Inelastic K⁺p Reactions at Incident Momenta from 1.37 GeV/c to 2.17 GeV/c Citation Loken, Stewart Christian (1972) Inelastic K⁺p Reactions at Incident Momenta from 1.37 GeV/c to 2.17 GeV/c. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ZKRE-V355. https://resolver.caltech.edu/CaltechTHESIS:05092016-111120208 Abstract A number of recent experiments have suggested the possibility of a highly inelastic resonance in K + p scattering. To study the inelastic K + p reactions, a 400 K exposure has been taken at the L.R.L. 25 inch bubble chamber. The data are spread over seven K + momenta between 1.37 and 2.17 GeV/c. Cross-sections have been measured for the reaction K + p → pK°π+ which is dominated by the quasi-two body channels K∆ and K*N. Both these channels are strongly peripheral, as at other momenta. The decay of the ∆ is in good agreement with the predictions of the rho-photon analogy of Stodolsky and Sakurai. The data on the K*p channel show evidence of both pseudo scalar and vector exchange. Cross-sections for the final state pK + π+π- shows a strong contribution from the quasi-two body channel K*∆. This reaction is also very peripheral even at threshold. The decay angular distributions indicate the reaction is dominated as at higher momenta by a pion exchange mechanism. The data are also in good agreement with the quark model predictions of Bialas and Zalewski for the K* and ∆ decay. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics0 Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Barish, Barry C. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1972 Funders: Funding Agency Grant Number Caltech UNSPECIFIED Record Number: CaltechTHESIS:05092016-111120208 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05092016-111120208 DOI: 10.7907/ZKRE-V355 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9708 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 09 May 2016 20:54 Last Modified: 01 Jul 2024 22:00 Thesis Files Preview PDF - Final Version See Usage Policy. 28MB Repository Staff Only: item control page

INELASTIC K+p REACTIONS AT INCIDENT MOMENTA FROM 1.37 GeV/c TO 2.17 GeV/c Thesis by Stewart Christian Loken In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute.of Technology Pasadena, California 1972 1. ii ACKNOWLEDGMENTS This experiment was performed by a large collaboration: R. Juhala and E. Malamud of NAL; P. Schlein, and w. Slater of UCLA; D. Davies, and B. Barish, R. Gomez, J. Pine and m:e from Cal tech. All the members of the group participated in some part of the experiment, and helped in its successful completion. Many others contributed to the experiment as well. c. Rosenfeld of Caltech built the gas Cerenkov counter; Art Ogawa, H. Grau and B. Freidler of Caltech built the triads, and helped greatly in the installation of all instrumentation. B. Lockman, R. Poster, and J. Brown of UCLA, T. Boark and J. Cooper of NAL, and B. Garnjost of LRL spent many long hours taking pictures. The hospitality and assistance of G. Eckman and the crew of the LRL 25 inch chamber is also greatly appreciated. The film was measured and the data analysed at UCLA. The assistance of R. Lippman and the scanners at UCLA was invaluable in completing this project. I wish to thank Drs. Ricardo Gomez and Jerry Pihe for their advice 'and assistance throughout the experiment, and particularly Dr. Barry Barish, my advisor, for his guidance, help and encouragement iii during my graduate work at Caltech. For financial support I am grateful to Caltech. iv ABSTRACT A number of recent experiments have suggested the possibility of a highly inelastic resonance in ~P scattering. To study the inelastic K+p reactions, a 400 K exposure has been taken at the L.R.L. inch bubble chamber. 25 The data are spread over seven K+ momenta between 1.37 and 2.17 GeV/c. Cross-sections have been measured for the reaction KP ~ pK0 -rr + which is dominated by the quasitwo body channels K.A and K*N. Both these channels are strongly peripheral, as at other momenta. The decay of the A is in good agreement with the predictions of the rho-photon analogy of Stodolsky and Sakurai. The data on the K*p channel show evidence of both P-Seudo scalar and vector exchange. Cross-sections for the final state pK+~+ffshows a strong contribution from the quasi-two body channel K*A • This reaction is also very peripheral even at threshold. The decay angular distributions indicate the reaction is dominated as at higher momenta by a pion exchange mechanism. The data are also in good agreement with the quark model predictions of Bialas and Zalewski for the K* and A decay. v to geanie vi TABLE OF CONTENTS PART TITLE PAGE I. INTRODUCTION 1 II. DESCRIPTION OF THE EXPERIMENT 6 A. Beam 6 B~ Scanning and Measuring 8 C. System of Analysis Programs 9 SCANNING, MEASURING AND FITTING 11 A. Results of First Scan 11 B. Scanning Efficiency 14 c. Fitting Failures 15 D. Separation of Reaction Fits 16 III. IV. v. CROSS-SECTION DETERMINATION 19 A. Normalization 19 B. Beam Track Count 20 c. Total Cross-section 20 D. Comparison of Normalizations 21 E. K0 Escape Correction 23 F. Pion Contamination 24 SINGLE PION AND TWO PION PRODUCTION 26 A. pK 0~+ Final State 26 B. NK7T Cross-section 26 c. Two Pion Production 29 vii TITLE PAGE QUASI-TWO BODY REACTIONS 33 PART VI. VII. VIII. A. Single Pion Production 33 B. KA and K*N Cross-sections 36 c. K*..6 Interference 36 D. Two Pion Production 40 E. K"'~A Cross-sections 42 F. ~p Total Cross-section 46 PRODUCTION AND DECAY ANGULAR DISTRIBUTION 49 A. General Discussion 49 B. K0 A++ Differential Cross-section 52 c. A Decay Angular Distribution 57 D. K*p Differential Cross-section 64 E. K*+ Decay Angular Distribution 68 F. K*.A Differential Cross-:-section 70 G. K* and Ll Decay Angular Distributions 73 H. Quark Model Predictions 81 CONCLUSIONS 86 APPENDIX 88 REFERENCES 100 viii APPENDIX EXTRACTION OF PARAMETERS FOR QUASI-TWO BODY STATES PAGE PART A. Three Particle Final States 88 B. Dalitz Plots 88 c. Mass Conjugation 91 D. Cross-sections and Interference 92 E. Four Body Final States 94 F. Resonance Production Angles 96 ix FIGURES NUMBER TITLE PAGE 1. K+p Elas tic and To tal Cross - secti ons 2 2. Representative Phase Shift Solution 4 3. Beam Layout 7 4. Measured Topologies 12 5. Single Pion Production Cross-sections 28 6. Two Pion Production Cross-sections 32 7. Dalitz Plots for the pK 0 rr+ Final State 35 8. Cross-sections for Kil and K*N Production 39 9. Effective Mass Plots for the . pK+~+n- Final State 43 10. Cross-sections for K* ~ Production 45 11. K+p Partial Cross-sections 47 12. KA Production Angular Distribution 53 13. Legendre Expansion Coefficients for the Reaction K+p ....;> K0 A++ 56 14. Differential Cross-section for K+p, K°.l\++ 58 15. Decay Angular Distributions for the Reaction K+p ~ K0 ~ ++ 16. A++ Density Matrix Elements 17. Legendre Expansion Coefficients for the Reaction K+p--. K*+p 59 62 66 x NUMBER TITLE PAGE 18. Differential Cross-section for K+ p-» K*+p 67 19. Decay Angular Distributions For the Reaction K+p ~ K*+p 20. K* Density Mat r ix Elements 21. Legendre Expansion Coefficients for the Reaction K+p -? K* 0 A ++ 69 72 75 22. Differential Cross-section for K+p~ K*o~++ 76 23. Decay Angular Distributions for the Reaction K+p ~ K*o A++ 77 24. A and K* Density Matrix Elements 80 25. Test of Quark Scattering Relations 84 A.1 General Features of Dalitz Plots 90 A.2 Decay Co-oridinate Systems 98 xi TABLES NUMBER TITLE PAGE 1. Measurement Summary 13 2. Beam Track Length 22 3. Pion Contamination 25 4. Single Pion Production Cross-section 27 5. Two Pion Production Cross-sections 31 6. Ratio of Charge States in K+p Reactions 34 7. Cross-sections for KA Production 37 8. Cross-sections for K*N Production 38 9. K~d-A Interference 41 10. Cross-sections for K~~A 11. Legendre Coefficients for K0 A++ Production Production 44 55 12. A++ Density Matrix Elements 61 13. Legendre Coefficients for K*+p Production 65 14. K* Density Matrix Elements 71 15. Legendre Coefficients for K*o~++ Production 74 Aand K* Density Matrix Elements 79 Test of Quark Scattering Relations 83 1 I. INTRODUCTION The spectrum of baryon resonances established by experiment can be explained very successfully as states of three constituent particles, . called quarks, each of which has strangeness O or -1< 1 ). In this raodel, then, it is impossible to form baryon states with strangeness +1. Such a resonance (usually called a Z*) would have to consist of at least four quarks and one antiquark. A number of recent experiments have suggested a possible Z* resonance in K+p scattering. The total K+p cross-section(2) and elastic cross-section(3) are shown in figure 1. The elastic cross-section falls smoothly with momentum but the total crosssection data show a peak at 1.35 GeV/c and a second shoulder at about 1.9 GeV/c. A fit to the first bump suggests a resonance of 4 mb at a mass of 1910 MeV. The full width at half height is 180 MeV and the value of (J· + t)X is .3 where J is the spin and Xthe elasticity. The second bump corresponds to a .2 mb peak at 2190 MeV with a width of 120 MeV and (J + t)X = .03. Analysis of the elastic differential crosssection and proton polarization data also suggests 2 19. 1' I 1, Ii i iI i ,., II ' I f .~ ,.1 I t 'i '~I•· 1 '1' ·If u,.O{Q.( II ' 15. J mb. 1 f lj ~ t l J I 10. a: E,IQ.Sl1c; l I 5. •·5 1' • GeV/c Figure 1. K+p Elastic and Total Cross-sections 3 a possible resonance(4). Some of the solutions, from both energy dependent and energy independent analyses, indicate a possible resonance in the P3/2 partial wave at an incident momentum between 1.3 and 1.9 GeV/c. Figure 2 shows a typical solution for the P wave amplitudes. These analyses show that the resonance, if it exists, is very inelastic (elasticities vary between .1 and .45). They also suggest that the speed, the rate of change of phase shift with energy, is not consistent with resonance behavior. All experiments which show some resonance features share the common characteristic of a small elasticity. For this reason, it is of interest to study the inelastic channels in K+p system. In an earlier experiment, Bland et a1.(5), studied single pion production at incident momenta between .84 GeV/c and 1.37 GeV/c. They conclude that the first bump in the total cross-section is a threshold effect, resulting from the opening of the inelastic channel K~ • There is no indication of a Z* resonance over the momentum range of their experiment. It is the purpose of this experiment to extend the study of the inelastic K+p reactions. The lowest 4 . 'I -p!t.i /85 *'137 +.% 1\ .2 + lz. 1.21 /.ir . 8G. ;::r .ff(,, ~ l' _./" -.'i - .2 ·2 "Re Rr± Figure 2. Representative Phase Shift Solution. The P 312 Phase Shift has been fitted to a resonance amplitude. The parameters of the resonance are: Mass, 1950 ± 100 MeV (momentum, 1.34 GeV/c); Total Width, 180 ·MeV; Elasticity, .28; Kato et al. (4) 5 momentum 1.37 overlaps the data of Bland. The maximum momentum of the beam transport system was 2.2 GeV/c. In this experiment we will consider single pion and two pion production. Quasi-two body processes contribute significantly to the pion production reactions and much of the analysis will be devoted to these states. The details of the experiment will be described in parts II - IV. In parts· V and VI, cross- sections are discussed for single and two pion states, and for the quasi-two body channels respectively. The production and decay of reson- ances in the quasi-two body reactions is studied in part VII. The angular distributions are com- pared with data at higher and lower momenta to look for evidence of possible direct channel effects. 6 II. A. DESCRIPTION OF THE EXPERIMENT Beam The experiment consisted of 400 K pictures taken in the Lawrence Radiation Laboratory 25 inch hydrogen bubble chamber. These were distributed over seven K+ momenta, spaced between 1.37 GeV/c and 2.17 GeV/c. The K+ beam was produced at a platinum target in the Bevatron external proton beam, and was transported to the bubble chamber in the Bevatron beam K9 (see ~igure 3)(6). The beam momentum was restricted to about ±1% by slits at the first horizontal focus. Two stages of mass separation, each consisting of an electrostatic separator,. and· slits at the vertical focus, were used to remove most of the pions. A Freon 12 gas threshold Cerenkov counter was installed in the quadrupole magnet directly in front of the bubble chamber. This counted pions at all momenta except 1.37 and 1.52 GeV/c, and gave us a continuous monitor of beam contamination. The pions amounted to about 5% at 2.17 GeV/c but were less than 2% at all momenta below 1.94 GeV/c. A marker light was flashed in the chamber for each frame where one of the particles was a pion. ao. B B. ~ G. .,,98 us. -<"',.. :a.o ~ - nf"£,f:J'._~88~ 8 Q. · • -e-_.. tJU.JU T c B .o.Q. LJ S Qo . B. °B. H~e s, · . ~Baf:JEJe= °'~· - L] Q·Q. ~ "°" '&Dl\C ... 11 Q . Q. _ Figure 3. Beam Layout ' The elements of the beam are denoted by: E.P.B. ,· Bevatron External Proton Beam; B.D., Beam Destroyer; R.D., Rapid Deflector Magnet; T., Production Target; B., Bending Magnet; Q., Quadrupole Magnet; Sx., Sextupole Magnet; s., Electrostatic Separator; v., Vertically Apert uring Slits; H., Horizontally Aperturing Slits; B,C,, Bubble Chamber. -...J ' 8 A triad of scintillation counters set on the pion image at each mass slit allowed us to continuously check beam steering and focussing to maintain a constant flux at the bubble chamber. For each pulse, we took a picture only if there were more than five, and less than fifteen tracks in the chamber. To further control the number of tracks in each picture, a beam destroyer magnet was installed upstream of the production target. This was pulsed rapidly when eleven tracks had been counted at the scintillation counter in front of the bubble chamber. The proton beam was deflected from the production target into the beam dump. B. Sca.rming and Measuring The film was simultaneously sca.rmed and measured on SMP's (Scanning and Measuring Projectors) at UCLA. The SMP facility consists of five projectors connected to an IBM 360/44 computer. The computer filtered the track measurements, checked for continuity of vertex and track data points, and output track points . in each of three views at about 3 cm. intervals. 9 C. System of Analysis Programs The output of the SMP system was processed through the TVGP-SQUAW-ARROW system developed at LRL to produce a data summary tape containing the physical quantities describing each event which are needed in the subsequent physics analysis. TVGP (three view geometry program<?)) inputs on SMP tape and performs a space reconstruction of each track. A curve is fitted through the measured track points by a least-squares method, and the angles and curvature are determined for each track. In order to properly account for the energy loss by ionization, the fit for each track is tried for each possible particle. The output of TVGP consists of azimuth, dip, and inverse projected momentum at the beginning and end of each track, for each possible mass, as well as the errors for these errors. Film setting errors and Coulomb scattering errors are included. The kinematic fitting program SQUAW(B) inputs a TVGP output tape, and fits the measured momenta to kinematic hypotheses. The fit was done by minimizing the X2 function subject to the analytic constraints appropriate to the hypothesis being tested. The four energy-momentum conservation equations provide four 10 constraints minus the number of unmeasured variables involved in a given hypothesis. These constraints are introduced by the method of Lagrange Multipliers, and an iteration procedure is used to minimizeX2. SQUAW outputs the fitted quantities and errors for each hypothesis which achieves a confidence level greater than 10-5. ARROW(9) inputs a SQUAW output tape and selects those events of interest for a particular analysis. The output data summary tape consists of identification information for the eVEnt, and the four vectors for the reaction fit. 11 III. A. SCANNING, MEASURING AND FITTING Results of First Scan All the film taken in the experiment was carefully scanned for interactions of interest (see figure 4). Topologies we have studied include all events with more than two outgoing visible tracks, and all those with a visible decay of a neutral particle. If all reaction particles are detected in the chamber, these correspond to the reactions K+ _,. v+rr+lT- .3 prong K+p -> pK 0 7T+ 2 prong V K+ p -» pK+7T+7T- 4 prong We have, in addition, required that the incoming track have approximately the same curvature; ' ·a nd that the production vertex fall within a volume which corresponded to about 80% of the chamber length. The reason for this fiducial volume cut was to ensure a constant detection efficiency, and to provide as long a decay length as possible for the neutral K'.s. Table 1 lists the number of frames scanned at each momentum, and the number of events of each tqpolo€zy found and measured on the first pass. All these events have been processed through the system Figure 4. Measured Topologies a. Three prong, b. Two prong V, c. Four prong 13 TABLE 1 MEASUREMENT SUMMARY prong V 4 prong 638 1270 1792 51157 695 1195 1340 1.94 54679 678 1270 1165 1.81 53643 734 1234 906 1.67 50936 660 1058 432 1.52 48257 802 1069 160 1.37 54298 677 807 71 momentum frames 3 prongs 2.17 53319 2.07 2 14 of programs described in the previous section, and the results of this fitting will be discussed in a later section. We will first consider the efficiency of this scanning in finding events of each desired topology. B. Scanning Efficiency Because we wish to measure production cross- sections, and the dependence of these cross-sections on incident momentum, it is important to determine the efficiency of the scanners at finding events. To do this we have re-scanned about half the film at each momentum. We have then compared these two scans frame by frame. If we assume that the scanning losses are random, we can easily measure the efficiency of each scan. first pass Ni = €1 No second pass Nz = €z No both passes Niz = €1E2 No efficiency of pass 1 = €1 = Niz ~2 This efficiency is used to correct the number of events of each topology found in the first scan. The average correction for this inefficiency was 15 5% and showed no systematic dependence on momentum or topology. C. Fitting Failures Approximately 10% of all measured events cannot be successfully reconstructed in the TVGP. program, These are events for which the scatter of the measured points from the fitted curve is outside the predicted errors from Coulomb scattering and the measuring machine errors. Some of these events have been studied on the scanning table and appear to be associated with crossing tracks or low contrast film. About half of the failing events have been remeasured and the passage rate is again approximately the same as the first pass. The failure rate is constant over all momenta and is the same for all reactions so that there is no correction to the cross-sections for this inefficiency. Some of the measured events did not pass any reaction fit in SQUAW. These had too large a -X2 in the final fit, or took too many steps to find a x2 minimum. 'When these events were re-measured, the passage rate was consistent with the first measurement rate. 16 The ratio of events fitting the possible reactions in the second measurement was in good agreement with the first measurement results. To correct the cross- sections for this fitting inefficiency we assume that the events that were not re-measured, and those that failed the second fitting were distributed in the same ratio. The correction varied slightly with momentum and averaged 21% for events with a neutral decay V and 14% for the four prong events. All three prong events found on film fitted the tau decay hypothesis and no correction was made for fitting. D. Separation of Reaction Fits Because of the measurement errors in the momenta and angles, many events can pass more than one reaction fit. The program ARROW, discussed briefly in section II, is used to sort out the reaction fits and to determine the most probable hypothesis. For the events which have no missing neutral, that is the four constraint fits, ambiguities between possible fits are not a serious problem. 5% of these had more than one fit. Fewer than For events with one missing neutral the reduced kinematic constraints allow more ambiguities. Here about 35% of the 17 measurements fit more than one reaction hypothesis. From many previous bubble chamber experiments , two guidelines have been established to separate hypotheses. If an event corresponds to a reaction where there is an undetected particle in the chamber, it cannot pass a reaction fit with four kinematic constraints. For hypotheses with the same number of constraints, the x 2 of the fit is a reliable criteria for choosing the correct reaction. From these guidelines we have created a badness function B defined by B = x2 + 5 * (number of kinematic constraints) This corresponds qualitatively to the confidence level but is more heavily weighted in favor of the four constraint reactions. We will always assume that the correct fit is the one given by minimum badness. In order to evaluate the effectiveness of the event selection, we have scanned a sample of events and have compared the predicted ionization for each track with the darkness of the track in the chamber In all cases where the reactions could be distinguished by ionization, the fit favored by this was the fit 18 favored by the badness criterion. It was never possible to rule out the preferred fit by ionization. IV. A. CROSS-SECTION DETERMINATION Normalization The number of events in each reaction final state has been determined from the kinematic fitting. To determine the cross-sections for the reactions of interest, we must also know the number of inciden~ K+ 's and the number of protons. · The primary normalization for this experiment is the decay of the K+ in the tau mode, rr+rr+rr-. Using the tau decays, the cross-section is given by the formula C) = N interactions) * _Ji_ * _Aa_ N decays 'JC'[ !°H NA where AH(atomic weight of hydrogen) = 1.008 NA(Arogadro's number) = 6.0225 * 10 23 mole-1 J°H(density of hydrogen in chamber) = ,0608 g cm-3 1'f =(3 Y = 'P beam / M~ C = 2.998 * 1010 cm sec-1 "r(K+ lifetime) = 1.235 * 10-8 sec B(K+ branching ration into the L mode)= .056 Since the tau decays were measured at the same time as the reactions and with the same fiducial 20 volume criteria, this normalization should be free of any systematic bias. To check this we have also used a beam track count, and a total cross-section scan. B. Beam Track Count We have scanned every fifth roll of film, and have recorded the number of tracks in every tenth frame. In this scan, care was taken to ensure that the tracks counte d were only those that could have had measureable events. No off-beam or off-momentum tracks were included. Using the data from this scan, we have determined the average number of tracks per frame at each momentum. This does vary slightly with momentum because of variations in beam tuning and in Bevatron operating conditions. The total beam track length has been determined at each momentum using the total number ,of frames, the fiducial volume length and the track average. c. Total Cross-section When the second scan was made to measure the scanning efficiency the two-prong events were also recorded. For this sample, then, we have all events of all topologies, and hence have measured a total 21 cross-section, By comparing this to the accurate total cross-sections measured by counter experiments(2), we have determined the beam track length at each momentum. D. Comparison of Normalizations The beam track lengths determined by the three different methods are given in table 2 along with the uncertainty in each determination from counting statistics. The results are in good agreement, and appear to show only random fluctuations. We have decided, however, not to combine the results of the three methods. The beam track count has two disadvantages which make it less reliable. It is difficult to be sure that the tracks counted really correspond to the tracks for the measured sample. In addition, the method requires an accurate knowledge of the length of the chamber used. The fiducial length was defined by marks on the chamber lens, and when the beam moved vertically, the effective length changed. In principle the total cross-section should be a reliable normalization since the length cancels out as it does with the tau decays. The total cross-section scan, however, is much more sensitive 22 TABLE 2 BEAM TRACK LENGTH track count momentum tau decay GeV/c x 10 6 cm C>toio.L x 106 cm 2.17 24.2 ± .9 23.6 ± .3 21.9 ± 1.1 2.07 21.6 ± .8 21.5 ± .3 20.2 ± 1.0 1.94 20.0 ± ~8 20.6 ± .3 20.1 ± 1.0 1.81 23.6 ± .9 26.0 ± .3 20.4 ± 1.0 1.67 19.8 ± .8 18.5 ± .3 .. 1 8.8 ± .9 1.52 19.5 ± .8 21.8 ± .4 17.1 + .9 1.37 13.9 ± .6 13.2 ± .4 14.9 ± .8 x 10 6 cm 1 2J to scanning biases than the tau's, It is very difficult for scanners to detect the very small angle deflections of forward scattered particles, and the momentum transfer to the proton is too small for it to be seen in the chamber. On the other hand, all the K+ decays having only one charged particle in the final state look very much like scattering events, We have not measured this scanning efficiency, and have not fitted the two prongs in TVGP and SQUAW, so that it is impossible to know the effect of these difficulties. The tau decays were measured at the same time as the reaction sample, and the scanning efficiency has been determined in the second pass. They have all been fitted in the TVGP-SQUAW programs so that all fitting inefficiencies tend to cane.el. We have, therefore, used only the tau decays in determining the cross-sections presented in this paper. E. Ko Escape Correction For those reactions corresponding to the two- prong V topology we must make a correction· for the unseen K0 events. This includes the factor for the number of K0 's that are in the K0 L state, and a factor for the K0 s ~ 7T+rr- branching ratio, 24 We must also include a correction for the K0 s's which escape from the chamber. This factor has been calculated by averaging over the K° momentum, and angles, and averaging over the fiducial length for the production vertex. The correction varied from 6% at 2.17 GeV/c to 2% at the lowest momentum. The dependence on momentum and angle is small and can be ignored in the calculation of differential crosssections and angular distributions. F. Pion Contamination Even after the two stages of mass separation, there still remained a few pions in the K beam, The pion contamination at each momentum is given in table 3. In order to determine the effect .of this in our event sample, we took one roll of film at each momentum with the beam tuned on TT's. These frames were scanned, measured and fitted in the same way as the K+ data. Table 3 shows the results of this scanning and fitting. For all momenta, .we have an effective pion contamination of at most 1%. This will be neglected in all the subsequent analysis. TABLE 3 PION CONTAIVIINATI ON momentum Cerenkov 4 c fit GeV/c counts passage 2.17 4.3% 2.07 ITevents frame K events frame ef.f uctive pion contamination 10% 2.3 1% 3.2% 10% 2.4 .8% 1.94 1.2% 8% 2.7 .3% 1.81 1.0% 5% 2.8 .2% 1.67 .8% 1% 2.6 1.52 ----- 0 3.1 0 2.9 .o .o .o 1.37 N \J'\ 2q V. A. SINGLE PION AND TWO PION PRODUCTION pK0 TT+ Final State In this experiment we have studied only one single pion reaction The cross-sections for this final state are presented in table 4 and in figure 5. The data from previous experimentsC5, 10) are also shown in the figure. The cross-section rises sharply with beam momentum from about .8 GeV/c, and reaches a peak at about 1.3 GeV/c, very close to the bump noted earlier in the total cross-section. Above 1.4 GeV/c there is a very smooth fall off with momentum. B. NK77 Cross-section To better understand the behavior of the single pion production channels, we have included in figure 5 the cross-sections for the two reactions not measured in this experiment. K+p -~ p~7To 4 nK+7T+ At any momentum where all final states are 27 TABLE 4 SINGLE PION PRODUCTION CROSS-SECTIONS 2.17 cJ (K+p --:» K0 p 17+) mb. 3.06 ± .31 2.07 3. 21 ± • 25 1.94 4.09 ± .35 1.81 4.29 ± .33 1.67 4.54 ± .40 1.52 5.52 :t .37 1.37 5.85 ± .61 momentum GeV/c • 28 8. mb. 6. 5. 4. 3. 2. 1. • 2• GeV/c 2.5 Figure 5. Single Pion Production Cross-sections The open symbols refer to this experiment, the solid to reference 10 • 29 measured, we can easily obtain the KN// cross-sections. We have used an interpolation technique where data were . not available and the results are also shovm in figure 5 As in the case of the pKOn-+ we see a sharp rise and then a more gradual fall off with increasing momentum. The threshold for single pion production is a momentum of .520 GeV/c. The cross-section, however, is very small until about .8 GeV/c, the threshold for production of the quasi-two body channel K~. We will return to this point in our discussion of all quasi-two body channels in the next section. c. Two Pion Production In this experiment we have measured three reactions having two pions in the final state. K+p ~ pK+rr+7T~ pKo7T+1To ~ nKOn+n+ There are two other final states, K+p ~ pK+7T 0rr0 --.:,. nK+7T+7To These have two neutrals in the final state and 30 cannot be separated in the reaction fitting programs. The cross-sections measured in this experiment are given in table 5 and are presented in figure 6. The figure also includes data from a number of other experiments(11). The threshold for two pion production is .82 GeV/c, whereas the threshold for the quasi-two body channel K*~ is 1.74 GeV/c. The data suggest that this channel contributes very strongly to the two pion reactions. If we assume that these reactions are dominated by this simple state, we can estimate, using ClebschGordan coefficients, the contribution from the two unmeasured reactions. This has been done in figure 6 to calculate the KNrr"cross-section. TABLE 5 TWO PION PRODUCTION CROSS-SECTION momentum C)(K+p - > K+p7T+TT-) cJ(K+p - > K0 p11+770) cJ( K+ p -> K0n 1T+7T+) 2.17 2. 28 :!: .16 1.75 :!: .2 • 38 :t • 06 2,07 1.82 ± .11 1.19 :!: .15 .43 :!: ,06 1.94 1.72 :!: .12 1.50 ± .17 ,34 ± .04 1.81 1.47 :t • 09 1.20 :!: .1.5 .26 ± .04 1.67 .88 :t .07 .64 ! .09 .17 :t • 04 . 1.52 .24 ± .08 • 26 :t • 0.5 • 04 :!:" ,02 1.37 .09 :!: .03 .12 +- .03 ,04 :t .02 \,.) .._,, 32 4. mb. 3. 2. 1. 1. 2. • Figure 6. Two Pion Production Cross-sections Open symbols refer to this experiment, solid to reference 11 • GeV/c • 33 VI. A. QUASI-TWO BODY REACTIONS Single Pion Production We have already guessed, from very simple threshold considerations, that the reaction is an important part of the single pion production. There is, in addition, the reaction which would also be expected to contribute. In table 6 we present the predictions of charge independence for the various charge states in the KN 7Tfinal state. The final state pKO-rr+ is richest in both KA and K*N production. This is the cleanest sample from the point of view of fitting ambiguities and is the only reaction we will consider in this analysis. Our interest in the states Kband K*N makes the choice of variables for a Dalitz plot obvious. Figure 7 shows plots of p7r+ effective mass squared versus K0 ~+ effective mass squared. The presence of both resonant channels is evident at all momenta. 34 TABLE 6 RATIO OF CHARGE STATES IN K+p REACTIONS single ·pion production charge state quasi-two body channel KA K*N 9 2 2 1 1 0 two pion production charge state quasi-two body channel K·:f b. p~17+17- 18 pKOrr+11 o 13 nK0 n+n-+ 2 nK+71+71o 1 pirr71 ono 2 35 """' ........ •••1· •· .. •• ••••••• b. a. "'""'' c. .... ....... ''"'" ....... d. . Figure 7. Dalitz plots for the Reaction K+p-),PK 0 TT' ... at four momenta: a.) 2.17 GeV/c b.) 1.94 GeV/c c • ) 1 • 67 Ge V/ c d. ) 1 • 37 Ge V/ c 36. B. K.6 and K*N Cross-sections The properties of Dalitz plots and the extraction of parameters for the quasi-two body channels are reviewed in appendix I. Using these techniques, the cross-sections for the reactions K+ p -->.> K0 ..6++ -> K0 p rr+ --? K*+p -;;> K0 n-+p have been calculated and are given in table 7 and table 8. Using the Clebsch-Gordan coefficients of table 6 we have also calculated the total K~and K*N crosssections, which are shown in figure 8. Data from other experiments(l~· are also presented in the figure. The behavior of the KN1Tcross-section is clearly dominated by these two reactions, not only in the threshold region but at higher momenta as well. At 2.5 GeV/c it appears that the two channels account for about 80% of the sing+e pion production. c. K*.6 Interference In the region of the Dalitz plot where the K* and A bands cross, it is impossible to determine whether the reaction proceeded through the K~or 37 TABLE 7 CROSS-SECTIONS FOR K~ PRODUCTION momentum GeV/c (}( K+p -> K0 ..6.++) mb. C)(K+p--» Kil) mb. - 2.17 1.13 ± .17 1.51 + .23 2.07 + .19 1.70 :t .25 1.94 1.27 1.53 :t • 23 1.81 1.86 :!: .27 2.48 :!: .36 1.67 2.56 :!: .47 1.52 1.92 ! .35 2.22 :!: .39 . 1.37 3.14 :!: .46 2.04 ! .30 + .61 4.18 - 2.56 + .52 38 TABLE 8 CROSS-SECTIONS FOR K*N PRODUCTION c:JCK+p -> K*+p) cJ' (K+p --9 K*N) .92 ! .14 1.38 + .21 1.11 ! .18 1.66 + .27 1.30 ! .20 1.94 : .30 1.81 1.46 : .21 1.67 1.52 ! .25 2.19 +- .31 2.28 + - • 37 1.52 1.88 ! .29 2.82 ! .43 1.37 2.02 ! .32 3.02 ! .47 momentum I 39 8. 6. 5. I mb. 4. l l r 1 lr1 d ¥ \ Tl 3. 2. 1. fr lf lf fY-1 rfr lf / .5 1.0 1.5 2.0(GeV/c)2.5 Figure 8. Cross-sections for K.6 and K*N Production. Data for KA shown by i and! ; K*N by f and ~ • The open symbols refer to this experiment, solid to reference 10. 40 K*N intermediate state. In this case the Dalitz plot density is not just the sum of the squares of the production amplitude, but can include an interference term. In appendix I we have discribed a technique for measuring the contribution of such an interference term. 9. The results of the analysis are shown in table There appears to be no net enhancement in the inter- ference region and we will neglect interference in the discussion of the K~and K*N final states. D. Two Pion Production The only quasi-two body reaction with a threshold in the range of this experiment is the reaction The ratios for the possible charge states have been calculated using isotopic spin Clebsch-Gordan coefficients. The reaction which is richest in reson- ance production is K+ p ~ pK+77+77This is the four constraint fit for our four prong sample and is for this reason the cleanest reaction to study. This reaction will be used for all our 41 TABLE 9 K*-~ INTERFERENCE CONTRIBUTION TO THE pK 0 ~+ FINAL STATE momentum cross-section GeV/c mb. 2.17 2.07 +.15 + - .07 -.04 + .08 - 1.81 +.31 + .11 +.12 + .08 1.37 +.16 + - .17 1.94 - - 42 studies of the K*..6 state, but first we should return to the assumption made in determining the KNrr7Tcrosssection. The table of coefficients shows that the ratio of the three measured reactions K+p - ? pK+7T+7TpKo,j-+7To nKo7T+rr+ is 18 : 13 : 2. This is in good agreement with the observed cross-sections, and strengthens our confidence that we can account for the unobserved reactions by this argument. E. K*A Cross-sections In figure 9 we show plots of po+ effective mass versus ~~- effective mass. The cross-sections for the reactions have been determined by the method described in appendix I. These are shown in table 10 along with the total K*A cross-section. In figure 10 we present all available data for K*4 production up to 2.5 GeV/c, along with the KNmrcross- 4J ·.;,..... il&l :tf, . : _ _ _ _ _ _I b. a. .. ... ~ ~ ~ ~ ----- ·---c. d. Figure 9~ +Effective Mass Plots for the Reaction K p-.pK rr rr at four momenta: a.) 2.17 GeV/c b.) 1.94 GeV/c c.) 1.67 GeV/c d.) 1.37 GeV/c 44 TABLE 10 CROSS-SECTION FOR K*~ PRODUCTION momentum GeV/c (~p 4> K* 0 1::1++) mb. (~p -'> K*~) 2.17 .77 ! .12 1.53 : 2.07 .61 :!: .09 1.21 + .18 1.94 .59 + .09 1.18 + - .18 1.81 .49 + .08 .97 +- .15 1.67· .29 : .04 .57 - .08 1.52 ~06 ! .. 04 .12 ! .08 1.37 o. o. . .23 + . mb. 45 4. 3. mb. 2. I 1. 1.0 2.0 Figure 10. Cross-sections for K*A Production. Open symbols refer to this experiment, solid to reference 11. 46 section from figure 6. Again we see that the two pion production is related to the production of the K*b state. Throughout the momentum range studied, K*~ production accounts for about 50% . of the KN7777 final state. F. K+p Total Cross-section We began our study of the K+p reactions to investigate the structure observed in the total crosssections. It is interesting, then, to see how the features of the total cross-section are related to the cross-sections for single pion and two pion procuction. In figure 11 we show one e again the KN7T and KN7T7T cross-sections. We also include the elastic cross-section from figure 1. The sum of the three curves is shown at the top, and is compared with the measured total cross-section data. The curve repro- duces the measured cross-sections very well, and seems to account for the features observed earlier. The first bump in the total cross-section appears to be associated with the rapid rise of the KN~ crosssection. This has been studied in more detail by Bland et a1.(5) who found no evidence for a resonance in this momentum range. They conclude that the structure is due entirely to the onset of the 47 19 15 10 5 1.0 Figure 11. ~P Partial Cross-sections. 2.0 48 inelastic channels. The second bump in the total cross-section appears just above the rapid rise of the KN7T7T crosssection. It is tempting to conclude that this feature is also related to inelastic thresholds. This possi- bility will be considered in the remainder of this paper. 49 VII. A. PRODUCTION AND DECAY ANGULAR DISTRIBUTIONS General Discussion The production and decay of resonances in quasi-two body reactions can provide very useful information about the reaction mechanism for the process. In the case of the reactions studied in this experiment, considerable information has already been accumulated at higher energies, showing the dominance of peripheral mechanisms. The data of Bland on single pion production shows the dominance of exchange processes at momenta down to threshold. The dominance of peripheral mechanisms is not, however, inconsistent with the existance of direct channel resonances. The two descriptions are, in fact, complementary, and are related in an average way by finite energy sum rules. The presence of a Z*, then, would give some local variation about the ave rage contribution from the perpheral mechanism. Such an effect is already known to be small and is very difficult to analyse quantitatively. 50 The data on reactions where resonances are known to be important can serve as guidelines in our discussion of the K+p scattering. For example the reaction 11+p -"'> -rr o A++ is analogous to our reaction K+p-;> Ko A++ This reaction has been studied over approximately the same momentum range as this experiment(lZ). The differential cross-section shows a very striking effect from the dominant F3 7 (1950) partial wave mt momenta about 1.5 GeV/c. A Legendre polynomial fit has been made to the angular distribution and shows that the coefficient A6 becomes large and negative in this range. The odd coefficients As and A7 change sign at about 1.5 GeV/c, indicating a rapid phase change in the dominant amplitude. This resonance is known from elastic phase shift analysis. It has a rather large elasticity (.4), and the branching fraction into TT~ is about 50%. In addition the high spin enhances the effect both because of the factor (J+t) and because the contribution to the Legendre expansion is in the .51 highest order terms. The effect of higher partial waves appears to be negligible at these energies. It is much more difficult to see the effect of resonances in lower partial waves. Since any partial wave (angular momentum L) contributes to the Legendre coefficients of order 2L and less, a good knowledge of the highest partial waves is required to separate the effect of any lower wave. This information has come from the phase shift analysis of the elastic scattering data(1J). The Kp system in this momentum range is very similar kinematically to the the same partial waves. 7T p case, and allows The best candidate for a Z*, however, is the P13 partial wave. These facts make it impossible to answer the question of the existance of the Z*. The knowledge of the inelastic differential cross-sections can, however, provide additional constraints for the partial wave analysis of elastic differential cross-sections and polarizations, and should help reduce the number of ambiguous solutions. In the next sections, we will consider in turn each of the quasi-two body channels. 52 We will use the normalized Legendre expansion coefficients to describe the differential crosssection at each momentum. This, combined with the data of Bland(5), will provide a complete, and model independent description of the inelastic channels from threshold to 2.2 GeV/c. All suggested resonance phenomena are within this range. The remainder of the discussion will be in terms of the exchange model description. Using the spin density matrix elements we can compare our data with data at higher momenta, and can study in a qualitative way the mechanisms in each of the quasi-two body channels. B. K0 A+TDifferential Cross-section The qualitative .features of the production angular distribution are shown in figure 12 where -6cMis the angle of the A measured with respect : to the incident proton in the production center of mass. As expected, the reaction is strongly peaked in the forward direction. This peaking becomes more pronounced as the energy increases. A quantitative measure of these effects is shown by the energy dependence of the normalized 53 100 100 Cll ~5 <l> 50 > <l> 1 0 cos Bc11 b. 1 100 m ~50 <l> 50 > <l> _J . ,· . -:r ! COS -0-CM c. -1 0 cos -0(;'1 1 d. Figure 12 • ~ Production Angular Distributions at four incident momenta: a! . 2•17 GeV/c , b. 1.94 GeV/c , c. 1.67 GeV/c , d. 1.37 GeV/c · 54 Legendre coefficients. W (cos-& ) = 1 +L A1 P1 (cos~,..,) CM I These are shown in table 11 for the momenta of this experiment. The behavior of the first four coeffici:e nts from threshold to 2.17 GeV/c is shown in figure 13. The region up to 1.58 GeV/c has been studied by Bland et al( 5) and the A production and decay were found to be in good agreement with the predictions of the rho-photon analogy of Stodolsky and Sakurai< 14). In particular, the production distribution shows a sin 2 -e-&H dependence corresponding to A2 = -1. At higher momenta, the reaction becomes more peripheral as other partial waves become important. The data from this experiment is in good agreement with that of Bland over the range of overlap, and show a smooth variation with incident momentum. It has been pointed out by Bland that the simple rho-photon analogy fails to explain the magnitude or energy dependence of the differential TABLE 11 LEGENDRE COEFFICIENTS FOR K° b. ++ PRODUCTION momentum GeV/c 1.37 1.52 1.67 1.81 1.94 2.07 2.17 A, .97±.06 1.19:!.06 1. 40 t. 07 1.52±.07 1.72t.07 1.77t.08 2.03±.07 A2 -.22 .10 • 22 • 09 .51 .11 .83 .12 1.19 .11 A3 -.37 .12 -.19 .11 -.07 .13 .06 .15 .17 .14 A.., -.30 .13 -.46 .12 -057 .14 -.36 .16 -.76 .16 .05 .21 -.14 .21 As -.09 .15 -.59 .14 -.77 .17 -.18 .17 -1.09 .17 -.22 .24 -.74 .23 A, -.17 .17 -.23 .15 -.39 .19 -.10 .19 -1.35 .19 -. 52 • 25 -.76 .25 A7 -.16 .18 .o4 .16 -.27 .19 -.24 .20 -1.00 .21 -.65 .29 -.56 .27 Ai -.11 .19 -.13 .17 -.29 .20 -.44 .22 -.35 .24 -,66 .29 -4.o .29 Aq -.19 .20 .o4 .19 -.03 .21 -.55 .23 -.11 .25 -.48 .30 -.05 .30 A,o • 02 • 21 .15 .19 .53 .24 .47 .25 • 01 • 27 -.29 .32 -.02 .33 1. 44 .15 1.97 .12 I 76 e 19 1.03 .18 \J\ \J\ 56 .l 2. A1 l /. 'I f l II I f o. fI f £ i '·' /.'}. .8 E---z'1 2.0 I G~Vlc: I /. A2. f o. II I II f I1 I I ft /. A '3 f o. I I f If If I A~ o. I _, I If 1d I I I Figure 13. Le~endre Ex~ansion Coefficients for the Reaction K p _.., KO L\ +. 57 cross-section. The data requires very large coupling constants, and the reaction becomes more peripherial faster than predicted from the simple (° exchange hypothesis. The data at higher momenta(15) show a sharp forward peak, explain~ d by absorption of low partial waves, or by a Regge model. The differential cross-sections shown in figure 14 show a smooth variation with energy. All momenta show approximately an exponential dependence with an increasing slope. This behavior is confirmed by measurements at higher momenta. The statistics of the experiment are not sufficient to permit a meaningful comparison with possible exchange models. In the next section, however, we will consider the relation of the exchange mechanism to the decay angular distribution. C. A Decay Angular Distribution We describe the decay of the A in the Gottfreid- Jackson coordinate frame. The characteristics, shown in figure 15, appear to be independent of momentum. 58 ILJ\ -I a. o cos. 0 b. ~::~J~ 1V\ _ J. - - - - . ..- _, 0 I GOS~ I I 0 180 f1 c• 0 cose 0 e. -I 0 0 cos~ d. 180 Ji ·· ~ . _, I 3~0 lfO ¢ Figure 15. Decav Anf1lar Distributions for the Reaction ~P -+K 0 ~ + at five momenta: a. 2.17 GeV/c, b •. · · 2.07 GeV/c, c. -1..94 GeV/c, d. 1.81 GeV/c, e. 1 • .3 7 Ge v/c • · 60 To analyse the momentum dependence in detail, we parameterize the distribution by the Epin J/2 density matrix elements. The angular distribution then is given by W(cos-&,9f) = 3 [/:J3 sin + fJ,, ( 1/3 + cos -&) 2 I "[f;i I ' -.n- {i-i Re 2 sin -& cos2 ¢ -1.r Re (?ii sin2e cos¢ ) where A,= t - f" The values of the density matrix elements have been evaluated from the moments of the angular distribution. The results are shown in table 12 and are compared, in figure 16, with data at other momenta ( lO). As mentioned previously, the rho-photon analogy of Stodolsky-S kurai(14) provides a simple description of the N~A coupling. the f They suggest that exchange reactions 77+p -'> 7To .A++ K+p-} K0 A++ have the same features as A photoproduction which TABLE 12 1:1 ++ DENSITY MATRIX ELEMENTS momentum f>:n Re {°3-• 1.37 .379±.027 .230±.031 .059±.025 1.52 .365 .028 .127 .030 .074 .026 1.67 • 278 • 032 .122 .031 .015 .033 1.81 .329 .032 .245 .032 • 016 • 027 1.94 .232 .037 .208 .033 .069 .031 2.07 .319 .035 .279 .034 -.033 .031 2.17 .314 .036 .179 .037 .oo4 .034 Re. f31 °' p 62 I. -I- - --------t /. Re fs-1 - - - I- - I .ir i I i l JI ! ii - - - - I - - - - - - - -I o.........-------------------------------------------1. :2. 3. ~. s. I. t- \<_e \31 ~ o.~~--~I~·-=-I~I~ir,,_.,r~1t-=-l~11H~'F-------,;I1r-------~----------?1/. 2. 3. 'I. Gev,, Figure 16. b. ++ Density Matrix Elements. The dotted line corresponds to the Ml prediction of Stodolsky-Sakurai discussed in the text. 5. 63 is known to be dominated by an Ml transition. The rho-photon analogy then predicts a decay angular distribution W(cose-, ¢) = 1 g1T [ 2 + 3 sin~- 3 sin2.e-cos2¢} In terms of the density matrix elements, the Stodolsky-Sakurai predictions are .375 Re 13.,= Re f3• = o. • 218 These values are shown in figure 16 and appear to be in good agreement with experiment over a wide range of incident momentum. This result is actually more general than simple exchange. In contrast to the case of 11° .L:~/+ production, which allows only(° exchange, · the K° L::. ++ state can also be formed by exchange of the Az. An analysis(16) of the complementary reaction TT +p4 "]b.++ which allows only A2 exchange, also appears to be 64 consistent with the predictions of Ml dominance. The data at higher momenta(15) do show a deviation from the simple Ml predictions for ftl~ .1 (GeV/c) 2 • We have found no consistent variation of the density matrix elements with momentum transfer. However, because of the kinematic cut off in t, we cannot determine values for It I '- • 1 • D. K~p Differential Cross-section The production angular distributions for the reaction show the same qualitative features as previously described for the K0 A++ production. The coeffi- cients of the Legendre expansion are given in table 13. In figure 17 we summarize the energy dependence of the production angular distribution from threshold to 2.17 GeV/c. The data confirm the smooth behavior and the increasingly peripheral nature of the reaction. The differential cross-sections shown in figure 18 suggest an approximately exponential .- TABLE .13 LEGENDRE COEFFICIENTS FOR K*+p PRODUCTION momentum GeV/c 1.37 1.52 1.67 1.81 1.94 2.07 2.17 A, .89±.09 1.16±.07 1.10±.08 1.42±.08 · 1.64±.08 1.49±.10 1.82±.08 A2 .16 .12 .48 .11 • 50 .12 .89 .14 1.27 .14 1.01 .15 1.63 .15 A3 .o4 .15 .12 .13 .17 .15 • 35 .17 .66 .17 .49 .19 • 85 • 20 A'i -.30 .16 -.24 .15 -.07 .17 • 01 .18 .08 .20 -.14 .24 • 24 • 24 As -.14 .18 -.36 .16 .o4 .19 -.16 .10 -.15 .22 -.46 .26 -.26 .26 A, .23 .20 -.30 .18 .15 • 20 -.17 .22 -.32 .25 -.17 .28 -.26 .28 A1 • 05 • 21 .10 .19 • 22 • 22 -. 21 • 24 -.18 • 27 • 22 • 30 .12 • 31 Ag -.08 .23 • 23 • 21 .19 .22 -.34 .26 • 00 • 28 .39 .32 -.02 .33 Aa, -.10 • 24 • 01 • 22 -.18 • 24 -.34 .27 .03 .30 • 81 • 34 .03 .35 AIO -.22 .26 -.09 .23 -. 61 • 28 .15 .32 .65 .34 -.15 .38 °' \J\ 0 • 26 66 2,. A, . 1I I I /. II f 1I 1 I I I 0. Al /. rf I f o. f I f f I A3 /. o. A'1 /. '- 0. f I- I I Ht{ ! rI 1 I I I Figure 17. Le~endre Expansion Coefficients for the Reaction K p -;. K*+p. / 67 /. .5 /.52. ,3 ·' ,+- -i--+ - '·r .' +-t-+....±_ ':_~[+ :·37 G•v,, . f-t~,t . I 0 .Z. .+ .'1 .8 -t I. 0 1.2 (GeV1c) l.'4 I u. 2 Fi gure 18. Differential Cross-section for K+p~K*+p The curves are to guide the eye . 68 dependence on momentum transfer. The slope i ncreases with energy, in agreement with observations at higher momentum( 15). The details of this reaction are more complicated than the Ko~++ production since both natural and unnatural parity exchanges are allowed by angular momentum-parity· conservation. The analysis of data at higher momentum(15) has shown the dominant contribution to be from exchange. TT and w The statistics of this experiment are not sufficiently good to permit a detailed study of predictions from exchange models·. In the next section, however, we will study the relative contribution of the exchanges over the momenta of this experiment. E. K*+ Decay Angular Distribution The general features of the K* decay (figure 19) are also very similar to the A decay. We again measure the angles in the Gottfreid-Jackson frame, defined now by the incoming ~. and we parameterize the angular distribution in terms of the spin 1 density matrix elements. 69 'U> 2.0 -,_.,·; ___ _ ........_ _, _, ' 0 0 cos 6 - 1n-· JI'"'I _, •cs l~O 0 Cos~ ? a. b. J~R _, _, 0 cos.<; Cos B ' c. d. I{() luJ~ .....__.,..,.:....-_ j 'l -I o cos -6' 0 180 p """ e. Figure 19. Decay Angular Distributions for the Reaction K+p -> K*+p at five momenta: a. 2.17 GeV/c, b. 2.07 GeV/c, c. 1.94 GeV/c, d. 1.81 GeV/c, e. 1.37 GeV/c. 70 2 W( cos&, it) = 4-rr [f 00cos -e- + f" sin 1'" 1 -r-, sin~-o-cos2¢ - fi Re . fo sin2'9' cos p ] with fii = 12(1-:-fJ' The results are given in table 14 for the momenta of the experiment. In figure 20 we show the values of these density matrix elements from threshold to 5 GeV/c. There seems to be very little dependence on momentum over this range. The two exchange mechanisms, TT and w , mentionned previously contribute to the density matrix elements fog and fu respectively. The relative exchange contributions appear to be independent of momentum, in contrast to s i mple absorption model or Regge predictions. We would expect to see changes particularly at small momentum transfer. However, because of the kinematic cut-off in t our data sample is not sufficient to determine the density matrix elements for low t values. F. K* fj,. Differential Cross-section The qualitative features of the angular distribution are very similar to those of the TABLE 14 K* DENSITY MATRIX ELEMENTS momentum f°oo 18·-· 1.37 .163±.039 .169±. 044 R~ f10 -.115±.026 1.52 . • 201 • 040 .228 .038 .045 .021 1.67 .141 .042 .230 .o4o -.085 .022 1.81 • 231 • 043 .176 .041 -.051 .027 1.94 .286 .048 .278 .041 -.079 .028 2.07 .203 .047 .215 .042 -.089 .027 2.17 .264 .054 .244 .047 -.075 .027 --.J ..... 72 /. foo ·5 1f1u 1 Uif1 I I r 0·'--~---1._;;_~~~---L~~~~~£""-~~~--'1--~--~---'I. o 2 .o '3. O ~- o S. o Ge\'l, /. fi-1 .5 I If I! i i o. /. 1r !i f ;(. f 3. \ - .S' Figure 20. l I K* Density Matrix Elements. I.{ . 5'. 7J other channels. This is indicated in the coeffi- cients of the Legendre expansion shown in table 15, and in figure 23. The data again show a very smooth variation, becoming more peripheral at higher momentum. The reaction has been studied in detail at higher momenta and found in reasonable agreement with models based on 7T exchange but none of the models successfully accounts f or all the features of the data< 1 7). The differential cross-sections shown in figure 22 provide new information on the threshold region for this reaction and is qualitative agreement with expectations from data at higher momenta. We have not made a quantitative comparison with theories but will in the next section discuss the TT exchange interpretation in connection with the K* and ~ decay. G. K* and b.. Decay Angular Distributions The decay cosine and the Treiman-Yang angle are shown in figure 23. In contrast to the reactions just studied, there is a very flat distribution in the Treiman-Yang angle¢, consistent TABLE 15 LEGENDRE COEFFICIENTS FOR K* 0 .6 ++ PRODUCTION momentum GeV/c 1.52 1.67 1.81 1.94 2.07 2.17 .68t.53 .68±,15 1.12:t:.08 1. 24:!:, 07 1. 51.±. 07 1. 67±. 06 A2 .93 .66 • 60 • 20 • 77 .12 • 98 • 11 1.42 .11 1.79 .10 A3 -.04 .87 • 30 • 24 .55 .15 .62 .14 1.00 .15 1.23 .13 A" .08 .95 .54 .28 • 23 .18 .49 .16 .49 .17 • 82 .16 A5 -1.18 1.06 • 29 • 31 • 27 • 20 • 35 .18 • 32 .19 .69 .18 A<, .66 1.03 -.09 .34 .39 .22 .34 ,20 • 25 • 21 .40 .20 A, .72 ,89 • 27 • 36 .40 .23 • 20 • 22 • 24 • 22 • 28 As -.14 ,97 .48 .40 ,54 .24 • 27 • 23 .24 .23 .40 .23 A"' -1.31 1.24 -.05 .43 .40 • 26 • 05 • 24 .39 .25 .19 .24 • 56 • 27 -.08 .24 • 23 • 26 • 21 • 24 A, A,o . .80 1.40 .o4 .44 .22 --,J +:- 75 2. ! A, I I I. O· I f f 1.2 ~.o / . (, i 2.'i i Az /. f o. I I A'3 II J. fI I 0. A., /. fII I 0. /.2 / .(, 2.0 I 2.'i Figure 21. Legendre Expansion Coefficients for the Reaction K*p -'> K*o b. ++. 76 I + /. 0 .5 .3 dcs dt 1. 0 .5 .3 rnb. (Gcv,,}· 1.0 .5 .3 1.0 .5 .3 0 .2 .4 .6 .8 1.0 l.4 -t Figure 22. Differential Cross-sections for K+p ~ K*~ The curves are to aid the eye. 77 n I 'fo Ju~ 2D _, . --µ 0 0 Ll+r cos..e- _, ~.~ ui, 0 190 ¢ I 'i• p co.so , oa. _, ~o be -1 6 COS-& •' 0 I cos e- K 18'0 -1/0CJ <-(7/ 0 ,, 3~ ~ o I p' . ·w--S--I -/ o A C<JS~ 0 cos~ I • b. o u 180 ¢ 0 cos -& I K JI () ISO ii 0 (/ Figure 23. Decay An.e:ular Distributions for the Reaction K+p -i> K* 0 .6 T+ at four momenta: a. 2.17 GeV /c, b. 2.07 GeV/c, c. 1.94 GeV/c, d. 1.81 GeV/c. 78 with the dominance of .7 / exchange. This is in agreement with the data at higher momentum(l?). We have determined the density matrix elements for both the K* and .6 (table 16) and compare the results with data at higher momenta in figure 16. The data appear to be rather independent of momentum up to at least 3 GeV/c. The predic t ions of simple pion exchange can be obtained by ordinary addition of angular momentum. This gives K*i ?oo = 1. As fi-1 = o. Ref = o. 0 Refi.,= O. The density matrix elements, averaged over momentum transfer, do show a consistent deviation from these predictions. Data at higher momenta show that the predictions are well satisfied for small momentum transfer (/ti~ .1(GeV/c)2 ) but our data are not sufficient to make any conclusion on this. TABLE 16 8 AND K* DENSITY MATRIX ELEMENTS Re f'3-1 Re F~1 ('oo , p,_, Re f'o .129±.043 -.025±.042 .047±.049 .599±.067 .013:t.044 -.030±.037 1.81 .127 .029 -.034 .026 -.012 .029 .659 .040 - .047 .028 -.080 .025 1.94 .158 .027 .014 .024 -.027 .026 .616 .037 .010 .026 -.086 .022 2.07 .139 .028 .037 .025 -.031 .027 .604 .037 .018 .027 -.066 .022 2.17 .099 .025 -.018 .021 -.064 .025 .619 .032 -.030 .024 -.048 .020 momentum (J-03 1.67 • -..J '° 80 /. I. i f i: U1 f33 o. ffEEi 1 I l 1} l I I foo I x s.o 2.0 (). 2.o 3,0 G~ "ic /. /. /. I. Re~, Re~o 0 Figure 24. r l I Is I a. I l JU r 1 .Lland K* Density Matrix Elements. l I 81 H. Quark Model Predictions Bialas and Zale~ski(lB) have derived a set of relations between single particle decay angular distributions and joint decay distributions for the reaction The observable predictions given for decay distributions are presented as predictions about ·tensors formed from .the moments of the angular distributions. = /2 . i +1 F ( K* ) <Y,..,(K*) > T:~ (K*, 6) = 11 1 1 ,./2.1 +1 F(6) 2.2 2 <Y;( L~d) ' T M Ill ( K* , 6 ) = F (K*) = -M F(6)=-/f.: M where Y2 (K* or.6) =spherical harmonic evaluated at K* and 1 /:::;. decay angle < ) means an average over the decay angular 82 distribution. To describe the quark scattering process, there are four independent spin nonflip and four spin flip amplitudes. The particle- particle scattering amplitude is the coherent sum of the constituent quark-quark scattering amplitudes. The tensors can be written in terms of these amplitudes and if the quantization axis is taken as the production normal, we get the following relations. I. oz 20 T00 (K*, A ) = f2' Too ( K*, A ) II. T::(K* ,A ) = .6. Tzo(K*,A ) 2 III. 2.2 1 0:/. Toz.(K*, A ) ={7£ ToiK*, A ·) IV. 20 T:;(K*, A) 1 1 = 2/6'-n T:(K*,A) Since II and III are in general complex, there are six equations. We have combined all data at four momenta 1,81, 1.94, 2.07 and 2.17 GeV/c. The results are shown in table 17 and indicate rather good agreement. Figure 27 shows a summary of the test of these relations up to 5 GeV/c. The values of the tensors are rather TABLE 17 TEST OF QUARK SCATTERING RELATIONS EVALUATED IN THE TRANSVERSE HELICITY FRAME equation left side right side difference I .104±.013 .105t.027 -.001:t.030 Re II -.012 .023 .006 .006 -.018 .023 Im II -.040 .022 -.034 .005 -.006 .023 co \.....) Re III .003 .021 .020 .010 -.017 .023 Im III -.062 .021 -.057 .010 -.005 .024 IV .177 .029 .100 .018 .077 .034 84 .I Re II I .2 fI ti !I f 0 .05 ! 0 l ! Re III Im II .OS -.os IV Im III •I -.os a. b. c. a. b. c. Figure 25. Test of Quark Scattering Relations for three momenta. a) 1.8 - 2.2 GeV/c, this expt. b ) 2. 5 - 3. 2 Ge V/ c , ref. 17. c ) 5. 0 Ge V/ c , ref. 17 • The figure shows the difference 1 , left side I , and right side I . 85 independent of momentum and in all cases, the equalities are well satisfied. 86 VIII. CONCLUSIONS We have studied single pion production and two pion production in K+p reactions. We have studied in detail the two reactions K+p -> K0 p rr+ -~ K+p rr+rr - The cross-section for the K0 p77"+ reaction shows a smooth fall off with increasing momentum, as had been suggested previously by Bland(5). This final state shows strong production of the quasi-two body states KA and K*N as at lower momenta. There is no evidence of significant interference between these two channels. The pK+ TT+ TT - cross-section rises rapidly above 1.7 GeV/c, the threshold for K* A production. About half the events at each momentum correspond to this quasi-two channel. This rapid rise in the two pion production cross-section appears to contribute to the second bump noted in the total cross-section. We have studied the production angular distribution and decay properties of the quasi-two 67 body channels. All the observed features are in agreement with data at higher and lower momenta, indicating the dominance of peripheral mechanisms. It is impossible from the inelastic reactions alone, to place limits on the contribution of direct channel mechanisms. Our data on quasi-two body production, however, does provide additional information on the partial wave structure of the K+p interaction, and should be useful in resolving some of the aJlbiguities in phase shift analyses of elastic scattering and polarization data. 88 APPENDIX EXTRACTION OF PARAMETERS FOR QUASI-TWO BODY STATES A. Three Particle Final States We wish to consider the states K0.6++ and K*p which decay into the final state pK 0 ~+. The parameters of interest include the production cross-sections, possible interference between the K* and~ } differential cross-section for resonance production, and the resonance decay angular distribution. A three body state is completely specified by five variables. We will always take one of these to be the invariant mass of the system, or equivalently the beam momentum. The choice for the other parameters cannot be made so easily, and in the following discussion we will consider a number of choices each revealing some different aspects of the reactions. B. Dalitz Plots One very useful choice of parameters is the invariant mass of the two body systems. In the reaction 1+2~.3+4+5 we can choose for example 89 M34 = (p3 + P4) 2 2 2 M45 = (p4 + P5) The third choice is not independent 2 2 2 2 2 M35 = S + M3 + M4 + M5 - M34 S = (p1 + pz) 2 2 M45 = constant The remaining two variables could be chosen as the polar angles of the plane defined by the three out going particles. For the purposes of this discussion, however, we will average over these angles and consider a plot of M~ 5 . versus M34, suggested first by Dalitz(l9). Figure A.1 shows a number of general features of the Dalitz plot. The boundary corresponds to the limits imposed by energy and momentum conservation. Within this boundary, events should be uniformly distributed if there is no dependence of the reaction amplitudes on the masses of the out going systems. Resonances in the out going systems show up as heavily populated bands in the plot, for the~ and K*. a~ shown in the figure In the region I, where the b. and K* bands cross, we can have events which go through 90 ~ ..., "'\S E· 2.fi " II .) ~ v ~ \..::;:; B (I.I . r. ~ 2 .0 M2..'i5 A' (GeV)2. ~ 1-5 A /o-+-~~~~~-..--..-~~~~~~-r~~~~~~~.,-~- .2 .6 ). 0 Figure A: .1. General Features of Dalitz Plots The s i gnificance of the labled regions is discussed in the text. 91 either resonance system, and we must also consider the interference of these two channels. c. Mass Conjugation To study the A and K* channels, and the possible interference, we will use the mass conjugation technique suggested by Eberhard and Pripstein( 20) • . For any effective mass squared for particles 3 and 4, there is a range of effective mass squared for the system of particles 4 and 5. This range is given by 2 -- A + B cose-4 M45 where cos e 4 is the decay cosine of particle 4 measured with respect to the direction of the system of particles 3 and 4 in the production center of mass. A and B are constants which depend only on the total energy and the effective mass M~~- The Dalitz plot limits correspond to cos -04 = :t 1 or to M;5 = A :!:' B, and the line AA' corresponds to cos-f>-4 = o. The region II shown in figure I.1 is related to the region I simply by changing the sign of cos e- 4, or by the interchange of particles 3 and 4. For any resonance of specific parity then, the density of points in I and in II should be equal, except for the presence of the resonance in the other two particle 92 system. In order to get a sample of events in the M j4 band, we replace the events in region I by pseudo . events from region II with the two particles 3 and 4 interchanged. Similarly we can generate pseudo events from the M~ 5 band by taking events from region III and interchanging particles 4 and 5. This procedure should give us a decay angular distribution for one resonant state without the effect of the other resonance. To check the validity of the reflection procedure, we can compare the reflections of regions where the other resonance does not contribute. This technique provides a way to extract resonance decay parameters, without knowing anything about the reaction in the interference region. It also provides a method for testing whether there is any interference between the two reaction channels. D. Cross-sections and Interference The mass conjugation technique provides a way to determine the production cross-sections for quasi-two body channels in the reaction. We can simply add up all events in a resonance band, using events from the conjugate region to fill in the band where the other resonance, or interference could 9J be a problem. We then subtract the uniform phase . space background by interpolating between the two regions outside the resonance band. This has the advantage that it requires no assumption about the exact form of the resonant amplitude, or about the interference of the resonance amplitudes in the region where the bands cross in the Dalitz plot. We can also use the technique to determine whether there is an interference between the two amplitude. Region II and region III each contain events from one resonance, plus events from the phase space distribution, whereas region I contains events for both resonances, phase space and interference. The number of events for the phase space distribution events Nps can be determined from the region where neither resonance can contribute, so that we can easily determine the number of events due to interference effects. NINT = Nr + Nps - (NrI + NIIr) We can then calculate an interference crosssection and determine whether the interference is constructive or destructive. E. Four Body Final States The four body system can be specified by eight para.meters. We will always choose one of these to be the total effective mass. In many cases we will choose specific variables of interest and average over all others. We will consider effective mass distributions as we did for the case of the three body system. For the reaction 1+2--.lJ>J+4+5+6 there are many possible choices of effective masses. Since we will be interested in two body resonances, however, we will consider a plot of Ms6 versus M34. We choose here to plot the effective mass rather than effective mass squared, since it makes the calculation of kinematic boundaries slightly simpler. The range of M34 is given by where Ecm = total energy in center of mass. For fixed M34• the range of MS6 is given by 95 The boundaries of the plot then form a right triangle, and the effective mass plot will be referred to as a triangle plot. For the Dalitz plot, we had the result that a constant reaction amplitude gives a uniform density in the plot. There is an analagous result for the triangle plot. N(M34, M56) d.M34 d.M56 = Pcm P34 P56 dM34 d.M56 Ecm where Pcm = momentum of M34 or M56 in the center of mass P34 = momentum of M3 or M4 in the M34 system P56 = momentum of M5 or M6 in· the M56 system This given a distribution which varies smoothly over the plot so that resonances in the two particle systems will show up as heavily populated bands. It is much easier to study the quasi-two channels here, since there is no interference to worry about We can consider all events in the region of the triangle plot where M.34 and Ms6 are within the region of resonance mass for the K* and L:. • The effect of events from the phase space distribution can be removed by interpolation, as was done in the three body case. F. Resonance Production Angles A fUrther specification of the J particle or 4 particle system could include the production angle of a 2 particle resonance in the center of mass. This choice is independent of the invariant mass specification considered in the previous section. For the description of the quasi-two body reactions in this paper we will always define the production angle with respect to the incoming particle which forms the resonance. For A production, we measure with respect to the proton, for K* production, with respect to the irr. The choice of production angle is equivalent to specifying the state by the four momentum transfer squared. t = (Pin - Pres) 2 In the description of the differential crosssections for quasi-two body reactions, , both of these descriptions are considered. G. Resonance Decay Angles There are two co-ordinate systems in common use to analyse the decay angular distributions of 97 resonant states. The Gottfreid-Jackson system or t-channel frame is shown figure A.2. Here we choose the positive z axis along the direction of the incoming proton or K in the ~ or K* center of mass. The y axis is chosen as the perpendicular to the production plane, and the x axis is defined by the right hand rule. The angl.e s of interest are the polar angles (-&,¢) in this co-ordinate system, commonly called the Jackson angle and Treiman Yang angle respectively. The other system commonly used is the Helicity frame shown in figure I.2. The z axis is chosen as the direction of motion of the resonance in the production center or mass. The y axis is again the production normal, and the x axis is defined as before. These two co-ordinate . systems are parallel for zero momentum transfer, but are rotated about the y axis as the momentum transfer increases. In the case of the three body final state we have already noted that the decay angle in the Helicity frame is directly related to the effective mass of the other two particle system in the Dalitz plot. In the Jackson frame the relation affects both the ()IA p (K) ', el a. -~ Tr b. Figure A.2. Decay Co-ordinate Systems a. Gottfreid-Jackson system for decay of the 6. , (K* ) ; b. Helici ty Frame. r cos-e- and¢ distributions. To eliminate this bias in all angular distributions we use the mass conjugation technique already described. 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