Nucleon Structure Functions from νμ-Fe Interactions and a Study of the Valence Quark Distribution Citation Purohit, Milind Vasant (1984) Nucleon Structure Functions from νμ-Fe Interactions and a Study of the Valence Quark Distribution. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4RBK-RV59. https://resolver.caltech.edu/CaltechTHESIS:04082013-111418042 Abstract Data were taken in 1979-80 by the CCFRR high energy neutrino experiment at Fermilab. A total of 150,000 neutrino and 23,000 antineutrino charged current events in the approximate energy range 25 < E v < 250 GeV are measured and analyzed. The structure functions F 2 and xF 3 are extracted for three assumptions about σ L /σ T : R = 0., R = 0.1 and R = a QCD based expression. Systematic errors are estimated and their significance is discussed. Comparisons or the X and Q 2 behaviour or the structure functions with results from other experiments are made. We find that statistical errors currently dominate our knowledge of the valence quark distribution, which is studied in this thesis. xF 3 from different experiments has, within errors and apart from level differences, the same dependence on x and Q 2 , except for the HPWF results. The CDHS F 2 shows a clear fall-off at low-x from the CCFRR and EMC results, again apart from level differences which are calculable from cross-sections. The result for the the GLS rule is found to be 2.83 ± .15 ± .09 ± .10 where the first error is statistical, the second is an overall level error and the third covers the rest of the systematic errors. QCD studies of xF 3 to leading and second order have been done. The QCD evolution of xF 3 , which is independent of R and the strange sea, does not depend on the gluon distribution and fits yield ʌ LO = 88 +163 -78 +113 -70 MeV The systematic errors are smaller than the statistical errors. Second order fits give somewhat different values of ʌ, although α s (at Q 2 0 = 12.6 GeV 2 ) is not so different. A fit using the better determined F 2 in place of xF 3 for x > 0.4 i.e., q = 0 in that region, gives ʌ LO = 266 +114 -104 +85 -79 MeV Again, the statistical errors are larger than the systematic errors. An attempt to measure R was made and the measurements are described. Utilizing the inequality q (x) ≥ 0 we find that in the region x > .4 R is less than 0.55 at the 90% confidence level. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Hitlin, David G. Thesis Committee: Fox, Geoffrey C. (chair) Sciulli, Frank J. Boehm, Felix H. Hitlin, David G. Defense Date: 6 October 1983 Funders: Funding Agency Grant Number Caltech UNSPECIFIED Department of Energy (DOE) UNSPECIFIED NSF UNSPECIFIED Record Number: CaltechTHESIS:04082013-111418042 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04082013-111418042 DOI: 10.7907/4RBK-RV59 ORCID: Author ORCID Purohit, Milind Vasant 0000-0002-8381-8689 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 7585 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 08 Apr 2013 20:16 Last Modified: 04 Nov 2025 21:46 Thesis Files Preview PDF - Final Version See Usage Policy. 29MB Repository Staff Only: item control page

Nucleon Structure Functions from v -Fe Interactions 11 and a Study of the Valence Quark Distribution Thesis by Milind Vasant Purohit In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1984 (Submitted October 6, 1983) Acknowledgements All the members of the CCFRR collaboration contributed to these results and to them I am most grateful for their splendid efforts. My principal association has been with Dave MacFarlane who produced the twin thesis and with Mike Shaevih and Frank Sciulli, my ad...,-iser. I am deeply indebted to both the professors for their guidance over the years and insightful help with the analysis. I would also like to thank them, especially Frank Sciulli, for reading this thesis and their comments on it. I consider myself lucky to have had Dave for a colleague; his abilities and good cheer make an excellent combination. The rest of this group has always been very cooperative, in particular Bob Blair by his patient explanations and by blazing the trail. Petros Rapidis was a good mentor during the ion chamber calibration run, as also Keith Jenkins. I have also benefited much from contacts with Gene Fisk, Arie Bodek and Taka Kondo on various occasions. I wish to thank Dr. R .M. Barnett of SLAC and Dr. J.F. Owens and his colleagues at Florida State for providing us with their QCD-fl.t programs and counsel. I also wish to thank my friends and all the people at Caltech, Fermilab and Nevis Labs who have gone out of their way to help make my itinerant graduate education easier. Prof. David Hitlin very kindly agreed to act as surrogate adviser at Caltech. Preparation of my thesis was aided tremendously by the Nevis staff, in particular Mr. A. Vermeulen and Ms. J. Therrien. I am grateful to Cal tech for financial support during my Ph.D. years and the American taxpayer who, through the good offices of the U.S. DOE and NSF, made most of this research possible. Most of all, I thank my parents for their encouraging, patient support of the endeavours of a son so far away and my sister for her dependable hospitality and intimate friendship . .. Abstract Data were taken in 1979-80 by the CCFRR high energy neutrino experiment at Fermilab. A total of 150,000 neutrino and 23,000 antineutrino charged current events in the approximate energy range 25 < Ev < 250GeV are measured and analyzed. The structure functions F 2 and :r:F3 are extracted for three assumptions about crL/crr: R=O., R=O.l and R= a QCD based expression. Systematic errors are estima-ted and their significance is discussed. Comparisons of the :r: and Q2 behaviour of the structure functions with results from other experiments are made. We find that statistical errors currently dominate our knowledge of the valence quark distribution, which is studied in this thesis. xF3 from different experiments has, within errors and apart from level differences, the same dependence on x and Q2 , except for the HP'WF results. The CDHS F2 shows a clear fall-off at low-x from the CCFRR and EMC results, again apart from level differences which are calculable from cross-sections. The result for the the GLS rule is found to be 2.83±.15±.09±.10 where the ftrst error is statistical, the second is an overall level error and the third covers the rest of the systematic errors. QCD studies of xFa to leading and second order have been done . The QCD evolution of xFa, which is independent of R and the strange sea, does not depend on the gluon distribution and fits yield ALo = 88±}:3 _-Jrb 13 MeV ._ The systematic errors are smaller than the statistical errors. Second order fits give somewhat different values of A, although as (at Q~ = 12.6 GeV2) is not so different. A fit using the better determined F2 in place of xF3 for x > 0.4 i.e., assuming q = 0 in that region, gives 114 +ss MeV ALO- 266+ -104 H Again, the statistical errors are larger than the systematic errors. An attempt to measure R was made and the measurements are described. Utilizing the inequalit.y q(x)~O we find that in the region :r: > .4 R is less than 0.55 at the 90% confidence level. To my parents .. Contents Page Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation and Physics 1 1.2 Experiment E616 6 1.3 Organization of Thesis 6 Chapter 2. The Neutrino Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 .. 2.1 The Dichromatic Beam 8 ' 2.2 Beam Flux Monitors . 13 2.3 Ion Chamber Calibration .17 2.4 The Cerenkov Counter . 26 Chapter 3. Flux Analysis . . . 34 3.1 WBB Parametrization . 34 3.2 Beam Centering and Other Cuts . 36 3.3 The Beam Monte Carlo . 36 3.4 Flux Parametrization . . 36 'll vi Contents . 39 Chapter 4. The Lab E Detector . . . . . . . . . . . . . . . 39 4.2 Layout, Fiducial Volume and Alignment . 39 4.3 Spark Chambers . . . . 41 4.4 Scintillation Counters . 43 4.5 EHAD Calibration . 47 4.6 Electronics . 47 4.7 Triggers . 47 4.1 Requirements Chapter 5. Event Analysis . 51 5.1 E 11 . 51 5.2 (}j.l . 61 5.3 Ehad . 62 5.4 Vertex, Oh, Acceptance, etc. . 63 Chapter 6. Extraction of Structure Functions . 66 . 66 6.1 An Overview of the Physics 6.2 Total Cross Sections . . . . ' . 67 6.3 Cuts, Resolutions and Binning: . 70 6.4 Method or Extraction . 73 6.5 Corrections . 75 6.6 Systematic Errors . 90 6.7 Results and Comparisons . 95 Chapter 7. Physics Analysis and Conclusions 107 7.1 Weak Interactions 107 7.2 The Parton Model 109 7.3 Quantum Chromodynamics 115 yjj Contents 7.4 Tests of the Quark-parton Model 120 7.4.1 F 2 Comparisons . . . . . . 120 7.4.2 The Gross-Llewellyn-Smith Sum Rule 122 7.5 QCD Analysis . . . . . . . . . . . . . . . 126 7.5.1 Leading Order Non-singlet Analysis 126 7 .5.2 M S Non-singlet Analysis . . . 134 7 .5.3 Non-singlet Analysis Summary 137 7.5.4 Singlet Analysis . . . . . .. . 137 7.5.5 Comparison with Other Experiments 139 7.6 Extraction of R . . . . . . . . . . . . . . -; 140 7.6.1 R from y-dependence of Differential Cross-sections 140 7.6.2 Limit on R 143 143 7.7 Conclusions Appendix A. Beam Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Appendix B. Event Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Appendix C. Differential Cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . 154 ' Appendix D. Multiple Scattering . . . . . . . . . . . . 159 Appendix E. Event Counting for Total Cross-sections . . . . . . . . . . . • . . . . . . . 161 Appendix F. Models for Fitting Structure Functions Appendix G. R=0.1 Structure Functions . . . . . . . . . . . . . . . . . . . 165 . . . . . . . . . . . . . . . . . . . . . . . . . 168 Appendix H. Tables of Statistical and Systematic Errors . . . . . . . . . . . . . . . . . 170 Appendix I. List or Collaborators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Ch.upteT 1 Introduction §1.1 Motivation and Physics Just as Rutherford's experiments proved that the positive charge in the atom is concentrated in the nucleus, electron experiments have shown that charge in the proton is not. uniformly diffused over the nucleus but instead is concentrated in very small regions. These experiments were pioneered by Hofstadter and others at SLAC and MIT. The electron is a good probe because it is a point particle and is well understood in the framework of the highly successful theory, QED(ll. In 1967, a SLAC-MIT collaborative effort began which used the SLAC linac to study the scattering of electrons from nucleons at different energies and angles(2). The incident electron beam had energies ranging from 4.5 GeV to 20 GeV. The incident beam energy spread was less than 0.5%. Two spectrometers were used to scan the scattered beam in energy at fixed angles. As the sampled energy was decreased at fixed angle, the cross section did go through an elastics peak and resonances as expected. The elastics peak had the characteristic l/q4 dependence expected from the photon propagator; in additidn it showed a rapid fall with rf ascribed to a form factor. Beyond the resonances was found the deep inelastic scattering cross section, which was surprisingly fiat in q2 except for the 1/ q4 from the propagator. Such a cross section suggests a point scatterer since the form factor is the Fourier transform of the charge density or scatterers. Such behaviour of the cross section has since come to be known as "scaling" and was first explained by Bjorken and Paschos( 4l and by Feynman(3). Since then interest in the structure of the nucleon has grown considerably, especially with the identification of these point scatterers, or partons, with quarks. The behaviour of quarks within the nucleus is expected to tell us more about QCD, the heir-apparent to the mantle of theory of strong interactions. It is also expected that experiments performed with leptons as probes of the nucleus will further confirm the by now accepted "\Veinberg-Salam-Glashow theory of electroweak interactions. It is precisely because we have so much faith in the theory of weak interactions that leptons are used as probes of the nucleus. And because, as far as is known, leptons are point 1. Introduction 2 particles. Charged leptons suffer from the disadvantage that their interactions with quarks are principally electromagnetic, giving rise to the 1/q'* photon propagator. On the other hand, they are easier to use as a tool since momentum selection of charged particle beams is simpler. The experiment described in this thesis utilized neutrinos as probes instead; this was facilitated by the advent or energy-selected neutrino beams first suggested by Sciulli(s), Peterson<6 l and others. Neutrinos have no electric or colour charge and consequently interact only weakly with the quarks in a nucleon, a fact that enables us to study -the Q2 behaviour or nucleon structure without the large 1/Q 4 modulating effect of the photon propagator. Neutrino interactions with the nucleon are shown in figure 1.1. The standard model or electroweak interactions describes the lepton vertex completely, the hadron vertex is unknown. By studying the x-distributions of cross sections we hope to learn more about the nucleon. For example, the integral of the structure function 2xF1 (=q+q), is approximately 1/2. This leads us to believe that the remaining half of the nucleon momentum is carried by something other than quark-partons and these particles are expected to be the colour force carrying gluons. Since there are three valence quarks in the nucleon according to standard hadron spectroscopy, we expect the valence quark structure function, xFa , to peak at roughly a third of the available momentum, i.e., around 1/6. This indeed seems to be the case. In the picture accepted now, valence quarks are accompanied by gluons that bind them and a sea of quark-antiquark pairs that are constantly being created and annihilated. Figure 1.3 illustrates these structure functions . The scaling hypothesis is violated by QCD processes shown in figure 1.2. They change the relative momentum fractions carried by the valence quarks, sea and gluons as the nucleon is probed to shorter and shorter distances (Q 2 increases: see fig . 1.2). It is the aim of this ex periment to study the structure of the nucleon well. An introduction to the theories relevant to neutrino-induced charged current interactions is presented in the first part or Chapter 7. Briefly, in the ~ark-parton model point-like quarks are confined by forces within nucleons but behave as though they are free when probed by high energy particles. Quantum numbers or the quarks are assigned in a way that explains the spectrum or hadrons. To simplify calculations the quarks are frequently assumed to be massless; this cannot be done for the more massive quarks {charm and beyond). QCD attempts to explain the interactions between quarks on the basis or a property known as "colour" . Colour charge is responsible for the attractive forces between quarks and their consequent confinement within hadrons. The property or asymptotic freedom explains the basic parton model assumption of interactions with free quarks. Thus, deviations from the parton-model predictions are small corrections at high energies. Quantitatively, the predictions about charged current analysis that can be tested are: i. From the quark-parton model: a. The total cross section should be linearly proportional to the incident neutrino energy, Ev. b. They-dependence or the differential cross-sect ions is dictated by spin considerations {see Appendix C). For example, the differential cross section for antineutrinos on quarks is 1. 1. Motivation and Physics 3 HADRON SHOWER Figure 1.1. Neutrino-nucleon interactions y ~ ~ .g q q Figure 1.2. Basic QCD processes 4 1. Introduction C! I oO 1'· ci C> "'ci ~ ...., ci D C! ··-·~·>'~ · · · · . . . ~I ~-..__---no;;?-- -.,.·-',....--~---: .Do 1 0,26 ........ O.l>O .. • O.'tu >--- l.U __I 0 .,.; 0 ai IO' n ~.h~o~--~~~~------'o'.o~·u'~·----~--~u."'l~ti---r~---.1~~ ::c Glnon Distribution I • ~f I "'! o r- - I I .I · Q• = 70GoV" 10 .,; 1 C!.koo. ---........-.~ ~~--~==u~~~to~~==----~u!;~~ . ----~--71~ 0 :X Figure 1.3. Structure functions 5 1.1. Motivation and Physics (1.1) e. There are no spin-zero constituents in the nucleon, therefore 2xF1 ( = q to F2 (~ q q 2k). The only contribution toR, defined by + + + q) is equal (1.2) comes from target mass, transverse momentum and binding effects, and R"' 1/Q2 , i.e.,(3) (1.3} d. The total number or quarks in a proton is 3 (the Gross-Llewellyn Smith sum rule}: (1.4} e. The scaling hypothesis! 3•4) i.e., the structure functions are functions or x alone, not x and Q 2 • f. + F2 in electron and muon scattering experiments is not q q since the virtual particle exchanged there is the photon. The photon coupling amplitude is proportional to the charge or the quark, hence (1.5} 5/18 is the mean square charge or quarks in a nucleon. The factor in curly brackets is a small correction. ii. From QCD: a. The structure functions depend on Q2 as well as x; to leading logs in the perturbative expansion the dependence is like 1/ ln(Q 2 }. QCD prescribes the evolution or structure functions (see chapter 7). The evolution or xF3 is described by an independent integra-differential equation while that or F2 is coupled to the (as yet unknown) gluon structure function. fJxFa = !( F} fJ In Q2 x a (1.6} (1.7) 6 1. Introduction b. R, t o leading order, behaves as 1/ In Q 2 , is large at small x and small at large x: (1.8) In addition to the above, Regge theory predic~s that as x tends to zero xFa behaves like .jX. §1.2 Experiment E616 This experiment was done at the Fermi National Accelerator Laboratory located in Batavia, lllinois (near Chicago). The experiment consisted or a large steel target on which were impinged neutrinos and antineutrinos or energies ranging from 30 GeV to 300 GeV. Charged current interactions change the neutrinos to muons when they leave a nucleon and also produce a shower or hadrons. The neutrinos can also interact without transforming into a muon, the study of these so-called ~neutral-current" interactions is vast and complicated by itself and does not form part or this thesis. Thanks to some earlier experimentsC7 l done by this group at FNAL, the detector was essentially completely built by the time the run started in early June 1979. In the 8 months that followed we used a total of 5.87xl0 18 protons from the main ring. The neutrinos produced from the decay of secondary pions and kaons created by the interaction of the proton beam with a target gave rise to approximately 150,000 charged current neutrino events, the antineutrinos producing a smaller total or 23,000 events. The antineutrino data and about half the neutrino data have already been subjected to analysis for purposes or measurement or the total cross section - this work is described in an earlier thesisC8 l. Since E616 is a high statistics experiment with a dichromatic beam the data not only yield normalized cross-sections, but can also 1le exploited for a relatively accurate measurement or structure functions. While this data set was being analysed our group carried out another experiment to investigate the possibility of neutrino oscillations.< 109l Another experiment is planned Cor the Tevatron at Fermilab, it will, in main, study structure functions at the much higher energies possible there. Combined with E616 data all these data add up to an invaluable bank for the study or the nucleon and QCD. §1.3 Organization of Thesis As mentioned earlier, a whole Ph.D. thesis has been written on the measurement or total cross sections in E616. An earlier experiment (E356) by the same group has also produced a thesisC 9 l. Consequently the description of the apparatus for the detector and the monitors requires no more repetition and can be looked up in the above-mentioned references. This thesis will state the bare minimum in these areas except to describe recent advances in understanding (such as with the C-counter) and to emphasize those aspects of the beam monitoring that are particularly relevant to structure function analysis and/or have not received adequate treatment 1 1.3. Organhation or Thesis elsewhere. Also included is a discussion or the calibration or our ion chamber, in which the author played a major role. Another thesis( 10) is being written on similar material from E616. Much or the analysis has been done in common and it is not possible to unambiguously divide it in two. Two methods or analysis were used by this group with this author consistently preferring maximum likelihood techniques. This applies to the extraction of structure functions and extraction of R. For the Gross-Llewellyn Smith sum rule and QCD analysis we restrict ourselves to xF3 . The reader may find more detailed descriptions or some topics in one thesis as opposed to the other, re.H.ecting the history of the analysis. Most of the topics are clearly outlined in the table of contents and the reader should be able to navigate his/her way through with facility. ' Cha'PteT 2 The Neutrino Beam §2.1 The Dichromatic Beam High energy neutrino beams can be produced by the decay of charged pions and kaons generated in the interaction of hadrons with hadrons. At Fermilab part of the 400 GeV proton beam in the main ring is extracted and impinges on a BeO target 10.5 in thick. Typical intensities during the E616 running period were around 10 13 protons per pulse. The proton beam was extracted in two spills: in a fast spill or "' 2ms and a slow spill of "' 500ms. We used only the fast spill (which minimizes cosmic ray events) for antineutrino running, the data taking not being rate limited there. Typical per pulse secondary beam intensities were about 5 x 109 for negative settings and 2 x 10 10 for the positive setting!. The proton beam and the dichromatic train are shown in figure 2.1. Listed below are total proton and secondary beam intensities achieved during the ten energy settings at which we took data. The proton beam produces many charged hadrons on ineraction with the target, chiefly pions, kaons and protons. Charged pions decay 99.97% of the time into muons and muon neutrinos; charged kaons do the same only 63.5% or the time. More interesting is the fact that these are two-body decays, so the energies of the decay products are fixed in the centre-of-mass frame. Therefore we isolate pions and kaons in a narrow momentum range to obtain neutrinos in a narrow energy range. The desirability of a thin momentum slice must be balanced against high-flux requirements. Since neutrinos from kaon decays have much higher energies t han those from pion decays (see App. A), the neutrino beam is really dichromatic. The pions and kaons are removed by a 20 ft steel and aluminum beam dump, the muons range out in 930m of steel and earth. 9 2.1. The Dichromatic Beam HORIZONTAL. -2 DICI--4ROMATIC TRAIN 0.~0 •O 0 -t" VERTICAL. -40' -zo· o· 20' 40' 60' 80' roo· 120' 140' 160' reo· 200' ~ Figure 11.1. The proton beam target and dichromatic train. MANHOLE ' BERM (SHIELD) ENCLOSURE 100 N30 TRAIN LABE NEUTRINO EXP. ION CHAMBER SWIC RF CAVITY ._30m~ L-60m-+--~340 m-~----- 910 m - - - - - - l TRAIN DECAY PIPE SHIELD Figure 2.2. Secondary beam line. 1---30m~ v DETECTOR 10 2. The Neutrino Beam Eser (GeV) -250 -200 -165 -140 -120 120 140 165 200 250 Total - ves Total +ves Total Fast Spill Open slit Protons Secondaries 12 (X 10 ) (x10°) 1121000 202000 531000 264000 3i9000 270000 288000 283000 199000 232000 110000 484000 673000 138000 207000 1155000 301000 2068000 556000 5243000 2516000 1252000 1312000 9623000 3828000 10880000 Cl. Slit Protons (X 1Q12) 93800 25600 22700 32800 15500 17000 11600 15100 45800 56700 181000 146000 327000 Slow spill Open Slit Secondaries Protons 12 (X lOg ) (x10 ) 0 0 0 0 0 0 0 0 0 0 151000 642000 187000 837000 295000 1641000 353000 2444000 542000 4894000 0 0 1529000 10460000 1529000 10460000 Cl. Slit Protons (X 1012 ) 0 0 0 0 0 17600 12900 15100 44500 42100 0 132000 132000 Table 2.1. Accumulated beam intensities over running period Here it seems appropriate to mention different kinds of neutrino beamsJ 12 l The simplest kind of beam, or course, would be one in which no attempt is made to select the sign, momentum or direction or the decaying mesons. Such a beam is clearly not very useful for any precise normalized measurements. Examples or beams that sign-select but still manage to use most or the flux of mesons are horn and quadrupole triplet beams. Simple focussing magnets can be used to collect more mesons into the forward direction than with a bare target beam. This improves the neutrino flux. Horn focussed beams have good sign selection and high flux, but have a relatively high flux of low energy neutrinos. The quad-triplet beam on the other hand compromises flux and sign selection in order to improve the relative content of higher energy particles. Our dichromatic beam (fig. 2.2) is sign-selected in a narrow momentum bite (Ap/p ~ ±9%) and has an angular divergence of ±.15mrad in the horizontal and ±.18mrad in the vertical direction. The secondaries are then allowed to decay in an evacuated decay-pipe 340m in length. As shown in appendix A, neutrinos from a thin monochromatic beam of secondaries will have a fixed, known energy for a given angle of decay. This means that as one goes away from the centre or the neutrino detector in Lab E, the energy of neutrinos falls off as a known function of transverse direction (see Appendix A). Such a dichromatic beam is very desirable since it is easy to measure its flux. Increasing the length of the decay pipe, the momentum spread or the angular spread all increase the flux at the price of uncertainty in knowledge or the intensity as a function or energy and radius . Shown in figure 2.3 is a scatter-plot of neutrinos observed as a function of radius and 11 2.1. The Dichromatic Beam - '~~-~U .: .! S 1. 0 I .·· ·.· ........ (.) Cl i-. ~-::-. liJ C\l I (!) ..__, ·. :·... . ·: ·. ··. .·.·. ·. . :. ·' I i. ..... , ·. . - -.ltJ f-1 tl) 0 r:: ci ;.£! 0 ...-. . .. . .:-·. ,.·. . •' ' ~ -=:. -·~. : •\ w .. . \.. ~ • _' ' •·. ·, I ;• . ·:. ·.; .·.··..·. -=,~:::··; • • '• ~ ·, • .. :··: ·:' . . ..-~ -~ .. ·;: .'·. • . .• ··..: .. '. . :· ' . . : .· : :· . . .... . . .. . ·.· Pions Figure £.9 E vs. r scatter plot; 1500 events at 250 GeV. Only 1500 events are displayed. energy with the E616 secondary beam set at 250 GeV fc at Lab E (muon triggers). Notice the clean separation of neutrinos from pion and kaon decay. Figure 2.4 is a projection of this scatter-plot at a .ll.xed radius (10 in < r < 20 in ), showing agaip the separation. For details beyond those in appendix A the reader is encouraged to read references 9 and 12. Particle production by a 400 GeV beam is described in ref.l3 which includes the particle content of the secondary beam (pions dominate negative energy settings and the lower positive energies, gradually giving way to protons at higher positive settings}. Pion and k:aon production cross--sections fall otr steeply going away from 0°, while their relative content does not show as sharp a variation. For this reason the production angle in the E616 beam is zero degrees. In order to minimize background from the decay of secondaries before sign and momentum selection, the proton beam strikes the target at an angle or 12.0lmrad. This background, called the wide band background is therefore mainly at low energies (hadrons produced at~ 12mrad have lower energies}. There is a like background from the large-angle decays of forward going hadrons. Measurement of these wide-band backgrounds is easily accomplished: one simply cuts off the secondary beam by closing a slit and the wideband neutrinos are the only ones to reach the detector. Their flux can be estimated well by using the neutrino cross--section measured by this and earlier experiments.l7 •9 •30 ) Another background arises from 3-body decays of kaons (K,..3 and K.3: K±-JJ±'v~ 7r0 , 12 2. The Neutrino Beam 100 7QQJ l/TT > Q.) (!) I[) a: w CL ._ (/) z w > w ,,nnrl IJnn 10 n II II II ,,,, Jl II I ,1,1 t'l' Jill II I lit till II,, lilt .,,.. I I .~ II II I 0 .._I II II II II II II I 'I I . 100 200 NEUTRINO ENERGY (GeV) Figure 2.4. 10 in <r<20 in Energy histogram; 250 GeV setting. 300 2.2. Beam Flux Monitors 13 K± __,. e± 1 v~ 1r0 ). This background is small, lower in energy with a larger spread and is estimated via linkage to the main neutrino flux by a Monte Carlo program. §2.2 Beam Flux Monitors As mentioned earlier, one or the better distinctions of a dichromatic beam lies in our ability to measure its tlux with relative ease. This may be accomplished either by the measurement or the secondary beam flux or by the measurement of the decay muons produced along with the neutrinos. The latter approach did not prove to be consistent and accurate, at least to the level at which it was pursued, and will not be discussed further. Both methods involve tying t~reads or information about total tlux, mean momentum, angular dispersion, beam composition, direction and background together into one consistent unit. Several devices were used to monitor one or more or these quantities, the dependence or the final flux as a function or energy and radius at Lab E being predicted by a Monte Carlo program that utilized all available information. Other programs were used to perform cross-checks and to estimate the total cross-section. Calibration (esp. of the ion-chamber) and analysis (particularly that of the Cerenkov counter) of all the devices was done by yet other programs. It is not possible, as mentioned earlier in the introduction, to go into every detail here - the cross-section theses( 8 ,9l are again to be referred to. As is usually the case, one of several devices used to measure a quantity ends up being its primary monitor, the rest serving as useful checks. The total flux comes primarily from the ion chambers at the expansion port (fig. 2.5) which is just before the decay pipe (see fig. 2.2), and the target manhole (towards the end or the decay pipe). It was mainly the expansion port ion chamber that was used, with the manhole ion chamber acting as a backup. The expansion port ion chamber (fig. 2.6) consists or three ion chambers in one: one to measure the total flux, one to monitor the beam direction, and a tb.ird for calibration purposes. All three consist or a central high voltage plate 18 in in diameter (the beam is contained within a 4 in radius at the expansion port and within an 8 in radius at the manhole) flanked by two similar signal plates. In the case of the calibration chamber the HV plate is separated from its neighbours by an inch; the HV plate in the flux-measuring chamber being separated by a 1/4 in from the signal plates. The signal plates in the beam-direction monitoring chamber are split in two at lines through their centres, with one plate split into top and bottom halves and another into east and west halves. The idea, similar to tracking devices in telescopes, is to ensure spill by spill that both halves receive equal amounts of charge and the beam is thus centered. The larger the gap, the more the signal collected, a total gap or 1/2 in being sufficient for regular flux monitoring. The inner diameter of the chamber plates is 16 in as opposed to 22 in for the manhole ion chamber (which has an outer diameter of 30 in ). The manhole ion chamber has a total flux gap of 1/4 in. The operating voltage on the expansion port ion chamber is 600 V. The plates are .003 in aluminum foils. There is also a 2 in gap containing an Americium source for relative calibration due to temperature and pressure fluctuations. In order to calibrate the ion chamber using particle counting techniques one must go to lower beam intensities than those of the real beam (typical secondary beam intensity varies J4 2. The Neutrino Beam , TITANIUM WINDOW C!RENI('OV c::o.JNTER (SI ot: VIEW) Figure 2. 5. The expansion port. ~~---~~· 15 2.2. Bea.m Flux Monitors ~ ~ 1/4" ~ 1"--{ HV RV RV T 1 18" Beam Signal Planes Calibration Signal Planes Y Signal Plane X Signal Plane Figure 2.6. The expansion port ion chamber. between 2 and 40 times 10" particles/pulse). Since one cannot use the total flux planes for calibration anyway (at low intensities noise is sufficiently high for them), it was decided to use a small model ion chamber instead. But more about ion chamber calibration later . .. The other major total flux monitors are the RF cavity at the expansion port (fig. 2.5) and the toroid in the proton beam. These were primarily used as checks on the ion chambers, with plots of the ion chamber vs. the RF cavity (XR) and the toroid (NT) showing a remarkable linearity (figs. 2.7,2.8). It is also relevant to point out that the expansion port and manhole ion chambers tracked each other to within 2%, a fine example or how well these devices work (fig. 2.9). A current. transformer in the beam was not used because of its sensitivity to halo particles in the beam. The RF cavity is a resonant cavity through which the beam passes. Since the beam is accelerated in the main ring by RF cavities, it has an RF structure, i.e., it comes in pulses separated by 19 ns, corresponding to a frequency of about 53 MHz to which the cavity is tuned. The cavity may be used to calibrate the ion chamber because its response is calculable directly from Maxwell's equations. In practice however the observed stability and the readout electronics limit this to 5%. Another major parameter of the beam is its angular dispersion. This is measured by two segmented wire ion chambers (SWICs), one at the expansion port and another at the manhole. These provided projections of the secondary beam in x and y views at two points 16 2. The Neutrino Beam cr x1..0 o.o e..o •• •z. rdu CHAri!lER l.O Figure 2. 7. Linearity of XR vs XI ~,. (/) Z•oo 0 0 ._ ~ ,....,-'·.~· ..,/ a:: Q_ ltC'O .. ,,.) .0 a lDO :&DC> ~DO .. vo 600 XI"'oo ' Figure 2.8. Linearity or NT VS XI ---- .. .. 00 ........ 2•oo 0 .. . .· \DO aDo ~DO Figure 2. 9. Linearity or MI VS XI XI .... .. .... 2.3. Ion Chamber Calibration 11 separated by 154m (fig. 2.10). The increase in width can be related to the angular dispersion or the beam. A check on the SWIC profile is the small1/4 in x 1/4 in scintillation counter in the expansion port. It could be placed at different positions in the beam, enabling us to spatially sample the beam. For a detailed description or all the flux devices and the accuracies to which one determines beam properties the reader is referred to reC.8, particularly its appendix A. We continue with the ion chamber calibration and the Cerenkov counter analysis. §2.3 Ion Chamber Calibration The ion chamber was calibrated in several ways. One technique employed was to irradiate a copper foil, place it in the decay pipe and measure its residual radioactivity. For this purpose a 200 GeV proton beam was sent through the decay pipe with the target removed. After about 1013 protons the roil was removed and this led to a calibration constant or (3.45 ± .22) x 10- 18 Coulombs/particle. An old calibration against a 275 GeV pion beam using particle counting gave a result or (3.57 ± .18) x 10- 18 Coulombs/particle. A better calibration can be obtained from the RF Cavity. As mentioned earlier, this device has an accuracy of about 5%. The calibration constant varies from (3.31 ± .11) x 10- 18 for negatives to numbers around 3.7 x 10- 18 for positives. This systematically larger constant (about 10%) Cor positives puzzled us. One clue was the difference in proton content (zero ror negatives; between 25% at +90 GeV and 92% at +250 GeV Cor ~ositives) . The dE fdx energy loss due to protons is slightly different from that due to pions, however the energy loss from pions is Cl little greater . This means that the observed difference is opposite to that expected from dE/ dx losses. This phenomenon clearly presented a puzzle and a particle counting measurement was planned, this time with 1% accuracy. Using the expansion port ion chamber itself was out or the question, since its large size and G-10 separated gaps lead to leakage and noise levels incompatible with a 1% measurement at low intensities. A fairly similar ion chamber was built with 1/4 in gaps like the expansion port ion chamber, except for the use or ceramic spacers instead of G-10 and t he smaller diameter (6 in ). The signal plates were still made or .003 in thick aluminum foil. The windows were made or .022 in or titanium. The ion chamber (fig. 2.11) was put in the M2 beam line of the meson lab at FNAL in May, 1982. This meson beam was operated at different energy settings and with positively and negatively charged particles selected. The initial proton beam impinged on a 15 in target and particles were extracted at a production angle of 2.1 mrad. The beam composition or course varied with energy setting and sign selection and is tabulated below. 18 2. The Neutrino Beam A / I. I \ I \ \ \ I ' ~ ·' _,.I '"'·'· . r _,... "-,..,. -5 10 0 I!>J:(cml-10 -5 I\ i \ I I \\., .I -10 j '\ Target Manhole Beam Profile . \. / \ \ r ' '\ I iI ............... "-.... 0 Y(._) i\ \\ __,· I0 5 0 n , \ ·, ".,_'·'·.... I '·'- ·'" -10 . . \ Expansion Port Beam Profile J\ i 1\\ 10 .~· .I. \ ' '· . , ; \ ~ -I:> Figure 1UO. SWIC profiles. 0 ·.....'\. '·.... 10 20 Y(cml 19 2.3. Ion Chamber Calibration - HELIUM IN ~ HV e: ~022" TlTANIUM v IL ;!-" GAP ----v ~ .003 ALUMINUM - JERAMIC SPACERS / = ~: SIGNAL l_. HELIUM OUT Figure JUt. The M2 line ion chamber. EsET (GeV) +90 +90 +140 +200 +200 +200 +300 -90 -200 Proton Fraction 29% 29% 53~ 82% 82% 82% 97Jb -0% ,....,o% Calibration Constant (10 18 C/particle) 3.468±.050 3.468±.063 3.448±.050 3.468±.127 3.619±.050 3.679±.059 3.569±.050 3.380±.050 3.385±.050 Averaged Cal. Const. (10 18 C/particle) 3.468±.039 3.448±.050 3.630±.037 3.569±.050 3.383±.035 Table f .f . M2 line beam composition and ICH calibration results A!ter toying with various combinations or counters we settled on the configuration shown in fig. 2.13. The ion chamber was placed upstream or all the counters in order to 20 2. The Neutrino Beam count only primary particles. The scintillation counters were of varying thickness, with the fat counter (F) being used as input to a QVT analyzer. The beam spot size as seen in Polaroid tUm exposed to the beam was used to centre the various pieces of apparatus. The counter was situated 10 tt downstream of the .005 in thick beam pipe window in an effort to simulate actual conditions in the expansion port. The counters were plateaued and discriminator levels set to be above noise levels. The fat counter output went to a QVT analyzer which was triggered by a coincidence pulse. The idea was to try and determine the fraction of particles that arrived in pairs or higher-order bunches. A typical pulse height analysis by the QVT is shown in figure 2.12. (If 2 or more particles arrive in a single RF bucket of ~ 2ns durat ion, the counters with a width of 25 ns counted them as one). As suspected, the number of such coincidences never exceeded 1%, at least at the intensities used {from 2K to 800K particles per pulse). Helium gas was slowly injected into the chamber and bubbled out. A record of pressure was kept at all times. Output from the ion chamber itself was integrated and the total charge collected across a capacitor was converted into a voltage which was then amplified and input into an ADC. This ADC was sampled by sending interrupts between 80 to 100 times every spill from an on-line PDP 11/45 computer. Data acquisition and on-line analysis were also handled by this computer. The ADC, when calibrated against a very accurate DC voltage calibrator, was found to be extremely linear. The ion chamber calibration consisted, of course, of measuring the drop of the ADC output due to the passage or a beam pulse through the chamber and comparing it with the counted number of particles corrected for RF-bucket pairing, randoms and counter inefficiencies. We know from coincidence rates that counter efficiencies were all always greater than 0.9995. Singles and coincidence rates suffice to extract the randoms rate; this small correction was applied to all the data. Ideally, if one plotted the charge from the ion chamber as a function of time {fig. 2.14) the curve should be flat before and after the beam spill, with the slope of the drop in between being proportional to the beam intensity. Noise, leakage and even possibly pickup from magnet currents ramped in the vicinity lead to slight slopes for the curv~ before and after every spill {fig. 2.15). The true charge collected during a beam spill must be determined. At least three different ways of doing this were tried: fitting straight lines to the portions before and after the beam spill, subtracting an averaged no-beam curve and subtracting a "no-beam" fit to the curve spill by spill. Let AV be the measured voltage difference in volts (see fig. 2.14) and x = the number of ion p airs/em/particle, the gap size for the M2 ion chamber, d = the gap size for the exp. port ion chamber, Q - the total charge collected at the ion chamber, f = the cal. const. in pCoul./106 particles. D = Then, Q = (nxD)1.6xl0- 1 °C = .16nxDx10- 6 pC . (2.1) 2.3. Ion Chamber Calibration 21 U) r-----------~----------~.-----------,.-----------,~ - 0:: w co ::::::!: :::> co 1- z (\J_J -w z z ct I 0 ~ ct w Cl. (f) _, ~ w w co ct ' Cl. 1 :::> v U) 0 0 (f) _, w ~ \ - C> z (f) l ,..._...,. I § 0 ~ Figure 2.12. Typical QVT analysis of one pulse. 0 0 2. The Neutrino Beam 22 .005"TITANIUM WINDOW A1 C r• I • F f HV r- 1' BEAM • Az ,. 4 4 ~~-~~RI ~ BEAM PIPE ...____ 10'-o~ 14" 7 4 --,: 6}" - INTEGRATr1 .. ¥ c ~\ 1,\ I I J 0 M p E R lr-~ - - SCALERS COINCIDENCES INPUT QVT TRIGGER -l Figure !Z.19. The calibration run set-up. BEAM PULSE BEGINS t ' TIME- Figure 2.14. An ideal ion chamber voltage vs time cu rve. 23 2 .3. Ion Chamber Calibration . . .:... 8 • ....• .. .••• ..... .•... I I 0 CD - . .. 0 ~ f.D 0 .. ~ vw - 00::: ~ z .•. • 10 0 ~ a.. ::> a:: .. . ....• -.• • •• .• •• z - - .. -2 I 0 I 10 (7) 0I 0I ,.., I I ": _l_ .1_ C\1 I ~::I:>.J~ Figure :us. An actual ion chamber voltage vs time curve. 0 lt1 C\1 I 2. The Neutrino Beam Q I= (D/d)(n/106) = .16xd (2.2) Since the charge is collected across a capacitor of 995 pF and since the voltage generated is amplified by 500.3 before measurement, 12 - Q = 995x10- Fx.6.V = 1.989 .6.V pC. 500.3 (2.3) Thus, combining (2.1),(2.2) and (2.3), I= 0.4975 .t..v. (n/10 6 ) (2.4) Also note that if Pz = pressure of gas in the X I.Ch. PM= pressure of gas in the M2 I.Ch. Tz = temperature of gas in the X I.Ch. TM =temperature of gas in the M2 I.Ch. then I= 0.4975.6.V. Pz _TM (n/106 ) PM T" (2.5) The mean pressures and temperatures are used to effect the above conversion for Lab E purposes; the pressure variation during the calibration run is also incorporated in the analysis. Since both the X I.Ch. and the M2 LCh. were deep inside "caves", the temperatures varied very little. There were slight differences in the .6.V as obtained by the three different methods. Our suspicion fell on a spurious pedestal coming from noise, making .6.V depend on the intensity I in the form (al +b) where the first term is the true ~V and the b the noise that varied from method to method. Clearly, this problem could be solved by extrapolating a = (.6.V- b)/ I to infinite intensity (1/I=O) and indeed, all three methods tend to converge on one calibration constant. The QVT is a hardware histogramming device (a multichannel analyzer) that was triggered on the coincidence of the two channels Al and A2. Fig. 2.12 shows the typical QVT response to a pulse of beam. There is a small second peak due to 2 particles in one gate. These pairs may occur because 2 RF buckets occur in one 25ns period, because there may be 2 particles within a single bucket or because of the interaction of a particle with the beam pipe window or the 10 tt of air upstream of the chamber. In any case, the scintillation counters count them as one particle and a correction must be applied. The number or extra particles to be added can be expected to be proportional to the intensity and indeed a fit of the form itself for any or the above effects, (l.l77x 10-8 ! 2 153.7) was added to the intensity I. + 25 2.3. Ion Chamber Calibration T .. J. -- - "T'"' I .. I I l . I ,.0 ...... .-. ro I u i ~ [ --- I ~ •oI·'"o" .,.---..L-.-,2=-oc:-.-::!o---'--4.,-,0-:-.-:'-o----'----:li-:-:,0.,. . _. ,.0_ __....._80. u'----'--•. uO. 0 I Proton Fructlon Figure 2.16 The ion chamber calibration constant as a function of proton content. ' Finally, extra material (.042 in of titanium) was placed upstream of the ion chamber for 2 energy settings, +90 and +200 GeVfc, to see if any interactions with the material caused a change in the calibration constant. The final results of the analysis are tabulated in table 2.2. There is no measurable difference between material "in" and "out" for the two energy settings at which the effect was studied. This cannot explain the approximately 10% effects seen with the RF cavity. The only explanation that meets the bill is a proton-pion difference in the calibration constant. When the results of table 2.2 are plotted (fig. 2.16) as a function of proton content, one sees a definite increase in the calibration constant as the number of pions in the beam decreases. This is attributed to the lower cross-section of interaction for a pion as opposed to the proton (at approx. 200 GeV, the 1rp cross-section is ~ 24mb, the pp ~ 42mb). The low energy particles created in interactions (~ 17MeV) ionize very heavily due to their low velocity. Thus, one can conclude that protons have an ~ 6~ % higher response than pions in helium-filled ion chambers, and the calibration constants used for protons and pions are (3.63±.06) and (3.38±.05) x 10- 18 C/particle respectively. 26 2. The Neutrino Beam §2.4 The Cerenkov Counter The Cerenkov counter is a veritable cornucopia of information. In principle it is capable of telling us the total Hux, the Hux due to each particle type, the mean momentum of each particle type and even information on the momentum and angular dispersion or the beam. We shall see in this section how it is used to provide information about all but the total Hux, for which this particular Cerenkov counter cannot be used while simultaneously measuring the other quantities of interest. The principle of operation of a Cerenkov counter is simple; if a charged particle of speed f3 goes through a medium of refractive index n, and if {3 is greater than 1/n, coherent light is emitted at an angle Be with respect to the direction or motion of the particle, where Be is given by 1 (2.6) cosec= {Jn . For the highly relativistic particles we are considering, (2.7) The refractive index of gases is related to their pressure P by n~ 1 + teP. te for helium has been measured to be 4.37xl0- 8 jmmHg. (2.6) gives, for small angles (cos Be ~ 1- 8~ /2), (2.8) Combining (2.7) and (2.8) with (2.9) ' Many important consequences follow from this central relation, as we shall shortly discover. The secondary beam consists of different particle types all at the same momentum. From (2.9) we see that at a fixed pressure, P, and a fixed momentum, p, the lightest particles have the largest angles. If a lens or focussing mirror is used to focus this light, one obtains rings of different radii corresponding to each particle type. The angular separation or two particle types of masses m 1 and m2 is obtained by differentiating (2.9) to yield (2.10) Thus, in order to obtain as wide a separation as possible, it is best to minimize Be. The ring imaging possibility also gives us a means of clearly separating particle types: in the ring image plane is an annular iris that permits only one ring at a time, the light that passes through being measured by a phototube. Instead of varying the annulus, we vary the pressure or the gas (Helium), with a fixed annulus. The counter is shown in fig. 2.17. At one end is a spherical mirror of focal length 119.02 in and 10 in in diameter. It is tilted slightly and the Cerenkov light is rellectcd off 27 2.4. The Cerenkov Counter a couple of plane mirrors before going through the iris after which it is reflected off a fourth mirror (also plane) and goes through a lens and enters the pbototube. An alternate transducer, the photodiode, can be used with the help of a light splitting mirror. Since the typical number or particles in a pulse is very large, a weak signal is definitely not a problem (Cerenkov counters are capable of detecting single particles). Consequently the counter is short, with a radiative medium of length 74.3 in . The phototube accepts light roughly within the limits 3300 Aand 5000 Awith a mean accpeted wavelength of~ 4400 A. The iris has several apertures of which, for ring imaging, we used the annular aperture from .7 to 1. mrad (at the focal plane or the primary mirror). Bafiles before the iris cut down stray light and a shutter between the second and third mirrors can be closed to measure background light. Helium gas is used in the counter as the Cerenkov medium. Its pressure is constantly monitored by a pressure gauge. The ray optics of the counter in a plane containing the optic axis is shown in fig. 2.18. From the figure it is clear that if the beam were parallel to the axis but shifted a small distance from it, light would still be focussed at the same ring in the focal plane. Consequently not much attention is paid to the precise coincidence of the beam and optic axes. The counter is placed on a movable table and may be moved in and out of the beam (see fig. 2.5). The alignment of the directions of the optic and beam axes is, or course, very important and is done before every pressure curve is taken by remotely adjusting two orientations of the counter until the signal goes through a maximum, where the counter is then set. A Cerenkov curve is then taken by varying the pressure from zero by filling the counter with helium. A typical Cerenkov curve is shown in fig. 2.21. The number of photons emitted by a particle in a length L of medium in a wavelength range ['A,).. d'A) is given by + ._ dN = 21l'a . L.V. sm2 Bed'A . (2.11) Not surprisingly, the amount or light is proportional to the length, L. At small angles, it is also proportional to 8~. From (2.9), we can plot the Cerenkov counter response with respect to 8~ or, equivalently, pressure and the area under each of the bumps will, apart from diffraction effects, be proportional to the number or particles of the corresponding type. Before this is done however, various backgrounds must be subtracted, a procedure drscussed later in this section. The Cerenkov rings may be broadened for several reasons. There is chromatic dispersion in the gas which produces an angular broadening ~8 ~ ~8ew where w is a constant of the gas. Helium was chosen because it has a fairly small w (4.5%), does not scintillate and has the right magnitude of gas constant for our counter and iris dimensions. Another cause of broadening is the ......., 10% momentum bite or the beam. ~8 due to this is given by (2.12) For our counter, 1/8~ ~ 1.37x106 which means ~8/8 = ~pfp(= 10%) for 1 ~ 1170. Clearly, ~8/8 can be very large for kaons and protons (see table in Appendix A for typical values or /). From (2.10) we see that the momentum broadening and the angular separation maintain a fixed ratio. Angular dispersion of the beam causes a fixed broadening ......., .18mrad, which is the magnitude or the dispersion. In order to minimize this e[ect the angular separation or the particle types must be made large i.e., the Cerenkov angle Be must be made smalL Before the Cerenkov curves can be analyzed they must be corrected for various experimental effects. Since any pressure scan extends over many beam spills, the spill-to-spill 28 2. The Neutrino Beam Figure 1U1. The E616 Cerenkov counter. PRIMARY MIRROR o' SECOND MIRROR F' FOCAL PLANE Figure 2.18. Ray optics of the Cerenkov counter. 29 2.4. The Cerenli:ov Counter variations must be relatively normalized by one or the ion chambers. The expansion port ion chamber is so close to the Cerenkov counter that it, was reared that the halo or secondaries produced by beam-primary mirror interactions would make it unreliable. Consequently, the downstream manhole ion chamber was used. Another variable that can affect analysis is temperature. At 300K, a 3°C variation can cause " to change by 1%. However, the counter was in a cavity buried fairly deep and consequently temperature variations were not severe. They were monitored by a thermocouple affixed to the counter itself. One unfortunate consequence of pressure changes in the counter was a variation in alignment or the iris with respect to the central axis. This effect was measured after the run by a theodolite through a transparent hole in the beam window aimed at a cross-hair on the iris plate. The deflection t:..Oiri• in milliradians was fairly linear with pressure and could be parametrized as t:..Biri•(mr) = (.001318±.000050)P, P being in mm of mercury. Such a pressure dependent alignment can cause a very significant change in the amount or light collected, especially at high pressures. Since the correction can be easily calculated, this effect is safely accounted for. Studies were also made of the calibration of the pressure and total response of the counter. The counter is filled by opening a valve for a fixed short time repeatedly. This procedure can be shown to make the counter pressure P = Po(1 - e-an ), (2.13) where Po is the (fixed) pressure of the gas before the valve and n is the number of fills. This relation was directly checked against a pressure gauge that monitors the counter pressure and the two agree to less than 0.8%. After the end or the data taking run, a 200 GeV beam from the accelerator with an extremely small momentum dispersion (::::::: 0.1 %) was fed through the train, with the target removed. The iris of 0.7 mrad to 1.0 mrad aperture was replaced by a circular bole of 2.0 mrad aperture. In this mode, the counter functioned as a total light measuring device, and when its response was plotted against the manhole ion chamber, the linearity or the phototube and digitizer was checked and found good to 0.6%. The primary 200 GeV beam intensity was varied by almost an order of magnitude for this test. Inverting (2'.9), one can also use the 200 GeV beam to measure tc. The value thus obtained is 4.37xl0- 8 /mmHg, very close to the book value of 4.35x1o-s fmmHg. Another method of obtaining K. is to measure the threshold pressure at which light is emitted for the 200 GeV proton beam; " can be obtained by solving (2.9) for Be = 0. tc obtained this way agrees with the above method and is included in the average. Once the Cerenkov curves have been corrected for all the effects mentioned above, there are still some backgrounds to be subtracted: ( 1)Shutter-closed background: interactions outside the main body of the counter e.g., near the iris or phototube can cause extra Cerenkov light. This light is easily isolated by closing the shutter and taking Cerenkov curves. Such a Cerenkov curve is shown as the lower curve in fig. 2.19, and is subtracted from all raw curves. (2)Beam-material interaction background: Upstream or the gas is the counter window which is made of .020 in thick titanium. Including air and other material, there was an equivalent of .080 in or titanium upstream or the gas. Interaction or the beam with all this material produces additional particles 30 2. The Neutrino Beam 10 r---~~-----r-,--~~.~~--~.-----r------r-----~~--. CERENKOV DATA 9 ....•. 168 GeV, POSITIVE BEAM 8 !"'"" • -.. 6 r-·. .. -rr+ 5 ..:: >.... 4 -· . en 7 ..•····.. ·. .. . .... 3 ~ : z .:: :·.....··....·..........· : ~2 .. . . . .. . . . . ·.·. zw ~.... - ·.... ····. ... .. ......... .. .. .... .. ······.:. \ (!) - p .,.EX·T~R~A~ ~A~K·G~O~N~ • :J • • • •••• · (SHUTTER CLOSED) I 0 I I 200 - 400 600 800 PRESSURE ( mm Hg ) Figure 1l19. Cerenkov curve with a shutter-closed curve underneath. which in turn give rise to extra Cerenkov light. The easiest way to account for this is to add even more material, obtain another curve and subtract the pcoperly normalized difference as a background correction to the Cerenkov curves. Unfortunately beam-material curves were not taken at all energy settings and some interpolation had to be performed. (3)Light-scattering Background: After subtracting the above backgrounds it is found that some background light persists. One possibilty was scattering or light off dust on some mirror. Evidence for this possibility came from the monotonic increase with time of this background, observation of dust on the second mirror and a dramatic decrease in the background when the second mirror was cleaned with a nitrogen jet. It is possible in theory to accurately calculate this background (if the position, size, amount and albedo of dust were well known) by applying the optics or diffraction and scattering. Such an attempt was made< 17) and the qualitative conclusion was a background that rises from zero at zero pressure and ultimately flattens out, being dominated at high pressures by scattering. Since our knowledge of the quantity and properties or dust in the counter are necessarily limited, it was decided to try the simple form shown in fig. 2.20. Po is held fixed for all energy settings, L is allowed to vary from setting to setting.The resulting fits are then subtracted from each energy setting. (4)Counter Diffraction: The broadening or the Cerenkov ring due to diffraction (the counter has an effective 31 2.4. The Cerenli:ov Counter aperture or L8c where L is the counter length) is approximately A 118difl = LBc = 0.27 mrad. (2.14) This is obviously significant for our counter and in fact is the dominant broadening effect for the lighter particles (pions, muons, electrons). The number of photons in the range [A, A+ dA) and (8, 8 +dO) where 8 is the angle with the beam direction is given by(lS) 2 dN =21ro(~) dA dcos e A A • 28(sin,P) SID 2 tP (2.15) 2KP]. (2.16) where ,P(O) = ;~[1- {P + 82 - In the limit (L/A)-+oo, (2.15) reduces to (2.12). It is clear from (2.15) and (2.16) that there will be some Cerenkov light even below the Cerenkov thresh9ld. This is due to transition radiation when the beam crosses the counter windows and is the correct explanation for the mysterious zero pressure background that plagued earlier Ceren.kov analyses.!9 ) A!ter the shutter-closed and beam-material backgrounds are accounted for and removed, a Cerenkov curve looks like the one shown in fig. 2.21. Apart from diffracted light, particle fractions are directly proportional to the areas under their respective peaks. There is a small contribution to Cerenkov light from electrons (or posit rons) at the lowest pressures. These arise due to 0 production at the target with the 0• subsequently decaying into two photons which then pair-produce. Corrections are made to the particle fractions for this effect and also for the small changes in fractions due to decays before the counter (fractions are quoted at the target). The final results are to be found in table 2.3. Since the kaon and proton peaks are dominated by the momentum bite, their peaks can be translated into momentum probability distributions and their mean momentum is thus obtained. The pion peaks are broadened by iris acceptance, diffraction and angular dispersion. Consequently the energy of neutrinos from pion decays as measured in Lab E is used to estimate the mean momentum or the pions rather than the Cerenkov data. The final mean momenta are tabulated in table 2.3 below. 1r 1r 2. The Neutrino Beam 32 Pressure Figure 2.20. Form or the light scattering background. I0 I CERENKOV DATA AND FITTED MONTE CARLO FOR 165 GeV, POSITIVE BEAM x • DATA _,.MONTE CARLO .. ~1o 1 (/) z w .... z .... §10 _J 2 PRESSURE ( m"m Hg) Figure 2.21 Final Cerenkov curve with fits to individual particle type and light scattering background indicated. 33 2.4. The Cerenkov Counter I EsET I (GeV) -250 -200 -165 -140 -120 +120 +140 +165 +200 +250 1rjtotal I K/total .0419±.0018 .954±.006 .0486±.0018 .939±.009 .0615±.0019 .915±.009 .0647 ±.0025 .891±.012 .881±.014 .0600±.0032 .529±.022 .0545±.0033 .0485±.0024 .424±.014 .313±.012 .0388±.0018 .0263±.0013 .193±.007 .0773±.0027 .0124±.0007 111" mean p I K (GeV/c) 239.0 194.0 164.3 137.8 118.4 119.5 139.2 166.3 197.0 243.8 m~an p I J < e; > I (GeV/c) 238.0 194.6 165.3 138.9 119.6 122.4 142.2 169.8 200.6 247.0 (mrad) .16 .15 .13 .15 .16 .16 .15 .13 .15 .16 Ta.ble 1!. 8. Particle fractions, momenta and dispersions .. J< e; > I u,Jp I (mrad) .20 .20 .20 .21 .23 .23 .21 .20 .20 .20 (%) 8.7 9.2 9.5 9.4 9.7 10.1 9.9 10.0 9.6 9.4 Cho:pteT 3 Flux Analysis In this chapter we shall discuss how data from the monitoring devices is actually used, and how the final flux and various backgrounds are parametrized. §3.1 WBB Parametrization At every momentum setting, some beam time was spent on accumulating data with a slit in the secondary beam closed. The slit is prior to magnets D4 and DS (fig.1) and consequently the only neutrinos that can then reach Lab E are those that arise from large angle hadron decays or from decays of hadrons themselves at large angles. This background is thus primarily a low energy neutrino source and is known as the wide band background. Since the background is negligible except at low energies, an easy way to parametrize it is simply to count the number or events as a function or energy and radius at Lab E, and use the neutrino cross-section< 8 l to obtain the flux. Any systematic errors introduced by this technique are bound to be small ( < ""'1%), in any case much smaller than the statisical error. We assume that the background uniformly illuminates the detector at Lab E, and that the background is the same ror all negative energy settings. Since the beam dumpi.n g depends on proton fraction , the background is estimated separately for positive settings but all the negative energy settings are lumped together. Also, since bare target beam fluxes go through a peak( 12l at a low energy (approx. 15 GeV in our case), a form for the flux that peaks at a low energy is assumed. \Ve use the form (1 _ e-E/Eo) f(E) = { ( 1 _ e-E/Eo)e-(E-b)/c (3.1) The number or neutrino events in the energy range [E, E +dE), N(E)dE, is given by N(E)dE = k /(E) dEAn da(E) (3.2) 35 3.1. WBB Parametriution where, A= the area in which events are counted n d = the number of nucleons per unit area of the target u(E) =the assumed neutrino (or antineutrino) cross-section k = the overall normalization (varies with energy setting) Thus, integrating over energy for a given energy setting, N= f N(E)dE (3.3) = k{! f(E)a(E)dE} 5361.8x6.0221x 1023 xl0- 38 x(7r502 ) where a (E)= 0.669E (v)} 0.340E (i7) E in GeV. 5361.8 is the target density for the fiducial volume used in gmf cm2 • The fiducial volume is cylindrical with a radius of 50 in . The above equation can now be solved for k, giving k= N f f(E)a(E) dE 3.9432x106 (3.4) Hence, we first find Eo, b and c by fitting the energy distribution of wide band background events and then evaluate k. k must be normalized to the number of protons on target so that this parameter can be used for real open slit data correction. The final values for k are then normalized to 10Q secondary beam particles. The constants k, Eo, band c are tabulated below. .. EsEr(GeV) -250 -200 -165 -140 -120 +120 +140 +165 +200 +250 k 74.72 26.98 18.88 13.68 11.48 4.961 3.668 2.502 2.722 1.603 Eo 17.47 17.47 17.47 17.47 17.47 1.18 1.20 0.89 1.78 1.32 b 12.96 12.96 12.96 12.96 12.96 7.36 8.15 8.07 9.87 7.50 Table 3.1. Wide band background constants. c 12.63 12.63 12.63 12.63 12.63 21.01 14.68 21.81 19.26 22.97 3. Flux Analysis 36 §3.2 Beam Centering and Other Cuts During the E616 run, a constant effort was made to ensure that the beam centre coincided with the centre of the apparatus. The strictest off-line cut was a requirement that the fractional difference between the two split plates in the manhole ion chamber not exceed 0.1 for open slit data. This corresponds to a 1.2 in deviation from the nominal centre at Lab E. Pulses or very low intensity were also thrown out. U either the expansion port or the manhole ion chamber disagreed with the rest or the flux monitors for a run, that run was discarded. The digitizers for the ion chambers were calibrated pulse by pulse with a fixed current being fed in for 3 different lengths of time. The calibration constant and pedestal used were derived from a running average using all previous pulses, most recent pulses being weighted the most. Finally, the calibration corrected sum or ion chamber response for each run was written out on to a ftle for analysis use. The spark chambers in Lab E had a recovery time in tens or milliseconds, during which the apparatus was 'dead' for any possible additional events. The flux was measured for the duration or the 'live' periods by sending a pulse on a fast cable to the monitor electronics. Checks have been made on this procedure e.g., it was checked that the trigger rates during livetime and deadtime were equal. §3.3 The Beam Monte Carlo All the information extracted from the monitors was used to generate flux distributions at Lab E. We first use a program to generate rays or pions and kaons before the decay pipe using a production modell 8) at the target. Another program then 'decays' these rays using the measured particle fractions, angular dispersion and beam centre. The beam centre is measured by averaging the position or high energy pions at Lab E (since they are wholly contained). Alter producing all two and three body decays using the beam kinematics described in appendix A, distributions at Lab E are binned in 5 in radial bins around thll beam centre. Calculated in each bin are the total flux in the bin, the error on the flux, the mean neutrino energy and its error, separately for pions and kaons. Figure 3.1 shows the resulting flux distributions at Lab E for 1000,000 secondaries in the decay pipe. §3.4 Flux Parametrization To use the fluxes for analysis, it is necessary to have a fast function that returns the flux in a specified energy interval at a given radius at Lab E. Since the radial bins used are small (5 in ), the exact form or fluxes within a bin is unimportant. A form a- br2 was used b2 r 2 ) for the energy distribution. The total flux within for the flux distribution and Eo/(1 a bin and the average energy were normalized to the beam Monte Carlo output mentioned in the previous section. Two checks on the entire procedure were made. Firstly, the flux thus generated was combined with finer binned higher statistics output from the beam Monte Carlo + 31 3.4. Flux Parametrization 0 0 0 FLUX HISTOGRAM r--------r--------~------~--------~-------,--------~ ....-i 11" 2-body decays K 2 - body decays 0 rno Q)....-i •...-i F-. ttl '"d r::0 0 Q)O U)......i K 3-bod}' decays 0 ....-i 0 o.~o-o~--~----10-o~.o--~--~---2-o~o~.o ------~---3-o~o.o Ev (GeV) Figure 9.1. Flux distribution for 1000,000 secondaries at 250 GeV 3. Flux Analysis 38 at one energy setting. Also, the entire structure function analysis was done with the latter. No significant differences were round. Cho.pteT 4 The Lab E Detector §4.1 Requirements 1.1 km downstream or the decay pipe is a 1000 metric ton detector, roughly 60% or which serves as the target for the neutrinos. Any neutrino detector must necessarily be massive, given the small cross section or the neutrino-nucleon interaction. The neutrino can only interact with quarks in the nucleon through the w± or Z 0 as shown in figure 1.1. Only the charged current interactions are studied in this thesis, and thus there is always a muon in the final state. The detector must therefore be capable of measuring both the hadron shower energy and the momentum or the muon. Therefore the target steel is part of a calorimeter, with scintillation counters sandwiched between steel plates. To measure the momentum or the muons there is a toroid magnet spectrometer downstream or the target equipped with spark chambers. The vast majority or muons have su.fflcient energy to penetrate the entire spectrometer. The angle or the muon at the interaction vertex is measured by spark chambers il\. the target. The calorimeter counters double as trigger counters with the system or triggering designed to accomodate almost all charged and neutral current events. The enormous quantities of information produced in every event necessitate an on-line data acquisition system that writes all the data onto magnetic tape. §4.2 Layout, Fiducial Volume and Alignment A schematic diagram or Lab E appears in fig. 4.1. The target and toroid sections are clearly separated. The target consists of six carts on rollers. Each contains 14 counters and 6 spark chambers. The mean separation between counters is 8.16 in and that between chambers 18.79 in . Counters were separated by two steel plates ( 2.01 in x 10 ft x 10 ft ) and chambers by four . The toroidal magnet consists of 3 toroids. Each of these three is further divided into two half-toroids which have a set or chambers covering their downstream ends. Each half-toroid 40 4. The Lab E Detec tor COILS ._ I I I IO"DIA . STEEL SCINTILLATION COUNTER SPARKCHAMBER Figure 4.1. The Lab E detector AGN E T IRON \ I 4.3. Spark Chambers 41 is itselr made or 4 segments equipped with a counter each. Coils go in close to the axis of the toroid and generate a toroidal field averaging approximately 17kG. For reasons appearing below, the fiducial volume is restricted to the region between the 3rd and the 63rd counters (counting downstream). Meticulous counting of all intervening matter leads to a target thickness of 5361.8gm/cm 2 . This includes the steel plates, counters (6.5%) and chambers (0.8%). For accurate physics analysis the longitudinal and transverse positions of the counters and chambers must be well known. Alignment is first done optically, using a theodolite which leads to measurements good to .01 in . This is good enough for the z-positions of the chambers (the Lab E coordinate system has z increasing downstream, y increasing upwards). Translating the tranS\'erse position or a chamber as measured from outside into the position of a spark inside inYolves uncertainty much larger than the intrinsic resolution of a chamber(""' 1/2 mm). T_he sparks themselves must be used for chamber alignment . This is done by utilizing muons produced upstream of Lab E and going right through it as reference straight lines. §4.3 Spark Chambers There were two sizes of spark chambers in Lab E: 10 ft x 10 ft chambers in the target and 5 ft x 10 ft chambers in the toroid carts and in the gaps between toroids. Two 5 ft x 10 ft chambers with a slight overlap made up a single spark plane in the toroids . The chambers had wires spaced a little less than 1 mm apart in both transverse directions. Chambers were pulsed on a trigger with a voltage of""' 6.5 kV. Sparks occured where ionization had been produced in the chamber gas and currents in the nearest wires were caused. T-hese currents in turn caused an acoustic pulse in the magnetostrictive wire laid perpendicular to the wires (see fig. 4.2). At the time the chamber is pulsed, two fiduci al pulses are generated in each wire to serve as reference marks. The time taken by an acoustic pulse travelling at ""' 5mm/ J.l S with reference to the fiducial pulses was measured by a 20MHz clock. The pulses have a peak at the spark position and were differentiated to improve centre finding. Up to 16 sparks per chamber per direction could be measured in this way. The gas circulated through the chambers was about 89% Ne, 10% He and 1% C 2 H5 0H. Purity of the gas was maintained at all times by a purifier that removed contaminants (m ainly N2 and 0 2 ) by condensing them out of the system using liquid nitrogen temperatures. While the dead time of Lab E depended in part on the data taking of the computer ("-' 20ms), it was determined mainly by the spark chambers. The 5 ft x 10 ft chambers for example, had a HV recharge time of ""'6ms. l\"lore importantly , clearing fields were applied to all chambers to clear ions after a spark discharge. The DC field on the target chambers was 90V. the toroid chambers used 30V. In addition a~ 600V pulse clearing field of ,....., lOmsec duration was applied. The memory time due to t hese fields was reduced to ""'30ms. 4. The Lab E Detector - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -4_2 7. 1" i LUCITE ,.j fTHK . ilf S~RK CHAMBER WIRES 25 PER INCH MAGNETOSTRICTIVE WIRE Figure 4.2. A spark chamber 43 4..4. Scintillation Counters §4.4 Scintillatio n Counters As with spark chambers, the counters come in two kinds: liquid scintillation counters for the target and acrylic scintillation counters for the toroids. The target counters are 10 tt x10 ft x1 in in size. Mixed with the liquid scintillator is wavelength shifter which converts the UV light produced into blue light. From each corner to the middle or the two corresponding sides are shifter bars (see fig. 4.3) which serve to 'shift' the wavelength of light further to green and to collect the light for detection by a phototube at the corner. Wavelength shifter is necessary to reduce attenuation. There is an air gap between the bars and the counter to increase the total internal reflection of light on its way to the phototube. The toroid counters have the more complex arrangement with a total of 10 phototubes and 14 shifter bars per plane. As explained in chapter 5, the toroid counters were not used for hadron energy measurements and we will therefore concentrate on the target counters. The output from the phototubes for a minimum ionizing particle is a function of several variables. It depends on where through the counter the particle went, which counter it went through and when during the 8-month run this happened. Variation with real time over the course of the experiment was essentially a deterioration in counter gain of ,....._ 2.5% as measured by pulse heights from muons. Counter to counter gain variations were removed by requiring the mean hadron energy for events to be independent of counter location. To correct for an imbalance in the gains of the four phototubes for each counter, light from spark gaps was carried to the centre of each counter by light fibres. Phototube voltages could thus be adjusted periodically to get equal gains for all the tubes. By far the most important effect to account for is the variation of output with position. The important physical effects are attenuation of light in the counter, transmission across the air gap and attenuation in the shifter bar. The transmission across the gap is a function of two angles: 0, with respect to the plane of the counter and ifJ, from the normal to the bar (see fig. 4.4). Because of multiple internal reflections, one can integrate over 0 to get ~ T(¢) = /, T(ifJ, 0) dO, an 'averaged' transmission coefficient for every ¢. Now one considers all possible light paths into a given shifter bar. For the subset of paths between AD and AB for example, the path length for a given path characterized by an angle ¢ is AC /cos¢. One thus divides up all possibilities into dist.inct groups of ¢ allowed by Jack of total internal reflection. Reflected light paths such as AED are accounted for by extending the shifter bar BG to GB'. In each segment an average cos¢ is calculated from tables of J: T(¢)d¢ and J0¢ cos¢T(¢)d¢J. A factor e-AC/<cos¢>).. and a similar one for Ay combine to give for the ith segment. An additional exponential exp(- ri/>.bar) is inserted for attenuation in the bar. A6ar turns out to be large and the maps are insensitive to it, so it is fixed at a mean nominal value of 115 in . >., and Ay are parameters to be determined experimentally for each 4. The Lab E Detector 44 PHOTO TUBE SHIFTER BAR AIR GAP RIBS PHOTOTUBE 3 2 LAYERS MARVEL GUARD (II mills eoch ) -....~ • STEEL - ( . -=-'-WATER BAG fLUCITE LIQUID SCINTILLATOR 8 WAVELENGTH SHIFTER STEEL (SECURED TO TARGET CART) J r"i..:1L2"J Figure 4 ..9. A target sci otill at.ioo coun t er 45 4.4. Scintillation Counters 8' o', ' ' ',E G~--~~-------------------------, D c A Figure ...... Target counter light paths counter, as are x 0 and y0 • Almost all x 0 , y0 are less than 3 in from the centre of the counter and >.,., Ay ~ 70 in . Fits to the physical model were done using hadron showers, their mean position determined by the spark chambers; x 2 minimization is used. 46 4. The Lab E Detector 0 ci 0 ,_. I[) ..,/ tlll 1=1 ...... N ...... 0 .s~ ci,_. ./ 0 0 !i -, i ! ! X I PH _I ! .J t___...._,t'""'o=oc-c.0!1.,.._ _....--:1:-::5""'0:--: . u',.----"-~ :-.,.-:00. 0,.--- ....8G 0. 0--"-3 ;::-0:-:-:0.b Incident Pion Moinenh.nH (l.;uv/ c.;J Figure 4.5. Calibration: Eh vs. pulse height sum ~ ~ r-----,-----~----~r-----,------r-----.------~----- ~ ,_. I[) :r.: -=t. 0 ci ,_. u = O. 72+ .01-v'E ~ b I I ~ I .i I 0 -I lci I I 0 0 5.0 10.0 :------~---:-:-:::-- I 15.0 ~0 . 0 v'E(GeV 1 / ::!) Figure 4.6. Calibration: aE,. vs. ~ 41 4.T. Triggers §4.5 EHAD Calibration It is possible to move the target in Lab E so that another beam, the N5 hadron beam, goes through it. We ran a beam of incident pions through at 5 different energy settings: 25GeVJc, 50GeVJc, 90GeVJc, 200GeVfc and 250GeVfc. Means and sigmas for the hadron energy were determined at each energy setting. The best fit gave Eh = {0.2157 ± .0006)GeV x (pulse height in shower in terms of min. ionizing) and a e. = (0.72±.20) + (0.81±.03)y'E;;' (see figs . 4.5 and 4.6). §4.6 Electronics The output of every counter is used for two purposes: triggering and shower energy measurement. To cover the whole range of energies possible, ADCs of different ranges were used. The output of each tube went to a "low" ADC, the sum of the 4 target tubes in a counter to a "high", and a combination or various tubes to a "superlow". The highs saturated very fast and are used for minimum ionizing muons; the lows for pulse height measurements. IT a low saturated, the pulse height for that tube could be extracted using the superlows (see fig. 4.7). §4. 7 Triggers The apparatus is designed to trigger on charged current events, neutral current events and straight throughs. Neutral currents do not concern us. The straight throughs are triggered by muons that enter Lab E, fire the wall of veto counters, T2 and T3 (see fig. 4.1). The veto may at times fail; a fiducial cut at 80 counters eliminates such spurious events. The triggers that are most relevant are the two charged current triggers- trig.1 (muon trigger) and trig.3 (penetration trigger). The logic Cor these events is shown in figs. 4.8 and 4.9. Essentially, the muon trigger is meant for those events in which the muon is produced at a small angle such that the muon passes through enough of the toroid for its momentum to be measurable. The penetration trigger tries to cover the rest of the charged current events while maintaining a significant amount of overlap with the muon trigger. It requires the muon to penetrate at least 16 counters and has a modest energy requirement of 4 GeV to eliminate cosmic ray muons. The muon trigger requires that the muon originate in the target and fire either T2 or T3. The two triggers use independent logic and independent counters, thus allowing for constant monitoring of efficiencies in the overlap region. 4. The Lab E Detector 48 tube Triuer Level I s T c Faaia far>.Dut Target Front End Hlgh ADC ~er Level s T Toroid Front End Figure 4.7. Target and toroid front end electronics 49 4.7. Triggers ' Figure 4-.8. Muon t rigger logic "2] A R 0 0 o.3. .... ('!) Qq aq· .,... c;· I=' .,... .... G E T T (!) c+ .... Cl) 6 R 0 I=' ('!) ~ !0 ~ ~ ....,. .Q· T 0 ._. VETO ----------------------------------------~ TARGET--------------------------------------~ H EHAD > 10 GeV FAST SPILL S82 Sf S2 S3 SA24 SA I SA2 SA3 PENETRATION TRIGGER t.. Q ~ "'~ ~ M t1 c:f' ~ r'" t:7' o-:l '!'- Cha'PteT 5 Event Analysis The kinematics for a charged current event are described in detail in Appendix B. From that discussion it is clear that there are three independent variables which can be measured; they are the energy of the hadron shower, the energy of the muon at the event vertex and the angle the muon makes with the incident neutrino direction. It is possible to measure the mean angle of the hadron shower instead of the muon angle; for reasons cited below this was not attempted. Shown in fig. 5.1 is a typical muon trigger event. To analyze this event one must identify the first and last counter where the hadron shower is detected, recognize the muon track beyond the shower and track it through the magnetized toroids. To get the muon angle at the vertex, the track must be followed backwards through the shower using a multiple scattering matrix technique described below. §5.1 E~ ' The software for finding the muon momentum is by far the most complicated part of the analysis of this experiment. A crude vertex position is determined by a routine that picks up most or the sparks near the vertex and does a simple linear fit. Next, sparks from both wands on all the chambers are decoded and converted into position measurements. A least squares fit is then done using chambers in the target with only one spark. Now that a reasonable vertex position is known, we search for tracks by looking in angular bins away from the vertex for a cluster of sparks. Sparks are thus assigned to probable track candidates. Bad sparks are then eliminated if they lie outside a window determined from the spark scatter of a least squares fit. The longest track thus found is used to obtain a better vertex position and the whole procedure is then repeated. At this point bad sparks are removed and then an attempt is made to correlate tracks in the x and y views, if there are more than one in any, by correlating chamber hits. By now the track in t he target is well established and only the problems of recognition in the toroid and momentum determination remain. • 52 5. Event Analysis - - ----·--- ~- .x; I .... "' I l Il - J l I I J l J I --:::: : . : I I· -=It= -=- - .:: - - -- r s: I ...."5 . -' I l II I .,..,:z: u Cz .,.,_. ... ~ - .... a: <.:I ..... -' no .,.. ...... ., a: .,0 "' I .... j -VI -- - -o.. ;;, ,.., :r : I j i - -- - I l -<...J - .;; ~ ~ :r ex ..... <.:I ~ a: ~ :r .., ..... "'~ ~ . 0 OL 0,- J' Figure 5.1. A muon trigger event. CL- " 53 5.1. Ep 8 G .:;:; "" 0 0 .,. I r -"""' [ i[ II II I II Ii iI I: ] I I[ l ! ] ~ .:;:; .... :; N ....,-, ::r: _u II -:Z: .n- ...... - -... - : "' ~ \; : :: ! ;: -= --= -= = -= -= ' : . I. _. _. I -z ---= -en = :. \ ;;? ....crw .., ,;;; ':" ...."'cr : ~ IZ w .., ~ IZ :I ..., -~ ,;;; ~ N . X OL 0 cs;J,.::Nil )., 3L- 'JL cs;J,.::Nil X Figure 5.2. A penetration trigger event. ;:::.. CL- 54 S. Event Analysis Sparks are picked up in the toroid by trying to follow the principal target track into the toroid. Of course, this is impossible unless the vector momentum p is known. Neglecting multiple scattering and energy loss and assuming small angles with the beam direction it can be shown however that 2 dt/J (5.1) r dz = constant, where we use cylindrical polar coordinates around the beam direction. This follows from the toroidal nature of the magnetic field, B = B(r)¢ and the equation of motion. Actually the iron in the magnet is fairly saturated, so B(r) is a weak function of r; this does not affect the result (5.1) above in any significant way. Using sparks thus round a preliminary estimate of momentum may be obtained. This momentum is used to treat errors due to multiple scattering correctly and a still better estimate or the momentum is obtained. It is now time for the final fit. Three major processes affect the muon as it traverses the apparatus. Energy is lost due to the usual dE I dx losses at the rate, very roughly, of about 11.6 MeVI em of steel. It is straightforward to account for this effect and the action of the magnetic field. At the same time the particle undergoes multiple scatterings from nuclei leading to a small, random, uncalculable contribution to its path. As is common in all such random walk problems(20 l the most probable path remains the one without the scattering but the root mean square deviation from it is proportional to the square root or time. Consider the element shown in fig.5.3. It is possible to show(21 l, for small angles, that < 82 >'"" LIIPI 2 ; in fact, for the projection on to a plane, .015{;f V<8 2 >~--=ao IPI (say) (5.2) LRAD where pis the muon momentum and LRAD, the radiation length in steel (=1.76cm). Also, L <85> = 2a~ <8d> =0 L2 <52 >= -a~ 3 and L2 <d2 > =-a~ 12 ' {5.3) {5.4) {5.5) {5.6) Thus, to parametrize multiple scattering in any one view, two variables such as 8 and dare needed. It would seem that given two planes or perfect spark chambers before and after a 'scattering centre' it would be possible to determine these two variables exactly in both views and thus efficiently parametrize the track. This is true but for the magnetic field. We consider the entire target to be divided into 12 scattering centres (6 in the cart. closest to the toroid, 2 in the next, 1 each in the rest) and each half-toroid a scattering centre as well. Let Sz be the slope in the x-view, x 0 the intercept at zo, Zi the position of the ith scattering centre, p; the multiple scattering momentum kick, q, the momentum kick due to the 55 5.1. E,.. 0 u:i I I I A Multiple Scattering Element z0 0r;rj i- - ....... E-< ....... (/) 0 ~ P-. ...... rx:l ... =-- ·· · . .. . . (I) ~ rx:l ~ ..-1 i:> I ("JJ ~ ~ E-< - - 0 - {j d IE- L ---3:1 ~ 1- (Y'J I ~ 10 I .00 ~I . _L _i u.u BEAM DlHEC'l'lON ~.0 U.O 4. 0 I lU.U Figure S.S. A multiple scattering element. 0 ..... TOROID MOMENTUM FIT 0 0 oo ~C\1 DF > 10 11 ~ lZl -+-' ' 0 0 10 1=1 ...... Q) I :> .I cu ...... 0 0 r-. Q) ,.0 c) 0 0 _j I I 1 Predicted 'I El 0 :::J z c) 0 10 0 ~ .00 -Jx2/ J5f.' Figure S..f x2 in the Loroids. 56 5. Ennt Analysis magnetic field and x; the displacement (either d or o above). Then if p is the momentum of the muon, (5.7) i < j, L;i = 0 =1 i?:. j. The unknown parameters are !PI, xo , s., , p; and Xk . The p; are known almost exactly if we know the path in the magnets well. The problem is a linear least squares one except for IPii we use an old guess for 1/IPI and solve that part it.erativelyl 22 l. To form the error matrix < x;Xj > we use the expressions (5.2) through (5.6) a bove. The technique works very well , at least to trace the muon path, i.e., eliminate bad sparks, obtain the p; and x; and get a good estimate or the momentum. As 1/IPI was varied t o minimize x2 , the other parameters were varied as well. This led to a small bias in the momentum which was corrected by an unbiased routine that merely fit slopes, intercepts and momen tum . Appendix D describes some or the mathematical machinery of multiple scattering fits. A x2 distribution that results is shown in fig.5.4. T his distribution is for all the events that pass our structure function extraction cuts (about 68000). While the peak is at 1. as it should, the distribution has a long t ail on the high side which arises partly from a weak dependence or the mean x2 on the number of degrees of freedom and mainly from a few bad sparks. The deflection due to a uniform magnetic fie ld B(kG) after the muou traverses a distance d (em) is given by 8 = 0.3Bd (5.8) p where the momentum pis in MeVfc. (5.8) implies A.p -A.8 -p = -8- (5.9) Combining (5.9), {5.8) and (5.2) gives1 43l A.p 50 = ~:::;:;:::== A:i 9% p B..jdLRAD Cor Lab E . {5.10) Spark chamber efficiencies, spark position resolution and non-uniformitites in B actually make A.pfp closer to 12%. Some small fraction of muons do not go t hrough the magnet but range out due to energy loss. Their momentum is determined from their range to about 0.4 GeV, which is much 57 5.1. E 11 ~~XXXX XXX XX XXXXXX X XXX Figure 5.5 A range-out event. Shown are the sparks in two views and the counter pulse heights. The target ends at the right edge of the views. Notice the absence of counter pulse heights at the downstream end. Probability Dista nc e lY'O!Il e xr_}(:)L> Led pu;Jillo:n Figure 5.6 Spark distribution near vertex. 5. Event Analysis 58 better than 12%. These muons are easily identifiable as their tracks stop in the apparatus (chambers that should have sparks do not) and a string of counters with minimum ionizing pulse heights comes to an end at the same spot. A sample 'range-out' event is shown in figure 5.5. ' The muons may also suffer small-angle collisions in which they lose significant amounts of energy due to deep inelastic scattering, high energy 6-rays or bremsstrahlung before their momentum is measured in the toroids. Using an algorithm based on the observed distribution of energy loss by the muons in the counters, any set of contiguous counters with an excessive deposition of energy is identified and that energy is added to the muon energy. Effects due to such losses within the shower and hence indistinguishable from it are removed by a Monte Carlo simulation of the effect. Listed in tables 5.1 and 5.2 below is the magnitude of such losses in the form of the cumulative probability of energy Joss for a given energy muon. 59 5.1. Es,~ Loss Range (GeV) 0-2 2-4 4-6 6-8 8-10 10-12 12-14 14-16 16-18 18-20 20-22 22-24 24-26 26-28 28-30 30-32 32-34 34-36 36-38 38-40 40-42 42-44 44-46 46-48 48-50 50-52 52-54 54-56 56-58 58-60 O<Es,~<20 20<Es,~<40 40<Es,~<60 60<Es,~<80 80<Es,~<100 100<Es,~<120 9.172 4.855 0.875 0.159 0.020 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 27.359 18.100 6.995 3.047 1.202 0.468 0.203 0.070 0.049 0.042 0.021 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 39.270 29.374 14.698 8.631 5.299 3.266 1.893 0.963 0.541 0.260 0.184 0.119 0.065 0.065 0.065 0.043 0.022 0.022 0.022 0.022 0.022 0.011 0 .011 0.000 0.000 0.000 0.000 0.000 0.000 0.000 49.552 38.575 22.377 15.004 11.306 8.333 6.367 4.915 3.581 2.411 1.779 1.147 0.585 0.351 0.164 0.140 0.047 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 55.218 43.072 25.455 18.005 13.775 10.283 8.537 7.140 5.821 4.812 4.346 3.686 3.027 2.328 1.863 1.203 0.621 0.427 0.116 0.116 0.039 0.039 ' 0.039 0.000 0.000 0.000 0.000 0.000 0.000 0 .000 62.789 50.442 28.152 18.351 12.635 9.848 7.879 6.582 5.573 4.804 4.083 3.363 2.930 2.498 2.306 2.210 1.922 1.681 1.537 1.153 0.865 0.769 0.528 0.240 0.096 0.048 0.048 0.000 0.000 0.000 Table 5.1 1000 times the cumulative energy loss probability per metre of st eel: Part 1. 60 5. Event Anajysis Loss 120<E11 <140 140<E11 <160 160<E11 <180 180<E11 <200 200<E11 <220 220<E11 <240 Range GeV) 0-2 70.963 77.594 81.518 87.047 85.652 91.010 2-4 65.118 79.023 75.919 58.117 68.428 77.491 4-6 41.460 48.081 54.223 58.158 58.399 35.198 6-8 23.767 27.691 35.510 39.888 39.956 52.559 8-10 16.411 20.235 27.475 28.670 33.296 43.799 10-12 12.280 16.356 22.550 21.814 39.906 25.749 12-14 8.941 18.273 35.040 13.693 18.282 21.754 14-16 7.074 10.726 15.163 15.789 19.534 31.146 16-18 6.225 9.357 13.219 29.200 14.127 17.758 18-20 5.150 8.292 10.886 12.673 16.870 23.360 20-22 4.131 7.227 9.850 22.386 11.011 15.094 22-24 3.395 5.782 9.202 14.206 19.466 10.180 24-26 2.660 5.097 8.554 13.762 9.141 18.493 26-28 2.207 7.776 4.640 16.546 8.310 12.431 28-30 6.869 7.271 14.600 2.037 4.032 11.543 30-32 1.754 3.880 12.653 6.480 6.440 10.655 32-34 1.471 3.652 5.702 5.402 9.767 12.653 34-36 10.707 1.245 3.119 4.795 4.986 9.767 36-38 1.075 2.663 3.629 9.733 4.363 9.323 38-40 7.787 0.905 1.978 3.240 3.947 8.879 40-42 1.674 2.722 3.740 8.879 5.840 0.792 42-44 0.679 1.598 4.867 7.103 2.333 3.324 3.893 44-46 0.679 1.521 1.944 2.285 6.215 0.566 2.920 46-48 1.217 1.814 1.870 6.215 1.947 1.662 4.883 48-50 0.453 1.141 1.685 1.947 50-52 0.989 1.555 1.454 4.440 0.340 4.440 0.973 0.283 0.989 1.296 1.247 52-54 0.973 3.996 0.113 0.761 0.778 1.039 54-56 0.623 2.664 0.973 56-58 0.000 0.456 0.778 0.000 0.208 1.332 0.304 0.259 58-60 0.000 .. Table 5.2 1000 times the cumulative energy loss probability per metre of steel: Part 2. 61 S.2. 81' The initial track finding and fitting in the target is described in the previous section, here we shall concentrate on extrapolation of the muon track upstream into the shower. 95% of the shower is typically contained in 70 to 100 em of steel which means between 3 and 7 spark chambers. In order to extrapolate upstream into the shower, we first choose a best spark in each view arid each chamber on the basis of closeness to the track. Sparks with bad second differences (kink in track at spark) are then thrown out, the window being twice the expected scatter from zero or the second difference. A regular multiple scattering fit is then done, again using (5.2) through (5.6) to obtain the error matrix. The procedure is considerably simpler than the toroid case because the momentum is now known and there is no magnetic field. The z = 0 plane is first taken to be the downstream-most chamber in the shower. Solving the x2 problem gives an expected position for the muon spark there in both views with an error, say u0. Let the spark chamber resolution be s. Then the closest spark is picked up within a radius 2.80' from the expected position, where u = y'u~ + s2 , the sigma for the probability of a real spark occurring in the neighbourhood of the predicted position. This procedure is repeated with z = 0 being the next. upstream chamber and so on until the 'vertex' chamber is reached, if Ehad < 25 GeV. If 25 GeV < Ehad < 100 GeV we only go down to the 'vertex+ I' chamber and if 100 GeV < Ehad < 200 GeV only to 'vertex+2'. This energy dependent selection of chambers is due to the increase of background hadron sparks with hadron energy. How did we arrive at the above criterion! Assume, for the sake of simplicity, that there is a uniform background of n 0 sparks per unit area due to the hadron shower. Then the distribution or sparks around the expected posit ion will look like that in fig. 5.6. The probability of picking up a good spark is, approximately, the upper shaded area (n2) and that or picking up a bad spark n 1. n 1 can be determined by looking at large distances from zero. n2 will not be one because of the 2.80' cut and also because of spark chamber inefficiency. In table ' 5.3 we list n 1 and n 2 for different Ehad ranges and at different d istances from the vertex. The dotted line indicates that a cut was made when n 2 fn 1 was less than 2. The multiple scattering fit also predict s the error on the space angle 8w Clearly this should be proportional to 1/p, and there may be a slight worsening of the error with an increase or Ehad due to the higher probability of mistaking a hadron shower spark for a muon spark. Tabulated in table 5.4 are the coefficients a and b for a linear fit in different Eha d ranges: b u1 =a+-p a is indistinguish able from zero. As a rough summary, 68 ~ 100/pp mrad-GeV /c. (5.11) 62 5. Event Ana.J.ysis Quantity VTX IVTX+1 i\'TX+2 VTX+3 VTX+4 tvTX+5 VfX+6 VTX+7 0.027 nl/n 0.056 0.034 0.045 EHAD O<E,.. < 10 n2fn ~.674 0.797 ntfn 0.119 0.061 0.760 0.047 0.806 0.046 1.000 0.026 0.000 0.081 0.000 n2/n 0.492 0.691 --ntfn 0.172 : 0.092 0.733 0 .066 0.701 0.044 0.869 0.034 1.000 0.044 0.419 0.000 1.000 0.033 n2/n 0.261 : 0.558 ntfn [o.159 : 0.125 0.654 0.075 0.676 0.056 0.742 0.056 0.725 0.049 1.000 0.026 0.867 0.014 n2/n n1/n 0.185 : 0.480 0 .598 ---0.137 0.130 : 0.092 0.719 0.053 0.672 0.050 0.651 0.041 0.869 0.022 0.903 0.000 n 2 fn 0.127 0.664 0 .728 0.813 0.887 1.000 10<E,.. <25 '' 25<E,.. <SO 50<E,.. < 100 ' ' : 100<£,.. <200 I 0.206 : 0.513 Table 5.3. n1 and n2 in planes away from vertex. See fig. 5.6. EHAD Range (GeV) O<Eh < lOGeV 10<E,.. <25GeV 25<E,..<50GeV 50< Eh < 100 GeV 100 < E,.. < 200 GeV 200 < E,.. < 400 GeV a (mrad} .162 .257 .111 .111 .126 -.031 ' b (mrad-GeVfc) 84.35 80.25 105.38 106.23 129.22 156.10 Table 5.4.. Angular resolution fit parameters. See eqn. 5.11. A description or the counters and hadron energy calibration has been given in the previous chapter. Here we shall describe how the shower limits are located and how Ehad is obtained. A software bit is set for all target counters if at least 2 out of the following 3 conditions are satisfied: (i) The hardware bit is on and the pulse height is > 3 times minimum ionizing. (ii) The pulse height is > threshold. (iii) The 'high' pulse height is > threshold. 5.4. Vertex, 8h, Acceptance, etc. 63 Then we sweep all the counters starting from the downstream-most. Whenever the software bit is turned 'on' a new possible event is logged. When the bit turns 'off' in this sweeping process, the event is taken as beginning just before that particular counter if none or 3 counters upstream also have bits set. The first counter to fire in an event is designated 'PLACE', and the last, 'CEXIT'. Ir (CEXIT-PLACE+l) is~ 5, we define CEXIT to be also 'SHEND', the last counter to see shower particles. Ir not, another sweep is made from (PLACE-I) to CEXIT. If the sweep ends at the second counter upstream of the toroids, SHEND is set to 2. Otherwise, if 3 successive downstream counters have pulse heights less than 4 times minimum ionizing (and 3 even further downstream have an average pulse height < 4 min. ionizing), that counter is labeled SHEND. The shower thus lies between PLACE and SHEND, the muon continues to CEXIT (see fig.5 .1). The hadron energy is obtained by summing up the pulse heights in the shower, subtracting the contribution of the muon and multiplying by the calibration constant described in the previous chapter. As mentioned there, AEhatt = .72 + .81 ~- The question bas been asked "Do we miss any hadron energy in the longitudinal direction!" This can happen if SHEND is one or two counters too close to PLACE. The mean energy is computed (in terms of minimum ionizing) in different Ehad bins in the 6 counters downstream of SHEND and compared with that in 6 counters even furth er downstream. Only about (0.16 ± .03) GeV is being lost and is corrected for. Another worry regarding Ehad is the possibilty that fluctuations due to 1r0 decays in the shower might increase the error, AEhad · A method has been suggested(23) which involves using E~ = E;(l- ~E;) as the response of the ith counter instead of E;. ~ is defined by c <=~~~~==;==;=;;=~ JE; E; (in min. ionizing) (5.12) Shown in figure 5.7 is AEhad/ Ehad for different values of c for 90 and 250 GeV/chadron beams used for our hadron energy calibration. Clearly the preferred value is c = 0, i.e., we see no such improvement. This may be due to the coarseness or sampling (lOcm Fe) being enough to average over electromagnetic fluctuations. §5.4 Vertex, e~~., Acceptance, etc. Some miscellaneous aspects of analysis will be covered in this section. The z-position of the vertex is simply taken to be midway between the first counter of the event and the previous one. We considered the possibility that back scattering faked a more upstream z-position. An algorithm based on typical shower behaviour was employed to improve the vertex z position; it failed to show any significant change in cross-sections ( +0.2% for pions, - 0.7% for kaons; 3000 pions, 1900 kaons). The transverse position of the vertex is determined well by the shower probing algorithm - the error in the position is about 0.1 in in each view. Studies were also made of the possibilities of using 8h instead of 8JJ and recovering The best achievable value for AOh turns out to be PJJ from multiple scattering in the target. 64 5. Event Analysis q 0 .-----~------~------~-----.-------r------.-------r------, Cll q - ........... l() ~ ..... .__... '"7 41 !'£1 ~ 0 I ~ '0--. '0--. I 90 GeV .,.,._ I ~ ~ 0 a "-.... ..... '0--- - (I) +- ~ ~ 'i<.... ""'-.. ......_ 250 GeV 0 tO -0.050 c - / I ..,. / <I ./ I )/ ..,).... -1 ._.. -v- ..if"I I I ./ I ./ ....._ _.;r -+- -+"' .00 U.OGO Figure 5. 7. tl.Eh/ Eh vs. c (see text; eqn. 5.12) Projected circle at T2 RB L -' T2 Figure 5.8. Geometric acceptance problem. o.!o S.-t. Vertex, 8h, Acceptance, etc. 65 50mrad, at the highest Ehad· This corresponds to a low Oh region which means very large t:.8h/8h, ruling out the possibility of using eh. p,.. may be obtained from multiple scattering, in fact this method works well if the true p,.. is less than about 40 GeV. However, momenta larger than 40 GeV are estimated to be smaller than they really are, making such a technique highly unreliable. An interesting quantity to measure is the resolution of the target spark chambers. At higher momenta, this is as significant as multiple scattering. Using the x2 from the upstreamward extrapolation or the muon track into the hadron shower we find that for muon momenta above 40 GeV fc the x2 I DF is 1 if the resolution is set to 0.53 mm. This value is also confirmed by comparing predicted and observed spark positions while extrapolating upstream into the shower. A very important quantity is the geometric acceptance of the detector. For charged current structure function analysis only muon trigger events were used. Penetration triggers were used to complete the set by adding candidate muon triggers that never made it to T2 (see flg.4 .1) because of ranging out. Consider the straight line projection or the muon track into the toroids. Since the magnet is focussing, any muon will tend to be 'accepted' if this projection is 'accepted'. Our 'software trigger' then accepts only t hose events for which the muon projection is within a 110 in -side apparatus-centred square at T2, a 69 in -radius apparatus-centred circle at the front-face of the magnet (TF; see fig.4.1) and spends less than 30% or its time in the magnet hole. Since we pick up all rangeouts, the acceptance depends only on the muon angle e,.. and the vertex position. As described later, a cut is applied and only events with e,.. < .2 are used. The geometric acceptance is calculated as the fraction of times an event with a given e,.., v~, Vy would be accepted if V.z and ¢,.. took on all possible values with equal probability. ( v~, vy, V.z) is the vertex position. V.z can take on all values between the z-limits of the fiducial volume; ¢,.. can take on all values on [0, 27r]. The problem, as illustrated in figure 5.8, reduces to the intersection of a circle with a cirlce (at TF), a circle with a square (at T2) and a cone with a cylinder. ... Chapter 6 Extraction of Structure Functions §6.1 An Overview of the Physics Let us take a look at the result of Appendix C which is reproduced here for convenience: (6.1) It would be best if all the structure functions utr ,4- ;rf' ;t', sP, and kP could be independently and accurately measured as functions or x and Q2 ., From (6.1) it is clear though that only certain combinations or these are measurable. The three main functions or interest are 2xF1 = q+q xFa = q-7j and F2 = q (6.2) + 7j + 2k, + + or, equivalently, xF3 , F2 , and R (defined by F 2 = 2xF1 {1 R)/(1 4m2 x2 /Q 2 ))). The neutrino and antineutrino differential cross-sections provide two independent measurements, yielding F 2 and xF3 with an assumption about R, which, as we shall see, is expected to be small. The nature or the term containing F2 and R clearly indicates the necessity of using y-dependences in extracting R. Partly to get a 'feel' for the structure functions and the problems associated with their extraction, a brier description is presented here; excellent refcrencesf 2 5- 2 Q) may be found 61 6.2. Total Cross Sections elsewhere. At the very outset it should be noticed that xF3 is the 'valence' quark distribution while F 2 is the 'total' quark distribution. Clearly, if quarks were all a proton contained, then 1 / 0 F 2 dx would be 1. This means that a" should be G 2 s/2rr implying (6.3) The measured value is closer to 0.67 x 10-38 cm2 /GeV<30l, indicating that a little over half the nucleon momentum is carried by objects other than quarks. Current wisdom has it that these are gluons, carriers or the strong 'colour' force. Present understanding and calculational abilities prevent us from predicting the structure functions as functions or X and Q 2 • However, there are many clues. At the high Q2 values in this experiment the distance and time scales or the neutrino-quark interaction are small enough to justify the 'quasi-free' nature or quarks assumed by the parton model. Consequently quark distributions appear to 'scale', i.e., depend on x alone. 1 Since xF3 is x times the valence quark probability density, we expect / 0 xFafx dx to be 3. The 1/x forces xFa to go to zero as x-+0. In fact, valence quark contributions are expected to be nondifl'ractive in Regge theory, which predicts that Regge exchanges with a~ 1/2 will dominate, giving<29 l xFa"' Jx as x--.0. The sea arises from virtual pair production by gluons, and is consequently confined to low x. Quark counting rules<29 l and comparison with the electromagnetic form factor in turn indicate<26 l that F2 (x ),..__ (1 - x ) 3 as x-+ 1. Shown in figure 6.1 are the functions xFa, F2 and 7j that satisfy these predictions. QCD suggests that quarks bremsstrahlung to give gluons and gluons in turn produce more gluons or qij pairs (fig. 1.2). These processes are more and more visible as one goes to higher Q 2 • This means that the valence quarks should lose momentum, which appears in the form or extra sea quarks (mostly) and gluons (small increase). All 3 distributions (valence, sea, gluons) should have lower mean x values. In particular, the sea sh~uld see an increase at low x. These QCD deviations from the parton model are illustrated in fig. 1.3. One of the objectives of this experiment is to measure such scaling violations and compare the results with QCD. In the limit of very large Q2 (not reached by this experiment), only sea quarks are significant, and a" and av are equal. In the energy range or E616 however (30-+ 240 GeV), we only expect a slow fall in the neutrino cross-section slope and a slow rise in that from antineutrinos. The onset or charm production tends to reverse any fall with Q 2 • §6.2 Total Cross Sections As an illustrative example, let us calculate the total cross section of neutrinos and antineutrinos a.ssuming that the slope with energy is constant. The detailed references are, of course, (8) and (30). This section is meant as an introduction to the problem or extracting structure functions by way of the conceptually simpler problem of obtaining a cross-section. An explanation of the mechanics of event counting for cross-sections is included in appendix E. 68 6. Extra.ction or Structure Functions llJ..USTIU\TlON ONLY 0 co I ci _j 0 ~ 0 0 0 I ~--~--~=~----~~~~~~--~=L--~~~=;~----~--~~1 0.130 l.U Figure 6.1. F 2 , xF3 and qat fixed Q2 • The event sample is restricted to lie in the fiducial volume-defined between PLACE=20 (see §6.5} and PLACE=80 (§4.7}, which implies a target density of 5361.8 gmfcm 2 • Neutrinos from pion decays are only accepted in a beam-centred circle of 30 in radius at Lab E, while neutrinos from kaon decays are accepted in a beam-centred square with a 100 in side. This is necessary since fluxes are not well understood at large radii for neutrinos from pion decays (Chapter 3; Appendix A}. Only those spills in which the beam centre was within a 2.4 in sided box around the centre of Lab E are accepted. Also, if either the expansion port ion chamber or the manhole ion chamber disagreed with the rest of the flux monitors for any run, that run was discarded. It is useful to separate the neutrinos from pion and kaon decays because they have very different energies. Since the penetration trigger penetration requirement is 16 counters, the apparatus is fully efficient (after geometric inefficiency is corrected for) out to about 600 mrad. We thus safely make a cut at 370 mrad, the outer events being corrected for by a Monte Carlo as described below. The 'software' triggers used can therefore be summarized as follows: 6.2. Total Cross Sections 69 ~ ..... IO ..... 0 co 0 0 ~ 0 I 1b . . 1~ ... . I . . ·I ~ I >< .• . lC ci .... 0 ~ 0 0 c:q 0 0 ~ .00 0.40 0.80 0.80 y Figure 6.2. Effect of kinematic cuts around Ev = 120GeV. Muon trigger(S1) Penetration trigger(S3) Hardware trigger 1 e, < 100 mrad E 11 > 10 GeV if R.,•rtes < 5 in then 811 > 7.1 mrad Hardware trigger 3 e, < 370'mrad Eh > 10 GeV E 11 > 2.9 GeV (6.4) Clearly the S3 cuts are less restrictive and therefore define the kinematic boundaries as depicted in fio<T\ll'e 6.2. The angle beyond which the muon trigger efficiency cannot be geometrically recovered is the angle a line from the apparatus centre at z=-167 in (closest event to the toroids) to a corner of T2 (55 in square, z=142.5 in ) makes with the apparatus axis; this is 251 mrad. As described in appendix E, we can establish categories or events, containing either muon or penetration triggers, and thereby obtain the total number or 'pion' and 'kaon' events. Assuming that the neutrino cross section is cv(E) = kEx10- 38 cm2 , where E is in GeV, we can use our knowledge of the neutrino flux (/-.:,/k; chapter 3) to predict the number or 'kaon' and 'pion' events, e.g., 1 00 Nv(7t) = Cv(E)f-.:(E).Area.Target density.Avogadro's no. dE (6.5) 10 6. Extraction of Structure Functions Comparison with the experimental value gives k. A high statistics Monte Carlo generation of events is used to correct for smearing and unsampled regions which are the regions excluded in (6.4). Our final result, obtained after properly averaging over various energy regions to handle systematic errors correctly is( 30 l and Uv/E = (.669±.003±.024)x10-38 cm2 fGeV u;;/E = (.340±.003±.020)x10- 38 cm2 /GeV (6.6) Note that the dominant error is systematic, resulting from flux level and particle fraction uncertainties. The slopes as a function of energy are plotted in fig. 6.13. While both slopes are consistent with being flat, there is a slight tendency, at high energies, for the neutrino slope to rise and for the antineutrino slope to fall. This affects the Q2 behaviour or the structure functions, especially that of xFa, which is related to the difference of the neutrino and antineutrino differential cross-sections. Extraction or structure functions involves an increase in the dimensionality of the problem, since events must now be binned in x and Q 2 as well. We can therefore use only muon trigger events for which a muon energy is known. In the next few sections the structure function extraction procedure is described, with its cuts and corrections, and is followed by a discussion of systematic errors. §6.3 Cuts, Resolutions and Binning: From the cross-section discussion it is already clear that cuts must be made; the cuts made for the structure function analysis are listed below. ' Fiducial Cuts Kinematic cuts 1) 205PLACE580 2) Beam radius5 30 in for neutrinos from 1r decays 3) Beam position within a 100 in square for neutrinos from K decays. 1) 8" < 200mrad 2) Eh > 10GeV 3) Ep > 4GeV (6.7} The beam position requirements in the fiducial cuts reflect the limits beyond which only flux tails exist which are not well understood. PLACE cannot be greater than 80 (i.e., 81 or 82) because one might then be including straight through muons. If PLACE < 20, there are too few position measurements before the toroids and after the hadron shower for track finding to be fully reliable (see (iv) in §6.5}. As pointed out in the previous section, beyond Op = 25lmrad, the muon trigger efficiency falls irrecoverably and we thus make a conservative cut of 200 mrad. Below Eh = 11 6.3. Cuts, Resolutions and Binning: 25 GeV there are no calibration data and therefore the calibration curve is extrapolated only down to E11. = 10GeV. A minimum E,. requirement of 4 GeV is ample for the muon to penetrate the longest shower and be identified. Not mentioned above are the cuts which remove 'stray events' such as those with no track round (very few; see track finding efficiency below). Also eliminated are events with track fitting failures and track fitting x. 2 / df greater than 9. The correction for this is also described below. or course, only muon trigger events are used. Penetration trigger events which satisfy the 'software trigger 1' requirements (radius at magnet front face < 69 in , position at T2 within a 110 in -sided square and less than 30% of time spent by muon track when projected from vertex into toroid) and do not fire the muon trigger are presumed to be range--outs and are picked up and classified as muon triggers. Table 6.1 illustrates the effects or all the cuts as they reduce the raw number of events for each energy setting. Cut Raw evts, PLACE~ 20 Event Gate Muon trigger Good trg 1 Good fit X.~or < 9 Xtar < 9 85.2 8 > .0071 if r11 < 511 Eh.ad ~ lOGeV E,.~4GeV r6 5 30'' if from 1r In 100'' square if from K 05x<1 1 GeV2 < Q2 < 251.2 GeV2 -250 -200 -165 - 140 -120 120 250 140 165 200 4241 4507 4374 3972 2857 16241 20235 31785 38224 45822 4156 4441 4324 3950 2830 16000 19200 31330 37565 43854 3541 3821 3706 3344 2324 11487 13926 23806 29413 35697 3211 3443 3332 3002 2071 10043 12279 21237 26340 32287 3211 3437 3331 3000 2060 10033 12230 21091 26220 32182 3210 3426 3329 2999 2040 9979 12120 20749 25910 31932 3208 3425 3328 2999 2040 9955 12096 20699 25846 31836 3197 3418 3322 2996 2032 9862 11986 20525 25625 31578 3195 3417 3322 2996 2032 9860 11984 20524 25618 31566 1989 2034 1870 1519 889 5513 7565 14192 18810 24440 1875 1957 1810 1476 858 5456 7500 14067 18680 24232 1610 1690 1517 1193 653 4322 6289 12534 17090 22587 1576 1642 1482 1162 631 4157 6063 12054 16426 21756 1573 1638 1481 1160 631 4146 6048 12024 16374 21685 1468 1531 1365 1065 556 3881 5711 11499 15790 21009 Table 6.1. Effect of structure function cuts From the expressions (6.8) and (6.9) it is easy to calculate the resolutions in x and log 10 Q2 , given those in E,.,Eh. and 8,. (see chapter 5). The x-resolution worsens with increasing x but improves with increasing Q 2 . Listed in table 6.3 are the x and log10 Q2 resolutions for a fixed neutrino energy of 120 GeV. Clearly, 12 6. Extraction of Structure Functions at low x (x......, .015) the x resolution can be as good as 0.005 which helps the determination of the low-x behaviour of xF3 tremendously. Table 6.2 clearly shows that the low-x resolutions are dominated by the o,. resolution and the high-x ones byE,. (except at lower Q2 where the effect or the Ehad resolution eclipses that of the E,. resolution). The binning scheme could be adapted to the resolutions, but it was thought best to use the binning shown in table 6.7 to facilitate easy comparison with CDHS who use the same binning except for their log 10 Q2 bins, which are twice as fine. For the low-x behaviour or xFa, finer binning is used. X X-.015 (,} .. - 2.51 X=.045 Q 2= 3.98 X=.080 Q2= 6.31 x=.150 Q 2 ' 15.85 x=.250 Q 2 ' 15.85 x=.350 Q 2 : 25.12 x= .450 Q2 39.81 x=.550 Q2 = 39.81 x=.650 Q2 = 63.10 x=.850 Q2 = 63.10 resol. .006 .Oll .019 .031 .060 .080 .095 .125 .132 .190 %from %from %from EIJ Ehad OIJ 21.82 61.09 68.13 78.34 72.83 76.86 82.06 78.35 85.06 79.90 1.16 10.43 14.28 9.77 21.06 18.59 14.12 18.70 12.11 18.16 77.02 28.47 17.59 11.88 6.12 4.55 3.81 2.95 2.82 1.94 loglo Q2 %from %from %from resol. EjJ o,. Ehad 14.93 .164 3.84 21.23 .103 64.95 4.79 30.26 19.56 .098 75.67 4.77 .089 80.07 7.98 11.94 .095 88.40 4.13 7.48 4.87 5.23 .093 89.90 4.16 .088 6.45 89.39 .091 91.45 5.12 3.43 2.97 .085 89.70 7.33 2.27 .089 92.71 5.02 Table 6.~. Resolutions in typical bins and where they come from (E11 =120 GeV) . .. 6.-t. Method of Extraction 13 x-resolution x=.015 :r=.045 x=.080 :r=.150 :r=.250 :r=.350 :r=.450 :r=.550 x=.650 :r=.850 .883 Q"~ 1.26 .005 .016 .033 .075 .152 .245 .351 .468 .596 .004 .014 .028 .379 .481 .709 Q 2 = 2.00 .063 .125 .199 .284 2 .012 .391 .574 .007 .024 .053 .104 .164 .233 .309 Q = 3.16 .045 .321 .468 Q 2 = 5.01 .010 .021 .088 .137 .193 .254 2 .386 .010 Q = 7.94 .116 .162 .213 .267 .018 .039 .075 2 Q .225 .324 12.59 .015 .034 .065 .139 .181 .100 Q 2 = 19.95 .028 .056 .155 .193 .274 .087 .120 Q 2 = 31.62 .028 .047 .074 .134 .167 .237 .103 Q:l= 50.12 .040 .061 .087 .205 .115 .144 Q 2 = 79.43 .175 .070 .094 .120 2 .094 .142 Q =125.89 Q 2 =199.53 log10 Q"-resolution x=.015 X=.045 X=.080 x=.150 :r=.250 x=.350 x=.450 x=.550 x=.650 :r=.850 .128 Q"= 1.26 .131 .129 .130 .130 .130 .130 .. 129 .130 .129 Q 2 = 2.00 .124 .118 .118 .120 .120 .120 .120 .121 .121 .120 Q2 3.16 .203 .107 .111 .113 .114 .115 .114 .114 .115 .115 Q 2 = 5.01 .099 .102 .106 .108 .109 .110 .110 .110 .110 Q:l 7.94 .097 .094 .099 .102 .107 .108 .105 .106 .106 2 .105 Q = 12.59 .086 .093 .097 .103 .100 .102 .103 Q2 = 19.95 .084 .092 .097 .099 .098 .101 .103 2 .099 .096 Q = 31.62 .090 .082 .088 .092 .096 2 .093 .089 .076 .079 .083 .086 Q = 50.12 .086 .074 .078 .081 Q 2 = 79.43 .070 .075 ~ 2 =125.89 - - - - - - - Q 2 =199.53 - - - - - - - - - - - - - - Table 6.3. x and Q2 resolutions at E 11 =120 GeV. §6.4 Method of Extraction The equation to be solved is (6.1). Assuming that all correction terms are calculable, there are three unknowns in any small x and Q 2 bin (integrating over energy), which may be F 2 , 2:rF1 and xF3 . With only 2 equations, one of the unknowns must be fixed beforehand. The similarity of the term containing F2 in both the equations demands that xF3 not be fixed; it must be F 2 orR. F 2 being the most interesting quantity and R being small ("" 0.1), we chose to fix R while analyzing the data. The values chosen were 8. Extraction oC Structure Functions R = 0., 0.1 and .73(1- x)a.es ln(Q2f.242) (6.10) the last or which derives from calculations based on our results for the QCD evolution of F2(X, Q2 ). The number of events in a given x and log 10 Q2 bin is given by (6.11) A(v,., Vy, 811 ) is the acceptance and ¢v,v(v,., vy, E)dv,.dvydE the flux of neutrinos or antineutrinos in a small area dv,.dvy and a small energy interval [E, E +dE). v,. and vy are the transverse coordinates of an event. The acceptance A(v,., vy, 811 ) is integrated over all v.z: values within the fiducial volume and hence is not a function or vz. We now assume that F2 and xFa are linear in the bins used (see section on corrections below). Then, using the value at the bin centre as a fixed unknown, the neutrino and antineutrino equations may be integrated to give 2 simultaneous equations in 2 unknowns which are then easily solved. If F2 and xFa be the (constant) unknowns in a given bin, then the expected number or events is c} Nv = aF2 + ~ xFa + N-v = aF2 + bxFa +c (6.12) where the 6 constants a,b,c,a,b and care known from the integral6.11. The likelihood Lor seeing N 11 neutrino and N-v antineutrino events in the bin is then simply the product or two Poissons with the expected values or N 11 and N-v as the means: , (6.13) Maximizing L or, equivalently, minimizing -In L gives (6.14) and, It is obvious that (6.12) is indeed a solution of (6.14). To speed up the integration, the product A(v,.,vy,Bp)¢v,v(v,.,vy,E)dEdv,. dvy is evaluated first after integrating over ¢ 11 and is stored in tabulated form as a function or E,Bp, and r . This reduces the problem to a 4-dimensional integral of which at least one integration, over rbeam, is easily performed without including the y-dependent terms. The kinematic cuts described earlier are applied during the integration, thereby restricting it to allowed regions only. 6.5. Corrections 15 Another technique is used by D.B. MacFarlane as described in his thesis( 10). That technique eliminates two integrals by utilizing the fact that it is also possible to take A( v~, vy, 911 ) to the lett-hand side of (6.11) before the integrations (i.e., in the differential expression) and sum up weights (!/acceptance) instead or 1 for each event. It is easily checked that this leads to a larger fractional error for the left hand side or (6.11); the fractional error increases from 1/VN to (6.15) -where wand C1 are the mean and standard deviation respectively of the weights of theN events in the bin. A large dispersion of weights in a bin will therefore lead to a larger error on the extracted structure functions. However, since we keep A(v~, vy, 811 ) on the right hand side of (6.11) a 5-dimensional integral has to be performed as opposed to a 3-dimensional one otherwise. We effectively reduce the integration to a 4-dimensional one (see above) and with the help of some programming short cuts find that the integration takes less than 50% longer than the other technique. Apart from the completely independent computer implementations, other differences also exist between the two methods. We choose to parametrize the flux as a function of energy and radius at Lab E, instead of binning it in an energy histogram. This allows greater flexibility and ease of use. We also choose to make all the corrections in an additive fashion as opposed to multiplicatively in the other technique. It has been verified that none or these differences is quantitatively significant. §6.5 Corrections ' structure functions can be Several corrections, most of them tiny, must be made before extracted. These fall into two kinds: those that arise from experimental inefficiencies and those that are either theoretically motivated or require prior knowledge of the structure functions; the former are described fl.rst. L Veto deadtime correction An unbiased loss of events occurs because some of the flux is vetoed by the veto wall. The magnitude of the effect is small (~ 2%) and is tabulated in table 6.4. ii. x2 -failures There are two fits to the track: one in the toroids for momentum finding, and one in the target for the muon angle. In either case it was felt that events with a bad x2 fd.f. would lead to worse resolutions, hence only events with x2 fd.f. < 9 were accepted. The effect of this cut is ,...., 1% after kinematic cuts, as evidenced in table 6.4. A setting by setting correction is computed by making all possible cuts in a 30 in beam centred radius including the E 11 cut for events with a good x2 • 16 6. Extraction of Structure Functions Energy Setting (GeV) -250 -200 -165 -140 -120 120 140 165 200 250 Veto Deadtime correction 1.016 1.004 1.006 1.016 1.020 1.012 1.010 1.009 1.011 1.014 x2 > 9 correction 1.0025 1.0080 1.0014 1.0009 1.0250 1.0059 1.0123 1.0265 1.0169 1.0072 Table 6.4. Corrections (i) and (ii) described in the text. iii. Trigger inefficiencies One may ask whether the muon trigger is really 100% efficient. A way to determine this is to examine small-8,.. events for which both the muon trigger and the penetration trigger should be 100% efficient. Such an analysis has been performed for fast-spill eventsC42l with no significant difference from 100% detected. We have also investigated the o,.. behaviour or the muon trigger efficiency and the Ehad behaviour or the penetration trigger efficiency. Arter geometric efficiency corrections both are equal to unity in the expected regions. Beyond o,.. = 251 mrad the muon trigger efficiency shows a fall oft'. The penetration trigger efficiency is only relevant for the purposes or picking up rangeouts and reaches one well before the hadron energy cut or 10 GeV. These dependences are shown in fig 6.11. iv. Track finding efficiency The track-finding efficiency is virtually 100% . This was directly checked by asking how many muon triggers underwent successful track finding. The result if plotted as a function or PLACE is fiat at 100% except for PLACE < 14 where it rapidly begins to fall to zero. We impose a cut to eliminate PLACE < 20 and hence there is no correction to be made. v. Wide band background This correction is made by including the wide band .flux correctly scaled to the number or open slit protons and has already been discussed in chapter 3. vi. Cosmic rays When the structure function analysis cuts are applied to events in the cosmic ray gates, only 2 events (one at +200 GeV/c, one at +250 GeV/c) make it through. Allowing for a slow spilllivetime ratio or 3.7, this number translates into 7.4 events out of,..., 30,000 slow spill events. This is an extremely small fraction and hence the cosmic ray background will be ignored. vii. Flat cross-section correction TT 6.5. Corrections Due to experimental uncertainties the slope of the cross-section measurements with energy is not a smooth curve (references 8,30; figure 6.13). QCD scaling violations, charm production etc. make this curve not flat. Figure 6.14 shows a prediction from QCD using a Buras-Gaemers(·U) kind of fit (A = 0.1 GeV) and charm production assuming me = 1.5GeVfc2 • Since neither A nor me is really well known, our assumption to force the cross-section slope to be flat is adequate. This serves to remove experimental fluctuations. There are many ways of applying this correction. The most obvious is to use, at each energy, a multiplicative factor that changes figure 6.13 to a horizontal flat line. Another way is to separately alter the pion and kaon flux at each energy setting so that the pion and kaon decay neutrino cross-section is the same at all energy settings. A third technique, the one we used, involves adjusting the relative flux levels of each momentum setting so that the structure functions integrated over the measured x and Q2 range of that setting agree with the averaged structure functions in that range. A "' 0.5% correction was then applied for consistency with the cross-section measurement. The constants we multiplied the fluxes with are listed below. Presumably they remove errors in particle fractions i.e., the fluctuations in the cross-section slope measurement are mainly due to errors in particle fractions and partly due to inaccuracies in flux measurements (we fl.nd that the rms for the adjustment constants is roughly equal to the expected errors on particle fractions). Our technique would be the same as assuming a QCD cross-section slope if all the fractions in fig. 6.14 were 100% , since the model is an approximation to QCD. Since the total cross-section is insensitive toR, we do not expect the uncertainty in R to affect the adjustment constants. -Pions +Pions - Kaons + Kaons 250 GeV .963±.034 .964±.014 .901±.050 1.112±.012 200 GeV .964±.034 .973±.015 .896±.047 1.042±.014 165 GeV 1.04.3±.039 .947±.016 .964±.051 1.028±.016 140 GeV ,1.126±.055 .938±.022 1.071±.065 .986±.023 120 GeV 1.004±.064 .967±.026 .970±.081 .993±.030 Table 6.5 Flux adjustment constants- they flatten the cross-section in regions where structure functions are measured. All the following corrections require prior knowledge or the structure functions. For this reason we iterate. Starting from an approximate set or structure functions, corrections can be calculated, made and then the newly obtained set of structure functions used to repeat the process. Experience has shown that one iteration is enough. The strucuture functions used for corrections are really fl.ts, and we used two different models (see appendix F). For the purpose or smearing corrections and external radiative corrections, the empiri- 6. Extraction or Structure Functions 18 0 t'i Neutrinos, It:! ..-i X 0 F [] Bin ~entre Isoscalur Strru~~t sea Rawa ·alive ;...f;-:1 + z0 :It or. ...... E-< 0 pq Q2 =7.94 Charm mass=1.5 GeV X ~ ..-i (/) I (/) (/) 0 ~ 0 It:! 0 0 0 --- ~ 0 10 0 I .00 0.20 0.60 0.40 0 .80 1.0 X 0 t'i Antineutrinos, X 0 It:! [] ..-i + :It z0 ...... E-< 0 pq i[j 2 Q =7.94 ~ • ::1 Din centre Jaoscalur Stran.:lo oea Raw ·au-vo X Charm mass=-1.5 GeV ~ ..-i (/) I (/) (/) 0 0 It:! 0 ~ 0 I 0 ~ 0 10 0 I .00 0.20 0.60 0.40 0.80 X Figure 6.3 Magnitude or various corrections ror neutrino and antineutrino differential cross-sections. 1.0 19 6.5. Corrections cal model forms the kernel or a Monte Carlo which generates events. The generated variables E 11 , E11 and 811 are then smeared using the resolutions described in section 6.3. This Monte Carlo is very fast and ample events at each energy can be quickly generated. Figure 6.3 illustrates the following physics corrections in the form of their contributions to the neutrino and anti-neutrino cross-sections. Clearly F 2 and xF3 dominate at all but the lowest x values where the strange sea, radiative and other corrections begin to get large as z-+0. viii. Smearing corrections Ideally, an integral in an x and Q2 bin over x,Q2 and E, say (6.16) can be transformed into an integral over E 11 ,E11 and 811 as (Two other integrations that are performed over v., and Vy, the transverse event coordinates, are understood here). It is now easy to include the effect of non-zero resolutions. Ir the probability that a given variable, say q, truly lies in [q, q dq) when the measured value is r/ is R 9 (r/; q) dq, then the above integrals should change to + ' f(x, Q2 ,E)J(x, Q2 , E; E 11 , E11, 811 )RE,. (E11 ; E'11 )RE,. (Ell; ~,)R,,.(8 11 ; 8'11 ) (6.18) where the integrals over E~, E'11 and 8~ are over all space. Given the complexity or f(x, Q 2 , E), the problem of evaluating these 8-dimensional integrals with the available CP time was insuperable. We settled for the alternative technique of generating much higher statistics than the data using the Monte Carlo described above, and using the ratio of unsmeared to smeared events in every bin to multiply the number of data events in each bin. Ir ni be the number or events initially in a bin, with entering due to smearing and n~ leaving, the number or events after smearing in the bin will be n: (6.19) The smearing correction is then defined by . . n'· n '. n~ /i=-=1 - - + ni 1 ni n, (6.20) 80 6. Extra.ctlon or Structure Functions co ........ F2 Smearing ~ ........ + u X=.~.lliO X=.~5ll )( \D ~ ~ 0 X=.D1G X=.D4G X=. DBO X=.H)O X <> u .. X=.450 X=.!.l51J X= .GoO X C\l ........ 0 E-t ~ 0 ::r::: E-t 0 - - ........ - - ~ 0 ""'::r::: tX:! E-t ,__. 0 ~ 0 co 0 0 "<:ji 0 I 1.0 10.0 100. 0 Q2 (GeV) 1000.0 2 co ........ xl'3 Smearing X ¢ u ~ ........ + u Qf ~ ~ X ;:< 0 0 C\l 0 ........ E-t X=.0113 X=.Ot!G X=.DilO X=.HiO X= J!50 X= .350 X= A!.lO X= .550 X=.l350 ~ 0 ::r::: E-t 0 ........ ~ 0 ""'::r::: co E-t ,__. 0 ~ 0 co 0 0 "<:ji 0 I 1.0 10.0 100.0 Q2 (GeV) 2 Figure 6.,4. Magnitude or Smearing corrections. 1000.0 81 6.5. Corrections n1 is independent or ni and n~. However, ni is binomially split up into events that remain in the ith bin and those that leave the bin. Thus the error on nUni is the error on an e.fficiency, giving (6.21) By generating a large number or events this error is kept at a fraction of the statistical error but is nevertheless propagated into the statistical error on the structure functions. Kinematic, fiducial and acceptance cuts applied during the calculation of smearing corrections are identical to those used on the data. Corrections for large muon energy depositions within the hadron shower are simultaneously made by randomly simulating such processes utilizing the data collected outside the shower (see end of §5.1). ix. Bin centr~ corrections Within every :r and Q2 bin, the flux is not necessarily uniform, and the structure functions are not linear either. Further, parts of the bin may lie outside the kinematicaiJy allowed region after cuts. Therefore, the evaluated value or a structure function is only guaranteed to be the value somewhere in the bin. To obtain a value at the bin centre, say F 20 for F 2 , we must calculate the above effects using a fit. The integral within a bin, (6.22) The subscript zero refers to the bin centre. The derivatives are evaluated using the fit; the integrals Of (:r- Xo), (loglO Q2 -loglO Q~), (:r- Xo)2 , (:r- Xo)(loglO Q 2 -loglO Q~) and (log 10 Q 2 - log10 Q~)2 are evaluated using the correct flux and kinematic cuts. Thus we find Io = J dV{ · · · }F:w, where F2o is the unknown to be determined. As shown in figure 6.5, bin centre corrections are usually small, except in edge bins where cuts imply larger corrections. x. Isoscalar correction Easily the most straightforward of corrections, this one is also small ($ 5%). It is the ±(1- 2/)((l~y)2)(u~- dn term in (6.1). Clearly it goes to zero i f f = 1/2 ( < Z >= <A> /2) or if =de. For our target, f is close to 1/2 (f= 0.466). The models we use for corrections are explained in appendix F, where the difference between and de is described. The correction is plotted in figure 6.6. ue ue xi. Radiative corrections There are electromagnetic corrections to the basic weak interaction charged current process. 82 8. Extraction or Structure Functions ~ ..... F2 Bin centre X () u ~ + ....... !( <I• ~ ~ ~.: 0 ~ 0 ..... E-< !:::) 0 0 0 X=.01G X=.0-15 X=.OBO X=.150 X=.2JO X=.3:30 X=.450 X=.550 X=.650 ..... ::r: E-< !€_ 0 co ::r: E-< 0 ...... ~ 0 cc 0 0 "<:!' 0 1.0 10.0 100.0 Q2 (GeV) 1000.0 2 co ....... :x:F'3 Bin centre X <> ~ ....... D + .)( ~ ~ ID 0 N !:::) 0 0 !€_ 0 D E-< ' ..... v '"'0 X= .015 X=.O!l5 X=.080 X=.150 X=.250 X=.350 X=.450 X=.550 X=.650 ....... ::r: E-i ~ ~ ...... 0 ~ ~ 0 cc 0 0 "<jl 0 ~~;---l----1--~~__,___._,~j 1.0 10.0 100.0 Figure 6.5. Magnitude or bin centre corrections. 1000.0 83 6.5. Corrections ~ ,..... Isoscwar I I Fn :< 0 u ~ ,..... + !( ~ ~ 0 0 X=. 015 X=.01G X=.OBO X=.150 X=.~5 0 0 :-: ~ . :.1!5 0 X=A!)U 0 X= .G50 x~.t>s o C\2 ..--. - E--< 0 0 ~ ::r:: ..--. E--< ~ ::r:: co 0 E--< 1-i f- - co f- - 0 ~ 0 0 0 ~ 0 1.0 I I 10. 0 100.0 1000.0 co ,..... Jd'a Isoscwar :< 0 D ~ ,..... ~ ~ 0 0 ... C\2 + ){ a. le ;-;:: 0 X=.0 15 X=.04G X=.OBO X=.1GO X=.250 X=.3 5 0 X= A 5 0 X=.550 X=. 650 ..-t E--< 0 0 0 ::r:: ..--. E--< ~ ::r:: co 0 E--< 0 1-i ~ 0 CD 0 0 -::jt 0 ~----~--~~~~~~------~~--~~~~~~!~----~--~~~-~ 1.0 10. 0 100.0 Figure 6. 6. Magnitude of the isoscalar correction. 1000.0 84 8. Extraction or Structure Functions y Figure 6. 7. Diagrams ror radiative corrections. These involve photon bremsstrahlung from the muon and hadron legs or the diagram, selrenergy terms and interference terms which could include the '\V-boson. Some or these diagrams are shown in figure 6. 7. Radiative corrections are most important for the lightest particles (here muons), since they tend to go like In Q 2 fm 2 • or course a rigorous discussionC32.33 ) must include all diagrams and calculational results must be gauge-invariant. Unknown quark masses and unknown details or the hadron leg are potential problems. Howeyer, as de Rujula et al.C 33) show, most or the corrections are quark-mass independent. They show that to leading log order calculations may be perrormed because the lepton leg factorizes. The observed cross-section C111 &s, can be written in terms or the 'bare' cross-section aa, as Uoh = a Q2 a Q2 a aa + -ln-F(aa)+ - ln-H + -G 21T m~ 21T M2 21r (6.23) The lepton 'log', hadron 'log' and remaining terms are clearly separated. G is small, and even H may be neglected. F(aa) is a calculable function. 85 6.5. Corrections ~ r-----~--r--r-ror~~------r-~.-~,-~rrrr-----,--~~r-~OT~~ Radiative Fe ...-1 :< <> a + u •I• ., '"' o X=.OHi X=.OII(; X=.D80 X= .150 X=.l.lliO X=.3ti() X=.J!t)U X=.G5tl X=.!lliO 0 co 0 0 ..q< 0 1.0 10.0 1000.0 100.0 co ....... xF'a Radiative X o ..q. D ...-1 ~ ~ 0 D X=.OOO + X=.loO X ol! X=.l!50 X=.350 ~ X= .550 ~ o C\2 X=.01G X=.015 X= .450 X= .650 ...-1 b ~ 0 0 ::r:: ....... b ~ ::r:: 0 ~ b ,__. 0 ~ 0 co 0 0 ..q< 0 1.0 10. 0 100.0 Figure 6.8. Magnitude or radiative corrections. 1000.0 86 6. Extra.ction of Structure Functions The end result is dCT0 &. = daB dx dy dx dy + .!!__In s(1- y + xy)2 x 211" m; 1dz 1 + z {y8(z- Zmin)[ dCTB I - -]- dCTB } + 0( .!!__) { lo 1- z z(y + z - 1) dx dy x - x dx dy 211" (6.24} 2 y=y Here, xy z+y-1 _y=-:.......:;. z+y-1 __ - X = --:---~-:- (6.25) z Radiative corrections are large(-.. 10%) at very low and very high x, also at very low y. They are plotted in figure 6.8. Note that our Ehad > 10 GeV cut effectively excludes the very low y region. xii. Strange sea correction This is the (1- (i- y)2 )sN term in (6.1} above. They-dependence is exactly the same as for xF3 , but the sign remains positive even for antineutrinos. Hence this correction has virtually no effect on xF3 • Assuming that sN is some fraction of q, which can be iteratively well extracted, the only uncertainty is in the fraction. Dimuon data(34•35l suggest that sN must be half of u or d, a value which we used. This correction is obviously significant only at low x, since q -.. (1 - x )8 · 5 • It is shown in figure 6.9. xlil. Charm-mass correction A correction arises due to the heavy mass of the charm quark. Because of its mass, charm is not produced at low energies and production sets i~ as described in reference 24. Parton model kinematics prescribes a change from the variable X toe= x+m~ f2mpEh . Whenever a light quark changes into a heavy one, the differential cross-section remains the same if Q 2 >m~, except for the replacement everywhere of Q 2 by Q2 m'i£, where fflH is the mass of the heavy quark. Also, the structure function scales in where e= X+ m1£ f2mpEy is now the fraction of the nucleon momentum carried by the struck quark. These changes are included in the differential cross-section expression of appendix 3. The resulting effect is depicted in figure 6.10. + e, xiv. Fermi motion correction If nucleons are bound in the nucleus, but freely move within it, and they resemble a Fermi gas, we can calculate(36 •37l the effect of this motion on structure functions and total cross-sections. For example, for an average momentum < p2 >= (250MeV)2 , a number close to that for nucleons in iron nuclei, it is easy to show that s can increase by -.. 3%. Detailed calculations(37 l assume a fiat Fermi-gas-like momentum distribution from zero to the limit, say Kp, and then a tail that falls off as 1/p4 • Corrections to both F2 and xFa have thus been calculated. However the assumption of free nucleon motion is not valid for iron nuclei. Deuterons have a much smaller binding energy (-.. 81 6 ..5. Corrections ~ r-----~~~T-~~~~-----r--~-r-r~rTTT----~--~~~~-r~ F2 Strange sea X <> c + x a. X ;a o X=.015 X= .045 X=.OBO X=.160 X=.260 X=.350 X=.450 X=.660 X=.C60 0 co 0 0 0 """ 1.0 1000.0 100.0 10.0 co ...-i s Strange sea X <> c ~ + ...-i x e X >:e o t\l X= .015 X=.045 X=.OOO X=.160 X= .25 D X=.360 X=.450 X= .660 X=.650 ...-i 0 ..... ~ ~ I fl 1:1 • • II • 0 0""" 1.0 10.0 100.0 Figure. 6.9. Magnitude or the strange sea correction. 1000.0 e. Extraction of Structure Functions 88 F2 Charm. mass X <> c + x 6o X ;z o X= .016 X=.045 X= .OBO X=.lfiO X=.260 X= .360 X= .460 X=.650 X= ..B6D 0 co c) 0 ..q. c) 1.0 10.0 100.0 1000.0 ~ ..-i xFs Charm. mass X <> c ~ + ..-1 )( ~ ~ 0 t..) .. ~ 6o X ;:l 0 ..-1 X=.015 X= .045 X= .DBO X=.16D X= .260 X=.350 X= .460 X=.660 X= .85D E-< :::::> 0 0 ::X:: ..-1 E-< ~ ::X:: co 0 E-< 0 ~ 0 co c) 0 '<!' c) 1.0 10.0 100.0 Q2 (GeV) 2 Figure 6.10. Magnitude of the charm mass correction. 1000.0 89 6.5. Corrections MUON TRIG EFFICIENCYI I :.: .. - :I X :a: I 1 I I I I I I I I ±; X Cl) ........ .0 (tj H Cl) :> 0 H !=: --PI UJ 'dl ro HI 0Ell 0 - .00 100.0 I I ~ ........ ........ I •.-1 I 01 I II I I C\l 0 0 .......... ;:::s 51 0 10 - () Cl) ::tl I CDI 200. 0 J (tj () I H ...p Q.) s t::l !=: 0 •.-1 tlD Cl) H ~ t::l sD ~ ~ ....._... 0 Cl) a I 300.0 400.0 eJ.L (mrad) PEN TRIG EFFICIENCY 10 ~ r------~-----~~~~~~~=~~~~=r~~~~---------~------· I I ..,.. :0: ... z .... ..,.. I I ...p ;:::s 0 0 P::l 1--i () > a 0 Cl) ~ 0 10 0 t=l r:x::l t=l r 0 0 X ~ u :a: ~ I X I .00 II .....:1 % 5 .0 - ....-i ~ X ' :a: r:x::l I 10.0 15.0 EHAD (GeV) Figure 6.11 Efficiency or the muon trigger as a function of the muon angle and of the penetration trigger as a function or Ehad· 20.0 8. Extr&ction of Structure Functions PO 2.5 MeVfnucleon as opposed to ....... 8.7 MeVfnucleon in Fe) and the nucleons in them are expected to be closer to being free. Data from EMC and the SLAC-MIT experiments show that the ratio or cross-sections for iron and deuterium is markedly different from predictions (fig. 6.12, refs. 38,39). Aluminum data show virtually the same x dependence as iron data. Experiments have recently been performed{SO} to study the A-dependence or lepton-nucleon cross-sections and better theoretical understanding should result. For our purposes we shall be satisfied with the observation that the Q 2 dependence in both cases is minimal (fairly flat in Q 2 ) and therefore should not affect QCD stuclies. At this point in time it does not seem wise therefore, to make Fermi motion corrections and we have not made them. §6.6 Systematic Errors Estimation of systematic errors is limited by two important factors. One is the usual obstacle that systematic errors are only estimated for effects not completely understood. The other is the tendency or the statisitical errors to creep into the systematic errors if the data are used for their estimation. We try to guard against the latter as far as is possible. Some or the topics listed below are not necessarily 'errors', but are included as this is a convenient place to group them with the rest. The numbers in tables H .6 and H.7 (appendix H) are the errors in individual x and Q 2 bins with no attempt to remove correlations i.e., they are the 'diagonal terms'. i. Ehad shift From the hadron beam data used for Ehcld calibration we estimate a 0.41% possible error on Ehad· Along with a 0.25% error that could come from map corrections, a total error of 0.5% is estimated. Comparisons or the predicted neutrino energy as a function or radius at Lab E with the measured energy in y-bins indicates that the systematic error on both Ehad and E10 is less than or equal to 1% . " This 1% is propagated into the structure functions by evaluating them with and without a 1% shift in Ehad in the integrations. Instead of using the data for the number of neutrinos in a bin we use an integration or the 'Buras-Gaemers' model to predict these numbers, thereby eliminating the possibility of statistics dominating the result. We shall call this the 'unlimited statistics' technique for future reference. As can be seen in tables H.6 and H.7, this error is extremely small except at the bigbest x-values where it still is negligible compared to the statistical error. ii. E 11 shift The magnetic field in our toroids was measured at 3 different radii (10 in , 30 in , 50 in ) with a Hall probe. We obtained J B · dl using these data as input points and a computer program and estimate the uncertainty to be around 1% . Another experiment {E595)<78•70) used the same toroids for muon momentum measurement. They had a muon beam (with momentum known from bending magnets) go through the toroids and the mean ratio or the two momentum measurements at different momentum settings (between 100 and 278 GeV/c) was different from 1 by (0.7 ± 0.6)% . Taken together with the estimate in (i) 6.6. Systematic Enors SII 5 ~--~----~-----r----~----~----.-~--~---.-.--~--~1 <> EMC x Rochester-Mrr-SLAC "<I! .-1 l I I 0 CD 0 .00 0.20 0.60 0.40 0.80 1.0 X Figure 6.12 Confrontation or Fermi-motion calculations with experiment. I I CCFRR cross-section measurement (this expt.) -G~rrt-[~f~l!!~iiNeu~os ~ - Antineutrinos ---- - ,...:340 0 0 .00 I I 100.0 200.0 Energy (GeV) Figure 6.19 E616 cross-section slope as a function or energy. 300.0 92 6. Extraction of Structure Functions ~ ....-! -> Q) (.!) ~ I I I QCD Prediction (A=.l GeV) lO z.- 0 s Neutrinos - - 0 co 0 - lO t') I 0 0 ..-t ............ ~ '"J'lD t\l . 1b 0 ~ ~ - ------ Antineutrinos ------ ------- 0 ~ .00 I I I I 50.0 100.0 150.0 200.0 250.0 Energy (GeV) 0 ....-4 I I I Neutri:nos 0 co 0 - / / / 0 ~ 0 Qj .. / Fraction of cross-section from measured regions I - I -+-> 0 - Antineutrinoa / co 1- 0 ...... / -- -- --- 0 ~ - - 0 - - ~ 0 t\l 0 0 ~ .00 I I I 50.0 100.0 150.0 200.0 Energy (GeV) Figure 6.14 A QCD prediction (A= 0.1 GeV) for the cross--section slope; fraction of the prediction that comes from regions measured by this experiment. 250.0 6.6. System&tic Errors above, we conclude that there is possibly a 1% error on Ep and study its effect as for En ad above (i.e., using 'unlimited statistics'). Again, the error is very small and gets larger at high x where it is still smaller than the statistical error. Notice that a simple Taylor expansion of the structure functions may also be performed to obtain this and the previous error (x and Q2 are expanded in E,., enabling an expansion of the structure functions in E,). We have checked that both procedures lead to the same nlues for the errors. iii. Uncertainty in R There is no accurate measurement of R in our Q2 -range spanning all x. Hence we have extracted structure functions using three different values of R (see §6.5) and we take the standard deviation of the three values obtained in every bin as a measure of the error. or course, such an estimate is subject to statistical fluctuations and yet the effect is small except at low x for F2 as expected (see tables in appendix H). The structure functions for R = RQcD and R = 0.1 are listed in table 6.7 and G.1 respectively. lv. Extraction technique As mentioned in §6.4, structure functions have been extracted using two different techniques and with completely independent software. We fl.nd that the resulting integrals of F2 and xF3 differ by 2% and 0.3% respectively. If the difference between the two techniques weighted with statistics is plotted for all the available points, we fl.nd a mean shirt of 0.8% for F2 and 0.7% for xF3 with standard deviations of 2.1% and 5.6% respectively. Monte Carlo studies indicate an even smaller difference: about 0.5% for the difference in the integrals of F2 and xF3 . Clearly, statistics plays a limiting role here. It is emphasized that in all results we use a set of structure functions that is an average of the results of the two techniques. v. Model dependence The dependence of the extracted structure functions on the two models used for iterating the correction terms is found to be insignificant as evidenced in tables H.6 and H.7. Ti. SU(3) symmetry of the strange sea Analysis of our dimuon dataC35) indicates that the error on the 1/2 SU(3) assumption about the strange fraction of the sea is about 35% ()... = .5o±Jg). CDHS quote a smaller errorC34) i.e., >-. = .52±.09. Using our value and 'unlimited statistics' we fl.nd the errors on F2 and xF3 (tables H.6 and H.7). Because of the similar y-dependence of the strange sea correction term and the xF3 term we don't expect a sizeable error on xF3 • Even for F2 it is significant only at low x where it less tha n 2% . vii. Smearing corrections High statistics were used to calculate these corrections making them free of statistical errors. Errors on the resolutions of Ehad, E, and Bp and on the parameters contributed even less to the errors as was verified by ch anging them by one standard deviation. Therefore, no error is assumed from this source. viii. Angular dispersion or the beam The uncertainty in the angular dispersion of the secondary beam is discussed in the e. Extraction of Structure Functions 94 appenclix to reference 8. There it is pointed out that no obvious correlation exists among the angular dispersion errors at clifferent momentum settings. However it appears that, in the y-view at least, the measured angular clispersion is uniformly higher than the clispersion calculated by a Monte Carlo. Since this difference was used to estimate the error, we calculate the effect or the angular dispersion in two ways: assuming perfect correlation between all dispersions and assuming no correlations between them. The perfect correlation case is easy: we perform the integrals necessary for structure function extraction and use 'unlimited statistics' with and wihout all the clispersions ott by one standard deviation in the same direction. The uncorrelated case is handled by randomly throwing an angular dispersion at each energy setting using a Gaussian clistribution with the known mean and standard deviation and then evaluating the integrals. This procedure was repeated 26 times and the standard deviation or the resulting structure functions in each bin was taken to be the error. The two assumptions about the correlations bracket the error from this source; in summary we find that it is less than 2% and always much smaller than the statistical error. ix. Flux level uncertainties Both correlated and uncorrelated uncertainties in the fiux levels or neutrinos and antineutrinos contribute to systematic uncertainties in the values or F2 and xF3 . The correlated parts or the errors clirectly translate into errors on F2 and xF3 or the same magnitude. The error on F2 due to the uncorrelated errors on the v and i7 fiux levels lies between the two (since F2 is related to the sum or the differential cross-sections). However, the same is obviously not true or xF3 • As can be seen in table H.7, for x = 0.15 and x = 0.25 this error is roughly equal to the statistical error and therefore must be heeded in any higher statistics experiment. v Correlated 1.5% 2.0% v Uncorrelated Ion ch. calib. Connection to ion ch. calib. run to - changeover Ion ch. temperature Proton fraction E., error Veto deadtime Lab E livetime x2 losses Correlated 1.5% 2.0% + I% I% I% 0.5% 2.3% 0 .5% Uncorrelated 3% I% I% 0.5% 0.7% 0.5% Table 6. 6. Sources or systematic errors on fiuxes Most sources of Hux level uncertainties are described in reference 8. We list in table 6.6 all the errors assumed. The error due to temperature comes from the variation in fl.T. Results and Comparisons 95 the temperature or the Cerenkov counter (which is close to the ion chamber) during the running period. For the positive momentum settings, protons were a major fraction in the flux and the uncertainty in the proton fraction is added as a source of error. We have added a 0.5% error on the events lost due to a bad x2 (x2 /DF > 9) in the fluxes, a.s this is the most convenient place to include this error. Similarly, a 1% systematic error on Ev is also added here. The error on the Lab E livetime comes from a comparison or the live muon trigger rate at Lab E and the livetime proton fraction in the BCT. In summary, the overall error on the neutrino flux level is 3.9% , that on the antineutrino flux level is 4.3% and the correlation coefficient is 0.57. The uncertainties on the structure functions were calculated using the same technique as for the Ehad error, i.e., using simulated 'unlimited statistics'. x. Flat cross-section correction As described in §6.5, we correct for particle fraction errors using the constraint that the structure functions from different energy settings be the same. As for the uncorrelated angular dispersions we randomly throw the flux adjustment factors and use simulated unlimited statistics to estimate the errors shown in tables H.6 and H.7. These are, again, much smaller than the statistical errors. In a higher statistics experiment with precisely measured particle fractions, presumably there will be no need for such a correction. §6. 7 Results and Comparisons The final results for R=RQcD are displayed in figures 6.15 and 6.16 and in table 6.7. Appendix G contains the R = 0.1 results. In both cases a.s well as in all the analysis we use the average of the results from the two extraction techniques. The statistical errors on the structure functions are, or course, easily estimated from (6.12). We eliminate bins with a statistical error greater than 50% . Not unexpectedly, most of theie bins are also the ones with larger systematic errors. As is clear from the previous section, the magnitude of the various systematic errors is not the major limitation for us - it is the statistical errors. e. Extraction o! Structure Functions X .015 .045 .080 .150 .250 Q2 1.259 1.995 3.162 5.012 7.943 1.259 1.995 3.162 5.012 7.943 12.589 19.953 1.259 1.995 3.162 5.012 7.943 12.589 19.953 31.623 1.995 3.162 5.012 7.943 12.589 19.953 31.623 50.119 79.433 3.162 5.012 7.943 12.589 19.953 31.623 50.119 79.433 125.893 96 F2 T.28722 1.34295 1.53678 1.40248 1.58422 1.13355 1.35851 1.36325 1.54465 1.66153 1.53078 1.07091 1.24351 1.44496 1.48738 1.46410 1.54460 1.59056 1.57623 1.22757 1.18261 1.18013 1.28000 1.23488 1.23209 1.29159 1.26979 1.16089 0.88873 1.86830 1.02552 0.98540 0.93337 0.92345 0.95203 0.83999 0.74729 0.56844 AF2 .05054 .05850 .08391 .11610 .26887 .05026 .04696 .04445 .05858 .08052 .11347 .19650 .12804 .05454 .04635 .04393 .05137 .06787 .08978 .16582 .10895 .03634 .02828 .02545 .02815 .03614 .04625 .07534 .25122 .72375 .03873 .02647 .02412 .02640 .03339 .03383 .05295 .27936 xFa .16496 .36558 .30329 .43061 ' t:J.xF3 .05778 .05555 .07627 .10614 - - .44017 .61983 .60818 .50844 .69292 .64604 .63074 .14347 .08875 .06233 .07176 .09354 .11634 .18162 - - .61452 .76930 .66560 .65504 .78069 .77118 .66213 .18234 .10337 .06999 .06895 .08524 .09767 .16162 - - .73186 .68901 .87850 .85483 .79031 .84719 .79934 .70402 .13484 .07346 .04599 .04166 .04853 .05516 .07985 .23871 - - .62307 .79386 .79652 .76972 .79887 .74736 .65042 .16119 .07627 .04805 .04247 .04974 .04232 .05688 - - fJT 6.1. Results a.nd Comparisons X .350 .450 .550 .650 Q2 5.012 7.943 12.589 19.953 31.623 50.119 79.433 125.893 7.943 12.589 19.953 31.623 50.119 79.433 125.893 199.526 12.589 19.953 31.623 50.119 79.433 125.893 199.526 12.589 19.953 31.623 50.119 79.433 125.893 199.526 F2 !:1F2 0.88210 0.67603 0.64851 0.63772 0.62709 0.58736 0.59725 0.68446 0.49786 0.41907 0.39458 0.38217 0.34730 0.37452 0.27919 0.28974 0.24078 0.23066 0.20599 0.19837 0.19854 0.14983 0.11988 0.14994 0.12042 0.13803 0.11095 0.09837 0.09249 0.06768 .18935 .02730 .02250 .02381 .02743 .02812 .03720 .07624 .04114 .01975 .01866 .02039 .02115 .02649 .02780 .13247 .01712 .01550 .01434 .01807 .02046 .01905 .04019 .01762 .01044 .01507 .01453 .01613 .01350 .02048 xFa l:ixFa .58361 .55283 .46873 .51247 .56287 - .50031 .26579 .10834 .05959 .04743 .04784 .04116 .04531 .08257 - - .45220 .30636 .31889 .33956 .31687 .30267 .30152 .21779 .16832 .20865 .17112 .16888 .16956 .12987 .16284 ...17144 .12170 .07206 .05842 .09891 .07645 .06766 .04320 .03833 .03577 .03652 .03180 .13921 .07091 .04358 .02944 .03260 .03086 .02377 .04402 .07990 .03326 .03943 .03036 .02795 .01817 .02343 Table 6. 7. F2 and ::Fa with statistical errors for R=RQcD We shall defer the comparison or F2 from different experiments until its natural place among the tests or the quark parton model {§7.4). Comparison of the integrals of F2 and xFa with the total neutrino and antineutrino cross-sections is complicated by the fact that the former covers only a subset of the region in :: and Q2 covered by the latter. The total cross-section results include events out to larger angles and smaller hadron energies (§6.2 and appendix E). For this reason, we cannot make an exact comparison. Approximately, however, the results of the R=O and R=0.1 extractions can be used with the relation between the total cross-section slopes and the structure function integrals. In the scaling approximation and 98 6. Extr&etion or Structure Functions CCFRR 0 0 r---~---r~-r~~~----~--~~~-r~~--~---r~-r~rr~ 0 ...-i ...-.-m--'m.u...-1 1ffrJ litm = -:t "' 0 0 ...-i .. ~ • ~ • ~ e Ill ro ...-i ~ x=.045 (x12.5) • • a ~ ~ • • • • • • ~ 0 x=.015 (x25.o) • e f m = ~ Ill x=.OBO (x6.25) .. -G- I l!i • • • • • x=.150 (x3.75) 1 i x=.250 (x2.25) x=.350 (x1.50) ~ x=.450 0 x=.550 ...-i ci x=.650 0 ...-i 0 Ol~.-0--~~~~~1~0~.-0---L~~~~l~O~OW.-0--~~_J~~lO~OLWO.O Figure 6.15. The structure function F2(x, Q 2 ) for R=RQcD 99 6.1. Results a.nd Comparisons CCFRR 0 .... or----.~~rT~~~--~--r-~-rrrn----.--.-.- ~ 0· ........ x=.045 (x30.0) 0 0 ........ 1.--~----'aT--'Ifl--~-!t~(j! ir-~:-m:--R:i---l:lL__~ x=.150 (x7.50) I x=.250 {x5.00) ~ ! ~ • i fIi ; : : I ~ 0 x=.OBO {x15.0) x=.350 {x3.00) T x~.450 (x2.) x=.550 (x2.) ! ........ 0 x=.650 0 .....-1 0 Ol~.-0--~~~~~l~OW.-0--~~~~~1~0~0~.-0--~~~~l~O~OWUO.O Figure 6.16. The structure function zF3 (x, Q 2 ) for R=RQcD 100 6. Extraction or Structure Functions neglecting the effects or a non-zero charm quark mass, (6.26) where, (6.27) These approximate relations serve as a useful check on the internal consistency of data sets. For R=0.1 and f(s s) dxf f(q q) dx = .05, cl = 1. This corresponds to the assumption of a 1/2 SU(3) symmetric sea. In table 6.8 we make a comparison or the various available high statistics data sets. Different experiments make different assumptions about R as listed. To facilitate comparisons we have homogenized the data sets by making appropriate corrections (very small) in each case to obtain data with the following uniform assumptions in all cases: + + (i) A 1/2 SU(3) symmetric strange sea. (ii) A mass or 80 GeVfc2 for theW boson. (iii) Complete coverage or the 0 < x < 1 region. {iv) A zero mass for the charmed quark. .. All the data sets seem internally consistent, within errors, except for the CDHS data set where the integrals or the structure functions, especially the integral or F 2 , are not compatible with the prediction from the cross-sections. With this in mind, we now compare the xF3 results for the different data sets. As is clear from table 6.8, there are differences in the overall levels of xF3 from the four experiments, with our data and the HPWF data having the largest integrals followed by CHARM (12.3% ) lower and then CDHS (16.3% lower). The CHARM level difference is covered by systematic errors, and the CDHS difference is 1.4u away from zero, still plausible. If we take the prediction for J xF3 from the cross-sections as the best measure of J xF3 , then all the experiments are consistent with each other within errors. 101 6.7. Results and Comparisons q.,jE av/E R J F2 predicted from cross-sections J :r;F3 predicted from cross-sections J F2 from data (statistical errors only) J xFa from data (statistical errors only) CCFRR .669±.024 .340±.020 0. .466±.015 CCFRR .669±.024 .340±.020 0.1 .478±.015 CDHS .62±.022 .30±.013 0.1 .436±.012 CHARM .604±.032 .301±.018 0. .418±.017 HPWF .63±.02 .30±.01 0. .430±.010 .312±.030 .312±.030 .303±.024 .287±.035 .313±.021 .474±.003 .482±.003 .402±.002 .412±.006 .458±.003 .328±.005 .326±.005 .273±.003 .285±.012 .322±.005 Table 6.8. Comparison of integrals of data sets Apart from level differences, we can compare the x and Q2 behaviour of xF3 • First we make linear fits in log Q 2 to data in a given x bin (since scaling violations are expected to depend on log Q2 ). {6.28) 10 GeV2 is chosen as it is roughly the central point in Jog Q2 for the data. To make the comparison or the x-behaviour or the structure functions, we then fit a form to our xF3 data at Q2 = 10GeV2 : {6.29) The resulting fit parameters are A= 3.49±.61 a = .511±.069 fJ = 2.96±.24 (6.30) with a x2 /DF = 1.05. We then divide the values or xFa at a given X for a given data set by the value or this fit at that x. The resulting ratios are plotted in fig. 6.17. Systematic errors are not included on the points since we are ignoring overall level differences; they are shown at the right with the dashed lines indicating the ratios expected from the integrals of xF3 averaged over Q2 • Clearly, the HPWF data show a marked rise with x. Although this is in the right direction expected from the EMC effect<38 ) (since HPWF have a mixture of iron and scintillator as a target) we do not expect that effect to be quite as large as the observed variation. However, no definite quantitative statement can be made until the EMC effect has been studied in materials other than deuterium, aluminum and iron, the only substances for which data are presently available. 102 G. Extraction or Structure Functions o xF3 Comparison with CCFRR ci r---~----~r-~--~-=---~r---~-----~r-~~-----~~--~----~ Q2 =10 GeV2 X I ! I CDHS ~CHARM c HPWF - -I---I- - - - - - =---- :- "' frP=r=r = = I==== r= = = ~ -_:_ - - ~- - !_ o! I ~~ - 0 0 .00 I I I I 0.20 0 .40 0.60 0.80 1.0 X Figure 6.17 Ratios or zF3 from other experiments with our results at Q2 = 10 Ge¥-Z 2 Q Fractional Slope Comparison 0 ..... I 0 ~ (Xi xF3 l rrI-+-- +-± 0 0 - ............ ~ +- .. • CCFRR x CDHS <>CHARM c HPWF I ---- -- ---- T t I~ 1 l i ,..Cl 0 1.0 0 r I I ~ ..... I I I .00 0 .20 0.40 O.oO I I O.BO X Figure 6.18 Fractional slopes w.r.t . log10 Q 2 or zF3 from various experiments. 1.0 103 II.T. Results a.nd Comparisons The CHARM ratios suffer from poor statistics and are consistent with being fiat. The CDHS data show a slight rise with x, but not as marked as in the F2 case (§7.4). The ratios at high x, or course, track those or F2. We conclude that there is a rise in the ratio as X increases but keeping in mind the additional systematic errors not associated with level differences, the rise appears to be smaller than for F2. The fractional slope (b/a- eqn. 6.28) or xF3 in various bins is shown in figure 6.18. Again, the errors displayed are only statistical and again the errors on the CHARM data are too large to warrant any firm conclusions about that data set. Clearly, the HPWF data exhibit the largest scaling violation. That data set spans a range with slightly lower Q2 values (.7 GeV2 < Q2 < 125 GeV2), where non-perturbative effects are expected to be larger, but this does not seem enough to explain the observed magnitude, especially at high x where Q 2 values are higher. The CCFRR and CDHS data sets both show a smaller Q2 dependence and are clearly consistent with each other, except at x = .045, with CDHS showing a slightly larger scaling violation. A more rigorous QCD analysis or the scaling violations in our data are presented in §7.5. More graphic displays of the differences between the data sets are shown in figures 6.19, 6.20 and 6.21. In these plots, the dashed lines are the fits to our data and the solid lines are these same fits with the overall level adjusted so as to best fit the data set displayed. Thus the difference in levels or the two lines indicates the degree or x-dependent differences and the goodness or fit or the solid line indicates the level or agreement between the Q2 behaviours. There are insufficient data from the CHARM group at all their x values. The observations above about CDHS and HP\VF remain the same. We conclude that xF3 from the CDHS group is in agreement with our values for xF3 , but it will be interesting to see if this holds true when higher statistics data are available. .. 104 6. Extraction or Structure Functions 0 0 0 CDHS r----.--.--..-TO~.----.---,-.-..,.,.-----r--r-o-.o-rrn ....... x=.Ol5 (:x:lOO.D) 0 0 ....... :x:=.DBO (x8.00) x = .350 (:x:l.50) x=.450 x=.660 0 ....... 0 x=.65D 0 ....... 0 Ol~.-o--~~~~~1~0~.0----~~~~1~0~0~.0----L-~~~1~0~0LWO.O Figure 6.19 xF3 from the CDHS group with fits to our data (dashed Jines) and the same fits adjusted to the level of the CDHS data (solid lines) in our Q2 range. 105 6.1. Results and Comparisons ------- rr-----,i:tr-----11 - x=.200 (x3.00) IT -!- - x~.325 (x1.5) --- -- -- _,_ x=.525 (x1.5) 01.0 10.0 100.0 1000.0 Figure 6.20 xF3 from the CHARM group with fits to our data (dashed Jines) and the same fits adjusted to the level of the CHARM data (solid lines) in our Q 2 range. 6. Extraction or Structure Functions -0 0 0 ...--! 106 rr ! HPWF ~ l 0 0 ...--! x~.025 (x200.0) 1: 1 ! ! =:! ~ :x:=.075 (:x:40.0) :m :x:=.150 (:x:15.0) m m .m m -~--~--F • x=.25o (x7.5D) ~ :m:m:m =----.--:::::l-IL::::--::~=--m ! ~ - j - !. - ~ ~ :x:= .350 ( x4.50) ~ .:x:=.450 (:x:3.0) ! -~~ - - :x:=.550 (:x:2.) - ~ ! 0 -- ...--! 0 T T ~ __ x.:.650 0 ...--! 0 ~--~~~~~~~--~~~~--~~~~~~--~~~~~~ c0.10 1.0 10.0 100.0 1000.0 Figure 6.21 xFa from the HPWF group with fits to our data (dashed lines) and the same fits adjusted to the level or the HPWF data (solid lines) in our Q2 range. ChapteT 7 Physics Analysis and Conclusions In the first three sections or this chapter we present a brier theoretical outline or the physics or charged current neutrino interactions. Comparisons of the data with theory begin with §7.4. §7.1 Weak Interactions In 1934 Fermi proposed( 48 •49) a current-current interaction theory of weak interactions which together with universality, explained a whole range or weak processes. In this theory the weak interactions occured at a point, with one current interacting with the other directly, and not through a mediating quantum. Later, a couple of modifications had to be made. The intermediate vector bosons were introduced to mediate the current-current interaction. Since they were not observed in decays and high-energy neutrino interactions, they were assigned a high mass, ~ 5 GeV. Recent neutrino scattering experiments pushed this limit up to ""30 GeV. The massive nature of these bosons implies a short-distance weak interaction, as opposed to QED. The predktion and discovery of parity violation<50•51 ) further changed the theory. Only left-handed neutrinos and right-handed antineutrinos were allowed, and leptonic weak currents could only have a V-A nature. Subsequently, universality was extended to the non-leptonic (quark) weak currents by Cabibbo.C52 ) The weak charged current changed u-quarks into a state that is a mixture of d-quarks and s-quarks: de= d cosec+ s sinec Sc = -d sinec + s cosec This extension seemed to imply the presence or weak neutral currents, but the experimental absence of strangeness-changing neutral currents discouraged much progress. By this time Glashowl53) (1961), and then Weinbergl54) (1967) and Salam(55) (1968) had proposed a model or weak interactions. Glashow, lliopoulos and Maianil 56) proposed a mechanism in 1970 which 108 T. Physics A.n&!ysis &nd Conclusions completed the &c doublet by adding a fourth-quark, called "charm". This also explained the very small K1-.p+p- decay rate, expected to be or the same order as K+-.p+v11 • In ract this was used to predict me to be between 1 and 3 GeV. The Fermi t~eory involving point interactions could not handle higher-order corrections to processes like v 11 e--.v11 e-: in every order infi.nities appear and we need an infinite number or parameters to correct for them. For this reason the Fermi theory is non-renormalizable and in spite or its dimensionless coupling constant, so is the intermediate vector boson model. In 1971 't Hooft(ST) showed that gauge theories with massive fields introduced by spontaneous symmetry breaking, as in the standard model, were renormalizable. In the early 1970s neutral currents were discovered in high energy neutrino scattering experiments ( 1973, 1974) and by November 1974 the charmed quark had been inferred to exist. Hence by this time the standard model had been accepted. Today we believe that the w± and Z 0 have been observed<45 •46l leaving only the Higgs scalar to complete the triumph or the GSW theory. In this now accepted theory, the charged current couplings are still strictly V-A: Joe= IS 7l,.., (1- 'Ys) I+ U .,.., (1- 'Ys)M· ·L . "" ~ l IP 2 l IP 2 ,, 3 (7.1) l={e,p,r} u = (tt, c, t) (7.2) L = (d,s,b) (7.3) and M;i connects them (the Kobayashi-Maskawa< 58) matrix). For example the familiar Cabibbo mixing angle is now Mud = cos 01 = cos Oc Experimentally, cos0 0 = 0.9737±0.0025,47>. (7.4) (7.5) The neutral current couplings are no longer strictly V- A, they are in general some linear combination or V and A which depends on sin2 Ow, where Ow is the Glashow-Weinberg angle. sin2 Ow R:j 0.23 (7.6) The standard model is based on the observation that only left-handed components of charged current lepton waverunctions interact as weak charged currents, their right-handed counterparts do not. Of course, only left-handed neutrinos exist. This suggests consideration of the left-handed lepton and its neutrino as a doublet under some symmetry and the right-handed lepton as a singlet. In the quark sector, the left handed components or +2/3 quarks and their K-M rotated -1/3 counterparts form weak doublets, the right-handed parts all falling into singlets. This symmetry leads to a SU(2) gauge invariance and associated is a weak isospin. For example, the neutrino has J!fk. = + 1/2 and the lett handed lepton has J!fk. = -1/2. An 1.2. The Parton Model 109 SU(2) gauge theory has only 3 gauge fields however, and associating this triplet with w± and Z 0 gives the same V-A nature for both charged and neutral currents. Consequently, a weak hypercharge is introduced with a corresponding U(1) group. In order that thew± and Z 0 are massive and the photon remains massless the mechanism or spontaneous symmetry breaking is invoked. In the resulting model, the weak charge g turns out to be related to e by gsinDw = Also, the masses or the w± and e (7.7) zo come out as mw = e 2 sin Dw 1 r-;;:- (7.8) R:S 80GeV y.f2GF mw and mz = - - R:l 90GeV cos Ow (7.9) The mw along with a propagator term for the w± contribute a factor Q2 /m~)2 that modifies the current-current expression for charged current neutrino 1/(1 scattering. This fact and the V-A nature or charged current interactions are the most important features or the standard model relevant to us. + §~ .2 The Parton Model We now turn from the nature of the interaction vertices to the object being probed viz., the nucleon. Traditionally, the parton model is introduced via charged lepton scattering and we do the same. Consider the scattering diagram in fig. 7.1. The cross-section for this process is given by ' (7.10) The matrix element squared is 2 u(k' s')l"u(ks)u(ks)ivu(k' s')~ 4 <p,spll"IX><XIJ"!p,sp > (2rr) 6 (p+q- p') 4 4 (7.11) This can be separated into a lepton tensor L"" which is evaluated to be 2 Ll'" = 2[k~kv + k~kl' + q2 g~ow] (7 .12) and a hadron tensor W~'". Since we do not know the final state X, we use gauge invariance, Lorentz invariance and the symmetry of L,..v to write W"" in t erms of only 2 structure functions, 1 . Physics Analysis a.nd Conclusions 110 Figure 7.1. Diagram for electron-nucleon scattering. ' W1 and W2 which are functions of Q 2 (=-!(!)and v (= ko- k~): (7.13) This leads finally to (7.14) Bjorken argued<59) that the structure functions W1 and W2 are functions of x = Q2 f2mv alone in the limit in which Q2 -+ oo and v-+oo. This is the celebrated phenomenon or scaling in which the structure functions F 2 (= vW2 ) and 2xF1 (= 2mxWt) are functions of x alone and do not depend on Q2 • Feynman interpreted this<3 ) as evidence that the nucleon is composed or point scatterers, called partons. Scattering of leptons from these partons then imposes an elastic scattering condition, Q2 = 2m 9 v where m 9 is the mass or the parton. 111 T.2. The Parton Model It is instructive to work out the kinematics of parton scattering. If the parton carries a fraction { of the initial nucleon momentum p then {7.15) where m1 is the mass or the final state quark, giving (7.16) Clearly, {is identified-with x above (we neglect all masses.) By now, it has become natural to identify these partons with quarks, the constituents or a nucleon. The parton model or the nucleon assumes that the quarks are confined within the nucleon, but that this confining force acts over a sufficiently long time scale so it doesn't interfere with probe-parton scattering at high Q2 • The identification of partons as quarks is further established by the discovery that the partons have spin-1/2. The essential advantage o( the quark-parton model is that the wl and w2 are no longer specific to this process but are related to the %-distributions of quarks and antiquarks in the nucleon and all processes involving nucleons see the same structure functions. A relevant example is the very similar case of neutrino scattering. Here, the exchanged virtual boson is a w+, not a photon. The hadron tensor W'"111 now contains 4 more structure functions. Three of these contribute as (mr /m!) and are neglected; the parity violating fourth must be included however. The differential cross-section is cf2a(v,V) dO dE' = 2 G'lE' ( 21r2 1 + 1 ) Q2 /mw 2 { .28 2Wl sm 2 28 + w2 cos 2Tw3 (E+E').28} mp Sin 2 (7.17) The charged lepton and neutrino scattering structure fu'nctions W1, W2 and W3 may be abandoned in favour or 3 other variables. One conventional choice is the cross sections Cor the three virtual boson polarizations (two transverse: and one longitudinal: L). In both the charged lepton and the neutrino case one obtains +, - (7.18) (7 .19) (7.20) where the constant k is different in the two cases. In the electromagnetic case, a+ = a_ (parity conservation), and W3 = 0. We define the transverse cross-section as (7.21) T. Physics Anal:rsis and Conclusions 112 Collision -& After collision Figure 7.£ Collision or parton and photon in frame in which the 3-momentum or the parton is reversed. and Rby R=UL UT = W2(1 W1 + ~) _1 Q 2 (7.22) In terms of F2 and 2xF1 defined above (xF3 = vxW3 ), (7.23) For aL = O, this gives R = 0. Including transverse momenta or the partons one gets (7.24) m 2 x2 + Pl may be interpreted as a transverse mass squared = Pl· R becomes(63) 4<p1_> (7.25) where the average is over the parton density. We can examine what happens toR given that quarks all have spin 1/2. Both vector and axial vector currents conserve the helicity or massless particles. Now consider a frame in which the boson and parton are collinear and the 3-momentum or the parton is reversed, fig. 7.2. For the boson to be absorbed, and helicity conserved, only (anti)quarks with spin 1/2 can absorb a transverse boson (helicity=±l) and only spin zero quarks can absorb longitudinal bosons (helicity=O). Experimental observations(60•61 •62) support UL = 0, implying spin 1/2 quarks. This can also be expressed as the Callan-Gross relation, which in the limit 4m 2 x 2 /Q2 < 7.2. The Pa.rton Model 113 Figure 7.3. Spin considerations in a neutrino-antiquark collision. 1 gives (7.26) The question orR in QCD will be discussed in the next section. The quark parton model makes additional basic predections. The y-distributions or the differential cross sections (or angular distributions in the c.m. frame) follow quite naturally in the massless model. From (B.ll), ' (7.27) Consider a neutrino-antiquark collision, as in figure 7.3. The initial state helicity and final state helicity are dictated by the V-A interaction; they are both -1. The angular distribution is governed by ld~ 1 (8)1 2 = (1 +cos o•)2 /4. Similarly, the helicities for a neutrino quark collision are both zero, and lead to a jdg0 (8)1 2 or isotropic distribution. Thus, dav - dy = k[q(x) + (1- y) q(x)] 2 (7.28) Here, q(x) and q(x) are the x distributions or quarks and antiquarks. Actually, a nucleon has both valence quarks and a "sea" of quark-antiquark pairs, and all these come in more than one fiavour. These have already been described in the introduction. Here we merely point out that using the y-distributions above, it is possible to proceed as in appendix C and heuristically derive the cross section. It is clear from (C.23) that the total cross section rises linearly with neutrino or antineutrino energy, apart from scaling violations expressed as a weak dependence 114 T. Physics Analysis and Conclusions or I F 2 dx and I xF3 dz on Q 2 • The ::-distributions can, in the quark-parton model, be described as obeying sum rules, for example 1 fo F3(z)dz=3 the GLS sum rule (7.29) which merely expresses the fact that there are 3 valence quarks in a nucleon. The identification or zF3 with the valence quark distribution is clear from (7.20). Other sum rules which are similarly obvious include, for the proton, fol (u(z)- u(z)) dx = 2 (7.30) 1 fo (d(x)- d(x)) dx = 1 (7.31} 1 fo (s(x)- :S(x)) dx = 0 (7.32} Combinations of these give rise to the Gross-Llewellyn-Smith sum rule and the Adler sum rule. One important conclusion can be formed for the comparison of charged and neutral lepton-nucleon scattering. Since charged lepton scattering occurs via a photon, the parton distributions are multiplied by their charge squared, and F~- m. should be F2k. times the mean square charge in a nucleon: (7.33} What can be said about the x shapes or the parton distributions! As x tends to zero, one expects more and more quark-antiquark pairs to be produced, possibly by gluon bremsstrahlung followed by pair creation. Thus the sea dominates at z d 0, with the bremsstrahlung distribution dk/ E"' dxjx implying a number density"'.!. (7.34) X and a momentum distribution '"""' 1 i.e., approaching x = 0 as a constant. This is also explained by associating the sea with difl'ractive components (cr = 1; Pomeron exchange) and the valence quarks with non-difi'ractive components (cr R1 1/2; /, A2 exchanges). We then predict in addition that 1 (7.35) xF3""~x.-----.f"i as :t-+0 ,fX Quark counting rules<2g) and form factor arguments<3•26l suggest that F2(x)"' (1- x)3 and q(x)----(1-x)n (7.36) 5~n~7 (7.37) 115 7.3. Quantum Chromodynamics §7.3 Quantum Chromodynamics It is actually quite surprising that the parton model described in the previous section works so well; after all, it says nothing at all about the strong force between the quarks. The current candidate theory or strong interactions is QCD and as it turns out, deep inelastic scattering is one of the few processes where it makes definite predictions. There are many pointers to an additional degree or freedom in quarks, colour. If quarks did not possess this degree or freedom, the ratio R e+e- _e+ e--+ hadrons e+ e--+ p.+ p.- (7.38) would be a factor or 3 smaller than experimentally observed. Similarly, the rate for 1r0 -+'Y'Y can be calculated exactly; for no coloured quarks the rate is 1/3 smaller than observed. Baryon spectroscopy breaks down for tl. ++ (spin 3/2) unless an antisymmetric colour wavefunction is introduced since without it the wavefunction is symmetric under an interchange or any 2 u quarks. Other evidence for 3 colours includes the roughly 20% branching ratio for r- -+e-ilellr and r--+ p.-ilJj llr as opposed to the "' 33% expected without colour. QCD utilizes this extra degree of freedom and coloured quarks experience a strong force mediated by coloured gluons. Unlike SU(2)r., colour symmetry is exact, even for massive quarks. There are 8 gluons, all massless, and since they are themselves coloured (unlike QED where the photon is neutral), they couple to each other (fig. 1.2). Baryons and mesons are colour singlets. Evidence for gluons lies in gluon jets in e+ e- scattering and the fact that quarks carry only about half the momentum of nucleons- the other half is presumably carried by gluons. In QED, the coupling constant a is a function or Q 2 because vacuum polarization contributions modify even lowest order diagrams. o(Q 2 ) increases extremely slowly with Q 2 from the value ao at Q2 = 0, the dominant contributions giving, for Q 2 >m2 , " o(Q2 ) = 00 1- (oo/37r)lnQ2fm2 (7 39) · For example, at Q2 = 1000GeV 2 , a has changed by only about 1.7% The denominator in (7.39) becomes zero around Q2 ~ 10555 Gey2. In QED we can thus safely do all low energy calculations with a well defined Q 2 = 0 limit to start from. In QCD however, as(Q2 ) gets large as Q 2 -+0, and one must use the Q 2 -+oo region for perturbative theory. That is, QCD is an asymptotically free theory and at low Q2 confinement of quarks and gluons occurs. The unproven attribute of confinement explains why no free quarks or gluons are seen and asymptotic freedom is the reason for the applicability of QCD to deep inelastic scattering, where Q2 is presumably high enough. Since we do not have a definite mass-scale in the high Q 2 region to normalize to (unlike QED where Q 2 = 0 serves well), an arbitrary mass-scale must be introduced. Normally, this is exchanged for a parameter A and the dominant contributions to as(Q2 ) can be summed up to give 2 41r as(Q ) = f3o ln(Q2fA2) (7 .40) 116 T. Physics Analysis and Conclusions 2 flo= 11- -nf 3 'nf is the number or quark flavours. Since several excellent reviews of QCD exist< 6"" 68), we will not dwell on generalities any further and move on to perturbative QCD and its applications to deep inelastic scattering{ 68-T 3). Given that confinement is yet unproven, the absence or calculations to all orders in deep inelastic scattering and a lack or a clear point in Q2 beyond which perturbative QCD can be guaranteed to apply, why do we adopt this approach at all! Perhaps the most compelling reason is the closeness of the high Q 2 data to zeroth order QCD (quark-parton model) e.g., Re+e- is close to 3 I:e; and deep inelastic structure functions very nearly scale. Perturbative QCD also has been successful (albeit to varying degrees) in explaining jet production in e+ ecollisions and large momentum transfer processes. Perturbative QCD makes predictions for several high energy processes, but only prescribes their Q 2 dependence. The hope therefore is that parton distributions (q(x, Q 2 ), q(x, Q 2 ), G(x, Q 2 )) and their fragmentation functions be measured in a few processes and then be used to predict quantities in others. In other words, another check on the applicability of perturbative theory is the extraction of the same value or os(Q2 ) from different processes. Specializing to deep inelastic scattering, the processes involved to first order are shown in figure 7.4. The first process describes the creation or gluons and lower momentum quarks by bremsstrahlung. The second and third are pair creation of gluons (unique to non-Abelian theories) and q7j pairs. Consider the moments of a general structure function Y: (7 .41) Y may be F 2 or :r:F3 • In reality, all high energy processes are mixtures of high and low energy subprocesses making it hard to make predictions. In QCD we cannot as yet calculate all the Mn(Q 2 ), however, wonderfully we are able to separate the higb--Q 2 (perturbative) and nonperturbative parts of Mn(Q 2 ). This property is known as factorization and enables us to write M"(Q 2)= ~A~(JS2)c~( ~:, 92) (7.42) I The summation is over all the non-singlet, singlet and gluon pieces. J.l is the mass scale at which the theory is renormalized and g2 is related to os by (7 .43) In (7 .42) the A~(J.l 2 ) are non-perturbative and must be determined from experiment at some Q 2 = J.l2 • The perturbative parts C~ can be evaluated explicitly. For example, in the nonsinglet case we may write(60 ), to order g2 , 111 1.3. Quantum ChromodynamJcs q Figure 7. ..(.. First order QCD corrections to deep inelastic scattering. The coefficients dt;; 5 ,zr;;s and Bt;;5 a re all calculable and g is the effective coupling constant. The term 'singlet' refers to flavour. The sum of all quark distributions is a singlet distribution O:i(q, qi)); non-singlet structure functions, like xF3 , involve differences of quark distributions and the singlet contributions cancel. ' + These predictions can all be restated directly in density space. Before doing that we will mention that in leading order all parton densities are the same as in the naive quark-parton model. The ambiguities or renormalization scheme dependence only develop in next to leading order. There are many schemes in existence. The momentum subtraction scheme MOM, minimal subtraction MS, and modified minimal subtraction MS are the most common; we use MS throughout. (MOM seeks to minimize third order terms, while MS is closest to leading order, with a minimum or corrections.) One reason for using next to leading order calculations is the ambiguity in the leading order expression for as(Q 2 ): In general, 2 47r as(Q ) = f3o(lnQ 2jA2 +c) (7.45) For large Q2 this becomes 2 47r ( as(Q ) = f3o lnQ2fA2 1 - c lnQ2fA2 ) (7 .46) 118 1. Physics Analysis and Conclusions The second order expression in fact is (7.47) where f3t = 102- -n, 38 3 (7.48) The quark distributions at a given z receive contributions from bremsstrahlung of higher z quarks and pair production or qq pairs from gluons; gluon distributions from bremsstrahlung and g-+ gg diagrams. Let z denote the fraction of the parent's momentum carried by the resultant object. Then P119 , P99 , P, and P99 are the probability densities or "splitting functions" or the above processes. Defining qi(z, Q 2 ) as the density or a given flavour of quark (or antiquark) and g(z, Q2 ) as the gluon density, the leading order evolution equations are simply 2 2 dq,(z,Q ) =o(Q ) dy[ ·( Q2)P (~)+ ( Q2)P (~)] dIn Q2 2tr lz y q, y, qq y g y, qg Y (7.49) and dg(z,Q2) =o(Q2) fl dy[~qi(y,Q2)p (~)+g(y, Q2)P99 (~)] d lnQ2 2tr }., y ~ gq y y (7.50) t ' Utilizing the convention A®B =I: d: A(y)B(~) = B®A (7.51) the above equations (7.49) and (7.50) can be cast in matrix form The non-singlet equation is not coupled with the gluon density: (7.53) In leading order, (7.54) (7.55) (7.56) (7.57) 119 T.3. Quantum Chromodynamics The + definition implies a singularity at x = 1 which is removed by integration: 1 1 fo dx g(x)(!(x))+ = fo dx [g(x)- g(1)]/(x) 4 Cp=3· CA=3, nl T = - and 2 b=11CA-4T 1211" (7.58) (7.59) Interestingly, in the high x region (but not too close to 1 i.e., 1/(1 - x)> 1 and (as(Q 2 )/7r)ln(1/(1- x)) < 1) the valence quark equation is satisfied in the leading log order by a A(1- x)8 form, where(66 ) (7.60) In the same region, the gluon and sea densities fall by one and two extra powers of (1 faster than the non-singlet case. x) The ambiguities that arise in going from the leading log approximation to the nextto-leading order can be resolved in several ways. Most commonly, the quark number and momentum sum rules receive no corrections to all orders in as(Q 2 ) and the quark-densities are then process-dependent. \Ve may, for example, use an F2 which satisfies the Adler sum rule to all orders {this is motivated by the independent derivation of this sum rule from current algebra): (7.61) + In this definition F2 is still q q, but xF3 is (q- q) convoluted with an order a addition to a a-function. Of particular interest to us is the resulting GLS sum rule: (7.62) The most important result of a QCD analysis of our data is then the value of ALo and AMs• but all the preceding caveats and details should be borne in mind while interpreting these values. U the result agrees with all other experiments exploring the same region of Q 2 we may be past the higher twist and non-perturbative regions. One reason to settle on a value of A is the critical dependence of the proton lifetime on A in some grand unification models. The mass scale at which the falling o 5 (Q 2 ) meets the weak-electromagnetic a(Q 2 ) gets larger with decreasing A; since this mass-scale is the propagator mass for decays, {7.63) Any analysis leading to a value of A must correctly include non-perturbative and higher-twist effects. The former arise because of effects like primordial p.L (from quark T. Ph,ysics Analysis and Conclusions 120 confinement within a hadron's radius) which contributes,.... pj_ fQ 2 • Such effects are negligible at higher Q2 • Higher-twist effects rrom coherent phenomena ( diquark scattering, elastic scattering, resonance production etc.) add to (7.42) or (7.44) whole new series down by powers of Q2 . Higher twist moments are expected to increase with n (greater importance or higher-twist as x-+ 1). Consequently, structure functions are sometimes parametrized as F(x,Q 2) 2 [ ~ Fo(x,Q ) 1 + Q2 ( 1a- x) + Q4 ( lb- x)2 + ··· I (7.64) In any case, barring new calculational results, these terms are unknown. Some terms have been calculated, in the bag model for example, and have a negative sign leading to speculation that A might increase and not necessarily decrease with the inclusion or these terms. Finally, we examine QCD predictions regarding R. The longitudinal structure function FL is, to order as, (7.65) Since R is defined by R = FL/2xF11 R is expected to be small at large x and large at small x (from the integral). Also, apart rrom the parton model contributions to aL, we expect R to fall with Q2 like 1/ln Q2 . Integrating (7.65) over x one gets (7.66) where the bar indicates an average (or integral) over x. For example, assuming F 2 = G = 0.5 (at Q 2 = 10GeV2 , say), gives, for n1 = 4, R ~ .35as ' (7.67) §7.4 Tests of the Quark-parton Model 7.4.1 F2 Comparisons As mentioned in §7.2, F'!f."· and F2·m. are related by the simple relation 18 5 F'!f."· = -F2·m· (7.68) Actually, there is a correction term, which comes from the strange sea correction, leading to (7.69) T.4. Tests or the Qua.rlc-pa.rton Model 121 ~r-----~r-------~----~-------,------~~-----, - I I ~ - j - T -1- - ~- ~ :.ON 0 - i ~~..... ± o.o<s -r- I ! • ~ +-------_:- ~.ogs -r- r- - - - -!- - -!- - - - - - - - - - - - - - - 0.939 ± 0.056 ~ I ~IV) -- 0 - co 0 cCCFRR/EMC oCDHS/EMC 0 co 0 .00 I _[ 0 .25 0.50 X 0.75 Figure 7.5 F2 ratios with EMC. The numbers on the right are mean values of the ratios with systematic errors. We use this to convert EMC data<86) into a neutrino F2 ' and then compare that data set with CDHS data<81) and our data. Again, 've shall assume a 1/2 SU(3) sea and make corrections to the CDHS data set to include an 80 GeV W-boson propagator and slow rescaling with me = 1.5 GeV. Also, as in the xF3 case, we interpolate all data sets linearly in log Q2 to Q 2 = 10 GeV2. Since all three data sets involve an iron target, the comparison is independent of Fermi motion effects. Also, since EMC quote results for R=O. and R=0.2, we have used R=O.l in all three data sets. The ratio of our data to those of EMC and similar ratios for CDHS are shown in figure 7.5. On the average our points lie 9.5% higher than EMC but exhibit no x dependence. The combined systematic errors (3% for EMC and 5% for us) come close to explaining the level difference. A recent measurement<82 l of F~N indicates that the EMC data may be systematically low by as much as 4.7% . On the other hand, the CDHS ratios are not only 6.1% lower than 1, they also show a striking low-x fall off, i.e., are not explainable by overall level systematics. Note that the strange sea correction can at most be ,...., 12% at x = 0, with an error of,...., 2% using the CDHS value<34l for the strange fraction of the sea. The correction gets much smaller as x increases and q(x) domi nates q(x). T. Physics Analysis and Conclusions 122 The reader interested in greater details or F 2 comparisons is referred to ref. 10. 7.4.2 The Gross-Llewellyn-Smith Sum Ru]e We now examine the integral of F 3 (x), which is expected to be 3 in the quark-parton model. QCD predicts, in next to leading order, (7.62) The obvious thing to do is to sum up xF3 fx times the bin width for all x bins. The problem with this approach is illustrated in fig 7.6. or the total integral, equal to 3, about 1.5 comes from the region x < .06 of which about 0.5 is from x < .01. As mentioned before, our good angular resolution enables us to make a meaningful attempt to measure xFa below x = .06. However, as xF3 tends to zero, the statistical errors get large, leading to large errors for F 3 dx . Further, since most of the integral is from low x, it is also from low Q 2 (Q 2 < 6Ge"0!), thereby casting doubt on a QCD conclusion. J; 123 7.4. Tests of the Qua.rk-parton Model 0 t\l ~ '"d t') ~ .,..M ~ ~ ..... 0 0 0.60 LU 0.80 X Figure 7. 6. I: d:r F3 illustrates the concentration at low :r. 0 ~ CD ci t\l fx1F 3 dx :xFs 0 ~ ..... ~ 0 0 0 c:i 0 ~ ..... ~ 0 ~~~--~~~~~o!~.o ~1--~~~~~~~o k.~ 1----~~~~~~1.~.o 1 I 0.001 X Figure 7. 7 in fine :r bins at Q2 = I: :rF3 3 GeV2 with the fit (7.70). Also shown is F 3 d:r from the fit with points from the simple summation t echnique superimposed. 124 1. Physics Analysis and Conclusions X xFa .005 .015 .025 .035 .045 .055 .065 .075 .085 .095 .120 .160 .200 .240 .280 .350 .450 .550 .650 .850 -.012 .324 .466 .462 .603 .690 .553 .544 .790 .829 .796 .721 .959 .865 .780 .602 .390 .236 .187 .021 fxFa Error .307 .092 .062 .064 .066 .083 .100 .123 .126 .148 .075 .093 .097 .103 .111 .073 .064 .054 .047 .009 I Fa 2.339 2.364 2.148 1.961 1.829 1.695 1.570 Error .016 .016 .015 .015 .015 .015 .015 .015 .015 .015 .015 .015 .014 .014 .013 .012 .010 .008 .006 .003 .365 .365 .362 .357 .353 .347 .340 .334 .329 .321 .313 .281 .252 .214 .179 .148 .087 .048 .025 .007 1.484 1.412 1.319 1.231 .966 .786 .594 .449 .338 .166 .079 .036 .007 Error .623 .095 .072 .068 .065 .064 .062 .060 .058 .056 .054 .047 .041 .036 .032 .028 .019 .012 .008 .003 Table 7.1 xFa interpolated to Q 2 = 3 GeV2 in fine x bins; cumulative integrals of xFa and Fa from x = 1 to lower limit or each bin obtained by simple summation, with statistical errors. 1 As shown in table 7 .3, we get our best estimate of / 0 Fa dx around Q2 = 3 GeV2 • We therefore make fine bins in x at low x and interpolate our data to Q2 = 3 GeV2 after making linear fits in log Q 2 , as usual. As is clear from table 7.1, a simple summation of xFafx from x = 1 leads to good errors for all except the lowest x bin. There we must use a fit. As mentioned before, a Regge theory prejudice suggests a form ,....._ a..(X for xFa as x--+ 0. vVe have tried several forms and feel that it is valid to use all x provided a high-x behaviour term is included. The results of these attempts are displayed in table 7 .2. Clearly, including all x with a (1 - x ).8 term, i.e., {7.70) xFa(x) = Ax<l'{1- x)tl does not significantly shift the result. We can therefore state that h 1 Fa dx = 2.83± .146, Q 2 = 3GeV2 , Figure 7.7 shows the fit to xF3 and the resultant tion superimposed. statistical errors only. (7 .71) f; Fa dx with the values from simple summa- 7.( . Tests or the Quark-parton Model Fit limit.s 0 1 0 .06 0 .06 Integr limits 0 1 0 .1 .02 0 .06 0 0 .02 0 .01 .06 0 0 .02 .01 0 125 A Q fJ 3.91±.73 .577±.064 2.86±.27 3.55±2.09 .575±.177 0 2.78±.17 0.5 0 J; Fa x2 /DF 2.83±.15 2.83±.14 2.84±.15 2.79±.28 2.80±.28 2.80±.25 2.92±.10 2.93±.09 2.92±.10 12.2/17 1.58/4 1.76/5 Table 7.2 1 0 Fa with statistical errors by making fits of the form (7.70). f The regions for the fit and for the integration are specified. The rest of the integral is done by the simple summation technique. Fit parameters without errors indicate that they were fixed. The binning in xis the same as in table 7.1. Estimating a systematic error for the above number requires knowledge or all correlations between errors. The statistical error indicates that such an effort is not warranted. Instead, we assume that all systematic errors except the 3.2% correlated overall error on the v and I7 fluxes enter in the same way as the statistical errors, i.e., as uncorrelated errors. Weighting the ratio or these errors to the statistical error with f Fa in every bin, we find the contribution from this source to be 70% of the statistical error. Thus we conclude that at Q 2 = 3 GeV.Z, ' (7.72) where the first error is statistical, the second is the correlated level error and the third includes the rest or the systematic errors. At the 1a level we can say that A < 525 MeV. It is noteworthy that at this time this is the best available value for the GLS rule. The measurement derives mainly from low x, and hence a reasonably high Q 2 implies a high energy experiment. We have such high energy data with high statistics and good angular resolution, which dominates the x-resolution at low x. Other results include that of CDHS who quote< 89 l 3.2±.5, and CHARM< 92) (2.66±.41). These values are from data averaged over Q2 • ABCLOS report< 107) 2.89±.45 in the range 1 < Q 2 < 10 GeV2 . CDHS use a fit below x = .005, CHARM use one below x = .01 and ABCLOS use a fit below € = 0.1, where € is the Nachtmann variable (7 .77), not too different from x at these x and Q2 values. Only CHARM do not assume a ..jX behaviour as x-+0, but instead fit the power of x, like us. Our values for the power or x at low-x are quite consistent with 0.5 (table 7.2). Also notice that the power of (1 - x) at high-x from the global fit is consistent with 3, as expected on theoretical groundsJ 3 •26 •29 l 126 1. Physics A.na.J;ysis and Conclusions ! 01 Fa dz Q2 (GeV2) 2 3 2.63±.142 2.83±.146 2.95±.167 3.03±.180 3.09±.198 3.18±.224 3.27±.265 4 5 6 8 10 Table 1.8 J; Fa with statistical errors by making a fit of the form (7.70) in the region 0 < z < 1. At high Q 2 (Q2 > 5 GeV2 ) the integral involves large extrapolations in the important low-z regions and therefore becomes unreliable. We emphasize that no meaning should be attached to the Q2 variation, this table only shows where the minimum error is to be found. §7.5 QCD Analysis 7 .5.1 Leading Order Non-singlet Analysis The QCD evolution of the non-singlet structure function zF3 is independent of the largely unknown gluon z-distribution. Also, the extraction of z F3' is, as we have seen, independent of R and the strange sea. Therefore, analysis of zF3 is an excellent and unbiased way to extract the QCD parameter A. We therefore proceed to quantify the observed scaling violations of xF3 (and briefly, of F 2 ) in the framework of QCD. The literature is replete with articles of interest(64-TT, 8 3- 104). There are several approaches to the problem, including fitting moments of structure functions to QCD expectations and fitting functions known to satisfy QCD moment evolution expressions, at least for some first n moments. We choose to evolve the structure functions directly using the Altarelli-Parisi(92l equations. The procedure used to determine A begins with parametrizing the structure functions zF3, F2 and the gluon distribution Gat some Q~: zF3(x, Q~) =a3x 63 ( l - x)" 3 F2(x, Q~) =a2(l- x) 02 (1 12x) 0 G(x, Q~) =ac(l- x)" (1 +'rex) + (7.73) 127 1.5. QCD Analysis Either :zF3 or the coupled pair F2 and G (or both, simultaneously) are then evolved using the general formalism described in §7.3. Target mass corrections, if to be applied (see below), are made at this point. The resulting structure functions are compared to the measured values, with errors, to form a x2 which is then minimized with respect to the parameters at Q~ and A. The leading order evolution equation for xF3 at fixed :z is, 2 1 r d xFa(x, Q ) = as ~[1 d (1 + z ) {=-F3 (:_ Q 2 ) _ F3 ( Q 2 )} _ F3 ( Q2 ) d (1 + z )] d lnQ 2 2tr 3 ~ z (1- z) z z' x x, x x, lo z (1- z) 2 2 (7.74) The prescription or Georgi and Politzer(24 • 104) may be applied to make target mass corrections. In this prescription, the observed xFa is reconstructed from an evolved one, say F(x,Q 2 ), by (7.75) 1 where v = ~======== 4m~x2 fQ2 and { = ----;======= {7.76) }1 + 2x (7.77) 1+ }1 + 4m~x2fQ2 is the N achtmann variable. However, as pointed out by Devoto et at.<83 ), the expressions for target mass corrections are inconsistent, to order of 1/Q2 terms, with the equality of the regular moments or F and the Nachtmann moments of the corrected function. We resolve this problem by restricting our analysis to regions where these effects are small. We cut out the high-x region (x > .7) where non-perturbative effects, Fermi motion, bin centre corrections and smearing corrections are large as well as the low Q 2 region (Q 2 < 5GeV2 ) where non-perturbative effects are large. The Q 2 cut effectively eliminates the low-x region and we therefore also cut out x < .03. The change in xF3 over the resulting Q 2 -range at any given x-value from (7 .75) can easily be estimated. In every x-bin, we find it to be less than ,...., 10% or the observed change and less than 3% of the observed change for x < .5. We have also verified that the value or as changes by"' 3% due to the inclusion of {-scaling as in (7.75). Therefore, we have decided not to make any target mass corrections in the xF3 analysis. We also do not apply the GLS rule constraint to the fits since the low-x region is excluded. To further minimize non-perturbative effects,(GS} the low W2 region (W2 < 10 GeV2 ) is eliminated (W2 = Q 2 (1 - x)fx m;; it is the square of the invariant final state hadronic mass). In all our analysis we have assumed that there are 4 flavours of quarks. + 128 7. Physics Analysis and Conclusions The result or the leading log analysis is 3 13 ALo = 88:t*~ _-f.r~ MeV as = .204±.079 for Q~ = 12.6 GeV2 b3 = .672±.058 ca = 3.29±.24 aa = 4.34±.24 x2 = 44.2 for 45 DF The x2 indicates a good fit. x2 is plotted versus A and versus as in figures 7.8 and 7.9 (the curves labeled xF3 ). Notice the striking asymmetry of x2 around the minimum in fig. 7.8 -this arises due to the nonlinear nature of the A dependence of as. On the other hand, the x2 plot versus as is quite symmetric and quadratic since d xF3 / dIn Q 2 is proportional to as (7.74). In recognition of this fact we feel that at low values or A it is better to quote statistical and systematic errors on as rather than A, where as is evaluated at Q 2 = 12.6 Gey2 (roughly the mid-point of our Q2 -range). We shall henceforth use the symbol a~ for as(Q 2 = 12.6 GeV2). In figures 7.10 -7.12 we see that the strongest correlation of a~ is a negative correlation with the power of (1- x) indicating some feedthrough of the x-dependence into the Q2 evolution. ' Despite the absence of x < .03 data in the fit, the best fit values above correspond to (7.78) This value is at Q2 = 12.6 GeV2 and is in good agreement with the value quoted in the previous section for Q2 = 3 GeV2 by a global fit (7 .72). The statistical error from the global fit is also 0.15, indicating that adding a Q 2 -dependence parameter A does not affect the error on the integral of F 3 , as expected. 1.5. QCD Allalysis 129 ... QCD ITL' Lea din ~ 0 cc CCFRR 'rTil 0 ~,_1 u:i 3 /'1 2 ~ 1 4U cl ..!. 1D '1j Q) !-. liS ;:J -I .I 0 Qi ci rn 1D ...... ..c1 () ~ ..q. •H3 d.f. 1D 0 ci ..q. .00 I I I 200.0 -'100.0 600.0 A (MeV) Figure 7.8. x2 vs. A, leading order. 0 fit 0 cc CCFRR 0 u:i xF3 /F 2 1D '1j 46 d.f. Q) !-. liS ;:J 0 Qi ci rn 1D ...... ..c1 () ~ ..q. 1D ~ 0 ..q. .00 0.10 0.20 0.30 0.40 Figure 7.9 x vs. as, leading order. The curves labeled xF3 /F2 use xF3 below x=.4 2 and F2 above. o.Gu 130 7. Physics Analysis a.nd Conclusions 0 co 0 CCFRR an l- 0 0 l- 0 t') ,.0 an co 0 0 co c:i an an 0 .00 0.10 0.20 0.30 0.40 0.50 as Figure 7.10. Correlation between ba and a~; xFa, leading order. CCFRR ~ t'J 17) 0 0 Er5 co C'd .00 0.10 0.20 0.30 0.40 Figure 7.11 Correlation between c3 and a~; xF3 , leading order. Displayed in both figures are la and 2a contours. 0.50 7 .5. QCD Analysis 131 Systematic effect Flux smoothing Flux level errors Angular disp. Angular disp. (uncorr) Systematic error in Ehad Systematic error in E,. Different R assumptions 1/2 SU(3) changed to SU(3) Different models for correction terms in F2, xF3 extraction n1 = 4 changed to n1 = 3 Inclusion Of (1 +'}.,X) term Change in Q~ (12.6-+ 5 GeV2) xF3 xF3/F2 .6-o~ .6-o~ .027 .047 .018 .028 .014 .004 .002 .008 .010 .006 .024 .011 .011 .009 .015 .002 .006 .001 .0003 .001 .003 .002 .001 .Oll Table 1.4 Effect of systematics on o~ and changes in o~ due to other assumptions in leading order. Systematic errors on o~ are as shown in table 7.4. The sources of error have already been discussed (§6.6); here we only touch upon points pertinent to the QCD analysis. The 'unlimited statistics' technique is extended to QCD by extracting o~ for the 26 data sets and using the standard deviation as an estimate of the error. This applies to the first three errors listed, which are among the largest. Clearly, xF3 is most sensitive to uncorrelated flux leYel errors. The two assumptions about beam angular dispersion serve to bracket the error from that source. Again, the error due to R comes from analyzing data extracted with three different assumptions for its value (6.10). The error is small, as expected, despite the fact that real data (limited statistics) are used. The effect of changing the assumption regarding the strange fraction of the sea is also small, again as expected. The models used for structure function extraction also make little difference. We have also tried to change certain assumptions regarding the QCD analysis. Changing the assumed number of quark flavours to 3 and changing the value of Q~ (starting point for evolution) have no significant effect. The functional form for xF3 at Q~ (7.73) was '}3X) term but again the change in o~ is slight. modified to multiplicatively include a (1 We conclude therefore that the most significant problem in extracting a value for A from xF3 presently is statistics, since the combined systematic error on o~ is .06, as compared to .08 for the statistical error. This corresponds to systematic errors of +113 MeV and -70 MeV on our value for ALO· In a high statistics quad-triplet run at the Tevatron, which would cover roughly the same energy range as this experiment, the uncorrelated error on the flux levels may be further reduced by using da / dy at y = 0. Thus the systematic errors can be reduced too + 132 7. Physics Anal,ysis and Conclusions when statistics increases, even for an unnormalized wide-band run. Following a standard technique(83), we have analyzed our data in the same non-singlet framework using F 2 or 2xF1 instead or xF3 in the high-x region. The major justification for this procedure comes from the fact that 7j = 0 when x > .4. Table 7.5 lists values of F2 and 7j in x bins and the CDHS values(81 ) or 7j, which they determined by adding to their narrow-band data another 155,000 Ii and 35,000 v wide-band events with Ev > 20GeV. Clearly, above x = .4 both data sets imply ij = 0. Notwithstanding the difference in the ij values from the two experiments at low x, which arise mainly from overall level differences, we can set q = 0 for x > .4. This is because adjusting the values of 7j for the level difference in cross-sections beyond x = 0.4 does not change the result ij = 0. The Q 2 range over which the two data sets are averaged for table 7.5 is virtually identical except for x < .4, where the CDHS data displayed cover a range about 0.1-0.2 lower in log10 Q 2 • In the region of interest the errors on the CDHS q values are much smaller than our F2 errors, thereby facilitating the procedure of Cl,lmbining F2 or 2xF1 and xF3. X .015 .045 .080 .150 .250 .350 .450 .550 .650 F2 1.36±.033 1.37±.023 1.49±.022 1.24±.013 0.93±.012 0.64±.011 0.38±.009 0.20±.007 0.11±.006 q .477±.023 .354±.021 .341±.021 .159±.012 .034±.012 .033±.012 .0021±.0092 - .00094± .007 4 - .0018±.0061 Table 7.5 CDHS q .344±.016 .321±.013 .277 ± .010 .155±.0058 .054±.0040 .013±.0027 .00076± .00197 - .00128±.00160 - .. Values of F 2 , q and the high-statistics CDHS q averaged over Q 2 with statistical errors. The result or combining F2 for x > .4 and xF3 for x < .4 and doing a non-singlet analysis is the following set of values for the parameters: ALo = 266±g! ±nMeV as = .291±.047 for Q~ = 12.6 GeV2 b3 = .635±.049 c3 = 2.90±.13 a 3 = 4.29±.22 x2 = 50.0 for 46 DF One important point is served in doing this analysis. We get an idea of how much the systematics are improved (except those due to R) by a gl ance at table 7.4 which also lists the 133 7 .5. QCD Analysis Cll td 0 0? CCFRR co >< C\i '1:j tr.l ~ ..... 0 c:o '...::; Cll "<t C\i Cll C\i .00 0.10 0.20 0.30 0.40 0.50 O:g Figure 7.12 Correlation between and 2a contours. J; dx F and a~; xF leading order. Displayed are la 3, 3 MS QCD fit q 0 c:o CCFRR 0 l() 10 '1:j Q) h ro & tiJ ...... 0 0 lO ,..q u q LD "<t q 0 "<~'.00 0 . 10 0. 20 0.30 0 .40 Figure 7.13. x2 vs. as; MS analysis of xF3. o.ou T. Physics Analysis &nd Conclusions 134 errors from this analysis. As expected, the relative insensitivity of F 2 to the flux smoothing procedure and to the 4% level fluctuations reduces those errors dramatically. The error on a~ is .035, which translates into errors of +85 MeV and -79 MeV on the value of A. '"V However, in doing such an analysis we are confronted with the uncertainty in R. Since q = 0 implies xF3 = 2xF1 , we should use (7.79) instead or F2 in the high-x region. Fixed values of R and QCD values of R induce minor Q2 variations, but the 4m;x 2 JQ 2 term induces large (up to 10% ) variations in 2xF1 with respect to F2. This changes the value of A, or as, significantly as seen by the result of a fit with 2xF1 used instead of F2 for x > .4 and R = RQco: '"V ALo = 426:!:U~ MeV as = .355±.046 for Q~ = 12.6 GeV2 b3 = .627 ±.047 ca = 2.76±.13 aa = 4.45±.22 x2 = 51.9 for 47 DF Perhaps the wisest approach in attempting to substitute F 2 for xF3 at large x is to wait until data are available in regions where 4m~x 2 JQ 2 < 1. 7.5.2 MS Non-singlet Analysis We have also done a second order analysis or the xF3 data. We choose to work in the MS scheme; the value of A in a different scheme can be obtained from the MS result. We have used the programs or Duke et ai.,(83) and of Barnett!84). The latter parametrizes xF3 at Q~ as in the leading order case while using the second order evolution equation, or course. The other approach, that or Duke et al., uses a definition or parton densities that makes them 'universal' i.e., the same densities apply in all processes. In this method, the parton densities are first evolved and then the structure function is constructed from them at any given Q~. Effectively, this is a two-step procedure. In the Barnett approach the coefficient functions that are convoluted with the parton densities to obtain xF3 in the two-step procedure are absorbed into the xF3 evolution equation (7 .80). In order to get the same value for AMs from both procedures, the parametrization of xF3 at Q~ in one case must be different from the parametrization of the parton densities at Q~ in the other, and both must be the true functional forms. Since those forms are unknown, both Barnett and Duke et al. have chosen to parametrize their distributions at Q~ by the same functional form, the one in (7 .73). Thus, the result is two different AMs values. However, the difference in as (Q2 = 12.6 GeV2 ) is small 7.5. QCD Analysis 135 in comparison to the statistical error and both values of a~ agree with the leading order result (table 7.6). The zFa evolution equation in second order is 2 d x:~~zQ~ ) = 2 as~~ ) {<2 + ~ ln{1- z))zF3 (z, Q 2 ) 1 { 4((1 + z 2 )~Fa{t,Q 2 )- 2zFa(x,Q 2 ))dz ls 3 (1- z) + 2 ) - xF (x,Q 2 ))W(z) as~Q1r ) lst dz(.:.Fa(.:.,Q 3 Z Z 2 + (7.80) as~~ ) xFa(z, Q2 ) los dz W(z)} 2 - where, as(Q 2 ) is the second order expression {7.47) and 2 )(1n(l- W(z) = -2{30 ((1 + z 3 {1- z) z) _ z ~) + (9 + 5z) _(I+ z)) 4 4 2 z) 3 + 2 z) lnz- -(1+z)ln 1 2 +16( - -2(1+z )lnz ln(1( ) - ( -z-5(1-z) ) 9 1-z 1-z 2 2 (I z ) 2 11 67 rr2 40 ) +2 ( {I )(In z+-Inz+---)+2(1+z)ln z +-(1-z) -z 3 9 3 3 + 2 + z ) (- In z - -) 5 - 2( 1 - z)) + -16((1 9 (1-z) 3 2 4(1+z {* (-lny) ) +g 1 +z Jl/zdy 1 +Y +{l+z)lnz+2(1-z) {7.81) The x2 versus as curve for the Duke et a!. technique is displayed in figure 7.13. The ' it is emphasized, for the fit is plotted through the data in figure 7 .14. The best fit parameters, two techniques are for xFa in the Barnett case and for the universal parton distribution in the other. ·we do not show the {fractional) systematic errors in the MS case which would merely be a duplication of the fractional errors in the leading log case. Parameter A as(Q 2 = 12.6GeV2) ba ca aa x2 /DF Leading Order 88:!:i8;j .204±.079 .672±.058 3.29±.24 4.34±.24 44.2/45 MS Barnett (ref.84) 120::!:1()6 .176±.062 .631±.059 3.36±.22 5.05± .35 44.0/45 MS Duke et al. (ref. 83) 193::!:156 .201±.070 .664±.058 3.54±.20 5.94±.46 43.5/45 Table 7.6 Best fit parameters resulting from first and second order (M S) fits to zFa . 136 T. Physics Analysis and Conclusions CCFRR 0 or---.--.-.-.. .on,.---.-~~-rrrrnl----.--.-..-rrnn 0 .-l '! ;J; I x=.045 (x30.0) ! - - : I - l. r T -I- -!x=.OBO (x15.0) -! j"" -llL -m- -m- -m- + r _._ --~ ..A_ _._ i x=.150 (x7.50) -i x=.250 (x5.00) t-v~-y-L~x rTI i. __ 1 ofM 0 + x=.350 (x3.00) I +--! T -f- + x~....o (x2.) 1x~.550 1 (x2.) - tT~i x~.850 - 0 8ot~.-o--~~~~~1~o~.o--~--~~~1~o~o~.-o--~~~~1~o~o~o.o I I Figure 7.14. xF3 data with M S fit (dashed lines). 131 1 .5. QCD Analysis 7 .5.3 Non-singlet Analysis Summary The hypothesis or no scaling violations (A = 0) is clearly ruled out by all the leading order and second order results (see figs. 7 .8, 7.9 and 7.13). Also, an attempt was made to look for higher twist effects by the multiplicative inclusion of the term (1 h/Q 2 ) in the xFa evolution, but the values or h thus obtained had large errors, with h consistent with zero and no improvement in x2 • For example, an MS analysis yields h = 26±23. + After Barnett<98 ) we have also attempted to look for W2 dependence in as. Our present statistics prevent us from corroborating his evidence that the value or as falls if the lower limit for W2 is raised. The same conclusion is reached with a search for Q2 dependence (table 7.7). Q 2 cut (GeV'2) 5 5 5 5 10 15 25 W2 cut ALo x2 /DF AMS x2 /DF (GeV'2) 10 20 30 10 10 10 10 (MeV) 88±111 24±93 131±254 88±111 135±181 62±115 410±512 44.2/45 34.8/38 27.3/32 44.2/45 31.7/36 29.8/29 16.7/21 (MeV) 193±206 67±163 197±346 193±206 144±197 64±124 327±477 43.5/45 34.6/38 27.5/32 43.5/45 32.2/36 30.1/29 17.5/21 Ta.ble 7. 7 Variation in ALo and AMs due to Q2 and W2 cuts. The MS values are for the technique of Duke et al. The errors are quadratic errors; despite appearances they do not really permit A = 0, e.g., the,true errors for the leading order value in the first row are +163 MeV and -78 MeV. We conclude that the determination or A from xF3 is presently limited by statistics. Higher statistics experiments will need more precise fiux calibrations as the uncorrelated error on v and i7 fluxes leads to significant uncertainties in xF3 . 7.5.4 Singlet Analysis The QCD analysis or F 2 is the subject or another thesis( to); we merely summarize the situation here. The evolution or F2 depends on the gluon distribution, and its extracted values depend on the value orR and the strange fraction of the sea, particularly at low x. By using the momentum sum rule: (7.82) 1. Physics Analysis and Conclusions 138 0 c::i 0 <0 = 0 .9 'YG MS Analysis 3.5 ~ 0 .......... 0 !> (!) 0. "<i' ~ ..._.,. ~ ~~~~i'!'· 0 c::i -r 0 OJ ~r ;n----J...--;2' .-..Jo'-----J..--4.0'-----J...--.u"·.-;;' u'------'---,u. u''----J.-._,l..,O-. .00 I .J I I J Figure 7.15 Best fit points in the A-ce plane, for different values of 'YG· The shaded area indicates the reasonable range of values for the parameters co and 'YG· we put a powerful constraint on the gluon distribution. The regions used for the F 2 analysis in x and Q2 are the same as those for xF3 except we eliminate x < .1 and thus decrease the sensitivity to R and the strange sea. A remains highly correlated t~ other parameters, especially to co whlch is poorly determined. The statistical precision and lower sensitivity of F2 to scale errors enable us, however, to extract a value of A for various assumptions about the gluon distribution. The reasonable assumptions "fo~O and 4~ co~8lead us to a range of values for A with different R assumptions (table 7.8). Figure 7.15 shows the correlation between parameter values and A for R = RQcD and helps delineate the allowed range. The best fit A values for the three different R assumptions are listed in table 7.8. As noted earlier, most systematic errors on A are smaller for F 2 - the values in leading order are ±25 MeV for the flux smoothing procedure, ±30 MeV from flux level uncertainties, ±10 MeV from the beam dispersion and ±15 MeV each from the hadron and muon energy calibration errors. However, as evidenced from table 7.8, the The uncertainty from the gluon distribution is estimated at ±50 MeV. In future a more precise value of A from xF3 could be used in the F 2 analysis to measure the gluon distribution. At present, our value implies, albeit weakly, a high value for co i.e., a soft gluon distribution. Using the CDllS measurement of q, we find that the value and error on A do not change significantly, but a much better determination of 139 T.5. QCD Analysis the gluon distribution is possible. The gluon distribution now has no freedom beyond x = .25, and is essentially the same as the CDHS distribution in that region, but the difference in the F2 integrals from the two experiments forces their gluon distribution to be much larger at low-z. R=O.O R=0.1 RQcD ALo AMS" 360± 100MeV 200± 90MeV 300 ± lOOMeV 390 ± llOMeV 230 ± 100MeV 340± llOMeV Table 7.8. F2 fits with ca = 4.6 , 1o = 9.0 7.5.5 Comparison with Other Experiments Other experiments have attempted to measure A (see table 7.9), notably EMC( 8~ 81) who obtain AMs = 173±~~4 MeV from F 2 using iron data, in good agreement with our result from xF3. CDHS{SI) obtain a value of AMs = .2±:~ GeV, from a non-singlet analysis of their xF3, again for iron and also in good agreement with our result. Clearly, the values for A from F2 lie within errors of the xF3 results, perhaps because of the somewhat large errors. We can however rule out the older high A values obtained by BEBC(88) at lower Q2 (ALo = .74±.05 GeV) and the older CDHS Buras-Gaemers style analysis<89) which led to ALo = .47 ±-1±.1 GeV. In fact, at the 90% confidence level, using statistical errors, we can state from our zF3 analysis that .2 < ALo < 420 MeV The BFP muon experiment< 82) at Fermilab, also in our Q2 -range, reports a preliminary result with leading order A values between 160 MeV and 230 MeV, depending on assumptions made. The CHARM collaboration bas done a fit{llO,lll) of the non-singlet structure function and obtain a leading order result that is consistent with ours: A =187±~~g±70MeV. Recent results from angular energy correlations in e+e- experiments indicate values of A in the same range: the Mark-J group reports< 90) AMs = 180±:g MeV and AMs from twophoton scattering has a comparable value< 91 l: AMs ~ .2±.1 GeV. The Mark-J result depends on Monte Carlo assumptions about fragmentation, however. Thus it appears that the highstatistics neutrino and muon experiments outside the low-Q 2 region all have values of A in the 100 - 300 MeV range, an encouraging sign. Values of A from neutrino and muon experiments agree to a remarkable level and singlet and non-singlet analyses lead to the same result within errors. The e+ e- results are similarly heartening, especially the higher Q2 result, but error bars prevent us from concluding that the similarity in A values from different experiments at different Q 2 values implies that perturbative QCD has been proven beyond doubt. 140 7. Physics Analysis and Conclusions Experiment Order BEBC,loo1 lower Q~ CDHS,l89 l old result CCfo'RRpurp2 CCFRR, xFa CCFRR, xFa EMC,l87 l iron data CDHS,l111 l xF3 BFP,l 82l preliminary CHARMlllO,lllJ' :r:Fa Mark-J{90 l Two-photonl91 l (JADE) LO LO MS A (MeV) 740±50 470±100±100 340±110 LO ss+}~ 3 MS MS MS LO LO MS MS :175 i3 193+n~ 173:!:~~ 4 200:!:~88 160 to 230 1s1±in±10 180+~g 200±100 Table 7.9 Comparison or A. values from different experiments with emphasis on values from non-singlet distributions. The two older values (first two columns) are higher than recent results which all agree within errors. Our M S value is for the Duke et al. method, with statistical errors only. §7.6 Extraction of R 7 .6.1 R from y-dependence of Differential Cross-sections The dependence or the differential cross-sections on R is not very strong (6.1). Further, F2 and Renter as one term, identical for both neutrinos and antineutrinos, and the small value of R makes it harder to measure. We have to utilize the different y-dependence or R and F 2 in (6.1) for extraction. A simple technique would be to fit a second order polynomial to ydistributions in x and Q2 bins and extract F2 , Rand xF3 from the coefficients of the powers or y. Given the level of our statistics, we have chosen the maximum likelihood technique instead, which is essentially an extension or the structure function extraction method. Each x and Q2 bin is now further divided into 20 y bins, and the number of neutrino and antineutrino events in every such bin is accumulated and corrected for resolutions, as before. If there are neutrino events in the i 1" y-bin and n, antineutrino events, then n, ni =a, F2 + bi 2xF1 + c, xFa + di ni =ai F2 + bi 2xF1 + Ci xFa + di (7 .83) 141 T.6. Extraction orR a, b, c and d have to be obtained, as in the structure function case, by integrating over the appropriate coefficients (see 6.11). The integration would require even more CP time to maintain the same accuracy in these integrals as before, since we have carved out 20 bins out of each x and Q2 bin. This becomes impossible to handle by simple techniques and therefore we resorted to a Monte Carlo integration. The integration uses stratified sampling with partitioning to reduce variance. We have verified that summing the integrals over the y bins returns the integrals in each x and Q2 bin evaluated for structure function extraction. We now form a likelihood for the numbers or v and v events in y bins and maximize it by varying F2, 2xF1 and xFa. R is then extracted using (7.23). We check to ensure that xF3 thus obtained agrees with xF3 extracted with an assumption about R (as it should because of its independence of R). The results still suffer from poor statistics and it is impossible to detect any meaningful Q2 dependence orR in an x-bin (see fig. 7.16). Averaging Rover Q2 , w.e obtain the values listed in table 7.10. The errors are still large. We can only say that R appears lower in the x > .4 region than in the x < .4 region- the averages for 0 < x < .1, .1 < x < .4 and .4 < x < 1. are .12±.08, .16±.09 and -.65±.13. Averaged over all x, we obtain R= - .01±.05. x-bin R 0 <X< .03 .03 <X< .06 .06 <X< .1 .1 <X< .2 .2 <X< .3 .3 <X< .4 .4 <X< .5 .5 <X< .6 .6 <X< .7 .7 <X< 1. (stat. errors) .04± .15 .27± .16 .08±.12 .30±.13 .20±.16 -.12±.17 -.21:!;.27 2.7±5.7 - .77±.14 1.7±10.3 Table 7.10 Values of R in x bins are averaged over Q2 • The errors are statistical only. or course, we have completely ignored systematic effects in obtaining the above numbers. They may therefore be misleading. R is sensitive to anything that affects y-distributions. Therefore flux levels and the energy dependence or cross-sections have to be better understood. We shall leave this task to some future student. Lacking an estimate or systematic errors, it is improper to draw conclusions from a comparison of our R value with other measurements in the same Q2 -r ange; suffice it to say that the values from c.mo! 105 l, SLACI 106 l , EMCI 6 1l and CDHSI62 l lie between 0 and .5 and are consistent with each other because of large errors (see fig7 .17}. 7. Physics Ana.lysis a.nd Conclusions J42 .2 < XI < .3 I _______ j I I I I IT -r- X - - - i I·r _I_ ..L- I _I I 0 J (;") I '--;:-----'----'---'----'---'--'---'.--::'-:-!-----l-----.._____.___-'---'---'--:-'. 1.0 10.0 lUU.U 2 3 Q (GeV/ Figure 1.16 R in the bin .2 < x < .3 as an illustration of the fluctuations and errors. 10 l'I I 0 0 10 0 Statistical errors only - r- 0 o o ' CCFRR CDHS oil cmo ll EMC - I l..... 0 I J lD C\l 0 I I .00 0.25 0.50 0.75 X Figure 1.11. R from various experiments. 1.0 143 7.7. Conclusions 7.6.2 Limit on R An attempt was made to obtain an upper limit on R from the condition that q(x)~O. If q(x) is set to zero, only t'vo unknowns remain in every x and Q2 bin viz., xF3 and R. Ignoring correction terms and using (7. 79), equation 6.1 gives (7.84) Integrating over flux and acceptance, the left hand side represents the number of v and II events in a bin. The rest or the procedure follows the details of the structure function extraction technique. xFa, as before, comes from the difference or the numbers of v and I7 events: nv- nv= k <(1- (1- yf)> xFa (7.85) where k is the overall constant and <> denotes integration over bin limits. Along with the equation that comes from summing neutrinos and antineutrinos, we get nv + nv) <(1- (1- y)2)>- <(1 +(1- y)2)> + <y2 > 2m~x2 = 2R <(1- y)> ( nvnv Q2 (1 + 4m~z2 jQ2) (7 .86) Now the left hand side is almost zero, since R is small, leading to large errors on the extracted limit for R. Thus we are again limited by statistics - the limit, if averaged in the region x > .4 is .55 at the 90% confidence level. The CDHS collaboration(8 l) used narrow-band and wide-band data in the region .4 < x < .7 to obtain R~ .04±.03. We conclude that the measurement of R has serious statistical and systematic problems that require further investigation and content ourselves for the present with extracting structure functions with three different assumptions about R (6.10), all of which are consistent with our attempt to eY,tract R . §7. 7 Conclusions Data were taken during the run of experiment 616 at Fermilab using neutrino and antineutrino dichromatic beams at five momentum settings: 120, 140, 165, 200 and 250 GeV/c. The flux of neutrinos was monitored, allowing a normalized measurement. The absolute level of fluxes was calibrated to 3% with further uncorrelated uncertainties of 1.4% and 3.2% for neutrinos and antineutrinos respectively. With 5.9x 10 18 protons on target, we measured a total of 150000 neutrino and 23000 antineutrino charged current events using the Lab E detector. The 640 ton instrumented target was used to measure the angle of the outgoing muon and the energy of the hadron shower. Downstream, a toroidal spectrometer measured the muon momentum. After making model independent acceptance corrections the data were analyzed to extract the structure functions F2 and xF3 or nucleons in iron (with no Fermi motion corrections) under three different assumptions about the value of R . We covered the range 0 < x < .7 T. Physics Ana.lysis a.nd Conclusions 144 and 1 < Q 2 < 250 GeV2 • Corrections were made for resolutions, the strange sea, the onset of charm production, radiative effects, and a correction for the mildly non-isoscalar nature of the target nuclei. Statistical and systematic errors on the result have been studied. Statistical errors dominate uncorrelated systematic errors for both xF3 and F2 . Correlated flux systematic errors are 3.2% but do not affect determinations of quantities that come from a combined analysis of all bins. e.g., of A. The major conclusions from analysis of the structure functions can be summarized as follows: (1) Both F2(x, Q 2 ) and xFa(x, Q 2 ) have been compared with results from other experiments. Apart from level differences expected from total cross-section differences, the low-x behaviour of the structure function F2 is markedly different from the CDHS measurement, but is in agreement with that of the EMC and BFP collaborations. In a comparison of xFa as a function of x, only the HPWF results show a striking dissimilarity. (2) Being the best experiment to measure the GLS sum rule we find at Q2 = 3 Gey2. This is consistent with the quark-parton model value of 3 and QCD expectations with A < 525 MeV. (3) The QCD evolution of xF3 , which is independent of R and the strange sea, does not depend on the gluon distribution and fits yield ALo = gg+163 +113 MeV -78 -70 The systematic errors are smaller than the statistical errors. Second order fits give somewhat different values of A, although cxs (at Q~ = 12.6 Gey2) is not so different. (4) A fit using the better determined F2 in place of xFa for x > 0.4 i.e., assuming q = 0 in that region, gives Again, the statistical errors are larger than the systematic errors. (5) QCD fits to F 2 with a particular choice of the gluon distribution and R give(IO) ALo =360± 100 MeV AMs =340±110MeV Variations in the gluon distribution parameters indicate an additional rms variation of ,..., 50 MeV. The combined systematic uncertainty from the flux and from assumptions about R is comparable to the statistical error. Using the constraint q ~ 0 for x > .4 enables us to make a significant determination of the gluon distribution. T.T. Conclusions 14.5 (6) An attempt to measure R was made. The value of R suffers from serious statistical and systematic uncertainties. The value is shown in figure 7.17 as a function of x. The upper limit on R from the condition q{x)~O is 0.55 at the 90% CL in the region x > 0.4. During the course of this analysis there has been an ongoing effort to upgrade the Lab E detector. Drift chambers will replace the spark chambers used in E616. This will permit a higher data taking rate, with the detector essentially not limited by deadtime (""' 10 psec per trigger) at projected trigger rates. With the availability of a 1000 GeV proton beam it will become possible to study the neutrino and antineutrino cross-sections at higher energies. Another high-statistics dichromatic run is planned (E652). The higher energy will make possible an accurate measurement of the GLS rule in a high Q 2 region for the first time. The most interesting question will be the Q2 dependence of structure functions - has A changed with the increase in Q2 ? It will be important to measure flux levels with higher precision for F 2 comparisons. Higher statistics will improve the A value from xF3 • A better value of A in the present energy range is needed to be able to draw any conclusion about a change in its value as one goes to Tevatron energies. Since our numbers are statistically limited, an extra four to five times the present data will be very useful. An experiment to measure neutrino oscillations with the Lab E detector( 109l and the same beam collected roughly an equal number of events during 1981- 1982. Further, a wide-band run at the Tevatron (which will mean neutrinos between 100 GeV and 300 GeV) can add an extra million or so events. Such a data set will have to use either our cross-sections or cross-sections measured in the future using a dichromatic beam. Even so, the systematic errors on A from xF3 will come mainly from the cross-sections, and therefore not be the limiting factor nntil several times the present number of events have been collected. Appendix A Beam Kinematics The flux of neutrinos at Lab E is a function of position and energy setting. As mentioned in the main text, for a dichromatic beam, there is a correlation between radius and neutrino energy; here we derive this and other illustrative quantities. Most or the decays are 2-body decays ( 1r± ~ JJ±<vt; K± ~ JJ±<v~). The few 3body decays that produce, on the average, somewhat lower energy muon neutrinos are a small correction. For a 2-body decay, the energy or the neutrino in the zero-momentum frame is given by • • mM mP E = p = -2- ( 1- ( mM )2) , where mM = the mass or the parent meson. For pions, p• is 29.79 MeV, for kaons it is 235.53 MeV. The lab energy or neutrinos depends, or course, on the energy of the decaying meson in the lab frame and the centre-of-mass decay angle of the neutrino. Defining the z-axis to be the direction of the meson momentum, let rt be the angle the neutrino makes with the z.-axis in the zero-momentum frame. Then, ' Clearly, the maximum and minimum lab energies or the neutrino will be Emin= 'YP•(1- /3) and Ema:z= 'YP•(1 + /3), spanning almost the whole range from zero to 2"fp• . The lab angle 0 corresponding to o• given by coso• = cosO - {3 . 1- f3 cosO' q/ = ¢. (A.l) 141 A. Beam Kinematics Thus, at a fixed radius at Lab E, 2 EM mM ( 1 ( m,. ) ) = mM_2_ mM ( 1- {J2 ) 1- {3cos0 2 = EM( 1 2 (~) ) ((mtt/Ett)) mM 1- /3 cosO (A.2) The maximum radius attainable for a neutrino occurs when 0 = 0, giving in the limit PM>mM, (A.3) The factor ( 1 - ( :;::, Y) is 42.7% for pions and 95.4% for k:aons, clearly showing that k:aon neutrinos have larger energies. Equation (A.2) also clearly establishes an energy versus radius relationship for neutrinos at Lab E. If the beam were infinitely thin, monochromatic and decayed at a point, fixed cosO would correspond to a fixed energy E at Lab E, and so energy would fall off with radius (r). Such an ideal E vs. r plot is shown in figure A.1 for all our secondary energy settings. Continuing with this simplification for a moment, let R be the distance of the decay point from Lab E. If the dimensions of Lab E are always small compared toR, cosO= R JR2 + r2 ::::s 1 - r2 2R2" ' Substituting this in (A.2) it is not hard to show that Em a., E = 1 +(r/R)2'72' (A.4) which gives us the energy vs. radius expression. Notice the characteristic r-dependent fall-off which comes from the (1- f3 cosO} term in (A.2}. The full width at half-maximum is given by TI'WHJW 2R = - . '7 (A.S} Since pions are lighter than k:aons, they have larger '7, and thus their energy falls off much more rapidly with radius. The other important factor in the kinematics is the flux variation as a function of angle. Since both pions and k:aons are spin-zero (there would be no spin selection anyway} the decays in the rest frame are isotropic. If C(O, ¢>}is the probability density for decay in [0, 8 +dB) 148 A. Beam Kinematics 0 ~ r-------~---------r--------~--------r-------~--------~ C\l Kaons 0 c:i t.n 0 0 .00 25.0 50.0 '(G.O Radius at Lab E (in) Figure A.1. Ideal Energy-radius relationship at Lab E and[¢,¢+ d¢} (spherical polar coordinates}, then (A.6) (A.7) Combining with (A.l) and (A.6) gives C(B ¢) = (1- p2) ' 41T(l - p cos8)2 (A.8} The quantity (1 - p cosO} occurs twice in the denominator as opposed to once in (A.2}; thus the !lux falls off even faster with radius at Lab E. Indeed, rewriting (A.8) as F(O} F(r) = (1+ (r/R)2'Y2)2' (A.9) we see that the F\VHM or the fiux occurs when TrwHW = (2 VV2r.: R 1.29R 1)- = - - . 'Y 'Y (A.IO) A. Beam Kinematics 149 0 0 10 0 0 ..qo 0 >< 0C'J ~ ....... 1'<-t 0 0 CIJ 0 0 .,..... Pions 0 ~ .00 25.0 50.0 Radius at Lab E (in) Figure A.f. Ideal Flux-radius relationship at Lab E A plot of flux vs. radius for an ideal beam is shown in figure A.2. ' The real neutrino beam differs from the one described above in four major respeets: (i) The beam decays over an extended region in z, in fact the decay pipe is 351.5m long and at a mean distance of 1113.7m from Lab E. A factor e<"-"2 )/'J>. • must be multiplied into the flux expressions above and integrated over the z..limits (z 1, z2 ) of the deeay pipe. >. • is the centre-or-mass decay length. In fact, the R in above expressions is now z..dependent. In as much as the z..dependence can be separated, the fraction of particles that decay in the decay pipe are ( 1 - e-(" 2 - "1 )/'J>. •). The fiducial volume of Lab E itself has a spread in z of 12.7m, small compared to the above dimensions. >. • for pions is 7.804m and is 3.709m for k:aons. Listed in table A.l are various interesting parameters for the beam. Since the decay pipe length is a significant fraction of the distance to Lab E, there will now be a range or energies at any radius r at Lab E instead of a fixed energy as given by (A.4). This fractional spread in energy is easily shown to depend on the fractional z-spread (=351.5/1113.7) as 6E E 6R = R 2 ·1 +R2fr2'Y2 (A.ll) J50 A . Beam Kinematics Thus this effect introduces a tl.E/E from zero at r = 0 to 37% for pions at 250 GeV and r = 30 in . Similarly, (A.9) may be differentiated to give tl.F tl.R 4 F=R.1+R2fr2'2 · (A.12) (ii) The beam has a momentum spread or ±9%. The effect on the radius and flux is straightforward: E as obtained in (A.4) develops the same ±9% spread, the flux spread is less than ±1%. (iii) The beam has an angular spread of between .1 and .2 milliradians. Clearly this affects both the energy and flux relations. For kaon neutrinos at 250 GeV the flux FWBM angle (see table A.1) is 2.55 mrad, large compared to the angular dispersion, and consequently the effect is small for kaons. For pions, this effect begins to become significant at the higher energies. Differentiating (A.9) it is easy to show that tl.F F tl.B 4 tl.B 2 ~ T 1 +R 2fr 2 , 2 (A.13) and differentiating (A.4), tl.E E ~ T1 +R2fr2'2 (A.14) (iv) Two major background sources exist: the background of muon neutrinos from Kp3 decays and from the wide band background. The latter is described in section 3.1. The decay K± -+ JJ±!vt 1r 0 has a branching fraction of 3.2% and contributes a small number of low energy neutrinos that are estimated by the beam Monte Carlo (see chapter 3). Table A.l of secondary beam related quantities Energy settings for secondary beam: 120, 140, 168, 200, 250 GeV. Momentum acceptance ~ ± 9.4%. Angular divergence, horizontal~± 0.15 mrad vertical~ ± 0.19 mrad Length of decay pipe = 351.5m ~ Gaussian width= 101.47m. Mean distance to Lab E = 1113.7m. EseT = Energy setting = FWH}.f radius for the energy versus radius at Lab E. FWHM radius for the flux versus radius at Lab E. Br .rwHu = FWHM angle for the flux versus angle distribution. rJJJ.rwau Tr.rwHu = 151 A. Bea.m Kinematics The following spreads are computed at a radius of 30 in for neutrinos from pion decays and 70 in for neutrinos from kaon decays (the limits or acceptance cuts). (tl.E/E)z: fractional energy spread due to finite decay length. (tl.E/ E), :fractional energy spread due to angular dispersion of beam, assumed to be 0.18 mrad. (tl.F f F)~ : fractional flux spread due to angular dispersion of beam, assumed to be 0.18 mrad. Neutrinos from 1r decays EsET (GeV} TII.I'WHM TI'.I'WHM Br.rwHM 120 140 165 200 250 (inches) 102.0 87.4 74.2 61.2 49.0 (inches) 65.64 56.26 47.74 39.38 31.51 (mrad) 1.50 1.28 1.09 0.90 0.72 (t:..Ef E)z (tl.E/ E), (t:..FfF)~ (%) (%) (%) 4.68 5.83 7.21 8.93 10.94 13.53 16.85 20.81 25.79 31.59 27.05 33.70 41.62 51.58 63.17 ,.., Emc.r. DF (GeV) 859.8 51.2 1003.1 59.8 1182.2 70.4 1433.0 85.4 1791.3 106.7 (%) 5.10 4.39 3.74 3.09 2.48 Nelltrinos from K decays EsET (GeV) Tli.FWHM TI'.I'WHM BI'.FWHM 120 140 165 200 250 (inches} 360.8 309.2 262.4 216.5 173.2 (inches) 232.2 199.0 168.9 139.3 111.4 (inrad) 5.30 4.54 3.85 3.18 2.54 (t:..Ef E)z (t:..Ef E), (t:..F/F), (%) (%) (%} 2.39 3.10 4.04 5.37 7.20 2.95 3.84 5.00 6.65 8.91 5.90 ,7.67 10.00 13.30 17.83 ,.., Emc.z 243.1 283.6 334.2 405.1 506.4 (GeV) (%} 114.5 32.29 133.6 28.41 157.4 24.69 190.8 20.86 238.5 17.07 DF Appen.di:t B Event Kinematics Consider the diagram below. ' Figure B.1. Charged current process The 4-momentum or the incident neutrino is k and that or the outgoing muon, k' . The proton is initially p, the final hadron shower r/. (B.l) (B.2) B. Event Kinematics 153 v =p · (k- k') = m,(Ev- Ep) = m,(Eh- m,) y =-p·v k = m,(Eh- m,) Eh Ep ~ ~ 1- mpEv Ev Ev .. (B.3) (B.5) Ir the struck quark has negligible mass but produces another quark with possibly significant mass m 9 , -Q 2 or, Neglect + + xp) = m; + x m; = m; · 2 (q 2xq · p 2 xm; to give 2 (B.6) Neglecting mg, m, gives (B.7) or (B.8) AJso, from (B.l), (B.9) Thus, or directly from (B.l), • 8 SID h IPpl . 8 1- y • 8 = -,- , SID p ~ - - S I D p • Ph Y (B.lO) It is easy to see that (l _ y) = (1 + coso•) . 2 This follows trivially if we assume that all centre-of-mass energies are equal to f and all particles are massless. Ir 1 be the boost from the centre-of-mass to the lab frame, Ep = 1f + 1Hose• Ev = 1f+1f, from which and (B.5) the result is obvious. (B.ll) Appen.di:r. C Differential Cross-sections Presented here is the expression for the differential cross-sections for neutrino and antineutrino scattering off nucleons, central to extraction or structure functions. A brier explanation precedes the formula. The total cross section for a neutrino-quark collision, barring spin considerations, is 2G2 s (C.1) Both the neutrino and the quark are spin-1/2 particles, glVmg rise to 4 possible situations. However, standard theory tells us that only lett-handed neutrinos exist, and at high energies, interact only with lett-handed quarks and right-handed antiquarks. We also consider the possibility of spin-zero constituents or nucleons. So all possible situations can be summarized as in fig. C.l. & shown in appendix B, ( _ y) = (1 1 + coso•) 2 .. (C.2) IC x be the fraction or the nucleon momentum carried by the struck quark, the differential cross-section for neutrino quark collisions is cPa" G2 s - - = -·x·q(x) dxdy 1r (C.3) The s in (C.1) must be replaced by sx, the square or the centre-of-mass energy for the neutrino-quark system as opposed to that or the neutrino-nucleon system. Since only lefthanded quarks can interact, a factor of 1/2 appears. q(x) dx is the probability of finding a quark with momentum fraction in [x, x dx). Because J = 0, there is no y factor in this case. In general though, one must include the appropriate power of (1 cosB.)/2 or equivalently, or (1 - y). For example, the antineutrino quark cross-section can be written + cf2ali G2 s -- = · X q( X) • (1 - y )2 dxdy 1r + (C.4) 155 C. Dilrerential Cross-sections Antineutrinos Neutrinos q II 8;a = -1/2 8;a = 1/2 q II J=O I1 q J=1 1 s, = 1/2 s, = 1/2 ((l+coso•)/2) I1 q J=O s, = 1/2 s, = -1/2 1 k J = 1/2 ((l+coso•)/2) ]=1 2 s, = -1/2 s. = -1/2 ((1 +coso• )/2} k II 8;a = -1/2 s. = 0 J = 1/2 ((I +cosB.)/2) I1 8;a = 1/2 s, =0 2 Figure C.!. Spin considerations diagram Let u, d, s and c represent x times the probability density of finding au, d, s or c quark in [x, x+dx). The subscript v refers to valence quarks only. The superscripts p,n and N refer to the proton, neutron and nucleon respectively. k(x)dx is x times the probability that a spin-zero object lies in the interval [x, x + dx). Since the target is not exactly isoscalar, we introduce z !=<->. A (C.5) ' We assume eN = eN = ue = d~' de = u~ 0,} (C.6) (C.7) and Also, the strange sea is expected to have the same x dependence as tr and d, therefore t s=s= -~ · q, (C.8) 2+t where t is a constant expressing the fraction of "SU(3) symmetric" content due to the strange sea. 1 (C.9) (u= d= 2+ t - -v For a "half SU(3) symmetric" sea, t = 1/2 (C.lO) C. Dilrerentia.l Cross-sections 156 Ignoring charm mass corrections ror the moment, we then get 2 G s. {(d N + s N ) + (1- y)2-N -rPav = u + {1 - y)k } , dxdy 1r (C.ll) (C.12) Clearly, dN = f(df, + (/) + (1- f)(d~ +d)= !elf,+ (1- /)u~ + (/ (C.13) f(u~ +uP)+ {1- /)(u~ + u") = fu~ + (1- f)d~ +uP (C.l4) 2xFt= q+lf=u~+d~+2l' +2sP } F2= q + q + 2k = u~ + d~ + 2dP + 2sP + 2k xFa= q- q = u~ d~ (C.15) uN = Defining + we can reduce (C.13) and (C.14) to dN = uP u + dPu + (1- 2/) (uPv - dP)u + dp = ~ + ( _ D 2 2 2 1 ( 2/) Uu - d ) u + dp 2 Similarly, (C.l6} (C.17} Using (C.15) and (C.l6) we may rewrite (C.ll} as .. rPav = G2s {xFa +(1- 2/)(uu- du)+ {2xFl - xFa + sN} dxdy 21r 2 N} 2 +(1-y) {2xFt- xFa -s +(1-y)(F2-2xFt)} 2 2 2 2 = G s {xF3 ( 1 - ( 1 - y) ) + F2(l- y) + 2xFt · Y 2'1f' 2 + (1- (1- y) )sN + (1- 2/)(uv- du)} (C.l8} 2 2 Target mass effects make the (1- y)F2 into (1- y- mxyf2E}F2. (C.l9} Thus, since R is defined by (C.20) 157 C. Dilferenti.?J Cross-sections rflav,v G2s ( y2 (1 - - = - { 1-y+-· dx dy 211'" 2 + 2m2x2 /Q2)) F2± (1- (1- y)2) xFa (1 + R) 2 2 ±(1- 2/)(( ~ y)2}u~- ~) + (1- (1- y) )sN} 1 (C.21) Also, in reality me =j:. 0, making it necessary to recalculate (C.21). This has been done<24) and the prescription is that (i) emust be used instead of where e= (Q + m~)/2mEh (see Appendix B). 2 X (ii) Only the region where { :5 1 is allowed. (iii) The y distribution coefficients 1 and (1 - y)2 become {(1- y) + x; }e(l- e> respectively. For our purposes it will be simplest to rewrite (C.21) in such a way that only a "correction term" gets added. In what follows the subscript eimplies that the quark density in question is a function of{, not of x. For ex ample, (C.13) may be written as dN = (fd~+(l- f)u~+dN )cos2 Be+Ud~+(l - f)u~+dN )e sin2 8e(l- Y+ xy )8(1- {) (C.22) e One can add and subtract (fd~ + (1- f)u~ + dN) sin2 Be; the final expression works out to be rflav,v = G2mpEv dx dy 1r(l Q2 /mw )2 + {(1- Y+ f · (1+2m2z2/Q2))F. ±(l-(l-y)2)xF (t+R) 2 + a 2 ( l - (1- y) 2 )sN ±(1- 2/)((l~y)2)(u~- d~) +R11 ,r;(x, Q2 ,E)+ C 11,r;(x, Q 2 , E, m e)} (C.23) where the isoscalar and strange sea correction terms are explicit, R 11,r;(x, Q2 ,E) is the radiative correction term, and the charm mass correction terms are: Cv = 2[- sin2 Be(/dv+(l- J)uv+dN)+Udv+(l- /)uv+dN)e sin2 Be(l- Y + =[-)8(1- {) - cos2 8eSN +sN, cos2 Be(1- y + =(-)8(1- OJ (C.24) and, - 2 C;;- = 2[- dNsin Be 2 - SN cos 8e + =[-)8(1- {) +sN, cos Be(l - y + =(-)8{1- OJ +ilN, sin2 Be(l - y 2 (C.25) C. Dilferentia.l Cross-sections 158 The l/{l+Q 2 fm'tv) 2 in (C.23) comes from theW-boson propagator which we neglected in earlier expressions for simplicity. Notice that here we neglect W 4 and W 5 since they come in as -(::r. Ap'Pendi:t D Multiple Scattering Multiple scattering enters both the track fitting procedures - in the determination or p 11 and or 811 at the vertex. The two are quite different problems; the physics and logic is discussed in Chapter 5. Here we present the mathematical mechanics starting with the simpler case or 811 • Both problems are linearized i.e., the displacements (or, in general, measurements), written in vector form as y, are linearly related to the parameters c by (D.l) y=Ac Ir we knew the true set or parameters c, t: would be the 'error' vector where (D.2) c=y-Ac The covariance matrix or the measurements y is ' (D.3) For the general case or correlated errors and unequal weights, u ij is a symmetric matrix and the problem is to minimize S = (y- Acf (u- 1 )(y- Ac) (D.4) with respect to the parameters c. This minimization is satisfied by (D.5) In the shower penetration case (811 ), we merely use (5.2), (5.3) and (5.4) along with i y; =Yo+syz;+ I:<ot+B~cz~c;) k=l (D.6) 160 D . Multiple Scattering where Yo =they-view intercept .sy = the y-view slope z, =the z..coordinate for the ith scattering centre o1e = kth multiple scattering induced displacement 81e = kth multiple scattering induced angle Zlei =distance between scattering centres k and i, to get i i a,i =<(y,- y~)(y;- yJ)> =< L L(o~e + B~ez~e,)(ol + Olzl;)> {D.7} le=ll=l . ~ 2 Lk L~e = LJ ao[3 + T(z~e; + z~ei) + ZlciZiej} 2 (D.8) le=l L1c is the thickness or the kth scattering centre, z~e; is 0 if k > j and y 0 is the (straight line) path without multiple scattering. The parameters are merely y0 , sy, x 0 and s,.; no attempt is made to recover all the o1c and O~e. A is thus a trivial 2x2 matrix and the main problem is to invert a. Calculating a~ , including dEfdx losses, is easy since p11 is known. For the muon momentum tracking, we need not only the slopes and intercepts but also all the o~e, 81e and the muon momentum. As mentioned in section 5.1, the problem is first linearized by iteratively using the last estimate of IPI· The matrix A is enormous- it couples, in general, 36 angles, 36 deltas, and all the sparks in both views to the slopes, intercepts, 1/lpl, and the 36 angles and 36 deltas (there are 18 scattering centres). Most of the CP time is spent in matrix multiplication of A in spite of some or its blocks being diagonal. It is necessary to obtain all 01e, 81c because the magnetic field is a function of position and good tracking helps ' get the correct magnetic kicks and eliminate bad sparks. Note that for both fits an intrinsic spark resolution, s2 , is added to the diagonal elements aii. For the linearized problem, one application of (D.5) gives the best estimates of the parameters. In the case of Bp, we repeat the procedure getting closer to the vertex by adding one plane at a time, as described in section 5.2. For pp, there is no hadron shower to contend with, but we iterate to improve Pp {the problem is artificially linearized). A-ppendix E Event Counting for Total Cross-sections In this appendix we describe the separation or events into various categories for evaluating cross-sections. For events with En < 10 GeV, we can calculate a maximum angle. ' y 1 e2 ,.....2mx--I - yEv ema~ But = ( 2m E~in = 28.5 GeV )1/2 (E.1) ema~ Rj 189 mrad. (E.2) 1 ( 10/Emin) v 1 - (10/ E~in) E~in The angle beyond which the muon trigger efficiency cannot be geometrically recovered is the angle a line from the apparatus centre at z=-167 in (closest event to the toroids) to a corner or T2 (55 in square, z=l42.5 in ) makes with the apparatus axis; this is 251 mrad. Hence (E.l) indicates that for En < 10 GeV, all events fire the muon trigger and the muon energy is measured. At every radius, an energy Esep can be defined that separates neutrinos from pion and kaon decays. One can safely assume that if En > .85 Esep, the event is from a neutrino from kaon decay. This leaves the region 10 GeV < En < .85 Eup in which to distinguish pion and kaon induced events. In this region it is possible to show using the energy versus radius expressions outlined in appendix A, that at every radius the maximum possible muon angle from kaon-decay-neutrinos is smaller than the maximum needed for Cull geometrically corrected acceptance. Consequently, events with Ev > Eeep are identified by 81 as kaon-decay-neutrino events. For Rbeam <30 in and Rbeam > 30 in , their muon trigger geometric weights (WMUZ) can be added up to give the total kaon-decay-neutrino events. The penetration trigger geometric weights (WPEN) are summed up for all 83 events with Rbeam <30 in . We can thus make 6 categories of events: E. Event Counting for Total Cross-sections 162 No. Weight summed Category 1 WMUZ. Eh < 10GeV, R <30 in, E < E••P• S1 2 Eh < 10GeV, E > Eup, S1 WMUZ 3 WPEN 10 < Eh < .85E••P• R <30 in , S3 4 10 < Eh < .85E,.p, R <30 in, E > Esep, S1 WMUZ 5 10 < Eh < .85E••P• R >30 in , E > E••P• S1 WMUZ 6 WMUZ Eh > .85E••P• S3 Table E.1. Event categories for cross section analysis. Clearly, L Cat.1 + L Cat.3- L Cat.4 neutrinos from kaons = L Cat.2 + L Cat A+ L Cat.5 + L Cat.6 neutrinos from pions= A few words about corrections to cross-sections: The correction for unsampled regions described in §6.2 is typically ,..... 1% and is never larger than 2.6%. The basis for our Monte Carlo is a parametrization of structure functions which is also used to make the isoscalar correction {,..... - 1.7% for neutrinos, 1.1% for antineutrinos). The wideband events are directly eliminated by using events !rom closed slit runs scaled to the correct number of open slit protons. The assumptions made for the wide band background are that it has no position dependence and that, because of the way the beam is dumped, it is the same for all antineutrino settings but different for neutrino settings. Table E.2 lists various features or this background. + Momentum 2:WBB 2:WBB-NOR 2:WBB-Nclt NcR Setting Rescaled to {GeV/c) Open slit Protons -250 462.5±82.5 38.8±6.9 .125±.125 38.7±6.9 -200 12.1±4.0 .5±.25 11.6±4.0 262.3±90.4 -165 0. 188.2±71.3 10.3±3.9 10.3±3.9 227.3±62.9 -140 20.0±5.2 18.8±5.2 .15±.15 -120 117.5±43.4 .16±.16 9.2±3.4 9.4±3.4 120 66.5±12.9 504.1±97.8 80.3±10.3 13.8±7.8 3·0 .9±5.9 22.7±10.5 8.2±12.0 109.0±159.5 140 597.7±211.5 165 35.9±12.7 56.6±8.7 20.7±9.2 876.5± 113.1 200 176.2±15.0 55.3±17.1 120.9±15.6 1550.0±241.0 250 189.0±15.9 50.3±14.5 138.7±21.5 E Open slit With WBB cuts 3066 3664 3607 2949 2054 11496 14479 23123 27998 33338 WBB Fraction {%) 15.1 ±2.7 7.2 ±2.5 5.2 ±2.0 7.7 ±2.1 5.7 ±2.1 4.4 ±0.9 0.8 ±1.1 2.6 ±0.9 3.1 ±0.4 4.'6 ±0.7 Table E.£. Wide band background fraction analysis. Using the cosmic ray gates, we find less than ,....., 0.1 % of fast spill events are due to E. Event Counting for Total Cross-sections J63 cosmic rays. Neutrinos during slow spill have a significant cosmic ray background however ("-' 2.8%), which is subtracted for each momentum setting and category using the properly scaled number of cosmic ray events from cosmic ray gates. We know that events are not being double counted because the DSTs contain events in strictly increasing order. To summarize, table E.3 contains the number or raw events, the corrected number or pion and kaon events, and the predicted number or pion and kaon events with a/ E 1 x 10- 38 cm2 /GeV. The final result (with statistical averaging alone) is (.691±.0036)xl0- 38 cm2 fGeV 38 Uv = (.685±.0036)xl0cm2 fGeV 38 av = (.339±.0034)xl0- cm2 fGeV. Uv = and (slow spill) (fast spill) This agrees with the detailed calculations involving more correct averaging or energy settings (using the estimated systematic errors) that led to the published resuit(30) or ' and a 11 = (.669±.003±.024)x 10-38 cm 2 fGeV av = (.340±.003±.020)x 10- 38 cm2 fGeV E. Event Counting ror Total Cross-sections Momentum Neutrino type Setting -250 1T K -200 1T K -165 1T K -140 1T K -120 1T K 120 1T K 1T 140 K 165 1T K 200 1T K 250 1T K Corrected# 'Predicted' # or events or events 2100.13±52.90 6433.05 449.74±23.66 1489.56 2453.75±54.70 7283.42 520.46±25.26 1674.97 2227.52±51.65 6393.15 542.04±25.50 1661.65 1948.66±48.30 5465.21 449.52±23.47 1335.91 1339.34±40.23 3673.43 247.56± 17.58 761.49 7268.38± 100.16 10540.67 2301.68±56.10 3417.80 9474.95±122.29 13551.28 5376.74 3654.11±69.95 21984.93 14715.43±148.45 10851.70 7638.39±104.36 24805.42 16777 .58± 158.68 10773.90±124.48 15503.35 16682.83± 156.72 24986.88 15989.53± 147.35 21954.82 164 Cross section x10 38 cm 2 /GeV) .326±.0082 .301±.0158 .336±.0075 .310±.0150 .348±.0080 .326±.0153 .356±.0088 .336±.0175 .364±.0109 .325±.0231 .689±.0095 .673±.0164 .699±.0090 .679±.0130 .669±.0067 .703±.0096 .676±.0064 .694± .0080 .667±.0062 .728±.0067 Table E.S. Events for elementary cross-section analysis. ' Ap-pendb: F Models for Fitting Structure Functions Structure function extraction requires some iteration since the correction terms depend o~ the structure functions themselves. We have used two models to fit the structure functions and these are described below. In the empirical model, we are motivated by standard forms for 2:r:F1 and :r:Fa at fixed Q2 , namely 2:r:F1(Q 2 = 10GeV2 ) = d(1 + e:r:)(1- x)f :r:Fa(Q 2 = 10 GeV2 ) = a(1 + c'd(1 - :r:)):r:6(1- xY (F.l) (F.2) In the above, a, b, c, d, e and f are parameters. c'd reflects the faster (1 - x) fall-off expected for d-quarks( 3 l). At fixed x, in the restricted regions or Q 2 explored by any experiment, it is reasonable to fit straight lines to 2xF1 and xF3 as functions of log Q2 : b 2:r:F1 (fixed :r:) =a+ b log10 Q 2 /10 = a(1 + -log10 Q 2 /10) a (F.3) If bfa is plotted as a function of x, one finds in generalll parabola with its minimum pointing towards -:r:. Thus, a Q2 -dependence of the type (1 + (g - h,fii) log10 Q2 /10) is used to multiply (F .1) and (F .2) above, giving the forms 2:r:F1= d(1+ex)(1-:r:)f(1+(g-h,fii}log10 Q 2 /10) } :r:Fa =a(1 + c'd(1 - :r:))x 6(1- x)c(1 + (g- hvfx)log10 Q2 /10) (F.4) where a, b, c, d, e, f, g and h are parameters obtained by a simultaneous fit to 2xF1 and xFa. We fix c'd with the requirement that there be twice as many u quarks as d quarks in a proton. This leads to 2c'd= (f(b)f(c+1)/f(b+c+1))(1+g log10 ~)-(r(b+1/2)r(c+1)/f(b+c+3/2))h log 10 ~ (f(b)f(c+2)/f(b+c+2))(1 log10 ~ )- (f(b+ l/2)f(c+2)/f(b+c+S/2})h log 10 ~ (F.S) +g J66 F. Models tor Fitting Structure Functions The constants after the final iteration were, for R = 0.1, a = 2.083±.218 b = 0.482±.421 c = 2.459±.134 d = 1.654±.203 {F.6) e = 0.283±.190 f = 2.440±.106 g = 0.445±.038 h = 1.043±.075 The other model is a QCD-inspired fit like that of Buras and Gaemers(·U}, which we have altered slightly in order to keep the number of paramters to a minimum. It should be emphasized that this fit is not an attempt to extract a value for A from the data, but rather is just another model used as a basis for the corrections to be applied in structure function extraction. Both models yield a x2 of approximately 210 for 152 degrees of freedom for their best fits and thus neither seems intrinsically superior. For our published results we use the 'Buras-Gaemers' model. In this model the valence quark distribution is approximated by {F.7) where and 17i(s) = 17io + S7Jil, i = 1,2. (F.8) Again, we have chosen to separate the {1- x) dependence of u and d quarks. As before, this implies that (F.9) vVe assume that the gluon distribution has the form (F.10) where Ao is fixed by the momentum sum rule: (F.ll) F. Models for Fitting Structure Functions 161 The sea or antiquarks and quarks is modeled by two moments as xS(x, Q 2 ) = Ps( 1 - 1)(1- x)(l/c>.)- 2 <x>s (F.12) where Ps =<S(Q 2 )>2 and < S(Q2 )>2 1 <x>s (F.13) <S(Q 2)>a < S( Q 2 ) >,. are moments or the sea distribution, defined by 1 <S(Q 2 )>,.= fo dxx"- S(x,Q 1 2 ). (F.14) Because or the coupling or the Q2 evolutions or F 2 and G, < S(Q2) >n= ~D&">(Q 2 ) + !_D~"l(Q2 ) 4 4 (F .15) where D~">(Q 2 )= <S(Q~)>n e- 7 •-; D&"l(Q2 ) ={(1- o,.) < q 5 (Q~) >n -fJ,. < G(Q~)>,.}e-'1+8 +{on < ~(Q~) >n +fln < G(Q~) >n}e- 7 ~ 8- < zFa(Q~) >n e-,•• (F.16) o,., fJ,., '1", '1+ and 1::. are known constants and < q5 (Q~) >,.=< S(Q~) >,. + < xFa(Q~)>,. (F.17) Thus, the parameters are A, A, 1J1o, 1J11, 1J2o, 1121, < S(Q~) >2 and < S(Q~) >a. This model was fit for all three values of R and the final values of the parameters are quoted in table F.l. For this fit we did not throw out points with.. large systematic errors as in the QCD fits described in the chapter 7; nevertheless, the value or A is consistent with our final result. Parameter R=O. R=0.1 R = .73{1- x)3 · 6 l:S /ln(Q 2 /.242 ) A (MeV) A 202.1±133.6 1.838±.113 0.4382±.0277 -0.1507 ±.0540 2.285±.0648 1.849±.4278 0.1671±.0055 0.0202±.0019 194.7 ±126.0 1.679±.129 0.4003±.0341 -0.2066±.0685 2.271±.0813 1.861±.4309 0.1453± .0056 0.0167 ±.0011 {QCD) 247.4±114.58 1.702±.105 0.4079±.0273 -0.2156±.0557 2.225±.0642 1.742±.3174 0.1494± .0053 0.0180±.0012 7110 7111 7120 7121 <S(Q~)>2 <S(Q~)>a Table F.1. Best parameters for Buras-Gaemers model; different R values. APl>endi:t G R==O.l Structure Functions :r .015 .045 .080 .150 Q2 1.259 1.995 3.162 5.012 7.943 1.259 1.995 3.162 5.012 7.943 12.589 19.953 1.259 1.995 3.162 5.012 7.943 12.589 19.953 31.623 1.995 3.162 5.012 7.943 12.589 19.953 31.623 50.119 79.433 F2 1.25582 1.30798 1.49922 1.36980 1.54774 1.13397 1.35019 1.34836 1.52800 1.64677 1.51965 1.06799 1.24675 1.44714 1.48609 1.46027 1.54216 1.59194 1.58351 1.24093 1.19379 1.18550 1.28368 1.24077 1.24165 1.30537 1.29031 1.18841 0.91729 AF2 .04923 .05698 .08192 .11360 .26328 .05011 .04659 .04393 .05793 .07982 .11269 .19609 .12790 .05451 .04624 .04377 .05127 .06791 .09018 .16773 .10906 .03639 .02832 .02554 .02834 .03650 .04696 .07711 .26105 xF3 .17085 .37642 .30911 .43427 AxFa .05995 .05721 .07770 .10700 - - .44840 .62746 .61455 .51304 .69616 .64685 .63006 .14579 .08976 .06288 .07227 .09385 .11640 .18132 - - .62005 .77359 .18380 .10383 .07019 .06910 .08531 .09766 .16152 ' .66828 .65696 .78149 .77123 .66191 - - .73272 .68936 .87819 .85393 .78867 .84621 .79932 .70531 .13496 .07349 .04599 .04163 .04845 .05511 .07983 .23941 G. R=O.I Structure Functions X .250 .350 .450 .550 .650 Q2 3.162 5.012 7.943 12.589 19.953 31.623 50.119 79.433 125.893 5.012 7.943 12.589 19.953 31.623 50.119 79.433 125.893 7.943 12.589 19.953 31.623 50.119 79.433 125.893 199.526 12.589 19.953 31.623 50.119 79.433 125.893 199.526 12.589 19.953 31.623 50.119 79.433 125.893 199.526 169 F2 1.87372 1.02767 .98905 .94128 .93629 .96817 .86187 .77668 .59753 .88153 .67718 .65216 .64518 .63736 .60015 .61857 .71853 .49761 .42063 .39816 .38788 .35372 .38524 .29228 .30656 .24130 .23222 .20861 .20174 .20319 .15649 .12753 .14995 .12099 .13931 .11247 .10015 .09539 .07128 ~F2 xFa .72552 .03878 .02655 .02431 .02674 .03391 .03467 .05556 .29864 .18938 .02735 .02262 .02406 .02784 .02871 .03848 .08329 .04115 .01982 .01881 .02067 .02151 .02722 .02978 .14743 .01716 .01559 .01451 .01835 .02093 .02045 .04488 .01762 .01048 .01519 .01471 .01641 .01427 .02238 - - .62169 .79181 .79395 .76636 .79476 .74495 .64935 .16070 .07607 .04792 .04233 .04956 .04222 .05738 - - .58121 .55024 .46626 .50918 .55930 .49863 .26618 .10778 .05929 .04723 .04765 .04092 .04520 .08603 ~xFa - - .44968 .30443 .31655 .33689 .31495 .30254 .29994 .21663 .16722 .20694 .16959 .16740 .16856 .13031 .16111 .17002 .12085 .07132 .05793 .09799 .07628 .06724 .04292 .03809 .03554 .03635 .03242 .14598 .07035 .04326 .02921 .03231 .03059 .02440 .04641 .07902 .03294 .03925 .03014 .02771 .01844 .02425 Table G.t . F2 and xF3 with statistical errors for R=0.1 A-p'Pen.dix H Tables of Statistical and Systematic Errors % .015 .045 .080 .150 Q2 (GeV2) F2 1.3 2.0 3.2 5.0 7.9 1.3 2.0 3.2 5.0 7.9 12.6 20.0 1.3 2.0 3.2 5.0 7.9 12.6 20.0 31.6 2.0 3.2 5.0 7.9 12.6 20.0 31.6 50.1 79.4 1.287 1.342 1.536 1.402 1.584 1.133 1.358 1.363 1.544 1.661 1.530 1.070 1.243 1.444 1.487 1.464 1.544 1.590 1.576 1.227 1.182 1.180 1.280 1.234 1.232 1.291 1.269 1.160 .888 Stat error (%) 3.92 4.35 5.46 8.27 16.97 4.43 3.45 3.26 3.79 4.84 7.41 18.34 10.29 3.77 3.11 3.00 3.32 4.26 5.69 13.50 9.21 3.07 2.20 2.06 2.28 2.79 3.64 6.49 28.26 Shape uncorrel (%) .67 .57 .38 .24 .37 .66 .76 .75 .52 .26 .29 .43 .64 .65 .75 .79 .57 .27 .28 .47 .69 .69 .75 .78 .59 .32 .24 .44 .08 Shape correl (%) 2.22 1.95 1.15 .72 .90 2.11 2.41 2.50 1.70 .74 .83 .69 2.04 2.09 2.37 2.58 1.85 .71 .82 .77 2.21 2.19 2.39 2.55 1.91 .83 .71 .85 ·.15 Flat xsec correction (%) .69 .70 .89 1.31 1.55 .71 .69 .64 .64 .99 1.25 1.45 .72 .70 .68 .62 .60 .87 1.12 '1.22 .76 .70 .66 .60 .56 .73 .91 .85 .91 Level error (%) 3.86 3.93 4.00 4.08 4.12 3.57 3.66 3.73 3.79 3.87 3.93 3.92 3.55 3.51 3.58 3.67 3.73 3.79 3.86 3.84 3.77 3 .55 3.52 3.59 3.63 3.68 3.74 3.62 3.43 Ehad di.ff (% ) ·.18 ·.12 -.09 ·.05 .01 -.17 ·.16 ·.12 -.08 -.06 -.01 .03 ·.09 -.08 ·.10 - .07 -.04 -.03 - .00 .03 .01 .07 .07 .05 .05 .06 .05 .05 .04 H. Tables or Statistical and Systematic Errors X Q2 F2 (GeV2) .250 .350 .450 .550 .650 3.2 5.0 7.9 12.6 20.0 31.6 50.1 79.4 125.9 5.0 7.9 12.6 20.0 31.6 50.1 79.4 125.9 7.9 12.6 20.0 31.6 50.1 79.4 125.9 199.5 12.6 20.0 31.6 50.1 79.4 125.9 199.5 12.6 20.0 31.6 50.1 79.4 125.9 199.5 1.868 1.025 .985 .933 .923 .952 .839 .747 .568 .882 .676 .648 .637 .627 .587 .597 .684 .497 .419 .394 .382 .347 .374 .279 .289 .240 .230 .205 .198 .198 .149 .119 .149 .120 .138 .llO .098 .092 .067 Stat error (%) 38.73 3.77 2.68 2.58 2.85 3.50 4.02 7.08 49.14 21.46 4.03 3.46 3.73 4.37 4.78 6.22 11.13 8.26 4.71 4.72 5.33 6.09 7.07 9.95 45.72 7.11 6.72 6.96 9.10 10.30 12.71 33.52 11.75 8.67 10.91 13.09 16.39 14.59 30.26 JTI Shape uncorrel (%) .74 .71 .77 .76 .57 .30 .24 .55 .56 .77 .74 .77 .66 .47 .18 .37 .52 .75 .76 .70 .54 .28 .23 .59 .58 .75 .72 .57 .37 .18 .46 .33 .75 .73 .59 .42 .18 .34 .57 Shape correl (%) 2.36 2.26 2.44 2.50 1.75 .68 .73 .89 -.84 2.46 2.36 2.52 2.20 1.18 .53 .90 .56 2.39 2.47 2.37 1.57 .62 .70 .94 - .83 2.44 2.42 1.83 .82 .57 .96 .21 2.44 2.43 1.99 1.02 .52 .87 .66 Flat xsec correction (%) .76 .68 .63 .57 .54 .72 .81 .72 .78 .73 .64 .59 .53 .59 .75 .70 .69 .65 .60 .54 .53 .69 .72 ' .64 .77 .61 .55 .51 .63 .73 .64 .72 .60 .55 .51 .59 .74 .68 .67 Level error (%) 3.85 3.49 3.45 3.54 3.60 3.65 3.73 3.71 3.49 3.77 3.41 3.44 3.52 3.57 3.64 3.73 3.71 3.44 3.38 3.45 3.52 3.57 3.66 3.74 3.64 3.36 3.40 3.47 3.52 3.60 3.71 3.72 3.33 3.35 3.42 3.48 3.55 3.67 3.73 Ehad diff (%) .30 .40 .36 .28 .25 .25 .19 .12 .07 .70 .78 .68 .56 .52 .48 .32 .19 1.28 1.23 1.04 .92 .89 .72 .44 .25 2.00 1.79 1.57 1.49 1.35 .92 .56 3.16 2.98 2.64 2.47 2.35 1.77 1.14 H. Tables or Statistical and Systematic Errors :z; Q2 F2 (GeV2) .015 .045 .080 .150 .250 1.3 2.0 3.2 5.0 7.9 1.3 2.0 3.2 5.0 7.9 12.6 20.0 1.3 2.0 3.2 5.0 7.9 12.6 20.0 31.6 2.0 3.2 5.0 7.9 12.6 20.0 31.6 50.1 79.4 3.2 5.0 7.9 12.6 20.0 31.6 50.1 79.4 125.9 1.287 1.343 1.537 1.402 1.584 1.134 1.359 1.363 1.545 1.662 1.531 1.071 1.244 1.445 1.487 1.464 1.545 1.591 1.576 1.228 1.183 1.180 1.280 1.235 1.232 1.292 1.270 1.161 0.889 1.868 1.026 0.985 0.933 0.923 0.952 0.840 0.747 0.568 Stat error (%) 3.9 4.4 5.5 8.3 17.0 4.4 3.5 3.3 3.8 4.8 7.4 18.3 10.3 3.8 3.1 3.0 3.3 4.3 5.7 13.5 9.2 3.1 2.2 2.1 2.3 2.8 3.6 6.5 28.3 38.7 3.8 2.7 2.6 2.9 3.5 4.0 7.1 49.1 172 E" d.itr (%) 0.39 0.42 0.41 0.36 0.27 0.33 0.37 0.33 0.28 0.24 0.16 0.06 0.20 0.16 0.16 0.11 0.04 0.00 -0.06 -0.15 -0.08 -0.26 -0.34 -0.38 -0.45 -0.50 -0.54 -0.64 -0.76 -0.82 -1.12 -1.19 -1.18 -1.21 -1.27 -1.25 -1.27 -1.37 R error (%) symmetry (%) Model error (%) Extraction techniques (%) 2.10 2.57 2.78 3.19 3.96 0.24 0.85 1.60 1.97 2.19 2.98 3.98 0.31 0.11 0.51 1.16 1.59 1.74 2.49 3.37 1.12 0.38 0.26 0.72 1.14 1.34 1.91 2.72 3.37 0.43 0.23 0.17 0.51 0.86 1.04 1.65 2.57 3.34 -1.51 -1.74 -1.94 -2.22 -2.50 -0.65 -0.98 -1.26 -1.44 -1.69 -2.01 -2.17 -0.40 -0.49 -0.75 -1.02 -1.19 -1.36 -1.67 -1 .83 -0.35 -0.37 -0.51 -0.70 ' -0.82 -0.93 -1.12 -1.17 -1.10 -0.23 -0.22 -0.29 -0.41 -0.49 -0.55 -0.68 -0.74 -0.66 1.67 1.90 1.63 1.08 0.26 2.04 1.89 1.92 1.84 1.51 0.82 0.70 1.82 1.60 1.50 1.51 1.66 1.65 1.09 0.95 1.51 1.14 0.96 1.09 1.39 1.55 1.24 1.00 1.03 0.56 0.43 0.50 0.75 1.15 1.40 1.08 0.81 0.94 0.15 1.65 2.38 -0.78 -7.09 -1.50 0.55 -0.33 0.22 0.48 0.16 -13.41 1.14 -0.42 -0.68 -1.42 -0.22 1.60 -0.96 4.31 0.95 0.40 0.48 0.87 1.04 1.95 -0.25 -2.17 9.05 4.67 0.73 0.72 0.18 1.84 1.73 1.61 1.53 -3.66 SU(3) H. Tables or Sta.tistica.l a.nd Systematic Errors :z; Q2 F2 (GeV2) .350 .450 .550 .650 5.0 7.9 12.6 20.0 31.6 50.1 79.4 125.9 7.9 12.6 20.0 31.6 50.1 79.4 125.9 199.5 12.6 20.0 31.6 50.1 79.4 125.9 199.5 12.6 20.0 31.6 50.1 79.4 125.9 199.5 0.882 0.676 0.649 0.638 0.627 0.587 0.597 0.684 0.498 0.419 0.395 0.382 0.347 0.375 0.279 0.290 0.241 0.231 0.206 0.198 0.199 0.150 0.120 0.150 0.120 0.138 0.111 0.098 0.092 0 .068 Stat error (%) 21.5 4.0 3.5 3.7 4.4 4.8 6.2 11.1 8.3 4.7 4.7 5.3 6.1 7.1 10.0 45.7 7.1 6.7 7.0 9.1 10.3 12.7 33.5 11.8 8.7 10.9 13.1 16.4 14.6 30.3 113 Ep diff (%) -1.83 -2.15 -2.15 -2.11 -2.15 -2.18 -2.08 -2.05 -3.33 -3.45 -3.36 -3.34 -3.40 -3.31 -3.12 -3.11 -5.26 -5.22 -5.13 -5.19 -5.16 -4.84 -4.65 -8.09 -8.19 -8.05 -8.09 -8.15 -7.72 -7.29 R error (%) 0.45 0.13 0.25 0.55 0.80 1.04 1.82 2.70 0.15 0.16 0.38 0.63 0.81 1.29 2.26 3.09 0.12 0.29 0.51 0.68 0.97 1.81 2.78 0.07 0 .22 0.40 0.56 0.76 1.39 2.37 SU(3) symmetry (%) -0.14 -0.13 -0.18 -0.24 -0.27 -0.31 -0.38 -0.39 -0.06 -0.08 -0.10 -0.12 -0.13 -0.16 -0.18 -0.16 -0.03 -0.04 -0.04 -0.05 -0.06 -0.06 -0.07' -0.01 -0.01 -0.02 -0.02 -0.02 -0.02 -0.03 Model error (%) 0.22 0.12 0.38 0.73 1.18 1.17 0.69 0.41 0.15 0.05 0.44 0.90 1.20 0.77 0.52 0.42 0.43 0.10 0.62 0.99 0.80 0.46 0.42 0.75 0.27 0.41 0.82 0.72 0.32 0.10 Table H.1 . F 2 errors. See note at end of appendix. Extraction techniques (%) 4.70 0.23 3.18 0.71 -0.32 3.60 2.41 1.13 1.30 0.57 3.13 0.97 -1.57 0.71 1.47 5.43 1.60 2.72 4.87 5.75 9.21 -6.62 -3.89 1.66 3.28 0.49 -1.60 -4.64 2.24 -0.56 H. Tables o! Statistical and Systematic Errors :r Q2 :rF3 (GeV2) .015 .045 .080 .150 .250 1.3 2.0 3.2 5.0 1.3 2.0 3.2 5.0 7.9 12.6 20.0 2.0 3.2 5.0 7.9 12.6 20.0 31.6 3.2 5.0 7.9 12.6 20.0 31.6 50.1 79.4 5.0 7.9 12.6 20.0 31.6 50.1 79.4 .1649 .3655 .3032 .4306 .4401 .6198 .6081 .5084 .6929 .6460 .6307 .6145 .7693 .6656 .6550 .7806 .7711 .6621 .7318 .6890 .8785 .8548 .7903 .8471 .7993 .7040 .6230 .7938 .7965 .7697 .7988 .7473 .6504 Stat error (%) 35.0 15.2 25.1 24.6 32.6 14.3 10.2 14.1 13.5 18.0 28.8 29.7 13.4 10.5 10.5 10.9 12.7 24.4 18.4 10.7 5.2 4.9 6.1 6.5 10.0 33.9 25.9 9.6 6.0 5.5 6.2 5.7 8.7 114 Shape uncorrel (%) 4.01 3 .23 2.01 1.04 4.65 3.36 2.80 2.05 0.82 0 .86 1.00 4.05 2.98 2.49 1.89 0.72 0.69 0.92 3.47 2.72 2.22 1.63 0 .74 0.49 0 .74 0 .15 3.26 2.57 2.03 1.39 0.57 0.43 0.75 Shape correl (%) 0.61 0.82 0.77 0.90 0.54 1.33 1.77 1.26 0.69 0.88 0.84 0.73 1.45 2.05 1.58 0.67 0.83 0.88 0.98 1.62 2.17 1.79 0.81 0.73 0.93 -0.05 1.25 1.86 2.34 1.80 0.71 0.71 0.96 Flat xsec correction (%) 4.10 3.63 4.26 5.72 5.10 3.15 2.32 2.29 3.06 3.71 4.31 4.37 2.74 1.92 1.81 2.30 2.75 3.09 3.57 2.38 1.65 1.43 ,1.63 1.88 1.82 1.88 3.14 2.10 1.45 1.23 1.40 1.42 1.33 Level error (%) 15.35 12.85 11.80 10.82 18.26 11.89 9.07 8.23 7.78 6.99 6.52 15.81 10.54 7.84 7.00 6.72 6.02 5.57 13.16 9.38 7.02 6.10 5.79 5.27 4.73 4.36 11.80 8.47 6.38 5.55 5.26 4.74 4.31 Ehad difl' (% ) 0.08 0.05 0.05 0.08 0 .28 0.19 0 .12 0.07 0.08 0 .08 0.07 0.38 0.29 0.18 0.13 0.12 0.09 0.07 0.54 0.47 0.33 0.25 0.23 0.18 0 .12 0.06 0 .89 0 .79 0.59 0.47 0.45 0.34 0 .19 H. Tables or Statistical a.nd Systematic Errors % Q2 :rF3 (GeV2) .350 .450 .550 .650 7.9 12.6 20.0 31.6 50.1 79.4 125.9 12.6 20.0 31.6 50.1 79.4 125.9 199.5 12.6 20.0 31.6 50.1 79.4 125.9 199.5 12.6 20.0 31.6 50.1 79.4 125.9 199.5 .5836 .5528 .4687 .5124 .5628 .5003 .2657 .4522 .3063 .3188 .3395 .3168 .3026 .3015 .2177 .1683 .2086 .1711 .1688 .1695 .1298 .1628 .1714 .1217 .0720 .0584 .0989 .0764 Stat error (%) 18.6 10.8 10.1 9.3 7.3 9.1 31.1 15.0 14.1 12.0 10.5 11 .5 10.5 46.2 32.6 25.9 14.1 19.1 18.3 14.0 33.9 49.1 19.4 32.4 42.1 47.8 18.4 30.6 115 Shape uncorrel (%) 2.93 2.29 1.69 0.98 0.32 0.54 0 .62 2.58 1.94 1.25 0.52 . 0.37 0.71 0.63 2.87 2.19 1.48 0.73 0.31 0 .60 0.38 3.05 2.43 1.70 0.92 0.34 0.51 0.67 Shape carrel (%) 1.62 2.22 2.24 1.30 0.51 0.93 0.63 2.04 2.35 1.72 0.68 0.68 1.01 -0.78 1.92 2.32 1.95 0.95 0.54 1.01 0.27 1.88 2.25 2.08 1.18 0.51 0.91 0.74 Flat xsec correction (%) 2.55 1.69 1.27 1.16 1.35 1.14 1.09 2.00 1.41 1.16 1.29 1.18 0.99 0.98 2.29 1.60 1.25 1.25 1.29 1.01 0.96 2.45 .. 1.80 1.37 1.28 1.40 1.11 0.96 Level error (%) 9.92 7.18 5.75 5.23 4.87 4.37 4.03 8.14 6.21 5.38 5.04 4.50 4.11 3.81 9.11 6.81 5.66 5.22 4.71 4.22 3.92 9.58 7.47 6.03 5.44 4.97 4.37 4.02 Ehad ditr (%) 1.29 1.10 0.87 0.76 0.71 0.47 0.25 1.75 1.45 1.23 1.17 0.95 0.56 0.31 2.60 2.29 1.96 1.82 1.66 1.10 0.66 3.83 3.56 3.11 2.86 2.72 2.02 1.28 H. Tables or Statistical and Systematic Errors X Q2 xFa (GeVZ) .015 .045 .080 .150 .250 1.3 2.0 3.2 5.0 1.3 2.0 3.2 5.0 7.9 12.6 20.0 2.0 3.2 5.0 7.9 12.6 20.0 31.6 · 3.2 5.0 7.9 12.6 20.0 31.6 50.1 79.4 5.0 7.9 12.6 20.0 31.6 50.1 79.4 .1649 .3655 .3032 .4306 .4401 .6198 .6081 .5084 .6929 .6460 .6307 .6145 .7693 .6656 .6550 .7806 .7711 .6621 .7318 .6890 .8785 .8548 .7903 .8471 .7993 .7040 .6230 .7938 .7965 .7697 .7988 .7473 .6504 Stllt error (%) 35.0 15.2 25.1 24.6 32.6 14.3 10.2 14.1 13.5 18.0 28.8 29.7 13.4 10.5 10.5 10.9 12.7 24.4 18.4 10.7 5.2 4.9 6.1 6.5 10.0 33.9 25.9 9.6 6.0 5.5 6.2 5.7 8.7 116 Ep diff (% ) 0.67 0.67 0.63 0.59 0.42 0.46 0.47 0.44 0.41 0.38 0.34 0.11 0.15 0.20 0.19 0.16 0.16 0.14 -0.35 -0.36 -0.29 -0.29 -0.32 -0.30 -0.33 -0.40 -1.24 -1.24 -1.11 -1.07 -1.11 -1.03 -0.97 R error (%) 2.20 1.05 1.77 0.64 0.10 0.37 0.61 1.26 0.69 0.36 0.13 0.06 0.20 0.48 0.78 0.63 0.34 0.16 0.05 0.13 0.23 0.38 0.48 0.24 0.09 0 .07 0.08 0.04 0.15 0.27 0.29 0.17 0.06 SU(3) symmetry (%) 0.48 0.47 0.35 0.18 0.41 0.41 0.41 0.40 0.21 0.06 0.01 0.27 0.28 0.29 0.29 0.20 0.05 0.00 0.17 0.16 0.17 0.16 0.11, 0.02 -0.01 -0.01 0.09 0.09 0.09 0.10 0.07 0.03 -0.01 Model error (%) 3.26 1.60 2.34 1.06 1.11 1.06 1.13 1.99 1.40 0.41 0.10 1.59 1.02 1.07 1.53 1.51 0.50 0.08 1.64 1.34 0.91 1.19 1.49 0.72 0.15 0.00 2.33 1.22 0.99 1.25 1.49 0.69 0.17 Extraction techniques (%) -35.16 6.75 -12.16 -2.19 18.87 -0.43 2.32 -8.77 3.96 -3.12 8.55 12.34 8.14 . 0.13 -0.31 1.36 -7.05 -15.51 0.72 -1.16 1.89 0.97 -2.09 -3.79 2.18 -24.75 -1.65 1.61 -2.49 -0.35 2.40 3.35 3.89 H. Tables or Statistical a.nd Systematic Errors z Q2 zFa (Ge¥-2) .350 .450 .550 .650 7.9 12.6 20.0 31.6 50.1 79.4 125.9 12.6 20.0 31.6 50.1 79.4 125.9 199.5 12.6 20.0 31.6 50.1 79.4 125.9 199.5 12.6 20.0 31.6 50.1 79.4 125.9 199.5 .5836 .5528 .4687 .5124 .5628 .5003 .2657 .4522 .3063 .3188 .3395 .3168 .3026 .3015 .2177 .1683 .2086 .1711 .1688 .1695 .1298 .1628 .1714 .1217 .0720 .0584 .0989 .0764 Stat error (%) 18.6 10.8 10.1 9.3 7.3 9.1 31.1 15.0 14.1 12.0 10.5 11.5 10.5 46.2 32.6 25.9 14.1 19.1 18.3 14.0 33.9 49.1 19.4 32.4 42.1 47.8 18.4 30.6 ITT Ep ditr (%) -2.34 -2.24 -2.09 -2.07 -2.09 -1.90 -1.81 -3.72 -3.52 -3.41 -3.45 -3.30 -3.02 -2.96 -5.72 -5.56 -5.37 -5.38 -5.33 -4.89 -4.63 -8.73 -8.72 -8.46 -8.42 -8.47 -7.91 -7.36 R error (%) 0.08 0.05 0.19 0.27 0.23 0.14 0.09 0.06 0.08 0.20 0.19 0.19 0.06 0.02 0.21 0.07 0.09 0.24 0.22 0.08 0.03 0.17 0.09 0.07 0.17 0.31 0.12 0.05 SU(3) symmetry (%) 0.04 0.05 0.05 0.05 0.03 0.01 0.00 0.02 0.03 0.03 0.02 0.01 0.00 0.00 0.01 O.Dl 0.01 O.Dl 0.01 0.00 0.00 0.00 0.00, 0.00 0.00 0.00 0.00 0.00 Model error (%) 1.86 1.30 1.29 1.55 1.35 0.48 0.26 1.36 1.47 1.49 1.58 0.86 0.25 0.16 2.39 1.72 1.33 1.85 1.24 0.34 0.19 1.77 1.22 1.33 1.77 1.76 0.46 0.12 Extraction techniques (%) -0.42 -1.30 0.13 1.48 -6.07 -0.01 -2.97 9.00 -2.60 2.04 5.10 -0.05 -3.50 5.76 -1.31 6.19 7.84 -9.71 -0.92 3.02 -3.67 31.12 10.41 15.51 17.35 -26.72 6.09 0.26 Table H.2. xF3 errors. The errors on some quantities were obtained as rms values after generating 26 data sets (see text in §6.6). The rest were obtained by subtracting the base data set from one obtained after shirting the relevant quantity by 1cr: hence the signs. The errors in all bins are only the diagonal terms, i.e., no bin-to-bin correlations are quoted. + List of Collaborators R .E. Blair1 , Y.K. Chu, D .B. MacFarlane, R.L. Messner2 , D.B. NovikoW, M.V. Purohit California Institute of Technology, Pasadena, CA 91125 F .J . Sciulli, M.H. Shaevitz Columbia University, New York, NY 10027 H.E. Fisk, Y. Fukushima4 , B.N. Jin5 , Q.A. Kerns T . Kondo\ P.A. Rapidis, S.L. Segler, R.J. Stefanski, D .E . Theriot, D.D. Yovanovitch Fermilab, Batavia, ll. 60510 A . Bodek, R.N. Coleman6 , W.L. Marsh6 University of Rochester, Rochester, NY l'l1627 O.D . Fackler, K.A. Jenkins Rockefeller University, New York, NY 10021 1. 2. 3. 4. 5. 6. Now at Columbia University, New York, NY 10027. Now at SLAC, Stanford, CA 94305. Now at Hughes Aircraft Co., El Segundo, CA 90245. Now at the National Laboratory for High Energy Physics, Tsukuba-gun, lbaraki-ken 305, Japan. Now at the Institute for High Energy Physics, Peking, P.R. China. Now at Fermilab, Batavia, IL 60510. References 1.) H. Frauenfelder and E.M. Henley, 'Subatomic Physics', Prentice-Hall Inc., Englewood Cliffs, N.J., 1974. 2.) J.I. Friedman and H.W. Kendall, Ann. Rev. of Nucl. Sci., 22 (1972) 203. 3.) R .P . Feynman, 'Photon-Hadron Interactions', W .A. Benjamin Inc., Reading, Mass., 1972. 4.) J.D. Bjorken and E.A. Paschos, Phys. Rev. 185 (1969) 1975. 5.) F. Sciulli et a!., 'Neutrino Physics at very high energies', Fermilab Proposal 21, FNAL (1970). .. 6.) V. Peterson, 'A Monochromatic Neutrino Beam from a High Intensity 200 BeY Proton Synchrotron', LRL Report UCID-10028, (1964). 7.) B. Barish eta!., Phys. Rev. Lett. 35 (1975) 1316. B. Barish et a!., Phys. Rev. Lett. 39 (1 977) 1595. 8.) R. 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