.7 mm. 2.4.3.2 Drift In a drift system, a charge particle (in our case, a muon) traverses a cell produc- ing ionization that drifts towards the anode (the wire) or cathode (cell walls) due to the applied electric field. By measuring the length of time for the ionization to reach the anode wire and knowing the drift velocity of the gas, one can determine the distance of closest approach of the charged particle. In the FMMF spectrometer, the time of traversal of the charge particle was determined by the coincidence of two acrylic scintillator planes. The scintillator planes were located in the gap between the second and third 24' toroids and the gap between the first and second 12' toroids. 41 If there were two back to back (separate layers, but adjacent wires) clean, isolat- ed, drifted hits in a given drift station, these hits were combined for a position resolution of 2 mm. This combination was done within the muon fitting process because one needs a rough fit of the charged particle to resolve the left/ right ambiguity. (See Figure 2.9.) 2.4.4 Event Display Figure 210 shows an event display of a charged-current event from the 1987 run. The neutrino beam is incident on the detector from the left. The left side of the display shows the calorimeter elements. The flash chamber hits for the three separate views are shown in the three larger panels. The two views of the proportional planes are shown above and below the flash chamber displays. One can clearly see the neutrino interac- tion vertex, the hadronic shower, and the outgoing muon in both the flash chambers and the proportional planes. The two views of the spectrometer are depicted at the right side of the display. The rectangular outlines show the positions of the toroidal magnets while a hit in the drift system is displayed as ”+." Hits that are grouped together as a cluster of hits are indicated by a surrounding circle. The fit of the muon in the spectrom- eter is indicated by the curved line connecting the spectrometer clusters. One can see how one measures a charged-current event using the FMMF detector. The energy trans- ferred to the hadron system is measured in the calorimeter. The angle through which the lepton was scattered is inferred using the trajectory of the muon in the calorimeter. The energy of the muon is measured in the spectrometer. From these three measure- ments, one can fully reconstruct the event kinematics. The next section discusses, in detail, how one measures an event. 2.5 W To fully reconstruct the kinematics of a deep inelastic neutrino event, one needs to measure three quantities; the energy transfer, v, the energy of the muon (i.e. outgoing lepton) E ,1 and the polar angle of the outgoing muon relative to the incoming neutrino, 42 drift distance / O Figure 2.9. Left/ Right Ambiguity. Drift systems with only two layers suffer from an ambiguity in hit position The drift time corresponds to the closest ap- proach to the sense wire.The figure shows two hit cells and the corresponding ”drift distance” and the four possible trajectories through the two cells which in turn correspond to four different positions at center wall between the two drift layers. The ambiguity is resolved in the fitting process. 43 RLN $26 EVENT 312 ‘ ll . —“O---"OO ah" as...» .. .I ..I - ll l-O U ‘ U . U D I —- ' ' ' ' ' 7 l ' i l l ' ' o ' I . i | o s l -= . ° - .. II - I Iii-lrlllljz.."!j ' - ' I I - = - I I I ' I I ' 1 I I i ' l ' ' ' . ' ' l l 14w! . 1 ‘ _ ll . o .' iuiii. ' " :’;;;’: .i‘ff. “£1.32"; . :a'fistogi‘ .;.:.,s..1joo -~’- 4" I- . 41%....mmeen — r I- -- ~- -- "'111'51:_."1'1‘1:".'1.’31 j ... I' _. " "‘ "' '- ' ‘ . L—I h.— hi I— . ‘ c . ...” t . ; . ...,Egr: - ’. ' .2125)?" _ . . -':-:.g - .32. 71:01 _,.-.-l' #353... ’35 ...u "v ... ..... r.- L. 1 ,3 L 26511” :95, . I. _ Y 3%: “.....firggj - _ l __ ll . ' "-°."-. ' "'1- - ."I' '1 1 ' 31..” - . .' . .0... . - "" "' "‘ .' ~"‘ "F's...” .. ."' "' FT .... "'— "‘ £535 ' " 1' {“323 .. 1...“--.— U .7 ~31.) * gall-1". :- fqtt :’5 FE ’- 1, I— l- I— XI. "L1 .f- _ _ _ :'_' :: C: :: f}. Ln Figure 2.10. Event Display. The figure shows a charged-current event from the 1987 run. The left side of the display shows the displays of the flash chamber hits and the proportional plane pulse height. The right side of the display shows the two views of the spectrometer. One clearly sees the vertex of the neutrino interaction, the resulting hadronic debris, and the track of the out- going muon in the calorimeter. The spectrometer hits of the muon are shown along with the fit of the muon in the spectrometer. 44 9”. From these three ”measurables," one can reconstruct the physics variables: the neu- trino energy (EV), the fraction of the lepton’s energy transferred to the nucleon system (1]), the fraction of the nucleon’s momentum carried by the struck parton (x) and the square of the four-momentum transfer (Q2). To reconstruct the kinematics of a deep inelastic event, one must determine any three of the physics variables. In practice, one always reconstructs all four and then uses the appropriate variable for a given analysis. These four variables are discussed in detail in chapter 1. Reconstruction of E. and y is straight forward. By definition, E V = v + Eli and y = 5%. For reconstruction of Q2, one uses the relationship, Q2 s 415.15,, safe a” ), which neglects the muon mass. Finally, x is reconstructed by the relationship, 2 x= Q , 2Mv where M is the mass of the struck nucleon and we have assumed the nucleon is at rest. The analysis chain for this thesis involves a number of preliminary steps before the event can be "measured”. A flow chart for this analysis chain is shown in Figure 2.11. The first of these preliminary steps is to find and reconstruct the event interaction vertex. Once the event vertex is found, a search light algorithm is employed to find muon candidates. The calorimeter flash chamber tracks of the muon candidates are then fit to a straight line. The fit to the calorimeter track gives the muon angle. The calorimeter tracks are also used as the input to the fitting of the muon in the spectrometer. The fit in the spectrometer measures the muon energy and charge. The calorimeter is then used to measure the energy transferred to the nucleon system, v. The next sections discuss vertex finding, muon finding and the techniques used to measure Eu, 6,, and v. Start Event Analysis Current Event Try and Fit Muon M UDRV package [ICCQ II ‘ Event Determine Fit Charged Current Shower Event - Energy l 1 End Event Analysis Figure 2.11. Event Analysis Diagram. The figure shows the event analysis chain in the form of a flow chart. Included are all of the required branches. Optional branches such choice of fiducial volume are ignored. 46 2.5.1 Vertex Finding. The vertex finding algorithm is based on energy deposited in the calorimeter. The algorithm starts by searching for a hadron shower in the calorimeter using the pro- portional planes. Searching with the proportional planes has two virtues: the coarser segmentation of the proportional planes allows a more efficient search for the shower and the flash chambers are susceptible to effects of ionization left behind by previous events, untriggered upon events, cosmic rays or other ”out of time” events which would confuse the vertex finding. The algorithm uses the latches of the proportional plane and finds a shower by looking for at least two proportional planes in a row with latches on An estimate of the 2 position of the vertex is made using the 2 position of the most upstream proportional plane of the most upstream group of planes with latches on Once the hadron shower is found by the proportional planes, a search for the vertex is made independently in each of the three flash chamber views. The search in flash cham- bers starts at the downstream end of the shower and ”walks” towards the front of the detector until the beginning of the shower is found and the algorithm has located the lateral vertex position Once the vertex has been found in each view, the final result is constrained to be a single point in threespace. The vertex finding routine has been found to be greater than 99% efficient at finding a vertex for events with v > 10 GeV (Mattison 1986; Mukherjee 1986). 2.5.2 Muon Finding and Fitting The muon track finding is done by a software package call the MTF (Muon Track Finding) package. The muon finding and fitting is done in a three step process: seg- ments in individual views are found, the segments from the separate views are then combined into tracks and finally the muon candidates are selected and fit. In each flash chamber view, a search is made for long tracks originating from the vertex. The search is done using angular bins that originate at the vertex. In the angular bins that contain a 47 potential muon candidate, hits are combined to form continuous segments which orig- inate at the vertex. The segments in the individual views are then ”3-view-matched” to form a track To be considered a muon candidate, a track must travel 5 meters in the z (beam) direction before exiting the calorimeter (either out the back or one of the sides). Finally, the track is fit to a line using a least-squares technique. 2.5.3 9. The muon angle measurement results directly from muon calorimeter fit. As- suming that the neutrino was traveling parallel to the z axis, the muon momentum polar angle, 6”, is the angle through which the lepton was scattered. The angular resolu- tion is estimated from Monte Carlo studies to be, 74 mR - GeV 0'9“ z E I p which is consistent with previous estimates made in studies using cosmic ray muons. 2.5.4 B p The muon energy is measured in the muon spectrometer and then corrected for dB Zr- loss in the target-calorimeter. The spectrometer fitting algorithm projects trajectories for different muon energies (and charge) into the spectrometer based on the calorimeter fit of the muon The calculation of the trajectory includes the mean muon g;- in the spectrometer iron These trajectories are then used to select the correct combination of hits in the spectrometer and to estimate the muon momentum. The projection process is then repeated and the 12 for each of the projected momentum is calculated and the x2 surface is fit to a parabola. The minimum of the parabola defines the fit momentum. Greater detail on the muon fitting can be found elsewhere (Strongin 1989; I-Iatcher 1993; Brock 1992). The calorimeter energy loss is accounted for by an ”integration” calculation us- ing the muon momentum entering the spectrometer as measured as the starting point. 48 The calorimeter muon track is divided into steps. The algorithm starts where the muon exited the calorimeter and adds to the measured muon energy the calculated mean gal-:— for the step based on energy of the muon as it leaves the step. The algorithm then steps back to the vertex to calculate the final muon energy. The calorimeter correction to the spectrometer fit is shown in Figure 2.12. The mean correction is 3.1 GeV and is directly correlated the amount of calorimeter material traversed and thus with the vertex 2 posi- tion Monte Carlo studies using a full simulation of the muon spectrometer and in- cluding parameterizations of the cross sections for discrete energy loss mechanisms (See Chapter 3 for details) such as nuclear bremsstrahlung and pair production show that the fractional resolution in E u is, o J:- = 14%, Pu for E p > 50 GeV. The resolution worsens at low energies due to multiple scattering in the iron toroids. 2.5.5 v The energy transfer from the lepton to the hadron system, v (nu not to be con- fused with neutrino) is measured directly in the calorimeter. As we saw in Chapter 1, v is defined as, v 5 EV — E”. vis sometimes referred to as E}, (or Ewmn) but 15;, is really the energy part of the outgo— ing hadron system 4~vector, P’ = (E, , P). One often finds in the literature statements such as, E, = E, + E, or y = §—:. Obviously, in this context, vand E), are being used syn- onymously, albeit incorrectly. In a neutrino target-calorimeter, one measures the kinetic energy transferred to the nucleon system. (The rest mass of the struck nucleon is unde- tectable because baryon number conservation requires that one baryon still remain after the interaction.) This is simply v. Events per 100 MeV 49 2000?- H 0‘ 8 l H N 8 l . . O LLJ L l LJJ LJLJLI l l I l l I 11 l l l O 1 2 3 4 5 6 Correction to Muon Energy (GeV) Figure 2.12. Calorimeter Energy Loss Correction to Fit Muon Energy. Plotted is the distribution of the difference between the final reconstructed muon ener- gy and the result of the spectrometer fit. The sharp edges of the distribution are due to the fiducial volume requirements imposed in the data analysis. 50 The FMMF detector is unusual in that it contains two separate detectors in its calo- rimeter, the flash chambers and the proportional planes. As discussed above, these two different detectors have very different properties. The primary advantages of the flash chambers are their very fine granularity and sampling. The primary disadvantage is that at higher energies the chambers are subject to saturation. The finer sampling in principle should lead to intrinsically better resolution for the flash chamber when compared to the proportional planes but satura- tion results in a severe degradation of the resolution, especially at higher energies. In contrast, the proportional planes sample much more coarsely than the flash chambers, but, being sampling analog devices, their resolution shows the typical 1/ W proportionality. In addition to the above problems, all large detectors can be sensitive to environ- mental effects. The qu data runs involved the gathering of data over periods of eight to nine months. The analysis for this thesis required a stable, known calibration of the calorimeter over an entire eight or nine month exposure. Both the flash chambers and proportional planes showed time-dependent behavior. These time dependent behav- iors are much better understood and handled with the proportional planes than with the flash chambers because of the well understood nature of proportional detectors and the self monitoring hardware built into the proportional plane system (1' artaglia 1984; Tartaglia et. a1. 1985). The one final problem that requires mentioning is that of calibrating a device as large as the FMMF detector. During both the 1985 and 1987 runs, a test beam of pions, kaons and protons of a known momentum was continuously brought into [ab C and used to calibrate the m detectors. This was extremely useful in understanding the detector, but at the same time was extremely limited because the calibration beam was only incident on the front of the detector. For E594, an algorithm (SHOWER) was developed that corrected the flash chamber 51 response for saturation and variation in detector response in a microscopic fashion. This algorithm was then recalibrated using the Ev versus radius of the DiChromatic narrow band neutrino beam (Mattison 1986). The SHOWER vscale was used as is for the 1982 data. In 1982, no attempt was made to combine the flash chamber calorimetry with that of the proportional planes. For the on; data, it was found that the SHOWER algorithm was unsta- ble over the long exposures. In addition, at the higher energies of the cm beam, saturation was a much more important effect. For all of the above reasons, it was necessary for the 1985 and 1987 data to devel- op a new method of measuring v and insuring the stability of the calibration. The requirements for a new method were: 0 Long Term Stability - the calibration must be consistent over an entire exposure. 0 Detector-Wide Uniformity - the calibration must be consistent over the entire fiducial volume of the detector. 0 Sensible Functional Form - the final calibration must have a smooth functional form that is sensible, for example, a quadratic in some form of corrected hits. 0 Inclusion of the Proportional Planes - for the better resolution of the proportional planes at high energies. A new calibration method was developed based on the above principles. The new calibration algorithm used an innovative scheme for measuring and then compensating for the variation of the response of the flash chambers. Using the proportional planes, showers in a small range of energies were selected from the neutri- no data. This data set was used to measure the response of each flash chamber module using the shower transition curves. The measured response was used to correct the ”raw” hits observed in a given module. A scale for the ”corrected raw" hits was then obtained using the test beam data and later adjusted using the charged current neutrino data set and the neutrino Monte Carlo. Finally, the flash chamber measurement was then combined with proportional plane measurement based on the measured resolu- tions. This method is described in detail in Appendix B. 52 The fractional resolution of the flash chambers and proportional planes and of the combined measurement is shown in Figure 113. As one can see the proportional planes exhibit the typical behavior with a fractional resolution of 3% + 1 16% / W where- as the flash chambers exhibit large effects of saturation with constant fractional resolution above 100 GeV. The final combined fractional resolution is consistent with, 3360/ +4770 v 3v. 2.5 EimLEmLSamk. The final event sample for this thesis consists of fully reconstructed charged cur- rent events, which requires a found vertex, a found and momentum analyzed muon and a measured v. Additional cuts are made to insure that the event is well measured and that the acceptance is well understood. The cuts applied and the resultant event sample are discussed in this section. 2.6.1 Cuts The cuts were of three types, acceptance cuts, measurement quality cuts and physics cuts. The acceptance cuts are made to compensate for effects such as trigger and algorithm inefficiencies. Measurement quality cuts are made to eliminate events which are questionably measured. Physics cuts are made for various reasons such as insuring that an event is a deep inelastic event or for comparing the data to Monte Carlo only in kinematic regimes that are well understood theoretically. 2.6.1.1 Acceptance Cuts There are a number of standard acceptance cuts: a requirement that the event vertex be within a defined fiducial volume, the requirement that v> 10 GeV and a com- plicated requirement on the calorimeter track of the fit muon. The fiducal volume was defined as 32 < LVEST < 401 (8 < LVEsT < 401 for 1982 data) and CCEDCE > 250.0. The first cut is on the 2 position of the vertex based on the flash fractional resolution 53 0.28 0.24 "-. 0.20 A Flash Chamber 0.16 v— 0.12 __ Combined ‘ ‘ ~ c \ Proprotional Plane ~ § ‘ ~ Q s 43:...___ 0.08 — 0.04 - T 0.00.11111111rlrlrrlrrralrrrrlrnn111111111 40 80 1 20 1 60 200 240 280 Energy Deposited in Calorimeter - v — (GeV) Figure 2.13. Fractional v Resolution as Function of v. Plotted is the v resolution, as a function of v as measured in the test beam. The fractional resolutions, ‘3', for the Flash Chambers, the Proportional Plane and the combined measure— ment are shown The open squares show the measured fractional resolution for the flash chambers with the dotted line showing the fit functional form. The open triangles show the measured fractional resolution for the propor- tional planes with the dashed line showing the fit of the resolutions. The solid line shows the combination of the fractional resolutions of the flash chambers and the proportional planes. 54 chamber of the found vertex, LVEST. The second cut is on the transverse position of the vertex and requires that the event vertex be no less than 25 inches (125 cm) from the detector edge in any flash chamber view. The vcutisbased on the standard Pm trigger. The trigger has been measured to be fully efficient at 10 GeV. As no attempt has been made to model the trigger efficiency, a hard 10 GeV cut has been imposed. The HiE trigger introduces a second threshold. For the HiE trigger, the trigger turn- on was measured using PIH triggers from the 1985 run. The measured v distribution for all P'I't-l triggers was plotted and the same distribution was plotted for the em triggers that were also HiE triggers. The ratio of these two distributions gave the trigger turnon shown in Figure 22 This function was then fit to the integral of a Gaussian where the width and mean of the Gaussian are the free parameters. This function is simply the error function. The form was chosen because the trigger is based on the amount of ener- gy seen by the proportional planes and if the amount of energy deposited is above threshold the trigger is taken. Since there are resolution effects, the integral of a Gaussian is the appropriate form. The error function gives a good fit to the distribution as shown in Figure 2.2. The fit was then used in modeling the HiE trigger in the Monte Carlo. The MTF package has been found to be excellent at finding tracks. The MTF pack- age was originally written to enable simple Monte Carlo simulation. For this reason, the package uses two simple criteria (only one of which is important for this analysis) to determine which tracks are those of muons. A track is classified as a muon, if the track is 10 meters or longer in z, or if a track exits the calorimeter and the track is 5 meters or longer in z. The first criterion is relevant for stopping tracks and is thus not important for this thesis. The second criterion is obviously relevant for this thesis. The problem is the definition of ”exiting” the calorimeter. Exiting through the rear of the detector is easy to define but exiting the sides is more difficult to define. The problem is that the software must determine which chamber was the last chamber hit before exiting and from this 55 chamber the z traversed is determined This is a difficult pattern recognition exercise. Chamber inefficiencies, phantom hits from cosmic rays and other instrumental prob- lems cause inconsistencies in the application of the criterion. For this reason, an additional cut is imposed on the muon track to insure the same criterion is used for both the data and the Monte Carlo. This cut involves projecting the exit point of the muon from the calorimeter using the muon’s fit trajectory and requiring that the 2 position of this pro- jected exit point be 5.5 meters downstream of the vertex 2 position. This is an easy requirement to implement in software and it eliminates uncertainty in the muon calo- rimeter acceptance. Finally, the acceptance of the spectrometer is modeled by a full detector simula- tion of the muon spectrometer. This simulation will be discussed in the next chapter. 2.6.1.2 Measurement Quality Cuts Measurement quality cuts are applied to the fit muons. The standard cuts were: 0The muon must traverse at least 175 centimeters of spectometer toroidal magnet iron. 0The muon charge must be consistent with the expected neutrino in the 1982 exposure. These cuts insure that the muon momentum was well measured. 2.6.1.3 Physics Cuts There are two standard physics cuts. The cuts are applied for different reasons but are closely related. The first of these cuts is on the square of the hadron system invariant mass, W2. W2 is defined as, W2 = P’2 = (P+q)2 = M2 +2P-q+q2 = M2 +2Mv—Q2. For this cut, aneventis required tohave W2 > 25 GeVz. This cutis made toinsure that the event is a true deep inelastic event, not the quasi-elastic excitation of a nucleon resonance. The second of these cuts is on the square of the 4-momentum transfer, Q2. This 56 cut is used only for comparisons between data and Monte Carlo because the quark distributions used as input for the Monte Carlo are derived from data with a Q2 > 5 GeV2 and the evolution of the quark distributions to low 4-momentum transfers (Q2 < 5.0 GeVz) is not well understood theoretically. 2.6.2 Sample The final data sample for this thesis includes data from the three exposures pre- viously discussed. This data set consists of roughly 110,000 reconstructed, accepted charged current events. Approximately one half of the data was taken during the 1987 exposure. The other half of the data was split somewhat evenly between the 1982 nar- row band data, the 1985 low bias events (the low-bias event sample consisted of em in the beam gate and the PIT-l, HiE and 2MU events in the tail gate) and 1985 HiE events (only in the beam gate). Table 2.1 gives event sums for each of the individual data sets and other useful information. 57 Table 2.2. Event Statistics Data Set Triggers Reconstructed In Charged Fit Final Events Fidicial Current Charged Event Volume Events Current Sample Events t—Ifir +165 N33 33,510 26,830 12,317 8,540 7.257 5345 -165 N BB 44,263 24,718 9,926 5,347 4,741 2,803 +200 NBB 31,810 22,455 10,323 6,848 5,788 4,544 +250 NBB 29,066 19,400 8,430 5,525 4,641 3,790 1985 75,078 35,661 32,050 23,482 20,925 17,682 Low Bras 1985 HiE 60,115 28,691 25,941 25,941 16,263 16,004 198? 239,187 118,866 112,183 112,183 75,209 60,441 Low Bras Table 2.2. Event Statistics. The table shows the numbers of triggers, reconstruct- ed events, events in the fiducial volume, charged current events, charged current events with a fit, and events in the final sample and how they are partioned between the three exposures and various setting and triggers. For an event to be reconstructed, the vertex must be found. This requirement along with the fiducial volume requirement eliminates any contamination from cosmic rays. Charged Current Events are events where a muon was found. Fit Charged Current Events are events where the muon was found and fit in the spectrometer. For an event to be part of the final event sample it must have passed all the previous cuts and had W > 2.5 GeV2 and v> 10 GeV. The final data sample contains 110,609 events. Chapter 3 Monte Carlo Simulation 3.1 lntmdnstinn To correct the data for acceptance and smearing, a means of calculating the com- plicated effects of the experimental resolutions and the acceptance is needed. This complicated calculation is done using Monte Carlo techniques. The VLIB Monte Carlo Simulation used for this analysis consists of four parts: the neutrino beam simulation, the interaction simulation, the event simulation and the event analysis simulation. Each of these parts of the Monte Carlo simulation is discussed below. 2.2 W The neutrino beam simulation starts with the parameterization of the Atherton production spectrum (Atherton et a1. 1980) by Malensek (1981) adapted for 800 GeV (or 400 GeV) protons on beryllium oxide. The beam line simulation program DECAY TURTLE (Carey, Brown, and Iselin 1982) propagates, and decays the produced pions and kaons through the simulated optics of the beam lines. When a pion or kaon decays, the simu- lation determines the kinematics of the decay and calculates the trajectory of the neutrino produced. From the trajectory of the neutrino, the program calculates the neu- trino position in the FMMF detector and creates Ev vs radius histograms for the various parents, and, in the case of kaons, various decays. Beam files are formed from the saved histograms along with information about relative fluxes of the neutrinos from each type of decay. A beam file consists of a list of neutrinos with their radial positions in the de- tector (the z position is chosen later), their energies, and their parentage. The interaction 58 59 portion of the Monte Carlo then takes the appropriate beam files as input. The beam simulation is extremely detailed. The interested reader is directed to the thesis of E. Gallas (1992). 3.3 I | l' S . l |° The interaction simulation consists of two primary sections, a simulation of the Fermi-motion of nucleons within the complex nuclei that make up the FMMF target calo- rimeter and the actual simulation of the neutrino—nucleon interaction. 3.3.1 Fermi-Motion Simulation The simulation of the Fermi-motion of the nucleons follows the method of Bodek and Ritchie (1981). To properly simulate the Fermi-motion of the nucleons, one first randomly selects the nucleus of the struck nucleon and whether the nucleon is a neutron or a proton. Based on the choice of nucleon and nucleus, the kinetic energy of the nucleon is thrown according to the prescription of Bodek and Ritche. Then the mo- tion of the nucleon is randomly oriented in space. The average kinetic energy of a nucleon is approximately 250 MeV. Figure 3.1 shows the thrown kinetic energy distri- bution. The Fermi-motion simulation determines the motion of the nucleon in the labo- ratory frame but to properly use the theoretical framework described in Chapter 1, the nucleon must be at rest. Thus, before the simulation of the neutrino—nucleon interac- tion, one selects a neutrino in the laboratory frame and then Lorentz boosts the neutrino-nucleon system into the nucleon rest frame. Then one simulates the interac- tion and the resulting system is Lorentz boosted back into the laboratory frame. Fermi-motion smears the reconstructed x distribution particularly, at high x, be- cause one assumes that the nucleon is at rest in laboratory. Figure 3.2 shows the effects of reconstructing the x distribution of including the motion of the nucleon versus mak- ing the assumption that the nucleon is at rest. As one can see, Fermi-motion results in 7000 U! m 8 8 I'llIUIIIIUI'Y‘UUIIIIYIIUTU‘IVII'IIITT ‘3 Events per 10 MeV i 2000 1000 rrrnlrrrr+rTrr4w 0 (L2 (L4 (16 (18 1 pm (GeV) Figure 3.1. Fermi Momentum Distribution. The figure shows the generated distribution of [Emil for the nucleons in the target calorimeter. Note the narrow peak at fife,“ =0 due to the presence of hydrogen (which has no Fermi-motion) in the target. The integral of the figure is normalized to 100,000 events. 61 4 10 _ D b b. I 3 10 I "'=-.Wnsmeared x I Eventsper0.025 H O 10 :- generatedx C p- 1 :- F l l l l l l l l l l l 1 1 1 l l 1 1 J41 l l l i 1 1 l l I l l 1 l l 1 L 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x Figure 3.2. Effects of Fermi-Motion 0n Perceived 3: Distribution. The figure shows a comparison of the generated x and ”Fermi-motion smeared” x distributions. The solid line shows the generated 3: distribution. The dotted line shows the ”Fermi-motion smeared” distribution. The Fermi-motion smeared x is calculated in the laboratory frame assuming that the nucleon is at rest. The integral of the figure is normalized to 100,000 events. Note the logarithmic scale. 62 reconstruction of x's greater than 1.0. 3.3.2 Neutrino—Nucleon Interaction The interaction simulation uses a leading-order quark based cross section. The cross section is calculated based on the species of the struck nucleon. The cross section calculation does the radiative and slow rescaling corrections but does not include be- yond leading-order QCD diagrams such as quark-quark or quark—gluon effects. Appendix C gives the details of the physical cross section calculation, radiative correc- tions and the slow rescaling correction The interaction simulation starts in the rest frame of the nucleon. The invariants x and y are selected. The selected x and y, the boosted neutrino energy and a set of quark distributions are then used to calculate the Vp or vn differential cross section. Based on the cross section and the weight of the selected 2: and y, the event is rejected or accepted. If the event is accepted, an additional muon leg radiative correction is made by generating a photon to be lost by the muon, which is presumed to be emitted collinearly t0 the muon. The momentum four-vector of the photon is then absorbed into the had- ron system and correspondingly removed from the muon. Finally, the four-vectors of the final state are Lorentz boosted back into the laboratory frame. 3.4 ExenLSimnlatinn The interaction part of the Monte Carlo generates the final state, which consists of the muon four-vector and the hadron system four-vector in the laboratory frame. The vua software package simulates the event from these two four-vectors. Whether an event becomes a part of the final event sample is determined almost solely on the characteristics of that event’s muon and the geometry of the FMMF detector. If one wished to parameterize the muon acceptance, it would be a function of the vertex position, the muon angle, the muon energy, the charge of the muon, and possibly other factors. While it might be possible to determine such a parameterization, it is likely that 63 the parameterization would still be insufficient or too complicated to allow its use in making the acceptance corrections necessary for this analysis. In sharp contrast, the event acceptance effects of the hadron shower are straight-forward and a simple pa- rameterization of the shower energy resolutions allows one to adequately make the acceptance corrections. For these reasons, vus divides the simulation of the event into a muon simula- tion and a hadron simulation. For the muon simulation, vus does a complete detector simulation of the muon tracking and fitting. In contrast, VLIB bases the hadron shower simulation on resolutions measured from the test beam data. This section concentrates on the exact simulation of the muon. The next section will discuss the simulation of the event measurement including both the measurement of the hadron system and the muon. 3.4.1 Muon Simulation The muon simulation consists of two distinct parts: calorimeter tracking and the spectrometer tracking. In both parts, VLIB divides the muon track into small steps in 2. In each step, VLIB calculates the energy loss and the multiple scattering. At the end of each step, the muon's energy is reduced and its direction altered appropriately. VLIB divides energy loss into two parts. losses greater than 1 MeV are treated as discrete processes while losses less than 1 MeV are treated in a statistical fashion using the limited Bethe-Bloch formula (Particle Data Group 1990). VLIB simulation of discrete energy losses includes four processes: knock on electrons (8 rays), nuclear bremsstrahl- ung, pair production, and nuclear interactions. For the last three processes, VLIB uses the parameterizations of the cross section by VanGinnekin (1986). Figure 3.3 shows the probabilities of a muon experiencing an energy loss of T in an interval dx, citric—1d]? due to one of the loss mechanisms mentioned above. For small losses, 6 rays dominate. The three other processes have cross sections that do not fall as rapidly and these processes dN dxdT I I 10 -\ . Nucleon Scattering ":-._\ 10 I l I 111111 I l l l lllllj l I I 111111 1 l 1 111111 1 l I It?! -2 -1 2 10 10 10 1 10 10 Energy of Discrete loss, T (GeV) Figure 3.3. Differential Cross Sections for 8 Rays, Bremsstrahlung, Pair Pro- duction and Inelastic Nucleon Scattering for 100 GeV Muon Traversing Iron. The figure shows the calculation of the differential cross sections, £3177 for 8 rays (dotted line), bremsstrahlung (dashed line), pair production (dashed / dotted line), and Nucleon Scattering (solid line). The cross sections are plotted as a function of the energy of lost by the muon (T). 65 dominate at large discrete losses. These large energy losses can cause serious resolution problems. To simulate the discrete loss of energy by a muon in each step through a materi- al, VLIB calculates the probability that the muon undergoes a discrete loss due to each mechanism. Then, based on the calculated cross section, VLIB determines whether the muon underwent a discrete loss. Finally, if the muon underwent a discrete loss, VLIB determines the energy of the loss. Once VLIB determined the amount of energy lost due to discrete processes, it cal- culates the energy lost due to the limited S—f— and then reduces the muon energy appropriately. This process results in an energy loss distribution exemplified by Figure 3.4. The statistical treatment of losses below 1 MeV results in the sharp lower edge while the simulation of the discrete processes results in the extremely long high loss tail. High energy discrete losses create an asymmetric resolution function for the muon energy as seen in Figure 3.5. One should also note the long tails in the distribution which result in extremely poorly measured events. The distribution also has a very non-Gaussian shape. For this reason, the importance of the simulation of the discrete loss processes can not be underestimated. 3.4.1.1 Calorimeter Tracking For purposes of propagating the muon through the target calorimeter, VLIB treats the calorimeter as a homogeneous volume with the average properties of the target cal- orimeter. VLIB makes no attempt to model the response of the flash chambers or the proportional planes to the passage of the muon. The energy loss and multiple scattering of the muon is treated as described above. The routine propagating the muon through the calorimeter saves the muon's initial (vertex) position and momentum and its final position and momentum. For the calorimeter tracking, VLIB defines the final muon posi- tion as the point where the muon stops or exits the calorimeter. 66 -1 10 -_— : : r- -2 10 :- E ~EI§ ’ -3 10 :- c —4 10 _— i [I l l L l l I ll 1 10 Energy Lost byMuon, T(GeV) Figure 3.4. Spectrum of Energy lost by 100 GeV Muon Traversing 20 cm of Iron. Using method outlined in text, this figure shows the Monte Carlo calculation of the spectrum of energy lost (I) by 100 GeV muons as they traverse 20 cm of iron. The bin size is 1 MeV. 67 10 - S I O .. b . “- . g LL] 102 L C Mom ptrue Figure 3.5. Monte Carlo Muon Momentum Resolution The figure shows the distribution of reconstructed muon energy to true muon energy for 1987 Monte Carlo events. Note the long non-Gaussian tails typical of muon resolution functions. The integral of the distribution has been normalized to 100,000 events. 68 3.4.1.2 Spectrometer Simulation A unique feature of this analysis is the full simulation of the muon spectrometer and the muon fitting. To this end, the simulation of the spectrometer is much more complicated than that of the calorimeter. VLIB continues the propagation of the muon through the spectrometer if the muon exits the calorimeter and if it hits the front face of the first 24’ toroidal magnet. The spectrometer simulation is much more complex than that of the calorimeter. Initially, qu propagates the muon through the spectrometer in the same manner as used in the calorimeter but with the additional complication of the toroidal field in the magnet iron. The routine used to propagate the muon through the spectrometer is also used in the muon fitting package used to analyze the data, but for event generation, multiple scat- tering and discrete energy losses are thrown. VLIB tracks the muon as it passes through the iron and field of the magnets, the air gaps, the lead shot (used to fill the holes of the toroidal magnets), and the position measurement stations. The tracking routine saves the position and momentum of the muon at each position measurement station. After tracking the muon through the spectrometer, the hits in the spectrometer are simulated. vus simulates the hits based on information obtained from the data. For the simulation, the efficiency of each measurement station was studied using the data. The typical measurement station (a pair of back-to-back planes) was 98—99% efficient. The spectrometer studies also included the frequency and resolution of back-to-back drifted hit clusters as opposed to single hits clusters or multiple hit clusters. The fre- quency and distribution of noise hits were also measured. The frequency of noise hits varied by as much as 35% between measurement stations with the 24' planes being the most susceptible to noise. From this information VLIB simulates the response of each of the muon measurement stations. As a cross-check, after analyzing the muon as dis- cussed below, we repeated the efficiency, resolution and noise measurements made for the data on the Monte Carlo and compared the results. The Monte Carlo reasonably 69 simulates the behavior of each of the actual measurement stations. Figure 3.6 shows a comparison of spectrometer hit distributions from the data and Monte Carlo. 3.5 Writs As discussed above, the simulation of muons in VLIB differs greatly from the sim- ulation of the hadrons. As discussed in Chapter 2, for each neutrino interaction one must first determine if the interaction was a charge current interaction and then attempt to measure the muon angle, charge and momentum along with the energy transfer v. Let us examine the simulation of the event measurement. 3.5.1 Event Classification and Muon Angle In the data, the muon track finding (MTF) package does the initial classification of the event by searching for muons. If one or more muons are found, we classify the event as a charged current event. The MTF package has a set of criteria that it applies to any track it finds to determine if the track is as a muon. For this analysis, a track must traverse 5 meters of calorimeter along the beam direction to be classified as a muon. VLIB simulates this behavior by imposing the same requirement on thrown muons using the true vertex and the true exit position. One reconstructs the muon angle from the found slopes of the muon in two per- pendicular views. In the data, the MTF package fits the average trajectory of the muon over the entire length of the muon's track To simulate the averaging of the trajectory of the track, VUB uses the line connecting the vertex and the end point of the muon track as the ”average" trajectory of the muon. The slope of the ”average” trajectory in each view is then smeared by the resolution of the least-squares fitting algorithm. 3.5.2 Muon Fitting Once MT'F finds a muon, the reconstruction program attempts to fit the muon in the spectrometer. Analysis of the data and Monte Carlo uses the exact same fitting package with the Monte Carlo using the previously generated hits. If the muon is fit O 7 10‘ 9000 _ 1 woo E— '— g 7000 .=. ... w" .600 =_ 8 gm E— § Lu .... E— 102 3000 :— I- ‘E “'1 ++++ + 5 2000 __— 1000 :— 10 1 1 I 1 1 1 1 I 1 1 1 1 l 1 1 1 1 0 b g 1 l 1 g —50 -25 o 25 50 o 2 4 Residual (cm) Number of Hits ;_ + 320 E— 4000 E. H E : 2” L- E 3000 :— O E H E N 20° r 2500 t— : g- E E. 160 :— 42 2”) I:— - & E grsoo :— § 12° :— DJ 1000 E— LL] ‘30 :- 0 : 1 1 - i l . I—Ll l 0 .P g+f 44 1 1 1 1 l 1 0 0.5 1 1 5 2 400 0 400 Hit Size Distribution (cm) Distance/mm Muon Hit to Next Nearest Hit (cm) Figure 3.6. Comparison of Data and Monte Carlo Drift Hit Distributions. The figure shows comparison of the typical drift hit distributions from 1987 data and Monte Carlo. In all four plots, the data is shown as the points with error bars while the Monte Carlo is shown as the solid outline. Upper left corner shows the fit residual (i.e. hit position - fit position) for all hits. Upper right corner shows the distribution of the number of hits. Lower left shows the hit size distribution. Hit size varies based on whether a hit is formed from a clean back-to-back hit, single hit or a group of hits. lower right shows signed distance from hit used in muon fit to next nearest hit. Noise results in the asymmetry seen in the distribution. 71 successful, the fitting package (after correction for energy loss in the calorimeter) re- turns the muon’s measured energy and charge. From the measured charge of the muon, we determine whether a neutrino or anti-neutrino interaction occurred. 3.5.3 Hadron Energy VLIB simulates the reconstruction of the energy transfer by use of the resolutions presented in Chapter 2. Using the true v, qu calculates the resolutions of the both the flash chambers and proportional calorimetry. From the calculated resolution, VLlB deter- mines the response of both types of calorimetry. The responses are combined in the same manner as the data to produce the measured v. 3.5.4 Comparison of Accepted Integral Distributions of Data and Monte Carlo To complete this chapter, let us compare some distributions from the data to the same distributions for the Monte Carlo. Figures 3.7—3.10 show comparisons for the v, Eu, 6,), and E V distributions for 4 different data sets. In addition, it is instructive to exam— ine the distribution of the iron traversed by fit muons which is shown in Figure 3.11. The comparisons involve an additional cut beyond the standard cuts described in the previous chapter. In general, low (22 physics is not well understood theoretically and the theory presented in Chapter 1 is no exception. For this reason, for the comparison between data and Monte, one makes an additional requirement that the reconstructed Q2 is greater than 5.0 GeVz. This moves the Monte Carlo out of the region of theoretical uncertainty. The agreement between data and Monte Carlo shown in Figures 37-311 is very good. In the ”iron” distribution, one notes that there seems to be a small discrepancy between the data and Monte Carlo as to the ratio of events that traverse the entire spec- trometer to those that exit from the 24‘ wings. From this discrepancy, one estimates that the Monte Carlo correctly models the acceptance of the detector and cuts to 1.3%. This uncertainty in the knowledge of the acceptance will be included in the systematic errors presented in the next chapter. 72 73 2250_ u 2 5 1987 : 1985 .— 600:- 1750§- % E 1500i 0 500’? : Ln - = 4001 glzsort b : 1000?;- 3.300;. : 4.. I: 750; . E §2005 250:— 10°? 0'- l l l l I l l o'- o 200 400 o v-GeV v-GeV 600-— . r 1985 200;- +250 > 500; HtE > 1755— N33 (‘5 E 6 1565 2 ’— “2 1255— A 3005— grooé 1‘3 : 52 E E 2002- g 75': 9 - :3 E. “'1 100E. LL) 505 ; 255- 0 l l l l L l l o l l l l 0 200 400 o 200 400 v—GeV v—GeV Figure 3.7. Comparison of Data and Monte Carlo v Distributions. The figure shows the v distributions for accepted neutrino events from the 1987 (top- left), 198510w-bias triggers (top-right), 1985 HiE triggers (bottom-left), and 1982 +250 narrow band (bottom-right) data sets. The data is shown as the points with error bars while the Monte Carlo is overlayed as a solid outline. The Monte Carlo is area normalized. The HiE triggers show the effects of the high trigger threshold. 74 2250; 1987 700; 1985 ;> 2000E > 600% qu U 1750 @ E Bias to 1500 “3 500; grzso £400- 3 1000 .53 300 § 750 § 9 200 EL] 500 FL] 250 100 o 1 l l l o l l l l o 200 400 o 200 400 Eu-GeV Eu-Gev 700: 1985 2003.. +250 : HiE E N33 > 600 % 1755 K3 L') 150: to 500 to E is 4... t. ‘25; h. S. 100 m V) g 300 g 75 LS 2‘” :3 5° 100 25 l O 1 1 1 1 l 1 o 7 1 1 1 4 -1 1 P1 0 200 400 o 200 400 Ep-GCV Eu-GCV Figure 3.8. Comparison of Data and Monte Carlo Eu Distributions. The figure shows the Eu distributions for accepted neutrino events from the 1987 (top- left), 198510w-bias triggers (top-right), 1985 HiE triggers (bottom-left), and 1982 +250 narrow band (bottom-right) data sets. The data is shown as the points with error bars while the Monte Carlo is overlayed as a solid outline. The Monte Carlo is area normalized. Events per 5 milliRadians Events per 5 milliRadians 1987 5 H N N 8 § § 8 III‘VHIIIUIUUIY § IILIUUTUIUII 1 I L 02 04 9” — Radians . O 0.2 0.4 all f" Radians 900 1985 800 700 600 500 400 300 200 100 o . . . w 0 75 800 3 1985 a mo Law § 600 Bras 3 500 E Ln 400 t 300 Q. .52 200 § 166 LL] 1 o l L l 0.4 U3 : a 180; +250 g 160; N33 g 140;- E 1203— E 100: Lo 5 '6 8°: “- 605 23 E 40_ § 265 i LL] 0 o 0.2 0.4 Hfl-Radians Figure 3.9. Comparison of Data and Monte Carlo 9p Distributions. The figure shows the 6“ distributions for accepted neutrino events from the 1987 (top- left), 198510w-bias triggers (top-right), 1985 HiE triggers (bottom-left), and 1982 +250 narrow band (bottom-right) data sets. The data is shown as the points with error bars while the Monte Carlo is overlayed as a solid outline. The Monte Carlo is area normalized. Events per 5 GeV Events per 5 GeV p1 H Figure 3.10. Comparison of Data and Monte Carlo EV Distributions. The figure shows the Ev distributions for accepted neutrino events from the 1987 (top- left), 198510w-bias triggers (top-right), 1985 HiE triggers (bottom-left), and 1982 +250 narrow band (bottom-right) data sets. The data is shown as the points with error bars while the Monte Carlo is overlayed as a solid outline. The Monte Carlo is area normalized. The distributions shows the typical double banded n/ K structure. For the HiE triggers, the high v threshold I'IIII'ITIIIIIIIIIII IIIII'III 1987 200 l 400 EV-GeV § I'IIIIITIII'IIII IIII III IIIIIII'III 11111 200 EV-GCV 1985 L l 1 1 400 600 76 H b O EventsperSGeV t-I H B 8 8 8 8 8 IIII'IIIIITIIIIIIIIIIIII'IIII'IIII'IIII O EventsperSGeV Ca 8 8 ”8‘ 8 ii 8 i 88 1985 Low Bias l l l l . 200 400 600 EV— GeV +250 NBB 200 400 600 Ev-GeV results in a data set that is almost entirely ”kaon” neutrinos. 0cm Events cm 77 fl perl Events per 10 4 1987 1° 1985 10 Low gm3 Bias 103 O H 102 $102 . , :2 10 g“) ' ‘ ' i to 1 1 rllllllllllllllLL llllllllllMllll 0 250 500 750 1000 0 250 500 750 1000 Iron Traversed — cm Iron Traversed — cm 4 10 1935 3 +250 HiE £10 N33 103 u o I?“ 2 10 Q. (I) t E10 l 10 9 LL] 1 1 l I | I I l l l l l l l 0 250 500 750 1000 0 250 500 750 1000 Iron Traversed - cm Iron Traversed - cm Figure 3.11. Comparison of Data and Monte Carlo Iron Distributions. The figure shows the distributions of iron traversed in spectrometer by the fit muon for accepted neutrino events from the 1987 (top-left), 1985 low-bias triggers (top-right), 1985 HiE triggers (bottom-left), and 1982 +250 narrow band (bottom-right) data sets. The data is shown as the points with error bars while the Monte Carlo is overlayed as the solid outline. Note the logarithmic scales. The Monte Carlo is area normalized. Chapter 4 Structure Function Extraction 4.1 Introduction For this thesis, neutrino-nucleon structure functions have been extracted. The analysis presented in this thesis assumes the average of the world's results for the total neutrino-nucleon (0"N ) and anti-neutrino—nucleon (6W) cross sections. Starting with 2 assumed total cross sections, the double differential cross sections, 11, are measured. The analysis then extracts the structure functions using the meaftiiyd double differ- ential cross sections and a parameterization of R(x,Qz). This chapter describes the measurement of the double differential cross sections and structure functions. The method for determining the systematic errors is then discussed. The final section of this chapter presents the measured cross sections and structure functions. 4.2 Diffemfialfimsafiecfimfatmfian To first order, this analysis measures the double differential cross section, 2 VN VN Ely—(2:13] , by assumrn' g the total cross section, 0 V the data for smearing and acceptance by the Monte Carlo techniques. This procedure and calculating the corrections to has been adopted because there was no monitoring of the flux of secondary particles in the decay region of the Quadrupole Triplet Beam. In neutrino experiments using narrow band neutrino beams, one does extensive monitoring of the number and spectrum of secondaries produced which allows one to directly measure the total cross section The lack of flux monitoring has necessitated the use of an assumed total cross section Once the total cross section is known, the double differential cross section (in a given Ev interval) is measured from the ratio of number events in the appropriate x, y, 78 79 E y bin to total number of events in the E V interval. It can be shown that the differential cross section is, 1 dzo'v" _ 0"” 1 number corrected data events in x,y,Ev bin E v dxdy E v AxAy number corrected data events in E, interval = 0V" 1 DATAiikMCfi‘“ mcfwd Ev AxAy MCQW“ DATA k MC?“ (4.1) where i, j, k refer to the appropriate x, y, Evbin, DATA,“ is the number of accepted data events in the given x, y, E V bin, MC g:“”‘“ is the number of accepted Monte Carlo events in the given x, y, E Vbin, and MC 5,?“ is the number of thrown Monte Carlo events in the given x, y, Eybin. Similarly, DATA k, MC:‘“’""’, and MCI” refer to the total number of events in the neutrino energy interval. In the above equation, the third fraction is the corrected number of events in the x, y, E V bin while the last fraction is the reciprocal of the corrected number of events in the EV energy interval. 4.2.1 Total Cross Section The assumed total cross sections are a combination of the three newest, highest statistics measurements of the total cross sections. The three sets of results from the CCFR (Auchincloss et al. 1990), CDHSW (Berge et al. 1987), and CHARM (Allaby et a1. 1988) collaborations are consistent with the total cross sections rising linearly with energy. W”) = 0.6762i00140 (03332100088) x 104’8 cszeV"1. The mean is V 4.2.2 Correction for Target Neutron Excess The above total cross sections are for scattering off an isoscalar target (i.e. number of protons = number of neutrons). If the target contains more neutrons than protons, the total cross section for neutrinos is enhanced because of the excess number of d valence quarks (as compared to an isoscalar target) while the total cross section for anti-neutrinos is reduced. The FMMF target calorimeter is nearly isoscalar, N—Z SEN+Z = 2.97%, 80 where N is the average number of neutrons per nucleon and Z is the number of protons per nucleon For the FMMF detector, N = 10.2 and Z = 9.8 which should be compared with N = 29.9 and Z = 26 (with 6 = 6.89%) for the typical iron calorimeter detectors. The slight non-isoscalarity of the FMMF target requires a small correction to total cross sections noted above. Then the calculation of the differential cross sections requires a correction of the opposite sign to account for the differences in the shapes of the valance quark distributions. If one assumes the simple quark-parton model, it can be shown that the correc- tions to the total cross sections are of the form, UVFMMF = OVN + O.correction (42a) O.VFMMF = OW _ O.cor‘rection (42b) where, 2 0mm“ = digijflub) - d(x))dxdy.+ (4.3) This correction, for the FMMF target, to the total neutrino (anti-neutrino) cross section is roughly one (two) percent One then substitutes GVFMMF in place of GVN in formula 4.1 resulting in, 1 d2 OVFMMF OVFMMF 1 D AT Ankh/ICE?" Mcziccepted (4 4) E, dxdy " E, AxAy M0337“ DATAkMCfm° ' Now, one must correct for the difference in the shapes of the u and d valance quark distributions. In the same manner as the corrections to the total cross section, one corrects the differential cross section. If one takes the double derivative of equations 4.2, using the definition of ammo" in Equation 4.3, one finds, + The u and d quark distribution referred to here are those measured in the proton This analysis assumes that the proton and neutron form an isoscalar doublet. From this assumption, one concludes that uneumn= dpmton and dneutron= “proton . 81 d2 avN d2 OVFMMP G: ME = - 5.5—v _ d dxdy dxdy n x(U(x) (96)) (4.5a) = 5_L_V. _ d . dxdy dxdy + ,, 3411(1) (26)) (4 55) and applies this correction to complete the calculation of the neutrino (anti-neutrino) nucleon double differential cross section 4.: Wm As we saw in Chapter 1, the differential cross section can be expressed in terms of three structure functions, dzo'VN'V” GZME My = in {32xFr+(1-y+7“g)1=2t(y-%y’)xF3}. To extract xF3, one takes the difference of the neutrino and anti-neutrino differ- ential cross sections. The difference yields dzo'” _ [12de = 262MB { __ dxdy dxdy 7! y jy2}xF3(x,Q2) The extraction of F2 requires that one eliminate xFl by making an assumption about R(x,Q2)§:_—L. The definition of R yields, '1‘ F2 _1+R_ 1+R 2xF1 1+‘3—: 1+il‘ézzi’ allowing one to eliminate xF 1. In this analysis, we use a parameterization of the meas- urement of R made at SLAC by Dasu et al. (1988). The parameterization given by Dasu et a1. is, R(x’ Q2) = [1111(1 2x)?“ + 0.11(1 —2x)"‘9‘ 08(Q /A) Q where A is assumed to be 200 MeV. This parameterization is based on ”an empirical parameterization of the perturbative QCD calculations of R" with the addition of a second higher twist term. To extract F2, one takes the sum of the neutrino and anti-neutrino differential 82 cross sections. The sum yields, dzo'" dzav" ZG§ME Mxy y21+AE2 2 = 1- A F I o a dxdy + dxdy 2r { 3“ 213, + 2 1+R + 2(x Q) (46) The A term in Equation 46 corrects for the non-cancellation of x133. As we previously saw in Chapter 1, xFa'” at foN due to the contributions of the sea quarks. One finds that the correction to F2 is, A = 4(y -ijy2)(4s(x,Q2)- 4c(x,Q2)). The average x, y, yz, By, and Q2 are calculated from the Monte Carlo in the appropriate x, y, 15., bin and then used to calculate the structure functions from the sums and differences of the double differential cross sections. After the structure functions are calculated for a given x, y, EV bin, in each x bin, one combines the structure functions from y, EV bins covering the same Q2 range. 4.4 W515 Systematic errors arise from a number of sources. In this analysis, systematic errors could arise from two general types of errors: an incomplete knowledge of the acceptance of the experiment and errors in the calibration of the scales used in the event measurement. 4.4.1 Acceptance The acceptance of this analysis has been extensively studied. The acceptance (of a given bin) is simply the ratio of the final number of events accepted in the bin to the total true number of events in the bin. To insure that the acceptance of the analysis is well understood, the Monte Carlo (described in Chapter 3) includes a full simulation of the muon spectrometer and of the mechanisms of catastrophic energy loss from muons traversing matter. The acceptance for a given bin is then calculated from the Monte Carlo simulation as the ratio of the number of Monte Carlo events accepted to the number of Monte Carlo events thrown. 83 Figures 3.7—3.11 showed comparisons between data and the Monte Carlo simu- lation of some integral distributions. In general, the agreement between the data and the simulation is impressive. A close examination of the distribution of the iron tra- versed by fit muons shows a small discrepancy. From this discrepancy one estimates an uncertainty in the knowledge of the acceptance as 13% of the acceptance calculated from the Monte Carlo simulation The error in the knowledge of acceptance results di- rectly in a 1.3% systematic error in cross sections. This systematic error is then added in quadrature with systematic errors from other sources. 4.4.2 Measurement Biases Systematic errors can also result from errors in the calibration of scales. These types of errors result in measurement biases. From the various calibrations, we have estimated the possible variations in the calibrations. The possible variation in the v calibration has two distinct components, a ped- estal error, and a scale error. A pedestal error could result from either an incorrect subtraction of the calorimeter noise or improper extrapolation of the calibration scale to low energies. A scale error could be the result of a calibration error due to a mismeas- urement of the momentum of the test beam particle or an error in the calibration procedure. From the scale fitting done using the neutrino data (as described in Appen- dix B), we have conservatively estimated the possible errors in the v calibration as a pedestal of 1.0 GeV and a scale of 2.0%. In contrast, the pp calibration only has a possible scale error which could be due to the a mismeasurement of the magnetic fields of the toroidal magnets. From a fit of scale similar to those described in Appendix B where one allows pp to vary in addition to v, the error in the p” calibration is estimated at 2.0% and is highly correlated with the errors in the calibration of v. The correlation of the scale errors results from the final part of the calibration procedure where the neutrino energy vs radius relationship is used to determine the final calibration constants. A possible error in the 0” calibration differs from those in p” and v. One calcu- lates 9,, from the muon slopes in two orthogonal views. Any error in alignment results in an increase in the resolution but not in a systematic bias. The physical size of the cal- orimeter and theflashchamber cells (and their spacing) actually sets the 6,) scale. From the physical size of the calorimeter and limitations in survey techniques in Lab C, one estimates the maximum possible error in the 6,, scale as 0.05% (1 cm over 20 m). In prac- tice, the size of any possible 0” scale bias allows us neglect 0“ as a source of systematic error. The systematic errors due to calibration uncertainties have been studied in two similar methods which give comparable results. Both methods involve varying the v and p” measurements and observing the effects on the final cross section and structure function measurements. The primary method involves varying the smeared Monte Carlo values. The second method involves varying the calibration constants for the data. Initially, one varies the calibration constants and does the complete cross section and structure function analysis. Each variation in the calibration constants is treated as a separate experiment. The set of different variations forms an ensemble of experiments. Based on a large number of variations in the calibration constants, one then directly calculates the systematic errors from the variations in the differential cross section or the structure functions. To account for correlations, in addition to varying each of the calibration constants individually, the calibration constants were varied simultane- ously. The details of the method of calculating the systematic errors is explained in detail in Appendix D. The systematic errors reported in this thesis were calculated from the simultaneous variation of the scales. 85 4.5 smarter: As discussed above, this analysis assumes the total cross sections (and the energy dependence) for neutrino nucleon and anti-neutrino nucleon scattering. The as- sumed total cross section was derived from an average of the world's data on the total cross section The error for neutrinos is 2.1% and for anti-neutrinos is 2.4%. The results of the propagation of the total cross section errors is presented as a separate scale error for both the differential cross sections and the structure functions. 4.6 Results To calculate the differential cross sections and structure functions presented in this thesis, data from 1982, 1985, and 1987 were analyzed on an event by event basis as described in Chapter 2 The cuts described therein were applied. The data naturally di- vided itself into seven different data sets, the four narrow band settings from 1982 (three neutrino settings, and one anti-neutrino), the 1985 low bias triggers, the 1985 High Energy triggers and the 1987 data (all low bias). For each of these data sets, the data was binned into neutrino and anti-neutrino bins, 12 x bins, 5 y bins and 5 neutrino energy bins. The binning is discussed below. For both data and Monte Carlo, the number of accepted events in each x, y, Ev bin and in each EV interval were accumu- lated. Because the y bins do not cover the entire y interval from 0.0 to 1.0, the sum of the number of events in the x, y bin in an Ev interval does not equal the total accepted event in an EV interval. In addition for the Monte Carlo, the same sums are accumulated for all throw charged current events using the thrown values of x, y, and E v. Table 4.1 shows the event statistics for both data, Monte Carlo accepted and Monte Carlo thrown The statistics for the Narrow Band Data sets are combined. Table 4.1 also shows the mean E V, and Q2 for each data set. In addition, to the bin and interval event sums for the best values of the calibra- tion constants, bin and interval sums are also accumulated for the variations of the 86 Table 4.1 Final Event Sample 1982 1985 1985 1987 Data Set Low Bias Low Bias HiE Low Bias TOTAL Data Events 2803 2319 1128 8821 15071 2 (15.) (GeV 2) 75.6 132.0 240.5 127.1 130.0 E (Q2) (GeV) 11.0 17.4 37.1 16.0 16.9 'E Thrown MC 48995 54778 54778 61904 — Accepted MCJll 27620 34496 3270 40025 — Data Events 13679 15363 14876 51620 95538 g (Ev) (GeV) 109.2 165.6 267.4 166.9 174.0 :3 (Q2) (GeV 2) 21.4 32.0 64.8 31.7 35.4 g Thrown MC 144675 325823 325823 372613 — Accepted MC 106959 225817 52590 264296 — Table 4.1. Final Event Sample. The table shows the characteristics of the final data set. The statistics have been tabulated for reconstructed events that pass all cuts and divided between neutrinos and anti-neutrims. The table also shows the average neutrino energy and 4-momentum transfer. In addition, the Monte Carlo Simulation statistics are shown For the simula- tion, the numbers of thrown and accepted events are shown The three 1982 Narrow Band neutrino data settings have been combined and entered as "1982 Neutrino” data set. 87 calibration constants as described above in the section on systematic errors and in Ap— pendix D. The systematic errors can be calculated from either Monte Carlo or data. The systematic errors presented here are based on the results from the Monte Carlo study but are consistent with those from the data study. 4.6.1 Binning An unusual binning scheme is chosen for the differential cross section measure- ment. While for the differential cross section measurement, any y, E y scheme would be acceptable, for structure function extraction it will be important that one is able to com- bine y, Evbins. For each x interval, each y, E Vbin represents a range in Q2. It is important that the y bin for a given EV interval is chosen so that there are y bins in the adjacent EV interval that encompass the similar (or better still, the same) Q2 intervals. For this rea- son, a logarithmic binning scheme has been chosen so that the same Q2 interval that is covered by bin (i,j,k) is also covered by bins (i,j-1,k+1) and (i,j+1,k—1). Table 4.2 shows the binning limits for x, y, and E V. 4.6.2 Final Cross Section Results The data for this thesis comes from 7 different data sets. The differential cross sections are calculated for x, y, E Vbins in each data set. The Monte Carlo statistics are at least a factor of 3 greater than that of the data for all data sets. For data from the 1982 narrow band running, wrong sign events (for example, events with a 11+ for a neutrino setting) were ignored. Once the differential cross sections were calculated for all appro- priate bins as outlined above for each data set, the results were averaged. From the Monte Carlo, one calculates systematic errors (as discussed above), and the mean true y, Q2, and EV (which are required for structure function extraction) for each bin in each data set and then the results are averaged in the same manner as the cross sections. The results for the differential cross sections are shown in Tables 4.3 and 4.4. 4.6.3 Final Structure Function Results The structure functions are calculated as outlined above from the differential cross sections presented in Table 4.3 and 4.4 and are shown in Table 4.5. In addition to the statistical errors, systematic and scale errors are also presented. 89 Table 4.2 Binning Limits for Structure Function Extraction N x limits y limits EV limits 0 0.000 0.030 16.0 GeV 1 0.030 0.060 32.0 GeV 2 0.060 0.120 64.0 GeV 3 0.100 0.240 128.0 GeV 4 0.150 0.480 256.0 GeV 5 0.200 0.960 512.0 GeV 6 0.250 — — 7 0.300 — — 8 0.400 — — 9 0.500 — — 10 0.600 — — 11 0.700 — — 12 1.000 — — Table 4.2. Binning Limits for Structure Function Extraction The table slows the limits of the bins used in structure functions extraction Note the logarithmic binning of y and By. Table 4.3 Anti-Neutrino Differential Cross Sections x = 0.015 1121 Statistical Systematic Scale 3’ E v (GeV) 15 dxdy Error Error Error Events 0.360 26.2 0.7353 0.2326 0.0628 0.0046 10 0.720 26.1 0.6499 0.1102 0.0319 0.0041 32 0.180 49.6 1.0680 0.2372 0.1454 0.0067 22 0.360 49.6 1.0233 0.0801 0.0509 0.0064 158 0.720 49.5 0.9058 0.0536 0.0281 0.0057 270 0.180 91.1 1.0449 0.1216 0.0992 0.0065 75 0.360 90.3 1.1104 0.0806 0.0477 0.0069 184 0.720 90.7 1.0736 0.0513 0.0352 0.0067 401 0.090 177.8 1.2385 0.2484 0.0779 0.0078 25 0.180 177.1 1.0466 0.1568 0.0617 0.0065 47 0.360 179.0 1.0896 0.1094 0.0392 0.0068 117 0.720 178.7 1.4739 0.0641 0.0424 0.0092 462 0.180 325.8 1.9084 0.4244 0.0593 0.0119 21 0.360 325.7 1.1699 0.2004 0.0412 0.0073 34 0.720 325.6 1.6287 0.1253 0.0698 0.0102 150 Table 4.3. Anti-Neutrino Differential Cross Sections. The table shows the mea- sured differential cross sections, —::;c—dy, (in units of 10"“cm2/GeV) for x ,Ey, v bins. For each x bin, the table shows the y, E bin, the differential cross section, the statistical error, the systematic error, the scale error and the number of events in the x, y, E V. The cross section measurement and the determination of the systematic and scale errors are discussed in the text. 91 Table 4.3 continued Anti-Neutrino Differential Cross Sections x = 0.045 1121 Statistical Systematic Scale 7 Ev (GeV) E dxdy Error Error Error Events 0.360 26.0 0.8671 0.2768 0.1105 0.0054 11 0.720 26.2 0.9067 0.1336 0.0232 0.0057 41 0.180 49.6 1.7454 0.2935 0.1430 0.0109 35 0.360 49.6 ll 1.2496 0.0889 0.0610 0.0078 187 0.720 49.6 0.9393 0.0563 0.0238 0.0059 259 0.090 91.7 1.2924 0.4882 0.1575 0.0081 11 0.180 89.9 ll 1.6200 0.1577 0.0943 0.0101 106 0.360 90.1 1.3572 0.0933 0.0347 0.0085 205 0.720 90.4 1.0572 0.0551 0.0418 0.0066 341 0.090 179.3 1.6573 0.2880 0.1121 0.0104 38 0.180 179.1 1.4236 0.1735 0.0589 0.0089 69 0.360 178.5 1.6376 0.1333 0.0335 0.0102 146 0.720 178.9 ll 1.0702 0.0587 0.0373 0.0067 309 0.180 325.9 N 1.0830 0.3209 0.0768 0.0068 12 0.360 325.7 1.3637 0.2200 0.0319 0.0085 39 0.720 325.5 1.1221 0.1103 0.0716 0.0070 95 Table 4.3 continued Anti-Neutrino Differential Cross Sections x = 0.080 1121 Statistical Systematic Scale 7 E v (GeV) E dxdy Error Error Error 0.360 26.7 0.5386 0.2185 0.1203 0.0034 0.720 26.3 0.7391 0.1122 0.0305 0.0046 0.180 49.6 1.6923 0.2514 0.1157 0.0106 0.360 49.6 1.2677 0.0793 0.0343 0.0079 0.720 49.6 0.8627 0.0490 0.0195 0.0054 0.090 90.7 1.3271 0.3665 0.1695 0.0083 0.180 91.0 1.6057 0.1372 0.0611 0.0100 0.360 90.3 1.3167 0.0804 0.0393 0.0082 0.720 90.4 0.8549 0.0439 0.0256 0.0054 0.090 176.9 1.5483 0.2450 0.0930 0.0097 0.180 177.8 2.0598 0.1992 0.0493 0.0129 0.360 177.8 1.3106 0.1023 0.0408 0.0082 0.720 178.7 0.8144 0.0465 0.0331 0.0051 0.090 325.7 1.4266 0.4376 0.1007 0.0089 0.180 325.7 1.0507 0.2784 0.0786 0.0066 0.360 325.7 0.8926 0.1593 0.0539 0.0056 0.720 325.5 0.7209 0.0758 0.0665 0.0045 281 17 135 253 349 41 107 160 276 11 16 31 87 93 Table 4.3 continued Anti-Neutrino Differential Cross Sections x = 0.125 id’_0 Statistical Systematic Scale 3’ Ev (GeV) E dxdy Error Error Error Events 0.360 26.2 0.8889 0.2293 0.1550 0.0056 16 0.720 26.3 0.5741 0.0965 0.0205 0.0036 32 0.180 49.6 1.6030 0.2262 0.1423 0.0100 49 0.360 49.5 1.0172 0.0650 0.0358 0.0064 225 0.720 49.5 0.6455 0.0390 0.0143 0.0040 247 0.090 91.2 1.2309 0.2970 0.1123 0.0077 18 0.180 91.0 1.3451 0.1087 0.0711 0.0084 150 0.360 90.9 1.0884 0.0645 0.0171 0.0068 269 0.720 89.9 0.5811 0.0323 0.0201 0.0036 304 0.090 179.4 1.4074 0.2130 0.0737 0.0088 57 0.180 178.8 1.5495 0.1491 0.0375 0.0097 105 0.360 177.6 1.0660 0.0823 0.0197 0.0067 159 0.720 179.2 0.5485 0.0334 0.0161 0.0034 242 0.090 325.8 II 1.4965 0.4144 0.1799 0.0094 13 0.180 325.9 ll 1.3636 0.2681 0.0768 0.0085 27 0.360 325.7 || 0.9767 0.1398 0.0297 0.0061 47 0.720 325.6 M 0.5313 0.0561 0.0132 0.0033 81 Table 4.3 continued Anti-Neutrino Differential Cross Sections x = 0.175 1}}: Statistical Systematic Scale 3/ Ev (GeV) 13 dxdy Error Error Error Events 0.360 26.2 0.7595 0.2038 0.0699 0.0048 13 0.720 26.1 0.5391 0.1037 0.0148 0.0034 24 0.180 49.6 1.5096 0.2234 0.0895 0.0094 48 0.360 49.5 ll 1.0036 0.0670 0.0198 0.0063 208 0.720 49.6 0.4258 0.0331 0.0128 0.0027 149 0.090 90.9 1.2295 0.3333 0.1348 0.0077 17 0.180 90.8 1.2365 0.1049 0.0435 0.0077 136 0.360 90.7 0.9710 0.0629 0.0211 0.0061 227 0.720 90.1 0.4725 0.0306 0.0093 0.0030 218 0.090 178.2 1.3588 0.2074 0.0650 0.0085 42 0.180 178.4 1.0914 0.1184 0.0417 0.0068 81 0.360 178.0 1.0229 , 0.0798 0.0254 0.0064 154 0.720 179.1 0.3835 0.0292 0.0089 0.0024 155 0.090 325.9 1.0117 0.3098 0.1327 0.0063 11 0.180 325.9 1.3759 0.3102 0.0931 0.0086 21 0.360 325.7 0.8667 0.1372 0.0291 0.0054 38 0.720 325.4 I] 0.3958 0.0506 0.0303 0.0025 60 95 Table 4.3 continued Anti-Neutrino Differential Cross Sections x = 0.225 1 1’1 Statistical Systematic Scale ll Ev (GeV) E dxdy Error Error Error Events 0.360 26.0 0.7443 0.2589 0.0681 0.0047 12 0.720 26.2 0.2888 0.0716 0.0155 0.0018 14 0.180 49.5 0.9464 0.1816 0.0839 0.0059 28 0.360 49.6 0.7212 0.0572 0.0194 0.0045 149 0.720 49.5 0.3082 0.0279 0.0070 0.0019 I 107 0.090 91.6 0.5440 0.2856 0.1068 0.0034 I 13 0.180 90.6 1.1168 0.1001 0.0336 0.0070 119 0.360 90.6 0.7276 0.0529 0.0143 0.0046 179 0.720 90.9 0.2648 0.0213 0.0064 0.0017 137 0.090 177.2 1.1913 0.2032 0.0943 0.0075 38 0.180 177.5 0.8455 0.1112 0.0421 0.0053 56 0.360 178.2 0.6116 0.0658 0.0149 0.0038 89 0.720 178.9 0.2356 0.0229 0.0070 0.0015 95 0.090 325.8 2.2550 0.5791 0.1335 0.0141 15 0.180 325.9 0.8411 0.2136 0.0812 0.0053 15 0.360 325.6 0.7384 0.1182 0.0303 0.0046 37 0.720 325.4 0.1932 0.0318 0.0116 0.0012 35 96 Table 4.3 continued Anti-Neutrino Differential Cross Sections x = 0.275 if; Statistical Systematic Scale ‘ ll Ev (GeV) E dxdy Error Error Error ‘ Events 0.360 26.1 0.7082 0.2221 0.0586 0.0044 . 11 0.180 49.5 0.7173 0.1494 0.0505 0.0045 i 22 0.360 49.5 0.6790 0.0573 0.0147 0.0042 , 129 0.720 49.5 0.2615 0.0268 0.0066 0.0016 1 85 0.180 90.3 1.1045 0.0981 0.0364 0.0069 3 121 0.360 91.1 0.5142 0.0454 0.0123 0.0032 ' 121 0.720 90.5 0.2212 0.0211 0.0059 0.0014 . 99 0.090 179.0 0.6754 0.1333 0.0656 0.0042 ) 26 0.180 178.6 0.7238 0.0993 0.0349 0.0045 . 51 0.360 178.6 0.5134 0.0579 0.0261 0.0032 g 78 0.720 179.1 0.2224 0.0215 0.0060 0.0014 ; 93 0.180 325.7 0.8670 0.2075 0.0689 0.0054 g 18 0.360 325.8 0.5310 0.1065 0.0283 0.0033 26 0.720 325.6 0.1186 0.0251 0.0090 0.0007 5 19 Table 4.3 continued Anti-Neutrino Differential Cross Sections x = 0.350 lip—0 Statistical Systematic Scale 7 EV (GeV) E dxdy Error Error Error I Events 0.360 26.0 0.6946 0.1763 0.0636 0.0043 I 17 0.720 26.1 0.2154 0.0508 0.0105 0.0013 15 0.180 49.6 0.6941 0.1113 0.0343 0.0043 39 0.360 49.5 0.4986 0.0355 0.0182 0.0031 177 0.720 49.5 0.1303 0.0130 0.0036 0.0008 82 0.090 90.9 0.7474 0.1915 0.0639 0.0047 22 0.180 89.8 0.7754 0.0606 0.0455 0.0049 155 0.360 90.2 0.4096 0.0279 0.0178 0.0026 196 0.720 90.4 0.1206 0.0101 0.0031 0.0008 112 0.090 178.4 0.8962 0.1212 0.0579 0.0056 I 53 0.180 178.3 0.7128 0.0757 0.0262 0.0045 84 0.360 177.9 0.4480 0.0396 0.0149 0.0028 118 0.720 177.4 0.1057 0.0108 0.0031 0.0007 80 0.090 325.7 0.7350 0.2069 0.0556 0.0046 12 0.180 325.7 0.4318 0.0986 0.0513 0.0027 20 0.360 325.6 0.4253 0.0688 0.0176 0.0027 36 0.720 325.4 0.1095 0.0161 0.0045 0.0007 37 98 Table 4.3 continued Anti-Neutrino Differential Cross Sections x = 0.450 _1__t_1:g_ Statistical Systematic Scale 7 Ev (GeV) E dxdy Error Error Error Events 0.180 49.6 0.3062 0.0727 0.0435 0.0019 18 0.360 49.5 0.3014 0.0282 0.0177 0.0019 104 0.720 49.4 0.0558 0.0084 0.0038 0.0003 39 0.090 91.8 0.5049 0.1524 0.0812 0.0032 11 0.180 90.3 0.4891 0.0459 0.0367 0.0031 107 0.360 91.1 0.2676 0.0233 0.0147 0.0017 124 0.720 89.9 H 0.0586 0.0072 0.0026 0.0004 50 0.090 177.7 ll 0.5523 0.0932 0.0441 0.0035 34 0.180 177.6 ll 0.4067 0.0534 0.0288 0.0025 55 0.360 178.3 H 0.2080 0.0262 0.0054 0.0013 64 0.720 179.1 0.0524 0.0073 0.0023 0.0003 41 0.090 325.8 ll 0.5684 0.1432 0.1179 0.0036 15 0.360 325.7 0.2535 0.0488 0.0161 0.0016 25 0.720 325.6 I] 0.0642 0.0118 0.0031 0.0004 25 Table 4.3 continued Anti-Neutrino Differential Cross Sections x = 0.550 _}__d_’g Statistical Systematic Scale 7 Ev (GeV) E dxdy Error Error Error Events 0.180 49.6 0.1497 0.0512 0.0279 0.0009 10 0.360 49.6 0.1415 0.0186 0.0132 0.0009 54 0.720 49.6 0.0287 0.0061 0.0016 0.0002 16 0.180 90.5 0.2897 0.0323 0.0277 0.0018 78 0.360 89.3 0.1198 0.0139 0.0099 0.0008 70 0.720 90.1 Jl 0.0415 0.0057 0.0009 0.0003 42 0.090 178.2 ll 0.2422 0.0595 0.0409 0.0015 16 0.180 178.5 N 0.1517 0.0292 0.0302 0.0009 31 0.360 wail 0.1075 0.0166 0.0071 0.0007 39 0.720 178.8 0.0201 0.0042 0.0019 0.0001 17 0.180 325.9 0.2662 0.0729 0.0191 0.0017 14 0.360 325.8 ll 0.1261 0.0341 0.0125 0.0008 16 0.720 325.4 0.0208 0.0054 0.0014 0.0001 14 100 Table 4.3 continued Anti-Neutrino Differential Cross Sections x = 0.650 1121 Statistical Systematic Scale 3’ Ev @917)“: E dxdy Error Error Error Events 0.360 49.6 0.0518 0.0097 0.0078 0.0003 27 0.180 89.6 0.0922 0.0142 0.0106 0.0006 42 0.360 89.8 0.0607 0.0085 0.0037 0.0004 49 0.720 89.9 0.0095 0.0018 0.0005 0.0001 18 0.090 178.0 0.1139 0.0262 0.0140 0.0007 18 0.180 175.7 0.0545 0.0156 0.0207 0.0003 13 0.360 179.4 ll 0.0207 0.0059 0.0054 0.0001 16 x = 0.850 _1_d_’0_ Statistical Systematic Scale y Ev (GeV) E dxdy Error Error Error Events 0.360 49.6 0.0067 0.0010 0.0006 0.0000 41 0.720 49.6 0.0015 0.0004 0.0001 0.0000 10 0.090 91.7 0.0059 0.0033 0.0018 0.0000 10 0.180 88.7 0.0107 0.0016 0.0016 0.0001 47 0.360 91.1 0.0034 0.0005 0.0005 0.0000 45 0.720 90.6 0.0009 0.0002 0.0001 0.0000 16 0.090 178.3 0.0118 0.0033 0.0020 0.0001 14 0.180 179.1 0.0073 0.0016 0.0015 0.0000 20 0.360 179.4 0.0027 0.0006 0.0003 0.0000 22 0.720 179.0 0.0005 0.0001 0.0000 0.0000 11 0.180 325.4 0.0058 0.0024 0.0005 0.0000 11 0.720 325.6 0.0006 0.0001 0.0000 0.0000 17 Neutrino Differential Cross Sections 101 Table 4.4 x = 0.015 1311 Statistical Systematic Scale ll Ev (GeV) E dxdy Error Error Error Events 0.360 26.0 1.2096 0.2352 0.1234 0.0076 27 0.720 26.0 0.9833 0.1029 0.0494 0.0062 89 0.180 49.8 0.6169 0.1225 0.1470 0.0039 39 0.360 49.7 1.2205 0.0710 0.0868 0.0076 291 0.720 49.7 1.1519 0.0491 0.0423 0.0072 534 0.090 92.3 0.4909 0.1894 0.1218 0.0031 15 0.180 92.1 H 1.3664 0.0919 0.1122 0.0086 222 0.360 91.7 1.4450 0.0621 0.0587 0.0090 533 0.720 91.7 1.4568 0.0409 0.0522 0.0091 1222 0.045 186.7 1.5292 0.4215 0.1893 0.0096 15 0.090 184.9 1.1107 0.1214 0.1407 0.0070 91 0.180 185.9 1.5442 0.0998 0.0650 0.0097 245 0.360 185.9 1.7569 0.0730 0.0622 0.0110 571 0.720 185.7 1.6706 0.0377 0.0491 0.0105 1851 0.045 336.7 II 1.4856 0.3328 0.1690 0.0093 20 0.090 336.5 1.1487 0.1849 0.0804 0.0072 39 0.180 336.3 2.1238 0.1819 0.0478 0.0133 139 0.360 336.7 1.8061 0.0949 0.0492 0.0113 358 0.720 336.5 1.8567 0.0551 0.0788 0.0116 1077 Table 4.4. Neutrino Differential Cross Sections. The table slows the measured differential cross sections, %g§, (in units of 10'38cm2/GeV) for x, y, EV bins. For each x bin, the table shows the y, E V bin, the differential cross sec- tion, the statistical error, the systematic error, the scale error and the number of events in the x, y, EV. The cross section measurement and the determina- tion of the systematic and scale errors are discussed in the text. Neutrino Differential Cross Sections 102 Table 4.4 continued x =0.045 _1_ 3:1 Statistical Systematic Scale y Ev (GeV) E dxdy Error Error Error Events 0.360 26.1 1.3089 0.2446 0.1488 0.0082 30 0.720 26.1 1.5824 0.1469 0.0525 0.0099 107 0.180 49.7 2.0579 0.2451 0.1645 0.0129 71 0.360 49.7 1.4601 0.0796 0.1021 0.0091 337 0.720 49.7 1.5993 0.0587 0.0597 0.0100 711 0.090 92.7 1.2214 0.2710 0.0991 0.0076 26 0.180 92.2 1.8020 0.1075 0.1233 0.0113 288 0.360 91.8 1.9131 0.0724 0.0728 0.0120 686 0.720 91.9 1.8685 0.0466 0.0754 0.0117 1532 0.090 185.3 1.9157 0.1639 0.1307 0.0120 140 0.180 185.1 2.0241 0.1160 0.0941 0.0127 307 0.360 185.8 || 2.0551 0.0798 0.0741 0.0129 652 0.720 185.9 M 2.0309 0.0431 0.0716 0.0127 2079 0.045 336.7 || 1.3134 0.2874 0.1551 0.0082 22 0.090 336.2 || 1.6587 0.2236 0.1177 0.0104 59 0.180 336.4 ll 2.1006 0.1762 0.0594 0.0131 142 0.360 336.6 2.2904 0.1100 0.0430 0.0143 427 0.720 336.7 I] 2.2165 0.0611 0.1076 0.0139 1240 1(B Table 4.4 continued Neutrino Differential Cross Sections x = 0.080 ld_’a_ Statistical Systematic Scale 3’ Ev (GeV) E dxdy Error Error Error Events 0.360 25.9 1.2640 0.2086 0.1138 0.0079 42 0.720 25.9 1.6462 0.1300 0.0462 0.0103 147 0.180 49.8 1.5447 0.1836 0.1104 0.0097 83 0.360 49.7 1.8831 0.0799 0.0808 0.0118 541 0.720 49.7 1.7612 0.0542 0.0536 0.0110 988 0.090 91.3 1.5057 0.2839 0.1113 0.0094 36 0.180 91.8 1.8583 0.0956 0.1247 0.0116 377 0.360 91.8 1.9018 0.0630 0.0601 0.0119 888 0.720 91.9 1.9244 0.0420 0.0653 0.0120 1971 0.045 186.7 1.2751 0.3351 0.1505 0.0080 17 0.090 185.8 1.9775 0.1454 0.1353 0.0124 186 0.180 185.6 2.1692 0.1054 0.0903 0.0136 433 0.360 185.9 2.2055 0.0713 0.0514 0.0138 937 0.720 186.1 2.0315 0.0376 0.0630 0.0127 2677 0.045 336.7 2.1039 0.3430 0.1431 0.0132 43 0.090 335.1 1.9678 0.2132 0.1107 0.0123 90 0.180 335.7 2.2705 0.1627 0.0606 0.0142 215 0.360 336.4 2.1808 0.0945 0.0504 0.0136 524 0.720 336.7 2.0011 0.0503 0.0936 0.0125 1474 104 Table 4.4 continued Neutrino Differential Cross Sections x = 0.125 _1_ £1 Statistical Systematic Scale ll Ev (GeV) E dxdy Error Error Error Events 0.360 26.0 1.2799 0.2037 0.1105 0.0080 46 0.720 26.0 1.4474 0.1153 0.0324 0.0091 143 0.180 49.8 1.8680 0.1781 0.1109 0.0117 113 0.360 49.7 1.8469 0.0701 0.0608 0.0116 668 0.720 49.7 1.8585 0.0521 0.0423 0.0116 1169 0.090 92.4 1.9448 0.2888 0.1130 0.0122 47 0.180 92.0 1.8731 0.0841 0.0924 0.0117 498 0.360 91.7 1.9517 0.0573 0.0424 0.0122 1134 0.720 91.9 1.8741 0.0378 0.0564 0.0117 2286 0.045 186.7 1.6175 0.3656 0.1583 0.0101 25 0.090 185.2 1.9353 0.1358 0.0654 0.0121 211 0.180 186.0 1.8730 0.0886 0.0465 0.0117 448 0.360 185.7 1.9210 0.0613 0.0447 0.0120 966 0.720 186.0 1.9205 0.0336 0.0428 0.0120 2982 0.045 336. 1.7663 0.2793 0.1842 0.0111 43 0.090 335.8 1.9441 0.2015 0.1004 0.0122 98 0.180 336.5 1.8198 0.1250 0.0813 0.0114 217 0.360 336.4 2.0339 0.0807 0.0413 0.0127 624 0.720 336.6 1.9523 0.0451 0.0772 0.0122 1725 5...... Kg Neutrino Differential Cross Sections 105 Table 4.4 continued x = 0.175 _1___d’_0_ Statistical Systematic Scale I ll Ev (GeV) E dxdy Error Error Error Events 0.360 26.2 1.5932 0.2476 0.1031 0.0100 45 0.720 26.0 1.8352 0.1557 0.0628 0.0115 129 0.180 49.8 1.5938 0.1706 0.0863 0.0100 92 0.360 49.7 1.8220 0.0727 0.0577 0.0114 612 0.720 49.7 1.7324 0.0524 0.0271 0.0108 1027 0.090 92.7 1.6344 0.2526 0.0991 0.0102 49 0.180 91.8 1.7386 0.0818 0.0517 0.0109 I 463 0.360 91.8 1.9232 0.0582 0.0363 0.0120 1066 0.720 91.6 1.6636 0.0373 0.0407 0.0104 1893 0.045 187.0 1.5323 0.3322 0.1536 0.0096 23 0.090 185.6 1.7337 0.1277 0.0675 0.0108 190 0.180 186.0 1.9088 0.0925 0.0501 0.0119 428 0.360 185.6 1.7147 0.0584 0.0271 0.0107 856 0.720 186.0 1.7460 0.0334 0.0374 0.0109 2541 ‘ 0.045 336.7 1.5139 0.2342 0.1385 0.0095 I 45 0.090 336.2 1.6374 0.1735 0.0915 0.0102 93 0.180 336.4 1.6315 0.1210 0.0527 0.0102 187 0.360 336.0 1.7272 0.0759 0.0396 0.0108 II 524 0.720 336.6 1.7169 0.0432 0.0575 0.0107 II 1479 Neutrino Differential Cross Sections 106 Table 4.4 continued x = 0.225 _1__c£g_ Statistical Systematic Scale ll Ev (GeV) E dxdy Error Error Error Events 0.360 25.9 1.4644 0.2439 0.0722 0.0092 39 0.720 26.2 1.7409 0.1678 0.0565 0.0109 107 0.180 49.9 1.4423 0.1744 0.0566 0.0090 81 0.360 49.7 1.6614 0.0709 0.0316 0.0104 540 0.720 49.7 1.5672 0.0544 0.0227 0.0098 796 0.090 92.7 1.0182 0.2120 0.1121 0.0064 33 0.180 91.8 1.5380 0.0764 0.0517 0.0096 412 0.360 91.7 1.6106 0.0540 0.0246 0.0101 880 0.720 91.8 1.4526 0.0370 0.0253 0.0091 1490 0.045 186.2 1.2136 0.2904 0.1068 0.0076 22 0.090 184.5 1.4452 0.1160 0.0565 0.0090 164 0.180 185.9 1.5740 0.0817 0.0365 0.0099 376 0.360 186.1 1.5986 0.0575 0.0253 0.0100 775 0.720 186.1 1.4508 0.0316 0.0230 0.0091 2006 0.045 336.3 1.5227 0.2526 0.1349 0.0095 38 0.090 336.6 1.4080 0.1638 0.0787 0.0088 78 0.180 336.2 1.4688 0.1215 0.0779 0.0092 150 0.360 336.7 1.4131 0.0674 0.0534 0.0088 441 0.720 336.7 1.4345 0.0406 0.0469 0.0090 1194 107 Table 4.4 continued Neutrino Differential Cross Sections x = 0.275 _1_d_’0‘_ Statistical Systematic Scale 7 E V (GeV) E dxdy Error Error Error Events 0.360 26.0 1.3213 0.2352 0.0795 0.0083 34 0.720 26.2 1.1280 0.1454 0.0532 0.0071 78 0.180 49.8 1.3536 0.1568 0.0390 0.0085 87 0.360 49.7 1.4444 0.0687 0.0320 0.0090 439 0.720 49.7 1.3479 0.0551 0.0352 0.0084 586 0.090 92.7 0.8982 0.1844 0.0809 0.0056 34 0.180 91.9 1.3372 0.0747 0.0504 0.0084 327 0.360 91.7 1.3614 0.0500 0.0296 0.0085 739 0.720 91.9 1.2699 0.0366 0.0198 0.0079 1184 0.045 186.8 1.4349 0.3355 0.1405 0.0090 23 0.090 186.5 1.3901 0.1194 0.0521 0.0087 144 0.180 185.1 1.3501 0.0769 0.0383 0.0084 319 0.360 185.2 1.2021 0.0493 0.0189 0.0075 606 0.720 185.8 1.2179 0.0301 0.0214 0.0076 1594 0.045 336.5 0.7804 0.1647 0.1166 0.0049 26 0.090 334.1 1.3287 0.1637 0.1512 0.0083 75 0.180 336.5 1.4306 0.1145 0.0570 0.0090 166 0.360 336.1 1.3249 0.0665 0.0344 0.0083 401 0.720 336.6 I] 1.2520 0.0383 0.0287 0.0078 1033 108 Table 4.4 continued Neutrino Differential Cross Sections x = 0.350 131 Statistical Systematic Scale 7 E v (GeV) E dxdy Error Error Error Events 0.360 26.2 1.0577 0.1656 0.0603 0.0066 46 0.720 26.1 1.1470 0.1151 0.0416 0.0072 101 0.180 49.8 1.1251 0.1101 0.0475 0.0070 108 0.360 49.7 1.0115 0.0417 0.0483 0.0063 586 0.720 49.7 0.9945 0.0365 0.0273 0.0062 721 0.090 91.8 0.9053 0.1358 0.0607 0.0057 55 0.180 91.7 1.0661 0.0463 0.0522 0.0067 537 0.360 91.8 1.0094 0.0308 0.0370 0.0063 1067 0.720 91.9 0.9029 0.0231 0.0192 0.0057 1488 0.045 186.6 1.0864 0.1992 0.0691 0.0068 31 0.090 185.7 0.9335 0.0680 0.0352 0.0058 196 0.180 185.9 0.9389 0.0452 0.0432 0.0059 440 0.360 185.7 0.9642 0.0323 0.0374 0.0060 891 0.720 186.0 0.8168 0.0181 0.0160 0.0051 1969 0.045 336.6 0.9782 0.1438 0.1017 0.0061 51 0.090 336.0 0.9412 0.0910 0.1252 0.0059 110 0.180 336.1 0.9968 0.0679 0.0672 0.0062 222 0.360 336.0 0.9521 0.0391 0.0465 0.0060 604 0.720 336.6 0.8152 0.0223 0.0181 0.0051 1288 109 Table 4.4 continued Neutrino Differential Cross Sections x = 0.450 _1_ 31 Statistical Systematic Scale 7 E V (GeV) E dxdy Error Error Error Events 0.360 26.1 0.5368 0.1241 0.0737 0.0034 20 0.720 26.3 0.3090 0.0766 0.0696 0.0019 33 0.180 49.7 0.7251 0.0851 0.0668 0.0045 78 0.360 49.8 H 0.6068 0.0327 0.0417 0.0038 356 0.720 49.8 0.5821 0.0315 0.0295 0.0036 360 0.090 92.3 0.7959 0.1394 0.0830 0.0050 42 0.180 91.5 0.6387 0.0356 0.0639 0.0040 337 0.360 91.3 0.5831 0.0235 0.0362 0.0036 644 0.720 91.9 0.4857 0.0179 0.0210 0.0030 753 0.045 187.5 0.6075 0.1464 0.0783 0.0038 22 0.090 185.8 0.6734 0.0544 0.0585 0.0042 164 0.180 185.8 0.5419 0.0344 0.0423 0.0034 257 0.360 185.8 0.5186 0.0232 0.0327 0.0032 519 0.720 185.9 0.4359 0.0138 0.0178 0.0027 1010 0.045 335.5 0.4117 0.0869 0.0595 0.0026 26 0.090 335.5 0.6630 0.0732 0.0574 0.0041 84 0.180 336.4 II 0.6493 0.0534 0.0459 0.0041 155 0.360 336.5 ll 0.5531 0.0291 0.0357 0.0035 366 0.720 336.7 [I 0.4020 0.0154 0.0163 0.0025 690 110 Table 4.4 continued Neutrino Differential Cross Sections x=0.550 142—0 Statistical Systematic Scale ll Ev (GeV) E dxdy Error Error Error Events 0.360 26.0 0.2749 0.0873 0.0351 0.0017 13 0.180 49.7 0.3558 0.0581 0.0546 0.0022 42 0.360 49.7 0.3572 0.0242 0.0366 0.0022 223 0.720 49.7 0.2860 0.0226 0.0234 0.0018 164 0.090 92.4 %0.4701 0.0912 0.0652 0.0029 31 0.180 91.6 0.3359 0.0239 0.0423 0.0021 209 0.360 91.6 H 0.2824 0.0149 0.0284 0.0018 372 0.720 91.4 H 0.2245 0.0120 0.0136 0.0014 369 0.045 186.5 j] 0.4717 0.1195 0.0577 0.0030 17 0.090 185.1 0.3051 0.0338 0.0456 0.0019 87 0.180 185.4 0.2782 0.0218 0.0268 0.0017 176 0.360 184.8 0.2381 0.0146 0.0243 0.0015 279 0.720 185.8 0.1889 0.0086 0.0131 0.0012 508 0.045 336.3 0.3405 0.0699 0.0597 0.0021 25 0.090 334.3 0.2420 0.0354 0.0309 0.0015 n 54 0.180 336.5 || 0.2479 0.0285 0.0300 0.0016 87 0.360 336.6 N 0.2367 0.0167 0.0184 0.0015 212 0.720 336.5 I] 0.1624 0.0093 0.0085 0.0010 II 315 111 Table 4.4 continued Neutrino Differential Cross Sections x =0.650 __1__dlr_ Statistical Systematic Scale I ll Ev (GeV) E dxdy Error Error Error Events 0.360 25.8 0.1508 0.0528 0.0217 0.0009 10 0.180 49.9 0.1112 0.0281 0.0241 0.0007 29 0.360 49.8 0.1201 0.0116 0.0141 0.0008 112 0.720 49.7 0.0871 0.0112 0.0099 0.0005 64 0.090 92.5 0.2199 0.0482 0.0306 0.0014 22 0.180 92.0 l}0.1219 0.0114 0.0210 0.0008 120 0.360 91.4 0.1131 0.0080 0.0119 0.0007 210 0.720 91.5 0.0881 0.0065 0.0065 0.0006 195 0.090 185.3 1| 0.1281 0.0176 0.0197 0.0008 58 0.180 183.3 0.1194 0.0125 0.0162 0.0007 102 0.360 185.6 || 0.0886 0.0076 0.0102 0.0006 153 0.720 186.0 || 0.0691 0.0046 0.0067 0.0004 244 0.045 337.4 II 0.1041 0.0333 0.0223 0.0007 19 0.090 335.4 || 0.0598 0.0147 0.0227 0.0004 30 0.180 334.8 N 0.1083 0.0143 0.0160 0.0007 62 0.360 36.2 H 0.0799 0.0077 0.0100 0.0005 114 0.720 336.4 [I 0.0654 0.0049 0.0033 0.0004 191 Neutrino Differential Cross Sections 112 Table 4.4 continued x =0.850 _1__d’_0_ Statistical Systematic Scale I! E v (GeV) E dxdy Error Error Error Events 0.180 49.7 0.0174 0.0033 0.0027 0.0001 33 0.360 49.7 0.0125 0.0012 0.0019 0.0001 118 0.720 49.8 0.0109 0.0014 0.0012 0.0001 63 0.090 92.5 H 0.0338 0.0062 0.0037 0.0002 31 0.180 91.1 0.0135 0.0012 0.0025 0.0001 141 0.360 91.4 H 0.0105 0.0007 0.0013 0.0001 232 0.720 91.9 H 0.0075 0.0006 0.0006 0.0000 190 0.090 177.0 0.0075 0.0012 0.0017 0.0000 66 0.180 186.1 0.0114 0.0011 0.0019 0.0001 103 0.360 185.4 0.0063 0.0006 0.0010 0.0000 133 0.720 185.5 0.0062 0.0004 0.0006 0.0000 259 0.045 336.7 0.0152 0.0034 0.0028 0.0001 21 0.090 334.0 0.0156 0.0022 0.0018 0.0001 51 0.180 334.0 {j 0.0100 0.0012 0.0013 0.0001 ll 79 0.360 336.0 0.0065 0.0006 0.0010 0.0000 || 123 0.720 336.5 I] 0.0056 0.0004 0.0005 0.0000 II 192 113 89.6 05 mo gorges“. 05 one 8265a 5:038 05 mo 58988 05 :0 £500 no". 2x3 05 00m 885 in O R 05 E 8:96 0:530: 98 053808. we non—Es: 2.: 2865 5:200 93 E 9:. 022.58 0.8 9.05 28a 05 0238893 485an 05 command 5 5300 we £5 5 03:08am on .9 .836 .e 5 see ease as Sod Beebe 6.8854 253m 9: as needed. 3:895 Encoded E we use... aefi am sad 88d aaaad demkd Ndaad eezd menad adada mead mead aamm at 88d Nemad Ndead «mead aaaad maaad 88d add? 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Chapter 5 Results and Conclusions 5.1 Introduction In Chapter 4 the extraction of differential cross sections and structure functions from the FMMF charged current neutrino—nucleon scattering data was described and the measured cross sections and structure function were presented in a series of tables. This chapter will discuss the structure functions in the context of QCD evolution, make comparisons of the FMMF structure functions with those of other experiments, describe the fitting of a set of parton distribution functions (PDF) based on the FMMF structure functions and finally present a series of measurements of AQCD from the (22 evolution of the FMMF structure functions. 5.2 W In this section, the FMMF structure functions are compared with the qualitative expectations of QCD and with the measured structure functions from other experiments. 5.2.1 Q2 Evolution An important prediction of the parton model is that structure functions should be constant instead of falling rapidly as a function of Q2. This is the phenomenon known as scaling and was an early indication that the nucleon had an internal structure. QCD modifies the predictions of the parton model. QCD predicts that the structure func- tions should evolve as a function of Q2. The Altarelli-Parisi equations (Altarelli and Parisi 1977) desribe this Q2 evolution. The qualitative prediction of QCD is that as Q2 in- creases, the observed number of soft (low x) gluons and quarks from the quantum sea 117 118 increases. This results in the evolution of the structure functions as a function of Q2. The low x structure functions will grow as a function of Q2 while the high x structure func- tions will shrink and the ”intermediate” x structure functions will be constant. Figure 5.1 shows a schematic of the Q2 evolution of parton distributions. Figure 5.2 plots the FMMF structure functions and their statistical errors as a func— tion of Q2 for the various x bins. The structure functions show the behavior expected by QCD. At low x, the structure functions grow as a function of Q2. At high x, the structure functions decrease as a function of Q2. As least qualitatively, the measured structure functions agree with the predictions of QCD. 5.2.2 CDHSW A comparison of the FMMF structure functions and those of CDHSW (Berge et al. 1991) is presented in Figure 5.3. The CDHSW structure functions are derived from an extremely large data set of 640,000 reconstructed and accepted neutrino events and 550,000 anti-neutrino events. The CDHSW data were taken using the CERN SP5 magnet horn wide band beam. Since the CERN 5135 could deliver only 400 GeV protons, the avaage energy of the neutrinos from the CERN horn beam was significantly lower than that available at the Tevatron. The horn beam did have the advantage of being a sign- selected beam (i.e. the beam consisted of only neutrinos or anti-neutrinos). This allowed the collection of the large anti-neutrino data set. The CDHSW detector consists of a large magnetized iron scintillator sampling calorimeter with interspersed drift chambers. The muons were immediately focused by the magnetized iron absorber plates and the muon tracks were reconstructed using drift chambers. The CDHSW structure function analysis was done in the same manner as this analysis using assumed total cross sec- tions. The major differences between the two analyses is that CDHSW makes no correction for Fermi-Motion and that CDHSW uses the assumed form of R, 119 3.0 oil'" I Y 2.5 2.0 xq+ xfi 1.5 I I T I I I I T Tfi ..T-...l..‘J...+o 1.0 0.5 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ‘1 x Figure 5.1. Effects of (22 Evolution. The figure shows the combined quark and anti-quark distributions, xq+ x'q' , as a function of x for different values of Q2. The solid outline shows xq+ xq for Q2 = 5 GeVz. The dotted outline shows xq+ xq for Q2 = 250 GeVz. Note how the sea quarks dominate the high (22 distribution. The Pumas—acorns parton distributions were used for this figure. -4 2.5 D 2.0 0 mm. 't 2.0 .0 D 0 d 1.5 D U - ‘l 1.. I- D D 3.0.068 U _ 05 1.1 t- o q 10 0.0 r— 1.0 D D D l3ns.-0.000 .1 1.50 .— D O 4 1.2 x-O.128 a D D 1.25 I— -1 0.. D 1.“) r- "I I.” 0.78 v- D D D D D 0.0.170 "'35 0.!) i— am 1.” I— - . .1 0.78 1.25 .- x-0225 1.“) u— D d 0.00 D D D D 0 1.0 0.78 t— D O 0.00 .. D D ”'03,. 0 0.0 0.. I- x—OSEO 0.7 I- - 0.. U D 0.6 - D o 0 0 0.0 0.5 - 0.4 — 0‘ __ D a 0.0.650 “ 0" D 0 °—3 - 0.0.500 0.2 _ D —I 0.2 D D -l 0.3 D 0.1 .0 —1 0.2 0.0 v— 0.000 _ 3.0.000 _ 0.: 0.015 .. D D x-0.050 _ 0.0 .010 _ ° 0 O 0.003 .— O 0.000 - .1 3 10 10 (22 (GE’W/e) 0.1 0.015 0.01 0 10 -| 0.4 ¢ —J 0.3 _ + ¢ -4 0.2 ¢ ”.018 T 0.1 d 0.0 - 0 ¢ -. 1.20 .. am 0 ¢ 0 q I.” — D D Ont-0M _‘ 0.78 — ¢ 0 0..” — ‘ D D ”.12. a a _t 0.28 — O - 1.25 .- -l 1.00 —t 0.?! 3.0.228 -: 0...) D D D D -« 0.20 '- q 1.0 _ + ¢ u 0.0 '- 0 03-027. _ -t 0.6 III-0.350 - ¢ 0 0.‘ D .— - o -l 0.‘ " M _ ¢ 0 0 x-OJIO _J 0.2 _- O D -l 0.3 0 D 0 q 0.! " x-om -t 0.1 ... D D —I 0.0 '- x-Ouo L- ¢ 0 a L -I 3 10 Q2 (0312/0) Figure 5.2. The PM Structure Functions. The figure plots the FMMF structure functions as a function of Q2 for all x bins. Errors shown are statistical only. 2L£5 1.55 113 ()15 204 ZLCD 106 1.2 (313 2L() 1.!5 1.2 2L() 108 1.!5 104 1.2 1.() 121 x=0.015 IIIIIIII'IIIIIIIII'IIII (3 hi E o fill IIII IIII IIII IIII IIII IIII IIII IIII ITII I l l I l l I t IIIIIIIIIIIII'IIII'IIIII (3 0| 1 1111;“! 1 11 x=0.080 "2 § 1.0 ‘93 C) C) 0 «a IIIIIIIIIIIIIIIIIIIIIII (304 x=0.125 "2 x=0.125 1 .0 § an!” o... W EJ:Fj‘FfJ!EJ (3:4 1 10 102 10 1 10 102 Q2 (GeV’) (22 (GeV) IIIIIIIIIIIIIIIIIIlIIII IIIIIIIIIIIIIII III ‘1 H 0l u .1. Figure 5.3. Comparison of FMMF and CDHSW Structure Functions. Shown are the FMMF structure functions (0) and the CDHSW structure functions (El) plotted vs (22 for the various x bins. The error bars for the FMMF structure functions are the quadratic combination of the statistical, systematic and scale errors. The statistical errors are indicated by the cross bars. Note the logarithmic abscissa. 122 F xF 2 3 x=0.l75 ; 5 :‘li I? l 1111‘” l llllflll Ill x=0.225- EIIIIIIIIIIIIIIIIIIIII' E- x=0.225 "2 F I I 1.1 L'— 1°° .— f 5 E i 1 1-0 E- ; °° :— W : I 0.9 L 06 7 : I 0.8 E— 0-4 E" 1.00 1: 10° : : x=0.275 : x=0.275 0.95 :— 0-90 E" I I 0.90 :_ 4) 0.00 :— + (:3 p- I- 0.85 [_— ¢¢ wall) 0.70 [- 0? 0.80 E— 0.60 E— : {1' 0.8 '_' 0.7 " : x=0.350 ; x=0.350 0.7 L C : 0.6 — § : : 06 E- i t + $ " 0.5 v-— i 0.5 T. P ; C t- t- o.4 . 0.4 . ..1 2 —‘l 2 10 1 10 10 10 1 10 10 (22 (GeV) Q2 (GeVz) Figure 5.3 continued. Comparison of FMMF and CDHSW Structure Functions. 0.08 0.04 0.020 0.016 0.012 0.“)8 0.004 0.000 Figure 5.3 continued. Comparison of FMMF and CDHSW Structure Functions. F 2 x=0.450 D M. x=0.550 Ch ”‘10.. x=0.650 D D D was IAIIIIIIIII'IIIIIIIII'IIII IIIIIIIII'IIIIIIIII IIIIITIIIIIIIIIIII 5 x=0.850 F t: F p L i 5 t t. O t: t: _1 2 0 1 10 10 02 (Gevz) 123 0.6 0.5 0.4 0.3 0.4 . 0.3 0.2 0.08 0.04 0.020 0.016 0.012 0.008 0.004 0.000 in! 3 E x=0.450 E’ mm t t IIII'lIIIT1WIIIIIlIIII t 1 1 111ml 1 1 1 x=0.650I .023: '1! W. Jc=0.850I IIII'IIIIIIIIIIIIIIIIIII it Q lllllfllll 2 10 1 1111“]! 10 02 (GeV) OI EIIIIIIII'IIII'IIII'IIII fl pl 124 .....2)=———10§(;.::2)- As one can see there is general agreement between the two experiments. In F2 there is very good agreement in all the x bins. The CDHSW data with its much greater statistics and better acceptance (due to the magnetized calorimeter) is much more finely binned and in some regions probes higher Q2 than the data of this experiment. For xF3 the agreement between the two experiments is still good. The lack of anti-neutrino event statistics results in larger errors for the FMMF structure functions but, in general, the two data sets are consistent within their respective errors. In the x = 0.175 and x = 0.225 bins, the FMMF xF3 data seems to be systematically above that of CDHSW but even in these bins the differences seem to be within 2 standard deviations. 5.2.3 CCFR A comparison of the FMMF structure functions and those of CCFR (Quintas 1992) is presented in Figure 5.4. The CCFR data was taken at Fermilab in 1985 and 1987 using the Quad-Triplet Beam concurrently with the data taken by FMMF. The CCFR data set is roughly an order of magnitude larger than FMMF with 1,050,000 reconstructed, accepted neutrino events and 180,000 reconstructed, accepted anti-neutrino eventa The CCFR de- tector was located upstream of the FMMF detector in Lab E. The CCFR detector is a more traditional design than the CDHSW apparatus. The CCFR detector consists of a large iron scintillator sampling calorimeter followed by a muon spectrometer. The CCFR detector is approximately 3 times more massive than FMMF and could be triggered multiple times per neutrino spill. This resulted in the larger CCFR data set. The CCFR structure function measurement was done in a more traditional manner. With their greater statistics, CCFR extracted the relative neutrino and anti-neutrino fluxes and used the known total neu- trino cross section at low Ev to normalize the neutrino flux and to measure the ratio of VN dZOVN AW 0 A ,N . The differential cross sections, 2 , can be measured once the neutrino dxd logQ 2.5 2.0 1.5 1.0 0.5 2.4 2.0 1.6 1.2 0.8 2.0 1.6 1.2 0.8 2.0 1.8 1.6 1.4 1.2 1.0 125 F2 XFS 0.4 ’ x=0.015 E x=0.015 :- OAAA 0'3 :— f '— A I _E_ o“ 0-2 :— {AAT6 + : O : E_ 0.1 :— W. .o W :- x=0.045 ‘ a x=0.045 : 0.0 :- E0 aA‘gA °" E” A A}; 5 9.0 04 E $113 A i 0.2 E— E E ”mull ”mull ”Hm“ “I E g=0.080 “2 :— x=0.0801 :- 069 "° 2' :— 0.6 5. *QAAAAA E- 0.4 E— g x=0.125 ‘2 :- x=0.125 E 1.0 E— : . Q § :— 0.0 :- AAAA¢ : 5 *3 E AAQQA“A 0.6 Er :— ? 0.4 E— 104 1 10 102 10'1 1 1o 102 Q2 (cm (22 (Gev5 Figure 5.4. Comparison of FMMF and CCFR Structure Functions. Shown are the FMMF structure functions (6) and the CCFR structure functions (A) plot-ted vs (2" for the various x bins. The error bars for the FMMF structure functions are the quadratic combination of the statistical, systematic and scale errors. The statistical errors are indicated by the cross bars. Note the logarithmic ab- scrssa. 1.4 1.2 1.0 0.95 Figure 5.4 continued. Comparison of FMMF and CCFR Structure Functions. ..1 O ..1 F 2 x=0.l75 0900’: 3 A 4 x=0.225 ‘1‘ t4» AA '§ § x=0.275 4A ‘tj t *M x=0.350 IIIIIIIIIIIIIIIIIIIIII II IIIIIIIIIIIIIIII'IIIII “it: “As; A IIIlIIIIlIIII'IIII 1 11111;]! 1 1 1111111 4 1111M, l u N 10 10 Q2 (Gevz) 126 1 .2 1 .0 0.8 0.6 0.4 1 .2 1.0 0.6 0.4 LCD 0.90 0.80 0.70 0.7 0.6 0.5 0.4 x=0.175 5* 0 AAQAAA¢ x=0.225 Q 4aad%:fi IIIITIII'IIIIIIIII'IIIII III'IIIIIIIII'IIII'IIIII I I I l I I I I I I I I I IIIIIIIIIIIIIIIIIIIIIIII D D a» ..1 O 1.1 *1 N 9. I .1 10 10 02 (GeV‘) ’7 l t 0.6 0.4 0.3 0.4 0.3 0.2 0.08 0.04 0.020 0.016 0.01 2 DADS 0.004 0.000 2 5 x=0.450 :- t A r +41“ : A :- “AA E I ll 5 x=0.550 :— ‘t A E Q ‘3AA x=0.650 E. A E A30 .- A E 0A°A E .mnnl “Hm; ......“I 1 .1 x=0.850 =_ A E— “A; :— A"MA p—l 2 l 11 O 1 10 10 02 (Gevz) 127 0.6 0.5 0.4 0.3 0.4 0.3 0.2 0.08 0.04 0.020 0.016 0.012 0.008 0.004 0.000 3 E x=0.4501 E + :— A i “1 QA : f A 5 MM 1 1 1111.111 1 I 1 5 x=0.550 E A E- ? Q5 A : s 3‘54 i '_1_1_1.1.1111I_L.1_1.1.1.url_1_1_1.11111|_1_1_1. x=0.650t E E ’- 1 1 1 111 1 1 1o.1 1 10 1o 02 (Gevz) Figure 5.4 continued. Comparison of FMMF and CCFR Structure Functions. 128 and anti-neutrino fluxes are known. From the measured differential cross sections, can: extracts the structure functions. The other major difference between the CCFR and the FMMF analyses is that cm: makes no correction for Fe'mi-Motion. As one can see there is general agreenert betweer the FMMF and CCFR structure functions. CCFR ’s greater statistics result in their structure functions coveing a greate' range in (22 than those of FMMF. For F2, there is general agreement between the two experiments. For xF3, again thee is gereral agreenent but in the mid x bins, the FMMF structure functions seen to be systenatically larger than those of CCFR. 5312' D'I'l l' E I' E'll' The FMMF structure functions have been used as input for a parton distribution fitting program written by W.K. Tung (Private Communication, Morfin and Tung 1991). The fitting precedure uses a large package of routines that include Next-to-Lead- ing Orde (NLo) QCD and standard electroweak theory. The program allows the fitting of partons to the form: p,(x,Q§) = Ax“ (1 — x)’9 log’(1 — x), where Pi is the probability that whet probing with a probe of strergth Q3, one will find an ith parton in the interval x—->x+dx. The values of a, [3, and ycan be varied for each quark or gluon distribution individually or correlations can be made as desired. The norrnalizations (based on A) are, where possible, fixed by sum rules. Whee sum rules do not fix the normalization, the normalization then becomes anothe parameter in the fit. The parton distributions are then evolved to higher (22 using NLO QCD with the value of A— another free paramete of the fit. The standard 352 is then formed and minimized MS’ to determine the values of Am, Ai, 01,-, 3;, and )9. For this analysis, the freedom the fit was allowed was sharply limited due to the limited data used as input. The factor, log’ (1 — X), was set to 1 for all distributions. The charm sea was assume to be zeo. The other sea quark distributions wee assumed to 129 have the same shape, which was simplified to the form (1 — x)”. The relative normaliza- tion of the strange sea to the up and down sea was set by previous measuremerts. The gluon distribution was fit to the same form as the sea distributions with the gluon normalization allowed to float. The valence quark distributions are fit to the form xa(1 - xf. awas assumed to have the same value for both up and down valence quark distributions while for [3 no correlation betweer the values of [3 for the two distributions was required. The fit for the PDF was done using the FMMF structure functions with Q2 > 16 GeV2 and w2 > 16 GeVz. Table 5.1 shows the final values obtained from the fit. Figures 5.5 and 5.6 shows a comparisons of the parton distributions obtained from fitting the FMMFdataand thoseofmtRs-BCDMsa-Iarrimanetal. 1990) atQ2=16GeV2andQ2=50 GeVz. Thee is geneal agreenert between the PM parton distributions and the HMRS distributions but the FMMF PDF favor a lower average x distribution for the valance quarks and a harde gluon distribution. The FMMF gluon distribution has a much longe high x tail. In should be noted that the gluon distribution is not directly probed by neu- trino-nucleon scattering and so the derived xG distribution must infe'red from the evolution of the structure functions. The integrals of the FMMF valence quark distribu- tions are larger than those of the I-IMRS. This due the fact that the FMMF xFa structure functions are larger than those of the other neutrino experimerts. 5.4 Aoco fitting As previously discussed, QCD makes a quantitative prediction about the Q2 evolution of structure functions and parton distributions. The evolution of the structure functions is described by the Altarelli-Parisi equations (Altarelli and Parisi 1977). In this section we present a measurenert of AQCD using the evolution of xF3. A program originally written by Duke and Owens (Devoto et a1. 1983) and mod- ified by Oltrnan (1989) and other membes of the ccm collaboration was obtained from 130 Table 5.1 Results of Parton Distribution Fitting ll Parameter Value Overall Fit Are: 176.9193 MeV 1’ 67.0 Degrees of 49 Freedom xz/ DOF 1.37 Valence Distributions a 0810:0030 Dow“ Quark ll [3 493210.575 Valence Up Quark Valence [3 351810141 Gluon Idex 041300.051 ,8 066910257 Sea Quark [3 8.6091078] Table 5.1. Results of Parton Distribution Fitting. The table shows the results of the PDF fitting program. The table is divided into 6 parts reflecting the portions of the fit for which a given set of parameters are relevant. The box titled ”Overall Fit” shows the x2 of the fit and Ar; which controls the Q2 evolution of the PDF. The othe' parts show parameters for the individual parton distribution functions at Q} = 2.5 GeV2 131 1.0 _ 1.0 A 0-3 E'- Xd 0.8 X11 0.6 E— 0.6 b 0.4 0.2 0.071lllllllllllllll Ill] 0 0.2 0.4 0.6 0.3 r o 0.2 0.4 0.6 0.3 1 4.0 3.0 'Jlo‘l" UIUIUI UF'ITTUUIIIIU C Figure 5.5. FMMF Parton Distribution Functions at Q2: 16 GeVz. The figure shows the FMMF PDF. The FMMF distributions are shown as the solid outlines. For comparison, the mans-acorns I’DF are also shown as the dotted outlines. The top row shows the valence quark distributions. The center—left plot shows the gluon distributions. The center-right plot shows sea part of the valence quark distributions, xd(a xi ). The bottom-left plot shows the strange sea distribution, x§. .15 132 [IUUIILL 0.2 0.2 _ 0.0 JJlllllllll 0.0:llllllllllllll 11.1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 'l 4.0 .0 0.4 .- xGl xd 3.0 0.3 p- 2.0 0.2 L : r- . C 1.0 '— o.1 L C I i- .... :- o.o"rrrrlrr'r"1~. I 11 " O 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 5.6, FMMF Parton Distribution Functions at Q’= 50 GeV2. The figure shows the FMMF PDF. The FMMF distributions are shown as the solid outlines. For comparison, the HMRS—BCDMS PDF are also shown as the dotted outlines The top row shows the valence quark distributions. The center—left plot shows the gluon distributions. The center—right plot shows sea part of the valence quark distributions, xd(.=. x'fi). The bottom—left plot shows the strange sea distribution, x§. 133 Paul Quintas (1992). The program was used to fit do leading order fits of the non-singlet evolution of the m structure functions. The program fits the non-singlet structure function FM; to the form, FNs(x) = Ax“(l - xf(1 + 7x) + 3x5, at a fixed (22 = Q3. The Altarelli—Parisi equation (1.38a) is then regularized to obtain: 37: 8FN_S(x, t) = a, (t) at [3 + 4log(l - x)]FN5(x,t) + 2 f 113%[0 - y2)FNs(x/y,t) - 21=NS (x,t)], where t = log(Q2 / A2). Now the evolution of the non-singlet structure function is a direct function of t and can be measured. This method has the great advantage of not requiring knowledge of the gluon distribution but suffers from the larger errors assoc- iated with the statistical errors of xF3. At higher x, one hopes that the contribution of the quantum sea will disappear and the relationship, F2 5 xF3 will hold and one could substitute F2 for xF3 above an x of 0.3 or so while doing the non- singlet fit. The FMMF data has been used to fit for A. Three separate non-singlet fits have been preformed. The input structure functions which are treated as PM are: OxFafor 0.0 10GeV2.Fortheerrorsused in the fitting, the statistical, systematic and scale error were combined in quadrature. The results of the three fits are shown in Table 5.2. Frgures 5.7, 5.8 and 5.9 show the re- sults of the fits. The fit using only xF3 returns, v— 134 A = 176.9:t66.2 MeV. The large error is due the lack of precision of xFa, especially at high x. The f for this fit is quite good at 25.4 for 26 degrees of freedom. This fit indicates that within the precision of the data the non-singlet evolution is as expected from QCD. In the second fit, the xF3 data above an x of 0.3 is replaced with that of F2 and the fitting is done in the same manner. This fit returns, A = 20121155 MeV. The much smaller error is due the greater precision of the F2 structure functions. The f for this fit is 34.1 for 26 degrees of freedom. 50 while the error on A shrank, the overall fit is of a lesser quality. The probability of the xF3 only fit is 49.6% while the probability for this fit is only 13.3%. Inthethird fit, thesamer data thatwasusedinthesecond fitisusedbuttheng data is ignored. This fit retums, A = 177.8 :I: 11.8 MeV. Thef for this fit is 4.2 for 11 degrees offreedom with a very small error on the deter- mined A. 55 Conclusions The FMMF structure functions are a set of high statistics vN structure functions. As we have seen, the structure functions are in good agreement with the predictions of QCD. Comparisons between the FMMF structure functions and those of CDHSW and ccr-rr show general agreement The FMMF structure functions have been used as input to a parton distribution fitting program and the fitted I’DF have been presented. Finally a measurement of AQCD from the (22 evolution of the non-singlet structure function has been presented which is consistent with the expectations of QCD. The FMMF structure functions presented in this thesis are a valuable contribution to our overall knowledge of the structure of the nucleon. 135 Table 5.2 Results for Non-Singlet AQCD Fitting Structure Functions Used AQCD x2! nor XF3 for 0.00 < x < 0.70 176.9 :t 66.2 MeV 25.4/ 26 xFa for 0.00 < x < 0.30 201 2 :1: 15 5 M V 34 1/26 132 for 0.30 < x < 0.70 ‘ ‘ e ' F2 for 0.30 < x < 0.70 172.8 i 11.8 MeV 4.2/ 11 Table 5.2. Results for N on-Singlet AQCD Fitting. The table shows the results of the fitting for non-singlet A00} The first column shows the structure frmctions used as input for the fitting program. The input structure functions are treated as Fm and the leading order Altarelli—Parisi equation is used to extract AQCD The second column shows the returned value of AQCD and the estimated error from the fit. The third column shows the x2 of the fit. 136 0.2 p r. 0.1 r— r- ‘3}— 0.0 \ I ~ —£ —o.r _- N or t '3 fl \_ .3 —O.2 — '4?— . —g3__ -O.3 .— —o.4 _— 1. _0.5rrrrIrrrrirrrrlrrrerrLrirrrrlrrrr O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x Figure 5.7. Result of Non-Singlet Fit using Fw=xF3 for 0.0 < x < 0.7. Plotted is the fit of d £3222 from the FMMF structure functions vs x. The line shows the value of 112$- as determined from the fit using the Duke and Owens £110ng fitting program. 137 0.2 0 pl j I T I I I I T j A T / 4;.— I I I I .051 ._ N F- 01 _ w t- n 2 . 9:, “B —0.2 -—- r— _{J— I- AH —o.3 _- _o,4 ... _o.5lllllllllIllJLlllllllllllLLLLlAlll 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x Figure 5.8. Result of Non-Singlet Fit using FNs=xF3 for 0.0 < x < 0.3 and FNs=F2 for 0.3 < x < 0.7. Plotted is the fit of (1:021:22 from the FMMF structure functions vs x. The line shows the value of [1:102:22 as determined from the fit using the Duke and Owens fitting program. 138 0.2 I I U T I fTTI ._\\ —c3— —O.3 —O.4 IYFijIIYIIfYYTIIUTIIIFIT _0.5 l l l l l l l l l l l l l l l l 1 l l I l l L l l l L l l l I l l l O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x Figure 5.9. Result of Non-Singlet Fit using FNS=F2 for 0.3 < x < 0.7. Plotted is the fit of d_lio_P_g% from the FMMF structure functions vs x. The line shows the value of (1:02“st as determined from the fit using the Duke and Owens fitting program. Appendix A The FMMF Collaboration Fermilab Experiments 594 and 733 are like most high energy physics experiments in that they involve large number of physicists in a collaboration that varies and evolves over time. The mm: collaboration includes physicists from four institutions: Michigan State University, Fermi National Accelerator laboratory, Massachusetts Institute of Technology and University of Florida. The University of Florida did not join the collab- oration until 1984. Here we list the physicists involved in either E594 or E733. The FMMF Collaboration W. Cobau, M. Abolins, R. Brock, A Cohen, J. Ernwein, E. Gallas, R Hatcher, D. Owen, GJ. Perlc'ns, M. Tartaglia, J. Slate and H Weerts Michigan State University D. Bogert, R. Burnstein, S. Fuess, G. Koizumi, LG. Morfin and L. Stutte Fermi National Accelerator Laboratory J. Bofill, W. Busza, T. Eldridge, 1.1 Friedman, M.C. Goodman, HW. Kendall, V. Kistiakowsky, T. Lyons, R. Magahiz, A Mukherjee, LS. Osborne, R. Pitt, L. Rosenson, A Sandacz, U. Schneekloth, B. Strongin, F.E. Taylor, R. Verdier, ].S. Whitaker and GP. Yeh Massachusetts Institute of Technology 1K Walker, A White and I. Womersley University of Florida 139 Appendix B Hadron Calibration L1 lntmdrrsztinn Hadron calorimetry is extremely important in neutrino experiments. For charged current events, it provides one of the three standard measurables, the energy transfer, v, (with the outgoing lepton energy and angle being the other two) and for neutral current events, only the properties of the hadron shower are observable. Exposures to the Quad- Triplet (qu) Wide band beam required determination of v over the wide energy range from 0—500 GeV. In addition, due to the long periods over which the am data was taken, stability over time (and over the detector) is essential. For the Ql'B data, the analysis used the calorimetric information from both the flash chambers and the proportional planes. The results of the calorimetry using the flash chambers and proportional tubes were combined based on the experimentally measured resolutions. The basic scheme for calorimetry using the flash chamber involves counting the number of hit cells. Corrections to the number of hit cells are made to account for varia- tions in chamber response. The correct number of hit cells is then calibrated to determine a relationship between hits and v. Calorimetry with the proportional planes is similar in concept to that using the flash chambers. With the proportional planes, one sums the pulse height in the propor- tional planes and a calibration determines the relationship between pulse height and v. Again, corrections to the pulse height are applied to account for variations in individual plane response. 140 141 Inthecasesofboththeflashchambersand proportional planes, thefinaldeter- mination of v involves many more details and complications than the simple sketchy outline provided above. The purpose of this appendix is to provide the reader with a detailed understanding of the process of defining the final calorimetry algorithms used in this analysis and the actual process of calibrating the large FMMF detector. This appendix will discuss all aspects of the hadron calorimetry using the FMMF detector. The appendix is divided into 2 sections, the methods used to measure the had- ron shower energy and scale determination. This appendix will focus on calorimetry in E733 although there will be some discussion of the algorithm used in E594. This appendix is not for the faint of heart. The nitty gritty details of hadron calo- rimetry in the FMMF detector are discussed. It is hoped that this is a complete (and honest) discussion of the hadron calorimetry in E733. E2 Calcrimm The W target-calorimeter as discussed in Chapter 2 was constructed of two different types of detectors, flash chambers and proportional planes. Each of these de- tectors has its advantages and disadvantages. The flash chambers and proportional planes had very different characteristics and required very different methods for recon- structing the hadron energy. Any algorithm for reconstructing vfrom the raw data must satisfy 2 requirements. The most important of these is that is possible to calibrate the al- gorithm and that the resolution of the algorithm be reasonable. It is essential to know the calibration to at least 5% and hopefully to better than 1%. A second requirement is that while the uncorrected response of the detector is non-uniform in position and over time, the algorithm's corrected response must be uniform temporally, spatially, and over the large range of energies available in the (11's. For example, an algorithm, for which the returned response to showers of a fixed energy fluctuated by 120% (or even 15%) over the duration of an exposure, is of limited value. 142 In this section, calorimetry using the flash chamber and proportional planes is discussed in detail with emphasis on how one obtains the uniformity of response and high energy calibration that is needed. B21 Flash Chamber Because of the properties of flash chambers, there were many difficult problems in developing a hadron energy algorithm. This section will discuss the inherent prob- lems in hadron calorimetry using flash chambers and the algorithms used in determining the energy deposited by a hadron shower in this thesis. B.2.1.1 Properties of Flash Chambers The most important aspects of the flash chambers for a hadron calorimetry algo- rithm are: 0 The binary nature of the detector. 0 The susceptibility of the detector to environmental effects. 0 The extreme sensitivity to residual ionization. The binary nature of the detector makes a hadron calorimetry algorithm difficult because the response is subject to saturation, which can be extreme at the highest energies. The susceptibility to environmental effects introduces time dependent effects which must be minimized. The sensitivity of the detector to residual ionization (from earlier events or cosmic rays) makes calibration problematic. These properties will be examined in detail below. ELLLI Sam The flash chamber is a binary detector. The term, binary detector, means that for a given cell, all that is known is whether a cell is on or off. A hit (or on) cell means that at least one ionizing particle traversed that particular cell. There is no additional informa- tion about how many particles actually traversed the cell. A hit could mean that the cell was traversed by a single minimum ionizing particle such as a muon or that it was tra- versed by a single highly ionizing particle such as a target nucleon fragment or that is 143 was traversed by several ionizing particles. Any type of detector can be subject to the effects of saturation, but a binary detector is extremely sensitive to saturation. The standard method for hadron calorimetry is to simply add up the charge or light collected (possibly corrected for the variations in channel response) and the signal collected is proportional to the energy deposited in the detector. In a detector subject to saturation, one might envision that, in addition to correcting for the channel response, making saturation corrections that are dependent on the size of the signal. When doing hadron shower calorimetry with flash chambers, a large fraction of the hit cells in the core (or other part of the shower where there is dense energy deposition) of the shower are saturated. Correction for this saturation is the first challenge for a hadron calorime- try algorithm. There are two tactics that one can use for correcting for saturation; 0 A local method based on recognizing regions of saturation. 0 A global method based on a non-linear calibration. In a local method, one attempts to correct for saturation by recognizing, on a event by event basis, regions that are saturated and applying an appropriate correction to the re- sponse for that region. In a global method, one compensates for saturation effects by calibrating using a scale that takes into account shower saturation One can also use a combination of the two methods (when the local corrections for saturation are not ade- quate.) Algorithms of both types have been developed for the flash chambers and are used in this thesis and will be discussed later in this appendix. 5.2.1.12 W In a detector as large as the FMMF detector, it is inevitable that there will be varia- tions in the response of flash chambers in different regions of the detector due to construction differences, electronic differences, gas composition variations and/ or oth- er factors. In addition, it is known from studies using cosmic ray muons that the response of the detector (to the muons) is subject to environmental effects such as humidity and 144 gas composition. These environmental effects can be rapid (on the time scale of a few hours or a day) when compared to the time span of an exposure (anywhere from 6 to 9 months) and are not necessarily uniform over the entire detector. These effects pose the second significant challenge to any calibration algorithm. Since one can observe spatial and time dependent changes in the response of the detector using cosmic ray muons, one could envision using the response to cosmic ray muons to correct the detector response in a time and spatially dependent way. This has been done in one of the calorimetry algorithms using cosmic ray muons that were col- lected at the same time as the data. While correcting the detector response using cosmic rays is appealing, there is no fundamental reason to believe that corrections based on isolated cosmic ray muons (which are minimum ionizing particles) will be the same as those for the many highly ionizing particles in the core of a hadron shower. For this reason, one might envision correcting the response of the detector to showers based on some observable shower property. The longitudinal transition profile of shower has been used to measure the response of parts of the detector in a time dependent way and these measured respons- es have then been used to correct the data. Both the correction methods outlined above will be discussed below in the sec- tion on algorithms. 3.2.1.132 B . l l I . |° Residual ionization from closely spaced events is a major problem for calibration using the test beam. The problem occurs when two events closely spaced in time occur in the detector. Only one of the events need satisfy the trigger, but the second event leaves behind residual ionization that can cause additional hits that will add to the apparent energy deposited in the calorimeter. Because the second event does not need to satisfy the trigger, the trigger rate does not necessarily reflect this problem. In the neutrino beam, the problem of overlapping events is rrunrrmzed' ' ' because in the very rare case where two 145 neutrino interactions occur roughly simultaneously, the events will, in all probability, be spatially separated. In the test beam, overlapping events can cause major problems because the events, by their very nature, occur in only one region of the detector. High energy test beam events are the most problematic because high energy means large amounts of ionization In addition, the higher energy beams had much higher rates of incident particles. Spatially separated residual ionization is a relatively simple problem for the cal- orimetry algorithms. In the neutrino beam, one simply restricts the volume of the detector used for calorimetry based on the event topology. Both flash chamber calorim- etry algorithms calculate the deposited energy in a limited volume. In the test beam, because the events overlap spatially, the problem is complicated and there are no sim- ple solutions. This leads one to be cautious of all test beam results. 3.2.1.2 Flash Chamber Calorimetry Algorithms There are two standard flash chamber calorimetry algorithms. Both of these al- gorithms are used in this analysis. The first of these algorithms is called SHOWER. This algorithm tries to correct for saturation on an event by event basis based on the shower topology and uses the response to cosmic ray muons to correct for the spatial and time dependent differences in detector response. This algorithm was written by S. Fuess for E594 (Fuess et al. 1982, Fuess et al. 1984). The scale for the E594 data was then refit by T. Mattison (1986) for each of the Narrow Band data settings. The E594 data was reana- lyzed for this analysis, using the E733 muon finding and fitting package (appropriately modified) but the hadron scale information was retained and used as is, without the aid of the proportional planes. This algorithm is also used in various analyses in E733, but this analysis uses a new algorithm. The second algorithm is based on simply correcting raw hits for response effects and then using a non-linear calibration. This algorithm is called EHFC. This routine is used for all the E733 data (1985 and 1987 exposures) and the result is then combined with the 146 proportional plane measurement. Both these algorithms are described in detail below. 112.12.]. SHQWER The SHOWER algorithm is based on the concept of correcting the shower for satu- ration and response effects by using the measured response of the flash chamber to cosmic ray muons and topological properties of the shower. Figure 3.1 shows the variations in response of the flash chambers to cosmic ray muons. To make the corrections to showers, the response to muon is factored into two separate parts, efficiency and multiplicity. The quantity EFFICIENCY is defined as the prob- ability that the flash chamber will have a hit within a small road centered around the point where a muon traversed the chamber (ideally the efficiency should be 1.0 but in reality efficiencies are in the range 0.5 to 0.9 with the average being between 0.65 and 0.75.) The second quantity MULTIPLICITY is defined as the average number of hits in the same small road when a muon traversed a chamber and caused at least one hit (ideally, the multiplicity should be 1.0 but multiplicities range from 1.1 to 2.2 with an average of about 1.4.) Multiplicity is due to electronics effects. The response, the average number of hits produced per minimum ionizing particle, is, RESPONSE = EFFICIENCY x MULTIPLICH'Y. Tables of responses, efficiencies and multiplicities are made from the cosmic ray muons collected over the course of an exposure. The tables are divided into a master table and time dependent tables. There is one master table per exposure and it divides each chamber into ten cell bins. Thus each master table contains the average (over the exposure) response (or efficiency or multiplicity) for each of the ten cell bins in every chamber. The time dependent tables then break the exposure up into roughly day-long periods. The time dependent tables contain the corrections to the master table for each chamber. In the time dependent tables, the chambers are divided into four pieces, the Measured Flash Chamber Response 147 3 I 2 ' ’I. t. . "' n I - .. . .. f. I. .I : ...! II 3'. In} " II $‘I I 1 ._--" -..“"1.J'g.n,v~. Iat'h {Ls-'35.”... .‘ -. .' .' " . I o I I I I LI L I I I L I L I L I. I I I I I -f I I I I I I I O 25 50 75 100 125 150 3 : I 2 L . t g ' . . E I .I. II .I. : .... ' . ' . .I ..' v .I ' I . . . 1 ’5‘". .. '3' .I I ..w ......W h ' I ."”3‘ .IC“’. ... .3351. ‘ ." - I OI-ILLIII.IIILLIléIIIllI=11£.l.LlIIIL 150 175 200 225 250 275 300 3 2 I .. l .. .’ I... . ‘::. .. a" ”....’. ... ..I I. 1 I'HVW $1 '3’... ..." :' 'u‘.. ...'*. I . '5' ..I. :I" ...". 'I‘.‘ ” I a. I . . ' I " 0 IL I I I I I I I J .I I I I I I I I: I I I I I I I I I I I I 300 325 350 375 400 425 450 3 C 2 F I... . ' .. . . '- ~ 1 :'- -‘t-.v:-’a.'..-.~-‘ a??? .:-‘-."‘w“"“ :w-q. "3:.- ~"‘:.‘ b .‘ ' o ? I I I ItI I: I i I I I L L J I I I L I I. I I I I I L L I I 450 475 500 525 550 575 600 Chamber Number Figure B.1. Variation of Flash Chamber Response. The figure shows the variation in response of the flash chambers during one time period of the 1987 run. The response of the flash chambers was measured using cosmic ray muons. One defines response as, RESPONSE = M ULTIPLICITY XEFFICIENCY. Note that there are a number of dead chambers, i.e. those with RESPONSE = 0.0. Also note that response and the variation in response is dependent on the position in the detector. In particular, Chambers 1—108 show a lower average response and a smaller chamber to chamber variation in response. 148 outer two panels and the two halves of middle panel which are read out by the different amplifiers. This reflects the divisions of the chamber by gas supply and readout. The fi- nal time dependent value of the response, efficiency, or multiplicity is the product Of the master table value for the ten cell bin and the correction factor for the appropriate sec- tion of the flash chamber. SHOWER starts by binning all the hits in the event into the same ten cell bins dis- cussed above for the master tables. The ten cell bins are first corrected for multiplicity in an attempt to eliminate electronic effects. A correction for efficiency and saturation is applied based on a probabilistic interpretation of the number of hits in the ten cell bin and measured efficiency. Both the multiplicity and efficiency corrections were original- ly derived by Stuart Fuess on a statistical basis and are highly non-linear. The multiplicity and efficiency corrections are shown in Figures B2 and 8.3. After the corrections for multiplicity, efficiency and saturation, an additional cor- rection is made for dead regions. The corrected sparks are then summed in the shower region. The scale for the corrected hits was originally derived from test beam data taken in 1982 and then was adjusted by using the known B V vs radius relationship of the Di-Chromatic narrow band neutrino beam. E2422 EHFC For the qu wide band beam, a second hadron calorimetry algorithm was invent- ed. The algorithm corrected the data for non-uniformities in response Of the detector using information derived from the neutrino beam. Corrections for saturation were made an intrinsic part of the calibration. The new algorithm is divided into two parts, the correction of the raw hits and the establishment Of a calibration scale. The correction of the raw hits is based on Observed variations in the response Of the detector. The response Of the detector is measured using transition curves of neutri— no showers selected to be in a narrow range of energies as measured by the proportional 149 Correction Factor .0 .0 .0 0 P .0 .0 0) U \l 0.2 0.1 Hit 0'5 Density 0-25 1.5 1 ° Measured ° Multiplicity Figure B2. Multiplicity Correction The multiplicity correction applied by SHOWER as a function of multiplicity and the observed raw hit density. The number of multiplicity corrected hits is then calculated as the product of the multiplicity correction and the raw hits in the ten cell bin The density Of raw hits is defined as the raw hits divided by the total number of cells. fgvf‘gu I I 150 Correction Factor 0.75 0.5 Hit Density 0.25 Measured Efi‘iciency Figure 83. Efficiency Correction The figure shows the efficiency correction as a function of the bin efficiency and density of multiplicity corrected hits. The final number of corrected hits is the product Of the efficiency correction and the number of multiplicity corrected hits. 151 planes. For this analysis, a transition curve describes the longitudinal development Of a shower by measuring the energy deposited, $112; as a function Of z. The selection Of the showers is not critical because the shower transition curves vary logarithmically with energy. To calculate the response of a module, one constructs an average transition curve using many showers in a small energy range. For each of the showers, the vertex is found. The number ofhitsin theU, Xor Y chambers of each module is determined. Then, based on the vertex position, one determines each module's position within the shower (see Figure B.4.) Based on where the shower starts (relative to a fixed module) and averag- ing over a large number Of showers, one can measure the average transition curve for the selected showers using a single module. As an example, for all showers with a ver- tex in module 10, the average number of hits in module 13 gives the point in the transition curve three modules downstream of the vertex. Similarly, for showers having vertices in module 6, the average number of hits in module 13 gives the point seven modules downstream Of the vertex. Using the events with vertices originating upstream of a giv- en module, one can construct an average transition curve for each U, X, or Y module. By comparing the transition curves measured by different modules, one can measure the relative difference in module response and then if one divides the data up into time bins, one can then determine the variation in response over time. For the 1985 and 1987 data, showers between 40 and 60 GeV were selected using the proportional planes. The average transition curves for each U, X, or Y module were then obtained from those data. Examples of these transition curves are shown in Figure ES. From these transition curves, the absolute variations in response was obtained. Once the absolute response for each module in each time period was determined, all the ab- solute responses were averaged and each module’s relative response was determined. The relative module responses were used to correct the data and are shown in the fig- ures that follow. Figure B.6 shows the variations in response over time for a three 152 + .6... + + Average Response of Module + Module Position - Vertex Position Figure B.4. Construction of Module Transition Curves. Shown is a schematic of how the module transition curves were constructed. One selects showers of similar energies throughout the detector. A fixed module, represented by the dark band above, samples different parts Of the showers based on the position of the vertex as illustrated in the cartoons at the top of the figure. The position Of the shower vertex determines which part Of the shower is sampled for each event by the fixed module. Using many showers, one constructs an average transition curve measured by the fixed module. The arrows indicate which to part of the transition curves, each cartoon corresponds. The area under the curve measures the response of the module. 153 160 _ 120 - + : +++ 80 ’- +Jr+ ++++ : + + Hi 40 : ++ + Jr++14? l. + F: ++ “Ms“. +* A o I 1J+ I I I I I I I I I I I I I I I I I I I4 I I I I I 3 O 10 20 30 40 so 3 160 s - + N o I + + V) E 120 E. ++ H++ = L : + H 8. a. 80 :- +Jr +++ vs ,2 ”Ml g m 40 E- ++ H++++H H4. +4»++ ‘H’ g 0 F IJ‘I I J I I I I I I I I I I I I I I Ifil I I I I ..fm + T“ a? O 10 20 30 40 so \v 160 _ 120 ; ++t+++ 80 E ++ + ++++ E d M 40 E— + y++++++++++ + + O:HJ.+rr+IIIrILrIrrrrlrr F+++++T+tfiq+u O 10 2O 30 40 so Distance Downstream of Vertex (in Chambers) Figure B.5. Module Transition Curves. The figure shows some typical module transition curves for 3 different modules. Each module transition curve represents an average transition curve for 40—60 GeV showers as measured by a single module. One notes the difference in the responses Of the modules shown 154 1 .50 l .25 § TTIIIUI Uj‘IlIIIU P x) u. O O b — I- h— p — p — b — I. b H M M O fl N 00 IIUIII—VY C \I U UIIIIIIII O U Ur O i. t. .. _. .. ... . .. t .. .. ... H O Module Response él l I pr N M ‘é III! TI—UIIIIIU 0.75 VIII )- — p s— I- _- I- b b _ p L 0.50 N b 0‘ m H 0 pr N Time (Arbitrary Units) Figure B.6. Variations in Module Response as Measured by Transition Curves. The figure shows the variation in module response over time for typical modules as measured using the transition curves. The abscissa represents the duration of the 1987 nine month exposure in arbitrary units. 155 modules, as measured by using transition curves. As one can see, there are significant variations in the response of the shown modules over the course of an exposure. The variations in each module's response seem to follow the same pattern, starting low and then improving as the exposure progressed. The improvement in module response was due to the reduction in relative humidity in lab C from the summer start of the 1987 exposure through the fall and into the winter. Figure B.7 shows the variation in module response over the detector. The measured response varied from 65 to 130% of the aver- age response. In the EHFC algorithm, one corrects the Observed hits in each module based on the measured module responses. Figure B.8 shows the results of the corrections on 50 GeV test beam data. One uses the ”corrected” hits in the shower region of the detector to measure the shower energy. A scale for the corrected hits was determined using the test beam and then adjusted using the neutrino data. The determination of the scale and the firrther adjustment using the neutrino data is discussed below. B.2.2 Proportional Planes The proportional planes system including electronics is explained in detail in other references (T artaglia 1984 and Tartaglia et al. 1985) and was briefly described in Chapter 2. The most important aspects of the proportional plane system for the purpos- es Of this thesis, are: They are analog devices. Gains were adjusted so that there would be no saturation in the ener- gy range of the experiment. 0 There were radioactive sources mounted on the planes to allow con- stant monitoring Of the variations in gain of the individual channels. 0 Environmental conditions were monitored continuously so as to cor- rect for effects such as density variation of the gas mixture. 0 Pedestals of the electronics noise were continuously monitored. The properties Of the proportional planes along with the continuous monitoring of con- ditions that could alter the response of the detector allows one to correct the data for Module Response 1 .50 1 .25 .0 q u 0 "‘ "‘ E" . '8 'N u u M O O .0 q u .0 fl in u 0 O 1.11 N M § 0.75 0.50 156 : L — I-F’ — —— _— u t; — _ I I I I I I I I I I I I I I I I I I I 4 12 16 20 24 E. _I I I L I I I I I I I J I I I I I I I 4 12 16 20 24 :h 5 ———— — :1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 12 16 20 24 Module Number Figure B.7. Spatial Variations in Module Response. Shown is the variation in module response as a function Of module number for each of the three flavors Of flash chambers, U, X, and Y. Response was measured using the transition curve method. The data show are from one the 12 time slices used in the analysis Of the 1987 data. 157 sooo_ 5 + 32800;] || ++++ $26005 + ++ ++++ )— 3 I: + + +++ + a = + ++ £24001: H ‘5 I 132200:— 2000:11111lrrrrlrrrIIrrr11111.Illl 40 so so 100 120 140 Vertex Position 3000 E 2800"— g : m : + + "326OOE+I +++II++ +++ 0 :P + ++++-1—1-+++ + §24°°§ + ++ L)2200} 2000:111rrlrrrLlrrrrlmrrrIerrIrLr 40 so so 100 120 140 Vertex Position Figure 8.8. Effects of Response Curve Corrections. The figures show the average number Of hits versus vertex position (in flash chamber number) for 50 GeV test beam data. In the top plot, the average number of uncorrected hits is plotted versus vertex position In the bottom plot, the average number of corrected hits is plotted versus vertex position For the corrected hits, the hits in each module are corrected using the relative responses measured using the transition curves. 158 variations in the response. Let us examine how these corrections are made and how the final calorimetry algorithm for the proportional planes works in detail. 3.2.2.1 Monitoring The first correction applied to the data from individual channels was pedestal subtraction The pedestals were monitored by taking a random noise (pedestal) triggers at the start Of every data tape (approximately every four hours). From the pedestal trig- gers, one calculated average noise for each channel which was subtracted from the pulse height for that channel. The channel to channel variations in gain were monitored using the 22 KeV line Of Cd-109. Special calibration triggers were taken between spills during normal neutri- no data runs. Using the pedestal-subtracted pulse heights, the 22 KeV peak was found and the variation in the pulse height of the peaks was used to correct for the channel to channel variations in gains. The variation in gas gains over time can be compared to the change in gas density as calculated from the measured atmospheric pressure and gas temperature. It was found that the fractional variation in the gas gain was related to the fractional variation in the density by the relationship, fl = 4.4% M p where M is the gain and p is the gas density (T artaglia et. a1 1985.) In addition, the proportional planes were sensitive to variation in the composi- tion of the gas mixture. The 90% Argon, 10% Methane (P-10) mixture was commercially supplied in large tankers. An exposure Of 9 months in duration require 3 or more com- pressed gas tankers. The percentage Of Methane varied slightly from tanker to tanker. Gas chromatography was used to analyze the composition of the gas and corrections were made to compensate for the variations in gas composition 159 3.2.2.2 Calorimetry Calorimetry using the proportional planes is straightforward. The corrected pulse height (i.e. pulse height with pedestal, gain and tanker corrections applied) for all chan- nelsoftheplanesintheshowervolumearesummedtogetl'erandthesummed corrected pulse height is then converted into a measure of the energy deposited by a scale deter- mined using from an analysis of the hadron test beam data. 3.2.3 Corrections for Muon Energy Loss in Calorimeter Muons, as they pass through matter, lose energy. This energy must be correctly accounted for to reconstruct the muon energy as we have seen in Chapter 2. In addition, in charged current events, one must correct for the energy deposited by the muon in the hadron shower. Two methods Of making this correction are discussed in this section 3.2.3.1 Explicit Elimination Because of the extremely fine segmentation Of the flash chambers, one might hope to eliminate the hits in the shower associated with the muon, and then use whichever calorimetry algorithm one likes best on the muon eliminated shower. The problem is determining which hits are associated only with the muon within the shower. For 3594, muon elimination algorithms were used prior to the calorimetry algorithm. Details on the muon elimination algorithms can be found the theses Of A. Mukherjee (1986) and T. Mattison (1986). 3.2.3.1 Statistical Subtraction For the proportional planes, it is not possible to eliminate explicitly the energy left behind by the muon This requires a ”statistical" subtraction of the muon energy. The mean energy lost by the muon in the calorimeter is calculated using the BetheBloch formula (Particle Data Group 1990) and then subtracted from the measured energy de- posited in the calorimeter to give v. This procedure was used for both the flash chamber and proportional plane measurements of v in the QTB data. 160 313 Minimum In principle, the scale determination is a straightforward exercise using hadron calibration data. One brings a mono-energetic hadron beam into the detector and then the response (in this context, response means the number of hits or summed pulse height due to the shower created by a incoming hadron) of the detector to the hadron beam determines the scale. In practice, it turns out, that for the desired precision and for some Of the reasons mentioned above, one must do an additional adjustment to the scale us- ing the neutrino data, which is described below. Inthis section, wewilldiscussbothofthese determinations Ofthe scales and their consistency. 3.3.1 Test Beam During 1985 and 1987, the NH beamline was run almost constantly and provided test beams to the FMMF detector between neutrino pings, in the slow spill part Of the Teva- tron accelerator cycle. The NH (formerly NT) beamline transported a low intensity hadron beam to Lab C using a circuitous route. For many years, it was said, that the NH beam- line was the longest and most Optically complicated beam in the world. The beamline consisted of over 100 magnet elements and was over two miles in length The beamline was capable of transporting hadron beams from 25 GeV to 400 GeV and was also capa- ble Of providing a muon beam. The momentum ”bite” Of the beam was approximately 2%. The test beam trigger consisted of a scintillator telescope that required that parti- cles satisfying the trigger pass through the last bending element and were therefore of the proper momentum. The signal from the scintillator telescope was delayed and com- bined with the standard P'I'H condition so that the timing of the PIT-I and test triggers were the same. The momentum of the test beam was determined using the recorded magnet current and the angle through which the beam was bent as defined by the scintillator 161 telescope. 3.3.1.1 Method The test beam data were recorded on magnetic tape simultaneously with the neutrino data. Test beam data were then split Off from the neutrino data and analyzed separately. Data with known problems (either detector or beam related) were rejected. The data was analyzed by first fitting and then eliminating the track from the incoming hadron The vertex, muon finding and calorimeter energy reconstruction routines were then run on the event. Events were then required to pass a number of cuts to insure a clean sample. Some of the these requirements were: The fit of the incoming hadron track should agree with the known in- cident trajectory of the test beam. 0 Incoming track must point back to last scintillator paddle. 0 NO muon may travel along the incident particle trajectory. Beam particle must interact between flash chambers 33 and 100. After Obtaining a clean sample, histograms were made of either the corrected raw hits or of the corrected pulse height. The means and width (root mean square, RMS) of the distributions were calculated. In addition, the distributions were fit to a Gaussian The scale was then determined by fitting the relationship between the known beam en- ergy and either corrected hits or corrected pulse height. The flash chamber data is treated in the method outlined above but a separate scale for the corrected hits from each view was obtained. The final scale was a function of the hits in each view, where the Obtained scale for each view was used and then the mean of the results from the three views was used as the EHFC result. Once the scales were determined, they were used on the test beam data to test the consistency Of the fits and to determine the resolutions Of the different algorithms. 162 3.3.1.2 Results The data used for scale determination were taken at five nominal energies, 25, 50, 70, 100 and 250 GeV. Three of the data sets (50, 100 and 250 GeV) are very large and were taken over extended periods of time. Histograms for the corrected raw hits (in each view) and the corrected pulse height for a single calibration point are shown in Figures 3.9. As one can see, the distributions are very clean and nearly Gaussian with slight tails. For the proportional planes, the mean corrected pulse height was fit to a straight line. Figure 310 shows the results from each calibration point plotted versus beam en- ergy along with a linear fit. The proportional planes show no signs Of saturation For the flash chambers, the fit was more problematic due to the effects of satura- tion Figure 3.11 shows the results from each of the individual calibration points (for the three views) plotted versus energy. Even a cursory look at the data reveals the extreme effects of saturation A quadratic (and even a cubic) was tried as the parameterization Of the calibration data, but the fits were unsatisfactory. After trying a number of different functional forms, it was found empirically that the data was best described by a power law, h, = a,- + b,E‘ h.- is the number of corrected hits in the 1‘“ view, a, and b,- differ between views but 0 is the same for all three views. The fit for the three different scales was done simultaneous- ly. The calibrated results from each view are then combined in EHFC. A scale without a pedestal, (i.e. the values of ai are forced to be zero) was also Obtained. The result of the fit with a pedestal is shown in Figure 3.11. After the scales were determined, new distributions were made using the deter- mined scales. The means and RMS’s were calculated. Figure 3.12 shows a set Of these distributions using the determined scales. The test beam calibration procedure was found to be self-consistent. 163 '3’ TIIUIIIIIIIIII'IIIIIIIIIIUIIIIIUIII fl 8 II'YIITI' a a N O H 8 u N W IIIUIIUIIU S l \1 or IIIIITUI—IIIII II 8 Event per 10 Hits Events per 20 Hits '8‘ M (II O r r r r I r r r r 200 400 600 800 400 800 1200 1600 Corrected Raw Hits-LI View orrected Raw Hits-X View § § § § u N O Ilrlertlrvrr'v 3 II i 8 20 ITIIIIIIIIII 0 r 1.4 r r r r I r r r r l r r r r r I r r r r I r r 200 400 600 800 4000 8000 12000 Corrected Raw Hits-Y View Corrected Pulse Height 0 Events per 10 Hits 8 1 Events per 100 ADC counts 3 ""lT“'l""lr'r‘l""l""l C Figure 3.9. Raw Test Beam Distributions. Shown are the response corrected raw hits and corrected pulse height distributions for the 50 GeV test beam setting. The bottom—right corner shows the pulse height distribution The other three histograms show the correct raw hit distributions for U (top—left), X (top— rigflrfi)arxi1{(bcn1cunrleft)vde~vs.PJcmetiratthre)(yie~vlrastrvicetrsrnuunyiflaslr chwurdrersarstheelltnnCllfviewvs. Corrected Pulse Height 164 40000 f 35000 30000 25000 20000 1 5000 r 0000 5000 “'0‘. VVV_j VVVV VVVV VVjV TVVV TVVV VVVV VVrV . r 1 l l I l l 0'..IIIIIIIIIIILIIIIIILJLIIIIIIIIIIIIIIII 0 40 80 1 20 1 60 200 240 280 Test Beam Momentum (GeV/c) Figure 3.10. Corrected Pulse Height vs. Test Beam Momentum The figure shows the relationship between mean corrected pulse height and the momentum of the incident test beam. The dotted line is the result of a linear fit to the data. 165 5000 1- {-0, 4000 - X..-" U) °' '4‘; 1- E S h g 3000 — "s . ‘fi-A U h- .‘t’ " O _ , - ’L” ’ ’ U 0 Y , . 561’ , m 2000 - , . C" , § Q. 1- '. ,’$ : I 3 _ ‘0 , ,4 'r M .' . ”° I e ,0 v” r 000 _ . 5 7’9, " 6.” J6" . 'fi‘ , .1 1- ,- r57 7 0 I I I L I I I I I LI L I J I I I I I I I I I I I I I I I I I I I I I I I I 0 40 8O 1 20 1 60 200 240 280 Test Beam Momentum (GeV/c) Figure 3.11. Response Corrected Raw Hits vs. Test Beam Momentum. The figures shows the relationship between the mean corrected hits and the momentum of the incident test beam. The results for the three different flavors (U view-Squares; X view-Circles; Y view-Triangles) of the flash chambers areshownThelinesaretheresultsofthe fits discussedinthetext. 166 , IEntrioI 3215 __ 11311111.. 3215 % 3°° .- % 3°° :- O : ‘5 I N .- N 1. h 2m l-' I.) 200 1:- 5 3. E a. - y, _ h : 0 g 100 — g 100 — - 1- :2 : i: : 0.- LL IIIIIIII LIII O I_I_I IIIIIIII _L 0 25 50 75 100 0 25 50 75 100 Flash Chamber Response (Ge V) Proportional Plane Response (GeV) > 120 _ Edie» 1864 > 120 L lEntrioI 1864 ‘3 " \3 : D Z L) . 1 (2 30 L r: so :— 3. t a : 0 Ia I “a : 3 40 — E 40 1— 0 :‘x’ I 3’ Z In . Lu .. o o A. I I I I I I L I I I I 50 75 100 125 150 50 75 100 125 150 Flash Chamber Response (Ge V) Proportional Plane Response (GeV) 200 lEntri 2033 200 - [Erin-is 2033 t. r. E c o 150 1— ~o 1:: 3 2 i ‘0 100 :— :2 S? E 5 5 w 50 '— 0 > > '- Lu Lu : 1 0 "I 1 L 1 r l r L 200 300 200 300 Flash Chamber Response (Ge V) Proportional Plane Response (GeV) Figure 3.12. Calibrated Test Beam Distributions. Shown are the calibrated distributions for both flash chambers and proportional planes for three test beam distributions. The left column shows the flash chamber results using the corrected raw hits algorithm. The right column shows the proportional plane results. The rows from top to bottom show the distributions for 50, 100, and 250 GeV incident test beams. 167 8.3.2 Neutrino Data The scale determined using the test beam data was then applied to the neutrino data. It was immediately noticed that for EHFC, the test beam determined scale was inad- equate, especially at high v. In comparisons between 1987 neutrino data and Monte Carlo of the v distribution (Figure 8.13), the data were more sharply peaked then the Monte Carlo and at high v, the number of events in the data was significantly below that pre- dicted by the Monte Carlo. There were also were significant problems in the y distribution, a quantity which is very sensitive to vscale. The true y distribution for neutrino events is normally almost flat (there is a little (1—y)2 behavior due to sea quarks) but the accepted y distribution shows the effects of both cuts and muon acceptance. At low y, events are rejected be- cause accepted events are required to have v> 10 GeV. This requirement is trigger related and was discussed in Chapter 2. At high y, which corresponds to low muon energy, the effects of muon acceptance dominate. While this analysis has no explicit minimum 13‘, requirement, it does require that a muon must traverse 550 cm in the calorimeter plus an additional 180 cm of iron in the spectrometer (resulting in a total mean energy loss to the muon of between 4 and 5 GeV) and, thus, there is an effective low 13” cut. This effec- tive low E # cut results in the lack of events at high y. Finally, the ratio of the E” and V scales determines the mean of the accepted y distribution If the ratio is not 1.0 as one expects and needs, the shape of the distribution will be skewed. The data / Monte Carlo comparison of the accepted 3; distributions using the EHK scale is shown in Figure 8.14. As one can see, the shape of the data distribution as compared to the Monte Carlo is sig- nificantly skewed. This disagreement leads one to contemplate using the y distribution and the Monte Carlo to determine an adjustment to the hadron scale. This sort of adjustment is done using a complicated fitting algorithm using the Monte Carlo including the mea- sured resolutions. Procedures of this type are standard in precision neutrino experiments 168 2000 10, > > m u (D 1600 U 2 I!) to 10 g .200 _ g h... a ' :2 if g 800 I. g 10 u.) : m t 400 :- p 1 o E I J I I l I I I I l I I I I I I 7 o 200 400 V(G€V) O 200 40° V(G€V) p: 79.22 a: 62.1 N: 62.1 Ag: —7.514 $0.388 P=O.(X)0E+00 [1: 86.73 a: 73.1 N: 73.1 x = 870.7 N: 98 P=0.000E+00 300 3 200 ; 3 1 W l 5 : 5 : W | ... .. .2 - All I ° ‘ \ s W” *3 Q 400 Q10d ... _2001 IIlIIIJJII 1111114411717 0 200 400 v(GeV) 0 20° ‘00 V(G3V) DIFFERENCE <— 0.000013+00 (— -> 7.000 -> RATIO +-0.00(X)E+00(-— -) 84.69 —) Figure 13.13. Accepted v Distribution-EHFC Scale. Shown is the comparison of the data and Monte Carlo v distributions for the accepted neutrino events. v is determined using the EHFC scale derived directly from the test beam data. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper—left Corner shows Data and Monte Carlo distributions overlaid with a linear scale; Upper-Right Corner shows same overlay but with a logarithmic scale; lower—left Corner plots the quantity DATA-MC; Lower—Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. Events per 0.025 Data - Monte Carlo O p: u: 200 100 E I,” L'— : ++ + + +++ E" + I I I I I I I I I I I I I I I I I I I 0 0.25 0.5 0.75 1 0.5031 0 = 0.237 N = 0.237 0.5250 0 = 0.238 N = 0.238 '+ Jf { Jrl II | . E 'fiWH + + * : I I I I I I I I I I I I I I L I LJ I 0 0.25 0.5 0.75 1 DIFFERENCE (— 0.0000E+00 (— —) 0.00005+0 —) (— 0.00(X)E+00 (— —) 0.00003-1-0 —) 169 Events per 0.025 ‘< O III IIIIIIIIIIIIIIII 0.25 0.5 0.75 1 y A]; = -2.1927E—-0 :t 1.413E—03 P = 0.000B+00 x = Data / Monte Carlo ‘< 315.5 N = 40 P = 0.000E+(X) VIUIII +H+++++WHI I "WNW-r +1» I I_L + + IIIIIIIIIIIIIIII O 0.25 0.5 0.75 1 y RATIO Figure 8.14. Accepted y Distribution-m Scale. Shown is the comparison of the data and Monte Carlo y distributions for the accepted neutrino events. v is determined using the EHFC scale derived directly from the test beam data. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper-left Corner shows Data and Monte Carlo distributions overlaid with a linear scale; Upper-Right Corner shows same overlay but with a logarithmic scale; Lower—Left Corner plots the quantity DATA—MC; Lower—Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. 170 (see as an example, Oltrnan 1989.) In essence, a procedure of this type ties the hadron scale to the more precisely known muon energy scale. This procedure is explained in the following section and the results follow the discussion of the method. 8.3.2.1 Method As has been shown before, there is a correlation between neutrino energy (E V) and radius (R) (See Figures 2.4 and 2.6). One can use this correlation and the known energy of the muon to determine the hadron scale. T. Mattison (1986) used a fit in the spirit of the one used in this analysis to determine the hadron scale along with the mag- netic field of the 24' toroids for his analysis of 8594. The algorithm used for this analysis requires a Monte Carlo (preferably with a full simulation of the spectrometer, since the spectrometer sets the scale for Vin the ana- lyzed data.) For all neutrino charged current events in both data and Monte Carlo data sets, on an event by event basis, one determines whether the neutrino was the product of the decay of a pion or a kaon. This is done by using a ”separtrix”. The separtrix di- vides the E rR plane into pion and kaon halves. If a event falls in the pion half of the E V—R plane, it was assumed that the neutrino originated from the decay of a pion. Then the quantity, A = (v+E,,)—E,(R) was calculated, where Eu is the reconstructed muon energy and E V(R) is the neutrino energy calculated from the EV vs. R relationship of the 018 beam and whether the inter- acting neutrino was the product of the decay of a pion or a kaon. The data were then binned in y and (A)versus y was plotted separately for events of pion and kaon origin. The fitting procedure adjusts the hadron scale in the data by minimizing the 1;- like quantity, 2 _ ((A)data " (A)MC )2 Z - 3032i!“ 00am + disc . n,K In the data, v is parameterized as, 171 v=a+b~vo+c-vf, —8 where Va is the vas originally determined from the testbeam data and 6is the correction for the muon energy loss in the calorimeter and the average A was recalculated at each step. The minimization routine MINIUT (James and R005 1989), was used to do the mini- mization. a, b and c were the free parameters in the fit. 8.3.2.2 Results The results of the refitting of the EHFC scale are shown in Table 8.1. The fitting progam returns a scale that differs markedly from that found using the test beam data. The new scale has a significant pedestal, a 1% change in scale and a significant quadrat- ic term The quadratic term is positive indicating that the original EHFC scale may not have completely corrected for the effects of saturation. Figures 8.13 and 8.14 showed the v and y distributions for the 1987 data using the test beam derived EHFC scale. The scale derived from the test beam data seemed to be inadequate. After applying the results of refitting, the rescaled EHKZ provides a much more satisfactory scale. Figure 8.15 shows the integral v distributions of data and Mon- te Carlo. The agreement is quite good between data and Monte Carlo. The extreme deficit at high v is gone and the agreement at the peak is much more satisfactory. The y distri- bution shows the improvement in the scale even more dramatically. The y distribution comparison of data and Monte Carlo is shown in Figure 8.16. There is now good agree- ment between data and Monte Carlo. For the original EHFC scale (shown in Figure 8.14), the ratio plot (lower right-hand corner plot) was extremely skewed. For the new scale, the ratio plot is flat within errors. All this is a bit puzzling. Why would the flash chamber scale as determined us- ing the test data differ so dramatically from the scale determined using the neutrino beam? Clearly, the refitting makes the agreement between data and Monte Carlo im- prove greatly. The question that remains as to which scale is correct. The very nature of the algorithm used is to force the data's v and y distributions to agree with those of the 172 Table 8.1 Scale Refitting Results Original Scale n Results of Refitting Procedure ll Pedestal Slope Quadratic 2 EHFC scale from Test _4 Beam Calibration +0.74 1.01 8.41x10 172/97 00p 1985 Proportional Plane Scale -3.36 1.10 - 134/ 98 DOF Prop Plane scale from Test Beam Calibration +031 0'97 ‘ 138/98 DOF Table 8.1. Scale Refitting Results. The table presents the results of the refitting of the v scales using the y fitting method described in the text. For the flash chamber EHFC scale the refit is to a quadratic. For the proportional planes, the refitting is linear only. 2000 § § Events per 5 GeV 'é § 100 Data - Monte Carlo O 173 ,E- 10’ :1 > "' \1 r U z I In 10 L b : Q. ,_ '8 10 P : § . LL] 7 1 p I I L I I I I I I I I I I I I I I I I o 200 400 V(G€V) o 200 400 v(GeV) - 88.17 0: 74.3 N: 74.3 Ail: -l.277 $0.437 P=0.348E—02 = 89.45 0: 74.6 N: 74.6 x = 109.8 N: 98 P=0.l95 P I- ; a] g I .22 1 .. 1 ‘ I Ilia" Ili I” w_ E ; | . I \ : :. g _ : Q . :- J I I I I I I I I L I I I I I L L I I I I I I I I 0 200 mo v(GeV) o 200 40° v(GeV) DIFFERENCE (— 0.00(XIE+00 (— —) 67.00 —) RATIO (— 0.00(X)E+00 (— -) 97.36 -) Figure 8.15. Accepted vDistribution—Rescaled EHFC Scale. The figure shows the comparison of the data and Monte Carlo v distributions for the accepted neutrino events. v is determined using the rescaled EHFC scale derived from refitting the test beam scale using the neutrino data as described in the text. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper-Left Corner shows Data and Monte Carlo distributions overlaid with a linear scale; Upper-Right Corner shows same distributions but with a logarithmic scale; Lower-Left Corner plots the quantity DATA- MC; Lower-Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. Events per 0.025 Data — Monte Carlo 174 1400 L— E + m’ ‘ r" 1200 :I— + + I- + In 1000 :— g t c z 300 L'— t 10 E a. 600 E. g .1 Z 9 ‘00 .— l-LI 10 200 P O IIIIIIILJIIIJIIIIII IIIIIIIILIIIIIIIIII o 0.25 0.5 0.75 1 y o 0.25 0.5 0.75 1 II p: 0.5175 0:0.240 N=0.240 Ap=—6.563SE—0i1.40915—03 P=0.3l9E-05 11: 0.5241 0:0.240 N=0.240 x = 60.45 N: 40 P=0.l99E-01 I it: : 200 F- 2.4 _ .. 2.2 __ t 3 2.0 _ F L 108 10° f + ++ (3 1.6 P C 3 1.4 _ o ‘ g 12 1’ C ’r + + + + I + ++++H+ 310 L + ++ 1+++ 40° _ 0.8 . a ._ : Q . . -200 L 0.6 __ hIIIIIIIIIIIIIIIIIII IIIIIIIIIILIIIIIIII 0 0.25 0.5 0.75 1 y o 0.25 0.5 0.75 1 y DIFFERENCE (- 0.00008+00 (— -> 0.00me -) 11.4110 6- 0.000013+oo (— —) 0.0000E+0 —) Figure 8.16. Accepted y Distribution—Rescaled EHFC Scale. The figure shows the comparison of the data and Monte Carlo y distributions for the accepted neutrino events. v is determined using the rescaled EHFC scale derived from refitting the test beam scale using the neutrino data as described in the text. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper-Left Corner shows Data and Monte Carlo distributions overlaid with a linear scale; Upper—Right Corner shows same distributions but with a logarithmic scale; Lower—Left Corner plots the quantity DATA- MC; lower—Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. 175 Monte Carlo by modifying the v scale. In the next section, we will examine the propor- tional plane results and the test beam data in order to try and resolve these inconsistences. 8.3.3 The Self Consistency of the Determined Scales Now that we have v scales that seem to give consistent results between the data and Monte Carlo, the inevitable question is whether either of the EHFC scales is to be be- lieved? In this section, we will re-examine all the flash chamber data and compare the results with that obtained using the proportional planes. The re-examination of the EHFC data will include a re-analysis of the test beam data. 8.3.3.1 Proportional Plane Results The calibration of the proportional planes using the test beam data was in many ways much more satisfying than that of the flash chambers As we have seen, there was a linear relationship between the corrected pulse height and the incident beam momen- tum. In addition, when applied to the neutrino data, the agreement between data and Monte Carlo was self-evident. Figures 8.17 and 8.18 show the v and y distributions for the data using the EHPR scale obtained using the test beam data. In addition, Figures 8.17 and 8.18 show the Monte Carlo (with the proportional plane resolutions included) dis- tributions overlaid. The agreement between data and Monte Carlo is more than satis- factory for both the v and y distributions. The excellent agreement between the data and Monte Carlo using the test beam proportional plane scale leads one to examine the effects of the scale fitting program on the EHPR scale. Figures 8.19 and 8.20 show the v and y distributions using the rescaled EHPR resulting from the refit of the EHPR scale done in the same manner as the refitting of the EHFC scale. The data / Monte Carlo agreement using the rescaled EHPR is, if anything, slightly better than that of the original scale. This result gives one some confidence that at least the refitting for a new scale does converge on an appropriate scale. As a further test, the 1985 proportional scale was used as input for the refitting Events per 5 GeV Data — Monte Carlo 176 2000 — 10: i a». 1600 U 2 I!) 10 1200 F-l g I a) 300 '— § 1° I [-1.1 400 L t 1 o I I I L 1 L I L J I I I I I I J I I I I I o 200 400 v(GeV) 0 20° 40° v(GeV) 11: 88.08 0: 72.8 N: 72.8 Ag: -1.403 $0.431 P=0.112E-02 11: 89.48 O= 73.1 N: 73.1 x = 130.6 N: 98 P=0.155E-01 300 , : t .9. 5 4‘3 g 1 \ 3 3 Q ~2CX) L 1 1 1 1 1 1 1 L 1 1 o 200 400 V (GBV) DIFFERENCE 4— 0.00(X)E+00 (— —) 15.00 —) RATIO (- 000008-100 <— —) 26.33 —) Figure 8.17. Accepted v Distribution-EHPR Scale. Shown is the comparison of the data and Monte Carlo v distributions for the accepted neutrino events. v is determined using the EHPR scale derived directly from the test beam data. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper-left Corner shows Data and Monte Carlo distributions overlaid with a linear scale; Upper-Right Corner shows same overlay but with a logarithmic scale; lower-Left Corner plots the quantity DATA-MC; lower—Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. Events per 0.025 Data - Monte Carlo 177 1400 I.— 1200 :— + L' + + + “1' 1n .- N 1000 .- O E o‘ : a. 600 :— ~13 .... 5 0 LI I I I I LI I I I I I I I I I I o 0.25 0.5 0.75 1 y o 0.25 0.5 0.75 1 y p: 0.5178 0:0.240 N=0.240 Ap=-9.5341£—0:1:1.414E—03 P=0.156E-10 11: 0.5273 0:0.238 N=0.238 x = 95.02 N: 40 P=0.225E-05 300 200 L ; '3 r 6 100 I. 3 0 . E 1 + + + I g .l I ”1"" ' “4» +1» 77+ -100 L Q : _2mPIIIIIIIIIIIIIIIIIII _IIIJIIIIIILIIIIIIII 0 0.25 0.5 0.75 1 y o 0.25 0.5 0.75 1 y DIFFERENCE (— 0.0000E+00 (— -+ 0.00008+0 —> RATIO *— 0.000013+00 (— —> 0.00008+0 —> Figure 8.18. Accepted 3; Distribution—EHPR Scale. Shown is the comparison of the data and Monte Carlo y distributions for the accepted neutrino events. v is determined using the EHPR scale derived directly from the test beam data. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper—Left Corner shows Data and Monte Carlo distributions overlaid with a linear scale; Upper-Right Corner shows same overlay but with a logarithmic scale; Lower—Left Corner plots the quantity DATA-MC; Lower—Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. Events per 5 GeV Data — Monte Carlo 135% § §.. 200 -1(X) -200 178 L— 10’ I > P N :i u . C In 10 g ‘8. b m -_ "g 10 I 0 t u.) )- f 1 h I I I I I I I I I I J I I I I I I L J o 200 40° v(GeV) ° 20° 400 V (GeV) = 89.20 0: 73.7 N: 73.7 Agt=-0.4321 $0.434 P=0.320 = 89.64 0: 73.2 N: 73.2 x = 121.8 N: 98 P=0.519E-01 1- P P j. _ b h D Data / Monte Carlo I I I .L I I I I I I L I I I I I I I I I I I 0.1111'1111 r111 1.1 O o T‘U'l 2m .0. v(GeV) 20° 40° v(GeV) DIFFERENCE «— 0.00008+00 «- -) 19.00 -) RATIO «— 000008400 (— —> 26.54 —> Figure 8.19. Accepted vDistribution-Rescaled EHPR Scale. The figure shows the comparison of the data and Monte Carlo v distributions for the accepted neutrino events. v is determined using the rescaled EHPR scale derived from refitting the test beam scale using the neutrino data as described in the text. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper—Left Corner shows Data and Monte Carlo distributions overlaid with a linear scale; Upper—Right Corner shows same distributions but with a logarithmic scale; Lower-Left Corner plots the quantity DATA— MC; lower—Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. ”“5 Events per 0.025 Data - Monte Carlo 179 1400 L- E + 10’ 1200 E- + + : + K} 1000 L- o. 2 E Q 10 : a. 1 600 E- 3 E § 10 400 _ Lu 200 E o I I I I I I I I I I I I I I I I I I I 1 . o 0.25 0.5 0.75 1 y o 0.25 0.5 0.75 1 y p: 0.5201 0:0.240 N=0.240 Ap=-7.15588—0i1.412B—03 P=0.402E—06 p: 0.5272 0:0.238 N=0.238 x = 72.85 N: 40 P=0.IISE-02 300 5.0 __ : 4.5 _ .. 4.0 _ . 3.5 __ 20° _— 2 3.0 __ k 25 L" : 5 ' r 100 r- + 1 + 3 2.0 F" :1 g 1.5 0 1H.“ l IIhHI Ill. '1" E +- u- '1! : T I 1 ' H 1 1+ E‘- . ++ * + + . f 5 ~ + *t + -1°° r l Q r J P )- '- r- -2m-IIIIIIIIIIIIIIIIIIL LIIIIIIIIIIIJIIIIII o 0.25 0.5 0.75 1 y o 0.25 0.5 0.75 1 y DIFFERENCE (— 0.0000E+00(— —)0.0000E+0 —) RATIO (— 0.0000F.+00 (— —) 0.0000840 —.) Figure 8.20. Accepted 3] Distribution—Rescaled EHPR Scale. The figure shows the comparison of the data and Monte Carlo y distributions for accepted neutrino events. v is determined using the rescaled EHPR scale derived from refitting the test beam scale using the neutrino data as described in the text. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper-Left Corner shows Data and Monte Carlo distributions overlaid with a linear scale; Upper-Right Corner shows same distributions but with a logarithmic scale; Lower—left Corner plots the quantity DATA- MC; Lower—Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. 180 program. Again, the vand y distributions for the data using the 1985 EHPR scale are shown in Figures 8.21 and 8.22 with the Monte Carlo overlaid. Using this scale, there is little agreementbetween data and Monte Carlo. The refittingofthis scale results ina new scale which is used in Figures 8.23 and 8.24. Again, v and y distributions using the new scale for the data are compared with the Monte Carlo. There is again, good agreement between the data and Monte Carlo after the refitting. An additional point lends additional confidence that the refitting procedure does result in the correct scale. The rescaling for the proportional planes was done as a linear function of the old scale. From the rescaling and the original scale's calibration of pulse height to energy, one can reconstruct the new pulse height scale calibration. These three different pulse height calibrations are the same within the estimated errors of the fits. Table 8.1 contains the results for refitting of proportional plane scales in addition to those of the refitting m. Table 8.2 shows the final pulse height calibration for each of the three scales. The proportional plane analysis leads one to draw two conclusions. The first con- clusion is that, because the proportional plane scale from 1987 works so well, there does not seem to be any intrinsic problem with the test beam analysis. By this I do not mean that there is not a problem with the flash chamber data but that there is not some funda- mental problem with the energy of the beam or the selection of events. The second conclusion is that, since, for the proportional planes, both the refitting of the scales and the test beam analysis return scales in good agreement, the refitting algorithm works and works well. 8.3.3.2 Re-Analysis of Test Beam Flash Chamber Results The test beam was re-analyzed using the rescaled EHFC scale. The neutrino deter- mined scale is completely inconsistent with the test beam data. In the extreme, the mean of the rescaled distribution for the nominal 250 GeV point is returned as being 309.7 GeV whereas the mean of the unrescaled distribution is 254.5 GeV. Obviously, the EHFC scales Events per 5 GeV Data - Monte Carlo 181 2000 E." a : > 1600 r. as : U) 102 g .2... g. r: __ fi. : e 300 - 5 1° C Q : [-1.1 400 I: 1 O a I I I I I I ‘ I I I LII LLI I I I I 0 200 400 v(GeV) o 200 40° v(GeV) 11: 81.66 0: 67.0 N: 67.0 Atl=—7.955 i0.425 P=0.000E+(X) 11: 89.61 0: 73.5 N: 73.5 X = 473.5 N: 98 P=0.0(X)E+w 300 L E E 200 " 6 1 w 1 I II I- ” P . w H . : - t I' v w l! 100 I—- g L I“ r} t E " ” 1H t : \ ’ 0 . g P i . P : M 'It Q -100 _- J a I I I I L I I I I4 I I I I I I I I I I I I 0 200 400 V(G€V) 0 200 400 V(GCV) DIFFERENCE (—0.00(X)E+00(— —-) 4.000 -> RATIO +—0.00(X)Er~00<— —) 23.59 —> Figure 8.21. Accepted v Distribution—1985 EHPR Scale. Shown is the comparison of the data and Monte Carlo vdistributions for accepted neutrino events. v is determined using the 1985 EHPR scale derived directly from the 1985 testbeam data. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper—Left Corner shows Data and Monte Carlo distributions overlaid with a linear scale; Upper—Right Corner shows same overlay but with a logarithmic scale; Lower-Left Corner plots the quantity DA TA—MC; lower—Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. Events per 0.025 Data — Monte Carlo § § §§§ 200 O u: u: 200 100 -100 182 E" + 10’ ‘ fi— : ‘H++ E- “ 8+ «‘13 E + + o- 2 ._ + z 10 E a. 5' a . 1 § .. : DJ IIIIIIIIIIIIIIIIIII‘ 1 IIIIIIILIIIIIIIIIII o 0.25 0.5 0.75 1 y o 0.25 0.5 0.75 1 31 0.5079 0:0.238 N=0.238 =-2.0893B-0:|:1.47OE-03 P=0.000E+00 0.5288 0:0.238 N=0.238 = 257.7 N: 40 P=0.000E+00 :- . : + + .9, i -+++ H 1 _+ J. + I 41.1 I‘ll 1 g : ' ll 1 1'1 l + E 1 Wm +"**+ :. l J, 5 » + l 1 ° * Q - :IIIIIIIIIIIILLIIIII I-IIIIIIIIIIIIIIIIIIL o 0.25 0.5 0.75 1 y 0 0.25 0.5 0.75 1 y DIFFERENCE (— 0.00008+00(-— —-) 0.00008+0 -) RATIO (— 0.00008+00 4— —+ 0.00008+0 -9 Figure 8.22. Accepted 3; Distribution—1985 EHPR scale. Shown is the comparison of the data and Monte Carlo y distributions for accepted neutrino events. v is determined using the 1985 EHPR scale derived directly from the 1985 test beam data. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper-left Corner shows Data and Monte Carlo distributions overlaid with a linear scale; Upper—Right Comer shows same overlay but with a logarithmic scale; Lower—Left Corner plots the quantity DA TA—MC; Lower-Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. Events per 5 GeV Data - Monte Carlo triii O in 200 O -200 183 T 10' p E: i. .. L’J a I In 10 D h r A 3 :2 L g 10 C 14.1 r F r; 1 a I I I I I L I I I I I I I I I I I J o .... .... v (GeV) o zoo .0. v (GeV) 88.08 0': 72.8 N: 72.8 Ag: -1.747 10.431 P=0.4948-04 89.83 a: 73.3 N: 73.3 x = 137.3 N: 98 P=0.54OE-02 ' s. " L i- : 5 - .2 I g 1 _ \ - 5 : =1 _ l 0 1 I I I I I I J I I I I J I I I I I I I I I I I I I 0 200 400 V (GEV) 0 20° ‘00 V (68V) DIFFERENCE (— 0.00005+00(-— -) 15.00 —) RATIO 1— 0.00008+00 (— —> 26.80 —-> Figure 8.23. Accepted vDistribution—Rescaled 1985 EHPR Scale. The figure shows the comparison of data and Monte Carlo vdistributions for accepted neutrino events. v is determined using the rescaled 1985 EHPR scale derived from refitting the 1985 test beam scale using the neutrino data as described in the text. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper-Left Comer shows the Data and Monte Carlo distributions overlaid with a linear scale; Upper—Right Corner shows same overlay but with a logarithmic scale; Lower-Left Corner plots the quantity DATA-MC; Lower-Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. Events per 0.025 Data — Monte Carlo 1400 L- 1200 F— E ++ +" *++ 1000 :— .... =_ 600 E— 400 E— 200 : l- o IIIILIJIIIIIIJLILIJ o 0.25 0.5 0.75 1 p: 0.5178 0:0.240 N=0.240 p: 0.5272 0:0.239 N=0.239 300 200 L .. :_+ H * + 0 ?m%II+-u }'}H IH'. E W HM} l .1“) — _zm-IIIIIIIIIIIALIIIIJI o 0.25 0.5 0.75 1 DIFFERENCE (— 0.00(X)E+004— 400000540 —r (— 0.00(X)E+00§— —-)0.0000EI+0 —-) 184 Events per 0.025 ‘< Data / Monte Carlo ‘1: 10 10 1 O iJrrlrrrrlrrrrlrrrr 0.25 0.5 0.75 1 y Asa. = -9.4189E-0 :l: 1.413E—03 P = 0.261E-10 x: 93.63 N = 40 P = 0.345805 Y I U I H 14 v n v— I + ++ +++ LLIIIIIJIILLJLIIIII O 0.25 0.5 0.75 1 ll RATIO Figure 8.24. Accepted y Distribution—Rescaled 1985 EHPR Scale. The figure shows the comparison of data and Monte Carlo y distributions for accepted neutrino events. v is determined using the rescaled 1985 EHPR scale derived from refitting the 1985 test beam scale using the neutrino data as described in the text. The data is shown as the points with the error bars. Monte Carlo is shown as a solid outline. Upper—Left Corner shows the Data and Monte Carlo distributions overlaid with a linear scale; Upper-Right Corner shows same overlay but with a logarithmic scale; lower-Left Corner plots the quantity DATA—MC; Lower—Right Corner plots the ratio of the two distributions. The Monte Carlo is area normalized. 185 Table 8.2 Proportional Plane Refitting Results for Pulse Height 9. Scale Pedestal 810 e 1985 Pro rtional Plane chale +2.76 6.36x10'3 1985 Proportional Plane _3 ;. Rescaled —0.33 6.93x10 I Prop Plane scale from _3 Test Beam Calibration "025 7.09x10 1987 Proportional Plane _3 Rescaled +0.07 6.86X10 Table 8.2. Proportional Plane Refitting Results for Pulse Height. Presented are the calbration constants for the conversion of 1987 pulse height to GeV. Note that the calibration constants are very similar for the final three scales. 186 obtained using the two different methods are completely inconsistent. 5.3.12.1 WW From all of the above, one must answer that there seems to be little reason to believe the test beam calibration results for the flash chambers. The proportional plane calibration gives a self-consistent picture. The refitting of the proportional plane scales showed that the refitting algorithm worked correctly. The comparisons between the neutrino data and Monte Carlo showed that. 0 There are no significant problems with the Monte Carlo. 0 The test beam determined scales worked for the proportional planes but not EHFC. 0 The rescaled EHFC provided excellent agreement between data and Monte Carlo. All in all, it seem that one should have ignored the test beam data in the calibration of the flash chambers. But, why does the test beam data give the wrong scale? As discussed in Chapter 2, there are a number of experimental problems with flash chambers. The two largest are saturation and memory. From the re-analysis of the test beam data using the rescaled EHFC, saturation is not the problem with the test beam data. Using the scale obtained from the neutrino beam, ehfc returns that the test beam energy is larger than the known inci- dent beam momentum. This is the opposite of saturation. The problem is the flash chamber memory. Because flash chambers are constructed of a non-conducting materi- al (i.e. plastic) it is possible for static charge to collect on the inside walls of cells. For a conductor, this charge recombines; however with an insulator, the residual charge can be long lived. This problem was probably exacerbated when there were large amounts of charge deposited at high rates in the same location. This is exactly the unavoidable condition present for high energy test beam running. At high energies, trigger rates of hundreds of Hertz were common and the high energy showers deposited large amounts of charge in the same flash chambers repeatedly. This was probably the cause of the 187 immense difficulties of calibrating the flash chamber using the test beam. BA Conflnsinna Intheend,flierescaledEHPcscalewasusedinconjunctionwiththe EHPRscale derived from the test beam data for the analysis of the neutrino data. These two scales are consistent with each other and provide very good agreement between data and Monte Carlo. Together, the combination of EHFC and EHPR provides a scale that has good high energy resolution, that exhibits uniformity over the entire volume of the target-calorim- eter and that was unvarying over the long span of the am runs. Appendix C The Physical Cross Section C]. Muslim The simple cross section outlined in Chapter 1 makes a number of simplifica- tions that are not adequate for a complete, physically correct extraction of structure functions. The discussion in Chapter 1 makes two major simplifications: next-to-lead- ing order Electroweak corrections to the simple Feynman diagram shown in Chapter 1 are ignored and charm mass threshold effects are not included.The next-to-leading or- der Electroweak corrections go by the common name "radiative corrections.” One commonly models the charm mass threshold effect using the ”Slow Rescaling Model.” These corrections are important for this analysis because of the greater statistical precision of the extracted structure functions. The extracted structure functions have been corrected for both next-to-leading order Electroweak effects and for charm mass threshold. This appendix discusses these corrections. £2 E 1' IV C I' There are two next-to-leading order Feynman diagrams of importance to this analysis. The first of these diagrams involves the exchange of a photon (or Z boson) be- tween the outgoing muon and either of the quark legs. This diagram is called the Box diagram. The second diagram involves the emission of photon by the muon. Figure C.1 shows both of these diagrams. This section discusses the calculation of each of these radiative corrections and their effects on the structure function extraction. 188 189 Figure C.1. Feynman Diagrams for Radiative Corrections. Shown are the two next-to-leading order Electroweak Feynman diagrams for neutrino—quark scattering. The top diagram is the ”Box Diagram" involving the exchange of an additional photon between the muon leg and one of the quark legs. The bottom diagram is the ”Final State Radiation” diagram for the process I vcr—Wq - 190 C11 Box Diagram The exchange of a photon between the muon leg and one of the quark legs re- sults in an enhancement of the bare (leading order) cross section for neutrinos. (Because the effect of the exchange of a Z0 is smaller by the ratio of Chum/z, one can neglect its contribution.) This diagram has been calculated as a correction to the bare cross section. The standard method of expressing the results of this calculation is as a factor of the bare cross section, i.e. acorrected =K0bare~ (C-l) Wheater and Llewellyn-Smith (1982) have calculated the value of K for the case of vd and V u scattering. vd scattering contributes the majority of the cross section to neutrino—nucleon scattering. Whereas, Vu scattering is the most important process in anti-neutrino—nucleon scattering. For the case of vd scattering, Wheater and Llewellyn- Smith calculate that .. .0: $2.13 alzfi KW—1+fl{[log[ 5‘ ]+4] 9[3108[m§]+6(fl 4)}. (C2) 5, 5 2p, - p, = 2Mva. (C.3) Wher , Now to examine the size of the correction, let’s evaluate the expression C.2 at a typical E V and x. If we use E v=150 GeV, and x=0.25, then sdz70 GeVz. Evaluating C.2 with MW=80 GeV and m d=50 MeV, we get Kw, 1+3 [4.5+2.25]-l[310.2+—1-5.12] 7r 9 3 6 1 + 0. 00232{6. 75 - 0.85} = 1.0137 This means that the vd cross section is 1.4% larger than the bare cross section calculated from leading-order electroweak theory at the chosen energy. The correction shrinks as the 5,; grows because the first term in the correction shrinks while the second term 191 grows, thus resulting in a smaller value of K "1. Equation C .2 can also be used for V3 and if one substitutes the s quark mass for the d quark mass, C.2 should also hold for neutrino scattering off the strange sea. Sirlin and Marciano (1981) have also calculated de. Their result differs slightly from that of Llewellyn-Smith and Wheater in that their result is smaller by i (or the % term, in Equation C.3, goes to 2). One finds that using the Sirlin and Marciano formula results in a value of de=1.0131 for the values used in the calculation above. For the case of V u, Wheater and Llewellyn-Smith find, 0: M3, 3 4 2 s“ __1_ _g vu=1+;{|:l°8( s“ )+—2-]—3[§log(mi]+6( 2 4)“. (CA) Evaluating this expression in the same manner as we evaluated the expression for K yd, one finds, K7": 1+g [4.5+1.5]—3[310.2+l(-o.38)] n 9 3 6 = 1 + 0.00232{6.0 - 2.97} = 1. 0070 This correction is a little smaller then that for vd. Sirlin and Marciano have not made the calculation for the Vu case. For consis- tency, we have chosen to use only the calculations of Wheater and Llewellyn-Smith Figure C.2 shows the corrections to the bare cross sections as a function of Squark for the four relevant cases. C22 Final State Radiation The second diagram of Figure C.1 involves the emission of a photon by the out- going muon. In principle, this diagram represents a different interaction, vq—>uyq' than the bare cross section interaction, vq—mq’. The cross sections measured in our experi- ment are for the process vN—mX, which is the combination of these interactions. The fact that the second diagram in Figure Cl is divergent adds an additional _w-a.1_l-fi..: .. . —K Correction to Bare Cross Sec ' n 1 .025 1 .020 1.015 1.010 1 .005 1 .000 192 r- &.\. \.\. ’ \.\. I. \ ...:°-\.- ‘n \ '. \ .. \ \ \ ...:n‘. \ '--.~. \ .. b \ \ “\ux - \ ..':.\ E \ \ _ \ ‘23 t- " \ "-:.\ . .. \ \ \ °'. \ , \ \ ‘3‘ h \ \ \ '._.:.~\ I- \ '. \ \ \ \ 5‘ . r- \ \ v d .'§. v S . \ \ \ 'az. F \ \ .’\\ \ \ \\ t- \ \ V C \ ‘1 ‘ \ \ — \ \ \ r- \\ — \ \ V 11 .. \ \ ,. \ \ .. \ r 1 IllllLI _L 1 lllILLI r 1 11\r\11 2 3 10 10 10 G V2 quark Figure C2. Box Diagram Corrections to the leading Order Neutrino-Quark Cross Sections. The figure shows the corrections to the bare cross section as a function of squark due to the exchange of an extra photon The corrections are from a calculation by Wheater and Llewellyn-Smith. The dotted line is the correction to vd scattering, the small dashed line to Vu, the dashed-dotted to vs,andthelargedashedlinetch. 193 complication. The outgoing muon will emit an infinite number of infinitely small pho- tons thus the cross section for the process, vq-myq‘, is infinite. In principle, the radiation of photons by the quark legs should also be including in the next-to-leading order corrections to the cross section but one can argue that radi- ation of photons by the quarks is just another part of the physics underlying the structure functions and thus should just be included as part of the structure functions. This analysis uses this approach. de Rrijula et al. (1979) have calculated the leading log corrections to the bare cross section due to radiation of photons by the muon. The correction takes the form of a correction to the bare cross section. In this case, ”bare cross section” means the cross section calculated as if the muon charge was turned off. The relationship between the observed and bare differential cross sections, in terms of the outgoing muon’s final en- ergy, E u and the muon three-space angle, 0, calculated by de Rfijula et al. is, daobserved _ dabare + llog[ 5(1 - y + xy)2 ]1 deudo ’ dEde 2n #2 (C53) where, 1+221 do do 1: dz . )———lbm ———fl C.5b J0 l-z [29(2 -Z"“" dELdQ £ng /2 dEpdfl] ( ) and 2m = [15,, /1-:,][1+ E,(1— cos 9,, )/m,,]. (C.5c) The term s(1 - y + icy)2 is the square of the center of momentum energy of the muon. u is the mass of the muon, 0,, is the standard lepton scattering angle and z is defined by = E” /z. This complicated expression can be thought of as the decrease in the ob- served cross section due to the radiation of photons by muons with energy Eu and the increase in the observed cross section due to muons of the same angle but higher energy radiating photons. The variable 2 relates the photon energy to muon energy by, 194 ear-B. =E.:(1-z>. TheOfunctionandzm enforcethekinematiclimitonthe maximum energylossbythe muon while maintaining the same trajectory. Equation C.5 can be thought of as an effective radiator of a strength involving a, a kinematic logarithm and an integral over the bare cross section. Using the interpreta- tion of an effective radiator, the effects of the muon radiation can easily be included in a Monte Carlo by generating photons with the correct spectrum and normalization and removing the photon energy from the muon and adding it to the hadron system. The end result of the final state radiation is an effective smearing. The observed cross section differs from the bare cross section. For each event, energy is apparently transferred to the hadron system from the muon system. This results in an enhance- ment of the observed cross section at high 3] and low it. The high y enhancement results from the shifting of every event to a higher apparent 3] due to the final state radiation. The shift in energy decreases the observed Q2 (while EV and 6” remain constant, 13,, de- creases) and increases the observed v, resulting in a smaller apparent x for every event. It is also important to note that in the leading log approximation, the total cross section remains unchanged. 9.3 W For the process vd-9u+q, there are two possible flavors (neglecting the possibili- ty of top quark production) for q . The outgoing quark may be either a u or c quark. At high energy transfers, where v is much greater than the rest mass of the charm quark, the process vd-9u+c is Cabibbo suppressed while vd->u+u is Cabibbo favored. In con- trast, at low energy transfers, where v is significantly lower than the rest mass of the charm quark, the process v+d—>uc is completely suppressed due to the charm mass threshold and all vd charged-current scattering results in the creation of a u quark. The ”slow rescaling" model incorporates a threshold for the production of charm. *A 195 The slow rescaling model substitutes the quantity 5 for x in the calculation of the cross section for processes producing a c quark. The slow rescaling model defines 5 as, 2 g: “21:4"; y. (C.6) gisthenusedtocalculatethecrosssectionfortl'ieproductionofcharm. Let’ 5 look at the specific case of vd scattering. The bare differential cross section for vd scattering is simply, dzo‘” = 6,2,5 d dxdy 2n (erz), ((2.7) neglecting the propagator term In the slow rescaling model, one replaces the simple d(x,Q2) in equation C7 by, d(x,Q2) —> cos2 a, d(x,Q2)+(1- pg)“ o,d(§,Q’). (C.8) Expression C.8 can be generalized for vs scattering by replacing the d parton distribu- tion functions by s parton distribution functions and using the appropriate K-M matrix elements instead of the Cabibbo factors. For values of 6 greater than 1.0, all charm pro- duction is suppressed and the second term in expression CS is set to zero. 9.4 Apnlisatinn The three corrections for simple bare cross section discussed above were includ- ed in the Monte Carlo. Inclusion in the Monte Carlo simplifies the calculation of the corrections. The alternative to inclusion in the Monte Carlo is calculation of the correc- tions for each x, y bin. In reality, this correction would have to be done by Monte Carlo to include the effects of acceptance and would add an additional layer of complication to the analysis. Both the slow rescaling and the box diagram correction were included in the cal- culation of the cross section and the throwing of the Monte Carlo events. The final state radiation correction was implemented as alluded to above. After 196 the bare event kinematics were determined, final state photons were thrown and the energy transferred from the muon to the hadron system. Photons were generated with a spectrum, .‘1’_"_=1+(1-k)2 a log[5(1-y+xy)2] (C9) 2 I ' dk k 27: u where k is defined by Ekau. The spectrum of photons is the same as that in Equation C5. The spectrum is cutoff by the kinematics of the photon emission as defined by km = (1 - 2mm) where 2min was defined equation in C.5c. The mean energy radiated is, (15,) = 15,, jg“ kdkd—N ... 15 z—a-log[s(1 ' y + xy) ]. (C10) dk “ 37: 112 The average energy lost due to final state radiation is about 1% of the muon energy. A quick calculation of the integral, N = 1:,“ Z—Ide, for the number of photons emitted in the range Ito—9km and then setting it equal to N = 1, yields that ko<10_17. From this cal- culation, one sees that while the number of soft photons emitted is infinite, their total energy is negligible. For this reason, we throw only one photon in the Monte Carlo but we require that its energy be above k0 The energy of the single photon is removed from the muon and transferred to the hadron system. 2.5 Conclusions Radiative corrections and slow rescaling corrections have been made to both the measured cross sections and the structure functions. All these corrections have been irr- clude in the Monte Carlo to keep the analysis of the data reasonably straight forward, if not simple. Additional corrections to the simple model were described in Chapters 3 and 4. These corrections include Fermi-motion and non-isoscalarity corrections. ““313 ' Appendix D Systematic Error Analysis 2.1 Intrednsfinn It is essential to have an understanding of the systematic errors associated with the measured differential cross sections and structure function discussed in Chapter 4. While causes of systematic errors in the extracted cross sections and structure functions are well understood, their determination is not straight-forward. This appendix out- lines the procedure used to calculate the systematic errors for both the differential cross sections and the structure functions. 122W Systematic errors arise from two sources. The first source is measurement biases. The second source is an incomplete knowledge of the (for lack of a better word) accep- tance. In this context, acceptance refers to the effects of the various resolutions, the trigger, the pattern recognition programs, the analysis cuts, and the geometric acceptance of the detector and has a simple meaning. Acceptance is simply the ratio of the number of events in some bin that are reconstructed and pass all cuts to the true number of events in that bin. This section discusses the methods used to estimate the systematic errors arising from these sources. The systematic errors from the two sources are then added in quadrature. D.2.1 Calculation of Systematic Errors from Measurement Biases From the procedures used to calibrate the detector, one can determine the uncer- tainties in the scales used to measure events. Based on the estimated uncertainties in the 197 - ‘. J . . _.__ 198 scales, one must then calculate the uncertainties in the quantities derived from the mea- sured events. A commonly used method for estimating systematic uncertainties is to vary the scales by one standard deviation and use variations in the derived quantity as an esti- mate of the systematic error. This method has the virtue of being extremely simple to implement but is susceptible to statistical variations. This analysis uses an alternative method The calibration constants used in event measurement are varied based on the estimates of their uncertainties. The cross section and structure function analyses are then repeated for each of the variations in the scales. This results in an ensemble of experiments from which the systematic errors in both the differential cross sections and structure functions can then be measured. This method can also provide the functional variation in a derived quantity as a function of the vari- ation in a scale. When all the scales were varied simultaneously, the ensemble of experiments can also account for the correlations and anti-correlations between the vari- ations in the different scales. 0.2.1.1 The Method in Detail The variation in the scale parameters can be done in many different ways but to allow for the simplest interpretation of the ensemble, the parameters were varied using a Gaussian of the appropriate width The Gaussian was divided into N slices of equal probability and the mean value of each slice was used as the variation in the scale (See Figure D.1.) If more than one parameter was being varied, a matrix of variations was formed so that all possible combinations were included in the ensemble. For each combination of variations in the scale parameters, the entire data set was reanalyzed. In each bin, for each member of the ensemble, the value of the quantity of interest was calculated. The RMS of the derived quantity for ensemble was then calcu- lated. The systematic error was chosen to be simply this ensemble RMS. The ensemble gives more information than is available from simply varying a -e. 1' ~41? 199 0.40 0.35 dP dx TT‘IIYfi'lVTVU 0.30 VIII T I 0.25 'l 0.20 — 0.15 0.10 ' Differential Probability — 0.05 0.00 —3 Figure 0.1. Division of a Gaussian for Systematic Error Measurement. The Fig- ure shows a Gaussian divided into 9 slices of equal area (i.e. equal probability.) The mean of each slice is shown by the black diamonds (9.) The binning is chosen so that for one bin the average is 0 and so that there are the maximum divisions possible. The number of divisions are limited by computer resourc— es. ...r. ...... “lash—II KL?)- H“ 200 single scale by one standard deviation. The dependance of the uncertainty on each scale can be easily obtained. It is also possible, if desired to include correlations between the scale errors. D.2.1.2 Scale Errors In doing the detailed calibrations necessary for a structure function analysis, one determines both the event measurement scales and the uncertainties in the event mea- surement scales. This section discusses the magnitude of the uncertainties and how uncertainties were estimated. These uncertainties are also discussed in both Chapter 2 and Appendix B. As discuss before, there are three measurables used in the event reconstruction for this analysis, 0“, E“, and v. Letusexamine the scaleuncertainties ineachofthesein turn. D2221 9,. 9,1 is extremely important in the event reconstruction. For small angles, one finds, Q2 s (v+ Efl)Efl9:. As one can see, Q2 is proportional to 0;. For this reason, a bias in 9a would result in a large systematic error. One might imagine that mis-alignment of the calorimeter elements could result in a bias in 0“, To determine if there is an alignment problem, the data’s muon slope distributions in the two orthogonal views were compared with the same distributions from the Monte Carlo simulation. The agreement between the data and the Monte Carlo is excellent but the means of the data and Monte Carlo distributions differed slightly (~0.1 milliradians). These differences were then used as the input to a simple Monte Carlo to estimate the induced bias due to the possible misalignment. The Monte Carlo study showed that there in no induced bias just a small increase in the 0,, resolution (<10% of the resolution quoted in Chapter 2). This result is not surprising because of the 201 muon trajectory’ s azimuthally symmetry about the beam axis (i.e. there is no prefered 4),). and because 0,, is the result of the combination of the muon slope measurements in the 3 different views of the flash chambers. The azimuthal symmetry cancels out any alignment bias by for a given 0,, by averaging over all ¢,,. The question now becomes what are the other possible sources of bias in 0,,. Another possible source of a 0,, bias is a bias induced by the procedure that fits the muon slopes. The comparison of the muon slope distributions previously discussed argues strongly against a bias due to the fitting procedure since the data and Monte Carlo orthoginal slope distributions agree very well. A bias in 0,, could result from the uncertainties in the length and/ or width of the FMMF detector. The physical size of the detector defines the 0,, scale. The uncertainty in the 0,, scale is proportional to the uncertainty in detector size. Since there is no evidence that there is any bias in 0,, due to the muon finding and fitting routines, this leaves the uncertainty in the 0,, scale as the largest possible source of a 0,, bias known. To estimate the possible bias in the 0,, scale, one conservatively esti- mates that the length of the detector is known to 1.0 cm over the roughly 20 m length of the detector. This results in an estimate in the fractional uncertainty in 0,, of, 9%6-2 = 5.0x10'4, ,r due entirely to the uncertainty in the 0,, scale. This uncertainty is negligible and is thus not included in the systematic error analysis. D2222 E n The uncertainty in the E ,, scale is due primarily to uncertainties in the knowledge of the magnetic field of the iron toroids in the muon spectrometer and in particular the field of the 24' toroids. The uncertainty in E ,, is determined by expanding the fitting of the calibration scales using the neutrino beams, E V versus radius structure as detailed in Appendix B. In the expansion, a multiplicative factor of the measured E,, is included as 202 part of the fit. This fitting procedure results in only a slight modification of the E,, scale but the estimated error in the fitting is approximately i2%, which is used here. D2223 v The two largest sources of uncertainty in v are: improper elimination of noise in the calorimeter and errors in the calibration of the calorimeter. These two sources of uncertainty would result in two different types of variation in the hadron calibration scales. The uncertainty in the noise elimination and in the extrapolation of the calibra- tion to low energies could result in a constant offset or pedestal in the calibrated v. The uncertainty in the calibration could also result in v scale error. From the fitting procedure described in Appendix B, one can again estimate the uncertainty in the scales. Again, using the errors estimated by the fitting program, one finds that the uncertainty in the Vpedestal is :1 GeV and the uncertainty in the vscale is i2%. D213 The Final Ensemble The final ensemble used to measure the systematic errors is described in this section. The ensemble used to measure the systematic error could be derived from either the Data Sample or the Monte Carlo Sample. The systematic errors presented in Chapter 4 are derived from an analysis of the ensemble derived from the Monte Carlo. The same procedure and variation were used with the data with similar results for the systematic errors. The results for the Monte Carlo are used because of the greater statistics of the Monte Carlo sample. The final ensemble consists of 93 (or 729) different sets of calibration constants. The calibration constants varied were the E,, scale, v scale and the v pedestal. The varia- tion in E,, and vtook the forms, E n -> a n E ,, (D-l) v-> 6‘, +va. (D2) IF. 203 Table D.1 shows the variations chosen for each of the calibration constants. The final size of the ensemble was dictated by computer memory limitations. The odd number of bins allowed the 0 bin to correspond to no variation. Figures D2 shows the ensemble structure functions for three bins allowing the reader to gain a feeling for the dependance of the systematic errors due to possible measurement biases on the calibration constants. D22 Acceptance Uncertainties In making comparisons between the data and the Monte Carlo, one finds that there are a number of small discrepancies. From these differences, one may estimate the uncertainties in the knowledge of the acceptance. Based on the iron distributions shown in Figure 3.11, one calculates the uncertainty in knowledge of the acceptance as 1.3% of the acceptance. This error is applied to all bins. 123 Results The results of the systematic error analysis are presented with the measured differential cross sections and structure functions in Chapter 4. As mentioned before, the systematic errors are a quadratic combination of the errors associated with measure- ment biases and with acceptance uncertainies. 204 Table D1 Variation in Calibration Constants L. -1 0.96591 0.96591 -1.7046 GeV -2 0.98049 0.98049 —0.9756 GeV —3 0.98816 0.98816 —0.5922 GeV —4 | 0.99434 0.99434 —0.2832 GeV 0 1.00000 1.00000 0.0000 GeV +1 1.00566 1.00566 +0.2832 GeV +2 1.01184 1.01184 +0.5922 GeV +3 I 1.01951 1.01951 +0.9756 GeV ‘ I I 1.03409 1.03409 +1.7046 GeV Table D.1. Variations in Calibration Constants. The table lists the variations in the calibration constants. a,,, [3» and 5,, are defined by Equations DI and D2 For a discussion of the variation of the calibration constants, see the text 205 r- l- )- :,- -. ’é'i.‘ .‘Q {5 : :~ '2 ‘- ’2‘! 4. v‘ 1.4“: 33:93“ g, 1.4— 5“ 1.4—j ‘13”, . N pi, 3". «3331;; § '3» 9‘ : ‘ N :3 '1 :E‘" a it ’5 u. ’2 .3, 361.2,. -: u. _ , FL r; is its} 1 .3 p“. ..t ‘3 -,J -" )- r- ‘- s 1.2pgmgi 1.233 12:3; 2‘51: b . h. n 1'0 1111111111 1'0 1 I 1 I 1'0 111111111 0.96 1 1 .04 —2 0 2 0.96 1 1 .04 (1,, 6V(GeV) fly 1.4 L 1.4 1.4 E N t' N N " ”r Wit-1mg. "" name! ”'1 :I sin-use. 1.2 — 1.2 1.2 1— ~ )- : C 1.0 1.0 1.0 +— I I I I I I I I I I I I I I I I I I I I I I 0.96 1 1 .04 -2 0 2 0.96 1 1 .04 01,, 5v(GeV) By D P D 1.2 L 1.2 :— 1.2 :- N r-g' u N : ' -! .-:,..).,-. ...: '. N : - “t I” i “1 “eerie r! 1.0 ...." 7°- . i 1.0 Fe... ., ,. .44; 5‘: ..5 1.0 —£ s _ new " l' ' "5 '. ‘3‘. ‘ 533-N '- '- pg " o.s.1-....1....i 0811- r 1 . 0.8}...11..Uu 0.96 1 1 .04 -2 0 2 0.96 1 1 .04 al,, 6v(GeV) [30 Figure D2 Ensembles of Structure Functions. Shown are scatter plots of F, as a function of the varied calibration constants, a,,, By, and By. The rows show F, for three different x,Q2 bins. The columns show the dependance of F, on (from left to right) a,,, By, and 6.. The top row shows a bin which is very sensitive to 60. The middle row shows a bin with a relatively small systematic uncertainty. The bin of the bottom row shows an anti-correlation between a,,, and 0,,. The variations in the calibration constants was done at the dis- crete values shown in Table D.1. The horizontal widths of the variations in the calibration constants shown in the plots are an artifact of the plotting procedure. or La ‘P‘d 'I_‘ f I IIID‘. Appendix E Alternative Structure Functions 15.1 Introduction A number of assumptions have been made in the extraction of the structure func- tions presented in Chapter 4. Two of these assumptions have been modified and different sets of structure functions have been extracted. These structure function are described and tabulated in this appendix. E251 I E l' 'Illl -1° C S I' The standard assumption is that 0:; is a linear function of Ev. This assumption was used in Chapter 4. The value used for % was the average the results from the world’s three high statistics vN scattering experiments. The CCFR collaboration (Quintas 1992) have measured the increase in the neutrino—nucleon cross section as a function of the energy of the neutrino and fit % to the form A(1 — BE). It was found that, for the neutrino data, B = (0.4:t0.3)% per 100 GeV which is consistent with a linear increase in the cross section. For the anti-neutrino data, it was found that B = (331:0.7)% per 100 GeV. It should be noted that an additional increase in the cross section as a function of E V beyond the expected linear increase is not unexpected as the suppression of the produe tion of charm should decrease as EV increases. The entire structure function extraction process has been repeated using the val- ues for A presented in Chapter 4 but including the CCFR measured values of B for both the neutrino and anti-neutrino total cross sections. This set of structure function is tabu- lated in Table B.1. 206 _JJ'AE 207 EéSI I E I' 'l] 15] B 1' C I' The slow rescaling model of the suppression of charm production is a very sim- ple and intuitive approach to the problem. While the slow rescaling model is adequate for leading order QCD modeling, for next-to-leading order QCD calculations, the charm suppression is simply another part of the QCD of the parton distributions (W .K Tung Private Communication). For this reason, the structure function analysis has been re- peated with the slow rescaling corrections to the Monte Carlo cross section turned off. 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