..... 1...... .... . .u . PR _ nu . _ T. . a. 2;: .. 4 .c... iatéon for tbs Begree 9%? Pit. in I BE .. ........ X?! 4.2 ., .11., l..l( .. . .r.... 5.1.1:. ,..,,J...u;; u ... z. 7.. 7...!” E. uv. .. . me mums- eaeum: : ., m . . r; 1;. :...... . .. _. f v. , . . . . . l . .o . . - .. . 7 . . .. , a .. ...u.,\..........z.‘ :5 : (.1 . $.rrv... LI BRAR Y Michigan time University thesis entitled~ . ‘ . ‘ fig, ‘ ‘ .' . 1 Test of Pole Extrapolation Procedures * '7 i I inpp+A++NAt66<5cW 5/ presented by 'Lm‘m J . ms. WI] .....———‘-‘ .— < #4. ' 3 ('1 (I) 'i ABSTRACT TEST OF POLE EXTRAPOLATION PROCEDURES IN PP + A++ AT 6 GeV/C BY John Douglas Mountz The experimental apparatus in this thesis is designed to detect one pion production at a beam momentum of 6 GeV/c resulting from proton-proton collisions. Eighty percent of the events have the PNn+ final state. Thirty percent of the events have the well known A++ (3,3) 1236 resonance-neutron final state. This resonance occurs at a low M(Pn+) and momentum transfer square. Assuming a one- pion exchange model, this resonance is produced in this reaction with a virtual pion in the initial state and a on- shell pion in the final state. The same resonance occurs in n+P elastic scattering where the pion is on its mass shell before and after the resonance is found. The goal of this experiment is to test the accuracy of different models and extrapolation polynomials which can be used to obtain the on-shell cross section from the off-shell scattering data. It is sufficient to use form factor models if one wishes to extrapolate the total cross section. The models considered here model, and the I It is r. gelaticn procesF known to high a hflf,it Will a extrapolating . on-shell value ' such as 17-7 as: m. . .SLS E): John Douglas Mountz considered here are the Chew-Low model, the Durr Pilkuhn model, and the Benecke Dfirr model. It is useful to test the validity of the extra- polation process in this case because the on-shell data is known to high accuracy. If the method proves successful here, it will add credibility to cross sections obtained by extrapolating initial virtual states to the unphysical on-shell value where the on-shell data is not available, such as n—n and n-K scattering. This experiment provides 14 thousand A++ events in their raw form. The processing and corrections necessary to obtain an unbiased high quality A++ sample necessary for the extrapolation is the subject of much of this thesis. Due to the large number of events and the good quality of corrections, this thesis represents the most exacting test to date of the extrapolation technique for the cross section. The results indicate that the Benecke Dfirr model and the Dfirr Pilkuhn model are indistinguishable for the mass and t range considered here. The results also show that an At + Bt2 polynomial fit to the "to" values cal- culated using either of the above mentioned models will reproduce the on-shell value if the curve is extrapolated to the pole. The good fit requires no scale factors. If the same fit technique is done without requiring the curve to pass through the origin, the extrapolated cross section at t=0 is consistent with zero within the limits of the error. TEST In pay-L; TEST OF POLE EXTRAPOLATION PROCEDURES IN pp + A++ N AT 6 GeV/c BY John Douglas Mountz A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1974 Sable aSSiSta: analysiS. a.“ ‘ fl ‘M Suggeyfiig as a: data. 1 are in. ESSiStanCe d: 5 3h ‘dy Schd'd'm this th‘ESis . AC KNOWLE DGMENT S I wish to express my appreciation to Professor Gerald A. Smith for his patience, guidance, and motivation throughout the analysis of this experiment. Many thanks are due also to Professor R. J. Sprafka for his indispen-l sable assistance during the initial stages of the data analysis, and Professor 2. Ming Ma for his many helpful suggestions and comments during the physics analysis of the data. I am deeply indebted to the University of Notre Dame for providing the apparatus for this experiment, and to the many people at Argonne National Laboratory for their assistance during the data acquisition. I wish to thank Sandy Schauweker for cheerfully typing the rough draft of this thesis. ii III. IV . '1‘ ‘ I {If , \ m‘v \‘i. 4-1 Zez 4-7- Sea 5,1 GEC 5.2 28: 6.2 Ele 6.3 Str TABLE OF CONTENTS LIST OF TABLES . . . . . . . . LIST OF FIGURES. . . . . . . . LOSSES. Chapter I. INTRODUCTION . . . . . . II. HARDWARE. . . . . . . . III. EVENT PROCESSING AND SELECTION IV. ZERO VALUE AND BEAM MOMENTUM . 4.1 Zero Value . . . . . 4.2 Beam Momentum . . . . V. ACCEPTANCE . . . . . . . 5.1 Geometrical Acceptance . 5.2 Zero Acceptance . . . VI. PION DECAY AND SECONDARY SCATTERING. . . 6.1 Pion Decay . . . . . 6.2 Electromagnetic Scattering. 6.3 Strong Interaction Correction. . . VII. SPARK CHAMBER EFFICIENCY AND DE/DX 7.1 Spark Chamber Efficiency 7.2 DE/DX Efficiency . . . VIII. RESOLUTION AND AMBIGUITIES. . 8.1 Miss Mass . . . . . 8.2 Invariant Mass. . . . iii Page 13 20 20 25 29 29 33 44 44 46 50 54 54 55 63 64 70 Chapter 8.3 M0: 8.4 Amt IX. CROSS SEC X. ONE PION 10.1 Ki: 10.2 Ver “vendix A. LJEnt Pro: 8. 39am Home Co T‘anSfOr‘. & Jag and R Monte Carl P Q Spark Ct & Error C 1 Chapter IX. X. XI. REFEREN APPENDI Appendi A. '11 ['11 U 0 w 0 8.3 Momentum Transfer Squared Resolution 8.4 Ambiguities . . . . CROSS SECTION AND NORMALIZATION. ONE PION EXCHANGE . . . . 10.1 Kinematics. . . . . 10.2 Vertex Contributions Without Form Factors . . . . . 10.3 Dfirr Pilkuhn Corrections 10.4 Benecke Dfirr Corrections 10.5 Corrected Cross Section and Pole Extrapolation . . . . . . . . SUMMARY AND CONCLUSIONS . . . . . . . CES . . . . . . . . . . . . . . CES x Event Processing Failures. . . . . . . Beam Momentum Fitting Program . . . . . Transformation From M, t, eJac, ¢TY to P and n. eJac and ¢TY Biases. . . . . . . . . Monte Carlo Event Generation. Resonance Mass and Width Corrections Experimental Errors . . . Spark Chamber Efficiency . . Error Calculation . . . . iv Page 83 83 98 107 107 111 115 117 119 160 165 168 175 178 186 189 194 199 206 Trigger EEG and EVents Circe ( TEuta 1 Teuta 1 GM T. Final. Initia Beam F TYPica Pion o Pion 2 Break- ChEmic Coulom Stron LIST OF TABLES Trigger Breakdown . . . . . . BAG and ub/event . . . . . . Events Out of Crunch . . . . . Circe Output Events . . . . . Teuta Fit Reactions . . . . . Teuta Fits CT>3% . . . . . . Good Teuta Events After Cuts . . Final Events. . . . . . . . Initial and Final Zero Values . . Beam Fitting Program Test . . . Typical Event Weighting Process . Pion Decay Events . . . . . . Pion Decay Track and Fit Results . Break-down of Secondary Matter. . Chemical Break-down of Material . Coulomb Scattering Event Analysis. Strong Elastic Scattering . . . Page l4 14 15 16 16 18 18 19 24 25 30 46 46 48 49 49 51 i8. L1. L2. L3. \J . ch 0 11. 8.2. Elastia Prott Alas;- Strong Spark C DE/ZX C Prot: EXPerir Peak 6 Table Page 6.7. Elastic and Total Cross Sections for Protons and Pions on Carbon and Aluminum . . . . . . . . . . . . 53 6.8. Strong Interaction Correction Factors . . . 53 7.1. Spark Chamber Efficiency . . . . . . . 55 7.2. DE/DX Counter Characteristics for 6 Gev/c Proton. . . . . . . . . . . . . 57 7.3. Experimental Discriminator Curve. . . . . 60 7.4. DE/DX Counter Characteristics for Real Event Triggers . . . . . . . . . . 61 8.1. Results of Gaussian Fit to Missing Mass Plots 0 O O O O O O O O O O O O 75 8.2. Bubble Chamber Background Estimates Fit Parameters . . . . . . . . . . . 76 8.3. Mass Parameters for A++ (1236) Resonance . . 82 8.4. 4296 Mark 4-Teuta Fit Breakdown . . . . . 87 9.1. A++ Cross Section Factors . . . . . . . 99 9.2. Acceptance for PNn+ and PPN° . . . . . . 101 9.3. PNn+ Cross Section Factors. . . . . . . 104 10.1. Chi-Square and Multiplier for at, at + bt and at + th + cT3 fits . . . . . . . . 129 10.2. Chi-Square and Multiplier for a + bt and a + bt + ct2 Fits . . . . . . . . . . 150 A.1. Crunch Losses . . . . . . . . . . . 168 A.2. Circe Failures. . . . . . . . . . . 170 A¥3- Reactions Leading to 90% of Circe Failures . 171 A.4. Teuta Events Confidence Level <.03 . . . . 171 F.1. Peak and Width of M(P,n+) With Resolution Errors. . . . . . . . . . . . . 197 vi H | .azle F.2. F.3. 6.1. (3.2. 6.3. (:04. Table F02. G01. 6.2. 6.3. Page Breit-Wigner Fits to Unsmeared and Smeared Data 0 O O O O O O O O O O O O 198 Parameter for Predicted Fitted and Experimental Mass and Distribution . . . 198 Beam Chambers. 0 I O O O O O O O O 20]- Hodoscope Chambers . . . . . . . . . 202 Probability for iEE»Magnet Chamber to Miss Two Tracks of a Missing Event. . . . . 203 a. Beam Chambers b. Magnet Chambers c. Hodoscope Chambers. . . . . . . . 205 vii Figure L1. 11. 12. 13. $1. 4.1. One pi This d 8Xpe The HQ from alt dRC In: Dia and Che COordir Figure 1.1. 2.1. 5.3. LIST OF FIGURES Page One pion exchange diagram . . . . . . . 2 This diagram is an overall plan view of the experiment hardware . . . . . . . . 7 The Hodoscope counter array is shown in a front view with lightpipes indicated. . . 9 a. Diagram showing how the four layers of aluminum wires are applied to a chamber and where the magnetostrictive ribbons run 0 O O O O O O O O O O O O 12 b. Diagram showing details of the mechanical and electrical construction of the chambers . . . . . . . . . . . 12 Coordinate system used in Circe . . . . . 16 The twenty distributions shown in this figure are the fitted minus measured beam track positions for the 20 spark chamber planes used in this experiment. . . . . 22 Fitted beam momentum and angles using data runs triggered on single beam tracks. . . 27 Typical event weighting process . . . . . 32 a-d show the mass squared of the pn+ system, the momentum transfer squared, the Jackson angle and the Treiman-Yang angle distributions respectively . . . . . . 35 a-b Three dimensional plot of the zero acceptance region . . . . . . . . . 39 viii Figure i4. 11. 12. 703. 11. 12. 8‘30 L4. is. 17. a‘d P17; (shad acce;' Spark c The tri nator Single Perce nator Confide is m: Missing event 9° NE! Wi‘ Figure Page 5.4. a-d Final experimental data shown before (shaded) and after (unshaded) zero acceptance correction . . . . . . . 42 7.1. Spark chamber classification . . . . . 54 7.2. The triangles mark differential discrimi- nator curve points . . . . . . . . 59 7.3. Single and double energy loss curves and percent trigger loss due to discrimi- nator setting. . . . . . . . . . 63 8.1. Confidence level distribution after the x2 is multiplied by 7.2 . . . . . . . 66 8.2. Missing mass squared distributions after event reconstruction . . . . . . . 69 8.3. a. Neutron error distribution consistant with flat confidence level. b. Gaussian ideogram around 2 = 0.88 Gev2 using errors . . . . . . 72 8.4. a. Pion missing mass data and Gaussian fit for MZN and Ft. b. Neutron missing mass data and Gaussian fit for FN with MN2 = 0.88. . 74 8.5. Breit—Wigner plus phase space fit to bubble chamber data. . . . . . . . 79 8.6. Breit-Wigner fit to experimental mass distributions 0 O O O C O O O O O 81 8.7. a. Probability of finding an error E at momentum transfer squared t. b. Probability of finding an error E at momentum transfer squared t normalized so that fW(t,E)dE = CONSTANT . . . . 85 8.8. Spark chamber data (unshaded) and Monte Carlo events with induced errors selected by best confidence level (shaded) . . . 89 ix Figure 8.9. 9.1. 9.2. 10.2 10.3. 10.4. 10,5 Figure 8.9. 9.2. 10.1. 10.2. 10.3. 10.4. 10.5. 10.6. Page Distributions are plotted without ambiguity correct efficiency (unshaded) and after (shaded) the normalized correction is applied 0 O O O O O I O O O O I 95 A++ cross section at 6 Gev/c plotted with near by values. . . . . . . . . . 103 PNn+ cross section . . . . . . . . . 106 One pion exchange (OPE) diagram for PP+PTT+N O O O O O O O O O I O I 107 Exchange graph of scalar particle with mass X . . . . . . . . . . . . 118 a-b. Chew-Low extrapolation curves c-d. Dfirr Pilkuhn extrapolation curves e-f. Benecke Dfirr extrapolation curves . . 123 a-b. Chew Low extrapolated on-shell mass cross section and on-shell data c-d. Dfirr Pilkuhn extrapolated and on- shell mass data e-f. Benecke Dfirr extrapolated and on- shell mass data . . . . . . . . 133 a. do/dM experiment and Dfirr Pilkuhn prediction b. do/dt experiment and Dfirr Pilkuhn prediction C- dO/dcoseJac experiment and Dfirr Pilkuhn prediction e. do/doTy experiment and Dfirr Pilkuhn prediction . . . . . . . . . . 140 Double-Regge-Pole Model . . . . . . . 144 10.3, 10.9. 11.1. A4. (D Orfn (n (‘7 Figure 10.7. 10.8. 10.9. 11.1. A.1. a. C.L. model with a + bt + ct2 extrapolation polynomials b. D.P. model with a + bt + ct2 extrapolation polynomials c. B.D. model with a + bt + ctz extrapolation polynomials . . . . . a. C.L. extrapolated cross section at t = an and on-shell curve for a + bt + ct2 extrapolation curve b. D.P. extrapolated cross section using same criteria as above c. B.D. extrapolated cross section using same criteria as above . . . . . . a. C.L. extrapolated cross section at t = 0 for a + bt + ct2 extrapolation curve b. D.P. extrapolated cross section at t = 0 using same criteria as above c. B.D. extrapolated cross section at t = 0 using same criteria as above . . One pion exchange diagram for PP+A++N . . a-b. Vertex distributions for events lost in the target cut plotted along the beam direction x, where x = 0 is the target center, and R . . . . . . Experimental apparatus after target (a) and coordinate system (b) . . . . . . . The reaction PP+A++N in over-all center of mass 0 O O O I O O O O O O O O eJac and ¢TY defined in A++ center of mass. Beam and decay proton in A++ center of mass 0 O O O O O O O O O O O 0 xi Page 147 152 157 161 174 175 178 181 182 ’ Figure t4 Li AL Ed. Rl° ThIEe Vers H'l' N(”Cat app m. Noe... mas E3. “Sta Figure C.4. Page Projected view of planes described in Figure C.2 . . . . . . . . . . . 183 Momentum of A++ decay products perpendicular to beam proton in A++ center of mass . . 184 PP+PNfl+ event in P-n+ center of mass . . . 186 a-d. Original bubble chamber events (shaded) and Monte Carlo distributions which are the result of generation program (unshaded) . . . . . . . 192 Three dimensional plot of error in mass versus mass. . . . . . . . . . . 196 Notation and coordinate system used in this appendix. . . . . . . . . . . . 206 Notation used in calculating invariant mass error . . . . . . . . . . . 208 Notation used in calculating momentum transfer squared error . . . . . . . 209 xii —-— f... —— .—--s H In 19 Ethane f0r 8) Virtual Stat. Their paper Physic“ ta PhYSICal V‘. lfib are 3a to their ‘. .3 a that CHAPTER I INTRODUCTION In 1959, G. F. Chew and F. E. Low1 presented a. scheme for extrapolating physical cross sections using the virtual states existing in hadronic matter as targets. Their paper draws a connection between scattering data off physical targets, which are on the mass shell, and un- physical virtual targets. Physical targets existing in the lab are said to be on-shell because their mass is related 2 2 2 to their energy and momentum by M = E - P and this mass is a characteristic of the target. Virtual targets are thought to be associated with real particles or Regge Trajectories and have a negative M2 as defined above. In the scattering region, the mass of the exchange particle is also the momentum transfer squared from the target proton to 2 the neutron, denoted by t. The extrapolation scheme mentioned above proposes to extend the scattering cross section, measured as a function of t in the physical region, into the unphysical positive t region. This extra- 2 polated cross section can be evaluated at t = M , where M is the mass of the exchange particle, to obtain the on- shell cross section.3' 4 The rea The reaction considered in this thesis is PP+A++N. ++ 11 Resonance Bean} P 7T Target Neutron Figure 1.1. One pion exchange diagram. The one pion exchange diagram for this reaction is shown in Figure 1.1. The beam particle is a proton and the virtual target is an off-shell pion residing in the pion "cloud" of the target proton. The beam proton collides with the virtual pion and imparts a momentum transfer t necessary to place the virtual pion on its mass shell. The final on- shell proton and pion are then detected in the lab. This process has a high cross section at low t and (P, n+) mass due to the presence of the A++1236 resonance.5 The pole extrapolation technique is not necessary to obtain cross section data in the reaction described above. Experiments using pion beams colliding with protons have been done.6 The same resonance occurs in n+P scattering where the pion is on its mass shell before and after the resonance is formed. The on-shell cross section has been measured to high accuracy.7 It is useful to do a pole extrapolation experiment using off-shell n-P scattering data in order to check the validity of the extrapolation process against the known correct results. If the method proves successful here, it will add credibility to cross sections obtained by extrapolating initial virtual states to the unphysical on-shell value where the on-shell data are not available, such as n-n and n—K scattering.8 The experimental apparatus in this thesis was designed to detect one pion production at a beam momentum of 6 Gev/c resulting from pion-proton and proton-proton collisions. The apparatus was built by the University of Notre Dame in order to carry out pion-proton experiments.9 Subsequently, the experiment reported in this thesis on proton-proton collisions was carried out as a University of Notre Dame, Argonne National Laboratory, Michigan State University collaboration. Wire spark chambers using magnetostrictive wands were used to obtain the data. Scintillators were positioned so that only data from a certain event configuration would trigger the apparatus and cause a spark. When a spark occurred, scalars automati- cally digitized the location of the spark in the chamber and a computer was used to write the scalar information onto magnetic tape. The apparatus was triggered on events having one beam track and two outgoing tracks. Typically 15 events were written on tape during each 400 m sec beam burst. The experiment ran for 12 days at the Argonne 2.6.8. and 1.5 million triggers were recorded. The final sample of 50,000 single pion production events were identi- fied after reconstruction and event type fitting using the Michigan S The processing unbiased high is the subject tie final ever 0f the events neutron final MOD-proton m5 large number ‘ rectlons' thi The t 58 is Comp] Shape from a the Michigan State University C.D.C. 6500 computer facility. 'Fhe.processing and corrections necessary to obtain the unbiased high quality A++ sample used in the extrapolation is the subject of much of this thesis. Eighty percent of the final events have the PNn+ final state. Thirty percent of the events have the well known A++1236(3,3) resonance- neutron final state. These 14,000 A++ events occur at low pion-proton mass and low momentum transfer. Due to the large number of events and the good quality of the cor- rections, this thesis represents the most exacting test to date of the cross section extrapolation technique. The t dependence of the cross section at a given mass is complicated.10 The t distribution derives its shape from a combined contribution of the PNn vertex, the pion prOpagator,2 the PnA++ vertex11 and dynamical form factors associated with the interaction. An accurate extrapolation of the data cannot be done unless the t distribution is linearized by normalizing the data with different models. It is sufficient to use form factor models if one wishes to extrapolate the total cross section. The models considered here are the Chew-Low 12 13 model,1 the Dfirr-Pilkuhn model, and Bénecke-Dfirr model. The Chew-Low model considers only the kinematics of the one pion exchange where the Dfirr-Pilkuhn and Benecke-Dfirr models introduce additional form factors to help linearize the extrapolation curves. CHAPTER 11* HARDWARE The hardware for this experiment was constructed by the University of Notre Dame and will not be described in detail here.9 A proton beam of momentum 6 :_.5% Gev/c was focused and directed at a 6 inch long hydrogen target. The 20% pion beam component was identified by a Cerenkov counter. The counters used to trigger the apparatus are shown in Figure 2.1. A trigger was defined as E 3132 A(o>2) HiHj where D>2 means two particles must hit the DE/DX counter and HiHj means two separate hodoscopes must fire. The DE/DX counter was 1/8" thick pilot F scintillator. The hodoscope array, shown in Figure 2.2, was designed using Monte Carlo events of the type n'P+n+n-N to have a high two track acceptance. The gross features considered in the Monte Carlo do not change when considering the reaction PP+PNn+. *I am indebted to the University of Notre Dame for providing the apparatus and supplying the information used in this chapter. .Emouumms Honuusw muouos ow usons cosmooa can mcoH poom m we Houcsoo >oxcouoo one .uommfluu was GM can: muoucsoo cofiumHHHucwom may mo mcowumooa on» ucomoumou mocaa ofiHOm one .muonfimco 6 xummm ones o>H90fiuumouocmme mo cofluwmom on» ucomoumou mocwa oouuoo one .oumscumn ucoaaummxm may no 30fl> swam Hamuo>o on ma Emummwc mace .H.m madman 7 0- SCM-IOS k J i 0.00.00.00.000.00.0000...OOOOOOOOCOOQ. 2 l 1m. .msumunmmm may no momoo map was: muoucsoo numcoa Adam nodes can Houcoo on» Home menus“ o u N no uwamm muoucsoo nose v on» mum csocm .coumoflosw momflmunmfla spas Bow> ucoum 6 ca ozonm ma hound Houcsoo omoomooom one .~.~ ousmwm mmai PIG: ssuoug 92; :8 .9 :m vvvvv :m ewe em? BOIVWWIINDS I The c Dame and have wands. If 5 after a chard; breakdown Wou Chm“! Wires mat(Shaiostrio aCthated by rate of 20 M PrOportiOnal total data 1 10 The chambers were wound at the University of Notre Dame and have 48 wires/inch crossing magnetrostrictive wands. If 5 kilovolts was applied to these wires shortly after a charged particle passed through the chamber, a breakdown would occur resulting in current flow in the chamber wires. The acoustical pulses resulting in the magnetostrictive ribbon were used to turn off scalers activated by a common fudicial pulse and counting at the rate of 20 MHZ. The scalar data for a given plane were proportional to the spark coordinate in one dimension. The total data from the 20 planes were read onto tape by the Varian 620/i mini-computer. Figure 2.3-b shows the chamber construction. Unambigous spacial location determination for a two track final states demanded that some chambers have non-orthoginal wire orientation. This is shown in Figure 2.3-a. 11 1.A.xUmm N ~0~ XWJK an: .mumbfimco on» no newuosuumcoo assauuooao can Hmowcmnoms on» no mawmuoc mcflsocm Emummwo ‘l'Illllllllili .ssu mconnau o>fiuowuumouocmme may muons can nonfimno w on ooaammm mum mouws Escwssam mo mumSMH .bm.m ousmam “sow gnu 30: mcfl3onm Emummwo E----:K WNQVQK Ntwt dhmetu km<2~ '4 \DC .Mm.m ousmflm 12 Pb? :r rill. ................ H .— 1 m a 8 V. .N O i I. S. n... W 1m I 3 o? o s . S m. w. .332 n w 933535.53 e n n ma: 3:3 v a ... 0 do.“ w. 1 I HI WMIMI Md<4m rm1Vh LJU .2 Hfll- NN3% are con- sidered good fits and are summarized in Table 3.6. The events which survive the target cut, discussed in Appendix .A" are listed in Table 3.7. The last section of Table 3.7 resolves ambiguities by a simple best selection rule described earlier. :as Pas ll 18 Table 3.6.--Teuta Fits CT>3%. Positive Field Negative Field Total Pass 1 18896 34105 53001 Pass 2 1180 621 1801 Table 3.7.--Good Teuta Events After Cuts. Fit Mark Pass 1 Pass 2 2 5177 153 4 9859 277 2 + 4 2857 83 104 ‘8807 290 2 + 104 5731 132 4 + 104 15877 462 2 + 4 + 104 287 4 PNn+ 39325 11146 PPn° 9270 255 A++ 14283 402 Total 48595 1401 *7 In Chi sanle of eve. standard devi events by the even-ts befon inflection f Table 3.8.-- \ \ PRr+ {Pub L++ 19 In Chapter VIII it will be shown that a cleaner sample of events can be obtained if one applies a Circe standard deviation (S.D.) cut of 0.6 and scale up all Teuta events by the same number to conserve events. The total events before a S.D. cut, after a S.D. cut and with a S.D. cut plus scaling are given in Table 3.8. The S.D. cut correction factor is 1.29. Table 3.8.--Fina1 Events. No. Cut Sol Cut =.6 Sol Cut + Scaling Total Events 49996 38738 49996 PNn+ 40471 31908 41180 PPn° 9525 6830 8815 A++ 14685 12271 15837 The weighting program described in Chapter V uses a model hodoscope array to eliminate orbited orientations for two tracks which hit the same hodoscope or miss the hodoscope array. This cut was also applied to the experi- mental data for consistancy. This cut reduces by 270 events the A++ sample. This gives 12001 A++ events in the final sample. and the r determine VALUQS as CHAPTER IV ZERO VALUE AND BEAM MOMENTUM 4.1 Zero Value The 20 spark chamber planes were surveyed into place and the relative centers of the various planes were roughly determined with respect to the beam tracks. The initial values as determined by this survey are given in Table 4.1. The zero value is the center to start distance in inches. To determine the actual zero values for the 20 planes with respect to the beam, data runs were taken at the beginning, middle, and end of the experiment with the magnet off and the trigger set for single beam tracks. The data used for zero value are processed through the same spacial reconstruction routine as real event triggers. Only straight through tracks with exactly one spark per plane are examined. The Y and 2 views are separately fitted to a straight line. The final zero value is the average fitted beam position in the chambers and is given in Table 4.1. The widths and displacements of the fitted beam position with respect to the surveyed zero 'Values are shown for the 20 planes in Figure 4.1. These 20 21 thI . I . \ Ii . I-IAII bl!..'L iLi‘l g! «1....ILU .mononfi ma oaoom HounOquon one .pcoEwuomxo man» as now: nonmam nonfimno nuomm on on» now mnofluflmom noon» soon nonsmooa moans oouufim onu one ousmwm mHnu ca nzonm mnoHuznwuuowo kudos» one lb .H.e gunman - . 22 wpzu>m mo ammzsz ma 4¢u_Hmm> n. a”. a. .6... «no! a... I54! h h In: r l n- T l on I I. m: r .. 8 T 1 no T l a I 1 $.— P p n h ON— 2 g n. a: a. 8.0 fine: ".0 p h b b C‘t c ‘1“ .I L On r. 48 Y- 1 8 I l 3 1 I. 8 1 r 8 b n r n u we: 8 "O 2. .6. 3.0 2.: a... J 2 J on 1 on I 2. I I 8 L p t n a was 8 Q. .6. 5.- an; «.0 awf .— - iml‘ l o. 1 on 4 8 L 3 1 8 4 8 L 2. - - rP n ._ use 8 a. 2. 3. S... 2.. a: J...» P b .I 1 .. 2 I l on 1. a 8 e . s I l a .l I. 8 .l I. g h [P b h .uS. 8 a. 2. .6. 8.. a... a... d .‘I- u I- i .. l a. I. l 8 e . s r .. 8 I J an I L a h i 8. LP n p p suit on. a. 2. 3. 8.- 2.- a: r a. .. 8 .. u. u 8 .. 2. T 1 3 .I I. 8— L p n P . #3 on u. 2. .3. 8.. 2.. a: dd Ll‘ . - I‘Ilhl .. 4r L 2 r J on r.- l g r I 3 rl l a I I. 3 .. 1 2. b n in b n we... 8 mwruzn zm oumzmmmz $22.. 8:: mn wmxm #32335: a. 2. 3. 5.. 2.: «i .. .. 2 .. .. 8 fl 1 8 1 48 .. .. 8 T .. 8 .. a 2. . {A _ .. n . a“ . J. 8.... 2.... a... e. 1:1 1 r J a. u 1 8 .. .. B .. l 8 .. .. 2. r a 8 T .. a: n L - .11 On— it: 388 I; T N 8 r .1. e. _ e .11 Few e IL}! -. a... -1... I g a 9. “A Iii“ i. g mhzm>u .19 $9.32 2 482a“; muzuz~ zn ouuzmmuz 323. 8:: mm mmxm nEzoNonr 8. 8r 2: a: a. 2. 3. x..- 2.- «.- a. 2. 3. K..- 2.- a: n. 2. .6. 8.- 2f «.- a. 2. 3. 8... ~..- n.- b P b n . b n n b b ‘ ‘i‘ 3.- rc-l 1: d at 3 Hi. I. t O T 1! C - 23 b i—r - L n n n n n L 8 was : we... a use 2 use 2 US. 8. 1 8288 r 1 l TrT S. B..- ~...- ~.- .... 2. 3. 8r.- n...- a: a. a". a... S..- u...- t- a. 2. S. 8.. 2: «.- a. 2. S. 8.- 2.- n.- - p p n n b P a. p a. wad n. and 2 and Table 4.1." ‘h‘ Plane \ l 2 10 11 12 13 14 15 16 17 la 19 20 24 Table 4.1.--Initial and Final Zero Values. Plane Initial Zero Value Final Zero Value 1 7.537 7.537 2 7.793 7.796 3 7.537 7.540 4 7.801 7.802 5 7.769 7.766 6 7.547 7.541 7 7.540 7.540 8 7.801 7.797 9 16.475 16.470 10 16.787 16.782 11 16.785 16.787 12 16.588 16.586 13 14.766 46.738 14 46.366 46.367 15 43.290 43.291 16 18.958 18.967 17 43.218 43.220 18 18.955 18.953 19 47.014 47.008 20 46.478 46.477 Ir distri] ning o simila 25 distributions use the magnet-off data taken at the begin- ning of the experiment. The other magnet-off data give similar distributions. 4.2 Beam Momentum The beam momentum was determined experimentally to be 6.0 (i 0.5%) Gev/c using a dipole magnet and a momentum analysing slit.9 As a consistency check on the magnet field fit and the beam momentum determination, single beam tracks with the magnet on were tracked through the magnet and the momentum width and center was determined. Appendix B details the fitting program used to fit the beam tracks. .A Monte Carlo program was used to generate X-Y-Z values in the 10 chambers to simulate beam tracks as a check of the fitting program. The results of the fitting program are listed along with the input values in Table 4.2. Table 4.2.--Beam Fitting Program Test. Input Track Output Track P A 0 P A 0 Gev/c Rad Rad Gev/c Rad Rad 6.0 0. 0. 5.993 0.006 .000 6.0 .018 .024 6.010 .010 .020 2.0 -.03 -.08 1.999 .030 -.080 Only single perfect beam tracks were used when fitting the data. Figure 4.2 shows a plot of the fitted 26 _ _ .L \omn \ 1 KN» 1 \oom \ 1 UDOZQ m—O MOOZQ OQIADENNQ .mnomuu soon oamcflm no cosmonauu menu sumo means moamnm can enunosoe soon pounds \ \ tfibkfié§xrffim .~.¢ masons \. SN 27 mo. mz¢_o¢m “O. P “O.- _ r _ dc Hr— unozc mmo mzanomu 8.... mo.-. mo. 2.”. .0.- _ mo.- _ m— om mo” own '5 I. L r. _ It 1 mnozc nmzpaz~Nm mo.- mm. om mm“ Dow m.m m.m _ e\>uo zaezmzez 3.: T zahzmzoz zaum am am om" om_ onm ozm SlN3A3 beam momentu centered at This width ‘1 come from th These are m AS a field-beam m 6 Chamber SE Rammed CO< For all Pla a width of 28 beam momentum and angles. The beam momentum is seen to be centered at 6 Gev/c with a width (FWHM) of 0.5 Gev/c. This width is roughly consistant with error expected to come from the wire spacing using planes 8 through 20. These are the planes after the target. As an additional check on chamber center values and field-beam momentum values coupled with the possibility of a chamber sag or rotation, the Y and z fitted minus measured coordinates were plotted for magnet on beam tracks. For all planes, these values are centered on zero and have a width of 0.002 inches. CHAPTER V ACCEPTANCE 5.1 Geometrical Acceptance The apparatus as described in Chapter II is designed to have no uncorrectable acceptance losses for reaction number 3. The degree to which this is not true is the subject of section 5.2. The apparatus does have a limited acceptance for reaction numbers 1 and 2. This limited acceptance arises from the wider angle and slower momentum data which comes from 2 and l as compared to 3. By designing the spectrometer length and field to be unbiased only for 3, better resolution for this reaction can be achieved. . Not all events of type 3 can make a successful trigger. Due to the rectangular shape of the magnet having limits of :04 inches in Y and il3.5 inches in 2, not all events in the x-z plane survive to the hodoscope. However, the larger X-Y plane acceptance insures that wide angle events are recorded in this orientation, and the losses from the vertical orientation are related by a rotation about the beam axis. A weighting program has been written 29 30 to assign each event an acceptance weight equal to the inverse of the probability of detecting the event. The acceptance program assumes axial symmetry along the X axis. Each of the two outgoing tracks is rotated together about the X axis in 100 steps of 0.0628 radians and the number of times both pass through the magnet and hit the hodoscope is recorded. The acceptance is defined as the hits divided by the total number of steps. The magnet cuts made are Y = :04 inches and z = 113.5 inches. The hodoscope is defined as Y = $.44 inches and Z = :18 inches. Also, since actual events require two or more hodoscopes to fire, an event orientation is not counted as a hit if both tracks hit the same hodosc0pe. A typical event weighting process is pictured at the hodoscope plane in Figure 5.1. The event shown has the characteristics given in Table 5.1. The outer circle is the pion and the inner the proton. The solid lines con- necting the circles are various pion-proton relative position on the circles. Table 5.1.--Typica1 Event Weighting Process. Gev/c P Arad ¢rad Pion 1.46 .0076 .1208 Proton 4.39 .0298 -.0470 31 a \ m7: \ u\ < meOZ.N < .mmmuoum mcwuzmwos uc0>a Hmoamhe .H.m muamam mmmmmm ..................... Wflm\ mad/77V The total m an acceptandl Onl3 defines the analysis. '1 the moment t neutron, and Yang angles .! b€fore and a 33 The total number of hits for the event shown is 84, giving an acceptance of 0.84 and a weight of 1.19. Only events of reaction 3 are weighted. Appendix C defines the four independent variables used throughout this analysis. These are the invariant mass of the pn+ system, the moment transfer squared from the target proton to the neutron, and two A++ decay angles, the Jackson and Treiman- Yang angles. Figure 5.2a-d shows these four variables before and after weighting. The average weight is 2.35. 5.2 Zero Acceptance It has been found that at 6 Gev/c incident proton momentum, the apparatus described in Chapter II will have zero acceptance for some events of reaction type 3 regard— less of their orientation. This loss cannot be corrected back by the normal weighting procedure described in the previous section. In order to understand the nature of the zero acceptance region with respect to M, t, 8 and ¢TY' a Jac’ four dimensional grid of data can be generated covering the possible range of these variables. The function used to map M - t - 8 - ¢ points to proton and pion tracks is described in Appendix C. The acceptance for each event type can be calculated. This investigation reveals there is a bias against low Treiman-Yang angle for low mass and the bias gets larger with increasing mass. It shows there is a bias against low JacksOn angle, especially at high \ \ C \ 000: a . beam Cream: mquummooQ tome _ - bro_w1 woldtiuoom {Duo Aan Navy .ouumsmcs mcwunm 03 no» a can omomnm mcfiunmfims ouommn czosm mum mcoHusnfluumwo one .mamwauoommwu mcoflusnwuumflo mamcm wcmwccmsfloua on» can mamas comxonh on» “tonnage umwmcmuu Esucweoe any .Emumam +:m may no message name on» zone can .~.m magmas 34 35 m Nunn>mou cummaom mummzcm» zapzwzoz . m“. o r _ pro_u3 muchmuuua scum 2: com coo" Dom“ Doom comm Doom comm coo: SlNBAB JD HBQHHN ommcnom ”Hm.m.mm¢z mm._ m.“ fr , J pzo_uz muzmhmuuuc zomu :3 com com com oom— oom— coma comm oozm SlN3A3 JO HBQNHN hw‘oHUl NUZQHQLQUI :nLLn. LIL-“.4 LLIQcLhNLLrLc :ELL ADV N0» 36 o mbozc »¢uwo wzcy Z¢ZMMMP L cream: uuzmhmmuuc some 2: omm Dom own ooofi 0mm" com“ owe“ Doom SlN3A3 JD HBBHHN ubozc zomxucn mzmmou o _ _ com com room 1 82 1 com" 1 com" 1 comm bro—u: muzmpmuuua zowo 3 corn SlN3A3 JO HBQHDN mass. a great explaix region. of the mass. the ac bl the 0f the Carlo the l+ GESer 0f 1". °5tair The 2! EveRt: inVES1 37 mass. The bias gets larger at high mass, but does not have a great dependence on momentum transfer. Appendix D explains physically the cause of the zero acceptance region. Figure 5.3a-b shows the three dimensional location of the zero acceptance region as a function of 0, ¢ and mass. The indentation in the lower right corner is where the acceptance is zero. Figure 5.3a-b are at momentum 2 and 0.12 Gev2 respectively. transfer squares = 0.04 Gev In order to correct for the zero acceptance exhibited by the apparatus to certain data regions, Monte Carlo events of the type PP+Pn+N were generated at 6 Gev/c. The Monte Carlo program is described in Appendix E. The events in the A++ region are tracked and weighted for acceptance as described in section 1 of this chapter. Using a mass cut of 1.36 Gev, 1580.7 tracked and weighted events are obtained from an original Monto Carlo sample of 1756 events. The zero acceptance correction in this mass range is 1.1109. Events with the larger mass range up to 1.42 were also investigated. Out of 2001 Monte Carlo events, 1752.3 tracked and weighted events result from the acceptance correction. This gives a zero acceptance correction of 1.14. The zero acceptance correction is similar for data with the mass cut-off at 1.36 and at 1.42 because, although the higher mass does have a marked decrease in acceptance, there are fewer events on which this has an effect. The correction factor of 1.1109 was used for the pole extra- polation analysis. 38 :3 Q «at < .a now «>06 ~H.° can a nou.~>oo ao.o an 05Hm> u $58 .Oth ma. GUCMDQOOUfl 03H 0H0£3 mHnaOUO GOfiflMUGflUGw one mucwom .mmmmoa mocmumwoom manmuomuuoo o>mn once on» moaned .coflmmu mocmumwoom oumn on» no uon Hmcofimcmawo mouse .nuam.m mucosa derived the ini as a ft to scai honte ( for the Parame the ra fitted Cuer CUIVe 40 The actual shape correction factors used were derived in a manner which does not drastically depend on the initial distribution. An efficiency curve was derived as a function of the M, t, 6 and ¢TY variables in order Jac' to scale-up the distributions where it is necessary. The Monte Carlo data were broken up into 10 bins of equal size for the mass, t, 0 and ¢TY of the event and a four parameter fit of the form A + BX + CX2 + DX3 was made to Jac' the ratios of the ten bins before and after track-weighting. Figure 5.4a-d shows the effect of multiplying the fitted curve by the actual experimental data. The shaded curve is the original uncorrected data, and the unshaded curve is the final data corrected for the zero acceptance. 41 .c0auomuuoo mocmumuoom oumu Accomcmcsv umumm new Aooomnmv mHOMun csonm sumo Hmucm6«uomxo Hmcwm .a-u¢.m «Human 42 Nuna>uou oumczom mmumz¢MF zDPZUtoz Dummaam A_m.m.mm¢z m B. o w .8; m; _ . 1 8m \\\\ 8: 1 82 I I i 1 8» I I I. M II 1 83 m r n .. 82 3 k x. _- I 82 m a .- . .. 8m: 3 m \k T. 1 8% m. r \—- 1 88 s \. \ a r 88 r — 1 8a .. 8.3 r 1 8mm . coo: tt comm aza_wz ouaauuua crash: outauuua 3V 2: SlN3A3 JD BBQHHN (d) (C) 43 mmozc om" ycuwo ozap 2¢z_umH Am 0 p firm—u: onhmmoum AB omm oom own 000“ omm_ oom_ own“ Doom SlN3A3 so aaeunn wbozc zomuumfi mzmmou _ D ““V\ F mmmmmwwmmwvx Dos com 1 coal . 4 coal .4 Doom 1 1 Dora 1 comm _ comm proauz onhmwuua A3 SlN3A3 JO HBQNHN travel hodosc “hich may be be rej the Pi dECay carlo track four. siZe CHAPTER VI PION DECAY AND SECONDARY SCATTERING 6.1 Pion Decay Pions resulting from the reaction PP+PW+N will _ travel approximately 116 inches before hitting the final ' hodoscope. A decay of the type n++u+v will produce a muon which can usually traverse the apparatus, but its momentum may be sufficiently different from the pion to cause it to be rejected by the fitting programs. The correction for the pion decays will then be the fraction of pions which decay and make a wrong fit. This is written as: Fraction of Pions lost . Decayed pions Bad fits (6-1J due to decays Tracked events Decayed pions A track decay program is written which uses Monte Carlo PNn+ events as described in Appendix E. The pion is tracked from the target to the hodoscope in approximately four-inch steps. The probability for decay in each step size ls is then -ls - _ PDecay 1 exp( /Lm), where Lm - BC yTn (6 2) 44 For! a ra PDec pion char. is t: tinue 45 For a 2 Gev/c momentum pion, Lm is about 4400 inches. If a random number generated between 0 and 1 is less then PDecay' a muon is assumed to emerge isotropically in the pion rest frame back to back with a neutrino, each with the characteristic momentum of 30 Mev/c. The muon four momentum is transformed back to the laboratory and allowed to con- tinue tracking. The various X-Y-z positions for the proton and muon can be recorded at the chamber positions. The two outgoing tracks are then reconstructed and fit for event types by Circe and Teuta and a fitting efficiency is determined. These severely altered tracks take approxi- mately 20 seconds each for Circe processing alone, and the statistics on this analysis are restricted by computer time. Table 6.1 summarizes the events generated in this analysis. The total events generated was 15290. It was found that 4.03% of all tracked events decay, while if a A++ cut is made 4.66% of the events decay. Thisqhigher decay number results from the fact that the A++ cut restricts the sample to only slow pions which have reduced Y and pm. The final 12001 A++ events obtained from this experiment can be tracked on the individual basis. This gives an average decay probability of 4.56% for the pion tracks. Table 6.2 indicates the results of the Circe and Teuta fitting program on the 100 tracked decayed events. $73 5.; Dec Of ' S’a’a} dece Tab: 130 97 91 77 41 “h hult 46 Table 6.1.--Pion Decay Events. Tracked Events No A++ cut A++ out All Events 2508 1179 Decays 100 55 Of the 77 fits with a confidence level greater than 3%, 15 swap to Mark 2 and 21 swap to Mark 104. The combined pion decay correction is (4.66) (I%%) = 2.75%. Table 6.2-~Pion Decay Track and Fit Results. 100 Decayed Events 97 Pass Circe 91 Pass Teuta (confidence level >10-5) 77 Pass Confidence level cut of 3% 41 Fit as PNn+ 6.2 Electromagnetic Scattering When a charged particle penetrates an absorber, it may instantaneously experience electric fields as high as 1019 volts/mtr due to the nuclei of the atoms which make up the absorber. For thick absorbers, the chances are good that the charged particle will undergo a large number of small-angle coulomb scatterings in a process called "multiple scattering." In addition the particle may under- go a single relatively large angle scatter with a probe The t scatt nunbe are 47 probability given by the Rutherford scattering formula. The transition region from multiple scattering to single scattering is known as a plural scattering because the number of collisions is larger than 1 but not very large. A precise formulation for the electromagnetic scattering in the three physical domains described above has been used to obtain the scattering angular distributions for the second- ary charged particles in this experiment.19' 20' 21 The matter seen by the secondary particle can be divided into three regions. The first region includes the target, DE/DX and chambers 5 and 6 plus the associated air. The second region includes chambers 7 and 8 plus air. The last region only includes air. These regions and their associated material are listed in Table 6.3. This material can be summarized in Table 6.4. The x position of the material is assumed to be concentrated at the weighted average position of the material in the region. The Monte Carlo program described in Appendix E is used to generate good PNn+ events. The two outgoing tracks are tracked through the apparatus and are allowed to elastically scatter in each of the three regions of matter. The two final outgoing tracks have up to three scatters apiece as they traverse the system. The various Y and 2 values at the spark chamber planes are recorded and the event is processed by Circe and Teuta. The results of the analysis of 200 Monte Carlo events are given in Table 6.5. Table Regio 48 Table 6.3.--Break-down of Secondary Matter Region X Inches Description -74.56 Hydrogen target-~3" hydrogen -73.06 Hydrogen target wall and vacuum window--.02" CH2 -70.0 DE/DX--l/8" CH2 1 -70.0 50 mil tape with DE/DX ~66.4 Chamber Aclar--.03" CH2 -66.4 A1 wires--.0072" effective width for 2 chambers -60.0 Air--26" nitrogen -34.9 Chamber Aclar--.03" CH2 for 2 chambers 2 -34.9 Al wires--.0072" effective width for 2 chambers -28. Air--45" Nitrogen 3 10. Air--45” Nitrogen } Table Regic Tab] 200 199 199 199 186 49 Table 6.4.--Chemical Break-down of Material. 2 Region Average x Element 2 A g/cm2 nuclei Sm1023 H l 1 .659 3.95 1 -71” C 6 12 .45 .225 Al 13 27 .0495 .011 N 7 14 .079 .0339 H 1 1 .01 .06 2 _ -32" C 6 12 .0602 .0301 Al 13 27 .0495 .011 N 7 14 .137 .0587 3 10. N 7 14 .137 .0587 Table 6.5.--Coulomb Scattering Event Analysis. 200 Scattered Events 199 Pass Circe 199 Pass Teuta (confidence level >10-5) 199 Pass 3% confidence level cut 186 Fit as PNfl+ of 4 w] total . events Swappi scatte 50 The thirteen lost events which pass Teuta consist of 4 which fit as Mark 2 and 9 which fit as Mark 104. The total events loss is 7%. It is known that the Monte Carlo events with no modification will lead to a 4% ambiguity swapping of events. The total loss due to the coulomb scattering is taken to be the difference of 3%. 6.3 Strong Interaction Correction Events of the type PP+Pn+N will be degraded due to the strong interaction of the secondary proton or pion with the nuclear matter present in the experimental apparatus between the target and the final spark chamber. This section will estimate the magnitude of the strong inter- action loss using experimental results of proton and pion scattering on hydrogen, carbon, aluminum, and nitrogen nuclei. The corrections which results from each of the eight reactions above can be further divided into inelastic and elastic contributions. Good events which produce secondary inelastic strong interactions are entirely lost in the target or charge cut in Circe or a missing mass confidence level cut in Teuta. Elastic secondary strong interactions are also lost due to the magnitude of the scattering angle. One can estimate the nuclear form factor by -B|t| F(A:t)=p , with B=R2/4h2. This is the small argument exPansion of the form.factor expected for the diffraction form nent will firs trac elas Para 51 by a spherical black body of radius R. The optical model form is 2 2 J1 (Rm/'77:") F black body= W (6-3) Estimates of the B parameter were made using the experi- mental data and using only the first diffraction peak. It will be shown that the relatively low angle scatters in the first peak are still too distorting on outgoing event tracks to allow many events to be correctly fitted. P-P elastic scattering at 5 gev is known to fall with a B parameter of about 8.5' 22 The B parameter is about 90 for carbon and 100 for aluminum.23"26 For pions on carbon the 27 Elastic scatter events were 8 parameter is about 60. generated for protons and pions with a B of 10, 40, and 90 and the scattered events were processed by a Circe and Teuta. Table 6.6 summarizes the results of this analysis. Table 6.6.--Strong Elastic Scattering. Conf. Elast. Pass Pass Teuta lvl. Good 8 Track Scat. Circe conf. lvl. >10"S >.03 PNn+ 10 P 50 48 35 17 7 40 P 50 50 45 38 10 90 P 50 50 46 43 15 10 n 50 47 34 20 8 40 n 50 g 50 42 29 9 90 u 50 50 46 26 18 fl YIi—g 4 4 | gets 1 it nev ratio {Esme 52 Although the rate for elastically scattered events gets larger for the larger 8 characteristic of heavy nuclei, it never gets above 30% for protons and 36% for pions. The ratio of good PNn+ to the total good Teuta fits is always one-third. This is what one would expect by chance fitting of events among the three fit types PNn+, n+NP, and PPno. Also the actual good PNn+ fits must all be reduced by 4% which is the ambiant loss level of Monte Carlo events having no induced spark chamber error. Because of the reasons above and because the elastic scattering is approxi- mately one-third28 of the total cross section for proton and pions on larger nuclei such as aluminum and carbon, it will be assumed that all elastic scatters are lost. One must now obtain the total cross sections for P nucleus or n+ nucleus scattering where °T=°e1+°ine1‘ Table 6.7 summarizes the data used to obtain the fits of cross section to energy. The cross section reviews given by reference 28 and 29 and data from reference 30 are used. The probability for a strong interaction can then be calculated on an event by event basis. The matter con- sidered is summarized previously in Table 6.3 and 6.4. 0 tot 3/3 31 increases as A and one can infer from the data above the total P-nitrogen and n-nitrogen cross sections. Table 6.8 summarizes the strong interaction correction results. m-‘ I Reac. M 1““ 53 Table 6.7.--Elastic and Total Cross Sections for Protons and Pions on Carbon and Aluminum. Reac. E Gev OT mb oel mb Reac. E Gev OT mb oel mb P+C 1 370:? 112115 .442 366133 128:26 2.2 36718 107:6 n+C 1.0 316 82 3.0 390 1.2 351:36 105:22 10. 344 100 2.86 280112 66.6:7 20.6 355:7 P+Al 2.2 739124 236:1? n+Al .442 782:46 379:37 10.0 717 214 1.0 650 178 18.4 687310 2.86 588:?2 215111 Table 6.8.-—Strong Interaction Correction Factors. Track Strong Interaction Loss Proton .0377 Pion .0322 secti Will Plgm consj Uh". Figt firj 0r: rim CHAPTER VII SPARK CHAMBER EFFICIENCY AND DE/DX LOSSES 7.14§park Chamber Efficiency The efficiency will be calculated for the three sections of the apparatus separately and a total efficiency will be derived from these three sub-efficiencies. Figure 7.1 shows the beam, magnet, and hodoscope sections consisting of 4, 4, and 2 chambers respectively. (h 7" {41“ Climb" I 2 34 78 9J0 Mumbor- ' ' 55 ' Figure 7.1. Spark chamber classification. The filter program Crunch demands 3 or 4 chamber firings per track for the beam and magnet sections and l or 2 chamber firings per track out of the possible 2 firings after the magnet. Appendix G details the spark 54 chem resu syst acc Pa] am in: Pa CO 55 chamber efficiency calculation. Table 7.1 gives the results of the calculations for each section and the total system. Table 7.1.-~Spark Chamber Efficiency. Chamber Section Efficiency Beam .995 Magnet .932 HodosCOpe .987 Total .915 7.2 OE/DX Efficiency 32 takes into The DE/DX efficiency calculation account the Landau energy fluctuation of energy loss by a particle through the scintillator, scintillator efficiency and photon production spectrum, the light pipe efficiency and the photo tube efficiency. The Landau energy fluctuation curve for charged particles in matter is a statistical phenomenon because the collisions which result in the energy loss are independent of each other. The energy loss distribution is not symmetric but has a tail due to the infrequent collisions which result in large energy transfer. The half width at half maximum A0 of the low side of the curve is given by33, 34 . whe is t1 in t1 Pr0t< 56 _ 2 2 no — 2Cmec xb/B (7-1) , where C is Euler's constant, m is the electron mass, x e is the material thickness in gm/cmz, and b and B are defined in the references cited above. For a 6 Gev/c momentum proton traversing 1/8" of the scintillator used in this experiment the most probable energy loss is 0.48797 Mev with A0 = .0411 Mev. The pilot F scintillator used in this experiment has a conversion efficiency of 2.72% for energy loss to light. The light emitted peaks at 4000 A° and has a FWHM of about 200 A°. This light will propagate through the scintillator and light pipe resulting with 4% of the initial light arriving on the first photocathode of the photo-multiplier. On the average this means that of the most probable 489.7 Kev of energy lost in the scintillator, 13.3 Kev are made into photons with an energy centered at 3.1 eV. With a light pipe-photo tube efficiency of 4%, only 182 of the initial 4538 photons arrive at the photo- cathode. The dispersion in this case due to photon sta- tistics is 0.169 Mev.3S When one compares the width due to photon statistics with the 0.084 Mev width due to energy loss fluctuations in the scintillator, it is clear that the actual experimental width of the single track events is due almost entirely to the photo tube-light pipe efficiency. In practice one knows the experimental width and the Landau width and derives the photo tube—light pipe eff in tl’d Tabl Aver Conv Kenn 57 efficiency to be consistant with these numbers. Summarized in Table 7.2 are various specifications for a 6 Gev/c proton track. Table 7.2--DE/DX Counter Characteristics for 6 Gev/c Proton. Hardware Data Physics Quantities 1 ,2 Average Wavelength 4225. A° Photon Energy 2.95 ev Conversion Efficiency 0.0272 Number of Photons 4538. I Probable Energy Loss 0.4897 Mev Landau Dispersion 0.0841 Landau Width 0.0411 Mev Photons at Cathode 182. Scintillator-Light Photoelectrons 38. Pipe Efficiency 0.04 Photo Tube Dispersion 0.1694 Photo Tube Quantum Efficiency 0.21 Total Dispersion 0.1891 Applification Factor 0.49x106 Number of Stages 10 Figure 7.2 compares the differential experimental discriminator curve, given by Table 7.3, to the calculation. The experimental curve has a width of 1.6:0.l disc units and is centered at 2.95 disc units giving a ratio of width to center of 0.542:0.035. The width to center ratio for the calculated curve is 0.48. The width discrepancy is comparable to the error and does not significantly effect the overlap of singles to doubles. Using the efficiency determined above, one can calculate the pulse shape expected for a 4.5 Gev/c momentum' Figure 7.2. 58 The triangles mark differential discriminator curve points. These points are obtained by taking differences between successive values of the experimentally measured fraction of beam tracks given in Table 7.3. The solid curve is calculated as described in this chapter. 59 Table 7.3.--Experimental Discriminator Curve. 60 Disc Disc Setting Fraction of Beam Tracks Setting Differential 1 99.8 1.25 0.1 1.5 99.7 1.75 2.5 2. 97.2 2.25 12.5 2.5 84.7 2.75 22.9 3. 61.8 3.25 21.8 3.5 40 3.75 13.4 4. 26.6 4.25 7.0 4.5 19.6 4.75 3.7 5. 15.9 5.25 3.0 5.5 12.9 5.75 2.2 6. 10.7 6.25 2.1 6.5 8.6 6.75 2.5 7. 6.1 pro in Tab Z-Ie Ave: 410 CQ‘J ‘! Una 61 proton and a 1.5 Gev/c momentum pion as are typically seen in the experiment. This calculation yields the results in Table 7.4. Table 7.4.--DE/DX Counter Characteristics for Real Event Triggers. Hardware Physics Quantities Average Wavelength 4225. A° Photon Energy 2.948 eV Conversion Efficiency 0.0272 Mev Number of Photons 9116. Probable Energy Loss 0.9836 Mev Landau Dispersion 0.0593 Landau Length 0.0583 Mev Protons at Cathode 365. Scintillator-Light Photoelectrons 77. Pipe Efficiency 0.04 Photo Tube Dispersion 0.1192 Photo Tube Quantum Efficiency 0.21 Total Dispersion 0.133 Amplification Factor 0.49x106 Number of Stages 10 Figure 7.3 shows the results of the calculation along with the singles energy spectrum. Real events had the discriminator set so that 10% of the singles are counted. Integration under the doubles curve indicates that 6% of the real events are lost at this setting. 62 .mcauuom uouocwEAHUmao on moo wmoH bemoan» bemused one mo>uso mood houses oaosoo ocs oaocam .m.h magmas 63 m”. A>szv hmsonfi e— M. b. , » hi as s \ . \ Jo \\ .om 1 03 l 08 l .3: {so 3 . sonoum o\fioo no: i .ow . sououm p\soo .m .00H (“stun Kitazrqav) andeno °Itaecta t'tzuOJOJJta notic shift and t fider 7.2 ‘ fath diSt: £++ E Were erroI Strai funCt CHAPTER VIII RESOLUTION AND AMBIGUITIES 8.1 Missing Mass Early in the experimental data analysis it was noticed that the confidence level for Teuta fit events was shifted to be too high around 1 indicating x2 was too small and the errors were too large. It was found that the con- fidence level was flat after multiplying x2 by a factor of 7.2 which indicates that the errors are too large by a factor of 2.68. Figure 8.1a-d shows the confidence level distributions of all events together, the PNu+, PPw°, and A++ events after multiplying x2 by 7-2- Since b°th fits were one constraint fits, it was possible to multiply the error matrix as a whole by a common factor since one con- straint fits have an error which can be written as a functionbf one variable only. Figure 8.2a shows the 40471 neutron missing mass fits using the CIRCE geometric fit and the track fit information as provided by Teuta. A simple best selection criteria is made to determine the proper fit. Figure 8.2b 64 65 .oouuon uoc one wm can» .N.b an ooaamwuasfi no N x on» .cowoou ++< cw muc0>m .muso>o oemm .muco>o +ezm .muco>m Ham .0 .n .M mood Ho>ma oocoowmcoo o saws muco>m noumo cowusoauumfio Ho>oa oocooamcoo .auaa.m magmas 66 4m>u4 wuzmo~mzcu L m. _ mbmu zomhauz 2: co“ com can con com com com com SlN3A3 JO BHBHHN 4m>w4 muzmQHmzou m. L _ ll mp_u 44¢ .3 on: 8m 8: 8m own SlN3A3 JD BBQNDN com omofi com" 67 bm>UJ wuzuommzou m. o r3 cm .4 Do“ 14 mm" .1 om“ J mnfi _ zomomm chemo z~ zomhawz com 2: SlN3A3 JD HBGNUN 4m>u4 muzwoqmzou L ma _ I mp~m AOLHm 3 ON 0: cm am 00“ om_ oa~ ow" SlN3A3 30 HBBNUN 68 .muflm moms ocflmmwe scam Hounds: mmmm .o .muam mama mcflmmfle couusoc Hhvoe .o .20euosuumcooou ucm>m Houmo mcowusofluumwo oouooom mmoe ocammaz .~.m mucosa 69 max>wo mmcz m. w _ mmcz ozqmmnz Hoaam 2: cm“ oom om: cow own Dom omofi oom— SlN3A3 AD BBQNDN maa>wo mmcz m. _ mwcz ozmmmnz zomhsmz E own com“ omnw ooom omen com: omnm Doom SlN3A3 JO HBQHHN 70 shows the 9525 neutral pion missing mass fits obtained in this experiment . Figure 8.3a shows the error in missing mass squared distribution for all 40471 PNn+ fits where the Circe error matrix is made to be consistant with a flat confidence level as in Figure 8.1a-d. This error is calculated as described in Appendix H. The pion-proton track ambiguity is resolved using the Teuta confidence level criteria. Figure 8.3b shows a Gaussian ideogram using these errors centered at M: = 0.88 Gevz. The Gaussian ideogram width of 187 Mev compares with the experimental neutron width of 197 Mev as wi 11 be shown later. Figure 8.4a shows the best Gaussian fit of the form 0"(MM) = (Novm) 1*PL“M‘-:fi—m)z] (8-1) to the pion data with the mass constrained to be at the 9101! mass squared of M112 = 0.019 Gev2. The error bars are Statistical errors only. Figure 8.4b shows the best GanSsian fit to the neutron data, shown by the error bars. The fit results are summarized in Table 8.1. The widths are the 1/e half widths. 8.2 Invariant Mass The Breit-Wagner form used throughout this section t° analyse the experimental invariant mass distribution is given by Ir 0 71 .muouum means N >00 mm.o ~22 ocsouo Eoumooow cowmmsou .n .Ho>oa oocoofiwcoo poam ouw3 acoumwmcoo cowusoauumwo nouns couusoz .o .m.m musmaa 72 max>uo mmcz m.~ J. m. ms 81 _ mam 8m .5 cm: mwm _ mamaom mmmz wzmmmaz com 2: mau>mo.mouuu mm. a. «exam awumaom mmcx 02—mmaz 2: cm" com om: com own SlN3A3 30 338HDN 8m 89 com" 73 nufi3 Zn now new neammsew one eueo mmeE unwm o CM PM my .FN .26 n 2: nae con» 02 2 now new newmmsew one eueo wees mnwmmwe noam .n on .a .m enema mmaz oszmHz ZDMFDMZ _ T '0008 r 'OODfi 74 '0009 SlN3A3 30 HBQNDN '0008 can 020 000: 0200022 “no I 300001 on.“ omMH omflo omho OH.o ‘0 mmmz oszmHz DMMN Hm 00.0 00.0 00.- 00.mu LL . 0. L14 . .hu 1.0 .o N mm 8 em. 0 1: rem -mm . S rm 0 .o .2... 02¢ mqu 028qu run/u. E m 75 Table 8.1.-—Resu1ts of Gaussian Fit to Missing Mass Plots. Fit Type Variables M Fit-Gev F Fit-Mev Neutron M, P 0.908 199. Neutron F M: 186. Pion M, P 0.0227 164. Pion F M2 165. -—j‘ “1E' OWN) ‘ (M‘-m;‘)‘+PA, 13’1) 3 where r‘ = MJWPLL te-a) IT *- team Mo and Y are determined by fitting and R is the A++ radius take as 4.0 Gev-1. Q is the momentum of the proton in the A++ center of mass given by Q=R(Mp,Mn,M), where M is the A++ mass, Mn is the pion mass, Mp is the pro- ton mass and R is defined by equation (lo-21). An estimate of the 81/2 wave background was determined using bubble cham- ber data in a wider mass range. The 3-body phase space is R3: d3P\o\3?t a“?! = a?) R‘a 8&3‘5 1‘3 \3_q) 25.15:.QE3 163 1E5 M *where R2 is the two body phase space term. This gives (192 ..-. (“1‘ 9:1)113 ts-S) P3 16.3 M ‘where P3 is the A++ momentum in the over-all center of mass (and E3 is the center of mass energy. P3 is given by 76 P3=R(MN,M,E3), where MN is the neutron mass. The chain rule gives ([83 .__.. l... __ (e-e) M 0193 01M. H g. :3 U s. w This leads to the Lorentz invariant phase space given by dRa 1R1 AM £3 3 K8 1) A fit of the form 0T (M) = c (M)+A (Phase Space)is made to the bubble chamber data for 1.14 Mm mo mmmzzz 1.122 It Figure 8.6. 80 Breit-Wigner fit to experimental mass distributions. 81 M988 FIT .02 1 l .30 1 1.26 MRSS (PIPI) GEV l l 18 l. 10 00.: _ 0m m {N 00_ 00. .0 >mz 0N\mz oneumm mm0m0 ]. 00.0 82 to the actual data. The uncorrected and corrected mass parameters are given in Table 8.3. Table 8.3.--Mass Parameters for A++ (1236) Resonance. F.W.H.M.-Gev Peak Position-Gev Experimental 0.133 3; 0.006 1.226 3; 0.004 Gev Corrected 0.126 1 0.006 1.266 3; 0.004 For cross section purposes, one can get from the experimental fit Kin. Lim. 0"(M)dm 1.42 = 1.476 (8-9) M'I‘hres. 0’ (M) an 1.14 83 8.3 Momentum Transfer Squared Resolution The momentum transfer squared resolution can be infered from the Teuta output errors. Appendix H outlines the method of obtaining the momentum transfer squared error from the Teuta output errors. Figure 8.7a-b shows a plot of the error in t versus t where t is the negative momentum transfer squared. Figure 8.7a has t bins pro- gnortional to the experimental t distribution. Figure 8.7b lmas the curve normalized so W(t,E) is constant, where VV(t,E) is the probability of finding an error E at momentum transfer squared t. To a good approximation, one can say the error is 2% except at small t. This small error has an insignificant effect on the shape of the momentum transfer squared distribution . 8.4 Ambiguities In order to simulate the experimental ambiguity Problem, Monte Carlo events were generated as discribed in Appendix E and are given a Gaussian error spread about their central tracked values equal to the original input Circe production errors. This choice for the error parameter produces a x2 distribution for Monte Carlo events out of Circe equal to the experimental event x2 distribution centered at 0.6. If no error is applied, this Monte Carlo 2 . X dlstribution is centered at 0.1. 84 .BZGBmZOU u no AW.HVSH uenn om oonwaeenon u oonenom nommnenu Enunmeoe ne m nonno ne onnonnm no annanoenonm .u oonesom nmmmnenu EsunoEoE ne m nonno ne mnnonnm no munafioeoonm .a .m .s.0 mnsmnn 85 0 0 0 0’0’0’0 00000 000000 0'0’0’0’0’0’293 0 0 0 0 0 04- 2. 0.0.0 0’0’0’0’0’0’0’0 0 0 0 0 0 0 0 0 0 0 \ 000000000000\ *2, \ 00000000000000‘ 00000000000000 .. I / ’ M34400‘0‘0’0’0’0’0’0’0’ “ 4 Q § § \ ”0.0.9.0...0’00 0.0.30.3}? ‘ ’0’0‘0z0’0’ e ’05: ’. . O O. ' 86 . The circe output events with the induced error were processed by Teuta and the fit confidence levels were examined. All input events were known to be Mark 4. Because of the ambiguity resulting from simulated errors in the spark position in the chambers, not all Teuta fits are Mark 4 when a best confidence level selection rule is applied. The Monte Carlo analysis yields 5424 Teuta events with confidence level greater than 10-5, 4968 events with confidence level greater than 3% and 4296 ' events pass a standard deviation cut for Circe reconstruction of 0.6. The drop from 5424 Teuta events to 4968 events after a 3% confidence level cut represents an 8% lost. This is believed to be due to the Monte Carlo accuracy and will not be used to support the notion that an 8% confidence level correction is to be applied. Out of the 4296 original PNn+ events, there were 1977 A++ events if all were interpreted as Mark 4 fits. If a best confidence level selection rule is applied, as was done with the experimental data, only 1233 A++ events are obtained. This loss of A++ events is due to miscalling the Mark 4 fits as Mark 104 and Mark 2 fits. These results are summarized in Table 8.4. The 1233 A++ events from the 4 + 104 category come from interpreting Mark 4 fits as 4 and Mark 104 fits as 104. The 470 A++ events under the 104 category are how many of the original 1977 A++ events with the correct 4 interpretation are in the 104 category. The 25% ambiguity 87 Table 8.4--4296 Mark 4-Teuta Fit Breakdown. Mark Number PNn+ A++ 4 2485 1107 4 + 104 3552 1233 2 744 400 104 1067 470 level of this experiment will have an effect on both the cross section and distributions of physical quantities. The dominate process which can change distribution Shapes is the ambiguity between Mark number 4 and 104. The Shape changes occurring from ambiguities between Mark 4 Plus 104 and Mark 2 events were studied and found to be minimal. The total ambiguity related shape changes on the Pole extrapolation results were shown to be less than the statistical error in the data. The effect of event ambiguity on the Mpn+, t, eJac' ¢TY' proton momentum and Pion momentum distributions is shown in Figure 8.8a-f. Figure 8.8a shows the spark chamber data mass distribution u“shaded and the simulated ambigious Monte Carlo distri- l311tion shaded. Both distributions have a smaller A++ peak and have more events at high mass as compared to the bubble chamber data shown in Figure E.1-a. The A++ peak was shown to be entirely due to fast Protons and slow pions in the forward hemisphere. The 88 .>H0>fiuoommon mnOwusnnnumao Enunosoe none one EnunoEoE nouonm .weo .oeho .u .+emz one zone muem.m monnmnm .Aoooenmv Ho>oa monoonmnoo omen an omuooaom mnonno ooononn nun3 mnnm>o oHneo wunoz one Aoooenmnnv eueo nooeeno xnemm .0.0 onsmna 89 «11>on omamaam mummzcmh 22.20202 >m0 7:8: 00¢: m m; n . m. 0 0 0.0 m m; l _ 3 ---1«|II..|- lu’rfl\\.\§\\u\\\ , .......\\\\\\ I pL \ com I \ L com —\ . m I .\ 000 r J 0001 w\ N N \ . . 1 \ 0002.1... T .. 002 w \m e e r w 0000 m r .. 000mm \ e e 1 Wm 0000 m I L 0000 m. “a s s I m 0000 r J 0000 a. \ \ .. m 0000 u .0 0000 _ _ _ , _ _ _ 08: :00 0 9 cos: ._ 000 0000 :00 0 2 20$: .1 000 30 E 90 04020 020» zczmmm» :00 w m> xucmp m mwm 2: 00: 000 Down Down 000m 002m 000m 00mm 04020 zomuuco mz_mco 0 . u-c I SlN3A3.032178N30N 000 Down on SlN3A3 UBZIWUWBDN 000 Lx§\\;\g:25:3§:§\. 002m 1 1 000m 000m J 1 00m: _ 000: :00 w m> 2000» m mwm 7: 91 u\>wo zazmzoz 220 o.m m.m 0 RN. 1. I com I 1_ 1 com“ T 1 Down A co: r. 1 Doom I J Dorm l L comm _+ comm 23m u m> “gawk m mmm CV SlN3A3 032 I TBNHON 0.0 Q\>mo 222.020: 229.0 m _.m I T- T saw u m> V605 0 mom 2: 0mm oom 0mm coon 0mm" com” om: Doom SlN3A3 032 I TBHHON 92 ummodified Monte Carlo and bubble chamber data predict a two to one dominance of fast protons to slow protons for the entire PNn+ sample. The pions are predicted to have the Opposite momentum distributions. The actual momentum distributions from the ambigious data, shown unshaded in Figure 8.8e-f, show a dominance of slow protons and fast pions. The effect of the ambiguity is to swap protons and pions. Since originally most protons were fast, they turn ‘. into fast pions after being miscalled. When a mass plot is made, these one time fast proton-low mass events are now fast pion-high mass events. If a low mass A++ cut is made, these events are eliminated from the sample. As mentioned earlier, this swapping of protons and pions has almost no ' effect on shapes of interest in the A++ region, but has a large effect on the over-all cross section. The shape correction will be defined as . _ Original Wei hted A++ Events Correction Factor — Experimentally getermined Weighted A++ Events All original 1977 A++ events, plus the 104 new A++ events from the Mark 4+Mark 104 swapping, were weighted for acceptance to make sure that if the swapping has any strong kinematic dependence, these could be incorporated into the final correction factor. No strong dependence was found. The average weight factor for the 2081 events is found to be 2.38 giving 4748.9 weighted events. If a A++ cut is 93 applied using the Teuta confidence level criteria one gets 2924.8 A++ events. The mass, t, and eJac and ¢TY plots are divided into 10 bins and the ratio of original A++ to be fitted A++ is calculated for the 10 bins. The errors are assumed to be statistical only and a four-parameter fit is made to the ratios. Figure 8.9a-d shows the uncorrected and corrected mass, t, 6 and ¢TY distributions normalized Jac to have equal areas. The correction factor for the cross section is the ratio of actual A++ in the Monte Carlo sample divided by the number of A++ as determined by the experimental data analysis. The Monte Carlo events are known to have 1977 A++ in the initial sample of PNfl+ because the correct name of all these tracks is known from the generation process. The events are made to simulate the actual spark chamber events and are processed in the same manner, and one ends up with only 1233 A++ events. The correction factor is then 1977/1233. The correction factor above considers only event losses resulting from Mark 4 events being misinterpreted as Mark 104 or Mark 2 events. Monte Carlo studies were made using Mark 104 and Mark 2 events and the fractional event misinterpretation due to the ambiguity was shown to be symmetric for all fit types. This means for example that the fraction of Mark 4 miscalled as Mark 104 is equal to the fraction of Mark 104 miscalled as Mark 4. The cross 94 Figure 8.9. Distributions are plotted without ambiguity correct efficiency (unshaded) and after (shaded) the normalized correction is applied. Figures 8.9a-d are the Mpn, t, 6 and ¢ respectively. Jac TY 95 m _ Nuan>wco cm00co m~. _ m 0m0m20mb chzmzcz c V T F com coo" ccmfi cocm comm ocom l comm cxo_uz aazm cecaocmzm E coo: SlN3A3 JO BBQHDN cm00com .~0.0cwm0t mw._ m. \\N I 15 A\\\ ‘ ‘L_1 L L Fromm: 001m yh~co~mz0 E H cc: com com“ com“ oocm ccrw comm ocwm SlN3A3 30 BBQNHN 96 mocz0 F0cuo oo20» z0zmm0p cc V\ \ _ 4Hd\\ l l l Fromm: 003m pp~oo~cz0 03 cmm com own coo" cmwfi ocmc own“ ocom SlN3A3 JO BBQNHN mcoz0 zcmxc0fi w2_mcc c hrc_mz 003m >pnco~mz0 to com com com com“ com" com” ccHN ccrm SlN3A3 J0 BBQHHN 97 section correction factor for Mark 2 miscalled as Mark 4 or Mark 104 in the A++ region is 0.934. This will be discussed more in Chapter IX. CHAPTER IX CROSS SECTION AND NORMALIZATION The corrections from Chapters V—VIII can be tabulated to give a cross section for the final sample. The A++ cross section normalization factors are given in Table 9.1. Some of the errors derived above are purely sta- tistical. Others have other factors folded in. The pion decay error, for example, combines the 2% decay error with a 16% Teuta fitting error. The strong interaction error includes the error due to the posibility that for heavy nuclei, all elastic scatters may not be lost in the Proauotion programs and includes also an average error in the exPerimental cross section. The ambiguity over-correct correction is the best estimate of how many events are corrected twice, once due ‘50 one of the first four items on the list and again in the ambigI-Iity weight correction. The two zero acceptance corrections are due to the lower acceptance for high mass. In the pole extrapolation, 98 99 Table 9.1.--A++ Cross Section Factors. *— Correction Factor Strong Scattering Proton 1.0377 3'- 0.0018 Strong Scattering Pion 1.0322 1 0.0017 Coulomb Scattering 1.03 1 0.0012 Pion Decay 1.0275 1 0.005 DE/Dx Efficiency 1.06 i 0.005 Spark Chamber Efficiency 1.093 Forwards Backwards Symmetry 2.0 Confidence Level Cut 1.03 i 0.005 Acceptance = 0 MA”N — TOTRL CROSS SECTION * '7 I __ I i I 'T— 1 1 T I I I f I T 7 2. 5 11. P BERM GEV/C 104 PN1r+ acceptance is given in Table 9.2. Other correction factors that apply are identical to those in Table 9.1. The changes are listed in Table 9.3. Table 9.3--PN1r+ Cross Section Factors. Correction Factor Swap Weight 1.209 1: 0.024 Mark 2 Correction 0.956 1 0.01 There are 31908 PN0+ after a Circe standard deviation cut of 0.6 and 41180 after the correction factor of 1.2906 is applied. This gives OPN1I+ = 7.53 i 0.7 mb (9-3) This cross section is plotted in Figure 9.2 along with near by values.5 The line is a fit to the data of the form 0 22 a(|P1ab|)b. The fit values previously known with a = 45.9 and b = -1.06 are used. 105 Figure 9.2. PNn+ cross section. This experiment at 6 Gev/c along with other values. CROSS SECTION MB 20. C3 1 L Jr Lmi 1 L1 11.111! J 106 I I P+P -+ Pn+N TOTRL CROSS SECTION T I I 1 1 ST 1 I 17 S. 11. P BERM GEV/C CHAPTER X ONE PION EXCHANGE 10.1 Kinematics In 1959 G. F. Chew and F. B. Low1 presented a scheme for analyzing experiments so that elementary cross sections of constituents of complex targets can be obtained. They argue that residues of poles known to exist in field theory are related to measurable quantities in physical regions of scattering and the value of the residues can be found by extrapolating off-shell scattering data into the unphysical region to the pole. The diagram considered here is shown in Figure 10.1. P O t '/P i \(1 I '0 I P b i, 3 N Figure 10.1. One pion exchange (OPE) diagram for PP+Pn+N. 107 108 The cross section 0 is the transition probability per unit flux of incident particles where the flux is the product of the particle densities, 4EaEb, and the relative velocity v. One can write37 ---'I. 0.. = BEE—‘1...— “ A as PA .- 314330’3. “Ii-.651! 1mm 3 “(P *P '7. a n? 15.1%,, 00 I) If the initial state is denoted by {a>=|a,b> and the final state is denoted by IB> = I1, 2, 3> then IMI is Z EI|2. In this calculation one needs to write the 2 cross section in terms of M12, eJac, ¢TY' S and t where 2 _ u u 2 _ u u 2 _ u _ u 2 M12_(P1+P2)'S—(Pa+Pb)’t-(P3 Pb)and eJac and ¢TY are defined 1n Appendix C. In this chapter, the four-vector P11 of a particle is related to its three-vector 5 and energy E by P11 (E,i§). One can define Ku = Plu + P2u and note M1: 2 u u- = + . K K - K and also EaEbv MbIPaI. One can multiply the integral of equation (10-1) times 1, where - q “801?." :K“) A K SUR“ M...) AM”, (\0-2.) and substitute K“ = Pl“ + 92“ to get a ' 1343?..ch 0‘0. 414.134 U"): 1933.93 2;: S i K‘A???‘ Pa“- Pb“) ' S\ \u)? “91‘- K“) qu lT/lltSw‘miMnT. 0°73) 109 One can define KO by K11 = (KO,iK) which implies K2 = K - +2 . K . One can write sex-mt) = s [WP—rm; m.-J=—-=~m..n 00-4) G) Also remembering 43dx 0 (ax) = 1/a, one obtains after integrating over Ko 4’ -+ -+ - ‘ .— .13 P. .69 .130, W .. .. .. AO- - ““5 \¥“\1“): 8 1e. 1511.153 X \K“ "P‘z -Pa ‘PL ) O W13 )Ufi Pf- “)daKdMu/K1JK‘V-nn) \\D"5’) 2 remaining 6 functions in this expression. W where Ko =JM12 + K2 is implicitely implied in the two It is useful to consider now only the two body phase space term 3‘9 . 0:: it” 3“)? 9.“.73 m“) (\0“ 9) | 1 contained in equation (10-5). The integral over d3P2 can easily be done, and the remaining integral can be done by recalling Si‘amfl'mdx = {Ud/ fun “0"") where g(xo) = 0. Equation 10-6 can now be integrated to give 110 Clcill.a q M... W B) ‘where Q refers to the proton momentum in the A++ center of nmss. Equation (10-5) now becomes 3% 3 do. " ___\_‘___ L—Kd3 3‘5“) (Kit ‘3“ ‘P: _P‘ “)‘Ml An‘ Q‘n \‘°“) mum-w" 263 " ‘mu 1‘4: ‘where 2 + M12 has been replaced by EA“ The remaining ‘variables of integration in equation (10-9) can be evalu- ated in the over-all center of mass. One can concentrate on the two body phase space integral of} 433: 153 S“ )‘W MP3 - Eta—PA) UO‘W) This integral has exactly the same form as equation (10-6) except it is missing a factor of 2EA in the denominator. Equation (10-10) can be integrated to give P 4.0.9 2 EA 00"“) “ET??— . + + C where P is [31 + P2' or IP3I 1n the over-all center of mass . r + 2 2 ‘ and EA 18 (31 + P2) + M12. Combining equation (10-9) and (10-11) gives - ‘ __ d .n. P —- '- _ dw-axm‘mu“)? Mud 44-9-9 3!?“ \M\ (\0 n.) 111 One can expand t in the center of mass as given earlier in this chapter as t= mZ+M.".. ~71:ch + mum to... (Io-13) and it follows that dt # (\o-I‘l) 7-\Pq\ «16.050- One can write37 (Io—15') Combining equations (10-12), (10-14), and (10-15), and integrating (10-12) over all angles but cos 0 gives Oman 118 (m, P.) 11‘“) 10.2 Vertex Contributions Without Form Factors If one considers for the moment the particles in Figure 10.1 to have no spins, the invariant amplitude [MIZ will be a function of five independent energies37 sij = (P1 + Pj)2 (i, j = 1, 2, 3, 4, 5). When 1 and 2 are known to produce a resonance, one can assume a plausable form for T as 112 G T: To *T15a5,tb'5’1’\3q) 511‘1": +'\V\n.. \\O‘\'1) To describes the background term which when integrated by itself becomes the phase space term for the reaction. The term is neglected here because in Chapter 8.2 the phase space was observed to be 4%. The Breit—Wigner term and the coupling constant G in equation (10-17) will be replaced by the on-shell cross section. When spins are taken into consideration, one gets additional t dependence. The amplitude for pseudo-scalar exchange for a spin 1/2 to spin 3/2 baryon is ulniprfl G'Pna. “(him (\o-ie) Where v is a vector required for the expression to be Lorent z invariant. One can let v = Pa and one can write - 38 the spin 3/2 vector U112 (P12) as Mam.) n Wm.) .y o J2 M 111’“) X“ C} ‘i Pu.) ‘42; A? Pu.) "q 2‘ us“) luipn) = 4, P A (\O‘H‘) , «t Euzkfid Yit‘ipu) *J-l—S‘ Mtfldx‘i t KPH) u M £19...) (‘i t (7“) 113 * where E A(P12) is the spin 1 spinor and u)‘ (P12) is the spin 1/2 spinor. The vertex factor can be averaged over initial spins and summed over final spins to give11 GAP“ %Jimq*m.1f‘t \10‘20) APTr is the coupling constant for the p-TT-A vertex and is related to the on-shell cross section 0(M) . Qt is the off- Shell pion and proton momentum in the A++ center of mass. One can define the function R(M1, M2, M) as R < «1., m, m) = J[r\"- m.m~.1‘][m‘-m.-m‘T/Jn {to - 1‘) R(M1, M M) is the momentum of the two particles of mass 2' M1 and M in their center of mass with a center of mass 2 energy of M. Q in equation (10-20) can be written as t Qt: RKMq,t,M\z.) (\0-11) In order to incorporate the resonance part of the transition matrix properly, it is necessary to relate the off-shell scattering to the on-shell scattering. Following 39 one can calculate the ratio of off-shell cross Jackson 3'fiction to the on-shell cross section with a real pion in the cross channel. This leads to a form factor 114 do- _ (3;— :MafiMnJ‘L "t _\ .— _. L L 1 (\0-13 016'? Q KMQ*M|L) ‘A J ) Q is defined using equation (10-22) as Q: R \MQ,M)M|L) K10‘1‘i) Notice that as t+uz, the ratio is unity, but in the physical scattering region, the ratio is greater than unity and increases with increasing t. The lower vertex of the amplitude in Figure 10.1 can be calculated using Feyman rules neglecting for now the form factors. The helicity amplitudes associated with the lower vertex can be calculated from T —) (Sat t 1'. '- P N“ ’ 53))b33 NMMASKVa)XSM)5KPQ UO‘J-S) One can write the helicity amplitudes explicitely in the center of mass for scattering along the x axis in the x-z Plane. Equation (10-25) can be reduced to Tp+N7r = Gena. J(M.b-M3)2-t. Neglecting the proton-neutron mass difference giVes TP—nnt = G'PunJ‘t" (”'26) 115 GzpnN/4" is taken for the charged pion coupling as 29.2 One can combine the phase space factors, lower vertex factors and upper vertex factors along with the aPPrOpriate normalizations.2' 40' 41 «1‘0 QM... (Lu L1. 07m) .1 g ) a :3 1.. N9“ - 4124 Mn. 3.11“» P: H K it"“r. ‘ (MM “M ‘0 11) Considering the 1-2 or proton-pion system to be the to give A++ resonance, Q is the on-shell momentum of the pion in the A++ rest frame and Qt is the momentum when the pion is off-shell. M13 and Ma are the proton mass and Pa is the lab momentum of the beam. T is the momentum transfer square for the target-neutron system and 0(M12) is the on-shell cross section for n+-P elastic scattering. The expression before the curly brackets is the Chew-Low1 pole expression and the curly brackets come from the spin sum of the A++ production vertex. 10.3 Dfirr-Pilkuhn Corrections Dfirr and Pilkuhn12 utilized a technique well known to Nuclear Physics42 to arrive at a vertex correction which takes into account the lack of the angular momentum barrier for r\ I Figure 10.2. Exchange graph of scalar particle with mass X. One can write the jth partial wave projection of the Bethe-Salpeter equation with the above form as gj = 93.0 + Agj Kj, where gjO is the Born term and Kj is the jt partial wave projection of the propagator for the loop h integral indicated by the arrow in Figure 10.2 One can see the a schematic solution43 of the equation is g. = gjo/(l- J AKj) where if one considers this as describing a resonance propagating in the S-channel then the vertex coupling is that of the Born approximation type and the denominator can be made to look like (Mj2 - i Mj rj - S). In practice the form factors arise by obtaining the imaginary part of the denominator of the approximated solution. The penetration factor can be identified by comparison of this width term to the non-relativistic width to give vj(x) = 1/2x2 Qj(1 + 1/2x2), where Qj are the Legendre functions of the second kind. For a p-wave resonance, one obtains 119 l Elli! 1 _ V‘kx) = :71. “1X1 1""qu *0 q (lo-'33) It will be noted that these penetration factors go to 0 as x+m unlike the Durr Pilkuhn penetration factors which approach 1 as x+W. Also the form above will give a complex number when the argument of the log becomes negative. This does occur for non-resonance decays like P+n + N when the pion is considered on-shell. One can write the Benecke DUrr form factors for the A++ vertex as V‘ (Que RA) = —3H PM“ m Vuka R.) kw ) where Qt and Q are the off and on-shell pion momentums in the A++ center of mass given by equations (10-22) and (10-24) and the RA is obtained from Wolf to be RA = 2.2 Gev—1.3 10.5 Corrected Cross Section and Po e Extrapolation The double differential cross section was written (equation 10-27) in the Born approximation in the absence of form factors as -—--°w -L——r—~. Gm‘ U51— ‘ n" film-M "~. u) WM“ AFN-UM: 3 1'“ Mb Pa '1“ “”3? 3 u Q‘ \MQOMnJl‘Wl‘ Q ) Mo -35) 120 gz(t) is an addition form factor fitted to the data. One obtains for g(t) = (C - uz) / (C - t) a value for C of 2.3 Gevz.3 The M12 above is the mass of the resonance in Gev and 0(M12) is the on-shell mass value at M12. The Durr-Pilkuhn model modifies the vertex factors to give t—‘it ‘*RN%1 ,h 2. MID-36) 1+ RN 1M: and 'L 1. 2- 7- R (9. Qt —-)G( (31‘. \* ‘3, ,_ .. Q(___) cm) (a) \+Ro Qt U0 3‘!) The Benecke Durr cross section can be obtained for the resonance vertex by the following substitution: Z Z (3.2 (mm a ‘3; “(Gag .. Q( Q ) QK Q) V‘KQgRo) (\D ‘58) where 1x‘+\ v. a (W WM -\ m-..) The pole extrapolation can be illustrated by using the pole equation and the various off-shell effects are substituted as described above. At the pole, the cross . 4 section becomes 121 ‘L \ (5 [t z .55“. = m M; P7 :33 (efai‘mua Wm“) “°"“°’ and one can write for the quantity "to" 1 7- 3.1 a“... = N]: m my; “Kt-ML (“H”) Sam At ML 0. 9",“ To evaluate "to" one can divide the experimental off-shell scattering data into bins of mass and t. N is the number of events in a particular bin and S is the ub/events in the experiment; 5 = 0.00001113 mb/event as given in Chapter III. The expression fdet is taken over the portion of the M - t bin experimentally accessible after kinematical restrictions like tmin effects are considered. The average of the factors on the right hand side of equation (lo-41) are used to represent "to" at the average t and M point of the interval. The extrapolation form factors can be incorporated into the above expression to give a smoother off-shell dependence for the data. Various extrapolation polynomials are used for the Chew-Low formula given above or with the Dfirr Pilkuhn and Benecke Dfirr corrections. Figures 10.3a-f shows the "to" points and fitted curve for polynomials of the type at and at + bt2 for the Chew-Low, Dfirr Pilkuhn and Benecke Durr off-shell correction factors. Table 10.1 shows a summary of the x2 for these fits and also for fits of the type at + bt2 + Ct3. The column marked ZFac is a multiplication factor used to scale all experimental 122 .mm>nso coflumHommuuxm Hume mxomcmm .wm>uso coaumaommuuxm ccsxafim Hume .mm>uso coflumaommuuxm 304:3020 ownlm ow'u O a.” .m.oa musmflm 123 «fine ammo » 5%.? 3.- on; 2 an.“ .6”: m— m?! ".maq 3N." 2 mm.— rox... m. an? ....,.. a“. 3.1% 5%... a- a... a3 ":0 :58 .o a .W H 1 La 1 an; E 8.. as: 2 8.! t .9 F b C F 9.. 3.2 "so so; _ Han." 2 cm.— toum m— and: :1 E, mum. «mama»: inflame: not 9. an: "to 3.2m _ to T... 1.9 8.. 2 8.. H :3: 28$. 1 .3 r F C W 9. 9a: xx» .53 I on; 2 2.— 6mm 9 m9! La we a .u. 13: Hth g; ... 3 .IA 1.. n 4. H m .m 3.. E 3.. SE 2 was. 32. _ F _ u...- 9. 98 "to «58 _ am.“ Oh an.— .a: 2 as; Milk!) WIS-'1 124 an .ro cum—«ma»: 50%qu 8r... 2 can ":0 ca.» . o z 1 H h I 9 1. 2.... 2 8.. .9: n. 8% rs — p h m.) 9. 5.8. "so so; an; 0.. «a; Ex... 2 m9! 9. a...» "so so... I [D .6 j h 1 T9 He... 2 3.. H :9: m. 9...: :9 F t L p... 9. .28.. "some; t ta rs .m nu. a... E 8.. BE .2 m2: 2 C... Tlea 5 . z a“. a... #3...- 9. mac. nun. so.” . P 0: ‘h- 84 2 an.” in... n— an! I F _ ..I .62 uhxu 3.0—m H on." o» 2.— 5.: 2 8% .u mac .4 .u 9. oénq at aim .3 n50 $33 on; 2 an.— .Bum a. 904! b uhxu 58.6 a... P— .1.— .ch 2 mm; 'h 9 m—ZIIABD 6018-1 1 I rm 125 a“. sage”... a- H mam £5 ‘33 on; 2 an.“ .8»: 2. an"! F r 9. mam. «.th 533 H an; 2 an.— suu 0— mg . I. ‘ wane gunman.— éfiflt .3... «fine «a. row—WNW! 8... nave 5%— éfiflt 3.- an.” 2 on.» H .9: n.3,... & r h h 9. as... . .10 1.9 1m. H iv 2.. E 8.. H 8.: «.32 5a HHm Ha m. 9.. can. ":0 is.» .0 j L. .m. aw 8.. o. 8.. 6a.. 28:. an F u . .r a: Qua. ".80 g; 8.. E a... 5338! La MED .m .0 9. ad:— nhxu gum '3: " “it jot '01 wag-139 13-1 8.. E 2... .63.. 2 m9! 1%. . b L u... a: «.3 ":0 538 . 1.0 1 . m “w H a... E 2.. 6.: m. 9...: H3 I “PEI-An 126 a... 5.3... a3? 3 .. r b L an 9. .2... "553.» .0 j .fl j. H 9 1 am H 8.. E 9.. 6&2?! rm r . p h. .0 or 98.4. ":3 $83 3n.— 2 an; 61m 3 m9! am. sum”... Jams, 3.... L- at mdm «Law 503 . or aim. nhxw $33 H a... E 8.. H .9: «.35. T. eHm Hme+H¢ and mu: MD h SR”? 8.: 9. ..... «.2032» .o a... E 9... 8.... 29.6. 5.... P L L .} 9. can. a...“ $3.... . 8.. E a... :2... 28.! r... mmo .m .0 «Nun «manna .— lmwwmwg 3..” 9. «.3: £5 gun .0 ma: 2 an; 6.... 2me .m r t L w. H To .... s I"! 1w m Lyn m In. 3... a» an.— 6: 33.... .m 2: 127 5 . 3:. J3! s... _ In n s «.8 "so so.“ .10 S j. 1m 1m. 3.. 2 9.. 6.228.! rm — h _ p.. u as... "3053.» .0 j 1m .m 1m 3.. 2 2.. 5.: 285. .m a. $.55- m. a: can use .5.» re .19 H t u... H 1.... 8.. 2 9... 8:28... .5 r F C w n e. as... "so so.» .0 .m am. 1.“ H «a... E 8.. £523.... rm 2“. E 9. 2.3- ....th gum on; Oh on.— erm 0. mg r“ P L— — f at n.3— uhxu g; . H on; o» 3.— Eu... 9 was: mas .o .m (.0 9. «am. "so so... .0 j I. - m m A I H...” am H 8.. 2 i4 is. «.3! ..m r L . w. 9. 13 "5.3.... .0 TO I; _ Law I J» w 1.. a... 2 s... H .9... 38¢. rm .3 128 wave 5.1%. é? no ... 2. 1.... as so.» To T“... H am. 4... H on... 2 an. :9... 286. f». F b F u. w. a. 3.8 "so so... .To .m .m .m i. 2 2.. 8.... £3! rm 5%... $1.. 3.- an... . r 9 9. 98 use as.» H..0 a 1. . x... H 7m 8.. 2 8.. 8.: m. 83. La _ L L .1.. m. a. mi... . 0 j .m .m H jw. H a... 2 8.. 8.... a. an! rm HHm bxpm+H¢ a... 3%... £1 a... 9. .3: ".583... .o an.— D» on.— H 5.... n. as. 5,... r . . .r 9. «.3. "so 53.... .o H: E H a». 8.. E a... H 58m 3 m9! :5 mac .0 .m m .2: £5 58—.» 8.. 2 an. H 8.: 29.5. -m r P P “I. 0.2. "so in.” .0 j .... 93 15W. 1m" 91 m 1.... o... E a... 8.: «.3! .w H: 129 Table lO.1.--Chi-Square and Multiplier for at, at + bt2 and at + bt2 + cT3 fits. Fit Type 1 2 3 4 5 6 7 8 Fac x DP - AT 10. 12. 26. 10. 9. 12. 25. 31. .98 60. BD - AT 9. 8. 16. 5. 16. 21. 42. 45. 1.07 57. CL - AT 83. 114. 189. 160. 89. 68. 65. 41. .68 241. DP - AT + 8. 8. 1s. 5. 8. 9. 15. 17. .999 15. BT2 BD - AT + 8. 8. 15. 5. 8. 9. 15. 16. 1.002 13. BT2 CL -'AT + 7. 7. 13. 7. 8. 6. 14. 15. .998 129.6 8T2 DP - AT + 2. 7. 14. 5. 8. 3. 15. 14. .95 25. BT2 + CT3 80 - AT + 2. 7. 14. 5. 8. 3. 15. 14. .96 23.4 8T2 + CT3 CL - AT + 3. 6. 12. 6. 8. 3. 14. 14. .99 58.4 8T2 + CT3 130 points equally. Because of the uncertainty on the total cross section it was felt that a shape comparison using a scale factor to minimize the x2 between the extrapolated on-shell cross section and experimental on-shell cross section would be most revealing. The factor listed is the ratio of the actual scale factor to the known experimental correction factor given in Chapter IX. If this number deviates from 1 by no more than the 6% cross section error, the fit results should be regarded as valid as they stand. Only two sets of extrapolation curves have factors outside this 6% error; the B.D. -at and the C.L. -at fits. This would indicate that the linear extrapolation is not good even for the Benecke Dfirr model. This table indicates that 2 and the B.D. -at + bt2 fit types are the the D.P. -at + bt best. Not only are their multiplicative scale factors almost one, but their x2 are smallest for the extrapolated on-shell cross section values. Figure 10.4a-f shows the extrapolated on-shell values obtained from extending the fitted curves to t = u2 plotted as error bars. The curve is the on-shell data.7 Again one can see the excellent agreement between the B.D. at + bt2 and D.P. at + bt2 extrapolation results and the on-shell cross section. One can use the models to predict the off-shell scattering when the on-shell results are known. Figure 10.5a-d show the Dfirr Pilkuhn curves (solid lines) with the off-shell data (error bars). The four distributions a—d 131 are the mass of the P-n+ system, the momentum transfer squared and the A++ decay angles 6 and ¢TY respectively. Jac The experimental mass and t curves are reproduced fairly well by the off-shell t dependence given by the Burr-Pilkuhn form factor model. The Dfirr Pilkuhn OPE curves shown in Figures 10.5c-d do not follow the data points. Form factor models which modify the individual helicity amplitudes and density matrix elements can also be used to predict the decay angular distributions.44-46 The 47 for example, gives predictions which 46 absorbtion model, are in good agreement with experiment. The three polynomial extrapolations terms used above all require explicitely that the "to" curve is zero at t = 0. Several schemes have been proposed to account for the possible deviation of the O.P.E. differential cross section from 0 at t = 0. It has been proposed that48 conspiracy could occur between the pion Regge pole and an opposite parity pole. The idea is that in reactions between particles with spins, kinematic constraints require certain helicity amplitudes to vanish at t = 0. For a single Regge pole in that t channel, these constraints force some amplitudes to vanish when they are factorized. Conspiracy is when a set of Regge poles conspire to satisfy the constraints collectively instead of each being zero. Conspiracy can give effects similar to damping corrections in absorption models. Figure 10.4. 132 Chew Low extrapolated on-shell mass cross section and on—shell data. Dfirr Pilkuhn extrapolated and on-shell mass data. Benecke Durr extrapolated and on- shell mass data. 133 (a) 1.28 1.34 MRSS (P,PI+] 18 T1. 0.00m . Q.omH 1 _ o.omH 0.0m m: onHumm mmomu 134 (b) C. L. OPE RT+BTXT FIT 1 0.0mm . o.oom _ _ Q.QmH Q.QOH m: oneumm mmomu 0.0m 135 1 18 1.26 MRSS (P,P1+) r 1. 10 . o.oom . O.Q©H . . 0.0NH 0.0m m2 onpumm mmcmu 136 D. P. OPE RT+BTxT FIT _ . . . .4 0.0mm O.QQN Q.omH O.QQH 0.0m m: onpumm mmomu 9. u. u. 9. + .1 6p. I .%¢P .11 q. S W R. 1 0 0.0 137 1 0.00m . o.ooH . . o.omH o.ow m: onFumm mmomu _ 0.03 138 (f) B. D. OPE HT+BTXT FIT .92 r 1.3M 18 1.28 M888 (P,PI+} 1. IO 71. . 0.0mm 0.00m _ . 0.0mH 0.00H m: 20Hp00m 00000 . 0.0m 0.0 Figure 10.5. 139 dO/dM experiment and Durr Pilkuhn prediction. dO/dt experiment and Durr Pilkuhn prediction. do/dcose . .Jac prediction. experiment and Dfirr Pilkuhn dO/d¢ experiment and Durr Pilkuhn predic ion. 1 20000. 25000. 1 15000. 1 NUMBER OF EVENTS 10000. 5000. 140 0. P. OPE 00TH TOT NT l .18 1 1.26 1.3% 1.U2 MRSS (P,PI+1 40000. J 32000. 24000. NUMBER OF EVENTS 16000. I 1 8000. 141 0. P. OPE TOT HEIGHT C50 .00 MOM 1 1 1 1 0.08 0.16 0.24 0.32 ENTUM TRRNSFER SOURRED (6EV1xx2 142 .U Tlli 11“ T11 Tl. T. T11 mm. .1 -5. 001.. TI. 0 8w .1 T: .11 ..U no.1 T1 1 n... I. no I I R. I. l.- 1 no _ . fl . _ 4 .0000m .0000H .000NH .0000 .000: .0 mpzw>m 00 000232 JRCKSON RNGLE 143 D. P. OPE Tl. H G I E llfi T 0 T1 l l 100. T 50. 4 .0000m . _ . .0000H .000NH .0000 mezm>m m0 mwmzsz . .0001 200. 150. .0 T. RNGLE T. 144 It is found useful to eliminate the t in the 2 numerator which arises naturally in the limit t+u in the Double-Regge-Pole model (D.R.P.).49 The diagram is shown in Figure 10.6. One can define S = (P1 + P2)2 and p 7* N H/ a? I a‘n'\P2 p P Figure 10.6. Double-Regge-Pole Model. S pn+ = (q1 + q )2. The application of the D.R.P. model assumes an explicate 3 body final state so quasi-two-body states should be removed. It is necessary to have JSpn+‘= M(Pn+);2.0 Gev. One obtains good results by allowing 2 49 t+u . Even if one extends the analysis to low M(Pn+)p22 one can still get reasonable fits using appropriate modifi- cations of the t factor from O.P.E. Reggeized pion exchange give good results to decay curves for the P-n+ when the t factor is modified as described above.44 The modifications necessary to have do/dt not pass through 0 at t = 0 arise naturally from an absorption mode1.45' 46 out.47 This model considers several spin states and applies A general absorption model has been worked absorbtive corrections on the angular momentum decomposi- tion of the individual Born term helicity amplitudes in the standard S-channel helicity frame. 145 The absorption can be represented for the helicity flip term as - b bt' - ’- tt 2, a sor ion > {M11 1 A“.U+Bt$,«t) " A“ " "\fl and the non-flip term is 16.101 . V11?“ (‘3) 2. - absorbtion 9 1“) t Cksfifl where t ' IA“ t 7 M“ n is the net helicity flip. This model provides a good approximation to the decay distribution of the A++ in 46 PP+NA++. A simpler absorption model has been used to extrapolate in t and obtain the on-shell n-n density matrix elements.8' 50 The absorption corrections can be used to account for the descrepancy between the data and O.P.E. model curves. As mentioned above, many models can account for do/dt # 0 at t = 0. The absorption model predictions agree fairly well with experiments at lower energies.51 Figure 10.7a-c shows the extrapolation curves for "to" for the Chew Low, Durr Pilkuhn, and Benecke Dfirr models with a fit parameterization of the a + bt + ct2 type. The + sign in the lower left hand corner indicates the position of the origin (t,"to") = (0,0). The results of these fits are summarized in Table 10.2 along with results from a linear fit. It is clear that the linear fits not constrained to pass through the origin do not reproduce the on-shell values with a good x2. Also the 146 .mHmHEocwaom COwumaommnuxm .mHmHEocmaom coflumaommuuxm .mamfleocmaom cowuwaommuuxm m N N 00 + an + 6 £083 H0008 .o.m 00 + 00 + m spas H0005 .m.o no + 00 + m nuw3 H0008 .A.O .0 .n .M .n.o. 6956.6 147 a“. 5.3%... g.-. an. age”... a... a... gag”... g... a... 53.2%... a- uh uh I! .r 9. m... ":0 in... . 9. .3... "so an.» . . . 2 u 9. 3.... "an. 5.2... . 9. .5 "8.65.... 9. a... £23....» 9.9. MM... Mann”. 9. a... £29....» to 10 1a to I. _ z 7. h. um 1 a. 1 W m I“ I.“ 1.. .1,“ m 9 9 9| 9| 1 1 a. a. 8.. E 8.. Ha»... o. 3.. 8.. o. 8.. H 8.. 8 3.. :8. a. an... r... 5.... a. 35. a. .8. a. as. .m .9... m. a... 5.. . L h a.) _ P L m.) _ b L m.) P r L um... 9. «.8. use 5...... . 9. 3.... "an. 9.2... . 9. 18. ”so .53 . 10 in To 9. 0...- «8.53.... ...o I. m... .15 1.5 I‘M m A ,m .m aw. H. m... H .w. z... 2 m... 9.... o. 8.. 8.. a. a... a... 8 a... .9... a. an... E 6... m. 8.... 8. is. m. 8.... .u .9: a. as. rm .: .535... w... J .u .3 148 a“. 599...... 9.. w 9. 5.9" .50 93.... .10 9. a: ".29.... I“ 8 I 1m H 8.. 2 3.. as... 239...”... _ . r h .o 9. 3...... "ca 9....» _ 9. 2. "8.93.. 10 1m 1m .u z... E Q... 8... m. 39. fim - m. 9. 55 use 9.2... .0 9. 2 "8.9.2.... 1 15 1m m... 8.. 2 3.. 5.... m. 29. 5 r _ . .r 9. «.3. use 9.2... . 9. a... £293.» :0 a To 3m H nu H 2.. E 2.. 8.... n. 89. fa :h. hxpu+pm+¢ a: o.nu_ u: n.0n a: o.:m~ 36 some h twig no: "so in... . "8.9.2... Lu 8.. 2 8.. 8.... 2 89. La _ * F "so .5.» . a.» "8:53 In on.“ a» on.“ mac .n. .a .. 9. «.6: "tag... .0 9. 3:- £29....» 1 I. . 3 low -m a... m nu 8.. E 2... H :2... Ennifin . . . .w. 9. 5.... "so 993 _o 9. .3... £895.» 1 a... oh 3‘.“ mean an mm¢x 3. ‘h ‘9 GH-CIIA39 UHDIS“1 Qt r9 149 . 9. 3...." "so 9...... .10 9. a..- 08.9... T3 am a... H 8.. o. 3.. 8.... n. 39. .x — b h p. m 9. man... “so 9...... .0 9. .1... ".89.... a. 1m .fl. 10 1m a... 8 9.... 5.... 2 89. rm an... gunmen... Jain: 8.- P b ~ r 9. .23 0:0 93.... .0 9. 5.. £29.... 1 T8 1.. .w 8.. 2 8.. 8.... 289...“ p p . h .0 9. «.8. «:0 9...... . 9. ... 229...... 10 H 9.... 2 8.. 6.... m. 8.... .2... Fxpu+pm+¢ a... 0.23.. .92.? 0.- p b L W 9. .18. use 9.2... . 9. as... 0.2.83 10 .19 1m .9 8.. 2 9.... H as... n. 8.! La r F _ $ 9. .19.. «S 95.... . 9. ad ”.8938 10 I... 1... y... H 9.... E a... .5... m. m9... .3 h am 5...... $2 3.- .m 9. 3...... "so 9...... .0 9. 3... £89....» 1 ..._... 7mm m hm 1.“ 8.. 2 8.. H .9... a. 89. .m — F L “ma 9. 18 "an. 9...... W0 9. 3...- "8.958 n.. s 15w .m w H: a... o. 2.. S... 2 89. .m .3 150 scale factors are not in good agreement with l, with the exception of the Durr Pilkuhn model. The Chew-Low model requires the data to be scaled up by a factor of 4. Table lO.2.--Chi-Square and Multiplier for A + BT and A + BT + CT2 Fits. Fit Type 1 2 3 4 s 6 7 8 Fac x2 F-‘ DP - A + ET 5. ll. 21. 9. 9. 5. 14. 23. .96 57. BD - A + ET 7. 8. 16. 5. 14. 5. 19. 31. 1.18 67. 1 CL - A + BT 3o. 64. 88. 74. 33. 42. 41. 25. .26 20. DP - A + ET 5. 8. 15. 4. 7. 4. 13. 17. 1.034 10.5 +~CT2 BD - A + ET 5. 8. l6. 4. 7. 4. 13. 16. 1.03 10.8 4-CT2 CL - A + ET 5. 7. 12. 7. 6. 3. 14. 15. 1.005 13.6 +-CT2 In all three models if one allows the parabolic fit for the extrapolation, a good x2 is achieved and the scale factors are consistant with one within the limits of the cross section error. Figure 10.8a-c shows the extrapolated on-shell cross sections as error bars using quadratic extrapolation curves not constrained to pass through the origin. The error bars are larger than those shown in Figure 10.4 because both ends of the extrapolation curve are free to move. The smaller errors in Figure 10.4 are obtained because the fit curve was constrained to pivot around the Figure 10.8. 151 C.L. extrapolated cross section at t = M02 and on-shell curve for a + bt + ct2 extrapolation curve. D.P. extrapolated cross section using same criteria as above. B.D. extrapolated cross section using same criteria as above. 152 (a) R+BT+CT¥T FIT 1.26 18 1 o.oom fl C.DD. . _ 0.0m. 0.0m m: onpumm mmomu _ 0.0: MRSS (P,PI+) 250.0 200.0 1 1 1 150.0 1 CROSS SECTION MB 100.0 .0 50 153 D. P. OPE R+BT+CTXT FIT 0.0 1 1.18 1 1.3M 154 (c) R+BT+CTKT FIT r 18 1.20 MRSS (P,PI+1 I 1. 10 T. 0.0mm . 0.00m 1. J 0.0m. 0.00. 0: 200.000 mmomu . 0.0m 0.0 155 origin. The freedom of the extrapolated value was severaly restricted because the origin is close to the pion mass squared. The solid curve is the experimental on-shell cross section. Figure 10.9a-c is the extrapolation cross sections evaluated at t = 0 for the three models. The O.P.E. with form factors would predict zero because the Spin amplitude must go to zero in the forward direction. The quadratic fit extrapolated to t = 0 gives a cross section that is free from increasing or decreasing trends and is in general consistant with zero. However, even the absorption model predicts this cross section to differ only slightly from zero at these energies.51 Figure 10.9. a. 156 C.L. extrapolated cross section at t for a + bt + ct2 extrapolation curve. The straight line at 0(t = 0) = O is unmodified O.P.E. prediction. D.P. extrapolated cross section at t using same criteria as above. B.D. extrapolated cross section at t using same criteria as above. = 0 the II C ll 0 157 (a) 2 U. T 1 H ET. % Wu... Tl. r. C 1 Ln 1 D. I. m. CH . Sp“ _ T20: T . 1.. 1_ s. _ .3 m r . mo. a _ I. TI111 0 .1 .01 .1 Ta 1“ 0 00 0 0: 0 0w 0.0 0.0m: 0.0:: 0H» .0 0: 200.000 mmomu 158 (b) .QZ TI. 1 H ET. U. Wu. .r ._ r3 .w. 1 Pm TI... m. 0% r . mo. _ _ _ _ .1... _ a 0 1.3 _ _ . «M... _r ._ w _ J I. I1 1 0 . . a . L 0.00 0.0: 0.0m 0.0 0.0m: 0.031 0.1.... .E 0: 207800 mmomu 159 (c) .U2 T 1 c1. F.r| Mw . + 0 m T11 m am.w _ IL .09“ _ _ mew _ L Q. _ _ _ mm _ T1” Tlli n. . . . . 1H 0.00 0.0: 0.0m 0.0 0.0m: 0.03- 0 p .0 m onpumm mmomu CHAPTER XI SUMMARY AND CONCLUSIONS This thesis reports on the analysis of one pion production reactions resulting from proton-proton scattering at 6.0 Gev/c. The final data sample includes 40,000 events with the PNn+ final state and 10,000 events with the PPn° final state. The PNn+ sample includes 14,000 events having the A++1236(3,3) resonance-neutron final state. This resonance is defined by restricting l.l48 iterations 8 Both tracks negative 3874 27 10 One track positive, 60326 1070 other negative 91 >15" from target 1903 18 92 Momentum track 2< 42 0 50 Mev 93 Condition 91 + 92 6 0 94 Momentum track 3<50 243 6 Mev 95 Condition 94 + 91 24 1 96 Condition 94 + 92 3 0 97 Condition 91 + 92 + 94 6 0 99 Varied beam momentum 191 0 Total 71629 1180 171 Table A.3.--Reactions Leading to 90% of Circe Failures. Reaction Cross Section mb PP+PNn+n+n- 3.1 PP+PPn+n-n° 2.4 Table A.4.--Teuta Events Confidence Level <.03. Positive Field Negative Field Total Total <1o'5 10'5m MOM mcoflpsnfluumflp xmuum> .QIMH.¢ musmflm m a mmpzmu p 29am mmruz_ 174 31N3A3 JD BEBNRN mmmaoaae baa Hmommh on x I «upzwo hwomap some mmruzn SlNBAB JD aaaunn mwm34~¢m haw pwoamp APPENDIX B BEAM MOMENTUM FITTING PROGRAM APPENDIX B BEAM MOMENTUM FITTING PROGRAMS2 A typical view from above the experimental apparatus is shown in Figure B.1a where the x represents measured spark positions and the line signifies an orbited track. (a) (b) -4n (1! Chmbor Numb" 5'6 ' Figure 8.1. Experimental apparatus after target (a) and coordinate system (b). 9.Io X The track has three parameters, the curvature K E %, A and ¢, shown in Figure B.1b. Denoting the deviations of the measured and orbited track by di, the fit procedure is to minimize x2 = 2612 by varying the three tracks parameters K, A and 6, denoted collectively as 8. One gets \0) w) A°. A; = A°. *- Z eke—5;) (3“) 175 176 (ggi (0) will be denoted by Di The superscript (0) A. refers to the quantity evaluated using the intitial estimate for the three parameters K(O), 1(0), ¢(0) E BA(O) and A81 = 81 - BA(O) is to be solved for as the correction term that will minimize x2. One can expand X2 above using the Einstein summation notation. X7. = click} = AfA: *- lADzoind? +ABADiAAsAD1M (8‘1) Now set 31‘ . . """" :: 02 103‘ A? TQDi‘thtfiA 18‘s) )9! Denote y¥= 1011a; KB-” and GI): 101'! D1) A $) , 18‘s.) We find -\ = ':2%.G&j}{)9 (E3"C>) One recalls for this 3 x 3 case 1143013. Y= 143°‘(3—g)” (8-7) L-:Z;¢J:°°( édb)(.fl a¢ (031 — . 177 The sum is over all chamber y and z spark position. Also one gets F U 91: OK )“1 a A; m 5:) Z( a" AK )“1 c—T my”. .1) APPENDIX C TRANSFORMATION FROM M, t, BJac, ¢TY + + T0 P AND U APPENDIX C TRANSFORMATION FROM M, t, eJac, ¢TY {. + -> A TO P AND 0 Fr; .1 EV.- One must work in the over—all center of mass in i order to relate the neutron and outgoing A++ to the initial beam and target protons. The scattering kinematics are outlined in Figure C.l. lly' F) 90» P. 4’ X 2 F3. Figure C.l. The reaction PP+A++N in over-all center of mass. The particle labels used above denote the 3- momentum of the particle. Two assumptions must be made in QOing from the four-variables M, t, 6 and ¢TY to the six Jac 178 179 variables P and E. These are a result of the symmetry of the problem and can be added without loss of generality. The first one is that the reaction has axial symmetry about the beam axis and so all scatterings can be assumed in the Z Z X-Y plane leaving PN = PA = O. This means in the lab one has PZ =-?z. Also one can assume the beam is along the x axis. One can write Mpn+ and t in terms of the variables defined in Figure C.l as MPTV' =fk?“+1\“)2" ((3‘)) and t: (Pb“-P:)7‘= (pg—PCS)“, (6-1) Pu and flu above are defined as Pu = (EP, 13) and flu = (E i?). Also one has PAu = (EA' iPA) and Pbu = (Eb, 1"! u and P u. One can in) and similar definitions for PT N expand t defined above to give t= Mgmt -1E.. E. + awoum c... 6‘ (c. -3> m0 This expression can be solved to give the center of mass scattering angle in terms of MA and t to: 9m“; \t- nt—m'g~1€.:5)/(2\P.\\Pu) . ( 6.“ '-\) 180 One can define R (mu mm) =/E(n-m.)"-m‘;.][kn~m.)‘- MEI/m (95‘) which can be used to get the beam momentum [Pb] and the A++ momentum IPAI in the center of mass. If the total energy in the center of mass is JS) then \Pb\ = kab’Mf.J—$‘) 1C_b) and \PA‘ = kafl,(V\P“.,J?). “‘7) So far, the mass of the A++ and the momentum transfer have served to determine the direction the resonance will travel away from the origin in the over-all center of mass. Now it is necessary to transform all center of mass vectors into the A++ center of mass and make use of the decay angle information. In the A++ center of mass, Figure C.2 shows the relevant decay quantities. 181 Normal ‘1’ka Normal / Figure C.2. eJac and ¢TY defined in A++ center of mass. One defines CJ9519 = P .F’ QCL‘iB) sm (PTY -.- In “13?“?N).&?B ‘3) (t-m) \Exk'é’r .15ng (a 182 First one can make use of the Jackson angle. One can find the component of the decay proton parallel to the beam as shown in Figure C.3, given by Figure C.3. Beam and decay proton in A++ center of mass. R is the momentum of the proton and pion in A++ decay center of mass given by R= R 1MP,M1\,P’\P1\*) (Cs-H.) and NB is the unit vector given by NB = PB/IPBI evaluated in the A++ center of mass. 183 The components of proton momentum perpendicular to the beam involves the Treiman-Yang angle. Figure C.4 shows Figure C.2 from above and illus- trates the angle between the planes. ‘7? 6 6.- f 6. (6, 6 2 Figure C.4. Projected view of planes described in Figure C.2. \ V The vectors shown above are defined in terms of known vectors as below: a... 6x6 (Cu-B) N ~ _ ‘1"? “ PM R :3 RB “R; (CV-‘5‘) W a P. (c-vs) a 1196—) 3 an? {Cs-Ha) I We a?) 184 By definition, the Treiman-Yang angle is the angle between the two planes and is given in terms of the newly defined vectors as COS (bT‘ =K‘Ofit KC.‘ \W) ._.7 + The vectors K and N1 form a perpendicular system and one can find the perpendicular component of the decay proton from Figure C.5, 6.1 >46 Momentum of A++ decay products perpendicular Figure C.5. to beam proton in A++ center of mass. where a, PA. = R113“ 5\Y\ ¢TT “ICOS $11) \C ”1%) ‘P -? and so 3: P1 ,. P“ .3 -7 -? R \N\ SM¢71 "K £05 (.111 1'33 c.0391“) {Cr-10) 185 and “2" P kQ‘l‘) Both the n“ and P“ four vectors can be transformed into the lab to give the desired decay vectors. APPENDIX D eJaC AND ¢TY BIASES APPENDIX D eJac AND ¢TY BIASES A typical event, as viewed in the A++ center of mass, is shown below. The beam is along the x axis in the lab. TONIC-‘9t ‘ XA CfDTY Decay Proton Proton ‘\ ‘ I \ I \\ / \ ,’ ‘ / \ JOC,’ \ I ‘ / Plane of Target ‘ ,’ Proton and Neutron Y X . , < . \ Beam \\ \\ Neutratt‘ 2 Figure D.l. PP+PNn+ event in P—n+ center of mass. 186 187 For simplification, one can consider the beam direction as forward in the spectrometer, and the neutron to be at a large angle or backwards in the lab. The pion is the track which is most sensitive to acceptance and has in most cases the widest angle and is the slowest. The acceptance will be high if the pion is forward and fast, and low if the pion is wide angled or slow. The forward speed of the pion is smallest if the decay proton is in line with the target proton. In order to get the decay proton to the lab, one just does a Lorentz boost in the direction opposite the target proton. By lining the decay proton up in the A++ cm, one gets a fast proton in the lab. This gives a slow wide angle pion. From this one can see that if there are certain decay angles which have zero acceptance, this effect will be larger for a high mass A++ than one of low mass. Typically in the A++ center of mass, the proton and pion come off back to back with a momentum of 200 Mev/c. At higher momentum, a backwards pion in the A++ cm would correspond to a slower or wider angle pion in the lab. In the A++ center of mass, the beam proton, which defines the zero for 6 and the target proton, which Jac' defines the way back to the lab, are in general not in opposite directions. As momentum transfer increases, so does this off-set angle. If the Jackson angle happens to equal this off-set angle, the target and decay protons can still be anti-parallel and a minimum acceptance will occur. 188 This line-up will be precise only if the Treiman-Yang angle is zero, as will be discussed later. For t close to zero, where this beam proton and target proton are almost opposite in the A++ cm, then BJac equal zero is the situ- ation which makes the pion slowest. Also, for eJac close to zero, a 360° rotation for ¢TY does little to disrupt the alignment because it is just making a tight small circle about the target proton. Monte Carlo studies show there is a zero acceptance at 0 = 0 for all Jac ¢TY‘ As t increases, and the angle between the target proton and the beam proton becomes less anti-parallel, it becomes necessary for the Jackson angle to become larger in order that the pion comes off directly backwards in the laboratory. This backwards pion condition requires a coincidence between the planes of the target proton and the neutron. This is the place where ¢TY = 0. Monte Carlo studies reveal that as one goes to high momentum transfer, the Jackson angle where the acceptance is zero increases slowly, and the zero acceptance occurs only if ¢TY = 0. APPENDIX E MONTE CARLO EVENT GENERATION APPENDIX E MONTE CARLO EVENT GENERATION An event of the type PP+Pn+N can be completely specified by four quantities. The most useful for this -~§ study are the mass of Pn+, momentum transfer squared from the target proton to the neutron, and the outgoing scatter angles of decay for the P or n+ in the Pn+ center of mass, 8 The bubble chamber data was Jackson and ¢Treiman~Yang' used to generate these distributions because it is thought to have no strong biases. In fact, the bubble chamber data agrees within statistics with the present experimental data once a A++ cut is made. The bubble chamber mass distribution is scaled down from the 6.6 Gev/c beam momentum kinematic boundary to the 6.0 Gev/c boundary. The bubble chamber mass, t, eJac and ¢TY distributions are converted into event generation probabilities by an integral transform. If F(x) is any of the above distributions defined between x and X events can be generated in accordance L U' with this distribution by first generating a number W such 189 190 that W is a random number between 0 and IuF(x)dx. Then one must generate an event of value x where :Lis the solution to W(X) = IuF(x)dx. Thzse events can be transformed first to the over- all center of mass where the neutron and (P-n+) vectors are found using M(Pn+) and t, then in the (P-n+) center of mass where the pion and the proton vectors are found from M(Pn+), eJac and ¢TY' It is known that only (P-n+) com- binations associated with the beam projectile will trigger the apparatus so only beam vertex P-n+ events are produced. Appendix C gives the details of the above transformation. Once the four vectors for the event are back in the laboratory, the two outgoing tracks are rotated randomly about the beam axis in order to resupply the event with a symmetric degree of freedom not obtainable from the basic four variables which describe the event. Figure E.1a-d shows the initial bubble chamber distribution shaded, and final distributions derived from the Monte Carlo event generation described above unshaded. 191 .Aomomnmqsv Emumonm cofiumuocmm mo Dagmmu map mum £0Hn3 unawusnwuumwo oHnmo mucoz can AUmUMSmV muso>m Honfimnu manndn Hmcflmfluo .6-6.H.m magmas 1972 Nuan>uov ouucaom mmmmzmmh znhzuzoz a I o. o. w. o o .a o .n 3 .— a . q . . _ _ . . . . ’0 _ — _ _ p _ _ _ p \\ \3\\'\\\\ \ \ \ .\ \ \ \ \\\-‘\\\\\'\‘T 33>3 -\\\ ‘ “\ \ \ . . 'V \\ -'\ ‘ ‘ \ \ \ . .‘\.‘\\\\ .~.\ ‘ \ \‘ .‘\\“ “‘ ‘ ‘ K‘ .7 \\ ~ \\~ \ " -,\- ‘ ‘ V ‘. .» . \ _. mhco Acapoa ozc oommu wpzoz 7: \\ \ \ \ -\ \ \s \. \\ .\. \\x .C 1 OW .- .\ \ \x\x \ \\ \\ .\ \ own . co: SlN3A3 ~+~m.mummmnam mmmz V§§§§§§§§§§§§§§§§§%& \\ .\\\\\\\\\\\ \\\‘ .\ 3 J cpco Jcahum ozc ogmcu whzoz 2: ms om" mmw Dom mum om: mmm cow SlN3A3 mgoz¢ 025129.233 WEE zomuumn szou cm" 0 T. \ V \\\\ \ owm \xfi. o8 \ K \ om: 1 com .XR m 6 8... m \ 1 82 l \ s \ I \ 83 \ 1 89 JC \ I .. 8m t 82 r . t 82 \X 1 88 cows oosm memo Bantam 02¢ enema Maze: memo Eczema 02¢ cameo maze: 2: T: SlN3A3 APPENDIX F RESONANCE MASS AND WIDTH CORRECTIONS DUE TO EXPERIMENTAL ERRORS APPENDIX F RESONANCE MASS AND WIDTH CORRECTIONS DUE TO EXPERIMENTAL ERRORS Figure F.1a-b shows the invariant mass and the calculated error in this quantity. Figure F.1-b has the probability of finding the error E at mass M normalized so that the area under the probability surface at fixed M is constant. In order to see the effects of the error on a given mass distribution, one must first select the mass, then look up the probability for a given error P(E,M) from the mass-error distribution. One gets the new mass distri- bution by a Gaussian smear of width E at the fixed M of probability P(E,M). This can be expressed as val-la“ 0"(M‘)=S€ MENU-6P1“) e )3 )F 0 subject to the restriction that fdM'a(M') = l. The actual error at the resonance peak of 1.230 Gev is 5 Mev. The error at the half maximum points are 3 Mev at M = 1.176 Gev 194 195 .mmmE m9mum> mmmE CH uouuo wo uon HmcofimcmEfip mouse .btm.a.m musmwm 196 197 and 6 Mev at M = 1.298 Gev resulting in a width error of 8 Mev. This is summarized in Table F.l. The column marked data gets its peak and width directly from the data, and not from the Breit-Wigner fit to the data. The fit does not have precisely the right shape. Table F.l. Peak and Width of M(P,n+) With Resolution Errors. Data Breit Wigner Fit Peak (Mev) 1226. i 5. 1226. i|5. Width (Mev) 122. i 8. 133. i.8' The average error increases from 1 to 6 Mev between the masses of 1.14 to 1.26 Gev. Above 1.26 Gev the error is approximately constant at 6 Mev. In order to study the effect of the experimental errors on the mass distribution, the events are smeared and fit by a Breit-Wigner curve. The Breit-Wigner form used is given by r! g 3 0" m) "" $54493 P74 ”here P g Md“) [1' K F-I) [ \ + (Qflfl M is the mass of the P-n+ system and Q is the proton momentum in the P-n+ center of mass. R is the A++ radius and Mo and Y are varied in the fit. The fit parameters are summarized in Table F.2. 198 Table F.2.--Breit-Wigner Fits to Unsmeared and Smeared Data. F Peak(Gev) FWHM(Gev) 2 . Fit 6 Mo Y (Mo) X /P01nt (Gev) (Gev) (Gevz) Exp Fit Exp Fit Unsmeared 1.246 .753 .179 1.226 1.226 .122 .133 2. Smeared 1.247 .766 .182 1.227 1.226 .130 .135 1.5 As expected the peak is insensitive to the error but the width is directly related to the errors. Table F.3 gives the corrected mass parameters after the effects of experimental error have been removed. The row marked fitted used smeared and unsmeared fitted parameters to predict the correct mass parameters, while the row marked experimental uses parameters obtained from the actual mass distributions. Table F.3.--Parameter for Predicted Fitted and Experimental Mass and Distribution. Mo Y FWHM Peak (Gev) (Gev) (Gev) (Gev) Fitted 1.245 .740 .131 1.225 Experimental 1.224 .700 .126 1.226 Id APPENDIX G SPARK CHAMBER EFFICIENCY APPENDIX G SPARK CHAMBER EFFICIENCY The spark chamber efficiency calculation will be given in detail only for the magnet section chambers. The rest are calculated similarly. Denote by aN where N is 5 - 8 the probability that chamber N misses 1 track and BN the probability of missing both tracks. Let EN = “N + BN be the probability that chamber N does not see either th track. The probability that the N chamber sees both tracks is l - EN and the probability for a perfect event in the magnet section, abbreviated by P4 _ 4, is then Pg " IV U‘Et) Ur“) " L=S and is symbolized by m . The probability of firing 4 chambers on one side and 3 on the other is symbolized by [4t an :1: 61mmHumanitarian D 199 200 and is denoted by P and is given by 3-4 8 8 63-.= 2 «.13 wen. (av-2.) 3:5 J’s. jfl The probability P33 for both tracks to misfire in one chamber is _§, 8 F3.3\same chamber) =3 L 8‘ 1:1 “‘60 (6‘3) i=5 9;? J \ and the combined probability for P4_3+P3._3 where the P33 case has one chamber miss for both tracks is 9 9 EI6+P34X same chamber ) aiE;}§gk\-— Ej) Ker-q) since Ei = oi + Bi' Another type of 3-3 event can occur when different chambers miss a track as shown below. P}3(different chambers)= in Fig): *flEIIj‘flL—‘an: . The entire term can be expressed as 1 8 8 (different chambers)= V; :01; Z. d3 ESU‘E‘Q. (5'4) i“ j‘i?‘ 3" tsij The factor of 1/2 is necessary because the expression above P3—3 includes events where<3ne track fires 4 chambers and the other fires only 2 chambers, symbolized by 201 9..-;- .13? amp maintaining: These types of events will not pass the filter program. The total probability of getting an event is P4-4 + P4—3 + (same chamber) + P (different chambers) = P3-3 3—3 3 8 TI SW E") +2 E3 “11“)fy1Z“:_dj “51“63) ‘3 ’5' 3;; '.5 an‘ Kgfi 5'5 19‘“) As one can see from the magnet chamber event probability calculation care must be taken to be sure I certain event classes are not counted twice. With this observation made and the notation established, the results for the other sections will be summarized in Tables G.1 and G.2. Table G.l.--Beam Chambers. Condition Abbreviation Symbol Probability . 9 All 4 hit P4 -H—H— TAU—E t 9 I miss P3 ++++..... Z E; .TESI'Ej) (1‘ I I 202 Table G.2.--Hodoscope Chambers. .quw I r 3 ~= 9.- - =2 _- Condition Abbreviation Symbol Probability \O All fire P22 fl nc‘(\-e.‘) r.- to (0 One miss P21 jpd- flp-t-H :0“ R “‘53) One side -= J‘fl t R 3" IO to Two miss P + _ i 8' TI (Peg) 11 | . Same chamber 33‘ J:‘ 3#\ Iteration If one knew ai and Bi for the chambers, one could put them into the formulas above and get an efficiency. The individual chamber efficiencies are coupled to the over all efficiency and the solutions must be obtained simultane- ously. In the magnet section, the first approximation to di and B1 is just Ni(l)/N and Ni(2)/N respectively, where N is the total number of firings for the chambers and Ni(l) and Ni(2) are the single and double misfires respectively in chamber i. However, Bi estimated this way would be too low because of the constraint that at least three firings must occur per track in the magnet chamber region. In reality, Bi estimated as above is the probability that one chamber does not fire and all three others do fire. An exact expression for Bi is the sum of the probability that 203 chamber i miss two tracks and is seen plus the probability that chamber i miss two tracks and is not seen. Similar statements can be said for mi and Bi to give N;‘°’/NT *P? N.S. (Gr-‘7) on = Nim/Nr *6 P". a... \6- 6) 31 = Ni /N1' * {n.5, (Gr- ‘1) where P? N.S. is the probability for N sparks in chamber 1 to be missing and the event not to be seen. Table G.3 will summarize the probability of the ith magnet chamber missing 4 two track and the event not be seen. The notation Z (l—Ek) 4 i-j will be used to mean 2 (l-Ek). k=1 k#i ka‘j Table G.3.--Probability for iEE-Magnet Chamber to Miss Two Tracks of a Missing Event. Symbol Term :‘jE-I-H + :U:+-- B;:Ejfi\“en) :1: j: ags-J“ 2.x. 13.3w E.) i + 'T:—+fv"r-_:_- &‘§:‘33 quiEi':§_:*L 5': Ba 204 The sum of the above terms is Pi N.S. which can be written as BiF(aj,Bj) j#i. Similarly one could write out terms for P: and Pi N.S. and obtain the efficiency by solving the four simultaneous equations below. Ema = PM \d'HBX) * qug Lda,Bg) * P34. \dafix) (5‘|0) m U) o<3 = N3 /N1 “ P1 N.S’\°‘J,Bj) (CE-“0 . \1) 2.) 3| 3 N" /N1*F$\ u.s.\e<3,83) Ur‘u) NT = Flo/EM“? (Cy-“3) In practice it is possible to iterate the equations which rapidly converge to give an efficiency. Similar equations can be written down for the beam and hodoscope chambers and the final apparatus efficiency is taken as the product = E E E EApp B Mag Hod' The data No' Nio), Nil) and Niz) are obtained for the most of the input data which makes Teuta fit confidence levels greater than .03 and is present in the final summary tape. The results of the counting of total firings, one firing and no firings are listed for 42427 Teuta fit events in Table G.4a-c respectively. 205 Table G.4a.--Beam Chambers. Chambers Sparks Misses a 1 41273 1154 .027 2 40787 1640 .039 3 42016 411 .0096 4 40859 1568 .037 Table G.4b.--Magnet Chambers. Chamber 2 Sparks 1 Spark No Spark a B 5 32531 9322 574 .219 .014 6 36620 5682 125 .136 .0030 7 34027 8284 116 .195 .0027 8 40912 1504 11 .035 .00025 Table G.4c.--Hodoscope Chambers. Chamber 2 Sparks 1 Spark No Spark a B 9 37271 5060 96 .119 .002 10 37021 5388 18 .127 .0004 The total efficiency is then the product of the three sub- efficiencies. APPENDIX H ERROR CALCULATION APPENDIX H ERROR CALCULATION Missing mass 3 We can calculate the error in the missing mass for - 1 the reaction PP+Pn+N or PPn° as below. A diagram and a I coordinate system for the reaction is given in Figure H.1. Missing Mass ) 4’ x Figure H.1.--Notation and coordinate system used in this appendix. We can write 3 1. 1. °' . .. i (M ”)1: KPBi'Pf‘Pf P3) : (E 8+E‘-EI-Es) - Z k P. 0131‘ -pt ag)‘ U44) Substitute P“ 3 Peas) c954? 9x. PcosanQ (H4) 9": Pan) 206 tr. 207 gives 1. (MM); : 3M, + m; +2EBE.t ~21. 58 E1 ~15,E‘3—-3 5‘ EI-E’*Ea*figs *‘ ’- Pa 91.". (.03 lanes); £05103 4}.) ram), Sm 21.] " 1&9: [(108)‘; (.05 )3 cos {QR-(b3) *' Sm'A-rSm'ls] + 29.9.LQ..A. cos). (nos mb.-t.) *5m335m7m 1 \a-s) If one defines Z(i) = (-l,l,l) and considers the beam as track 1, one can write AMM‘ A )i. 3 2“” Infilp; 81‘3““); 9""‘3 9°$k¢i‘¢j)‘COSNs}nAfl + 2m in“) 2 9% Flinn?“ (MM: “05 “W ‘45) “W‘A‘W‘X‘J AMM‘ _ W A.) W .. %\‘1)2=\j)2p;PSV—cosmcosn‘oskw- ¢3)] + a“) ‘tud )- 9'. PkLcosMcM‘An 3"“ “’5‘4’0] kH-S) Also define @ij = Losk{cOs¢\3tos\¢;‘¢j)1-nn 3; Sm )3 “1-5) Where i, j and k are defined cyclicly as l, 2, 3; 2, 3, 1; or 3, 1 and 2. Using equation H-6 one can write 1. 3W“ .. 1.3:. \E, -E‘-e,)+19z@n+2?393 tun) a P. Eb 208 335‘: 1.3.1. k-E,-MP+E3)+1P® was 04-8) 8 \ ‘1 ‘3 3?; ‘- I. An“ = 2.33. \‘E\ ‘MP+51) ‘1P1®'_3 + 2R®|3 \H‘fl) .1 P3 5‘ The error in missing mass squared can be written as I .. , . (Error) 2 Z; E‘IJ V‘ VJ (H '- ‘0) MJ where Eij is the symmetrized error matrix of Circe and V: m‘.m‘,M‘.Mfmjmtmim‘,m‘ (a-..) AP‘ 9)| 311’. 3?; 3A). J¢t 3?} 333 J¢3 Invariant Mass This error involves just two tracks. The notation is shown in Figure H.2. P w Track 2 Track 3 7T Figure H.2. Notation used in calculating invariant mass ' error. Using the decay and coordinate system as shown above we get 209 'z _ t x 5“ Y 1‘: b ‘1' MW - \EuE.) 4?. +93) 49,393) -\P.+P,) UH!) Px“ Ptoshcos‘P, P‘3Pc—osls'm4’,9§= PM“) (H‘B) gives ‘ L 1 M p“... = M P +M n+ 2E;E3 ‘27; 73 [cos 2 1COSA3t08Wf§3y$mAz 5.")3‘] \H - H) One can calculate the vector V: 3n__P_L"u 3M” +.1'33Mgn* Na'mm: L—Mffi HHS) 3P ’3). 34., 6?; '3); 273—3 and get (Error)2 = Z 8. (3 Vi VJ k“- u’) where eij is the 6 x 6 error matrix for the 6 quantities involved. Momentum Transfer Squared The errors in this are calculated similar to the missing mass errors. The notation is defined in Figure H.3. FE Proton BeOfll F% /,//’E:/” \Pion Figure H.3. Notation used in calculating momentum transfer squared error. 'r'h. U 210 All the results from the missing mass calculations apply except for the following: t ._. {Mm)zvm;‘2Mp\E\-EL‘EI) kH-n) .§__ :: ELEQEE: - V‘ .Jfl (qu 9.1. = 9m " M 21.: - 3P; 3?... + P Ez. {H H) 3.: .... 2.313 +M93_E1 {Pg-no)