DEEP INELASTIC MUON SCATTERING AT 270 GEV By PhiTIip F. Schewe A DISSERTATION Submitted to Michigan State University in partial fu1fi11ment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1978 Cd.“ 6 3: ABSTRACT DEEP INELASTIC MUON SCATTERING AT 270 GEV By Phillip F. Schewe The nucleon structure function vwz for deep inelastic muon scat- tering at 270 GeV has been measured in an experiment performed at Fermi National Accelerator Laboratory. A large violation of Bjorken 2 scale invariance has been observed out to q =150 (GeV/c)2, greatly extending previous deep inelastic results. The data reported here is based on a flux of 1.5 x 1010 positive- ly charged muons incident on an iron target/calorimeter. The energy of the scattered muon is measured in a spectrometer consisting of iron toroid magnets and wire spark chambers. 2 and The values of vwz measured in this experiment for high q fixed x lie systematically above the values predicted by a partic- ular formulation of quantum chromodynamics (QCD). The data also lies above the values for vwz obtained by extrapolating previous deep inelastic data to higher qz. The possibility that this rise in vwz is due a threshold-like behavior in Hz (the hadron final state mass squared) is studied by calculating the scale breaking parameter b(x)=aln(vw2)/aln(q2), and by fitting the data to various functions of NZ. ACKNOWLEGMENTS Fermilab Experiment 319, on which this dissertation is based, was conducted by a large group Michigan State physicists: R.C. Ball, D. Bauer, C. Chang, K.N. Chen, S. Hansen, J.Kiley , I. Kostoulas, A. Kotlewski, L. Litt, and myself. All of these people have been of some help to me during the running and analysis phases of E319. In particular, I have worked very closely with, and received much help from, Bob Ball, Jim Kiley, and Dan Bauer. It is a pleasure to mention the help and friendship of Sten Hansen, now working at Fermilab. His electrical troubleshooting abilities and his good humor helped keep the experiment going on many occasions. Keith Thorne was very helpful in preparing a myriad of fits to the data. He is patient, thorough, and resourseful. I have had several useful conversations about theoretical and experimental aspects of our experiment with Professors Wayne Repko, Lawrence Litt, William Francis, K.w. Chen (thesis advisor), and Eliot Lehman. Mr. Francis has been very forthcoming with the (as yet unpublished) results of the complementary muon experiment E398. The crew and staff of the neutrino department at Fermilab deserve special thanks. During the running of the experiment we were in almost continuous contact with them, and their response to the needs of our experiment was always quick and courteous. (ll) Finally I would like to thank the administrative and typing assistance of Mr. Mehdi Ghods, Candy Gronseth, and Delores Sullivan. (1.11) TABLE OF CONTENTS Page List of Tables VT List of Figures viii 1. Deep Inelastic Lepton Scattering 1 1.1 Introduction to Lepton Scattering 1 1.2 Deep Inelastic Muon Scattering and Related Physics 2 1.3 The Quark-Parton Model 8 1.4 Bjorken Scale Invariance 11 1.5 Gluons and Scale Breaking 15 1.6 QCD , 21 1.7 Experiment 319 27 2 The Apparatus and Data Taking 29 2.1 Fermilab Muon Beam Line 29 -2.2 Tuning the Muon Beam 31 2.3 The E319 Apparatus 34 2.4 Target/Calorimeter 34 2.5 Proportional Chambers 40 2.6 Spectrometer 40 2.7 Trigger Logic 48 2.8 Computer 54 2.9 Running Conditions 55 3. Analysis of the Data 61 3.1 Alignment 61 3.2 Calibration of the Spectrometer 67 3.3 Data Analysis 77 3.4 Resolution 97 3.5 Acceptance 99 3.6 Data Distributions 99 4. Monte Carlo 122 4.1 Monte Carlo Philosophy 122 4.2 The Beam 124 (W) 4.3 Interaction in the Target 124 Coulomb Multiple Scattering 125 Energy Loss 125 Fermi Motion 125 Cross Section 126 Conversion to Iron 126 Radiative Corrections 127 Dependence on R=osloT 127 Wide Angle Bremsstrahlung 129 4.4 Ray Tracing 131 4.5 MCP Distributions 2 131 4.6 Data/MCP Comparison: Extracging F2(x,q ) 131 4.7 Systematic errors in F2(x,q ) 142 5. Results and Conclusions 151 5.1 Summary of the Data Sample 151 5.2 Normalization of F 154 5.3 Parameterizing ScaTe Breaking 157 5.4 F versus x 160 5.5 030 Predictions 162 5.6 F2(x,q ) Compared to QCD 167 5.7 Moments 180 5.8 Fits to the Data 182 5.9 Speculations on Scaling Violations 184 5.10 Summary and Conclusions 190 Appendix A F2(x,q2) for Various Values of x and q2 191 References 199 (A) (A) (A) (A) no (A) (A) w 0) N N N N N N N N N N N N O O O O O O 0 O O O O O O O I O C O O 01.th tooowos .11 .12 Loooummpwm LIST OF.TABLES Magnet currents in Neutrino Hall Calibration of the 1E4 Dipoles N1 Muon Beam Line Magnet Settings at 270 GeV Z Positions for Elements in the E319 Apparatus Target/Calorimeter Density Proportional Chamber System Spark Chamber Properties Iron Toroid Magnets Trigger Types and Notation Primary Data Tape Format Scaler Averages for a Single Run E319 Data Runs Final E319 Alignment Constants in cm. CCM (Chicago Cyclotron Magnet) Calibration Runs Apertures in E398 Walls Calibration of the Spectrometer Using the CCM Calibration of the Spectrometer Using Monte Carlo Data Beam Tape Format Digitizer Clock Counts Track Finding Cuts Secondary Tape Format Page 30 35 36 38 41 41 45 46 so 56 58 6o 68 72 72 78 78 82 83 86 98 #h-D-h-h-P 0301th >>U1010101010101¢ .10 (a) % Resolution 0 as a Fundtion of y=v/Eo (b) % Resolution 0 as a Function of w (c) % Resolution 0 as a Function of q2 Summary of Main Monte Carlo Features Monte Carlo-(Eo+.4%)/Monte Carlo (E0) Monte Carlo (E'+1%)[Monte Carlo (E') Monte Carlo (e+.4%)/Monte Carlo (e) Radiative Corrections (off)/Radiative Corrections (on) Wide Angle Bremsstrahlung (off)/Hide Angle Bremsstrahlung (on) Monte Carlo (R=0)/Monte Carlo (R=.25) Single-Muon Analysis Cuts Data/Monte Carlo Comparison Results Various Fits to the Combined b(x) Data E319 Values for b(x)=3(lnF2)/a(lnq2) Moments Power Law Fit to F2(x,q2) in Various q2 Regions Fits to F2 2 for fixed x Regions 2 F2(x,q2) Versus q F2(x,q2) Versus x for fixed q Regions (vii) 101 102 102 123 143 144 145 147 148 149 152 153 159 164 183 183 185 192 195 NNNNNNN \Josma-wm .10 .11 .12 .13 LIST OF FIGURES Pa e Feynman Diagram for Deep Inelastic Scattering and 93 Associated Kinematic Variables Other Kinds of Lepton-Hadron Scattering 7 Incoherent Scattering from a Single Parton with Momentum xP 14 Quark Structure Function 14 Nucleon Structure Function 14 SLAC-MIT Data (ref. 10) Showing Approximate Scaling in 16 the Modified Scaling Variable m'=2mv/(q2+m2) Deep Inelastic Scattering Without Gluons: F2(quark) = 6(x/z-1) ' ‘ 2 18 Gluon Correction Terms: F2(quark)=6(x/z-1)+gza(x/z)ln(%g) 18 Nonzero OS and R_related to Gluon Bremsstrahlung 19 u - Fe Scale Violations 20 Constituents of the Quark in QCD Renormalization 25 Gluon Pair Production of Quarks 25 Kinematic Region of E319 28 Properties of the Primary Proton Beam 30 Schematic of Muon Beam and Beam Detectors 32 Proportional Chambers and Beam Counters 33 Magnetic Field in IE4 Dipoles 35 E319 Apparatus 37 The Details of a Corner of a Spark Chamber 43 ‘Trigger Banks 49 (viii) (A) 0.) (.0 N N N o o o o o o wwwwwwwwwwmmwww QmVO‘U'I .11 .12 .13 .14 .15 .16 .17 .18 .19 Fast Logic for the Full Trigger Trigger Bank Logic Gate Logic Aligning PC2, PC1, and the Front Spark Chambers Conventions for Spark Chamber Coordinate Axes Layout of E398 and E319 Apparatus During the Spectrometer Calibration Spectrometer Calibration (a) 250 GeV (b) 200 GeV (c) 150 GeV (d) 100 GeV (e) 50 GeV Flow Chart of the Analysis Program VOREP Joining Spectometer and Beam Tracks 2 A High q event Idealized Momentum Reconstruction Percentage Acceptance: w vs. q2 Percentage Acceptance: q2 vs. v 0 Acceptance 9 Acceptance Acceptance in q2 w‘Acceptance Acceptance in y=v/Eo and Ratio=E'/Eo Acceptance in x=q2/2mv Acceptance in W2=2mv+m2-q2 Data Distribution: ZMIN (a) before cuts (b) after cuts Data Distribution: q2 (ix) Page 51 52 53 64 64 103 104 105 105 106 107 108 109 110 111 112 .20 .21 .22 .23 .24 .25 .26 .27 .28 .29 .30 .31 .32 .33 .34 .35 wwwwwwwwwwwwwwww bh-fi-fi #wN Data Distributions: W2 Data Distributions: P, Data Distributions: x Data Distributions: E' Data Distributions: E Data Distributions: ebeam Data Distributions: escatter Data Distributions: xbeam Data Distributions: ybeam Data Distributions: R(WSC 5) Data Distributions: R(WSC 1) Data Distributions: Rbeam Data Distributions: RMAG Data Distributions: w Data Distributions: xBjorken Consistency of Several Kinematic Variables for Randomly Chosen Runs The Effect of R=oS/oT on the Cross Section Radiative Correction Diagrams The "Effective Radiator" Method Contributions to the Radiative Corrections in the (EO,E') Plane Monte Carlo Distributions: ZMIN (a) before cuts (b) after cuts Monte Carlo Distributions: q2 Monte Carlo Distributions: W2 Monte Carlo Distributions: P, (X) Page 112 113 113 114 114 115 115 116 116 117 117 118 118 119 120 128 130 130 130 132 133 133 134 #hb-b-P-P-P-h-F-P-h-D-P-b mmmmmmmmm KOCDVO‘U‘l-PWN .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .21 .22 H Monte Monte Monte Monte Monte Monte Monte Monte Monte Monte Monte Monte Monte Con o x-qE Corrected Yield vs. Carlo Carlo Carlo Carlo Carlo Carlo Carlo Carlo Carlo Carlo Carlo Carlo Carlo Distribution: Distribution: Distribution: Distribution: Distribution: Distribution: Distribution: Distribution: Distribution: Distribution: Distribution: Distribution: Distribution: Flux ebeam escatter xbeam ybeam R(WSC 5) R(WSC 1) R beam RMAG O.) xBjorken urs of Constant Systematic Error in F2 in the Plane ’ The Scaling Violation Parameter b(x) F2 vs. X E319 b(x) QCD Predictions for F2 and for Quark Densities Measured F2 versus x for Fixed q Measured F2 versus q 2 2 for Fixed x First Moments of F2 for u-p, u-d, and p-Fe Scattering A Fit to F2 Using a Linear Rise Above W2=80 (xi) Page 134 135 135 136 136 137 137 138 138 139 139 139 140 150 156 158 161 163 168 169 176 181 186 CHAPTER I DEEP INELASTIC LEPTON SCATTERING l.l Introduction to Lepton Scattering Since the time of Rutherford,physicists have probed the structure of matter, and the behavior of physical forces, by performing scattering experiments. It is convenient to describe the relative probability for a particular scattering reaction to take place in terms of a "cross section." This geometrical equivalent is intuitively useful: the larger the cross section, the greater will be the equivalent profile which the target particle presents to the incoming projectile particle, and therefore the more probable the interaction. Rutherford expressed the differential cross section for the scattering of an alpha particle from a nucleon target in terms of the scattering angle 6 (solid angle 9), the energy of the incident particle, E0, and the atomic number of the target nucleus, Z: d0 2284 (1) 3‘7 453 sin4e/2 For the case of an electron scattering from a nucleus the elec- tron's spin must be considered. If we also account for the effects of relativity and nucleus recoil, the formula in (l) becomes: 2 4 2 2E %%-= Z geosae/Z {l-f— m0 sin 2'16/2} (2) 4E0 sin 6/2 This is the so-called "Mott scattering" of an electron with spin from a spinless point-like nucleus with mass m.1 Finally one must also account for the proton's spin, and the proton's structure (it is not a point-like object). The "Rosenbluth formula" describes the scattering of an electron from a proton with structure:2 2 G +q 2G 22/4m 2 {E 2 =( day) 4» 37m 2G tan 8/2} (3) do" dDMott{1+qZZ/4m 4m In this formula, GE is a form factor which describes the scattering of the electron by the proton's charge (which is distributed in some way throughout the proton), while GM is a form factor for scattering from the proton's magnetic moment. m is the mass of the proton and q2 is the momentum transfer squared. The evolution of equations (l) - (3) shows how new concepts, such as relativity or spin, can be incorporated into the basic scattering cross section formula. The next development to be discussed is the situation in which the lepton-proton interaction is inelastic. l.2 Deep Inelastic Muon Scattering and Related Physics The Feynman diagram and associated kinematic relations for inelastic muon-proton scattering are shown in Figure l.l. The matrix element squared can be given in terms of a current-current interaction:2 P=(m,0,0,0)= proton at rest in lab frame k=(E0,0,0,E0)= incident muon k'=(E',0,E'sine,E'cose)= scattered muon q=(v,0,-E'sine,E0-E'cose)= virtual photon v=q.P/m =E0-E' = energy transfer q2=(k-k')2 = 4EoE'sin26/2= momentum transfer squared W2= Mi =2mv+m2-q2 =hadron final state mass squared x=1/w =q2/2mv = Bjorken scaling variable elastic scattering: 2mv/q2=m = 1 inelastic scattering: 2mv/q2 = m= 1/x>1 Figure 1.1 Feynman diagram for deep inelastic scattering and associated kinematic relations * 2 IMI2 = [(E'Yvk) (E'Yuk)1 (4"; )2 [Z X 4 -2w6((2+9)2-W2)1 (4) _ 4nez 2 5 - LW (_:F?-) wW ( ) The first bracket represents the lepton part of the matrix element and is known from quantum electrodynamics. This is the advantage of using a lepton beam to probe the structure of the nucleon; since the muon does not interact strongly, its contribution can be calculated exactly leaving only the hadronic part to be measured: *. = | I = I + l_ .l LW (k yvk) (k Yuk) 2(kukv. kvku Guvk k ) (5) The second bracket in equation (4), representing a summation over all hadron final states, can be simplified using gauge and Lorentz invariance:2 “iv = E |p> 2n6<2-w2) (7a) = (Pu'qu €$§i(pV-qv Pi?) w2(qz.v> (7b) 9 q ‘ "12(511V- 112V) ”1(qzav) q W1 and W2 are structure functions roughly analogous to GM and GE in the elastic case, equation (3). They are functions of the two Lorentz 5 invariants v and q2. Although I will return later to equation (7a) while discussing the formulation of quantum chromodynamics, I will now just utilize (7b), which can be used to give an expression for the scattering cross section analogous to the Rosenbluth formula. This expression, for small scattering angles,is given by: 2 2 2 d o 2 8 cos 6/2 2 2 2 —-.—— (4 ,v) = [W (4 ,v) + Ztan 6/2 W (q .v)] (8) dE d9 4Egsin4e/2 2 1 This cross section can also be expressed in terms of equivalent absorption cross sections for the scattering of transversely polarized (GT) and longitudinally polarized photons (cs): 2 331% = I‘(oT+eoS) (9) r(q2,v) --—i§~l%~%-(T%E) = effective flux of virtual photons 4 q - 2 2 2 -l _ . . . e - [l‘+2(l'+v /q )tan 8/2] — Virtual photon polarization k = (WZ-m2)/2m The conversion between W1 and W2, and o and GT is given by: s W =-—l%—-o 1 4n a T (10) ” =Tk 7927“ *0) 2 4n a q +v T S The ratio R(q2,v) = oS/oT is a more useful function than N]. 6 With a little algebra, equation (8) becomes: 2 2 2 vW 2 2 2 6 + T? 39 (w) = “20°? 4/2 f [l +2tan26/2 (—T—£—‘ “+4 )1 (ii) 4E0 Sln 6/2 Present data34 give R= constant = .25:.l0 although there are indications 2 that R may vary with q and v. In the quark-parton model, a measure- ment of sz(q2,v) and its moments can be used to find the momentum distributions of individual quarks within the nucleon. There are other interactions which also probe the structure of hadrons. Besides up-+ pX, which I have been describing, the reaction epi+ eX should be entirely equivalent from muon-electron unversality.3 . . . + - . . . . . The annihilation process e e +»X is Similar to the ep interaction, only turned on its side, as shown in Figures l.2a and l.2b. In the 2 annihilation case, q > 0 is timelike, whereas for inelastic 9P 2 < 0. Figure l.2c shows neutrino scattering where the scattering.q hadron's weak current is probed by an intermediate vector boson W. The scattering cross sections analogous to equation (ll) for the annihilation and neutrino scattering respectively,are given by:4 2 2 34% (e+e"+X) = g9- m2 / vzlqz-l q4 e+e' + - 2 vW {2W1g‘e +-2'Ez- (1'9?) —-2—2—m-— sin26/2} (12) q v dzo 62 2 W (159-*1“) = 5; s [F2(l-y)+F]xy iy(l -y/2)xF31 (13) 6+9." + X q2>0 vP + TTX 5P+ fix qz<0 (C) Figure 1.2 Other kinds of lepton-hadron scattering 8 In the above expression m=proton mass, y=v/E0, and F3 is a third structure function necessitated by the violation of parity in the weak interaction. From crossing symmetry, we can relate the inelastic and annihilation structure functions: W e'e 1 (Ci2 .v) - w1ep e+e' 2 e 2 vwz (4 ,v) - vwz p(q ,-v) The reactions up-thx (with certain final state hadrons being measured), ep-+eX (with polarized beam and target), vp-va (weak neutral current), and pp-+pr (massive lepton pair produced) also help to measure hadronic structure. All of these interactions can profitably be studied, and related, using the language of the quark-parton model. l.3 The Quark-Parton Model The identification of the hypothesized (charged) pointlike con- stituents of nucleons, known as partons5 6,2 , with quarks, appears to be nearly complete, and I will use the words interchangeably. With this identification comes the best features of both theories; the ability to classify the hierachy of observed particles as well as making dynamical predictions about interactions. The standard quark- parton model of the proton is one where three "valence" quarks are accompanied by a "sea" of quark-antiquark pairs.7 In addition there are perhaps an infinite number of neutral vector gluons around to mediate the interactions between quarks, and, presumably, to bind them within the proton. 9 In studying how the partons are distributed within the proton, it is useful to consider a single parton, carrying a fraction x of the proton's total momentum P. The remaining partons (and gluons) together carry the rest of the momentum. xP p :} (1-x)P Quark density functions qi(x) can be defined such that qi(x)dx is the number of quarks of type i with momentum between xP and (x+dx)P. i can be any of the quark flavors (u,d,s,c) or antiquarks. q,(x) = qia‘e"ce(x) + qiea(x) (is) The total momentum carried by i-type quarks is the density times x, integrated over x from zero to one: L;xqi(x)dx. In the next section, I will show that the structure function 6W2, as used in equation (ll), is the sum of scattering contributions from all the quarks in the proton weighted by their quark charge ei: xq1-(x) (16) Using this equation, and the above convention for quarks in the proton, several predictions can be made (sum rules, cross sections, etc.). The agreement between theory and data tends to be good, but not perfect. l0 For describing scattering from neutrons as well as protons, it is convenient to define u=u =dn and d=dp=un. Then, the structure P functions for the nucleons become8: luP=£-l-i-l- §””2 9(u+u) + 9(d+d) + 9(c+c) + 9(s+s) (l7a) l pn _ 4_ - l_ - 5_ - .1 - §vwz - 9(d+d) + 9(u+u) + 9(c+c) + 9(s+s) (176) If we neglect charm and set ecabibbozo for the moment, the neutrino structure functions are6: l x VP 2 N II N A o. + :1 v :3 oo v l vn _ - 2”“2 - 2(U+d> (19) Some simple sum rules can be formulateds: no. of u quarks in the proton = (;dx(u-G) = 41dxuvalence = 2 (20) no. of d quarks in the proton = fo'dx(d-a) = fo'dxdvalence = l (2i) '95. vn vp = vwzep-vwze" ~ 3(u+fi+d+d) + %(s+§) 5 = z — (23) vNZVP_szVH 2(u+0+d+d) ll 1 _ - f~%§(vW2eP-vw en) = §gfdx(u+u-d-d) 2 = 37(dx(uvalence'dvalence) ='3 l.4 Bjorken Scale Invariance One of the most important applications of the parton model has been in deep inelastic scattering. First, because the lepton part of the scattering matrix element is known from QED, the structure of the nucleon can be measured directly. Secondly, since the muon does not interact strongly, it need not scatter coherently off all the con- stituents in the nucleon, but can concentrate its transverse momentum transfer on a single parton; in this way, relatively higher q2 is attainable than in a hadron-hadron collision with the same center-of- mass energy. Equivalently, for large enough q2 (large compared to the proton mass squared), the virtual photon's wavelength is so small that the photon begins to resolve structure at the level of individual partons, and no longer scatters from the nucleon as a whole. The contributions from two-photon exchanges has been shown to be small9 so that the impulse approximation of a single photon, scattering incoher- ently, is generally assumed when discussing inelastic scattering. Bjorken and Paschos built up their parton theory of inelastic scattering using a reference frame where the proton has infinite momentums. In this frame the constituent partons share the proton's longitudinal momentum while their motion within the proton is slowed down by Lorentz time dilation. The muon discovers the proton in a particular virtual state and scatters off a single parton, as in 12 Figure l.3. The time of interaction in the proton-muon center-of-mass system is: r = i/q0 = 450/(2mv-42) (25) The lifetime of the virtual state is given by: [ l-x l-x R = 0' /o' = 2 z (33) L T Q2 Zlong/Az 2log4Q2 2 l-x 2 Therefore

~ (34) 18 mu 0" quark Figure 1.7 Deep inelastic scattering without gluons: Fguark = 6(x/z-1) antiquark (c) (d) Figure 1.8 Gluon correction terms: Fguark=5(x/z-1)+gza(x/z)ln(q2/q§) quark gluon proton photon 1111 Figure 1.9 Nonzero o and p.L related to gluon S bremsstrahlung Thus the gluon-bremsstrahlung induced "Fermi motion" within the nucleon contributes a scale violating term to the cross section, provides for a nonzero value of R, and could help explain Drell-Yan processes.25 After the initial success of the scaling hypothesis at SLAC‘O, several experiments were conducted at higher values of q2 and v. The results of these experiments indicated that scaling is indeed violated, that is,that the structure function F2 does possess a q2 dependence for l4 l5 M i6 i7 fixed x. u-Fe , e-p , , and u-p data show scale breaking effects. Similar results in neutrino scattering are summarized by Perkins, Schreiner, and Scott.18 Figure l.lO shows the u-Fe results. In this figure, the ratio [Data events]/[Monte Carlo events] (which is proportional to F2) is plotted versus q2 q2 dependence is present. for constant m=l/x. A definite 20 pA _. p: + Anything 150 GeV and 56 GeV T l I l I (o) 1-4 “aw-32.6 ‘ :- 1 a C on .8 1" g .i C O u d 0 8 :5 5 2 '1 h D o 2 C O 2 \ 2 W ‘.‘ A E V n .A Q A on. V \\ Data/Monte Carlo (hosed on scaling in w') l.2-(L .. 1.05)- § { .( 17 g: '62" ‘h) I -i to. 1 .i 0.8“- '3.5 "1 J I 1 I J 1 I 2 5 IO 20 50 Wow/c)? Figure 1.10 p-Fe scale violation results 2l At first an effort was made to recover scaling by defining new scaling variables. Indeed, by using the variable w'=w+m2/q2 some of 19 the scale-breaking tendencies apparently disappear. But the 2 (=4O(GeV/c)2), and violations persisted to even higher values of q the breaking of scaling is now reasonably established. The demise of scaling has been an important development in the study of constituent theories of the nucleon. The field theory which seeKS‘UDexplain how these violations come about is known as quantum- chromo-dynamics (QCD). It is a gauge theory of gluon-quark interactions and calculates the gluon radiative correction terms illustrated in Figure l.8. It is thought by some that QCD will be the field theory which can explain the strong interaction and possibly unite it with the weak and electromagnetic interaction as well.20 1.6 999_ Equation (7a) expressed the tensor for the hadron part of the deep inelastic matrix element ( |f> = final state). Wuv = E 2w6(P+q-X) (35) But since 6(P+q-Pf)='fd4x ei(P+q-Pf)-x (36) and (38) 22 The commutation of the two currents is [au1 = au(x)av(0) - avmau term is zero from momentum conservation,2 so that (38) can be rewritten: (x). The integral over the second wW =-§; 144x eiq'x (39) In other words, WW is equivalent to the Fourier transform of the one-nucleon expectation value of the current commutator. A lot of theoretical work has been devoted to the study of equation (39).20 The right hand side of (39) can be expanded using 21 Wilson's operator product expansion. The operators in this expansion are characterized by a spin n (tensor rank) and by their ”twist" 22 (dimensionality minus two). Pursuing this technique, one arrives at an expression for the moments of F2 but not F2 itself. The nth moment is described in terms of spin-n operators only:23 M(n.qZ) = IdE t"‘2 En(€.q2)F2(€,q2) n=2.4,6.... <40) 2 En(€,qz) = (i-m4a4/q4)(i-+q2/v2)(i-+3 ("+‘lmy§"i"22)%—i) <41) (n+2)(n+3)(v +q ) In these expressions, a new scaling variable is introduced to account for the mass of the target proton and differs from x only at small q2:24 5 :16 (“1426212 -) 23 For larger q2 (=l0(GeV/c)2) a simpler formula for the moments can be used fi(n.qzi = (,dx x"'2F2(x,qz> (42) In expanding (39) and in formulating the moments, there are two approximations which are conventional in QCD. Firstly, for a reference qg S 3 (GeV/c)2 one need only keep the "leading contributions" from 2:>3, the running coupling constant twist 2 operators. Secondly, for q as(q2) = gZ/4n is less than 0.3 so that only the lowest order pertur- bation term need be kept. This leads to the QCD operator expansion for the deep inelastic structure function moments:22 f M(n.42) =kgo efi [e'5*(")1§ ABq3. The method for computing the gluon are mixed together by the non- distribution function, and the expression for the elements in the A matrix, are given in reference [22]. The method for finding quark density functions will be described in chapter five at which time a QCD prediction for F2(x,q2) will be compared with the present deep inelastic data. Figure l.ll shows how the interdependence of gluon and quark densities comes about. Radiated gluons can split into quark-antiquark pairs of "sea" quarks which in turn can radiate gluons. In QCD, the virtual photon in deep inelastic scattering probes this complex system and not just a single bare quark. In equation (32) I indicated that the result of gluon-quark interactions was to introduce a scale-breaking term gza(z/x)£nq2/q§. A typical diagram is shown in Figure l.l2 where the muon scatters from a sea quark with momentum zP which was pair produced from a parent parton (a gluon in this case) with momentum xP. At small values of x this scattering from a sea quark will exceed that 13 of valence quarks. Altarelli gives a detailed account of how such diagrams arise in QCD and how the quark and gluon densities are effected 2 by the logarithmic q term: dqi (Z’t)- as“) 192$ 31- - ‘27,— Iz x [41(X.t) qu(z/x)+G(X,t)PqG(z/X)] (46) 25 I I I i quark l I J.__‘__ “I antiquark W I l I l l gluon quark t=tO Figure 1.11 Constituents of the quark in QCD renormalization Figure 1.12 Gluon pair production of quarks 26 t 2n dG(z,t) = 0‘5”) 1.9112; 2 x 1 . qg(x,t)PGq(2/X) + G(x,t)PGG(z/X)1 (47) where t==£nq2/qg, f==number of quark flavors, and qi and G are the quark and gluon densities. The function qu(z/x) is the probability that a quark with momentum zP is contained in a quark with momentum xP, PqG(z/x) is the probability that a quark with momentum zP is to be found within a gluon with momentum xP (Figure l.l2). There are also terms for gluons within quarks and for gluons inside gluons: unlike photons in QED, gluons in QCD can interact with other gluons. Equations (46) and (47) show how the quark and gluon densities observed at momentum zP (gluon densities are measured indirectlyzz) are a function of parent quark and gluon densities at momentum xP (where there is an integration over x from 2 to one). Except for the gluon- gluon interaction (gluons carry color while photons do not carry charge), this heirarchy «<——— 14 sec. cycle time 1113 bunches in the main ring bunch separation 18.8 ns. intensity in o neutrino area = 1.3 x 10 protons/pulse extraction RF frequency = 53.1 Mhz one spill 19 ns. Figure 2.1 Properties of the primary proton beam Table 2.1 Magnet currents in Neutrino Hall Magnet OUT OVT OHT OFTI OFT2 ODT OPT OPT3 Setting(amps) 290 15 121 96.2 95.6 2777 3102 3177 Reading(amps) 281-284 15.5 117.5 92.5 92.4 2690 2978 3060 31 At the end of the decay pipe the charged particles are swept out into the N1 beam line. If a pure muon beam is desired, the remaining hadrons in the beam can be absorbed using polyethylene inserted into the gap of the bending magnets. During E319, 60' of CH2 was in place, so that the effective hadron contamination in the muon beam was roughly 10's. The energy-selected muon beam is then brought into the muon lab via a series of bending, pitching, and focusing magnets. Figure 2.2 shows the N1 muon beam line leading into the muon lab. 2.2 Tuning the Muon Beam Figure 2.3 shows the last leg of the muon's journey into the muon lab along with the proportional chambers and scintillation counters used to define the beam trajectory and momentum. lF3 and 103 in enclosure 103 are sets of quadrupole magnets used to focus the beam on the face of the E319 target. In enclosure 104 the 1E4 magnets steer the muon beam through its final bend (28.7 mr) and are used for finding the energy of each beam muon. In Figure 2.3 HA and HB are beam hodoscopes, arrays of 3/4" wide scintillator counters which help to locate the position of each muon. The beam counters B1, 82, and 8 define a preliminary beam trigger. 3 Besides the beam hodoscopes, several proportional chambers were used to accurately establish a linear trajectory before and after the bending magnets; these are located in enclosure 104 and in the muon lab. We also had the help of several E398 (the Chicago-Harvard-Illinois-Oxford u-p experiment upstream of our apparatus) chambers for this purpose. These are labelled by plane orientation (x or y). The magnetic field in the 1E4 bending magnets was calibrated using an NMR probe, a gaussmeter, and a very accurate pole-face magnet. The 32 95323 3.2530: r. o 8 05288 3.2839." n. .83 0. vamp n>> Home 5 28m a .mom 2 news n m . ioN- 0050 88 @000 Ovmm mmmm Ommv mtum .. m u u “ fl " r em om om em 88 . .692 o. 8;. >83 55.605 59030 560: .252, 0.0 i w / 9.25623 ho .: on . s.’ 200. 3. 96 h\ Scan 385 .0. 833.05 3262a 86050 N3. a. 4’ . TO. 1 i‘ I . r . 4’ no m. _ 81$. 8. v ./ A it, ./ No. .0. now 8“. 8838; I 958.05 83305 /w8 om. <1 / . on. no. . . no 7‘ 8mm 2:865 vm mmir Om \ mos—m: .692 a eo. actmzoom _ 9:865 mmOHowkmo 24mm 024 25 con—2 ~.~ 23.: 24mm 2022 ..._O o_h<2mIom 33 Proportional Chambers and Beam Counters Figure 2.3 3:02 am»: iIIIWVii 3:02 wm>3 III-V: He“ Had _ =;_m_ Hme ems Hme mFm fill— _ l-J — JNHIIU ,5 _ i. .1 Hii— _ _ a _ _ _ __ . n _W “V. mznromcmm How mew“ mwmm mznromczm Hos ”Wows n_ Wm nu «ma fl . om" l.l In ._F L++_ _+i. d+ =_ _ §_+.L+ic .. .i...i .. enm we ax ens emu eom en_ meoneeoaoewe muem . mwem_ muse scoz r>w so" no mnmdm 34 measurement of the field as a function of the longitudinal coordinate (along the beam axis) is shown in Figure 2.4. This gives the effective length of the magnet. Table 2.2 gives a fit to the magnet field as a function of magnet current. The momentum spread of the beam at enclosure 104 is about 2% while the measurement uncertainty in E the energy of 0’ individual beam muons, is about 0.4%. After enclosure 104, the muons travel straight into the muon lab, through the E398 apparatus (shutters are opened in the E398 hadron shield), and into the E319 target where the spot size is an oval about 15 cm wide (east-west) and about 12 cm high (up-down). The intensity, energy, and focus of the beam could be controlled from a console located in the muon lab, from which the currents for all of the muon beam line magnets could be adjusted. These currents, both the settings and the measured values, are listed in Table 2.3. These currents were used for a majority of the 270 GeV u+ runs although there were some variations. 2.3 The E319 Apparatus The B counters (3.5" diameter) and the C counters (7.5" diameter) shown in Figure 2.3 act as a beam trigger. The proprotional chambers PC5, P64, and PC3 record the coordinates of the muon's trajectory up to the E319 target. Following the target is the rest of the E319 apparatus which serves to detect scattered muons and measure their momenta. A complete layout is shown in Figure 2.5, while the z coordinate of each apparatus element is listed in Table 2.4. 2.4 Target/Calorimeter During the principal 270 GeV u+ running, the target-calorimeter consisted of 110 sandwiches each comprising a 20" x 20" x 1%" slab of 35 1 1 1 1 1 1 B(KG.) 10 _ - -7 edge of the shim 8 .- 6 ._ I=324O amps 4 1—- 2 .- 1 14 z linches l (J l r O 2 4 6 8 10 12 14 Figure 2.4 Magnetic field in 1E4 dipoles Table 2.2 Calibration of the 1E4 dipoles B(KG) = a12 + bI + c I=current(amps) Runs before 8/23/76 Runs after 8/23/76 a (-.5964:.6656)x 10"8 (-.1714-_1-_.6134)x 10-8 b .32892x10'2:.3015x 10'4 .33635x10‘2:.2658x 10'4 C —.O3107:30273 -.O32787:30273 xz/dof 0.10 0.10 36 Table 2.3 N1 muon beam line magnet settings at 270 GeV if Magnet 1W01 1W02 1WO3 1V0 1F0 100 101 1E1 1V1 1W2 1F3 103 1E41 1E42 1155: bend bend bend pitch focus focus focus bend pitch bend focus focus bend bend Setting(amps) Reading(amps) 0 4332 4832 25 370 370 4175 3862 120 3712 940 980 4319.98 0 4630 4190-4180 4630 106.25 361.5 353-350 4000 3715-3720 8.125 3540 918.747 955 4237.48 4234-4230 37 mum: mom: v: mam 8m: 3m: mum: 398.83 33 33m E a mmmmmmmmmmmmmmmmmw—. .— NZ. m.~ deemed LT / zuEonégEoE‘p ./ a 38 Table 2.4 z positions for elements in the E319 apparatus Distances, in cm., are measured from the muon lab zero reference stud E319 PC's 1. 649.765 Trigger SA 1148.9 2. 625.318 Banks SA' 1170.7 3. -235.346 SB 1427.8 4. -517.764 58' 1449.9 5. -3685.54 SC 1710.5 SC' 1731.9 E398 PC's 1. -15512.95 2. -8512.305 3. -6393.487 Beam I 1464.8 4. -6393.487 Veto II 1746.7 5. -3294.281 III 1972.2 6 . -3294.281 E319 upstream end -166 5:13 I -480 target downstream end +572 11 -400 total length 738 Magnets position length 1911.193 78.90 1822.770 77.95 1655.128 78.74 1565.593 78.58 1370.330 79.06 . 1282.700 78.03 . 1092.678 78.98 . 978.555 78.98 mNmm-PUNH WSC'S 2190.433 2086.29 1988.03 1761.49 1478.92 1201.42 . 1035.37 922.02 . 848.68 tomVOTU'I-PNNH £31213 upstream piece 61.6 cm. thick front edge: z=736 downstream piece 37.5 cm. thick front edge: z=870 84" high x 145“ wide 39 3.. 8 density of the target is calculated in Table 2.5. When a muon interacts iron followed by a 20" x 20" x scintillator counter. The effective in the iron, the resultant hadronic shower deposited a characteristic amount of energy in the scintillators, which were observed by RCA 6342 A phototubes (gain==.4x106). These signals were digitized by LRS 22495A analog-to-digital converters and used to determine the energy of the hadronic shower, in addition to finding the interaction vertex. Calibration of the calorimeter was achieved by directing beams of hadrons (90% pions and 10% protons) at fixed energy into the target and then measuring the total digitized signal. Also, by using a standard light pulse from a light-emitting diode attached to the face of each scintillator, the effect of a single minimum-ionizing muon could be simulated. The following results for an optimum voltage of 1400 V were observed: signal/noise = 26.4, anode current = 62 mA, and anode charge = 18 p0. The construction and calibration specifications are given in greater detail in the dissertation of 0. Bauer.28 The use of the calorimeter for finding hadron energy has been a disappointment so far. It was feared that the electrical noise from spark chamber firings had disrupted the ADC gate pulse. This resulted in an apparent discrepancy between the hadron energy as found by the calorimeter and that found using the spectrometer. Since these two measurements are redundant, it has been possible to proceed with the data analysis without the benefit of the calorimeter. Recently though, the calorimeter mystery has been solved; the problem was in the way ADC pedestals (digitized signal for zero input) were being assigned, and not a faulty gate signal. This means that calorimeter results will appear in all future analyses of the data, but not in this dissertation. 40 2.5 Proportional Chambers The proportional chambers, on loan from Cornell University, were used to observe the incident beam track and, downstream of the target, to determine the scattered muon's trajectory before entering the spectro- meter. Each wire was monitored continuously and its status (fired or not fired) sent to a latch where it could later be read by the computer and stored on tape. The PC latches were cleared by a PC reset pulse while a second pulse, the PC strobe, enabled the latches only during the brief instant following an "interesting" event, as defined by a fast pre-trigger. Some features of the proportional chambers are described in Table 2.6. Construction details can be found in the thesis of Y. Watanabe.29 2.6 Spectrometer HADRON SHIELD Before entering the spectrometer the muon must pass through the hadron shield, two slabs of iron used to protect the forward wire spark chambers from the hadron shower particles which frequently emerge from the rear of the target. These slabs were 61.6 cm and 37.5 cm thick and covered the whole face of the spectrometer. The presence of hadrons in the spark chambers remained a slight problem, although not nearly as bad as in the previous muon experiment, E26. WIRE SPARK CHAMBERS The E319 spectrometer consists mainly of trigger banks to signal a scattering event, toroid magnets for deflecting the muon, and spark chambers for recording the muon's trajectory. Each spark chamber module consists of two pairs of planes; one set of planes (x-y) ' 41 Table 2.5 Target/calorimeter density The E319 calorimeter consists of 110 Fe-scintillator sandwiches Fe 110 x 1§- = 523.9cm x 7.87 gm/cm3 =4123.09 gm/cm2 stint. 110 x —4 104.8cm x 1.032 gm/cm3 108.15 gm/cm2 Vinyl 110 x 2 x .015" 8.4cm x 1.39 gm/cm3 11.68 gm/cm2 Al.foi1 110 x 4 x .006" 2.64cm x 2.70 gm/cm3 7.13 gm/cm2 Air 110 x 34 = 104.8cm x .0012 gm/cm3 .13 gm/cm2 4250 gm/cm2 effective density=42509mlcm2 /738cm 2 5.759 gm/cm3 5.759gm/cm3 x 738cm x 6.022x1023atoms/mole x 56 targets/atom /55.85gm/mole no. targets/cm2 2.5 x 1027 target nucleons/cm2 (the target is not entirely iron) Table 2.6 Proportional Chamber system E§_ Planes Active area in cm. 1 x y 38.4x38.4 2 x y 32 x 32 3 u v w 19 cm diameter 4 u v w 19 cm diameter 5 x y 19 x 19 wire spacing = 2.0 mm. PC reset pulse = 10-15 ns. PC Strobe pulse 90-100 ns. Gas mixture .263 % Freon 1381 20.0 % Isobutane 3.92 % Methylal balance = Argon 5 kv. Typical voltage 42 covered with wires placed at right angles, and another set of orthogonal wire planes (u-v) oriented at 45° to the first pair, mounted immediately behind them in the same external aluminum frame. Each plane of wires was placed at a large voltage difference with respect to its mate. An event trigger would cause a spark breakdown in a polished brass spark gap which in turn caused a spark discharge between the two orthogonal wire planes along the path of ions left in the wake of an ionizing high energy particle. The wire in each plane nearest the spark carried the current to one edge of the chamber (the other edge was damped) where an acoustic wave was induced in a magnetostrictive wire lying in a trough running the whole length of the chamber. This wire was encased in a Plastic catheter which was filled with Argon to diminish corrosion. The catheter was mounted in a long narrow aluminum channel known as a "wand." This is positioned beneath the current-carrying wires of the chamber it- self. The acoustic wave, induced by the passage of the current at 90°, propagated toward the end of the wand where it was detected by a small pick—up coil and amplifier assembly. The signals from as many as eight sparks can be detected in this way. The train of pulses from each wand is sent along to a discriminator and digitized by comparing the time of arrival with an accurate clock signal. Knowing the physical distance between the fiducial wires at either side of the chamber (giving fiducial pulses), one can calculate the spatial coordinate of each spark in each wire plane. The digitized signals from each plane (x,y,u,v) and each chamber (l-9) are recorded on magnetic tape. There are thus 36 planes of spark chamber information, a complete record of the muon's passage through the spectrometer. A view of one corner of a spark chamber is shown in Figure 2.6 while general properties of the chambers 43 089“. .3620 foam 36. :3 2.83 ewe—8.5 323m a we .8586 36:32. of Rm 933... 3:3 2 “.3023 use .828 .62... .55 .830 sum... x .300. .233 Room .326 BEE 38. so: 3;: an 00.0 9.68m 885. 3:2 :9; 2.2... 855 so om...moo. 44 are listed in Table 2.7. For construction details, and diagrams of the associated amplifier and discriminator circuits, see the diSsertation of 0. Chang.30 IRON TOROID MAGNETS The analyzing magnets were wire-wound iron toroids. Made of four sections welded together at the outer edge, these magnets were run in saturation with an average field of about 17 Kg. This field, applied over the length of the magnet (80 cm), imparts a transverse momentum bend of about 0.4 GeV/c. The general features of these magnets are listed in Table 2.8. Construction details and the methods for precise field measurements are given in the thesis of S. Herb.3] VETO COUNTERS Halo particles, mostly muons in the beam at a radius larger than about 9 cm, were kept from triggering the apparatus by placing halo veto counters in front of the target. Muons in the beam at large radius tend to have a larger beam angle relative to the beam axis, they often miss the active area of the beam proportional chambers, and they have often suffered energy loss by interacting in magnets and beam pipes along the way. Such muons are unsuitable for studying deep inelastic scattering. A large counter array similar to the horizontally oriented trigger banks is placed directly in front of the target, and rejects muons at large radius. A smaller counter, 15" square with a 7.5" diameter hole in the middle is directly in front of the first halo veto. This counter lets in good beam particles, but vetos muons which pass just outside the useful beam area. The tubes used in all these counters were Amperex 56 AVP's. 45 Table 2.7 Spark Chamber properties - each module has 2 pairs of orthogonal wire planes at 45 degrees relative to each other - 25 mil Al plates 80" x 80" outer dimension - active area = 73" x 73" - Be-Cu wires .005" in diameter - wire spacing = .7mm. - distance between fiducial wires 184.15 cm for ch's h's 1,2,3,4,5 182.88 cm for c 6,7,8,9 -high voltage for each chamber module: f chamber|12 3 4 5|6 7 8I9I voltage(kV) $6 8.4 8.4 7.6 7.2|8.6 7.6 7.817.4J triggering process: NIM trigger "_” thyratron -—’spark gap --’ Spark break- 25 ns down between wire 120 ns. planes in chamber I II V 12 kV - from onset of trigger signal to spark gap break down = 220 ns. - recovery time of charging capacitor in spark gap box = 40 ms. memory time = 1.jisec (a clearing field sweeps out stale ions) - gas mixture: Ne-He 78-80 % gas purified in Ar 2-3 % 9 "Berkeley" purifier Alcohol .7 SCFH @ 80 F and recirculated Ar in wand catheters, N2 in spark gaps 46 Table 2.8 Iron toroid magnets 172.7 cm outer diameter, 30.5 cm inner diameter about 80 cm long saturation current = 35 A, 450 turns average field = [B(r) dr f r residual "degaussed" field 17.09 KG magnet 1,3,5,7 17.27 KG magnet 2,4,6,8 200 gauss each magnet = 7.87 gm/cm3 x 80 cm = 629.6 gm/cm2 spectrometer = 8 magnets x 629.6 = 5036 gm/cm2 field measured using (i) B-H curve was measured for a smaller toroid of the same type, and scaled up (ii) B(r) measured directly using a coil wound around one slab of the toroid; coil passed through the center of the toroid and small holes drilled in the body of the toroid slab radial dependence of the field known to within 1 % B(r) = A/r + c + Dr + Fr2 B(KG) r(cm) magnet f A C D F 1.3.5.7 | 12.20 1 19.92 | -.08357 .0004346 | 2,4,6,8 | 12.07 1 19.71 I -.0827 .0004301 1 47 The muons which are not scattered into the magnetized region of the spectrometer continue on down the beam axis through the holes in the toroid magnets. To add additional protection against an accidental triggering of the apparatus by such a muon, beam veto counters (12.5" diameter) were positioned in the beam region behind magnets 4, 6, and 8. These counters, called BV], 8V2, and BV3 respectively, vetoed the trigger whenever a signal resulted from the coincidence of BV3 with BV1 or 8V2. The use of these counters significantly reduced the accidental trigger rate. 0n the other hand, a good event (a suc- cessful muon scatter into the spectrometer) might be vetoed if a shower hadron, not in the original beam, were to exit the end of the target, survive the hadron shield, and then penetrate the veto counters. To give further protection against such "punch-through" particles, the toroid holes were filled with concrete plugs which should allow through only unscattered beam muons. TRIGGER BANKS The principal type of trigger used in E319 consists of a beam muon scattering in the target and proceeding into the spectrometer where it will register as a "good" event if it passes through the three trigger banks (counter arrays) located behind magnets 2, 4, and 6. In order to do this the angle of scatter must have been large enough for the muon to have missed the holes in the toroid magnets and also to have avoided the beam veto counters. Trigger banks SA', 58', SC' are arrays of vertical scintillation counters observed at either end by 56 AVP phototubes. Each of the five scintillation counters is 14.25" wide by %" other counters by %”. Immediately in front of these is another set of thick, and overlaps with the 48 arrays. SA, SB, SC are mounted in the horizontal position. Since these arrays have a square hole in the middle, correction counters with round holes were added to restore full azimuthal symmetry. The dimensions and layout of the trigger banks are shown in Figure 2.7. 2.7 Trigger Logic The essential components of the trigger were a beam trigger and a halo veto before the target, and a trigger bank signal and beam veto after the target. The notation used to describe the various triggers is given in Table 2.9. Not all of these triggers actually resulted in data being recorded and the spark chambers being fired, but scaler readings were kept and latch information maintained for each coincidence signal. Figure 2.8 shows the main trigger circuit. The electronic modules such as disciminators, gate generators, and logic units sat in powered crates which could be gated (or enabled) for the length of the whole spill ("spill gated") or only during that fraction of the spill when the computer was actually ready to record data ("event gated"). The distinction between these two types of gating is indicated in the figure along with the various delay times in nanoseconds. Figure 2.9 shows the logic circuits for the trigger banks in greater detail. Figure 2.10 is the logic diagram for the actual formulation of the trigger and for generating various gates. Below is a description of how a trigger comes about. (1) The Fermilab T2 timing signal enters the delay pulser (refer to Figure 2.10) which in turn puts out several timing signals. If the computer is not occupied and there is no "pinger veto" signal present indicating the onset of a sharp pulse of neutrinos for the bubble 49 Vertical Trigger Banks SA' 58' SC' - 56 AVP tubes - counters overlap 1/4" - all counters 14.25" wide x 3/8" thick 70" 56" 1 l . 1-_ h? .14-2:. ——4' l I 4.___70.25" .1 /////r “\\\\\\ Horizontal Trigger Banks 60" SA 53 SC / 28 ll 5 14" 28 I. \ 70" I 1‘4" I l 1 \ 70w ’ ’ / \ 30" / .JL. - — Figure 2.7 Trigger Banks 50 Table 2.9 Trigger types and notation evg = event gated spg = spill gated SA, SB, SC = horizontal trigger counter arrays SA', 58', SC' = vertical trigger counter arrays S = (SA+SA') - (SB+SB") - (SC+SC') HV = HVI + HVII = halo veto C = CI'CZ'C3 3 = 3104 . C . RV = Beam trigger SD any 2 or more counters in SA and SA', and in SB and SB' “dimuon trigger" SL = hits in outer lying counters = "large angle" trigger SS = hits in inner lying counters = "small angle" trigger H = pion trigger for calorimeter calibration = Bevg ' gnv Bevg ' S - BV = single muon trigger Bevg ' SD ° BV = full dimuon trigger Bevg - P = pulser trigger operating trigger for E319 at 270 GeV = o ‘B . o_ o Bevg S V + Bevg SD BV + Bevg P B104 ‘ B ' B2 ' B 1 3 BV = (Bv1 + sz) - Bv3 = Beam Veto 51 wuxr‘cth + >3: :2: 33¢ :3 ‘33 \No I .33 ”"0”“ % .‘vsoc o-.Os~¢ss “(otcu d 322:2: 2.»... .u H“. ”mm...“ 3:: . , w x :5“; d at: . a. a 2: 4 L33.» 9 .h . u . . . :92: ~_/ th<° 3:“ § J...- 03.! .313: \m: .0 Figure 2.8 Fast logic for the full trigger 52 5". ecu? Figure 2.9 Trigger bank logic 53 7'2 (lay ruastl CAP! r0 C DMT‘CL Ct” 710‘. t“ n( PA I (our: ,9 I- m r) A u TToA/ v! r: M 7! (9'5) =‘ BISOAI 0" PCS flaw“ 12 > ’3 fl PCS (:40) “3:7 "It (I!) I an 53.? (J'll . 1‘1: r O) a .1 Figure 2.10 Gate logic 54 chamber, the Spill gate is turned on and stays on for the duration of the spill, about two seconds. This enables all spill-gated modules. (2) A successful scattering event will generate a NIM-level trigger. An early quick trigger, B o (SA + P), has already cleared and enabled the proportional chamber latches. Now the more thorough NIM trigger, B-S-BVVFB°SD-BV3+B°P, is formed in the gate control box. This signal fires the spark chambers, begins the time-digitizer clocks, starts the process of latching counter and scaler information, and generates a TTL-level trigger for other specialized tasks. (3) Now the event gate is turned off, vetoing any new-arriving information. The computer begins to read all of the latched data and other information modules via a branch driver and the CAMAC data acquisition system. This takes about l0 msec, during which time the event gate remains off. Of course, the scalers which monitor the incident muon flux are also gated off; it's as if the entire experiment was turned off while the computer was busy. Actually it is the spark chamber recovery time, about 40 msec, which establishes the amount of dead time and not the computer. (4) When the built-in dead time counter has elapsed, and the computer and Spark chambers are ready again, the interrupt is lifted and the event gate is turned on. The experiment is "active" again. Events continue to be recorded until the end of the spill, signalled by another Fermilab timing pulse, and the spill gate is turned off. 2.8 90mputer The computer used for the on-line superintending of the experiment was a PDPll-45 with a 32K memory. This computer was interfaced to the CAMAC hardware via a BDflll branch driver . For a 2 second spill and 55 a deadtime of 40 msec, it was possible to handle as many as 50 triggers per spill. The block of data for each event was stored on disc, and when time allowed, was written onto nine-track magnetic tape. Approxi- mately l04 triggers could be written onto a single tape. There were 768 words per event and 4 events per buffer. The data block format for each event is shown in Table 2.10. Logging data is not the only function of the on-line software. The computer accumulated run information continuously. When time allowed, between spills for instance, this information could be displayed on a CRT or printed out on paper. Many of these accumulated diagnostics were regularly printed as part of the end-of-run procedure. The infor4 mation available concerned all aspects of the apparatus: spark chamber spark distributions for each wand, histograms are made of fiducial positions and behavior, and the number of sparks on each wand; hit distributions and hit multiplicities for all proportional counter planes; DCR latch information giving hit information for each counter in all the trigger banks; calorimeter counter pulse heights and the equivalent number of ionizing particles; and an event display which showed a plan view of the whole apparatus with the appropriate sparks displayed. 2.9 Running Conditions The majority of running time during E319 was devoted to 270 GeV muons. The trigger rate was sufficiently high that some care had to be taken in optimizing the shape of the main ring acceleration cycle. Although a high trigger rate was desirable, each trigger was followed by a 40 msec deadtime (while the spark chambers recovered) during which time the incident muons on the target, including those that scattered, 56 Table 2.10 Primary data tape format flogg§_ Contents # words used 1-15 1.0. block 15 16-87 24-bit scalers 72 88-179 E319 PC's 92 180-215 E398 PC's 36 216-220 DCR's 5 packed 221-228 TDC's 8 packed 229-456 ADC's 228 packed 457-464 unused 8 465-761 NSC digitizers 297 762-768 unused 7 768 words/event 57 were ignored. We can define the number of event-gated triggers per spill as Tevg' pg per spill, f as the duration of the spill, and d (40 msec) as the dead- If we define TS as the number of spill--gated triggers time then T r = SP9 (49) ev T 9 1 + SEf9*d The optimum trigger rate was achieved with a flat top (length of spill) of 2 seconds with a main ring cycle time of l4 seconds. There were several important indicators of the quality and con- sistency of each run (a "run" was usually a full tape's worth of data-- l0,000 events--or a fraction thereof). These quantities, for a typical run, are shown in Table 2.ll. They were recorded by hand from visual scalers in the lab, as well as written on magnetic tape along with the other data for each event. Some of these scalers, or ratios of scalers, need some explanation. BDERR is the number of branch driver errors, caused by malfunction of the CAMAC reading process or by the computer itself. We took data primarily during the summer of 1976, which was very hot. The number of branch driver errors rose almost linearly with the outside temperature. evg'BVdelay itself. Remember that BV=(BV1+BV2)-BV3 is the beam veto The effective incident flux of muons was given by B and not by Bevg signal. BV is BV delayed by 60 ns. which is approximately 3 r.f. delay buckets. In magnitude it should be the same as BV since the number of muons in any r.f. bucket should be a constant. There are two main 58 Table 2.11 Scaler averages for a single run Scaler Interpretation Average per run gngggfivg ++ standard B.p 8V9 trigger 7838 BDERR branch driver errors 111‘6 B°Bvdelay effective incident 7.831 x 107 (1'5 flux B-S-BVevg Single muon trigger 7383 B-SD-BVevg dimuon trigger 865 B-Pevg pulser trigger 376.7 B-s-B'V v 4 E‘_e‘9 event rate .90536 x 10' evg HV'Snv/Bspg halo 102.53% - -8 Bspg/SEM lJ/ P Yield 5.44 x 10 8 /no. of . . , spg . . inCident i: s 6 Spills per spill .50272 x 10 Bevg/Bspg dead time 46.56 % Bspg/Bspg(104) beam tune 68.38 % f average lux x average #targets/cm2 luminosity 2.0 x 1035 cm"2 per run 59 reasons why Bevg itself (the normal beam trigger) overestimates the usable incident flux: (i) BV has a non-zero accidental rate; that is, occasionaly, BV=l and 8750 even when there was no muon through the beam veto counters. This would kill an otherwise good event. We correct for this by substracting an appropriate amount of flux. (ii) A second muon coming in the same bucket as one which scattered successfully will fire the beam veto and kill the good event. Bvdelay effective flux accordingly: simulates both of these problems and can be used to correct the B-B delay = B - B'Bvdelay = B - corrections for (i) and (ii) (50) The halo is defined as the coincidence of a halo veto signal with a spill gated signal from the trigger bank divided by the number of muons in the beam proper (B spill-gated). The u/P yield is an indication of how well the whole muon beam line is tuned. For a given number of protons incident on the neutrino area production target, we tuned the magnets (Table 2.3) for maximum muon yield. SEM is just the Fermilab record of the number of protons sent to our experiment for producing muons. Dead time is the fraction of the muon beam which was actually used. Many of the muons in the beam passed unused because the computer was busy recording data (when the event gate was turned off). This dead time is related to, but not the same as, the "dead time" due to spark chamber recovery time. 60 The beam tune is just the ratio of beam muons into the target, Bspg’ divided by the number of muons which were in the beam as of enclosure l04 (B104=B1-Bz-B3). This ratio gives an indication of how well focused or parallel the beam was by the time it reached our target. There were several modes of running other than 270 GeV positive muons during E3l9. In order to check interference effects we used a 270 GeV u- beam. A sample was taken at l50 GeV as a possible check of energy dependent scaling effect or multimuon production. Another 270 GeV u+ sample was taken with two thirds of the iron target removed; it was hoped that this would facilitate the study of possible rate effects in the full-target 270 GeV u+ sample. Various calibration runs were made for the calorimeter and the spectrometer. The following table is a summary of running modes in E3l9 Table 2.l2 E319 Data Runs Type of Running Triggers Incident Flux 270 GeV u+ 1.47xio6 l.473xl010 u's 270 GeV u" 0.39xio6 0.365xio10 u's 6 6 0.418xio10 u's 0.162xio10 u's 270 GeV “T (l/3 target) 0.l4xl0 150 GeV u+ 0.29xl0 CHAPTER III ANALYSIS OF THE DATA 3.l Alignment The alignment of the apparatus elements produces, in effect, a system of absolute spatial coordinates for all proportional chambers (PC's) and wire spark chambers (NSC's) relative to the toroid magnets and the nominal beam axis. The establishment of such a coordinate system is crucial to the determination of the scattered muon's momen- tum. Any accidental offset, rotation, or physical defect in the chambers which would give an inaccurate representation of the muon's coordinate at any of the chambers before, after, or between the toroid magnets, must be corrected for. E3l9 run number 130 was made with the target removed and the toroid magnets shut off. Beam muons could therefore travel the entire length of the E3l9 apparatus in straight lines, except for some Coulomb multiple scattering in the toroid iron. The sparks registering in all the chambers were fit to a straight line. The residue Ax= xfitted ' xobserved is then histogrammed for each chamber. These "window" distributions show how much a particular chamber is misaligned. The intrinsic measurement error of the spark chambers is 0.l cm. An additional error is expected due to multiple scattering and is pro- portional to the amount of iron traversed by the muon. These errors 61 62 are symmetrical about the straight-line trajectory the muon would otherwise follow. If the chamber is misaligned the mean of the window histogram will be nonzero. This nonzero mean is used as an alignment shift for each PC plane and each NSC wand (x,y,u,v planes in the NSC'S). ' The alignment is run again with the new parameters; this process is iterated until the residues become acceptably small. The alignment procedure consists of several steps. First, the alignment of PC's 3 and 4 (the beam chambers upstream of the target position) is fixed. These chambers serve as the anchor for all sub- sequent alignments. Although the other beam proportional chamber, PCS, would ordinarily have been used to help locate the beam track, it was not utilized during the alignment since the heavy iron shutter in the E398 apparatus had been accidentally left in place. Due to multiple scattering in this iron (about six feet thick), and the great distances involved, PCS could not really contribute effective beam information. For run l30, only events with a single beam muon (about 80%) were kept. The muon beam track, established in PC3 and PCS, was extrapolated into the "hadron" proportional chambers, PCl and PC2, downstream of the target position, and into the four forward spark chambers (NSC 9,8,7,6). The rear spark chambers (NSC 5,4,3,2,l) could not be aligned with the rest since their center regions were deadened in exactly the central region where the beam passed through. In addition to having only one beam track, each acceptable event had to have sparks present in all four views (x,y,u,v) in at least three out of the front four spark chambers, and with residues smaller than 2.0 cm. In the case of multiple sparks in a single view, the one with the smallest residue was chosen. New alignment constants were 63 derived from the window (residue) histograms made for each wand, and the alignment process was begun again. The iterations were stopped when the mean of each window histogram (the amount by which the align- ment would have been shifted in the next iteration) was smaller than .00l cm. In this way the hadron PC's and the front spark chambers were aligned relative to PC3 and PC4. The layout of this part of the apparatus is shown in Figure 3.l. Each spark chamber module has four views. This built-in redun- dancy is desirable in reconstructing the proper trajectory of the muon in three dimensional Space. Therefore, it is important that in mini- mizing the "window" residues of all wands in a particular view, all the y wands for instance, that the internal relations among the four views of a single chamber module are not distorted. The following "match" residues were histogrammed for each chamber: = H:!._ = £15.- Axmatch ,7 x Al"match /2 u (50) = 211.. = X:§.- Aymatch ,/f y AVmatch ,7 V The coordinate axes, as they are used in E3l9, are shown in Figure 3.2. In this figure, one is looking downstream at a single spark chamber module. To determine how well the overall alignment was progressing, the spark positions were fit to a straight line and the resulting chi- squared was histogrammed. As the alignment converges, the window and match distributions should become more nearly centered along with a 2 decreasing average x per degree of freedom. The best indication of hadron hadron magnet shield shield magnet \\ m PC3 P02 PCl E” E .\ \\\\\ \ \x\\\\i\w NSC9 8 7 6 Figure 3.1 Aligning PC2,PCI, and the front spark chambers Aup / x\ V U Y \ / V. west looking downstream Figure 3.2 Conventions for spark chamber coordinate axes 65 a good alignment is the number of good events being found; previously misaligned chambers would gradually contribute true spark positions increasing the chances for acceptance of the event. The rear spark chambers were aligned by the same method, using instead straight lines found in the front spark chambers (9,8,7,6) to find expected spark positions in back (5,4,3,2,l). The data used for this part of the alignment consisted of runs ll3-l20, in which the muon beam was purposely defocused by turning off the quadrupole magnets in enclosure l03, sending a broad beam into the face of the spectrometer rather than down the nominal beam line. The trigger for these events was S-BV (a muon thrOugh the trigger bank but not through the beam veto). As mentioned above, one of the main problems encountered in the alignment procedure is the broadening of the window distributions due to multiple scattering. This problem was partially overcome by using high statistics, l0,000 events. Ne also tried to avoid multiple scattering by triggering only on muons traveling through the "bat wings" of the chambers, the eight triangular regions which stick out beyond the extent of the toroid magnets. But statistics were so low in these runs that they could not be used for alignment. A second, and more serious problem is that of the relative align- ment of wands within a single chamber module. Centering the window distributions in each of the four views within 0.l mm. can leave the match distributions off-center by as much as 3 mm. This is remedied by displacing the x and y wands by an amount Ax = a+bz and Ay = c+dz respectively, where z is the distance along the beam axis and a,b,c, and d 66 are to be found by minimizing the following expression: 2 _ g (u+v )2 + (u-v )2 + (y+az+b-x-cz-d _ u)2 wsc=1 7? ”7 X @ y+az+b+x+cz+d _ 2 «t V) (51) + ( This shift in the x and y views will leave all of the window distri- butions unchanged while it centers the match distributions. The remaining chambers, E3l9 PCS and the E398 proportional chambers in enclosure 104, were aligned using beam tracks from a regular data run (no. 363). At this point, the chamber planes are aligned relative to each other. It remains to establish the relation between this coordinate system and that of the toroid magnets and the beam axis. This is done by observing the reconstructed momenta for monoenergetic muons in the four azimuthal quadrants. Any misalignment, such as a rotation, dis- placement, or tilt, which remains between the chamber system, and the longitudinal axis of the toroids (oriented along the nominal beam axis), will result in an asymmetry in reconstructed momentum in the various quadrants. Several calibration runs using muons of fixed energy were used for this purpose. By introducing an overall shift and rotation in the four views which kept the relative alignment intact, the momentum asymmetry can be reduced and the chi-squared for the muon's fitted track through the spectrometer can be lowered. The final asymmetry in reconstructed momentum was 2.53%, which is within the statistical error of the 67 measurement. This final (absolute) alignment was accomplished by these shifts: AX 3.0 cm. Ay l.3 cm. A0 -.0l mr. A6 0.0 mr. A complete list of alignment constants is given in Table 3.l. These numbers represent the amount by which the spatial coordinate of each proportional chamber plane or spark chamber wand must be displaced from the raw data coordinate to give an accurate representation of the muon's true trajectory. 3.2 Calibration of the Spectrometer The calibration of the spectrometer is actually equivalent to a calibration of the analysis computer program which reads chamber and magnet information, and calculates from this the muon's incident energy, its scattering angle, and its outgoing energy (Eo,e, E'). Calibration is achieved by analyzing muon beams of known fixed energy, and adjusting the computer program until the reconstructed momentum nearly equals the known momentum. This calibration can be checked using a monte carlo (simulated) beam of muons which are analyzed in the same way as the data. Several runs were taken with small toroid magnets (inner diameter = l.5", outer diameter = 18") placed along the beam axis in order to deflect the beam muons outward so as to fall into the active area of the spectrometer; otherwise these muons would have travelled down the beam axis and through the field-free holes in the large spectrometer toroid magnets. One set of the small toroids, with a combined length 68 Table 3.1 Final E3l9 Alignment Constants in cm. Nand§ x y u v NSC l .2ll .742 .953 .52l 2 .324 .557 .508 .l48 3 .lll .6ll .663 .391 4 .34l .606 .375 .l36 5 .034 .l90 .429 .189 6 .l40 .069 .036 -.l42 7 -.l24 .057 .l44 .l22 8 -.020 .206 .255 .3l6 9 -.034 l.l22 .590 .327 x y E3l9 PC 1 0.637 0.688 E398 PC l 0.0 2 1.073 . -0.ll5 2 0.054 3 0.438 0.324 3 0.476 4 -0.090 l.284 4 0.0 5 0.l5l l.9l8 5 -0.435 6 0.0 69 of 48" along the beam axis, was placed between PC3 and PC4. Another set (total length of 97") was placed just downstream of PCS but upstream of the E398 Cyclotron Magnet. It was hoped that beam muons could be sprayed out in a conical, azimuthally symmetric pattern into the spectrometer. Unfortunately the muons were not bent out far enough or in sufficient quantities to make this type of calibration useful. In addition, the energy lost by the muons in the small toroids themselves was often difficult to measure, making a momentum determination unreliable. Instead of the small toroid magnets, the large Chicago Cyclotron Magnet (CCM) was used to steer muons outward from the beam axis into the face of the E3l9 spectrometer. This large magnet, which was once the cyclotron magnet at the University of Chicago, was the main analyzing magnet of the E398 apparatus just upstream of E319. Figure 3.3 shows the layout of the CCM and various walls within the E398 area in relation to E3l9. Calibration data was taken at several incident muon energies: 250,200,150,l00, and 50 GeV, and at several CCM current settings. These runs are summarized in Table 3.2. In this way the spectrometer could be calibrated in a wide range of energies (the expected kinematic range of the experiment and at several radial positions outward in the spark chambers. For these runs, the target was removed to decrease Coulomb multiple scattering and energy loss. Since the beam was purposely steered outside the active area of the E3l9 beam proportional chambers, in order for the muons to enter the spectrometer, a modified method was used for finding the beam energy. E398 PC planes l and 2 (upstream of the enclosure l04 bending magnets) and planes 3 and S (downstream) were used to define straight lines 70 cowuocnwpmo Lmuwsoguumgm mcu mcwsac mspmcoqam mfimm u=u w¢mm we “scam; m.m og:m_u .mpoom cg uo: mw m:_3agc mwgh Ase cw mmucmumwuv Emumam oucmgowmc an; :ozz cw mwumcwucoou age mgmnszz mom- m.~om- emwfl- owmfi- Nmmm- mmm mmm o _ CODE capoapcmu Aumcmmz gmumEo upm_;m N N , H H cocuopuxu -pomqm :ocum; a; mm as am omao_guv zoo 7l before and after the bend respectively. Unfortunately the PC-reset for these chambers was not working correctly for these runs, and the resultant chamber hit information corresponded to random muons in the beam. The beam energy for any of the calibration runs can therefore be established for the whole run, as an average over random muons in the beam, but not on an event-by-event basis. After being brought into the muon lab and deflected in the CCM, the muon passes through several iron and lead walls in the E398 apparatus before coming into the E3l9 spectrometer. Each of these walls has an aperture for admitting the normally-unbent muon beam. Some dimensions for these apertures are given in Table 3.3. After being deflected in the CCM field, some muons missed the apertures and passed through the walls, thereby losing energy, perhaps as much as’a few GeV. Finally the muon's energy in the E3l9 spectrometer is analyzed using the computer program VOREP which is discussed in section 3.4. The ability of the spectrometer to determine a muon's energy is limited by the Coulomb multiple scattering in the iron toroids. In the analysis process, it is the radius of curvature of the muon's tra- jectory through the magnetic field of the toroids which is of importance. A distribution of the radii of curvature for a sample of monoenergetic muons sent into the spectrometer would have a gaussian shape due to multiple scattering. Since the reconstructed muon energy is proportional to the inverse of the radius of curvature, the distribution of E' for the same sample of muons would be nearly gaussian with a high-energy tail. Table 3.2 CCM (Chicago Cyclotron Magnet) calibration runs 1E4 72 CCM RUN E0, Tape Events Current Current Date Shutter 467 150 297 5177 2306 2 4000 8/31/76 up - 468 150 298 5265 2306.5 3500 " up 469 150* 298 5178 2306.5 4500 " ‘up 470~ 200 299‘ 10065, 3072.5 4200 " up 471 250 300 7630 . 3840 4875 " up 472 100 300 2596 . 1538.7 2400 " up 473 -100 301 9963 1538.7 2400 " up 474 50 302 9994 770 1200 " up 475 250 303 10023 3841.2 4500 9/1/76 DONN 476 150 304 5407 . 2306.2 4500 " DDNN . 477 150_ 304 4746 2306 2 3500 " DONN 478 25 305 1217 392.5 600 " DONN Table 3.3 Apertures in E398 walls ‘ Fe1 20cm thick, all'muons through this aperture, z=-1320cm 961 41.3cm thick, aperture:40.6cm wide x 38.2cm high, z=-1224cm Fe2 (Rochester cyclotron magnet iron- used for hadron filter) . aperture: 160. 6cm thick x 90. 6 cm high x 90.6 cm wide .upstream edge: z=-892. 5cm P82; 2 slabs of Fe: l.27cm thick, aperture: 15.9cm wide x 13.4cm high Pb: 20.98cm thick, aperture: 19cm x 19cm, upstream edge: z=-605 cm 73 The following procedure was performed for each fixed incident energy: (1) (2) (3) (4) (5) For each event, the radius of curvature k is found. E'=A/k (where A is a proportionality constant) is immediately cor- rected for energy loss in the hadron shield. Energy loss in iron is computed using a fit to the CERN energy-loss table.33 Using the sparks found in NSC 9 and 7 (8 and 9 are too close together), a line can be extrapolated back upstream into the E398 apparatus. If the extrapolated trajectory is found to pass through one of the lead or iron walls (Table 3.3) the muon's energy is corrected accordingly. After all corrections have been made to E', the quantity l/E' is histogrammed. After chi-squared, radius, and angle cuts are made, the final l/E‘ histogram is fit to a gaussian function. The calibrated value of E' is taken to be the inverse of the fit- ted peak position of the l/E' distribution. The resolution of the spectrometer for this E' is the value of sigma (standard deviation) for the l/E' distribution. Figures 3.4(a-e) show the histograms of the quantity lDDO/E' for the five incident energies. Table 3.4 shows the results of the calibration using the CCM magnet. The runs using the CCM to deflect the muon beam are better than the small-toroid calibration runs, but they too involve calculating the energy loss of muons in iron and lead walls,and the extrapolation of tracks over great distances. As the final step in the calibration process, the monte carlo program MCP (to be described in chapter four) 74 cowuasa_pmu smumsoguumam any ¢.m mg:m_u. .m.¢ one 8.8 N.v o.v m.m m.m. 6.m . N.m Am<~\NAN<-xv-Vaxaflu Rm.mua.uvo. m um.memu x.u\SV\Hu.m .1 66=6>6 mmmm H58 :36 eowpatnwpau om ooH omfi com 75 300 .- 35“) CVMK‘ Figure 3.4 (b) 5' uni/55499131 0(E')'9.41 100 r- Event! 100 " so - . - ; moo/5' L 1 l l l I I J I I l 3.1 6.1 L5 1.7 3.9 5.1 5.3 5.3 5.7 5.3 5.l - 200 l i l l 1 Calibration run 468.469 1 ‘ 2900 «out: E'u/«l/E‘ule; 4 :(5)-9.0§ 76 5° 1 i I Calibration run 474 1 2797 event: , l i 5"1/<1/E'>“5.59‘.0€ 1 I a zoo . ' e , ' Figure 3.4 (d) Ari-9.3: f g i ' ‘ . l ‘ f l '1 150 - 1 l 1 1 1 i I I Events 1 i r 1 1 100 - ' ‘ ‘ ' i l I so - 4.1 ‘f l ION/6‘ I 1 I I I l 1 1 I 17.2 13.0 18.3 19.5 20.4 21.2 22 22.8 23.5 24.4 Calibration run 473 - 150 ' y . 1 —-,L 2174 aunts T E'-1/ “(E') EVENTS (EO'E')/Eo 471 250 248.4:1.0 243.533 9.5% 3488 2.0% 470 200 200.3306 199.333 9.4% 5528 0.5% 468 150 149.5304 149.332 8.9% 3098 0.13% 469 150 149.1334 148.633 9.1% 2954 0.35% comb 150 149.4304 149.032 9.0% 6052 0.25% 473 100 98.930.24 96830.2 9.4% 6055 2.6% I 474 50 47.56314 45.89308 9.3% 2665 3.5% Table 3.5 Calibration of the Spectrometer using Monte Carlo Data E(MC) E(reconstructed) 0(E) EVENTS (E(MC)-E(RE))/E(MC) 250 251.83: 17 1.8% 699. -0.7% 200 201.36:,21 1.6% 228 -0.7% 150 150.91:,08 1.4% 631 -0.6% 100 100.56: 08 1.2% 223 -0.6% 50 49.5133 04 1.2% 274 +1.0% 79 each event, fits the trajectory in a multiparameter fit to get the outgoing energy and scattering angle, and then writes the results on an output tape which can be scanned separately. Figure 3.5 shows the flow of the analysis process and the names of some of the major subroutines. I will now describe each of these steps in detail. INITIALIZATION Several input files are required by VOREP. These include a fiducial file for finding sparks in the wire chambers, and a pedestal file and peaks file for processing the calorimeter ADC information. Nhen all the preparations are complete, the analysis can begin. The first buffer on the data tape is read. It contains two events worth of information, each of length 768 words. All of the packed words, such as the scalers and ADC blocks, are decoded and loaded into special arrays. FINDING BEAM TRACKS If the PC-reset bit indicates that the proportional chamber infor- mation latched corresponds to the actual muon which caused the trigger, and not just a random beam muon, then the latches for PC3, PC4, and PCS (the "beam" PC's) are decoded. The hits in each plane (u,v, or w) of each chamber are found and converted into spatial coordinates. Clusters of hits on neighboring wires are averaged over. Then a three-way match is sought among the hits in the three planes. If no three-way matches are found, all two-way matches are formed. The window size for finding such matches is 0.5 cm. Next, the fired wires which have been matched to other wires within the individual chamber are compared to the matched wires in the other 80 i , Y 1 read the primary data tape . 1unpackbits | ngggry 2 events/buffer 768 words/event - #_BEAMPC read beam info. write a beam beam look for beam tape for use ta e tracks 3 _ in monte carlo p T 10 19:95“ # EVENT: Iconvert digitizer counts into NSC spark coordinatesl FETCH find tracks in each of the four chamber viewsl CHASER [match up the views: throw out bad tracks] MATCH - i fetch the spectrometer track candidates get best coordinates ith beam track candidates (x,y) for all chambers lmomentum reconstruction] FINAL of each event y_] oggggt 150 words/event 1 read another event 3 events/buffer rint diagnostics for the hole run lus scal Figure 3.5 Flow chart of the analysis program VOREP Bl beam chambers. Tracks with a 3-3-3 match (a match of 3 in PC3, PC4, and PCS) are examined first, followed by lesser combinations. In each case the extrapolated track candidate must proceed through the target, must make an angle of less than 0.25 mr. with respect to the beam axis, and must have a chi-squared (for a straight line fit) of less than l2.5. The measurement error in the beam PC's is 0.l cm. This beam information in conjunction with the E398 PC's, and the known magnetic field in enclosure l04, allows a measurement of the energy of the beam muon, E0. Each acceptable beam track is stored in an array. If the pulser flag is on, each track is also written onto a separate beam file which is used to generate monte carlo events. The format of this tape is given in Table 3.6. Interaction vertex candidates are formulated on the following basis: whenever four successive calorimeter counters give a reading of ten or more equivalent particles (ten times minimum ionizing), the first counter is deemed a potential vertex. Nhenever no such vertex is found, three dummy vertices are assigned at the center of each third of the target. The total number of beam-vertex candidates is the num- ber of beam tracks times the number of vertex candidates. FINDING SPARKS IN THE NIRE CHAMBERS The information from each of the four wands for each spark chamber module consists of eight words. These contain the digitizer clock counts corresponding to the arrival of as many as eight wand pulses (including fiducials). The fiducial file read in at the beginning of the analysis run contains the expected position, in terms of digitizer counts, of the two fiducials for each wand, for that run. Such a thorough record of fiducial positions was found to be necessary because .82 Table 3.6 Beam Tape Format 393g, . Contents l run number trigger number ex (beam) 0y (beam) x intercept (z=0) y intercept (z=0) x2(X) x20!) DCR packed with information on trigger type and PC reset straight line fit to beam track tomVO‘m-wa _a 0 EO (measured) of a large temperature dependence. On a hot day, the wands could expand and change the effective position of the fiducials. The train of spark positions, in terms of digitizer counts, is examined one by one. If the spark is within :l0 counts of the nominal second fiducial, it becomes the new second fiducial. Likewise, if it is within :l0 counts of the first fiducial, it becomes the new position for the first fiducial. If the spark position puts it outside either of the two fiducials, it is rejected. All other digitizer count readings are interpreted as real sparks which correspond to the passage of an ionizing muon through the chamber. The digitizer counts for these sparks are converted into real spatial coordinates (x,y,u,v). Table 3.7 shows a "wand dump," one of the on-line diagnostic displays which was written out during the run. This table shows the digitizer 0" -‘r 5| :0.‘ M MRHD DUMP EVENT NUMBER 83 Table 3.7 Digitizer Clock Counts 9439 - 1. if i 8888 T". i ‘3 1 8 8 8 8 8 338 4482 3838 a; 2 8 8 8 8 8 881 3384 3883 3 3 8 8 1 8 8 888 4388 3341 .f 4 8 8 8 8 818 3823 8838 3831 11 ' s 8 8 8 3 8 818 4888 3335 6 8 8 8 8 8 882 5199 ??39 3 8 8 8 8 8 381 3348 3318 ; 8 1 8 8 8 334 3438 3828 3321 El 4 8 8 8 8 8 8 338 4443 3313 ' 18 8 8 8 8 8 854 3382 3855 11 8 8 8 8 8 388 4333 3338 12 8 1 8 1 8 338 3142 3313 13 8 8 8 8 8 384 4848 3321 1' 8 8 8 8 8 P88 518? P223 5 8 8 8 8 8 384 3348 3328 1' 8 8 1 1 8 381 3433 3323 13 8 8 8 8 8 385 4434 3338 12 8 8 8 8 8 818 3484 33.: 12 8 1 8 8 8 383 4348 3333 28 8 8 8 J 8 814 S85? 3341 . 21 8 8 8 8 588 413: 4283 3388 1‘ 22 8 B 8 B 8 fl 8 ?36 3 3 13 El U 13 1 13 1 ‘5 4 3 5‘ 2 4 4 '3' 4 El 2 E' E: e 24 3:1 3234 3232 3323 3413 3888 3838 4828 j: 23 8 8 8 8 883 4138 4234 4383 E‘ 28 8 8 8 394 3423 3388 3938 4138 23 8 8 8 838 4383 4832 4888 3883 28 8 8 8 821 4312 43:8 4382 38:1 29 8 8 8 8 8 8 8 4183 38 8 8 8 8 8 8 1.8 3834 31 8 8 8 8 588 3811 4B°2 3345 32 8 8 8 5“8 3314 3884 39=1 3343 33 8 8 8 588 41r8 4834 8888 3336 34 8 8 2344 3318 3338 3232 4838 3384 33 8 3 8 318 3848 4284 4438 3834 38 1 8 :82 4382 4438 4884 4388 3384 III II H i" I" i .L. [‘11 H l.l] -\J ‘1 84 counts for all 36 wands (4 wands times 9 spark chambers). For most wands, the first and last numbers correspond to the two fiducials, while those in between should represent real sparks. A "clean" event with only one muon would then leave only a single spark in each chamber. For this particular event, extra sparks seem to be present in the up- stream end of the spectrometer (wands 2l-36) where particles from the hadronic shower can still sometimes be found. Wands 22, 29, and 30 appear to have been defective since they do not give any clear evidence of a spark. Near the rear of the spectrometer, the extra sparks seem to have died away. Wand 24 has an overflow of sparks (perhaps an edge breakdown problem). As can be seen from the wand dump, the number of digitizer counts from one fiducial to the other is about 7000. The physical distance between fiducials is about l84 cm. and the wire spacing is .07 cm. Therefore one finds that: 7000 counts/l84 cm. 38 counts/cm. = 3.8 counts/mm (52) 5.43 counts/wire spacing 184 cm./.07 cm./wire = 2629 wires (53) In real time, the distance between fiducials (the real time duration of the whole pulse train) is about 350 us. 5 pulse velocity = 184 cm./350 us. = .53 cm./us. = 5.3xl0 cm./sec (54) 350 ns./2629 wires = IK3us./wire = l30 ns./wire (55) 85 That is, if each wire carried a current (caused by a spark at that wire), the time between each wire would be 130 ns. The width of each individual pulse was measured to be about 300 ns. This means that sparks at neighboring wires could just possibly be resolvable. Once the sparks have been assigned coordinates, the only cut imposed at this time is that there be sparks in at least two views in at least two of the last three chambers. This insures that we can con- duct a hunt for muon tracks. FINDING TRACKS IN EACH VIEN For each of the four views (x,y,u,v) track finding begins at the back of the spectrometer by forming all possible straight lines in NSC's l, 2, and 3 (which sit behind the last toroid magnet). Taking into account the bending power of the toroids, sparks are sought in NSC4 and NSCS. At this point, a track candidate having the right polarity (curving "in" toward the axis rather than "out“), and having passed the cuts described in Table 3.8, will consist of sparks in the rear five spark chambers. Sparks in the front four spark chambers will be sought after the track candidates in the back have been matched to give three dimensional tracks. As many as 20 tracks can be retained. MATCHING TRACKS FROM DIFFERENT VIEWS A "matched" track is a three dimensional cOmbination of tracks from all four views. The match residuals Ax = x - 313- for each of /E the five rear spark chambers are examined for three views at a time. Firstly, the residual must be smaller than 0.5 cm for the match to be successful.‘ Secondly, matches which result in a location within a magnet hole are rejected. A trivial requirement is that the tracks 10. 86 Table 3.8 Track finding cuts maximum tangent at the back = TANTMAX = 125 mr. this corresponds to an energy out of about 25 GeV. the extrapolated trajectory in the magnetic region (i.e., at the "bend points") cannot be outside XBMAX = 100 cm. to be included in a track candidate, a spark must be within the allowable window for that chamber. The windows for chambers 1...9 = (.50, .30, .60, 1.0, 3.5, 3.5, 4.0, 3.5, 3.5 cm.). The window size for the hadron proportional chambers was 3.5 cm. tracks which cross the beam axis between bend points cannot also have an extrapolated position XB at the bend point of greater than the inner radius of the toroid=15.24 cm. a cut is made on the chan e in tangent over a two-toroid bending region. Tracks with A N 3_50 mr. are cut: this also is equivalent to a cut in E'. a cut on events that are obviously bending out: TAN < -25 mr. for x > 0 and ATAN>0. Tracks bending out only slightly will be retained reject tracks which are coincident with previous tracks, or are subsets of other tracks. for the same number of sparks, two tracks must have at least two sparks not in common. rank the tracks according to the number of sparks. No more than 20 tracks will be allowed. there must be tracks in at least 2 views for the event to be studied further. 87 being matched have the same polarity. Nhenever two views are being taken as reference (e.g. x and u), the match to a third view (in this case, y or v) must be successful in at least five out of the possible 2x5=lO matches. The maximum number of matched tracks allowed is 20. For each matched track candidate which is accepted, the contributions from the four views are converted into (x,y) coordinate pairs for each chamber. JOINING SPECTROMETER AND BEAM TRACKS The last step in the identification of the true muon trajectory is to join a beam track with a spectrometer track (sparks in NSC l-5) and then to add in the contributions from the forward spark chambers (NSC 6-9). The geometrical layout for this process is shown in Figure 3.6. All possible combinations of a spectrometer track with a beam track are formed. For each combination, the resultant curvature in the spectrometer is checked to see if the track corresponds to spurious low-energy particles or to halo muons. The best match-up of a beam- vertex candidate with a spectrometer track is kept for momentum recon- struction. The following two criteria were used to arrive at the best combination: (T) In Figure 3.6, 623 is the angle of bend from the front to the back of the spectrometer. In what amounts to an E' cut of about 25 GeV, we require that cosez3>»0.75. (2) As defined in the figure, 62 is the angle observed in the front chambers of the trajectory into the spectrometer, while 61 is the same angle found by extrapolating the spectrometer track candidate (sparks in the rear five chambers) towards 88 mxomsu Emma u=u swuosocaomam mcwcvoa m.m mg:m_u mumupc:uo gong» gmumsoguomam mmhmzomhommm gmpmecgaomam ecu gmaogga m=P_m>mgu cozs mumu_v:mo some“ Emma gmuweoguomam mga co gown use an umgzmmms mpmcm ocunmo M \ / \ mm GHVA/ / muouvucoo xmucm>uxumga Emma mg» u=u agopommmgg mcpmgosm msu an owns N gopmsosuumam may do “cog; mg» as mpmcm «sun 0 ummgau ms» ogmzou xomgu gopwsoguuoam mg» mcpumpoamgaxo an owns a Lmquoguomam any do use»; on» an mpmcm mg» n o 89 the front of the spectrometer and into the target. 612 is the difference of these two angles, or equivalently the difference between the predicted and observed angles in the front. If the beam-spectrometer track combination under consideration is the true one, the only reason 612 would be nonzero is the uncertainty in 62 due to Coulomb multiple scattering in the process of extrapolating the trajectory from back-to-front. Recall the formulas for multiple scattering, and for magnetic deflection in a magnet of length L: 0 (multiple scattering) = <02>U2 = 4%%§-,/Tl37- (52) a (bend) = 493] B-dl = e (53) E 23 _ .015 /L/l.77 , ’ 0 (mult. scatt.) - .03.fB-dl 623 - constant x 623 (54) Ne impose the cut sinelzlsin623<:12.5. This is essentially a halo cut. By dividing by the factor sin623, which is proportional to the multiple scattering, we can measure the departure of the measured angle 61 from the predicted angle 62, for reasons other than multiple scattering (e.g., that the muon did not originate in the target, but is instead a halo muon). Next, the hit positions in PC2, PCl, and the front two spark chambers NSC9, and NSC8 were filled in using the newly accepted beam- vertex-spectrometer track. These chambers did not contribute directly to the track selection process because of the errors introduced in extrapolating the spectrometer track all the way forward toward the 90 target. These forward chambers were beset with the extra hits asso- ciated with the hadron shower particles, and were also the poorest performing chambers. Nevertheless, when carefully selected, the sparks in the forward chambers were useful in the momentum determination during the multiparameter fit, where every additional point along the muon's trajectory contributed to a better fit. Figure 3.7 shows a 2 event taking place. The schematic of the E319 apparatus and a high q rear five chambers contribute sparks while the front chambers are less effective. One can see in this figure a beam track entering and the interaction near the end of the target. The vertical lines near the target indicate ADC information on the shower pulse height at each counter. MOMENTUM RECONSTRUCTION At this point in the analysis, a complete muon trajectory has been formulated: beam track, interaction vertex, and the curving path of the scattered muon as it bends through the magnetized regions of the spectrometer. Knowing the spark coordinates (x,y) of chambers before, after, and interspersed within the spectrometer, and knowing the mag- netic field in the toroids, we can find the scattered muon's energy, E', its scattering angle, a, and its interaction vertex. Along with the incident energy Eo measured separately in enclosure l04, these parameters specify all the kinematics of the deep inelastic scattering reaction. I shall begin my description of the momentum-angle reconstruction process by pretending that there is no Coulomb multiple scattering in the spectrometer. This idealized spectrometer, including several chambers and magnets, is shown in Figure 3.8. The incident muon enters 91 ucm>m a gap; < m.m mg:m_u N be... warez. z_ :puzu. no.9: ca.rp oa_mm ca.qm co.o. ow.»~ no.m. ca.“ no.ow- aa.~ww na.mn P h b ca-can-' co-sZ- I CO‘CS' oo-siL OO‘O NAU\>a¢V m.~¢~u~c .LE NmO.u® 3313H11N33 NI HIONHl o-sz >0w mmHn.w >aw mkmuom oc-és co-sl cc-odt c S 92 «N mu :0 .Puozsn—mcoumg EzucmEoE Ume paw—UH mN Nu mu 3 95m: H» p u xmggm> _\ mwxm Emma ex / p d \ Elm/.lll \ LG- P d _ _ . _ — \ .\+\ _ _ \ \ \ \ ow? \ Macaw... \ \ \ \ V cm: _\ umcmma \ um: “mamas smasmso xgmam om: “mamas \ .592... e x cowuowpmmu we m—mcmu. .g5~u>wa oom\>mw¢.uo 1 >mo com u u “mp Emu. x wxmfl x mo. >mw e. u flfiQ< t. x E Swanaa "magmas? Hogans some p oz» mo comupmoa ~ 2»? egg do covupmoa N J wN o u m—mzm m:_gwaumum Aouo~.oa.oxv u xmpgm> :oPHumgmu:_ 93 from the left, interacts at the point (xo.yo,zo=0), and scatters through an angle 6. After entering the spectrometer, it bends through an angle ¢ at the center of the magnet (the impulse approximation). The muon proceeds in this way through the spectrometer; bending through an angle ¢ at each magnet, and registering its path in the wire spark chambers along the way. The spark position (transverse coordinate x) at the first chamber (z=21) is easy to compute: x1 = x0 + 21 6 (6<€i) Now use the formula for ¢ found in Figure 3.7: xn = x0 + Zn 6 - [#gg-f D x El] g (zn - E1) (59) Since we know all the 2'5, 5'5, E, and the spark positions measured at each chamber, we ought to be able to invert the n equation (59) to get 94 p, 6(ex’ey)’ and (xo,yo). Unfortunately, we really do not know B; the field in the toroids has a complex radial dependence. Also, in real life the muon undergoes continuous energy loss and multiple scattering. Equation (59) is just too simple. The nonlinear multiparameter fit which is used in VOREP proceeds like this: (i) guess the initial values of p, x0, yo, ex, and By (these variables, on which everything depends, are called a],a2,a3, o4,a5); (ii) predict the spark positions in all chambers using a modi- fied version of equation (59); (iii) in order to test the quality of the fit so far, define a chi-squared function which depends on the re51dues Bx = xobserved’ and with proper allowance for xpredicted ' multiple scattering and energy loss; (iv) minimize the x2 with respect to the five variables oi; (v) solve for new values of the “i and make new spark predictions. Keep iterating until the values for the “i (i.e., xo’yo’ex’ey’p) change by an arbitrarily small amount. 2 The following expression is ngt_a good expression for x : 2 5": 2 5y i . . X = Z [(7;-) + (7;-)] 1 summation over chambers (60) i i i downstream of the target where 6xi=x residual at the ith chamber and o is the measurement error at that chamber. Because of multiple scattering in the toroids, spark predictions in some chambers (the back chambers for instance) will be worse than for others. Therefore, any expression for x2 should contain error terms which are correlated among all the chambers: 2 _ -l X - igj Yij (Bxisxj + Byidyj) (6l) iSj 95 In this expression the simple measurement error Oi has been replaced by an error matrix Yij which properly weights the correlation terms involving errors in chamber i and chamber j. The error in a measurement of a spark coordinate made at z=z1 due to multiple scattering in a piece of iron of length L, at z=E, is given by: Ax = ems (z.I - E) ._. 21/2.._q1_s_/T_— where ems (ems) p 1.77 (52) A typical correlation term would look like axioxj = Axiij = ems - (zi - E) - ems - (Zj - E) (63) The full expression for Yij will contain a summation over all th and jth spark chamber. The magnets which are upstream of both the i inherent measurement error of the chamber (Oi = 0.l cm) must also be included: =2 :8 (Z.-€)(Z.-€)+(S..O'2 ms mg] 1 m J m 1 . (54) Y.. 1J J 1 - 2_ 2 g -22 2 where Zi>€m’ zj>Em, and (ems) - (.OlS/p) L/l.77 l.lxlO /p (GeV) . Equation (59) turns out to be extremely complicated when multiple scattering and energy loss in iron, and the radial dependence of the magnetic field are introduced. Instead, the prediction of spark positions will be made using an expansion in powers of p']. As mentioned earlier, the quantity we actually deal with in the 96 reconstruction process is the radius of curvature k, which is related to the momentum: k=(qBo/3327.4)/p, (qBO=constant). The coordinate x and slope x'==dx/dz at each chamber is calculated in powers of k: 2 X II CD + clk + czk l I I l2. x CD + c1k + czk where co,c1,c2,c$,ci, and cé are coefficients of the expansion and which depend on the ai(xo’yo’ex’ey’p)' There are similar expressions at each chamber for the y coordinates. Using our initial guesses for the ai’ we can predict (x,x',y,y') at the front of the spectrometer. Since we know the behavior of muons in an azimuthal magnetic field, we can trace the muon's trajectory toward the back of the spectrometer. This provides us with a set of predicted sparks and launches the iterative procedure described above. We finally arrive at values for Xo’yo’ex’ey’ and p=E' We make a special effort to discover and correct for wrong sparks during the fitting process. By observing the residue 6x = xpredicted - xobserved for all the chambers, the sjgg_of one of the residues will sometimes be opposite that of all the other chambers. If, in addition, the sizg_of the residue is larger than a prescribed window, then we conclude that this spark was found erroneously (that it does not lie on the muon's true trajectory), and it is removed. The fit is then repeated. Usually the deletion of the bad spark significantly improves 2 the x for the overall fit, and gives a more reliable estimate for E' and B. 97 The last function of the analysis program is to write a secondary file containing the results of the spark selection, the momentum fit, and other useful information. The format for this file is shown in Table 3.9. 3.4 Resolution The reconstruction program described above is limited in its ability to find E0, E', and 9 by the nature of the apparatus used in E319. Since we used a nuclear target (iron) the Fermi motion of the nucleons in each iron nucleus has the effect of smearing the actual value of E0 in the nucleon rest system by as much as l3%. Furthermore, by using an iron spectrometer, multiple scattering limits resolution in E' to about 9%. The resolution in e is about 1%. The spectrometer cali- bration showed that the resolution in E' was relatively constant, about 9%, for an E' range of 50 up to 250 GeV. For E' below about 30 GeV, energy losses become more important and the calibration begins to break down. Above about 250 GeV the calibration again becomes suspect; the scattered muon's trajectory is relatively "stiff" and unbending, and this makes a reliable momentum reconstruction more difficult. The uncertainties in E0, E', and a can result in rather large 2, and w. Using a resolutions in derivative quantities such as u, q large sample of monte carlo events, made to simulate real data, we can see how big the resolution is. For each monte carlo event, the values of E0, E', and e are known for the nucleon rest system (without Fermi motion this frame would be the same as the lab frame); these I shall call the "physics" values of those variables. We also know the values of £0, E’, and 6 via the reconstruction process (just like for real data). A histogram of the quantity [vphysics'vreconstructedJ/vphysics 98 Table 3.9 Secondary tape format (energies in Gev, distance in cm.) 3932. 1 2 3 4-17 18-31 32-42 43-53 54-56 57-58 59-61 62-63 64 65 66 67 68-71 72 73 74 75 76 77 78 79-90 91-150 CONTENT run number x 100000 + trigger number E (hadron) from calorimeter spill number measured x in all chambers (PCS...NSCl) measured y in all chambers (PCS...NSCI) fitted x in most chambers (PCZ...NSCI) fitted y in most chambers (PC2...NSC1) (P ’P :P ) beam muon x y z (x,y) beam track at z=0 (Px,p ,pz) scattered muon (x,y)yscattered track at z=0 )2? (spectrometer track fit) degrees of freedom for spectrometer track ZADC Monte Carlo event weight (=1 for data) (x,y,ex,ey) at NSC 8 PBACK (E' at the back of the spectrometer) )E/DOF for the track in the rear seven NSC's packed word: number of fired wires in PC1,2 packed word: number of fired wires in NSC1-9 coordinates of PCS-1, NSC9-1 contributing to the beam track and the scattered track number of spectrometer tracks number of beam tracks DCR's and TDC's packed 16 bit ADC's 99 will have a Guassian shape, indicative of the Guassian processes causing the uncertainties (e.g., Coulomb multiple scattering). The standard deviation (square root of the variance) of this distribution is taken to be the “resolution" of the apparatus in the variable v; the same for the other kinematical quantities. Table 3.10 shows the resolutions of v, w, qz, and~x=q2/2mv for several values of y=v/Eo, and also for q2 and L0. 3.5 Acceptance The most striking feature of this apparatus, from an acceptance standpoint, is the bias against low-angle scattering. The field-free regions in the toroid magnet holes and the beam veto counters cause such events to be rejected. Muons which scatter at very large angles (>lOO mr.) and which pass outside the physical extent of the toroids (87 cm. outer radius) are also lost. The acceptance of the apparatus, as a function of one or more kinematic variables, is defined to be the number of accepted monte carlo events (events successfully reaching the rear of the spectrometer and passing other nominal cuts) in a certain kinematic range, divided by the total number of monte carlo events generated in that range. Figure 3.9 shows the acceptance in 2 plane while Figure 3.lO shows the acceptance in the qz-v the w—q plane. Figures 3.ll-3.l7 show the acceptances in single kinematic variables. 3.6 Data Distributions The data sample studied in this dissertation consists of approxi- mately l26,000 fully accepted and reconstructed data events, with a like number of monte carlo events. Figure 3.l8 through 3.34 show lOO histograms of this data for several important kinematic and recon- struction parameters. The analogous histograms for the monte carlo events will appear in chapter four. An overall comparison of data to monte carlo distributions, including averages of all important kine- matic quantities, will be given in chapter five. Several of the quantities histogrammed need explanation: --ZMIN is the 2 position at which the distance-of-closest-approach between spectrometer track and beam track occurs. It is taken to be the z coordinate of the interaction vertex. --x2 is the chi-squared per degree of freedom of the entire spectro- meter track for the multiparameter reconstruction fit. --(x ) are the coordinates of the beam muon extrapolated beam’ybeam to Z=O. ~~The radius of the muon‘s trajectory in NSCS, NSCl, and at the face of the front magnet (RMAG) is given in cm. Finally consistency plots of several important variable are shown in Figure 3.35. These plots show the average value of the particular variable plotted for randomly chosen runs from throughout the running period. 101 Table 3.10(a) (%) Resolution 0 as a Function of y=v/Eo " 0(v) 0(w) C(92) 0(X=l/w) all y 21.4 37.3 22.7 35.9 Q 6(0) 0(42) 3.160 42.8- 23.9 3.400 33.2 15.8 3.600 35.2 16.7 5.400 37.4 17.7 9.960 35.9 20.3 20.600 33.9 26.4 34.000 37.3 37.7 Table 3.lO (c) (%) Resolution 6 as a Function of q2 qZ O 0(X=l/w) 8.222 ’ 45.5 14.780 36.4 25.000 30.1 38.760 26.7 61.480 24.6 91.060 32.2 r 52 44 40 36 32 28 24 20 16 12 103 Percentage Acceptance: w vs q2 Figure 3.9 '70— 21 100 20 50 21 73 20‘ 66 20 79 19 85 22' , 90 20 91 92 23 88 93 43 I 21 81 98 74 45 20 74 96 96 76 39 I 16 IT 6 I 50 I _ 18 68 94 99 100 97 I 80 I 48 I 33 I 17 I 60 90 105 120 135 150 92(GeV/C)2 150 135 120 105 90 (GeV/c)";S . 60 45 30 15 104 Percentage Acceptance: q2 vs. v 58 64 100 50 20 53 68 39 64 33 100 89 65 54 51 41 100 100' 96 90 85 67 59 100 95 100 100 100 88 95 70. 100 100 f 100 100 99 96 95 80 ‘100 96 100 99 A 99 100 100 96 93 100 92 93 94 96 96 98 98 98 100 59- 64 71 77 79 85 87 91 93 95 13 18 20' 21 22 22 20 20 17 13 20 40 60 80 ‘100 120 140 l60 180 200 0 (GeV) Figure 3.10 80 60 4O 20 Figure 3.11 105 Figure 3.12 v Acceptance 8 Acceptance 11.11 lLLl 50 100 150 200 20 40 60 80 6(GeV)- 9(mr.) 106 8O — 60" 20 (a) (b) T 31] m 0mw com omm com ONH oe NAo\>mwv cm mm ma 85 83311.1 3 m 1 com coma ooem comm ooov some 6H.m ag=m_1 a m NH ma ow cu ll3 N5 in Z m; o; N. 3 >8 3 2 m. 11.11111L11 - . . . . “1.1.1. . _ - rL IlffrlrJ l1 8 11 1 S 1 2 F. J . , 1 8 Nw.m at=m_1 “N.m 61=a_1 11. 1 8 |_ 82 x 4 mu=m>m 1 x l.l a N (:3 . 1 8 Ea 1. cm oooH x mpcm>m ll4 NNN oNN mmN >68 NmN CNN NoN ONN 5N N6 1111. q . _ _ N . _ q >8 1 1 82: 1 1 NN.N 6.531 1 088 1 5N.m 8136?; mucm>m _ mucm>m . 1 88m 1 m.moN 1 A NV N mc w.NmH u m .LE me. u A©v mucm>m 53866 SE 1 8 688 SE 1 2 116 .26 o N- 81 .56 8 ~ _ — oon coco mwcws’w NN.M mLzmwn— . m NN m 61: _1 e-N.N- 1 1N1 on.N 1.1xv Emm comm 8» a mb=6>a 119 mw. om. em. . ma. c we cm em Na 1 N 4 11.11411 N _ N 111111r11 fi1 rJIH1rN 1H1 H1 11 c —. I. m gucm>m 11 NH vw.m mg:m_m mm.m mg:m_m I. mg 11 ON a caxeo.mx m oovm come CONN comm. OOONH oocefi l20 Hug Am.v AamU2 = 49%E- TT77' (66) where p is the momentum in GeV and L is the step size in meters. This is the familiar multiple scattering formula; single large-angle scatters were not included since this effect is small for lengths larger than about ten radiation lengths. The energy loss and multiple scattering simulation is carried at two uniformly spaced locations in the target leading up to the interaction vertex, and then again for two locations for the scattered muon as it leaves the target. The interaction vertex is chosen randomly along the whole length of the target. The x and y coordinates of the vertex are established before hand by the beam tape information, subject to slight changes brought about by multiple scattering. The muon's momentum 4-vector is transformed into the nucleon rest system. This is necessary since most deep inelastic phenomena are described in a "lab" frame where the nucleon is at rest. In this frame, the outgoing energy E' and scattering angle 6 are chosen randomly. The value of the target nucleon's "Fermi motion" is generated using a Fermi-gas model: 2 11p) = J, 2 l-Fexp[(P -Pf)/2MkT] l26 where Pf=Pmax=‘260 GeV and kT=.008 GeV. This momentum is oriented randomly in spatial direction. From the simulated values of E', E0, and 6, one can compute all the other userl kinematic quantities such as v, qz, w, and x. The weight for each event is proportional to the differential cross section: 2 F (X) 2 2 do: fl; £__ 2g1+vm EETEE' (dn)Mott v ["*2ta" 2 ( 1.1R )] (67) Before running the monte carlo program, called MCP, a large look-up table was constructed containing cross section information necessary for assigning a weight to an event with a given E0, E', and 6. Several remarks should be made about expression (67). Firstly, R=os/0T=.25:.10 represents the average of the SLAC results reported at 34 the Hamburg Photon-Lepton Symposium. Secondly, a scale-invariant form of F2=vN2 was used. This was done so that the contrast between a Bjorken scale invariant prediction, and our data (which was expected to show scale violating behavior), would be more evident. In particular, TO, the following formulas were used to derive F2 Fproton 3 4 5 2 (x') = l.062l(l-x') - 2.2594(l-x') + lO.54(l-x') (58) - 15.8277(1-x')6 + 6.7931(1-x')7 Fgeut”°"(x') = FBrOtO" [1.0172 - l.2605x' + .73723x'2 (69) - .34044x'3] 127 Firon(x.) = AZ- l:proton N I:neutron (70) 2 2 +1‘12 where 2:26, N=30, A=55.85, and l/x'=l/x+m2/q2. This formulation of F2 2 dependence is a fit to the data in Figure l.6 which shows little or no q for fixed x. As for the sensitivity of the cross section to the value of R which is used, Figure 4.l shows the ratio of dzo/dE'dQ computed for various R's, to that for R=.25. The average y=v/Eo for E3l9 was about 0.4, although a value as high as 0.8 was kinematically possible. Muon pair production and bremsstrahlung ("internal" bremsstrahlung) at the time of the deep inelastic collision are taken into account by 32 This using the-"effective radiator" technique of Mo and Tsai. process corrects the cross section for the reaction shown in Figure 4.2(a) with terms corresponding to the reactions shown in Figures 4.2 (b)-(d). "External" bremsstrahlung, taking place long before or after the nuclear collision, is handled in the energy loss mechanism described earlier. The sum of all these effects can be treated, to good approximation, like the "external" bremsstrahlung correction. The internal bremsstrahlung is equivalent to external radiation in two "equivalent radiators," one before and one after the interaction, with thickness tr = .b"(%)[4n(qz/m2) -11 b=4/3 (71) Figure 4.3 shows how the total radiative correction can be approximated by a single diagram (T is the length of the target scattering material). The effective length of the scattering material in which radiation of photons is important becomes Lg-i-tr, l28 1 1 1 1 1 1 1 1 1 1.20)- G (R) _ - C(R=.25) R=° 1.161" ' "1 1.12- - R=.1 1.08 .. N 1.04— R=.20 - 1 00 R=.25 0.96 __ R=.35 q 0'92 - R=.50 '1 0.88 -' . _ 1 1 1 1 1 1 1 1 1 0 2 .4 6 8 y=v/EO Figure 4.1 The effect of R=oS/ot on the cross section l29 Figure 4.4 shows how radiative processes confuse the measurement of the cross section at a particular point in the (E0,E') plane. Let point A represent a measured pair of Eo and E'; that is, Eo as measured before the incident muon enters the target, and E' as measured in the spectrometer. The actual scattering may have taken place at point B where E0=E': EO may have been degraded via bremsstrahlung as in Figure 4.2(b), with the effect of making an elastic interaction at B look like a deep inelastic interaction at A. Similarly, an elastic interaction with variables at point C could mimic a deep inelastic interaction. Other effects such as two-photon exchange, and a combi- nation of bremsstrahlung with inelastic scattering may give contributions from any of the points in the ABC triangle. The weight for each event is multiplied by a factor RC representing the correction due to con- tributions from elastic and inelastic scattering:32 2 2 d o d o (dE'd8)elastic + (dE'dQ)inelastic RC correcged corrected (72) d o (dE'30)inelastic uncorrected The last correction to the scattering cross section to be made was that due to wide-angle bremsstrahlung, the emission of a photon at a much larger angle than in the usual case already studied. This multiplicative correction to the event weight is of the form:35 do(wide angle bremsstrahlung) do(deep inelastic scattering) correction = l + (73) 2 2 2 201. GIwI ‘ ”"171’11-7Hx 2) 2 .9 III ,_ l30 (a) (b) (C) M Figure 4.2 L11'E7‘r”’,N—' Radiative correction \d““““~ diagrams (d) El .,«//// O 1 '1,» . .5 t1 “(’0’ Figure 4.3 O . . LE ‘ T/2 The "effect1ve rad1ator" o /2 tr method . Al T/2 tr El 1 . elastic C F1gure 4.4 scatt. Contributions to the radiative corrections in the (EO,E') plane ._-- observed (E ,E') A 0 B I ' vE O 131 where G(w)= -68.062/112 - 29.197/6 + .70671 + .0ll959w - .49948x10‘4112 and Z=26, A=55.85, y=v/Eo, and w=2mv/q2. 4.4 Ray Tracing After interacting and leaving the target, the muon is made to enter the spectrometer. In each magnet the trajectory tracing is done in three steps. At each step the muon's path bends in the magnetic field, undergoes Coulomb multiple scattering,and suffers energy loss. Spark positions are recorded at each chamber and given a Gaussian smear (o=O.l cm.) to simulate measurement uncertainties. Later, in the momentum reconstruction phase, certain chambers will be randomly "turned off" for various events to simulate chamber inefficiencies. 4.5 MCP Distributions Not all generated monte carlo events reach the end of the spectro- meter. Like real data, some of the hypothetical muons pass into the holes in the toroid and fail to hit the trigger banks. Others exit out the side of the magnets. For those muons which successfully traverse the spectrometer, a record is written on tape using the same format as for real data, and its momentum and scattering angle are reconstructed. If the event passes all the standard analysis cuts (see section 5.2), it enters the sample of events to be used in the comparison to real data. Analogous to the data distribution of Figures 3.18-3.34, the corresponding monte carlo distributions are shown in Figures 4.5-4.2l. 4.6 Data/MCP Comparison: Extracting VN9(x,q2) A ratio can be formed in each region of the x-q2 plane of the number of data events to the number of monte carlo events (corrected for incident flux): l32 N N N N N 1 N 1 N :8 cNS Fryer 3.. a: of Sr 8. ON 8 SN ocmm 11 mugo>u oomq 1. N2 1 r14111141111 NNN 1 1551 33 as: .8535 8% L . 1 N N 1 N N =Ne N .su aco ark are . saw coN cm. can . N a a. seen .1 u=u u=u mN=o>m # cocoN .1 Namgau No mace Nausea succumczoe 14 Na auto A3 3833: 25$ 1 31155 1. 38 283 N85 55 m.e meamNN l33 cow CNN oeN ooN om NN 4m 8m NN . o 1 a N a N N N _ >8 N326”: I. ”.0 I. 1 3 1 l ¢.N .l. 1 N.m 1 N.¢ acumNN m.¢ mcsmwm 1 N: . . 1 a N3 82 N .32 82 x 1 N: 82 x 1 mucm>m mucm>w NN 0N om cm l34 N.m ¢.~ mé m.o o o N N N l .1 ON 11 1 8. 1 .J .1 cm L Nx n8: .3 no: m.¢ mczmwm m.¢ mgszN oooN x 82 x1 om 3:08 1 mucw>m N: 3 on mm om ~l35 mmm omm mom ¢mm vow NmN CNN mmN wNN em Ne fi q N . N . N N >mu O . m mu: .m no: NN.e meszN cooN x1.eN oN.e eeszN .1 mucm>m mucm>m oooN oooN ooom coo- ooom coco l36 m6 Yo N N .LE. Nll L L L 11 low 11mm Lmuzmume no: I Emwnmv QUE . l X I1 NN 8N 137 mN Emmnh QUE e mgamNN macw>m 5N.e eL=NNN mucm>m acmN ooom oomv oooo comm ooom l38 5.6.3: 3... och x.| “N.v 0530—; muco>o 0N Nmumzvz v usson mu: 83 x! mm mucu>o_ l39 co Econ .39 a mu: «N.¢ ogaaNu mun—o)” oomN coca came coco cons comoN l40 o om N o N? 63:33 82 e 82 NNNN 8%: o: 2:3... . 1 182 x 82 x Lucm>m mucm>m NN 3 3 l4l R(x q2) = number of data events (x,q2) = DATA(X1921 number of monte carlo events (x,q2) MCP(x,q2) The most important result of the single-muon analysis, the derivation of the structure function F2=vN2, is obtained using R: F2(X192) = 1206421 - PSTEIWx) (741 . ' STEIN . . . . . In this formula F2 is the same function as in equation (70), the 10 I shall now scale invariant structure function dependent only on x. give the justification of this construction. The expression given earlier for the differential cross section can be expressed in terms of experimentally observed quantities: 2 2 F (U) ) 2 2 d o g 99_ 2 2 §_ l-rv lg dE'dQ (dQ)Mott v [l-+2tan 2 ( l+R )] (75) = event rate (E',Q) l . l AE'AQ ' luminosity acceptance where the luminosity is just the number of incident muons per time times the number of target nucleons per cmz. We can solve equation (75) for F2: 2 2 1111 +2tan2 % (111%,1—1)?‘ 2 _ . __ F2(x,q ) T data(E ’9) [AETAD luminosity - acceptance] (76) 142 The quantity inside the bracket is just the ratio FgTEIN/MCP(x,q2) if the following equation is true: 2 < d 0 > = -l-—- 2 1—939—-° Acceptance (77) dE Q Nacc accepted dE do events The averaging and summation implied in equation (77) is over all monte carlo events within the (x,q2) region in question. Figure 1.6 shows that FETEIN is a slowly and smoothly varying function of x for x less than about 0.5. The regions of x and q2, over which we compute F2(x,q2), are small enough that equation (77) is a good approximation. Used in this way, the monte carlo simulation of real data can be thought of as a sophisticated acceptance routine. The dependence on a particular model, such as the use of FETEIN, for finding the structure vwz, is STEIN F2 eliminated by using equation (74); in the numerator and denominator cancel out. 4.7 Systematic Errors in F2(x,q2) A possible systematic error in F2(x,q2) can arise from many sources. The greatest possibility for error comes from measurement uncertainties in E0, E', and 9. From the calibration runs, we have estimated that the uncertainties in these variables are .4%, l%, and .4% respectively. The effect on F2 of these uncertainties is shown in Tables 4.2, 4.3, and 4.4; both in the qz-y plane, and in the x-q2 plane. The change in F2(x,q2) due to an error in the measured E', for instance, can be calculated by tampering with the monte carlo: 11113 Table 4.2 .50 0.90 0.98 0.95 0.95 1.05 1.00 (a) .45 0.95 0.97 1.00 1.03 1.08 1.00 .40 0.35 1.02 0.97 1.00 1.02 .35 1.00 0.99 1.03 1.02 1.03 .30 1.00 1.02 1.00 1.00 1.02 Monte Carlo (50+.4z) x .25 Monte Car‘o (Ed) .20 1.03 1.00 1.00 1.07 ' 1.00 1.01 1.02 1.02 .15 - 1.02 1.02 1.03 .10 1.01 1.02 1.09 .05 1.01 1.04 o 0 I 43 54 86 107 123 q2 (GeV/c)2 (b) 150 1.00 128 1.19 1.19 1.00 107 -z 1.05 1.11 1.03 0.93 1.05 9 85 2 0.97 1.01 1.02 1.04 1.02 1.03 1.07 (GeV/c) 54 1.04 0.95 1.01 1.01 1.01 1.00 1.03 1.02 43 0.87 1.00 1.01 1.01 1.01 1.01 1.02 1.00 1.02 21 . 0.94 1.01 1.01 1.00 1.02 1.00 1.00 1.00 1.01 0 0 0.1 0.2 0.3 0.4 0.5 0.5 0.7 0.3 0.9 Y'VIEO 144 Table 4.3 .50 . 1.19 1.10 1.00 1.01 0.95 (a) .45 1.18 1.08 1.11 1.07 1.10 1.01 .40 5 1.02 1.01 1.02 1.00 1.01 1.02 .3 1.02 0.99 0.99 0.99 1.02 .30 0.88 1.04 1.05 1.02 0.98 Monte Carlo E'+l% x .25 n e r o 0.99 0.98 0.97 1.03 .20 15 0.95 0.97 1.00 1.04 ' 0.95 0.98 1.00 .10 0.95 0.99 0.99 .05 0.95 0.87 ° 0 21 42 54 85 107 128 2 2 Q (GeV/c) (b) 150 0.99 0.99 128 1.08 0.98 1.19 1.00 1.00 107 1.14 1.17 0.98 1.00 0.95 1.00 1.00 qz 86 2 1.05 1.03 1.02 1.04 1.00 1.08 1.01 (GeV/c) 64 1.17 1.02 1.00 1.00 1.00 0.99 1.01 0.99 43 21 1.04 0.99 0.98 0.98 0.99 0.99 0.99 0.98 1.05 0.95 0.95 0.95 0.97 0.97 0.97 0.95 0.94 0 0 0.1 0.2 0.3 0.4 0.5 0.5 0.7 0.8 0.9 Y'v/E 145 Tonte Carlo e 150 128 107 -92.. - 86 (GeV/1:)2 54 43 21 0 Table 4.4 .50 1.02 1.05 1.03 1.03 0.98 (a) ~45 1.01 1.01 1.03 1.05 1.01 1.01 .40 0.99 1.04 1.01 1.01 1.03 1.10 .35 1.01 1.00 0.97 1.02 1.00 .30 0.98 1.01 1.04 1.00 1.04 Monte Carlo (9+.41) x .25 1.01 1.01 1.00 1.05 .20 1.00 1.01 1.02 1.05 .15 0.99 1.00 1.02 .10 0.99 1.01 1.03 .05 0.98 0.93 0 0 1 43 54 85 107 128 q2 (GeV/c)2 (b) 1.00 0.99 1.07 1.00 1.04 1.10 1.03 0.97 1.03 1.00 1.00 1.03 1.04 1.04 1.02 1.05 1.11 1.02 1.01 1.02 1.01 1.01 1.01 1.03 1.00 1.04 1.02 1.01 1.01 1.00 1.00 1.01 1.00 1.01 1.00 0.99 0.99 0.99 1.00 1.00 0.99 0.99 0.98 0 , 0.2 0.3 0.4 0.5 0.5 0.7 0.0 0.9 .l y-v/a, 146 AF2 F2(E') - F2(E'-tAE') —-—-= . AE' = error in E' (78) 52 F2(Et) DATA FSTEIN _ DATA FSTEIN =_F'(—'TMC E 2 WC E +AE 2 (79) DITA FSTEIN MCF1E') 2 MCP E' 1 ' CP E' +AE' (80) The effect of switching on or off the radiative corrections or the wide angle bremsstrahlung are shown in Tables 4.5 and 4.6. The effect of changing R=osloT=.25 to R=0 is shown in Table 4.7. Figure 4.22 shows contours of constant systematic error (the errors due to £0, E', and 6 in quadrature) in the qz-x plane. In the kinematic region where the data exists, the possible sys- tematic errors are everywhere less than a few percent, except for x<0.l where they may be as large as l0%. In the last chapter I will discuss F2(x,q2) itself and also other possible systematic errors which can not be simulated by monte carlo, namely normalization errors due to the uncertainty in the muon flux, and errors due to analysis inefficiencies. 147 15515 4.5 .50 4 1.05 1.05 1.07 1.05 1.05 1.05 . s 1.04 1.05 1.05 1.05 1.05 1.03 (a) .40 1.04 1.05 1.05 1.04 1.03 1.02 .35 1.03 1.04 1.03 1.02 1.01 .30 Radiative Off 1.03 1.03 1.02 1.00 0.98 Corrections x . 25 Rad1ative 0" 1.02 1.01 0.99 0.97 Corrections .20 1 1.01 0.99 0.95 0.93 . 5 0.99 0.95 0.91 0.88 .10 0.95 0.89 0.84 .05 0.89 0.80 ° 21 43 54 85 107 128 42 (GeV/c)2 (b) 150 1.05 128 1.09 1.05 1.02 0.98 107 1.09 1.05 1.04 1.01 0.99 0.95 q? 85 2 1.08 1.05 1.02 0.99 '0.97 0.93 0.89 (GeV/c) 54 1.08 1.05 1.02 0.99 0.97 0.94 0.91 0.85 43 1.07 1.05 1.02 0.99 0.97 0.94 0.91 0.87 0.82 21 1.03 1.01 0.98 0.95 0.93 0.91 0.88 0.84 0.79 0 0 0.1 0.2 0.3 0.4 0.5 0.5 0.7 0.8 0.9 y'v/Eo q2 (GeV/c)z 144E} Table 4.6 .50 1.00 1.00 1.00 1.00 0.99 0.98 (a) .45 40 1.00 1.00 1.00 0.99 0.99 0.99 '35 1.00 1.00 1.00 0.99 0.99 0.98 ° 1.00 1.00 1.00 0.99 1.00 .30 Hide-Angle Off 1.00 1.00 1.00 0.99 0.99 Bremsstrahlung x .25 Wide-Angie 0" 1.00 1.00 1.00 1.00 Bremsstrahlung .20 1.00 1.00 1.00 1.02 .15 - 1.00 1.00 1.03 .10 1.00 1.02 1.07 .05 0.95 1.01 0 21 43 54 85 107 128 q2 (GeV/c)2 (b) - 150 0.98 128 0.99 0.99 0.98 0.95 1.00 107 5 1.00 0.99 0.99 0.99 1.00 1.03 1.08 8 54 1.00 0.99 0.99 0.99 1.00 1.02 1.05 1.00 1.00 1.00 1.00 1.00 1.01 1.03 1.08 '43 1.00 1.00 1.00 1.00 1.00 1.00 1.01 1.02 1.03 21 1.00 1.00 1.00 1.00 0.99 0.98 0.95 0.91 0.85 ° 0. 0.2 0.3 0.4 0.5 0.5 0.7 0.8 0.9 y'v/ E0 11151 Table 4.7 .50 . 5 1.00 1 00 1.01 1.01 1.03 1.03 .4 1.00 1.00 1.01 1.02 1.03 1.05 (a) .40 1.00 1 00 1.01 1.02 1.04 1.05 .35 1.00 1.00 1.01 1.03 1.05 .30 1.00 1.01 1.02 1.05 1.08 Monte Carlo (R-O) x .25 Wonte Carlo (Rel?) 1.00 1.01 l.04 l.07 .20 ‘5 1.00 1.02 1.05 1.11 ' 1.01 1.04 1.11 .10 1.03 1.09 1.14 .05 1.05 1.15 0 0 21 43 54 85 107 128 02 (GeV/c)2 (b) 150 1.03 1.05 128 ‘1.02 1.03 1.05 107 1.01 1.02 1.03 1.05 1.08 q2 85 2 1.01 1.02 1.03 1.05 -l.08 1.12 1.15 (GeV/c) 64 1.00 1.01 1.02 1.03 1.05 1.08 1.12 1.15 43 1.00 1.00 1.01 1.02 1.03 1.05 1.08 1.12 1.15 21 1.00 1.00 1.01 1.02 1.03 1.05 1.08 1.12 1.15 0 0 0. 0.2 0.3 0.4 0.5 0.5 0.7 0.8 0.9 y'v/Eo 150 m.o ~N.¢ «gamed to acmpa aux on» cw N39 cw cmcgm upaasmumxm accumcou 4o mczouzou . s OH NAU\>mwv Na 8. ow on em CHAPTER V RESULTS AND CONCLUSIONS 5.l Summary of the Data Sample The data reported in this dissertation represents about 90% of the 270 GeV 0* data. when the 270 GeV 0’ data (about 30% of the 0*) is fully analyzed, it will be added to the u+ sample; certain differences in the beam shape for the two data samples have to be studied first. The sample of monte carlo events was generated (with program MCP) in such a way that the effective flux would roughly match that of the real data sample. The monte carlo events were momentum analyzed just like the data and subjected to the same analysis cuts. These cuts are shown in Table 5.l. Before corrections were applied, the number of accepted monte carlo events was approximately equal to the number of data events. The fraction of triggers recorded on primary data tapes which are reconstructed and can pass all analysis cuts is about 12%. Table 5.2 shows a direct comparison of kinematic averages and other statistics for the two samples. Correcting only for flux (but not for other factors such as will be described in the next section), the number of accepted events past cuts is almost identical. Discrep- ancies between average values for data and monte carlo kinematic variables can be chiefly attributed to inefficiencies in the track finding program VOREP, and the divergence of the data from a monte lSl m 10. 11. 12. 13. 14. 15. 152 Table 5.1 Single-muon analysis cuts -366 cm < ZMIN < 672 cm. interaction vertex DMIN upmpa couumgaou ~.m mczmwu w.c 8.0 m.o m.o N.o 2.: _ . _ _ _ a _ a.o 5.: mofi x __Fam\x=_d i. o i.m.o O \11 o. O i. 1. o.H O O O O 0. 11 iiihnui.i .I o .omu.wvav N.H . 0 . k r. 1.5.“ xapw\mucm>m o_gmu mace: nmgamuu¢ x3—w\macm>m mama vaunmou< 157 MULTIMU-to-VOREP comparison, and N2 is the overall normalization factor (14%) representing the flux rate effects. There may, of course, be a class of events which is inefficiently reconstructed by both VOREP and MULTIMU. Ne estimate that the uncertainty in N](x,q2) may be as high as 5%. The systematic error in N2 is also believed to be about 5%. These errors, along with the errors in E0, E', and 0 discussed in the last chapter, can be added in quadrature to give a total systematic error for 0N2 of about 7-10%. This total may decrease somewhat as the calculation and correction of inefficiences become better understood. 5.3 Parameterizing,$ca1e Breaking One way of showing how the structure function F2 breaks scale invariance is to fit the data to a curve with an explicit q2 19 dependent term. It was first thought that such a term would be of the form N/(l-tqZ/A2)2. But this did not allow for a positive increase in F2 for increasing qz. It became convenient to parameterize scaling violations in the following way: 2 F2 k C CHIO O SLAC-HIT ‘ J; . d I i l I 1"(1‘x1 L 1 i -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 Figure 5.2 The scale-violating parameter b(x) 159 Table 5.3 Various fits to the combined b(x) data b = C1 + sz proton: C1 = .18497 :_.0115 C2 = .83179 :_.040 XZ/DOF = 1.18 iron: C1 = .18929 i .0098 C2 = .8787 :_.037 x2/DOF = 1.22 b = C11n(1/6x) proton: C = .11555 :_.0063 1 2 x /DOF = 4.45 2 X /DOF = 5.05 b = C11n(1/C2x) proton: C .11844 :_.0064 1 0 7.2189 ¢_.445 2 2 X /DOF = 4.16 iron: C1 .12227 :_.0057 C2 7.6334 :_.4208 2 x /DOF = 4.45 b = 01 + Czln(1-x) iron: 01 = .16895 :_.00987 02 = .5777 :_.0252 2 x /DOF = 1.062 160 in Figure 5.2. The best results occurred for a fit of the type b(x)=C]+C2 £n(1-x). This is the form we adopted when using the b(x)-type scale breaking factor. One additional note: by expanding the expression for F2(x,q2) in equation (82), one arrives at a formula reminiscent of QCD: F2(x,qz) = F2(x.q§) [1+b=10.9 = 19.9 =29.4 0.6” ‘1'} -1 0.5 _ 0.4 .1 0.3 ‘1 0.2 _ 0.1 - 1 0.7 —— - 50=6|.5 < 2>=9LO 0.6 -- q . '- 1 1 _ . . 1 . 0.2 04 0.6 162 As a preliminary check of the threshold hypothesis, b(x)=3£nF2/3£nq2 was calculated using all E319 data, and then again using only data for 2)2 which N25100120 (GeV/c The results are shown in Figure 5.4 along with the straight line fit to all the previous b(x) data, as in Figure 5.2. Except for the points at high x (small £n(l-x)), the values of b(x) below the N2=1OO “threshold" agree well with previous measurements, while b(x) calculated using, in addition, data above the threshold shows an unmis- takable rise above the fitted line. A few words should be said about the points which appear far below the line. The value of b(x) as a function of x is essentially the slope 2 of a straight line fit to a plot of Zan versus znq for a finite region of w(=l/x). In this case these points corresponded to the range 2=2.87. Each data point within the w region has its own average w, ranging from a low of 2.42 up to a high of 2.80. As in no other w region, the data points arrayed themselves in such a way that the points with largest average w (and lowest average x) were at lower values of £nq2, while points with small average m were consistently at larger values of 2nq2. Since the cross section grows with smaller x, no 2 2 than it matter what the value of q , the plot was higher at low £nq should have been, and the value of b(x) is therefore more negative than it should have been. The values of b(x) for all data and for data w2<100 is given in Table 5.4. 5.5 QCD Predictions Since QCD only makes predictions for the moments of F2, and not for F2 itself, some kind of inversion has to be performed. This involves a formula of the type: 0.2 163 I b(x)-3(1n 0N2)/a(ln qz) ? 0 all data (this expt.) .v2<100120 k Figure 5.4 E319 b(x) .386 .315 .265 .217 .165 .111 .058 .035 164 Table 5.4 E319 values for b(x)=alnF2/31nq2 2 ln(l-x) b(x) all data X_LQQ§_ -.49 4741.053 2.38 -.38 .05751. 039 .555 -.308 .155_+_. 028 3.55 -.245 .2351. 021 3.85 -.180 .314:,020 3.86 -. 12 2211.020 1.15 -.05 .1821.024 9.20 -.04 -.5353~_.125 2.72 b (x 1 w2<100+20 -.287:. 072 -.0577:. 074 -.0279:.050 .01141. 070 .1171. 090 -.0534_+_.173 -.453:.253 -.715:.305 2 DOF .789 .080 1.84 .821 T'only ,two points in each J~region 165 Fem?) = 73.717217 M(n,qz) (85) Since the n dependence of M(n,qz) is complicated, the integration can only be performed numerically. Using measured values of F2 at some q2=q§ from deep inelastic scattering (from which M(n,qg) can be computed), and inventing a particular expression for the gluon distribution within the nucleon, several authors have constructed numerical estimates of 2 36,37 the q and x behavior of F2 and of the individual quark densities. 2 These studies develop the q dependence of anusing QCD methods and the basic x dependence assumed in the simple parton model.38 The QCD model which will be discussed presently is that of Buras and Gaemers.39 By making certain assumptions about the n dependence of the moments M they are able to derive analytic expressions for the quark densities and for F2 as a function of x and q2. They define two valence quark densities: 2 _ 2 2 v8(xsq ) - uv(x:q ) + dv(x9q ) (86) 2 2 2 V3(X.q ) uv(X.q 1 - dv(X.q ) They also derive densities for the gluons (G), for the charmed sea (C), and for the non-charmed sea (S). Since G, S, and C are steeply falling functions of x, there is little contribution to the higher moments at large x. Therefore Buras and Gaemers use only the first two moments (n=2,3) in the inversion process and are able to derive analytic 166 expressions for G, S, and C in terms of x and the variable 2 2 —_ Zn/A . S'i’l‘z—yflz—zl- <1/c1o xS(x,§) = AS(§) (1-x) _ __ ”(:(E) xC(x,s) = AC(S) (1-x) (87) _ ._ flex?) xG(x,s) = AG(s) (l-x) _ 2 11602312) where, for example, AG(s) = MG(2,q )( 2 - MG(3.q ) _ MG(2,<12) ”6(5) = _—2- ' MG(3.<1 ) In order to formulate the valence quark densities, which have a larger effect at big x, the first 12 moments were utilized: _ _ n35) n36) xV3(x,s) = A3(s) x (l-x) (88) _. _ 1133(5) n35) xV8(x,s) = A8(s) x (l-x) Those parameters which are not given by the theory are gotten by fitting the experimentally observed moments of F2, which in this case 167 15 17 are those for ep and up inelastic scattering. The complete structure function is constructed from the quark densities, as in the parton model: p vw2(")(x.q21 = XET§3¥8(x.qz)-t%il3(x,qz)+%S(X.qz)+%C(x,qz)1 (89) The quark densities and F2(x,q2) calculated by these methods is shown in Figure 5.5 for q2=22.5 (GeV/c)2. 5.6 5:319(x,q3) Compared to QCD 40 and our u-Fe data, Buras has derived this particular parameterization for his model at q2=q§=2:41 Using some of the newer up data A = 0.4 GeV _ 5 XG - 2.41 (1-x) x5 = (1-x)8 - 3 0.7 2.6 xv - B(O.7,3f6) x (l-x) (90) _ 1 0.85 3.35 de ‘ 8(.85,4.35) X (1“) xC = O The curves generated by these formulas and the measured values of F2(x,q2) are plotted in Figures 5.6 (a)-(f) versus x for fixed q2 regions (the binning is slightly different than in Figure 5.3). For the sake of comparison, the QCD prediction and the CHIO (E398: Chicago- 168 0.8 0.7 —- 0.5 ‘ C 02 = 22.5 oev2 0.5 VW? 0.4 0.3 ——\\ I, “‘ 0.2 — , \ 015'! A \\\ J-X r1v \ .'\ \ ‘ng 0‘. 1 \ \\ 1 ..—.\--‘ ..... ‘x. .’ 3\~_\L:> _ ~ ~~. "" -—.._ - h.» _ O 0.1 0.2 0.3 0.4 30.5 0.6 0.7 0.8 0.9 X Figure 5.5 QCD predictions for F2 and for quark densities 169 5='8.53 .(GeV/c12 2 e O CHIO <9 >- 8-48 converted . 1. 0.6 +_ A SLAC =»8.56 t0 r0" 19 MIT (a) .241 I A l L 1 i I 0 0.2 0.4 0.6 0.8 Figure 5.6 Measured F2 versus x for fixed q 170 10=14.7 -> 0 _ CHIO =12.5 0.6.... A SIT-111$- <92>=13.8 __ (b) 0.1- ._ A\ 8 _ “~. 1 1 l 1 I l I l N 0 ' 0.2 0.4 0.6 0.8 . _ x Figure 5.6 continued 171 0.7" 20=24.8 ° CHIO =22.5 ._1 0 0.2 0.4 Figure 5.6 continued 0.6 , 0.8 172 0.7- 30=38.6 o CHIO =40 _ (d) _. L 1 1 1 1 1 1\.1 0 2 0 4 0.5 -0 8 Figure 5.6 continued 173- 0.7—- ' ‘ 2 50 = 61.1 (GeV/c)2 Figure 5.6 continued 0.7 174' 2<150 . 0.2, , ‘ 0.4 Figure 5.6 continued . 80 = 91.1 (GeV/c)2 1 l 1 L 1 1 1 1 1 ’ v0.6 0.8 175 40 Harvard-Illinois-Oxford) structure functions have been converted to u-Fe scattering using equation (70). There are even a few SLAC-MIT points15 at high x for the lowest two q2 regions. Also shown in these plots is a second curve representing QCD with A=0.5 and the following changes for the formulas in (90): xS )5 .9(1-x xG )4 2.1(1-x 41 These modifications were suggested by Buras to see if QCD could be made to agree with the data. The A=.4 curve is systematically below our data for small x; like Figure 5.3, the data rises above the curve below a certain value of x, as if some threshold had been reached. The A=.5 curve is much closer to the 5319 data, but is systematically above the CHIO data. The threshold-like behavior is not as evident in the 2 2 low q regions but does persist in the higher q regions where the A=.4 and A=.5 curves are similar. That it is possible to get better agreement in the lower q2 regions just by cranking up the sea quark distribution and changing A to 0.5, shows that such a formulation of QCD is still very tentative. _ This is demonstrated again in Figures 5.7 (a)-(g) where F2(x,q2) is 2 for fixed x (or w). The average w for each plot is plotted versus q given along with the highest and lowest values of w for any of the points used in that region. The two QCD curves drawn for each plot correspond to these high and low values of w for each region. Only curves for A=.4 are shown since the curves for A=.5 are not much dif- 15 2 ferent. The SLAC-MIT data is also shown, and lies mostly at low q . 176- Ax\an 3 cmxve so» No mzmem> we umgzmomz ~.m mgzmpm \ FL“ me.eu Ase—Vs ~m..ugem.e.s 3V QNA3V mcmmnxv mm. vuxav "HN :58; 0 3mm 0 the 2 cm“ I! E # gym cod N 3 ficogp op cmugm>zouv .mm.mu Ase—vs Gm.mnAzam£v3 NAo\>oc. MHA3V won. unxv e~.mu.a. N a m b .ziu<..m 2mm O 177 cm cm cm. a. d _ m c cm om a~ _ . 4 2 .e 8 ~38. m N N fife... :52; o mus: hziezm o mo_.nnxv ~_~.nnxv 1 —.@u AID—v3 1:! 5.?" Ago—v3 $812.32.... 3.0.3 22 o 8.5.235... vane... 23 o :5 3 . _ _ _ . b _ _ _ P1 . _ . coacpucou ~.m mgzmwu 17E! cm cm cm c. m on em cN o_ m N 1e _ _ 4 1a ‘ n _ a . . q 1“ . _ ~.o\>oc. Na Nwo\>ocv w: T. -. -.mus «.1. r em.m_us 1 .821- i 1. . t 2%.... #293 o ~QO.HAXV m~.v.u.za_vs __ x . a m o "A V N 3° 3 .1 mm.~_uA=a_evs m_u.av a_na me a A. .v 4 9i cm.,..uAesss 8.21.: 28 o E 3 r . _ _ _ _ _ _ _ p _ _ wmacwucou N.m mezmwm 179‘ "'== 1 1 1 1 1 1 (g) =22.7 m(high)=27.24 ' w(1ow) = 21.5 0'7- = .055 — 52 __ _ 0.5,. ._ 0.5-— . l ’11 111 1' 1 1 1 1 2 5* 2 10 20 50 .80 ’Figure'5.7 continued 180 An interesting feature in Figure 5.7 is the rather large rise in F2 above the QCD curves for increasing qz. This trend sets in as early as Figure 5.7(b) where =4.35. Even in 5.7(a), for =3.74, the 2 data is not decreasing with increasing q . Nhat this may suggest, again, is a threshold-like behavior in N2. On each plot, three arrows 2 2 have been drawn to indicate that value of q for which N =80, 100, and 120 (GeV/c2)2. As in Figure 5.3, the evidence for a rise in F2 in the vicinity of the arrows is not perfect, but is reasonably good. It will be very difficult to vary the quark density function, or A, in order to 2; none of the curves after having fallen at lower qz. get the QCD curves to approach the data at high q shown was able to rise with q2 5.7 Moments Figure 5.8 shows the first moment of F2: 2 _ ' 2 ' M(2.q ) - Io F2(x,q )dx (92) 42 for u-Fe scattering (E319) as well as u-p and p-d scattering. Also shown is the moment computed for the QCD structure function used in Figure 5.6 (A=.4), and the moment of the structure function employing the b(x) parameter (used in Figure 5.3), F2(b)=F2(x,q§)(q2/q§)b. 2 The moments at each value of q are given in Table 5.5, along with the n=3 and n=4 moments of the E319 data. The experimentally measured moments (n=2) in E319 rise with 2 2 increasing q . The moment of F2(b) rises only slightly with q , while the QCD curve falls. The proton and deuterium data do not extend far enough to tell what happens at high qz. In the parton model the n=2 moment of the structure function F2 is proportional to the mean parton 181 m:_emapoom mm1n new .u1:.a1:eoe Nu 4o pauses umewu w.m mesum om om ~.o\>oe. am we o“ m m _ . q q _ . no... “2552530 25 O o .. 59:22 2.8 0 .1 _. e a c Ame.x.~d eewmv "Any a Sawtoueoo o_=u .4 Acoc_ Leg emuumcgou mmczmv ego Sega: TEN... + + Av A? + + 1 11 ~.c O 1 + + + 1 .we 182 charge squared.5 One interpretation of a falling moment with increasing q2 is that neutral partons, such as gluons, could be more important at 2 high q ; either there are more of them or they assume a larger share of the nucleon's momentum. In contrast, the increase in the moments, observed in E319, is related to the other aspects of the data; namely the rise of F2 above a reference curve (QCD, or F2(b)), and the twofold behavior of b(x) computed with and without the data above w2=100s20. A few words should be said about how the moments are computed. First of all, x was used as the "scaling" variable rather than the more proper Nachtman variable (equation (40)) which takes into account various 2 mass effects; this was permissible since our lowest q region was 8.5 (GeV/c)2, well above q§=2. Secondly, the x axis was divided into three regions. In region 11, where data for F2 exists, the moment was found by Simpson's rule; just finding the area underneath the data points. For x below xmin (the lowest value of x for which there is data) the area computed was that for a trapezoid, the upper edge of which was a straight line given by the derivative of the power law fit to the data computed at xmi The coefficients for these fits to the n’ data in Figure 5.6 are given in Table 5.6. In region III, where x is above xmax (the highest x for which there is data), the function F2(b)=F2(x,q§)(q2/q§)b was used, making sure that F2(b) was adjusted to agree with the data point at xmax’ 5.8 Fits to the Data 2 The data in F2(x,q2) plotted against x for fixed q lends itself 2 to a power law fit in x. For F2 versus q for fixed w, several fits were attempted. Fit type III was a single parameter fit to the "standard" scale breaking curve F2(b)=F2(x,q§)(q2/q§)b times a normalization 183 Table 5.5 Moments 2b 2 2 E319 E319 E319 000 (9?) F2(x,qo) q n=2 n=3 n=4 n=2 qo n=2 8.53 .1740:.005 0428:0005 0299:.0005 .1594 .1591 14.7 .1739:.005. 0437:.0004 0291:.0004 .1557 .0175 24.8 .1800:00'8 0455:0005 0284:0004 .1527' .1580 38.5 .1932:015 0458:0009 0271:0004 .1504 .1591 51.1 .2035:.022 0507:0021 0257:0005 .1582 .1713 91.1 .2135:034 (0505:0053 0277:0015 .1554 .1735 Table 5.6 Power law fit to F2( x,q2) in various q2 regions 5 ‘ . F2(x,qz) =.£ a, (I-X)1 2 . 1:3 2 q 51 - a2 83 x /DOF 8.53 -2.835:1.355 8.243:3.14 -4.931:1.81 1.358 14.7 -3.320:.559 9355:1025 -5.524:.779 1.819 24.8 -3.24-3:.533 9551:1088 -5.801:.985 2.57 38.6 -1.045:.505 3473:1055 -1.559:.911. 1.153 51.1 , -1.255:~.770 4005:2051 -2.31o:1.537 1.74 91.1 —1.752:1.817 5.012:5.201 -3.482:5.195 1.229 184 constant N. Fit type II was of the form F2 = NF2(b) + A 0(w2-100) (93) where the second parameter A is the "strength" of a step-function which equals one for N2>lOO and is zero for N2<100. Use was made of a step function to simulate a hypothetical threshold in N2 at 100 (GeV/c2)2. A step function is a bit severe though: due to a shortage of data points, and our finite resolution, no such sharp rise in the data is visible. Therefore, for fit type I, the step function was replaced by a linear rise in q2 2 2 for the region between a N of 80 and 120 (20 on either side of N =lOO). This ought to represent the uncertainty in the 2 location of the would-be N threshold. The results of all these fits, for the various w regions, are shown in Table 5.7. Included there is x2 per degree of freedom for each fit. A particular fitted curve (type I) for the =7.25 region is shown in Figure 5.9. The curve follows the rise in the data for 8O F2=N F2(b) W <80 = + F2 F2(b) 2 2 F2 N F2(b) F =N F (b)+A+Bq 80 = 7.25 The curve is a fit of the type: 2 . F2=N F2(b) N <80 F2=N F2(b)+A+Bq2 80 q F2 AF2 2.87 25.77 .1548 .010 30.58 .1531 .011 35.36 .1417. .011 40.64 .1431 .012 48.28 .1206 .008 58.06 .1279 .009 74.79 .1283 .008 100.1 .1897 .019 3.56 16.40 .2114 .011 21.67 .2135 .011 26.55 .2132 .012 32.10 .1902 .011 ‘ 36.91 .1805 .011 42.75 .1987 .013~ 47.33 .2090 .015 52.89 .2119 .016 60.54 .1837 .011 77.85 .1802 .010 4.36 7.92 .2097 .015 11.99 .2287 .011 17.04 .2157 .010 22.05 .2141 .010 for fixed x (x=1/m) Table A.1 continued 193 q F2 AF2 4.36 27.43 .2333 .011 32.87 .2303 .012 37.93 .2541 .015 43.86 .2765 .017 48.68 .2435 .017 53.71 .2494 .019 62.20 .2912 .019 75.41 .2380 .017 5.4 8.06 .2529 .011 12.28 .2624 .008 17.53 .2515 .007 22.84 .3144 .009 28.22 .3422 .011 33.56 .3279 .012 38.52 .3204 .013 44.30 .3447 .016 49.45 .3193 .016 55.22 .2827 .017 59.87 .2799 .020 68.24 .3483 .026 7.26 8.22 .2890 .009 12.72 .3048 .007 17.98 .3743 .008 23.35 .3967 .010 28.67 .4300 .012 34.06 .4385 .014 39.92 .3826 .015 45.24 .3730 .018 50.84 .3809 .026 56.01 .4688 .051 Table A.1 continued 194 q F2 AFZ 11.0 7.24 .3605 .011 9.18 .3677 .010 11.27 .4005 .010 13.55 .4251 .011 15.58 .4626 .012 17.84 .4401 .012 19.94 .4439 .013 22.35 .4629 .015 24.17 .4554 .015 25.90 .4465 .022 29.04 .4715 .013 34.49 .4905 .021 22.3 6.02 .5153 .025 7.36 .4907 .011 9.37 .4816 .008 11.5 .4878 .008 13.53 .4789 .009 15.66 .5258 .012 17.92 .5134 .014 20.16 .5396 .019 22.40 .5236 .025 25.28 .6590 .041 195 Table A.2 F2(x,q2) versus x for fixed q2 regions q2 F2 AF2 10.9 .043 .4763 .008 .053 .4622 .007 .071 .4329 .007 .085 .4163 .010 .102 .4445 .010 .122 .4212 .009 .148 .3827 .008 .184 .370 .007 .206 .266 .012 .214 .259 .012 .222 .239 .011 .232 .2591 .012 .246 .2919 .015 .256 .2396 .009 .266 .2534 .015 19.9 .046 .5924 .043 .050 .5363 .027 .060 .5073 .012 .070 .4923 .013 .081 .4812 .012 .094 .4520 .010 .116 .4390 .010 .148 .3998 .008 .172 .3634 .015 .188 .3139 .012 .205 .3049 .012 .221 .2876 .012 .236 .2877 .012 .252 .2489 .011 .267 .2307 .010 .283 .2321 .011 .300 .2144 .011 .317 .2185 .012 196 Table A.2 continued qz F2 8F2 29.4 .075 .5354 .042 .090 .5298 .020 .105 .4351 .013 .132 .455 .012 .152 .4381 .023 .157 .4151 .021 .182 .3753 .019 .198 ' .3331 .015 .215 .3349 .017 .235 .3500 .017 .250 .2834 .014 .284 .2275 .012 .309 .2051 .011 .333 .1905 .011 .353 .1508 .010 .373 .1525 .011 197 Table A.2 continued q2 F2 AFZ 42.5 .113 .4993 .038 .131 .4211 .016 .148 .3913 .024 .163 .3829 .015 .190 .3403 .013 .221 .3252 .017 .248 .3082 .016 .275 .2531 .013 .306 .2268 .012 .337 .01760 .010 .370 .1590 .010 .402 .1169 .009 61.5 .152 .5601 .060 .166 .3349 .034 .180 .3329 .028 .193 .3498 .023 .224 .2979 .013 .265 .2442 .014 .299 .2063 .011 .336 .1675 .009 .381 .1378 .008 .430 .1131 .008 .506 .1240 .009 198 Table A.2 continued q2 F2 AF2 91.1 .268 .2373 .029 .325 .2066 .025 .380 .1628 .018 .450 .1431 .015 .552 .0606 .006 10. 11. 12. 13. 14. 15. 199 REFERENCES D.H. 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