ABSTRACT A STUDY OF THE REACTION 5n + n+n‘n‘ FROM 1.09 TO 1.43 GeV/c BY James Michael Mountz In a deuterium bubble chamber experiment 3, 4, 5, and 6 prong topological cross sections were obtained for an incident antiproton momenta from 1.09 to 1.43 GeV/c. Also determined were cross sections for the reactions 5n + n+n-w-, 5n + pow- and 5n + fon-. The differential cross sections for En + pow“ and 5n + for- were obtained. A test of the crossing symmetry principle was made by comparing the differential cross sections for the reaction 5n + pow- with those of its line reversed reaction fl-p + npo. Within experimental error the crossing symmetry principle was found to be valid for these reactions. In an analysis of the c.m. angular distributions for the po and fo mesons the spin state composition of the pn system was found to be predominately in a s = 2 state. The Dalitz plot was used to compare the predictions of the Veneziano model with the data. The overall agreement is good, but prominent whe 2.0 GeV/c. James Michael Mountz is good, but the areas of discrepancies become more prominent when the model is compared to data from 1.6 to 2. O GeV/c. in Part} A STUDY OF THE REACTION 5n + n+n'n' FROM 1.09 TO 1.43 GeV/c BY James Michael 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 I wish E. Zee Ming . the analysis gained from t eke the work. enjoyable. IV. Gerald A. Smi during the phj our scanning a curing the mea I am grat 1‘ v ““1119 the las ‘ r ‘3‘ the encouz ACKNOWLEDGMENTS I wish to express my appreciation to Professor H. Zee Ming Ma, for his patience and guidance throughout the analysis of this experiment. The motivation that I gained from his friendly and thoughtful advice helped to make the work involved in preparing this dissertation more enjoyable. Many thanks are due also to Professor Gerald A. Smith for his helpful suggestions and comments during the physics analysis of the data. I am thankful to our scanning and measuring staff for all their efforts during the measurement phase of this experiment. I am grateful to my wife Gail, for her devotion to me, during the last year of this research. I am also grateful for the encouragement that I received from my parents. ii LIST on TABLE LIST 0? FIGUR ChaptEr I. INTROD II. CROSS NNNNN O O C U‘IhWNH N o 2.5 TABLE OF CONTENTS LIST OF TABLES O O O O O O O O O O O 0 LIST OF FIGURES O O O O O O O O O O I 0 Chapter I. INTRODUCTION . . . . . . . . . . II. CROSS SECTION MEASUREMENTS. . . . . . 2.1 Scanning and Corrections . Measuring and Corrections. Fiducial Volume Length. . 2 3 4 Track Count . . . . . 5 The Interaction Likelihood 2 2 .5.1 Corrections for Unseen Elastics .5.2 The Interaction Likelihood Calculation . . . . . . 2.6 Topological and Reaction Cross Section . . . . . . . . . 1 Separation of 5p Contaminations 2 Topological Cross Section Calculations . . . .; . . 3 Acceptance Criteria_for Eyents _ for he Reaction pn + n n n . 4 pn + n n n Reaction Cross Section Calculation. . . . III. RESONANCE PRODUCTION. . . . . . . . 3.1 The pon-, fon- Resonance Production 3.2 Event Selection . . . . . . . 3.3 Resonance Cross Section Determination . . . . . . . iii Page viii b) OkOQChUJ 14 14 19 33 41 45 51 51 51 57 Chapter Page IV. DIFFERENTIAL CROSS SECTIONS AND COMPARISON OF LINE REVERSED REACTIONS . . . . . . 66 4.1 Motivation and Formalism . . . . . 66 4.2 Experimental Analysis . . . . . . 71 4.3 The Line Reversed Reaction Comparisons. . . . . . . . . 79 V. SPIN DETERMINATION OF THE ANTIPROTON NEUTRON SYSTEM . . . . . . . . . . 83 5.1 Formalism .’ . . . . . . . . . 83 5.2 Analysis . . . . . . . . . . 87 VI. THE DALITZ PLOT FOR THE 5n + n+n’n’ REACTION AND ITS COMPARISON WITH THE PREDICTIONS OF THE VENEZIANO MODEL . . . . . . . 91 6.1 Motivation, Theoretical Background . and Analysis . . . . . . . . 91 VII. SUMMARY AND CONCLUSIONS. . . . . . . . 117 APPENDICES Appendix A. Scanning Efficiency Determination . . . . 120 B. Measuring Efficiency Determination . . . . 130 C. Fiducial Volume Length Calculation . . . . 134 D . TraCR Count 0 O O O I O O O O O O 0 14~0 E. Proof That Equal Areas on the Dalitz Plot Correspond to Equal Probabilities in Phase Space . . . . . . . . . . . 147 LIST OF REFERENCES. . . . . . . . . . . . 150 iv Tab‘e 2.1 Table 2.1. 2.5. 2.6. 2.7. 2.8. 2.9a. 2.9b. 2.10. 2.11a. LIST OF TABLES The scanning efficiency, given in terms percentage of correct events . . . Measurement efficiencies for the Ed + p ... topological cross section. . . The average tracklength of the incident beam particles. . . . . . . . The number of beam tracks entering the fiducial volume . . . . . . . The intercept and slope parameters for dO/dt in the elastic cross sections. Experimental values used in calculating interaction likelihood OT/NT . . . Event type hypotheses . . . . . . The number of events in this experiment Slowest positive track project momentum characteristics (for the events shown Figures 2.2 and 2.3). . . . . . of SP in Chaaacteristics of the projected momentum distribution for the Slowest positive track (for events shown in Figure 2.4 ). . Probability that the measurer will measure a particular pp type event on the bas of the scanning criteria . . . . Contamination from the pp type reaction the 4-prong sample in millibarns. . is S in Page 10 13 15 16 19 28 . 28 34 35 Table 2.11b. Contan 6 pr 2.12. The 6a SECtl Table 2.11b. 2.14. 2.15a. 2.15b. 3.1. 3.2. 3.3a. 3.3b. 4.1. 4.2. 4.3. 4.4. 5.1. 6.1. 6.2. 6 prong sample in millibarns . . . . . The pd + p + . . . topological cross sections.p . . . . . . . . . . . En topological cross section and the screening factor . . . . . . . . . The initial contamination of the data. . . The number 9f_events accepted as the Ed + ps + n n n reaction, and the corresponding crgss section with the fiducial volume cut The number 9f_events accepted as the pd + pa + n n n reaction, and the corresponding cross section without the fiducial volume cut 0 O O O O O O O O O O O 0 Sample count . . . . . . . . . . . Resonance parameters for the reaction En + n w n from 1.09 to 1.43 GeV/c . . . . - +-- up and af values for pn + n n n for momenta 1.09, 1.19, 1.31 and 1.43 GeV/c . . . . The Do, n-, f0, n-; and n+n-n- statistical cross sections for the momentum range Of 1009 to 1.43 GeV/C O O I O O O O Invariant mass cuts.' . . . . . . . . Average differential cross sections . . . The differential cross section parameters . The comparison of the intercept and slope parameters from this experiment with its line reversed reaction . . . . . . . Spin state composition of the pn system . . Values of the p trajectory parameters. . . Percentages of each spin parity state and N, the ratio of the fitted to the total number of events . . . . . . . . . vi Contamination from the pp type reactions in the Page 36 41 41 42 48 48 57 58 59 60 74 76 80 82 90 106 108 Table A'l’Ae A-l-Be A-2. D-lo D-ZO Definition of the scan code numbers . . . Classification of major and minor errors . Results of code number count for the three and five prong events . . . . . Results of the code number count for the four and six prong events . . . . The total number of passed events is given in Table B-l-A; and total number of failed events is given in Table B—l-B. . The measurement efficiencies for the entire sample 0 O O O O O O O O O O 0 Events in measured sample which were unwanted quantified in terms of percentage of the total sample. . . . . . . . Beam count results. . . . . . . . . Summary of the track count . . . . . . vii Page 122 123 124 126 132 133 133 141 145 Figure 2.1. 2.2a. 2.2b. 2.3a. 2.3b. 2.4. 2.5a. 2.5b. 2.6a. 2.6b. 2.7. LIST OF FIGURES A typical bubble chamber picture . . . . . 1068 4-Prong events . . . . . . . . . 1143 4—Prong events . . . . . . . . . 413 6-Prong events. . . . . . . . . . S30 6—Prong events. . . . . . . . . Projected momentum distributions for the slowest positive track for a sample of 4 Prong events . . . . . . . . . . Probability that a positive pion with a spectator neutron from a four prong event will be mistakenly measured as a spectator proton. . . . . . . . . . Monte Carlo prediction for the slowest projected fl+ momentum distribution for Mark 16. O O O O O O O O O O O 0 Probability that a positive pion with a spectator neutron from a Six prong event will be mistakenly measured as a spectator proton. . . . . . . . . . Monte Carlo prediction for the slowest projected n+ momentum distribution for Mark 16 . . . The Ed + 3, 4, S and 6 prong topological cross section arising from the pn interaCtj-on O O O O O O O O O O 0 viii Page 22 22 24 24 27 30 30 32 32 38 .. .I~‘I' ee:loe Li Thegr sec: the Chara: .ch 1'? Chara: 3"”9 VV» The p: cc~w East H reac m. ~~ara< -CCE PH - The 7' C 1 CE} Figure Page 2.8. The 5n 3 and 5 prong topological cross section compared with the prediction of the impulse approximation . . . . . . 40 2.9. Characteristics of events satisfying initial agceptance criteria for the pd + P + n n n reactions . . . . . . S? . . 44 2.10. Characteristics of events satisfying final acceptance criteria for the pd + ps + n n n reaction . . . . . . S? . . 47 2.11. The pn + n+n-n- reaction cross section compared to the results published by Eastman, et a1. . . . . . . . . . 50 3.1. The n+n’ invariant mass distribution for the reaction pd + pSp + n n n . . . . . . 53 3.2. Characteristic of events satisfying final acceptange_criteria for the reaction pn + n 1T n O O O O O O O O O O O 56 3.3. The n+n- invariant mass distribution at 1.31 GGV/C. e o o e o e e e o o e e 62 3.4. The resonant cross section and comparison with higher momentum data . . . . . . 65 4.1. The elementary baryon exchange diagrams for the processes listed below . . . . 68 4.2. Differential distributions . . . . . . 73 4.3. The definition for 8 in the c.m. frame . . 74 4.4. Resonance region and control band regions . 75 4.5. The average differential cross section . . 78 5.1. The resonance produced differential cross section . . . . . . . . . . 89 6.1. Group+ 1 _data for (A) The Dalitz plot of s(1r+ n ) vs s(n n ); (b) The projection onto fhe mass squared axis. . . . . . 102 ix Anne Ll. 13. Q1. Q2 l1 Samplg 3(n ontc The cc data the Dalitz samp betw The CO} data Dali! Dalitz samp: betW1 Vertex {ESpq Geometi radii Vettex Figure Page 6.2. Sample 2 data for (A) The Dalitz plot of s(n fl ) vs s(n n )3 (B) The projection onto lhe mass sqaared axis . . . . . . 105 6.3. The comparison of the Veneziano fit to the data in sample 1 for Slices across the Dalitz plot. . . . . . . . . . 110 6.4. Dalitz plot for the Veneziano Model fit for sample 1 for (A) The absolute deviation between the model; (B) The Fit. . . . . 112 6.5. The comparison of the Veneziano fit to the data in sample 2 for slices across the Dalitz plot 0 O I O I O O O O O O 114 6.6. Dalitz plot of the Veneziano Model fit for sample 2 for (A) The absolute deviation between the model; (B) The fit. . . . . 116 C.l. Vertex position of all measured events with respect to the fiducial volume. . . . . 137 C.2. Geometric picture of a circle with the same radius of curvature as the beam . . . . 139 0.1. Vertex position for all measured events . . 146 ' ' .L‘H‘ In h . ...; “mental ac+' CHAPTER I INTRODUCTION In high energy physics one yearns to discover the fundamental principles by which elementary particle inter- action can be explained, or at least classified. Where scattering is concerned these principles may be quantified using the scattering amplitude formalism. Ideally a scattering amplitude not only represents the dynamical interaction but also is assumed to implicitly obey known and applicable symmetry principles. The basis for accept- ing a symmetry idea can sometimes be intuitive. The endowment of the scattering amplitude with these intuitive ideas should be experimentally tested with respect to the region of their applicability. Experimental tests of the dynamical ideas are no less important, but in reality it is sometimes difficult to obtain a clear separation between the intuitive and dynamical ideas. One such idea is the familiar principle of cross- ing symmetry. Though the theoretical implication of this principle is clear, sometimes one must invoke theoret- ically simplifying ideas in order that comparison with experimental data may be made. In this work we have used 1 the reactin h v a'p + 5 n f'5 IRIICDT the duality | dualistic V6 .l the {an + T- “ compare the Veneziano mo of the 5n 3;. Veneziano mo: A ma; obtained fro: Laboratory 3; beam of anti; C“EV/C. This feels of film “in? the 30- Katiml LabO: Most c 1301! which 110; :13 as mention Sivai HY in the the reaction 5n + Don“ and its crossed channel reaction n-p + Don for such a study. Another idea that emanates from the connection of the duality principle with strong interaction is the 1 model, and its novel application2 to dualistic Veneziano the 5n + n+n'n‘ reaction. The Dalitz plot is used here to compare the data with the prediction of the spin modified Veneziano model. In addition, the spin State composition of the pn system was determined separately from the Veneziano model comparison. A major portion of the data for this study was obtained from an exposure of the Brookhaven National Laboratory 31-inch deuterium bubble chamger to an incident beam of antiprotons of momenta 1.09, 1.19, 1.31 and 1.43 GeV/c. This resulted in a total of 150,000 triads on 72 reels of film. The remaining sample of events was obtained using the 30-inch deuterium bubble chamber at the Argonne National Laboratory. Details for these data have been published elsewhere.3 Most of this work deals with the Brookhaven data, from which topological, reaction and resonance cross sections were obtained in addition to the dynamical analy- sis as mentioned before. The Argonne data is used exclu- sively in the dynamics analysis, and in this capacity com- prises about out third of the total sample. CHAPTER II CROSS SECTION MEASUREMENTS 2.1 Scanning and Corrections The 72 reels of film from the Brookhaven exposure were scanned for all three and five prong events, plus those four and Six prong events with an obvious spectator proton track. By being obvious here one means that a track must be at least twice minimum ionizing in any of the three views. An event that fell in the above mentioned category was measured if its interaction vertex was visible in all three views and within the measurement region, as Shown in Figure 2.1. The subjective part of the event selection begins here, because the final measured sample contained only those events which the measurer felt belonged in the above mentioned category. Therefore in reality the aspired sample can never be completely obtained. In this manner a total of 37,716, 49,030, 17,808 and 16,845 events were found in the 3, 4, 5 and 6 prong topologies respectively. In order to determine the total number of events that occurred in the sample, one must know how many events were not recorded correctly, or not recorded at 3 .o Fig. 2.l.--A typical bubble chamber picture. The measure- ment region is in between the lines. all by the independent losses of e :issing in 5 rows the ny independ nt A 55. second scan: distributed this sample calculated . fiISt and 56 Stan table a. 3!: a. ‘he output "’33 all by the scanner. One way to do this is to perform an independent second scan of the sample. Assume that the losses of events in both scans are random, the number missing in a particular sample can be calculated if one knows the number found in common and separately from two independent scans of the sample. A sample of 12 out of the total of 72 rolls were second scanned. These rolls were randomly and evenly distributed among the four beam momenta. All events in this sample were analyzed and scanning efficiencies were calculated. In the case of a discrepancy between the first and second scan, the event was reprojected onto the scan table and was examined and classified by a third scanner. A full explanation of the details involved in calculating the scanning efficiencies list in Table 2.1 is given in Appendix A. 2.2 Measuring and Corrections A total of 121,399 events were measured. The track information for each event was reconstructed by the 4 program TVGP, and kinematic fits were performed by the program SQUAW.S A topological event count was performed on the output tape of SQUAW and events failing to pass TVGP or SQUAW were remeasured. The number of events counted on the final data tape was corrected for its measurement inefficiencies which are shown in Table 2.2. ”INN-iii IIIWaII-fcc (IEIII. h 16‘ QNQWVE +| 38. H mom. 38. 3m. :8. H 2%. 38. H 23. 24 .28. H New. 38. H NS. 38. H «.2. mmoo. H NS. :4 38. H $3. 38. H Sm. 38. H mg. 38. H HS. 34 88. H Es. 38. H 8m. 88. H TE. 38. H :3. 34 u i u n 3923 muonm m .mcoum m mcoum e mcoum m Esucmaoz .mucm>o HOOHHOO mo ommuamouom mo mfiumu GH cm>Hm .mocmHOHmmo OCHccmom onBII.H.~ OHQMB . .. _[ I 3"“ Table 2.2. Homentmn (F ‘7/ NEW C) x 1.09 Table 2.2.--Measurement efficiencies for the 5d + pSp + ... topological cross section. Momentum (GeV/c) 3'Pron9 4‘Prong 5‘Pr0n9 6-Prong 1.09 .938 .929 .912 .883 1.19 .957 .930 .916 .870 1.31 .947 .937 .908 .884 1.43 .956 .941 .916 .880 The details of the measurement efficiency calculations are discussed in Appendix B. 2.3 Fiducial Volume Length The target thickness that enters into cross section determinations is the average path length of the inter- action region. Since the measurement region is demarcated for the scanner by visual marks on the film, its precise location is unclear. In order to insure that every Observed event was recorded within the region over which the tracklength is calculated, one defines a fiducial volume by considering a sub-zone inside the measurement region. To calculate the fiducial volume length, an arc that has the same curvature as the beam is constructed through each measured vertex within the fiducial volume. The actual length of the beam track for that vertex is :11 the length of the arc lying within the fiducial volume boundary. Appendix C explains the geometrical ideas involved in the fiducial volume length determination. Table 2.3 gives the average tracklength for each momentum. Table 2.3.--The average tracklength of the incident beam particles. Laboratory Momentum Tracklength (GeV/C) (cm) 1.09 41.9 i 5 1.19 41.9 i .5 1.31 41.7 i .5 1.43 41.8 i 5 2.4 Track Count In order to calculate an experimental cross section one must know how many particles are incident on the target. In this experiment a count of beam tracks was taken over all reels using every 50th frame. The results are then scaled up to the total number of frames measured. The total numbers of beams are listed in Table 2.4 and the details are given in Appendix D. 10 Table 2.4.--The number of beam tracks entering the fiducial volume. Momentum (GeV/c) Total number of beam 1.09 168664 1.19 258432 1.31 497598 1.43 330805 2.5 The Interaction Likelihood The interaction likelihood, expressed in units of ub per event, is the cross section equivalent of an event. This ratio depends on the fiducial volume length, track count, and the total cross section. is determined for a particular experiment, the cross Once this ratio section for any event type can be calculated if the number Of events is known. If there are N TK tracks entering a fiducial volume Of length L of a liquid deuterium target of density p, and the total cross section for interaction is OT, then the total number of interactions N is given by T AO —D§— LOT NT = NTK(l-e ) where A0 is the Avagadroe number and the factor -pAo LO (l-e 2— ) T represents the fraction of the incident particles that will interact. (2.1) The 11 The interaction likelihood is given by O ..I = T (2.2) -p_O_ LOT NTK(l-e 2 ) 2.5.1 Corrections for Unseen Elastics A typical elastic event shows a slight kink on the incident beam track and a small stub indicating the recoil- ing of the target particle due to the momentum transferred to it by the beam. It is observed that a small stub is a more obvious signature of elastic scattering than a kink in the beam track. Therefore, it is more logical to deter— mine losses in elastic events based on a minimum observ- able stub length criterion. From the stub Size measurements, it has been estimated the minimum stub length that can be seen con- tains two bubbles, which corresponds to a length of .076 i .025 cm in Space. A proton with a range of .076 i .025 cm has a momentum of 70 fig mev/c. The momentum of a deuteron with a given range is related to that of a proton by the scaling law RD (PD) g2 RP(PP) where RP is the range of the proton with mass P . MP and momentum PP, and R is the range of the deuteron D with mass MD. The momentum of the deuteron PD is taken at PD = ED PP. Therefore the momentum of the deuteron with a range of .076 i .025 cm is twice that of a proton With a range of .038 + .012 cm. This gives a cutoff cszentum f C pending val: a... b-Kpllat. Extr abolation .‘c. - 1"»er Eat ' l 12 momentum for the deuteron of 112 :30 MeV/c. The corres- ponding values for the momentum transfer squared t are _ +.0007 '0049 -.0011 for deuterons. GeVZ/c2 for protons and -.0125 :'33%Z GeVz/c2 The total missing elastic cross section can be obtained from the expression: t t (2.3) Omissing = ORG D.» d: (pd elastic dt + é-Z 3% (pp elastic) dt + O(unseen pn elastic) where S is the screening correction as explained in Chapter 2.6.2. This expression must be corrected for high momentum target recoils which are not in the plane of the camera view and have apparent lengths less than the .076 cm cutoff. The O (unseen pn elastic) was determined by esti- mating the amount of En elastic cross section with kinks less than 80 in the laboratory, and without a visible spectator proton track. The differential cross sections dO/dt are param- eterized by dO/dt (t=0) e-bt where the parameters do/dt (t=0) and b are obtained from Ma, et al.,6 and from a compilation by the Particle Data Group.7 Using a linear extrapolation of these data to the mean momentum of this experiment, 1.27 GeV/c, one obtains the results given in Table 2.5. Table 2.5.-- Target Part: \ Neutron Proton Deuteron \ 13 Table 2.5.--The intercept and slope parameters for dO/dt in the elastic cross sections. Target Particle dO/dt (t=0) mb/(GeV/c)2 - b(GeV/C)"2 Neutron 617 i 27 14.2 i 6 Proton 617 i 27 14.2 i .6 Deuteron 1700 i 200 44. i 2 Using these values and equation (2.3) one Obtains omissing = 27:? mb (2.4) A correction for stubs not on the film plane was done for stub lengths from .076 cm to .167 cm. This corresponds to t values up to .02 GeVZ/c2 for the deuteron and up to .008 GeVZ/c2 for the proton. For stub length ranges of .077 cm to .167 cm the percent of the stub with a projected length less than .077 cm was calculated for 1.0 percent increment of increased stub lengths. The missing elastic cross section for each t bin due to short Stubs was determined and then summed. The higher recoil momentum events yield an additional 4 mb loss to the unobserved elastic events. The final unseen elastic cross section is 31:; mb. events is t lengthen t3J mseen part tribution t this amOunt the effecti L'si resting for count line a “wen-Clix D ( 14 2.5.2 The Interaction Likelihood Calculation Although the direct effect of unseen elastic events is to raise the number of recorded events and to lengthen the fiducial volume, one may reason that this unseen part of the elastic cross section makes no con— tribution to this experiment. If one therefore substracts this amount from the pd total cross section, one obtains the effective total cross section for this experiment. Using the effective total cross section and cor- recting for the attenuation of the beam between the beam count line and the fiducial volume line as explained in Appendix D one obtains the results given in Table 2.6. To determine the sensitivity of the quantity OT/NT with respect to the unseen elastic scattering cor- rection, one notes that if 50 mb of unseen elastic cross section is proposed, the value of OT/NT changes by only 1.6%. 2.6 Topological and Reaction Cross Section After each event is measured and processed through TVGP, the SQUAW program performs kinematic fits according to the event type hypothesis listed in Table 2.7. Ideally the best fit will be obtained when the correct hypothesis is suggested. In reality the systematic and statistical uncertainties in the track information may cause an incor- rect hypothesis to have a better fit according to objective 15 98. + so; 03.. m. H. mév 883 am + c.mfl 6.3: 34 mmo. H cm; 63.. m. H 5:. @334 am H ~63 ~43 24 30. H mm.~ 4mm. m. H 93 9:53 «N H 6.43 ed? 34 30. H om.m mmm. m. H 93 $33 an H mama m.oom mo; ucm>m\1 COHHOMHOHCH mo oEsHo> 80:33: 533305 EB 3663 35 SE a 3968 coHuomuoucH 9 .Im OIH buocoaxomue on» mcHuoucw mchmHEOIAUmVBo u 90 Apmv o Educoeoz 92\Bo bqogau mxomuu mo HonEdz .AOCHMMHE V coHuOOm mmOHO OHummHo commas OCH mmoa Acmveo coHuoom mmouo Hmuou pm ecu OH Henge mH mo wuHucmsv one .BZ\eo COOCHHOxHH COHHOMHOHCH any mCHumHSOHmo CH poms mooam> HmucoeHummxmnl.o.m OHnme Table 2.7.- I Topology 3 4 5 l6 Table 2.7.-~Event type hypotheses. Topology Reaction Mark number 3 Pd + PsppPfl- 3 Pd + PSPPPN"MM 5 pd + pspn+n-n- 8 pd + pspn+n-n_n0 9 pd + pspw+n’n’MM 10 pd + pspK+K-n" 30 5d + pSpK+K_n-MM 31 4 pd + pppn- 3 pd + pppn'MM 5 pd + pn+n'n' 8 pd + pn+n-n'n° 9 pd + pn+n7n7MM 10 pd + Nn+n+n'n‘ 16 pd + n+n+n‘n'MM l7 pd + pK+K'n' 30 pa + pK+K-N_MM 31 5 pd + pspn+n+n-n-n- 8 pd + pspn+n+n-n'n'no 9 pd + pspn+n+n'n'n'MM 10 pd + pspK+K'n+n-n' 30 pd + p K+K'n+n'n‘MM 31 SP Table 2.7.--Continued. l7 Topology Reaction Mark number 6 pd pn+n+n'n-n_ 8 pd pn+n+n'n'n-no 9 pd pn+n+n'n‘n'MM 10 pd nn+n+n+n-n-n- l6 Pd N+n+n+n_n—n-MM 17 5d pK+K’n+n’n“ 3o pd pK+K-H+H’N'MM 31 statisti< therefore well as 2 If these final sa: 1 re~E‘-ired “ u do 4. I:- 18 statistical tests. In order to obtain a pure sample one therefore imposes acceptance criteria based on physics, as well as statistical tests on all events in a desired sample. If these criteria are too stringent one must correct the final sample for the loss of good events. In this experiment an acceptable hypothesis was required to achieve a confidence level of greater than 10'4. In addition, the missing mass square must be within two standard deviations of the expected value. In the case of multiple fits, additional kinematic constraints were invoked to determine the correct hypothesis. Since the goal of this experiment is to study pn interactions the 4 and 6 prong events should have been obtained due to the presence of a spectator proton. Because of the uncertainty in calling a proton track by the twice minimum ionization criterion in the measuring procedure, as discussed in Chapter 2.1, a significant correction has to be performed to eliminate events from the pd + NS + 4 or 6 prong final states. P In Table 2.8 the raw topological count within the fiducial volume at each momentum is given. The 4 and 6 prong events are completely uncorrected for any nSp contaminations. 19 Table 2.8.--The number of events in this experiment. The even prongs include contaminations from pp reaction products. The fiducial volume cut as discussed in Chapter 2.3 has been invoked. Momentum (GeV/c) #3 Prong #4 Prong #5 Prong #6 Prong 1.09 3861 5289 1780 1495 1.19 5836 7381 2638 2208 1.31 10699 13533 4990 4262 1.43 7303 9364 3282 2784 2.6.1 Separation of pp Contaminations One consequence of the deuteron double scattering process8 is the production of very high momentum spectator protons. In this work, an attempt was made to measure the complete sample of these spectators. If a sample of pn events with high momentum spectator protons are examined on a scan table, one would see that in many cases these events appear very similar to those from pp annihilations. This similarity resulted in a significant contamination from pp interactions. The analysis discussed in this section deals with the events in this ambiguous region. In the 4 prong sample the most common type of track misinterpretation was due to scanners misjudging one of the positive pion tracks from the reactions pd + nsp + 2n+2n' + k(n°) as a spectator proton track, thus meas between t many of t events. track ior 20 thus measuring the event. Due to kinematic ambiguities between the above reactions and pd + pSp + n+2n‘ + k(n0) many of the pp events were incorrectly classified as pn events. This meant that a quantitative analysis based on track ionizations had to be made to determine the amount of contaminations. If one considers the projected momentum distribu- tion of the slowest positive track it is found that there is a significant difference between pd + pSp + n+2nf + no (called Mark 9) and pd + nSp + 2n+2n’ (called Mark 16), independent of kinematic information. In Figures 2.2 and 2.3 the projected momentum distributions are given for all events that are called Mark 16 by the acceptance criteria. Here the projected momentum is used because this is related to the ionization of the track as seen by the scanner, and this was used to determine if a measurement was to be made. A statistical analysis of these distributions Shows that although all were called Mark 16 by the acceptance criteria, the real Mark 16 events have on the average a lower momentum associated with the slowest positive track than do the pn events. This is expected because the twice minimum ionizing criterion corresponds to a cutoff at about 200 MeV/c for a pion while the proton can have up to ~1. GeV/c. To determine the amount and characteristics of true Mark 16 candidates that were called Mark 9 by the Fig. Fig. 21 2.2—A.--1068 4-Prong events. These events were cor- rectly classified by the acceptance criteria as being pd + nSp + 2n+2n‘. 2.2-B.--1l43 4-Prong events. These events were classified as pd + nS + 2n+2n’ by the acceptance criteria. After the ionization scan these events were found to be candi- dates for the reaction pd + pSp + n+2n'n0. MffV/C 1f)- FVFNTS/ 10- MEV/F (I'VE—N I‘S/ 1 GCL BEL 1 0- O I 22 90. _1 60. AL (R) EVENTS/ 10.MEV/C 30. a COI‘ iteria . I jW a F 1 200. 400. 600. 800. 1000. MOMENTUM MEV/C CD 50. _J :ion (81 11‘ .TTOO EVENTS/ 10.MEV/C 30. 0. O r 'T 200. 400. 600. 800. 1000. MOMENTUM MEV/C 23 Fig. 2.3-A.--413 6-Prong events. These events were cor- rectly classified by the acceptance criteria as being pd + nSp + 3n+3n’. Fig. 2.3-B.--530 6-Prong events. These events were clas- sified as pd + n + 3n+3n' by the acceptance . . 3 . . . criteria. After Ehe ionization scan they were found to be candidates for the reaction pd + pSp + 2N+3n'n°. V/L' H U . 10. Mt- L‘AVFNT 53/ c’U. I F’OLLI__ ’10. 1 IO. MFV/F c’f} . l L” VF‘N 78/ h.— F4). e COI' riterla 3 ClaS' :eptance :heY, 10.MEV/C 40 60. EVENTS/ 29 OO. 10.MEV/C 20. 40. EVENTS/ FD. 100. I 100. I 200. r 200. 124 nm 1 300. MOMENTUM MEV/C 1 300. MOMENTUM MEV/C (R) I I 400. 500. (B) m I* u 400. 500. acceptaz sample). scan tal tance c: of the i are not is seen Cf how . F “I? 25 acceptance criteria (the so-called "called 9 is 16" sample), a sample of 4 prong events was examined on the scan table irrespective of classification by the accep- tance criterion. In Figure 2.4 the projected momentum distributions of the slowest positive track are given. Although there are not as many events considered here, statistically it is seen that the pp events have the same shape regardless of how the event was classified. This is also true for the pn events. A complete statistical analysis is given in Table 2.9. The projected momentum distribution for the slowest n+ from a clean sample of Mark 16 events is com- pared to the Monte Carlo prediction corrected for pion decays in the chamber. It is found that these agree in shape for momentum below 80 MeV/c. A comparison of this is shown in Figure 2.5-B and 2.6-B. One expects this agreement to occur since all pions with low momentum satisfy the twice minimum ionizing criterion for the spectators. The existence of high momentum pions is due to the fact that tracks with large dip angles have higher apparent ionizations. The ratio of the experimental momentum distribu- tion to that of the Monte Carlo prediction gives the probability that a pion of a given momentum will be measured. These ratios are shown in Figures 2.5—A and Fig. 26 2.4.--Projected momentum distributions for the slowest positive track for a sample of 4 Prong events. 2.4-A Called 16 is 16 2.4-B Called 9 is 16 2.4-C Called 9 is 9 2.4-D Called 16 is 9 ( ~. \_) EVENTS FVFN ( E—VFNTS 15- I 27 30. if? Eflj_ (R) > L1.) , I J r-1 :1 (3 I I T’ I 0. 200. L100. 600. 800. MOMENTUM MEV/C In: (B) mr—1 [... Z LI.J > LIJ c5 I I? I I 0. 200. L400. 600. 800. MOMENTUM MEV/C Us- ‘ (C) (DH ...— Z LlJ > LLJ c5 1* I I r1 I 0. 200. L100. 600. 800. MOMENTUM MEV/C m'— (D) mF-i ..— Z LLJ > as W (5 I I I I 0. 200. L100. 600. 800. MOMENTUM MEV/C Table Raine. MeV/c 264. 1 5. 160. 1 4. R.M.S. deviation 152. 1 13. 119. 1 10. Number 6 prong 530 413 MeV/c 152. 1 4. 104. 1 3. R.M.S. deviation 81. 1 ll. 51. 1 9. Table 2.9-B.—-Characteristics of the projected momentum distribution for the slowest positive track (for events shown in Figure 2.4). Called Called Called Called 9 is 9 16 is 9 16 is 16 9 is 16 Number 4 prong 112 69 85 49 MeV/c 272.114. 289.119 148.1 12. 145.110. R.M.S. deviation 152.144. 159.156 107.1 34. 76.134. Number 6 prong 66 40 27 6 MeV/c 155.111. 136.111. 94.915.4 109.116. R.M.S. deviation 92.138. 67.134. 28. 127. 40.165. 29 Fig. 2.5-A.--Probability that a positive pion with a spec- tator neutron from a four prong event will be mistakenly measured as a spectator proton. Fig. 2.5-B.--Monte Carlo prediction for the slowest pro- jected n+ momentum distribution for Mark 16. The shaded area represents that from the four prong data sample. The Monte Carlo dis- tribution is normalized to the data below 80 MeV/c. NI'UM BIHS 1 (I MUMI IU-MFV/C ?/ F V/"N T ‘ Spec- '0ton. IO‘ diS‘ V.80 30 MOMENT 00 If: i 400 6 0. MOMENTUM MEV/C 160. J 120. EVENTS/ 10.MEV/C 400. 600. MOMENTUM MEV/C (R) IIlI—l vvvvvvvvvvvvvvvvvvvvvvvv [B] 800. 1000. l 31 If) 11 4 + $0 + H “ + DH 1' 3 i- I Z Um I Z .1 * 00 X v Fig. 2.6-A.--Probability that a positive pion with a spec- O tator neutron from a Six prong event will be Crfi- mistakenly measured as a spectator proton. 0 Fig. 2.6-B.--Monte Carlo prediction for the slowest pro- . jected n+ momentum distribution for Mark 16. D. The shaded area represents that from the six O prong data sample. The Monte Carlo distribu- tion is normalized to the data below 80 MeV/c. m‘. 03 \ > [LI 2 mo! \m m is 2 ill > m I U. Q‘\ U. 32 Ln Hi * + * (R) $0 ++ 0—4 ‘— 4 ++ m--* + Z 3 I— + + + Z + tum + Z .—. + + DD ++ + 2 ++ +47" .9 + + + ++++ 0 «703+ + + 'H O. l W174 ........ l .............................. :::::::l U. 120 290. 380. H80 800 MUMENTUM MEV/C 81 (B) " h I I U. 120. 280. 350. U80. 500. MUMENTUM MEV/C 2.6-A for this same the {3d + I in the sar ionizatior 0r . F "tie Fewer. found. T}: 5‘49 t0 the Table 2.11 aPPropriat‘ mO‘w‘ientmfl II Effect. Tl "estion 2.6 In Cal CrOss 5 give“ in Ta It ~ent 10Sses kg m topolm 33 2.6-A for 4 and 6 prongs respectively. One can also use this same probability curve to find how many events of the pd + nSp + 2n+2n' + kno where (k = 1, 2, 3, 4) are in the sample because the ratio depends only on the imposed ionization cutoff. Once the scanning bias per momentum bin is known, E the percentage contamination from any reaction can be found. These are listed in Table 2.10. The contaminations SP Table 2.11 are the product of these percentages by the 3,9,10 due to the pd + n + . . . type reactions given in § y appropriate pp cross sections extrapolated to this momentum range and corrected for the deuteron screening effect. The deuteron screening effect is explained in Section 2.6.2. 2.6.2 Topological Cross Section Calculations In the simplest situation, to obtain the topologi- cal croSs sections, one would multiply the raw event count given in Table 2.8 by the appropriate interaction likelihood given in Table 2.6, and correct for scanning and measure- ment losses, that is on = :3 Nn 1 'k * NT 8s,n€M,n In this work, the pp type reactions are excluded (2.5) frOMItopological cross sections determinations. 34 mes. mam. Hem. mas. 4mm. com. mew. cad. mv.H ems. was. Hem. ems. mos. mmm. mum. ham. Hm.a was. ems. Hem. mes. one. can. omm. mam. mH.H mus. emu. ham. mme. 0H4. omm. mam. mNN. mo.H AO\>OOV oempo oeaem em opvev ocmpv cemev oeapv ev Edusoeoz .OHHOHHHO oCHccmom OCH mo mHmmn can so ucm>m mom» mm HmHsOHuumm m ousmmoe HHHS umqumoE on» was» huHaHnmnoumII.oa.m canoe 35 Ill I run“! b... EH85 3H3. SHEEN mmHfia SHEEN SACHS? 21H SHEA SHE. ONHmTN SHEA o~.Hom.~ SoHsmm. Hm; SHSS 8H3. HNHmmN RHSH H~.Hoo.m SOHOS. 34 mmHomg: 3H2. SHEEN mmHmDm 8H3...” SACHS». 84 as . as as as as as €568 Hmuoe oevev opmev oemee cease :v Educoeoz .COHHQQO comb mm: mchmouom couousoo on» O» can COHHOOHHOU .mEHMQHHHHE CH OHQEMm mcoumnv OCH CH chHuomou mum» on may scum COHumcHEmusooII.1) + Ofin)/Opd (2.6) where OPP' O§n and 05d are the cross sections from reference 11. Table 2.13 summarizes the pn tOpological cross sections. Figure 2.8 compares these results with the predictions of the impulse approximations as expressed in equation 2.7. 37 Fig. 2.7.—-The pd + 3, 4, S and 6 prong topological cross section arising from the pn interaction. The symbols are E] (3 prong, this work), x (4 prong, this work), X (5 prong, this work), - (6 prong, this work). The symbols 8 andx are for the 3 and 5 prong data from Reference 3. (M SCF T I ON 5 CPOS 10.0 L 10. l CROSS SECTI 7.5 L 5.0 2.5 38 _0.0 0 1.5 2.0 2.5 3 0 LRBURRTORT MOMENTUM [DEV/Cl 39 Fig. 2.8.--The pn 3 and 5 prong topological cross section compared with the prediction of the impulse approximation. The symbols are X (3 prong, this work), X (5 prong, this work). The symbols - and E|(with wider error bars) are for the 3 and 5 prong results from the impulse approximation. ”.0... MB 3 CROSS SECTION ( 20. 10. I 40 1111 1111 1. 0 0 T’ I I I 1.25 1.50 1.75 2.00 LRBORRTORT MOMENTUM (GEV/C) 41 Table 2.12.--The pd + p8p + . . . tOpological cross sections. Momentum 3 Prong 4 Prong 5 Prong 6 Prong (GeV/c) (mb) (mb) (mb) (mb) 1.09 17.801.47 16.6811.43 8.451.32 6.411.55 1.19 17.651.43 l4.7911.32 8.341.24 6.431.47 1.31 16.901.39 l4.ll11.24 8.231.21 6.231.44 1.43 16.811.40 14.551l.23 7.881.22 6.051.44 Table 2.13.--pn topological cross section and the screen- ing factor. Momentum 3 Prong 5 Prong Screening (GeV/c) mb mb Factor 1.09 37.6 1 2.1 16.26 1 .88 1.09 1.19 35.701 1.8 16.26 1 .78 1.10 1.31 34.121 1.8 15.89 1 .71 1.10 1.43 34.8 1 1.76 15.48 1 .73 1.11 Nodd prongs N+1 Prongs N prongs N+1 Prongs O_ = O _ + O _ * s - O Pn (Pd) (Pd) (PP) (2.7) Here, good agreement is seen. 2.6.3 Acceptance Crigeria for Events for the Reaction pp + n+n'n‘ A sample of events satisfying the acceptance criteria, discussed in Section 2.6, for the reaction 42 +n’n‘ was separated from all other events. A pd + Psp + n partial ionization scan of this sample Shows an initial contamination as given in Table 2.14. Table 2.14.--The initial contamination of the data. Reaction % Contamination Fa 5d + psp + K+K-fl- 6% pd + nSp + fl+fl+fl-fl- 6% (Rd + psp + I+N’n‘)* 3% 9E TOTAL 15% *These are 4-prong events with spectator proton and n+ incorrectly switched in the fit. Initial confidence level, missing mass squared, and spectator momentum distributions are shown in Figure 2.9. One notes that the confidence level distribution is flat down to 2 1/2%. The missing mass squared distribution is symmetric around 0. The spectator momentum distribu- tion is compared with the Hamada Johnston wave functions prediction.12 Using only the events from the four prong sample (where spectator momentum was actually measured) the fit was performed over the momentum region between 100. and 200. MeV/c. Here good qualitative agreement is found. The area under the fitted curve was found to be equal to the number of data points. 43 Fig. 2.9.--Characteristics of events satisfying initial acceptance criteria for the pd + pSp + n n'n reactions (A) Confidence level distribution (B) Missing mass squared distribution (C) Spectator proton momentum distribution. The curve is from a prediction using the Hamada Johnston wave function. 44 C.) On! °‘ (8) N O \ | C) 3504 221‘ LIJ > LIJ W (3' l l l 1 l 0.0 0.2 0.4 0.6 0.8 1.0 CONFIDENCE LEVEL 500. (B) 250. I I I I I .16 -.08 0.00 0.08 0.16 MISSING M855 SQURRED (GEVxx2l 4 EVENTS/ .00416EVxx21 .0. 200. (C) 100. EVENTS/ (Q.MEV/C1 ‘ II II' Ev r' I 0. 80. 160. 240. 320. 400. SPECTRTOR MOMENTUM (MEV/C) 45 To eliminate the small contamination that exists in the sample, many combinations of cuts on the missing mass squared and confidence level distribution were tried, and the corresponding total contamination was studied, based on the ionization scan. The cuts which maximize the total number of events and minimize the contamination was a missing mass squared cut at 2 standard deviations from 0, and a confidence level cut at 2.5%. No spectator cut was needed. After invoking these cuts the contamination was found to be less than 4%; mostly due to the reaction pd + pSp + K+K'n‘. The confidence level and missing mass squared cuts gave an 18% overall reduction of the data. The final confidence level, missing mass squared, and spectator momentum distributions are shown in Figure 2.10. All distributions look representative of a clean data sample. The statistics for the final analysis sample are listed in Table 2.15. 2.6.4 ,6n + n+n-n' Reaction Cross Section Calculation The reaction cross section pn + n+n'n' was obtained by multiplying the number of events in this channel before the confidence level cut by the interaction likelihood given in Table 2.6, and correcting for con- taminations given in Table 2.11. Corrections for scanning and measuring losses as well as the deuteron screening effect were also applied. The cross sections are given 46 Fig. 2.10.--Characteristics of events satisfying final acceptance criteria for the pd + pSp + n+n‘n" reaction. (A) Confidence level distribution (B) Missing mass squared distribution (C) Spectator proton momentum distribution. 47 d O— 0“ (R) N 0. \ci LIJ > LlJ 0.0 0.4 0.6 0.8 1.0 CONFIDENCE LEVEL ~c5 NO- :“l (B) > 6 \ U? I'— Z U>J ‘ A'— I “I I L”C1.18 —.08 0.00 0.08 0.16 MISSING MRSS SOURRED (GEqu2) 8 a 1 SW (C) > UJ 2 3:6 HS... \ (D F- Z LIJ > UJCS r F l 1 l 0. 80. 160. 240. 320. 400. SPECTRTUR MOMENIUM (MEV/C) (of 48 Table 2.15-A.--The number of events accepted as the pd + pSp + n+n‘n’ reaction, and the corres- ponding cross section with the fiducial volume cut. Momentum 3 Prong 4 Prong Cross section (GeV/c) mb 1.09 210 164 2.07 1 .18 1.19 290 " 207 1.82 1 .14 1.31 443 364 ‘ 1.63 1 .10 1.43 245 227 1.49 1 .11 Table 2.15-B.—-The number of events accepted as the pd + pSp + n+n’n‘ reaction, and the corres- ponding cross section without the fiducial volume cut. The total number of events is 2871. Momentum 3 Prong 4 Prong Total (GeV/c) 1.09 285 217 502 1.19 383 261 644 1.31 588 483 1071 1.43 349 305 654 in Table 2.15 and shown in Figure 2.11, and are compared with the higher momentum data from Eastman, et a1.3 One Observes excellent agreement between these data. 49 Fig. 2.11.--The pn + n+n'n’ reaction cross section com- pared to the results published by Eastman, et al. The symbols are c](this work), and - (Eastman, et a1.) 1 3. 50 8 E E“? I :1 ii Q I E I; U III I ‘ IF I r c’1.0 15 ; 2.0 2.5 LRBORBTORT MOMENTUM (GEV/C) 3.0 __ EW'W 5! CHAPTER III RESONANCE PRODUCTION 3.1 The Don", fon’ Resonance Production The DO and f0 are resonance which dominantly decay into + into the two particle n n” state. If these resonances are + produced in the pd + pSp n n’n’ interaction, the invariant mass distribution of the n+ n" system will exhibit the characteristic Breit-Wigner bumps above an ever present background. Each bump should be located near the mass of the resonant particle, and the width of the bump inversely proportional to the lifetime of the resonance. The n+n' invariant mass distribution for events satisfying the final selection criteria corresponding to the pd + pSp + n+n'n' reaction is shown in Figure 3.1-A. The two bumps at 760 and 1280 McV/c indicate that a considerable amount of po and fo production is present in the data. 3.2 Event Selection As explained in Chapter 2.6.3, confidence level and missing mass squared cuts reduced the contamination in + the pd + pSp + n n'n' sample to less than 4%. Since the remainder of this work deals with the n+n'n’ final state, 51 52 Fig. 3.l.--The n+n' invariant mass distribution for the reaction pd + pSp + n+n‘n‘ (A) All events (B) Events with pSp < 160 MeV/c 53 180. (81 120. L EVENTS/ 20.MEV 60. OD. I r I I I 500. 1000. 1500. 2000. 2500. INV8RI8NT M855 (MEV) 120. l (B) 1 80. L ‘1. I l" l" f \I ‘1‘ 15“. L .‘ L 4 1 EVENTS/ 20.MEV Le- 0. r r* I I 0. 500. 1000. 1500. 2000. 2500. INV8RI8NT M855 (MEV) 54 one must now investigate the validity in the assumption that studying the reaction pd + pSp + n+n'n‘ will lead to information on the reaction pn + n+n-n-. A factor to be considered was discussed in section 2.6.2 concerning screening corrections. Another factor which is particularly important when discussing the dynamics of the interaction is the possible presence of double scattering effect. The incident antiproton may collide with the two constituent nucleons in succession or the product of the first colli- sion may scatter from the spectator nucleon. The effect is being studied in detail by Zemany, et a1.8 Since this complication is not a property of the pn interaction, the effect of this process must be removed. It is generally believed that such processes will alter the character- istics of collision products and/or that of the spectator nucleon. Collisions against the spectator nucleon increases its momentum due to the momentum transfer from the other participant. This distortion may be eliminated by impos- ing the additional acceptance criterion, that the proton spectator momentum be less than 160 MeV/c. Referring to "4- n‘ invariant mass distribution in Figure 3.1-B, one indeed sees that the po and fo peaks appear sharper than in Figure 3.1-A. The final confidence level, missing mass squared, and spectator momentum distribution given in Figure 3.2 reflect the high quality of this data sample. An event count for this sample is given in Table 3.1. 55 Fig. 3.2.--Characteristic of events satisfying final acceptance criteria for the reaction pn + n+n‘n- (A) Confidence level distribution (B) Missing mass squared distribution (C) Spectator momentum distribution 900 . 56 (R) N C) \ci (0 294 UJ > LlJ r"“~*LH-~._,~.i__.n_.__,..tlfilmfiflflmn,1 0' F I I I I 0.0 0 2 0.u 0.6 0.8 1.0 CONFIDENCE LEVEL ~c$ NO- §U° (B) > § 3.. 8% \ (n l— Ei Ed ”I I I I - 16 -.08 0.00 0.08 0.16 MISSING MRSS SQURRED IGEVxx2I at sf“ (C) > uJ Z I 384 \H (n I... Z LLJ > “JO f I I I 0. 80. 160. 2MB. 320. uoo. SPECTRTOR MOMENTUM (MEV/C] 57 Table 3.l.--Sample count. Total = 1919 events. Mom tum 6e33c 3‘Pr0n9 4-Prong Total 1.09 278 58 336 1.19 373 71 444 a: 1.31 573 147 720 -3: 1.43 340 79 419 3.3 Resonance Cross Section Determination The maximum likelihood method13 is used here. The data in Figure 3.1—B are assumed to be an incoherent sum of phase space background and resonant productions. The distribution is assumed to be proportional to the squared matrix element 14 ai Ri/Ni (3.1) MM IMIZ = (1 "' (11) + i=1 i "MN [.1 . . 2 2 With R- = I F3/2 I = I‘j/‘l (3.2) (Mj'E) ’ i Pj/2 (Mj-E)2+ rg/4 where Pj is the full width at half maximum of the jth resonance, with mass Mj. The normalization factor Ni is obtained by integrating Ri over the available phase space. In the fit only p0, f° production was assumed to exist, and the fit parameters were therefore a 0‘f, m p, pl Inf and Pp, Ff. The parameters obtained from the fit to the entire data sample from 1.09 to 1.43 GeV/c are listed 58 in Table 3.2 and shown as a smooth curve in Figure 3.1-B. Using the masses and widths thus obtained, the events at each individual momentum were fitted for up and of. These percentages were scaled to the total pn + n+n-n- reaction cross section to give the partial pow”, fon', and statistical production cross section contributions at each momentum. The results are given in Table 3.3. As an example of the quality obtained in the individual + momentum fitting process, Figure 3.3 shows the n n'n' fit for the events with a laboratory momentum of 1.31 Table 3.2.--Resonance Parameters for the reaction fin + n+n'n‘ from 1.09 to 1.43 GeV/c. Fit Parameter1 Values (MeV/c) M; 761. i 7. M; 1278. i 7. mp 165. i 11. (0f 176. i 13. Xz/(degree of freedom) 1.3 *The actual masses here should be 1.8% higher due to a small magnetic field problem. 1In this fit the masses and widths were constrained to be within 2 standard deviations of the Particle Data Group values. 59 mm.H ma H m.ms «H H c.mm m H m.m~ m¢.H am.H a H m.m~ o H v.Hm m.“ m.m~ Hm.H FH.H m H m.¢~ o H «.mq m H H.mm mH.H mo.H m H H.- m H m.~¢ m H o.om mo.H Aeocmwuw mo mmumwmv U\>mo x amoeumflumum mw aw Esucmfioz m w auoumuonmq .o\>mu mv.H can Hm.H .mH.H .mo.a mucmsoe new I=IF+F + cm MOM mmsHm> us can anI.¢Im.m magma 60 +1 3. 2o. 2. H «3. 2. H mam. S. H 34 $4 3. H 2m. 3. H 2m. mo. H was. 3. H 84 :4 3. H 03. S. H mi. mo. H mom. 3. H $4 $4 2. H 3;. .2. H 84 S. H s3. 3. H S.~ mo; Ansv Ages Ansv Ans. 0\>mo AHmoHumHumumv I=om + cm Inca + cm I:I:+= + cm Educmeoz I=I=+= + cm kuoa muoumuonmq .o\>mo mv.H on mc.H mo macaw Esucmeoe map How mcoHuomm mmouo HwOHHmHumum I:I:+: com “I: .om “I: .oa oneul.mum.m wanna 61 Fig. 3.3.—-The n+n‘ invariant mass distribution at 1.31 GeV/c. The solid lines give the shape and relative amount of each term in the fit. (A) (B) (C) (D) (E) (F) Final fit distribution 0 contribution O f0 contribution Statistical production contribution Breit-Wigner curve for 00 Breit-Wigner curve for f0 62 (0) _ , _ _ . .2: .8 do >mz.o: \mkzm>m nm 0 u. 2 m 0 IR. .09 .8 _o0 >mz. or \mkzm>u INVR >mt. o: \mhzm>w nm no In. «2 mm m. I. v. LL nxm rms Q. n” M” .m rmwm 93K nu v. mm _ d . .8 .8 .oo >mt. or \mkzm>m nm nu 1% mm 1. r. v. H ... 0M 0‘ . Imwa no on M” .M mem 93K nu V. mm 1 _ a .09 .8 .oo >wz. or \mhzm>m nm nu u. 2 mm .I v or...— nu" 0t 65 1g: an M" 0T mm 93K nu v mm — l .8 .8 Q... >mz. o: \mbzm>m nm 0 u. 2 mm 1. I v .E 0M 0‘ 6 nuuulw‘I‘ 1:; In. mm 0T [mm nEK n” v m _ . .2: .8 .oo 63 GeV/c. Figure 3.4 compares the pon', and Fon' cross sections with the higher momenta results by Ma et a1. One notes good agreement between the data. 15 64 Fig. 3.4.--The resonant cross section and comparison with higher momentum data (A) The symbols are [3(f0n' this work) and - (fow' from reference 15) (B) The symbols are [3(pow' this work) and - (pon' from reference 15) 65 1.5 [R] 110 l—‘B—l HHkH F4*H CROSS SECTION [MB] 0'5 I——9—-I I—I—«I F+fi H—I IIIIII I O_ III; 1 1 1.5 2.0 2.5 .0 LRBURRTURT MOMENTUM [GEV/C] O EBA" E [B] Z O :m H DO”) a H g I 80 III}; I (JO. I l I [I J .... I 1.5 2.0 2.5 LRBORRTURT MOMENTUM (GEV/C) CHAPTER IV DIFFERENTIAL CROSS SECTIONS AND COMPARISON OF LINE REVERSED REACTIONS 4.1 Motivation and Formalism The crossing symmetry principle asserts that the scattering amplitude describing a reaction remains the same in form when particles are replaced by antiparticles provided that signs of the four momenta are reversed.16 This basic property of strong interaction S matrix theory has been subjected to experimental tests in two body reactions such as fip + n'n+ and fip + K'K+.l7'18 This section deals with a further test of this principle using quasi two body reactions fin + pon' (4.1) and fin + fon' (4.2) and the line reversed reactions n'p + np° (4.3) n‘p + nfo (4.4) In Figure 4.1, if Ai(u,s) represents the amplitude for the 5 channel process (A) fin + Ron‘, and Bi(u,t) represents the amplitude for the t channel process (B) n’p + nRO for exchange of the ith Regge trajectory, where R stands 66 Lash |: 5 . 67 Fig. 4.l.--The elementary baryon exchange diagrams for the Processes listed below (A) Forward 3 channel process for fin + Row- (B) Backward t channel process for n'p + Ron (C) Backward 3 channel process for fin + Ron” (D) Backward t channel process for n+n + RpP 68 nmmm In. ADV £11. 0 AIILa >A> + Amv ATMM3(\4 Am a A? Hmmnw>mu wane Allalr>x2> Huccmao a .A mchmouo Housman s 69 for either the no or f0 meson, then it is knownl7'18'19 that there are two classes of trajectories for which gIAiIZ =1: lBilz Class 1: All the contributing trajectories have the same signature, or if they do not have the same signature the exchange of one of the trajectories is dominant. Class 2: Contributing trajectories are exchange degenerate. Similar arguments may also be made for Figures (4.1-C) and (4.1-D). In the case of p0 production one can exchange A6, NY and Na trajectories. For f0 production only the I=l/2 NY and Na are allowed. Although the signatures of A5 and NY are different, it is found experimentally20 in fl+p backward elastic scattering that Na is the dominant trajectory. The assumption made here is that the exchange of Na is the dominant trajectory. The assumption made here is that the exchange of Na also dominates the reactions considered in this analysis. In the f0 production the experimentally confirmed21 exchange degeneracy of the Na and NY is assumed sufficient to allow the cross channel comparison to be made. The same argument should hold in the po production case. Referring to Figure 4.1, the 5 channel differential cross section is given by 19 70 + my: " A u] A'l Jill? 2 I .2 I 35' '12:;1171733117-q2 1 A1 lcos (lieu)| I51" (geu)| S (4.5) where the kinematic singularities of the helicity amplitude have to be explicitly removed so that Ai may be analytically continued to the cross channel region of the Mandelstam diagram. Sa and Sb' Gu and 9 are the spins of the initial r1 h state particles, the incident c.m. momentum, and the “‘4 scattering angle respectively. A and u are the relative final and initial state helicities. (g i If ilAil2 = EIB.|2 as is expected here, then M 2 (10(5) , 2 do (4.6) + -—— — + —— (25a 1) (25b + 1) qS du (25c 1) (25d + 1) qt du where u clearly retains its meaning throughout this dis- cussion. Putting in the initial state spins, equation (4.6) is equivalent to (in + ROII') = )5 (332 do ("1" +PR°) (4.7.1) ‘15 31.7 s|s~ and (4.7.2) do (pn + Ron-) = 15 (31,2 92 (Tr-p + nRO) Hf QB du where q“ is the incident cm momentum for the HN reactions and q5 is the incident cm momentum for the fin reaction. 71 4.2 Experimental Analysis Data used in this analysis are divided into three groups. Group 1 include events from 1.09, 1.19, 1.31 and 1.43 GeV/c. Event selection for this sample has been discussed in Chapter 3. Group 2 data include momentum settings of 1.60, 1.75, 1.85, and 2.00 GeV/c, and Group 3 includes 2.15, 2.30, 2.60, 2.75, and 2.90 GeV/c. Both Groups 2 and 3 are from reference 15. Figure 4.2 shows the differential distributions for data which had n+1r- invariant masses within the resonance region described in Table 4.1. The variables, t' = t-tmin and u' = u—umin were used in order to remove the kinematic effects associated with the invariant mass spread within each resonance region. The variable R' stands for t' or u' , which ever is appropriate. The differential cross Seetion do/dt' is plotted if cose*3_0 and do/du' is plotted if (2036* < 0, where 6* is the cm scattering angle of the + — .. 7' 7' invariant mass system with respect to the incident p as Shown in Figure 4.3. In Figure 3.3 one can see that a randomly selected “+n‘ combination from a resonance region would not necessarily result from resonance production. To obtain differential cross sections for the production of the 90"“ and fofl- states, one must remove contributions due to nonl‘esonance background events. In order to determine the shape of the nonresonance differential cross section, Fig. 72 4.2.-—Differential distributions. The background is shown by the shaded area (A), (B), and (C) are for the reaction fin + pon‘, (D), (E) and (F) are for the reaction fin + fon’. 73 1. 1// .s R'tGEV/CIKIQ 1.112% .moxmxaéme m. \mazmfimtlm mm? ..1ma% N5. C\>mm: N. \mpzm>u .358 1. (B) 0. .Ts// u'IGEV/CMIQ o' (m .5 t‘tGEV/C) "2 941% .8 .0 m5. 826%“ \mazmsm 12% . .OmH (F) A 0.8 r .11 R'(0EV/C)xx2 0.0 T .09 d .oN Nan 8\>mm: H . N .\m._.zm>wA 9mm ...me HP...“ 0.8 (E) is 1 /. 1 .02 Na: 8\>.mwN OH NH N.\m . .om sz>m .Su\zv CD0.0 l .u u'(GEV/C)ux2 (0) _.III_ // I t'fGEV/01xx2 4 .QNH .Om Nun 8\>mm. N. \mhzm>m Laban 00.0 74 Table 4.1.--Invariant mass cuts. Description Mass (MeV) :30 resonance region 640 - 880 f‘) resonance region 1160 — 1400 QC) lower control band region 540 - 640 p»0 upper control band region 880 - 980 f’0 lower control band region 940 - 1140 150 upper control band region 1420 - 1520 R0 5 6 n 4$>000\mz r .u RWGEV/CJxx2 8.00 1 .5366 1.20368 m0 (B) 0.8 T J! ufiGEVVCIKn2 0.0 _ _ mid 0m.0 2.0 Nx18\>m59\mz .2366 00.0 NT .mmw. \. 1 (A) 0.8 1 .ll tWGEV/C)~I2 — K1 a 03.0 m~.0 H 00.0 anH0\>me\mz . union 00. CD0.0 N) (F) I 0 _ J H 0.R 0.0 0.0 m0 0.0 N11350:? 13363536314.) - 60.. RT on U . C ( ......w E E 0m .1 4 _ .. To me To 0.00 u N118\>mou\mz .5366 a...“ £1 Du. m C ol. l3/ .v E 4 . 4 0.“. 0.0 0.0 0.0 0.0 Na: A0\>w0. \mz .033 0.6 .3 TGEV/CIuI2 79 The data are parameterized by 3%. = a e b(R') (4.8) using the least squares method. The results are shown in Figure 4.5 as a smooth curve. Since the data were averaged over different laboratory momenta, the kinematic limit for R' at cos 8* = Ovis not a constant. The range of the fit was limited to the maximum R' value at 1.09 GeV/c. These are .8 (GeV/c)2 and .6 (GeV/c)2 for the 0° and fo respectively. Table 4.3 lists the parameters obtained in these fits. 4.3. The Line Reversed Reaction Comparisons In the spirit of Regge models, it is expected that exchange trajectories in the t' and u' channels are identi- cal. The data show that separately for pow- and £00. states, the slope and intercept values are indeed con- sistent for the t' and u‘ channels, thus suggesting that the data are consistent with such a conjecture. Therefore for the purpose of comparison with the line reversed reaction, the values of bR and a will be used. R The available backward 0N scattering datazz'23 limits the line reverse reaction comparison to the reaction in equation 4.7.2 where R0 is the 0° meson. The parameter- ization of these data was similar to the method used here, and thus can be directly compared. 80 mo.6mmm. HH.Hmme. He.6mem. m.em.HI 6.6m.HI e.em.HI on meo.6eem. mae.emme. mne.eeem. 6.6H.Hu m.66.HI m.ee.u e 16\>6ec\ee 16\>6ec\ee 16\>6ec\6e -16\>6ec 16\>6ee 16\>6ee 666666666 N m N s N u N m NI 5 N- u 6 6 6 n n n .mumuwfimumm coHuomm mmouo HMHHGmeMMHo mnBII.m.v manna 81 Table 4.4 gives the comparison for the three sets of data. The factor 1/2 (241;)2 is listed in column 6. Over- all, the data are in 9003 agreement, thus indicating that within the experimental uncertainties the crossing principle is valid in quasi two body annihilation processes. This further implies that the baryon exchange picture cannot be ruled out as a part of the annihilation mechanism, even at these relatively low energies. The good agreement between the direct and cross channel differential cross sections 24,25 indicate that any absorption corrections will have similar effects on both distributions. 82 .eoe + 6-: .86 6666: we. 6 .. swam: 6:6 .coHuowm mmouo :oHuoonoum me .Ipoo + am now cmumHH mH m moam> on» .cOmHummaoo uom+ oq numzxomn Hmuou may on OoNHHmsuoswu was moam> mngc 60. + 03. omd So. H moo. I- -.. 8:6. 0.6 See: + 6.: 08. H 02. 664 .68. H m3. 6. H «.7 6.???» BEN 28 NNone + an: wee. H 66.6.. I So. H 666. 6. H dd. 6. + 9.6 :6 84 +84 .666 + em Ao\>mov U 0 I WHOMWHWWHMM «WW0 \ NAM\>mov\nE N ma NA0\>mwv A>ouv hmmwwwwz coHuommm . a N H u mouwucH . . . unwououcH N 0 mmon omcmu m Nmumco a o wuoumuonmq wCHH muH £HH3 ucmEHuwmxm mHnu Scum oGOflHOMQH UGmH0>OH mumuwfimumm wmon 0cm ammoumucH on» no GOmHummEoo 0£BII.v.v manna CHAPTER V SPIN DETERMINATION OF THE ANTIPROTON NEUTRON SYSTEM 5.1 Formalism In order to better understand the reaction fin + w n-w- an angular momentum analysis was performed to deter- mine the spin state composition of the initial fin system. This determination is particularly important from the view- point of direct channel interaction. In this section the helicity frame is used to determine the initial spin composition by examining angular distributions of the resonance final states. This is justified since the 00 and f0 resonance production dominate the fin + n+1'w- reaction. Consider the reaction A + B + C + D. One can relate the helicity amplitude to the differential cross section by 26 Q3, 1 1 f 2 HH (Ea + '1 (Eb 1' '5 aged I c.d,a,bI (5'1) where the scattering amplitude is written as 83 84 J and T1 (i) A A is the transition amplitude. The function c, d, a, b dJA u(6) is a single variable rotation matrix defined in I reference 27. The letter 3 represents the cm energy squared and q is the initial cm momentum. The letter J is the total spin of the fin system in units of 1'1. The helicities of particles C, D, A, and B are written as Ac Ad )‘a and Ab I I respectively. The relative helicities A and u are defined _ as Aa-Ab and )‘c-Xd‘ J 5 + s - s - 5 J 27 c d T = nanbncnd(-1) a b T (5.3) --Ac’-Ad’-Aa’-Ab AC,Ad,Aa,Ab where n n n n are the intrinsic parities of the a, b, c, d _ Particles and Sc, Sd, Sa, Sb are their spins. If one now considers particle C to be either the 00 or f o meson, and particle D to be the 1(- meson, one knows that + the spin and parities of these particles are 1-, 2 , and 0‘ respectively. One further knows that the antiproton and rleutron have spin 1/2 and opposite intrinsic parity. This means "anb = -1, and equation (5.3) becomes T_)\ _A _x = c' a b “T‘; 1 A for both the 00 and fo mesons. The helicities c, a, b which must be considered for the antiproton and neutron are l'5‘:'~/2. The 00 may be :1, 0 and for the f° one has :2, i1, 0. The factor aficd lfcdab I2 for the 00 meson can be explicitly w): itten as follows, where the unconventional definition DJ = (2.1 + 1) (3J 3&1; q Au (9) 18 utilized. 8 5 E 2 __. abcd I fcdab I [z T J D J )2 + [z T J 0 J 12 + [2 T J 0 J 32 + J 1:159‘15191 J 1989;5091 J 131595 '15] [ZTJ DJJZ +[ZT‘J DJ]2+[£TJ DJJZ-i- J 19457}: 09] J 0:159'JS 10 J 0,35,35 00 (5.4) 2 2 In DJ12+EZTJ DJ] +[zIJ DJJ+ J 0,45,}; -1,0 J 0.45.4: 00 J -1 .3545 1,-1 J 2 2 J J 2 T 0 J J 0 J 1 + [2T D 1 + [z - [ET-'1 959% 0.") J “1”}595 '19'] J '19'150’15 0’ 1 T . J J = J(0) he symmetry relations for dim (6) are dbl! (e) d-u,-)\ _ l-u J . J _ J . J ____ J . ( IL) du,1(9)’ This means D1,l — D-1,-1'D1,-1 D-l,l’ an J ___ J = _ J ___ _ J . . d D0,l D-l,0 D0,-l D1,0 USing these symmetry relations equation (5.4) becomes: 2 J J 2 J J 2 J J 2 * E57125” 01.1] *[57-1.1s.-ls°I.-I] * [3701,1000] J J 2 J J 2 J J 2 (5.5) * [571.15.190.11 * [371.464.00.13 * [310.12.90.13] The last three terms of equation (5.5) have the same angular distribution, even though they correspond to different helicity amplitudes. Because the interest here 13 the initial spin state, and because the unique separation Of the last three terms is impossible, equation (5.5) is rewritten as: 86 J J 2 J J 2 t J 2 EJTI.k.-kol.1] + [ST-1,15.-15°1,-132 +[J5T0H15 1.0 0:120] + [33% o 13 (5.6) ‘where [gRqu,l 2 has been written in place of the last ‘three terms in equation (5.5). expression for the angular differential cross section of the po meson as: O 2 d (9 ) J J 2 + J J 2 ° ' '“*|[571.1:.-s°1.1 3 [574.35.491.43 [570.12.500.03 + :3 cos 9 [JRJD 00]] (5.7) U I?+ “won-at] (6.2) Where B is the Euler B—function and is defined by .. T'(x) HY) B(x,y) — r7x+§7 (6.3) aJICi'E is a constant. The functions d(s) is the Regge t-'-J==‘Eijectory in the 3 channel as previously defined, and P(x) :153 the gamma function. It is evident by examining equation (E5«-3) that the poles of the gamma function at O, -l, -2 ... ‘qu' lie on the real part of the 8 axis if a(s) is real. By ElJL-JLowing d(s) to be imaginary one may not only move these PQ les to an acceptable unphysical region, but also give a . . . I-:’]E>ropriate widths to the direct channel resonances. 93 The modified expression for d(s) can be written as A + 83 + ic J s-4M2 (6.4) where A and B describe a linearly rising trajectory, and C M is the d(s) = depends on the width of the 3 channel resonances. mass of the pion in this work. first applied Veneziano's formula to three- Lovelace In the analysis of the body final state scattering. . - + - - . . . . . reaction pn + n n n at rest the Initial spin parity state is assumed to be entirely 0-. 33 et al., has examined the reaction 5n + Bettini, ‘__ __ for laboratory momenta between 1.0 and 1.6 GeV/c +__ TI'TTTI’ land compared their data with the predictions of the A phenomenological approach was taken Veneziano model . They “hith regard to the spin parity state of the pn system. xfcrund that the Veneziano Model was in good agreement with their data if they assumed the spin-parity of the pn system Was entirely in a 2+ state, but contributions to the IPJ?<:cess by other states cannot be ruled out. In this analysis the data are compared to the Since the spin state composition for the VEEIleziano model. system has been determined in Chapter V, comparison can 5n k3‘33 made with Veneziano amplitudes appropriately summed over ktljlcnwn percentages of participating spin states. This will g.jL-‘re a stronger test of the Veneziano model since the spin 3 1t:WEIte is obtained independently from the model comparison. 94 The problem of constructing an amplitude describing the decay of a state of given spin, parity and isospin into three pions has been investigated by Zemach34 and Goebel, _ei al.35 Goebel has investigated the specific problem of constructing Veneziano amplitudes for definite spin parity In Goebel's derivation the full amplitude is states. ' ru‘ obtained by multiplying a spin factor times a scalar Veneziano amplitude. The spin factor consists of products The scalar Of the final state particles momenta vectors. Veneziano amplitudes differ for different spin parity States because the full amplitude is required to exhibit Regge behavior . In order to better understand how the Veneziano model will be related to the reaction 5n + n+1r-1r- discussed here, . . - + - one is led to conSIder the scattering process 1r 1r -.> 1r 5 l 2 3 4 Where particle s- with spin 8 represents the pn system. This process is shown below, with all four particles taken a S incoming. 53 channel channel t channel 95 The four momentum conservation equation reads: P1+P2+P3+P4 0 The Mandelstam variables are defined as _ _ _ 2 s - 512 — (Pl + P2) P3 _ _ 2 -"-:‘ I t — 523 - -(P2 + P3) 2 and u = 513 — H U) um CDhe dual nature of the Veneziano Model requires one to consider which channels have known resonances since resonant effects in one channel must be identical in the Ciuality sense to particle exchanges in the cross channel. (Dime can see that both the s and t channels are identical airid may have resonances, while the u channel with isospin III,IZ>=12,-2> cannot. If the pn system has the natural £3£>in parity quantum numbers (SP - 1-, 2+, 3-, ...), one may ‘vrzrite the full amplitude for the above mentioned scattering Process35 as 11 = - L M . ($12, 523) Const * [euasyP1aP28P3ysu (P3) (P1) V(s]2,523)+(1++3)] (6.5) where 31! is the polarization tensor of the pn system. euaBY = 0 if any indices are equal. -1 for odd permutations of the indices +1 for even permutations of the indices 96 L M is understood to mean the contraction of Sn ... (P3) (P1) the polarization tensor Su with L factors of P3 and M factors of P1. The relationship L + M = S-l is required to hold in order for the contracted product to have vector qualities, and the amplitude to be a scaler. The notation (1H3) will mean to symmetrize the previous term by inter- changing particle l and 3. V(s12 $23) is the appropriate I Veneziano Scalar amplitude for the particular Spin parity state being considered. The requirement for A(s12 523) to I exhibit Regge behavior restricts the Veneziano amplitude to the form r(S—L - 0(512)) F(5'M - 0(523)) ”512’ 523) g r(S+1 - (1(512) " W523” (6.6) If the pn system has unnatural spin parity quantum numbers (SP = 0', 1+, 2- ...) the amplitude may similarly be written A(sn, $23) = Const * S (P3)L (P1)M V(s]2,523) + (193) (6.7) Where now S = L + M is required. The requirement of Regge behavior gives the form of the Veneziano Amplitude to be in thi 8 case ‘ F(m - d(512)) F(n - d(523)) “512923) g (6.8) P(m + n ' p ' 0(512) ' 0(523)) 97 where m 3_ M+p; n > L+p; and p = integer _>_ 0. Further, one must have m l l and n _>_ 1 to avoid the poles eat d(sij) = 0. All spin parity state up to spin of 4 are considered with the exception of 0+, 1- and 3-. For a three Elion system with arbitrary total spin J, one may write the Iparity as P = [n]3 (-1)L(-l)£ where l is the relative cxrbital angular momentum of the n+n- system, and L is the (orbital angular momentum of the third pion with respect to ‘that system. Here n = -l, is the intrinsic parity of the Ipion. If the total spin J = 0 then L = l and P = [n]3 = -1, ‘therefore 0+ is forbidden. Now one is led to consider the natural parity states (IP = 1-, 2+, 3- ... of a fermion—antifermion system of 1:otal spin 8 and relative orbital angular momentum 2., The Iparity (P), G—parity (G), and charge conjugation operator £+S+I and (-1)2+8 respec- (C) are defined as -(-l)£, (-l) 1:ively, where I is the isotopic spin. Since there are no ssingle states one has 8 = l and C = P. The isotopic spin «c>f'the pn system is 1, therefore G = -(-1)J. For a system of N pions, one has G = (-1)N or here G = (-1)3 = -1. But Since G = (-l) (-l)J, J must be even and hence the states ‘1'-a 3-, ... are excluded. The spin tensor transition amplitudes for each of the allowed pn spin parity states will now be explicitly m» Essa-..., 98 written in accordance with the prescription given by Goebel and Zemach . JP=0' (L=0,M=0, p=T) T(0') =1 (6.9-A) .1"=1+ (L=1,M=o,p=1) Ti(1*)=p;+p}' (6.9-B) JP = 2- (L =1. M = 1o P = 0) _ 1 (6.9-C) Tm” I = [P31‘P1j + PHst] " ‘3' 5ij[P3iP‘J' + PHP3J] JP = 2+ (L = T, M = O) + + o ...) + . + + + + T”(2+) . [(plxp3)1 pg + (plxpaflng + [(P3xPl)T Pi + (P3xP])~j Pg] (609-0) JP = 3+ (L = 2, M = 1, P = 0) T‘3k(3+) = IP§P33P§ + P5P} 1 + PsapskP1‘J 1 + k ‘* ‘* l " 2 ° " ." 5 -y613[|P3|2 P] + 2P3 P1 ' P3] - 5 mil Pal Pi“ 2 . . F T-0 I' 2-0 .+ [EE3XP )1p3 + (p3xp1)361] * ( (523)) ( ($12)) (6.10-D) 1 1 P(3'O(S]2) ‘ a(523)) . 2 + . o o k 'l + k k-F . 3+) AU" =|[P;P‘3 g" + nggpg + P§P3P}] 1613[|P3| P] + 2P3P1 P3] 1+2j +-+j-l +21+$-3Pfl "2";61IJIP3I P1*2(P1 P3)P3] 555k['P3l P1 2(1 3) 3J r(1'o(512)) F(2'a(523)) + (]++3) (6.10-E) F(3-u(512) ‘ 0(523)) 100 The Dalitz plot is defined here to be the three dimensional figure described by the surface h (512' 523), where the two independent variables 512 and 523 are represented along two mutually perpendicular cartesian coordinate axis. An important property of this representa- tion, as proved in Appendix E, is that equal areas on the Dalitz plot correspond to equal probabilities in phase space. One therefore expects that the surface h will be proportional to the square of the transition matrix amplitude for values of 312 and 523 in the kinematically allowed region. The Dalitz plot comparison is especially well suited for the Veneziano model since the scalar amplitude has only two independent variables, aside from an overall scale factor. The Dalitz plot here will be symmetric along the diagonal since the two negative pions are indistinguishable. The data comparison has been separated into two samples. Sample 1 consists of the 1914 events (3828 points can the Dalitz plot) with laboratory momentum from 1.09 to 1.43 GeV/c. Sample 2 consists of 610 events (1220 Dalitz Plot points) with laboratory momentum from 1.6 to 2.0 GeV/c, and is taken from reference 15. Chapter V describes tzlie spin analysis performed with the data in sample 1. The events in sample 2 were analyzed in a manner similar to the phenomenological approach taken by Bettini. The alialysis here will serve to extend this comparison up to laboratory momentum of 2.0 GeV/c. Figure 6.1 shows the Fig. 101 6.l.--Group 1 data for A. The Dalitz plot of s(n+n1) vs s(n+n;) B. The projection onto the mass squared axis The Veneziano prediction is shown as a smooth curve . 102 e o e 0 0'0 ‘o‘.' o. ’00.“. no... a. o I 3 S (PI+,PI-—) GEVmu2 I— 2. I m .N i .o .omm .owi .omi .om mng>mo i-im..ai. m N..i>mo.i.\mtz:oo 103 Dalitz plot and its projection onto the invariant mass squared axis for the data in sample 1. One observes strong po and fo resonant bands at .57 and 1.64 GeV2, respectively. The minimum density areas, in between the resonant bands, on the Dalitz plot are also very evident. Figure 6.2 shows the Dalitz plot and its projection onto the invariant mass Fa squared axis for the data in sample 2. One here observes similar features as in the sample 1 data. The values of A and B in Equation (6.4) were obtained r x“ by simultaneous solving: :LE 2 2 = M = 1 = A + B M MS 0) ( p) and o(s=M§) = 2=a+B(M§) The parameters MD and Mf were obtained by fitting the + - O O I I 0 7’ 1r invariant mass distribution to Briet-Wigner resonance forms as described in Chapter 3.3. The values of MC) and Mf used for the sample 1 data were 760 MeV and 1277 MeV reapectively. The values of MD and Mf used for sample 2 d~a-‘ta were 712 MeV and 1260 MeV respectively. The para- rue1:er C in equation (6.4) was determined by requiring that the Veneziano model give the correct width to the po meson. Table 6.1 lists A, B, and C for the data in sample 1 and :Eiiauhmple 2. The values used here are compared to the values used by Lovelace, Bettini, and the normally) accepted Vfilues“ for the p trajectory. The model was expressed in terms of the Dalitz plot 9 § rameters as 104 Fig. 6.2.--Sample 2 data for A. The Dalitz plot of s(n+n1) vs s(n+n2) B. The projection onto the mass squared axis. The Veneziano prediction is shown as a smooth curve. 105 u ...: c on. .00 ”a .0. 0‘0. \ooo’oo. . 0 0| a. .0. o fiIHo-H o O o o 0.. ’0 O o o o son‘s-cocoa... o a” 0.00 o no... 0. '0 on... 0 to. o. ”a“? .0. o no on I o no. on o o. ‘ 0.0 0. on... 9000’. no..." 0 no. ..........,... ...... oo o. co Goat a o o 00 a... o. .0” o I... no... 00‘...- . O... O 0 ..~ ’~.. ‘ O O .0. oo‘of one... o. a O on o oo- o In. on. o o ‘00...." o '00. o... o. - ‘ho’o o a so”. 0‘0 0 on. o o. no o. oo o o 0. a on... 90 ole. W ”0 Oefioofio co 0‘ 0M 0 o o... o o o O O. 0' C .0. I O I O o o O .m N Nnn>wo AIHm.+Hi. m r 3. ,PI—) GEVnn? I 2. S (PI+ .ooH _ .mm. . Nnnn>moc a .mN \mpzoou 106 Table 6.l.--Values of the p trajectory parameters. Sample 1 Sample 2 Lovelace Bettini Accepted Value36 A .46 t .03 .53 t .03 .483 .65 .57 B .94 i .01 .925: .01 .885 .84 .91 C 025 i .003 .20 i .003 .28 .26 —- -'-‘E P P 2 2 = J J lTl 3p a IT (512: $23)| J- where IT]2 is the square of the total transition matrix P element and |TJ (312 523)]2 is the square of the transi- I P tion matrix elements given in equation (6.10). aJ is the percentage of each contributing spin parity. The fit was performed by considering Dalitz plot grid sizes of .5 (GeV)2. Approximately 60,000 events were selectively generated by a Monte Carlo routine, Sage 11,37 according to the number of events present at each momentum in the two data samples. These events were further weighted according to the magnitude of the square of the transition matrix P element. The normalization requirements are 2 Ingl2 = ij 2 Dataij where i and j are the Dalitz plot grid point ij P coordinates, and aJ are constrained to remain within two standard deviations of the values given in Table 5.1. Since the parity of the spin 2 state could not be deter- mined, contributions from both the 2+ and 2- states were assumed possible. The fit, projected onto one invariant 107 mass squared axis, for the data in sample 1 is given in Figure 6.1 as a smooth curve. Figure 6.3 shows comparison slices across the Dalitz Plot. Figure 6.4-B shows a Dalitz plot of the fit. Figure 6.4-A shows a Dalitz plot illustration of the absolute deviation between the Data and the Model. Overall one observes good agreement between the lfia data and the model. Obvious area of disagreement are when "“ 312 is large and $23 is small and vice versa. Figure 6.2 shows a comparison of the invariant mass f‘ squared predicted by the model, with the data for the 35 events in sample 2. Figure 6.5 gives a comparison for slices across the Dalitz plot. Figure 6.6 show the Dalitz plots for the model, and the absolute deviation between the model and the data. The area of disagreement seen before are also present here. The percentage of each spin parity state used in the fit, and the overall normalization is shown in Table 6.2. The amount of each spin state for the data in sample 2 was not constrained in the fitting process. The necessity for higher spin states is consistent with the expectation one might have had for this higher momentum sample. 108 Table 6.2.--Percentages of each spin parity state and N, the ratio of the fitted to the total number of events. 0- l 2 2 3 N Sample 1 0 .16 0 .68 .16 .90 Sample 2 0 0 0 .74 .26 .70 109 Fig. 6.3.—-The comparison of the Veneziano fit to the data in sample 1 for slices across the Dalitz plot. A. .o to .2 cev2 E. 1.9 to 2.5 Gev2 B. .2 to .8 Gev2 F. 2.5 to 3.0 cev2 C. .8 to 1.4 GeV2 G. 3.0 to 3.6 GeV2 0. 1.4 to 1.9 cev2 H. 3.6 to 5.0 Gev2 110 B H m m t T r. I t t I .2 “1' fl V m - 1 ..H 2P +, m _ a e . T _ _ _ _ an m m D m II\ II\ II I TI 1‘ T .2 u! I v E G T I - .H 2P +, m _ a _ e _ _ _ _ 4 .5 do .0: .0 do .0: o .8 .0: a do .9. 00 mu: 3%.: H {9238 ml... Sum: H {9238 w: Swot H .3538 mungmg 1.35:8 111 Fig. 6.4.--Dalitz plot of the Veneziano Model fit for sample 1 for A. The absolute deviation between the model and the data ‘ B. The Fit 112 an>wo HIHi~+Hm~ m S IPI+.PI—) GEVII? an>mo anis+Hic m 113 Fig. 6.5.--The comparison of the Veneziano fit to the data in sample 2 for slices across the Dalitz plot . .0 to .2 GeV2 . .2 to .8 GeV2 2 . 1.4 to 2.0 GeV2 2 A B C. .8 to 1.4 GeV D E. 2.0 to 2.6 GeV F . 2.6 to 3.2 GeV2 8. 3.2 to 3.8 GeV2 H. 3.8 to 5.0 Gev2 114 D H m m fir (l _I :l\ .l :l‘ I I .2 In v m I I I I .)_ 2m, + m _ _ _ a _ a 0.5 m m m m MI I I In...” fl V h m I I I I .1. 2m, + m _ a _ _ _ _ _ A 0.5 .0: .ON 0 .0: .ON 0 .o: .om o .9. .8 o wnl>moimwoim.\m.~238 mnl>u8mmoimwo HIHi.+Hmc m . o ... . O n o no . . " ... nu. - o o - O ' * .00.... . .. ..uo. oaas .0 I o o o a a on. no 0 O. o c 0 oo o 00 0. o a. .0 no. Io. I O. D I O .00. O .0 o o o. o .. a ....”u .. 0'... .0. O I O o O 0 o o a u. ... o o o '0’ o 0. 00". Q I I. Q I O O O 0.. o. o a. u a O O .0 o o o o O 0 o. O 0. new r' 3. T 2 5 (PI+,PI—) GEVIx2 -H d N H m N |— .H al>um AIHms+Hmc m CHAPTER VII SUMMARY AND CONCLUSIONS In this experiment a total of 121,399 antiproton deuteron interactions were analyzed. Data were collected using the 31-inch deuterium bubble chamber at the Brook- haven National Laboratory exposed to an antiproton beam at 1.09, 1.19, 1.31, and 1.42 GeV/c momenta. The cross sections were determined for the 3, 4, 5, and 6 prong topologies that had a proton spectator. The cross sections for the odd prong topologies agreed well with the cross sections from higher momentum data. An even prong event results when the spectator has sufficient momentum to produce a visible track in the bubble chamber. In this experiment, events with spectator proton momentum up to l. GeV/c were measured. Correcting for deuteron shadowing effect, topological cross sections for in interactions were obtained. These were found to be in good agreement with predictions from the impulse approximation. The 5n + n+n-n- reaction cross sections weredetermined by analyzing pd + p + n+n-w- events. 3P This reaction was found to be dominated by the production 117 118 of po and f0 resonances. Cross sections for pon- and fon- resonance states were determined and were found to be in excellent agreement with data from higher momenta. Not only do these cross sections add valuable knowledge to the subject of antiproton neutron interaction, but the excellent agreement with data from higher momenta further serves the purpose of establishing reputable credentials for the data for use in the dynamics analysis. The different cross sections §%r and 3%T-were obtained for both the p0 and f0 meson production. The slopes of the forward and backward differential cross sections were shallow, and consistent with being identical. In order to test the crossing symmetry principle the differential cross section for the reaction 5n + pon— was compared with its line reversed reaction fi-p +-np°. The agreement between the slope parameters and intercept comparison indicates that the crossing symmetry principle is applicable to these quasi two body reactions at this low momentum. The c.m. angular distribution for the po and fo were found to be nearly symmetric about cos 8* = 0. From these distributions the spin of the pn system was found to be predominately in a s = 2 state. The Dalitz plot for the invariant mass square of the fl+fl- system show strong po and fo bands. The minimum density areas in between the bands are also evident. The predictions of the Veneziano model was compared to the data in this experiment and data at 1.60, 1.75, 1.85, and l _ . 119 2.00 GeV/c from Reference 15. The Veneziano model was constructed by multiplying tensor spin-parity functions times scalar Veneziano amplitudes. Since for the data in this experiment the spin state composition of the pn system is known, the magnitudes of the spin parity functions are constrained to these values in the comparison. For the comparison with the higher momenta data, the magnitudes of the spin-parity functions were free parameters in the fit. The model gives good agreement with the data at the lower momentum region. Noticeable area of disagreement are at places where the invariant mass square of the (n+n1) system is large and that for the (n+,n;) system is small. This disagreement becomes more prominent at the higher momentum region. The overall agreement between the data and the Veneziano model confirms the validity of the model and its application to the 5n + n+n-n- reaction at these laboratory momenta. APPENDIX A SCANNING EFFICIENCY DETERMINATION APPENDIX A SCANNING EFFICIENCY DETERMINATION In order to determine the scanning efficiency an independent second scan was performed on 12 reels of film. All events from both scans were examined in a third scan and were classified according to the scan code numbers given in Table A-1. The results of the code number count for the add and even prong events are listed in Table A-2 and A-3 respectively. Consider a total sample of N true events from which N and N2 events were found in scan 1 and scan 2 respec- l tively. The number of events found in common is given by N EN = P P N (A-l) where P1 and P are the probabilities of finding a good 2 event in scans l and 2 respectively. 1' N2, and NION2 are written in terms of the code numbers one finds that the number of If the sets of N events in common is the sum of the events with code 7, code 6 and code 5. This will be written as 120 NlnN2 = 5 + 6 + 7 (A-2) Similarly, N1 = l + 3(1) + 5 + 6 + 7 (A-3) and N2 = 2 + 3(2)-+ 5 + 6 + 7. (A-4) where 3 the number of events which had (l)' [3(2)] represent a minor error in scan l,[2], and a major error in scan 2,[1]. The scanning efficiencies can then be determined for scan 1 and scan 2 as Since the events with code 8 were measured twice, to obtain total number of events N, one must correspondingly lower the number of events found in the entire 72 roll sample by the fraction of code 8 events found in these selected 12 roll sample. T1 122 wows» ownsmmoa mm3 ucm>m was w ooom mums muco>o cuon ocm mcmom cuon cw ocsom mm3 ucm>o one h uouuo Hosea m Tome H snow usn mcmom anon ca UGSOM mm3 ucm>m one o uouum Hosea m Toma N cmom pan mcmom anon cw canon mmz uco>m one m consou muoc II ucm>o xcso v nonuo one ca uouuw HOCHE m com .cmom oco as uouuo momma m on: uco>m one m uouuo on on: m cwom “a zoom CH uouum HOnmE m on: uco>o one N uouum o: to: H cmom am coon ca uouuw HmeE m can ucm>m one H mcflcmoz Hogan: moou .muoosaz oeoo snow on» no aoaoecemonII...fl.m ”cm mmaflfi ”5 HOW U500 “mg 2 080 MO muflsmmm'loN'q ”Ham.“ Nmm Hem m n vv mm o N 125 mom va m M NH 0 vv H mm Nmm mmw N m on me o N ooh mmo N m HH 0 mm H mm mom bNm mN m Nb me o N 0mm 5N0 mm m 0m 0 mv H Hm NMNH >00 0 o no MNH o N mm0H ham 0 o we 0 on H mm mONH ONm OH H mOH me o N vOHH ONm OH H mm o ONH H av th mvh NH @ Nm mm o N hum mvn ,MH m. no 0 No H ¢N Hmuoe h 0000 w 0600 m OUOU v 0600 m 0600 N mwoo H OVOU doom HHOK :11 6633.803 .~I< flame 126 emH H mmH o o m o o mN o H eN Ham 0 NHe N H mm 0 mm o a N emm m NHe N H mm o o as a H eN 0mm 0 eNN m N me o NN o m N mam H veN N m m o 0 mm m H «H oem o mHN m H mm o omH o v N NOOH 8H mHN m H as v o mHN a H eH Nae 0 Non m H mm H me o m N can m Nom m H 8H 0 0 mm o H OH NHNH 0 New N N meH o mmH o e N omoH NH New N N me o o mmH e . H OH mam o NNN o H mm 0 mm o o N mom m NNN o H 0H 0 o mN m H m mom 0 New a m oNH H mm o e N NNN a New a n ma N o omH e H m Hmuoe m «coo N oeoo o aeoo m «woo a maoo m «coo N 6600 H 6600 Scone snow HHom .mucm>m mcoum me new Hsom may uom unsou Hmnfidz opoo II] was No muHsmomII.mI¢ OHEEB 127 ONO O OOO H O OO O OOH O O N ONO O OOO H O Om H O ON O H OO OOH O OOH H N ON O NN O O N ONH O OOH H N O O O OH O H OO NOO O OOO O H OO O OO O O N ONO O OOO O H mm O O HO O H OO Omm O OON N O OO O NO O O N OOm m OON N N O O O Om O H Om OOO O OHO O N OO O NHH O O N OOO O OHO O N OO O O NOH O H Om HNN O OOH H O OO O ON O O N ONN O OOH H O OH O O ON O H ON ONO O NOO H O OO O NHH O O N OOO N NOO H O ON O O HO O H ON OON O OOH O O NN O ON O O N Hmoos O OOOO N mOoo O OOou O OOoO O OOOO N OOoo N OOoO H OOoO ozone :OOO HHOm I]. 63538: . OIO mHnOe 128 ovv h vmm m 0 Nm 0 0 v0 v H mm mmH O hNH O o VH O OH o o N ovH v hNH O O m 0 o o o H mm 0mm 0 00m N H ha 0 no . O v N mNm v m©m N H on O 0 mm v H mm HOH O ONH H N mH 0 MN 0 m N mvH N ONH H N o o O NH 0 H Hm NOm O mmm o N cm 0 Ho O v N va v mmm O N OH O O No c H H0 mmN O OOH m 0 cm 0 mN O m N mvN N mvH m O OH O O on m H mm 050 O mom N m cm O HO o v N NHN h mom N m ms O O msH v H mm 00m 0 mON H H 0v 0 mo 0 m N th H mON H H vH H 0 0m 0 H mv Hmaoe O OOOO N mooo O OOOO O OOOO O mOoo N OOOO N OOOO H Once Ocoua :OOO HHom ll .omscHucooII.MI¢ OHnt 129 55H 0 NMH N N ON 0 HN O o N OOH m NMH N N m 0 0 0H m H mm mvv o cmm m o No 0 mm o HO N HmuOB H. OHOOU N. OHOOU 0 $600 m OHOOU v 0000 m OHOOU N 0600 H OHOOU mcoum cmom HHom 11‘ .pmscHucooII.MI< OHQMB APPENDIX B MEASURING EFFICIENCY DETERMINATION APPENDIX B MEASURING EFFICIENCY DETERMINATION To determine the measurement efficiency each event on the original scan tape was classified as passed or failed, depending on whether it did or did not appear on ETE the final data tape. Table B-1 summarized the result of this classification. The measurement efficiencies given in Table B-2 are the ratios of the number of events on the final data tape to the number of events on the original scan tape. In these efficiencies one assumes that each counted event belonged in the desired sample. According to the scanning analysis some events measured were not in the desired sample, as defined in Table B-3. Since these events have a higher failure rate, the effect on the measurement efficiencies must be determined. One imagines the data containing two samples of events; a desired sample with N1 events and measurement efficiency 81, and an undesired sample with N2 events and measurement efficiency a The measurement effi- 2. ciencies in Table B-1 are the weighted average of these efficiencies, 130 1 Then, Ell = N1 [E(Nl + N2) - EZN;] represents the effi- ciency for the desired sample. Based on a sample scan analysis, 52 was determined to be .55 and the percent of undesired events was l.%. Table 2-2 lists the corrected measuring efficiencies. The overall effect for this experiment was .4%. i: 132 Table B-l.--The Total Number of Passed Events is Given in Table B-l-A; the Total Number of Failed Events is Given in Table B-l-B. Table B-l-A. Momentum gev/c 3 Prong 4 Prong 5 Prong 6 Prong 1.09 5063 6937 2285 2060 1.19 7567 9411 3347 2945 1.31 13634 17274 6379 5791 1.43 9427 12062 4164 3970 Table B-l-B. Momentum gev/c 3 Prong 4 Prong 5 Prong 6 Prong 1.09 355 558 234 281 1.19 376 753 323 453 1.31 823 1229 674 787 1.43 471 806 402 558 133 Table B-2.--The measurement efficiencies for the entire Sample. Momentum gev/c 3 Prong 4 Prong 5 Prong 6 Prong 1.09 .934 .926 .907 .880 1.19 .953 .926 .912 .867 1.31 .943 .934 .904 .880 1.43 .952 .937 .912 .877 Table B-3.--Events in Measured Sample which were Unwanted Quantified in Terms of Percentage of the Total Sample. Description 3-Prong 4-Prong 5-Prong 6-Prong No dark positive - 3.8 - 2.5 Junk .31 .19 .31 .19 Not beam I .25 .49 .25 .82 Out of fiducial volume .55 .28 .55 .39 Wrong primary event type .12 .41 .12 .93 Strange particles present .13 .26 .13 .12 Dalitz Pair — .22 .02 .51 No event found - .23 - .19 APPENDIX C FIDUCIAL VOLUME LENGTH CALCULATION APPENDIX C FIDUCIAL VOLUME LENGTH CALCULATION Figure C-l shows the position of the fiducial volume with respect to the vertex positions of all measured events. If one knows the radius of curvature of the beam, and the azimuthal angle ¢ of the beam at the vertex, the arc deter- mined by these numbers will intersect the fiducial volume boundary at two points. The radius of curvature is deter- mined by R : P/.03 B, where R is meters and P is the momentum of the incident particle in GeV/c moving at right angles to a magnetic field B in kilogauss. The origin of the circle thus described, and as shown in Figure C-2, is given by X = X + R sin ¢ v Y Y - R cos ¢ 0 v where (xv, Yv) is the vertex position for a measured event. By elementary geometry one finds that the circle will 134 135 intersect the fiducial volume boundary at J 2 X X = X - R - (Y -Y& and o o _ 2 1 . 0 Y — Y + R - (X -X )2 1f 0 < ¢ < 90 or o o w 1 _ _‘J 2 _ _ 2 . o 0 Y - Yo R (xO xw) 1f 90 < ¢ < 180 where Xw and Yw give the position of the fiducial volume boundary in the X and Y directions respectively. The positions of intersection between the circle and the fiducial volume boundary were found for each inter- action, and the arc length S = 2 R arcsin (D/ZR) was calculated, where D is the chord between the intersecting points. In this experiment the fiducial volume is defined by -24. < Y < 17.6 cm and -10 < K < 10 c.m. The average arc length is determined for each incident momentum setting and is given in Table 2-3. One minor consideration worth mentioning is the energy loss of the beam particle as it travels through the liquid. This will cause the particle to spiral, thus increasing the fiducial volume length. The momentum loss dp/dx for a 1.3 GeV/c antiproton in deuterium is .3 (MeV/c)/cm. The average particle will lose a momentum of 12.5 MeV/c travelling through 41.7 cm of liquid deuterium. This effect is much less than the experimental uncertainties, therefore has been neglected. LA 136 Fig. C—l.--Vertex position of all measured events with respect to the fiducial volume. The coordinate system corresponds to a front view of the bubble chamber. The rectangular box represents the fiducial volume. 137 so 9: u was GOMuwmom A xwuuo> EU .oal EU .OH M... X... EU .vNI M Econ 138 Fig. C-2.--Geometric picture of a circle with the same radius of curvature as the beam. The event vertex position is (X , Yv). The beam will intersect the fiduciaY volume at two points, labelled (XA, YA) and (X3, Y3) in this example. The rectangular box represents the fiducial volume boundary. 139 XA' YA) “ (xv: YV) \XJ'L) 2 2 J R - (YA-YO) "1 Y ) Q—L—R cose -—> 6 J R‘- (XB‘XO’T (x.. Y) APPENDIX D TRACK COUNT APPENDIX D TRACK COUNT Table D-l gives the track count for beam passing through the upstream boundary of the measurement region. Table D-2 summarized the results. In order to determine the corresponding track count for the fiducial volume, a correction must be made for beams entering the measurement region but interacting before reaching the fiducial volume. Figure D-l gives a two dimensional illustration of the vertex positions for all measured events, in relation to both the fiducial volume, and measurement region. The average distance between the two regions is 7.2 cm. There- fore it is necessary to calculate the attenuation of the beam through 7.2 cm of liquid deuterium. If No tracks enter the liquid then the number that survive at a distance -9A001/2' where p is the density of 2 is given by N = No e the liquid, A0 is Avagadros number and o is the total cross section. One must reduce the original beam count by the e-pAoz/Z to yield the number percentage of beam loss 1 - of beam tracks entering the fiducial volume as listed in Table 2-4. 140 141 Table D-1.--Beam Count Results. Reel Frame number Number of #Beam on basis Number of Estimated on Film frames on of counting frames total number film every 50th frame scanned of beam 1 1227 1224 187 24 9537.0 2 2089 2087 452 42 22460.1 3 2101 2099 447 42 22339.3 4 2092 2092 514 41 26226.5 5 2095 2096 417 41 21317.8 6 2090 2086 250 41 12719.5 7 2075 2073 301 41 15218.8 8 2085 2055 392 41 19647.8 9 2087 2085 298 41 15154.4 10 2094 2093 567 41 28944.6 11 2100 2099 649 42 32434.5 12 2081 2076 578 41 29266.5 13 2108 2108 396 42 19875.4 14 2102 2102 635 42 31780.2 15 2102 2102 518 42 25924.7 16 2091 2092 284 41 14490.9 17 2093 2093 592 41 30220.9 18 2104 2102 549 42 27471.9 19 2093 2092 431 41 21989.6 20 2090 2090 386 41 19676.6 21 2096 2095 386 41 19723.7 22 2098 2098 417 41 21338.2 23 2100 2100 365 42 18250.0 Table D-l.--Continued. 142 Reel Frame number Number of #Beam on basis Number of Estimated on film frames on of counting frames total number film every 50th frame scanned of beam 24 1 2198 2198 291 41 15600.4 25 l 2107 2107 575 42 28845.8 26 7 2087 2081 392 41 19896.3 27 1 2112 2112 393 42 19762.2 28 9 2114 2106 378 42 18954.0 29 3 2099 2097 364 41 18617.2 30 O 2096 2097 549 41 28079.3 31 1 2104 2104 593 42 29706.5 32 1 2101 2101 500 42 25011.9 33 2 2107 2106 512 42 25673.1 34 2 2097 2096 400 41 20448.8 35 18 2083 2066 462 41 23280.3 36 2 2100 2099 321 42 16042.4 37 2 2100 2099 401 42 20040.5 38 1 2091 2091 366 41 18666.0 39 2 2090 2089 300 41 15285.4 40 5 2092 2088 411 41 20930.9 41 1 2089 2089 251 41 12788.7 42 3 2092 2090 270 41 13763.3 43 2 2099 2098 272 41 13918.2 44 4 2098 2095 275 41 14051.7 45 1 2089 2089 210 41 10699.7 46 5 2092 2088 183 41 9319.5 143 Table D-1.--Continued. Reel Frame number Number of #Beam on basis Number of Estimated on film frames on of counting frames total number film every 50th frame scanned of beam 47 2104 2102 307 42 15364.6 48 2096 2095 344 41 17066.4 49 2096 2094 422 41 21552.9 50 2091 2090 432 41 22021.5 51 2109 2109 413 42 20738.5 52 1999 1999 462 39 23680.5 53 2086 2087 443 41 22549.8 54 2099 2098 386 41 19751.9 55 2093 2090 382 41 19472.7 56 2111 2110 309 42 15523.6 57 2092 2092 186 41 9490.5 58 2088 2089 169 41 8610.5 59 2094 2093 239 41 12200.6 60 2095 2094 210 41 10725.4 61 2096 2092 235 41 11990.7 62 2097 2096 210 41 10735.6 63 2089 2088 272 41 13852.1 63 2095 2093 238 41 12149.6 65 2111 2112 250 42 12571.4 66 2107 2107 196 42 9832.7 67 2132 2131 230 42 11669.8 68 2092 2091 232 41 11832.0 69 2088 2087 236 41 12013.0 Table D-l.--Continued. 144' Reel Frame number Number of #Beam on basis Number of Estimated on film frames on of counting frames total number film every 50th frame scanned of beam 70 1 2088 2088 199 41 10134.4 71 3 2084 2082 194 41 9851.4 72 2 2080 2079 203 41 10293.6 145 oo.oa mmmnvm mommm mvo mmmm mv.a 0H + H mm.oH mammmm novom vmm vmmoa Hm.H ow + 5H mH.m mommnm mmomm mmo Homm mH.H mm + av Hm.m mmmnna vammm mmo mmvm mo.H mm + hm mmfimuw poucsoo mxomuu ucmawummxm mnu oocsmom Aosmuu £00m mom mxomuu mo mo nomad: CH mmfimuw mo moEmum mum>m mcwucsoov Educoeoz amass: doom Hones: mmmum>¢ pmumswumm Hones: Hmuos mo Honesz mxomuu mo uwnasz .ucsoo xomua on» no aumsasmlu.~.o manna 146 Aé§KCamera view Measurement Region boundary Fiducial volume boundary / Y TEES?) 7.2cm# z I a; Fig. D.l.--Vertex position for all measured events. APPENDIX E PROOF THAT EQUAL AREAS ON THE DALITZ PLOT CORRESPOND TO EQUAL PROBABILITIES IN PHASE SPACE APPENDIX E PROOF THAT EQUAL AREAS ON THE DALITZ PLOT CORRESPOND TO EQUAL PROBABILITIES IN PHASE SPACE The phase space density of states for n bodies can be written in the relativistic co-variant manner as: 3P n d 3 n + + n pnm 411] 1r} 5 ( 1:] P1- - p) 5(1);1 £1 - E) (3.1) where P is the total momentum, E is the total energy and Pi is the momentum of the ith particle, E. l is the energy th of the i particle. If one considers a three body final state then d3P d3P2 d3P3 +. + + —__ 3" -P6E+E+E-E) “3(5) g! 2E1 252 2E3 6 (P‘ + P2 + P3 ) ( 1 2 3 integrating over d3P3 and writing d3Pi = pi dPi d i.one obtains 2 2 1,9091 ‘91 ““2 P2 (3.2) 93(5) 8 '5 EIEZE dPZ 6(51 +E2+E3-E) 3 . 147 148 If the quantization axis is taken along P1 then fd01+ 4n and dflz + 2n d(cos 612), where 612 is the relative angle between Pl and 32. In the P rest frame one has Pl + $2 + P3 = 0 so Pg = Pi + P; + 2P1P2 cos 612. The symbols P1, P2, and P3 will be used to represent the magnitude of the vectors P1, P2, and P3 respectively. Differentiating with in? respect to P3 one has 1 2 p3 d P3 . 2P1P2 d(cos 612) REE-4 _ 4‘ I [Hal—Ll. ..." 1 one can therefore replace d91P2 as 8.2 PldPledP2P3dP3 2 1 cth 2 dP1dP 2 and write equation E.2 as: deP1 Pde2 P3dP3 03(E)31t2f E] ??6(E1+E2+E3-E) Differentiating E2 = P2 + M2, one has dE = MI'U dP, so 93(5) - u2.de1dEsz3 6(E] + £2 + E3 - E) integrating over Ez gives 93(5) = "2 I dE1dE3 «3.3) The invariant mass squared of particles 1 and 2 is defined as 149 + + $12 ' (£1 + E2)2 - (P1 + P2)2 _ _ 2 _ 2 _ 2 2 In the P rest frame 312 - (E E3) P3 - E + m3 - 2EE3 Differentiating with respect to E3 one has ds12 = (~2E) dE3 = const dE3 similarly ds12 ~ dB 2 and ds23 ~ dE1 In terms of the invariant mass squared variables (E.3) becomes p3 ~ I d(s12) d(823) (E.4) LIST OF REFERENCES 10. 11. 12. 13. 14. LI ST OF REFERENCES G. Veneziano, Nuovo Cimento 57A (1968) 190. C. Lovelace, Phys. 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