T H E U N I V E R S I T Y OF M I C H I G A N COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Physics Technical Report INVESTIGATION OF DI-PION RESONANCES IN 3~,7 BeV/c x-p COLLISIONS Yong Yung Lee ORA' Procject.0Q4938 under contract with: Uo So ATOMIC ENERGY COMMISSION CHICAGO OPERATIONS OFFICE CONT.RA.CT NOo AT(11-1)-1112 ARGONNE, ILLINOIS administered through: OFFICE OF RESEARCH ADMINISTRATIO AN ANNARBOR May 1964

TA:BLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vi ABSTRACT vi i i CHAPTER Io INTRODUCTION 1 II. PARTICLES, RESONANCES AND THEIR INTERACTIONS 4 III o EXPERIMENTAL EQUIPMENT AND BEAM SETUP 9 3.1 Bubble Chamber 9 3.2 Beam Setup 10 303 Scanning Machines 14 3.4 Measuring Device 15 IVo DATA REDUCTION 17 4.1 Scanning of the Events 17 4.2 Measurement 18 4.3 Spatial Reconstruction and Kinematics 19 4.4 Event Identification 24 V. RESULTS 31 5.1 Introduction 31 5.2 General Properties of Resonances 31 5.2.1 Distribution of two-particle mass spectrum 31 5.2.2 The parameters of resonances 34 5~3 Interpretation of p Peak as Pion-Pion Scattering Resonance 35 5.3.1 Production mechanism 35 5.3.2 Decay angular distribution of p 43 5.3.3 Possible explanation of asymmetry in po 44 5.3.4 Extrapolation into the A2 = -1 pole 53 5.935 Discussion of l:p-' peak 58 5.4 Dalitz Plots and Isobar Formation 60 5.5 Study of f~-Peak and Determination of f~-Spin 63 5.6 Partial Cross Sections 68 iii

TABLE OF CONTENTS (Concluded) APPENDIX Page A. Dalitz Plot and Lorentz Invariant Phase Space 70 Bo Production and Decay Ratio of Nucleon Isobar (1238,3/2, 3/2) 74 Co Angular Distribution of p- Decay Assuming wo Exchange 77 REFERENCES 79 iv

LIST OF TABLES Table Page I. The Dimensions of Slits Used for the Beam 13 II. Statistics on Event Identification 27 III. The Resonances and Their Widths 35 IV. List of Fitted Coefficients and X2 Probability for Different Di-Pion Mass 49 V. List of Fitted Coefficients and X2 Probability for Different a2 55 VI. Types of Events and Cross Sections for 3.7 BeV/c Ec p Interactions 69

LIST OF FIGURES Figure Page 1. Beam layout at the AGS building~ 11 2. Schematic diagram for beam layouto 12 3~ X2 distribution for unambiguous events~ 23 4. Relation of bubble density and particle momentum ~ 26 5~ X2 distribution for ambiguous events between the two one-constraint fits~ 28 6~ A2 distribution for ambiguous events. 29 7~ A2 distribution for unambiguous events. 30 8. Effective mass distribution for pion-nucleon system~ 32 9o Effective mass distribution for pion-pion system ~ 33 10o Effective mass distribution for pion-pion system for different A2 regionso 36 11o Chew-Low plot of 4-momentum transfer~ 39 12. Four-momentum transfer distribution for p s and f0o 40 13 One-particle exchange diagrams for p production 41 14. Treiman-Yang angle distribution for p s and f 0 o 42 15 Space diagram of pion vertex in rest frame of final di-pion system~ 43 16o Distribution for cos 0 inside the p-meson region~ 45 17 Distribution of cos 0 for different n:A mass and for A2 < 2o2 5o0 18- Coefficients A, B, and C with B/2,JXF and possible phaseshift diagram, 51 vi

LIST OF FIGURES (Concluded) Figure Page 19. Distribution for cos 0 for different A2 -for the events inside the p~-regiono 54 20. Extrapolation of partial wave amplitude to A2 = -1. 57 21. Distribution of cos 0 for the events inside the p —region~ 59 22~ Dalitz plots for reactions (2) and (3). 61 23. Distribution of T+ and 3~ kinetic energy for events inside the N*+-regiono 64 24. Distribution of cos 0 for events inside the f~ mass, 66 25. Momentum vectors for 3-body final state in center-of-mass system. 71 26. One-meson exchange diagram for N* production. 74 27. One-particle exchange diagram in rest system of final di-pion. 77 vii

CHAPTER I INTRODUCTION Iu.ring the past few years, much attention has been given to resonances of elementary particles in high-energy physicso In fact many resonances including pion-nucleon, pion-hyperon, pion-kayon, and multipion resonances, have been found in this periodo1,2 Several pion-nucleon resonances, called nucleon isobars, have been found with different values of mass, spin and isotopic spin as well as many multipion resonances including pJ, n, and f. One can expect that more will be found in the futureo A great deal of study has been done on these resonances, yet there are many things still to be determined about them~ In this experiment, we took 60,000 pictures in the 20-ino hydrogen bubble chamber at the Brookhaven National Laboratory. Our beam was a 3.7 BeV/c separated E beam from the 33 BeV Alternating Gradient Synchrotron (AGS)o This report will deal with the reactions.- + p + i- + of + n twS + p + E + + 9 p These can be identified by two out-going charged-particle tracks~ A preliminary report on this experiment and a paper dealing with the fourprong events have been reported elsewhere 3>5 The complete report on the four-prong events and elastic scattering will be reported in separate

paperso The main things one can study with the above reactions are nucleon isobars with masses up to about 2.6 BeV and dipion resonances with masses up to 107 BeVo Since, -unfortunately, isobar formation in the reactions was very meager, very little was done with the isobars. The pion-pion resonances seen in the reactions are one centered at 765 MeV and another centered at 1250 MeV effective mass. They are p and fcO, respectively. The existence of the p was predicted by Frazer et alo,6 and soon thereafter several experiments showed evidence for a resonance in the T = 1 state (T is the isotopic spin) with a mass value of approximately 750 MeVo7-.9 Since then, many other experimenters have studied.i and found its quantum numbers to be of isotopic spin 1, spin 1, parity -, Gparity +. fo was first found by Selove et al, 10 and has since been studied by several other groupso3,1112 Most probably its quantum numbers are isotopic spin 0, spin 2, parity +, and G-parity +. This f -resonance might well fit into the pattern of Regge trajectoriesl4 l5 however its quantum numbers are not yet definite 16 We studied these two resonances by employing a pion-pion scattering model which was suggested by Chew and Low17 to analyze scattering on unstable targets, Recent experimental evidence12,18 shows that this method serves as a good model for analyzing pion-pion resonances. The fact that the target pion is virtual, however, makes the interpretation of such scattering somewhat indefinite~ Using this method an attempt was made to extrapolate the results into the nonphysical region in order to get

3 the real pion-pion scattering cross section.

CHAPTER II PARTICLES, RESONANCES AND THETR INTERACTIONS In the previous chapter we used the terms "particles" and "resonances" many timeso The detailed meaning of these words can be found in various articles~ The author thought, however, it is necessary to give a brief description of what is meant by these terms, and the quantities related to themo The word?"particle"' means an elementary particle, which is characteri.zed by being a subatomic particleO There are 30 elementary particles which have been known for some timeo These consist of 16 baryons and antibaryons, 7 mesons9 6 leptons, and the photon. The term "'baryon" refers to nucleons and to the group of particles which are heavier -than the neutron called hyperons. The term "lepton" refers to another group of particl:es which are lighter than the xrmeson and consists of muons, electrons) and massless neutrinos, along with their antiparticleso The family of "mesons" consists of the three charge states of the pions and four kinds of kayonso The kayons and hyperons are classified as strange particles. They are normally produced. only in strong interactions and decay only through weak interactionso The first so-calied "resonance" was discovered in the year 1952 'by Enrico Fermi and his colleagues. They found.t:hat the scattering cross section of pions on protons) measured as a function of energy, had a sharp peak for 12:38 millon electron volts (MeV) center-of-mass energy

5 in the pionproton system. Since then several other such pion-nucleon resonance peaks have been found. These peaks were found not only in pion-proton elastic scattering but also in the effective mass of the final state pion-nucleon pairs in which several other out-going particles were produced. The effective mass of a group of particles can be calculated by the equation (Effective mass) = (Ei ( Pi)2 where Ei and Pi are the total energy and the momentum vector of ith particle. It is a Lorentz-invariant quantity, which represents the total energy of the system in its center-of-momentum reference frame. In addition to the pion-nucleon resonances, many other kinds of resonances have been found, including pion-hyperon, pion-kayon, and multipion resonance s o There have been several attempts to classify this large number of resonances and particles into groups having similar properties. The currently most popular schemes are the so-called "eightfold way" and "Regge trajectory?" The eightfold way predicts that particles should fall into groups having one, eight, or ten members with the same spin and parityo The Regge trajectory hypothesis predicts that a given group should reoccur as another group with spin differing by two integer units. In this scheme there is no fundamental difference between a "particle" and a "resonance."' What is commonly referred to as a "particle" is simply a relatively stable (long-lived) member of the group and what is commonly

6 referred to as a "resonance:" is simply a relatively short-lived member. For a more complete discussion of these ideas the reader is referred to a recent article in Scientific American.25 These particles or resonances can be characterized by their mass and various quantum numbers. We discuss below some of these quantum numbers, since frequent reference will be made to them in this thesis. We first consider the spin angular momentum of particle, denoted by Jo A particle can have a spin of only an integral or half-integral multiple of PlanS2s constant i, which is the fundamental constant of quantum theory. The mesons have integer J-values while nucleons have half-integer valueso A second quantum number' in which we will be interested is the.isotopic spin T. This is not directly related to the spin angular momentum, although its mathematical behavior in quantum mechanics is very similar to that of the spin angular momentum. The idea o:f isotopic spin predicts that small surbgroups (isotopic multiplets) of particles should have approximately equal masses and that the behavior of the individual members of a multiplet under the strong interactions should not depend on their charge~ The number of members of a particle multiplet is given by 2T+l. We will also make use of the intrinsic parity of particles which can have values ~1:l (sometimes referred to as even or odd, or simply + or -). Thi s characterizes the behavior of the wave function of the particle under space inversion (i.eg, reversal of the directions of x, y, and z) The parity is conserved in strong and. electromagnetic

7 interactions but not in weak interactions. The meaning of the terms "strong," "electromagnetic," and "weak" interactions will be given later in this chapter~ The G-parity of the particles is only applicable to states in which both baryon number and strangeness is zero. This is conserved only in strong interactions. If there is more than one type of resonance between the same pair of particles, as in the case of the isobars, one usually gives the mass and the quantum numbers to identify them. For instance the isobar at 1238 MeV is denoted by 1238,3/2,3/2 or simply 3/2,3/2 which are its isotopic-spin and angular-momentum quantum numbers, respectively~ This isobar belongs to the same Regge trajectory as the 1720 MeV isobar (angular momentum 7/2) and in the eightfold way it belongs to a decuplet. We would like to devote the last part of this chapter to describing kinds of interactions between these particles. The strongest of these is called the "strong" interaction~ The range of the force is very short (about 10-13 cm), however, its strength is about 1039 times that of the gravitational forceo It is the force that holds nuclei together. The interactions involving strong forces conserve every quantum number mentioned above~ The next strongest interaction is due to the familiar "electromagnetic" force which. is about 100 times weaker than the strong interaction and has a much longer range. The "weak" interaction has strength of only 10-14 of the strong interaction, and is believed to have a very short range. The elementary particles can participate in all of these interactions.

The great differences in the strengths of these forces are responsible for the great differences observed in the lifetimes of particleso A particle which can decay through a strong interaction can be expected to have a lifetime of about 10-22 sec whereas if only electromagnetic decays are available, its lifetime will be several orders of magnitude longer. Weak interactions lead to lifetimes which are typically 106 to 10-1l sec. Particles with very short lifetimes will have experimentally detectable widths (ie.o, uncertainty in their mass) given by the uncertainty principle Am — h/T, where -i is Planck's constant (6.58 x 10-22 MeV-sec) and t is the lifetime in seconds. An experimental observation of this effect will be seen later in our data on the p- and fc-resonanceso

CHAPTER III EXPERIMENTAL EQUIPMENT AND BEAM SETUP 3 1 BUBBLE CHAMBER The particle detection apparatus used in this experiment is the Brookhaven 20 in. hydrogen bubble chamber. The shape is rectangular, 10xlOx2O in., with the beam entering from the 10xlO in. side. To take pictures two windows are placed on both 20x10 ino sides and a mercury arc is placed some distance behind the rear window. Four cameras are placed in front of the chamber with their optic axes perpendicular to the window and the intersections of these axes with the front window forms a 9 in. sq. During each expansion of the chamber the four cameras take a picture of the chamber when the arc flashes. The time between the expansion and the arc flash employed in this experiment is around 140 ~isec in order to produce the optimum bubble size. The magnetic field inside the chamber is perpendicular to the windows and its spatial and -time variation is negligible. There are 11 fiducial marks on the windows, four on the front window and seven on the rear window. The position of the fiducial marks is known to be better than 0.01 cm with respect to the optic axes of the cameraso A more detailed description of the chamber can be found elsewhere.19920 9

3~2 BEAM SETUP The 3~'7 BeV/c -beam used in this experiment was obtained from Brookhaven Alternating Gradient Synchrotrono The energy of the internal proton beam of the accelerator was 33 BeV and the intensity was on the order of 1011 protons per pulse. Only a small fraction of this was spilled onto our target, the remainder being used for another experimento A 3-sec repetition rate was normally used.. Since this experiment was parasitic, these parameters were varied because of the primary user's requirements o The beam was designed and set up by Brookhaven personnel, and details of the design and operating conditions are given in BNL internal reportso29122 A brief description of the beam as it was used for this experiment, however, will be given belowo The separated beam consisted of four sections: beam transport section, two separation stages, and a beam shaping section~ The total beam length was 270 fto Figure:1 shows the beam layout at the AGS building while Figure 2 shows a schematic diagram for the beam layout~ Our momentum of 357 BeV/c is the maximum accommodated by the magnetso The target used to produce the pions is of aluminum OD. ino high, 0o25 in wide and 2 in. long in the direction of circulation of the internal proton beam. The Rapid Beam Deflector (RBD) is fired causing a small part of the circulation protons to move into the targeto Since this experiment was parasitic, the RBD is fired early in the acceleration cycle in order to save most of the beam for the primary usero The

-?m mmmm mmm m mm m. mmm mmm mm -m -m -m m umm now mom -eIm go No! _ I I _ H = I ItI T ~~~I F-i l - -- i?- _ ~ _.. _ K +- -I - - - - - - - - - 0V- -T ittr~t Pi j tot ~_~igr -1 B-4a ao --- —— t at It - bulig-IFigure 1. Beam layout at the AGS building.

Transport First Second Beam Bubble Separation Separation Shaping Section Chamber Section Stage Stage SectionChamber Aperature Momentum Mass Momentum and Mass stop slit slit slit Bending Magnet uadrupoles Beam Separator Figure 2, Schematic diagram for beam layout.

13 secondary beam axis in the transport section made an angle of 80 with respect to the circulating beam in order to avoid the effect of the synchrotron magnetic field~ Slit 1 limits the angular acceptance of the beam to 40xlO-6 sr, and the bending magnet in this section gives rough momentum selection. Two beam separators consisting of crossed electric and magnetic fields serve as velocity spectrometers to separate out the desired particles21 A bending magnet in the second beam separating stage serves as a momentum spectrometer in the same manner as the first bending magneto The purpose of the beam shaping section is to modify the particle trajectories so that the incident beam is made of an appropriate size to fill the beam window of the bubble chamber. Twelve quadrupoles were used to focus the beam trajectories: two in the transport section, four for each beam separation stage, and two for beam shaping. Four slits separate the various sections. The dimensions of these slits are given in the table below. TABLE I THE DIMENSIONS OF SLITS USED FOR THE BEAM Slit Horizontal, ino Vertical, in. 1 8o0 oo5 2 0o3 0o0o45 3 0o3 0o2 4 0 03 0o2 Beam tuning was done after setting all bending magnets and quad

14 rupoles. The magnetic field in separator 1 was then varied by steps, and at each step separator 2 was varied until the n- intensity was optimized. According to the experiments of Cocconi et alo,23 and Baker et al24 at an angle of 80 and momentum of 3.7 BeV/c the composition of the negative beam is mainly pions with a few percent of K- and p-o After the two stages of beam separation there was negligible amount of K-, p-, and -.o The small remaining contamination of A- etco, could not give a significant effect as we normalize our cross sections to the known total cross section of it on protons. During the running of this experiment, the magnetic fields of the bending magnets were checked occasionally to be sure that the beam momentum had not drifted, and the corrector coil current was frequently changed to maintain constant number of particles that entered the chamber. This corrector coil is located just behind the bending magnet in the transport section and produces a horizontal magnetic field. This field deflects the beam vertically so as to adjust the intensity that passes slit 2. The actual beam momentum, as opposed to the nominal one that the currents in the magnets were set for, has been estimated by checking the X distribution of fitted beam momentum for the events fitted with four constraints and is 3.63 ~.02 BeV/c. 353 SCANNING MACHINES The films from the s20 ino bubble chamber were cut into 100ft strips and scanned at Randall Laboratory. The scanning machines project the

15 image on a green translucent screeno The size of the image is slightly larger than the actual size of the chamber. The machines have electronic view changers and film-driving deviceso They can also project two views at once and have a device which moves one image relative to the other so that one view can be superimposed on anothero This is useful if one wants to see if a track has stopped inside the chamber~ 3~4 MEASURING DEVICE Pictures to be measured are projected onto the same kind of screen as the scanning machineo These machines have both a low and a high magnification in order to give both an overall view of the picture and a detailed view of a small sectiono The lower magnification gives almost life size of the bubble chamber while the higher magnification gives about 2~8 times life sizeo All measuring is performed at the higher magnificationo The machine has a moving stage controlled by selsyns on which the film holders are attachedo The film is held down by a vacuum in order to insure flatnesso By moving the stage the measurer can position a particular point onto a fixed cross-hair attached to the screeno In this way one eliminates the nonuniformity of projection optics which can affect the accuracy of the measuremento That is to say, the coordinates of points on the film are measured directly from the motion of the film itself rather than from coordinates on the projected image, Two kinds of digitizing methods were employed in this experiment to

16 measure film motion~ The method used for most of the experiment was that of Moire fringe counting. A brief.description of the machine is as followso Optical heads consisting of four photodiodes and two gratings are attached to both axes of the measuring stage. The longer grating is attached to the moving part and the shorter grating and the photodiodes are on the fixed part of the stageso The two gratings are tilted relative to each other so that the grating lines make dark and light lines parallel to the measuring stageO As the stage moves the shadow of the two gratings move periodically, with one cycle corresponding to 01OOOl-ino of stage displacement~ A photodiode picks up these signal and counts the number of cycleso The advantages of this device are that one does not have to worry about "backlash of the screw which moves the stage and there is no worry of a poor contact or a broken blade such as can arise in a mechanical encoder system~ The third measuring machine, used for a small fraction of our analysis, makes use of a mechanical encoding device~ This consists of a rotating series of contacts which produces different combinations of electrical signals for different positions of the rotatoro The linear motion of the stage is coupled to the rotator motion by means of a highly accurate screw. The accuracy of all these machines is known to be better than a few microns on the filmo The film coordinates,, as indicated by readings on the digitizers, were punched automatically on IBM cards which serve as input to the geometry reconstruction programs~

CHAPTER IV DATA REDUCTI ON 4 o SCANNING OF THE EVENTS Two views of the 20-ino bubble chamber picture were used for scanning the events9 since not all four views were necessary to get the needed information~ On scanning the film, we made restrictions on the volume of the bubble chamber scanned, and the number of beam tracks ens tering this 'fiducial volumeo" Since we needed a certain length of primary and secondary tracks to maintain measurement accuracy, we rejected the events which happened outside of the fiducial volume, This fiducial volume -was taken such that the incoming beam tracks were not less than 4 5 cm in length and such that a minimum of 15 cm ofs forward-outgoing tracks occurred. for two-prong eventso Whenever more than 25 beam tracks entered the bottom of the fiducial volume the frame was ignored in order to maintain high scanning efficiencyo Scanning sheets were prepared for this particular experiment on which were noted such things as the number of nondecaying outgoing tracks (which we shall call prongs), decaying prongs, and the number of electron-positron pairs and '"V particles" pointing to the interaction vertex. Electron-positron pairs and V's could be easily identified by the curvature and bubble density of the tracks. Scanners also noted whether the tracks were stopped inside of the chamber or not9 and other relevant information such as the frame number9 event number if 'there were more than 17

18 one event in the picture, roll number, identification number of the scanner, and the date Every picture was scanned twice by different scanners independently. A physicist then checked every disagreement of the two scanners and also checked the number of tracks in any frame in which 20 to 30 tracks were listed by the scanner in order to make the 25 track cut-off accurate. Estimated scanning efficiency of this experiment was better than 95% for the nonstrange particle events in which we are interested. 4.2 MEASUREMENT The two Moire fringe-type measuring machines were employed to measure most of the events except for those few which were measured on the mechanical encoder machine~ The two best stereoscopic views were chosen in order to get the best result in the geometric reconstruction of the track~ The two best views vary from track to track and are those which have the biggest angle between the projection of intersection of the line between the two cameras on the front window and the projection of the tracks on the window. As described in the previous chapter, these devices punch out the coordinates of the three to ten points on each track, two fiducial marks (permanent marks on the front glass),, and other relevant information such as the sign of the charge of the particle, whether or not this track stopped inside the chamber, view number, track number, frame number, etco A remeasurement was done for those events which failed in either

19 the geometry or the kinematic programso Typical rejected events were those which had more than two bad point measurements or had the stereoscopic angle (described above) less than 15~ for any one track. This report contains all the events which did not fail by the third remeasuremento If there is no slippage in the encoder system, the coordinates of some point should be the same both at the beginning and at the end of a measurement~ It was necessary to check this for the fringe-type machines at the end of each measuremento For the mechanical encoder system, at the beginning of each day, the scanners checked to see if there was any backlash of the screw by comparing the coordinate of one point when approaching from both directionso The accuracy of these machines was about 3o05 for the fringe system and about 5p. for the encoder system when combined with the setting error of the measurer, as described in the previous chaptero 4.3 SPATIAL RECONSTRUCTION AND KINEMATICS After measurement, the IBM cards containing all needed information for events went through an IBM 7090 program so that possible errors in punched data could be detected and repunched in a different format to match the input of the geometry program~26 At this stage, there were quite a few mistakes in the punching, and these were corrected before getting into the geometry program~ The geometry program used here was written in the Fortran and Fap

20 languages at Brookhaven National Laboratory. This program (named TRED) calculates the initial momentum and azimuthal angle of the track in a projection plane parallel to the front window of the bubble chamber, and the dip angle with respect to this projection plane. The program then makes small corrections to these quantities taking into account the loss of energy of the particle along its patho Since the range-energy relation depends on the mass of the particle, TRED gives all this geometry information for five different particles, namely electron, pion, kayon, proton, and sigmao TRED also computes the errors associated with each. quantity of each track due to the experimental lack of smoothness and theoretical effect of multiple scattering.- All these quantities are written on a tape which serves as the input to the kinematic program. The only quantity which was not calculated by TRED was the beam momentum, since we knew this accurately from the beam magnet setting. The kinematic program (KICK) to analyze this experiment was orig27 28 inally written inBerkeley and has been modified at Brookhaveno This program examines each vertex point (beam-interaction point) according to a set of hypotheses. On every vertex point there are four constraint equations corresponding to overall conservation of the three components of momentum and also conservation of energy: ZEi = 0 and Z Pi = 0 where Pi and Ei are the momenteum vector and the total energy of a partidle at the vertex. We take a negative sign for an outgoing particle.

21 For two-charged-prong events KICK tests three reactions~ - + p + 31 + p (1) 4 Tr + p. + %0 (2) KICK first calculates the missing momentumenergy9 and mass according to the equations E Z Ei. AP = Pi (Missing Mass)2 = (AE)2 (AP)2 corresponding to each mass assignment for the charged particleso Since we were not looking for strange particles or leptons, the missing neutrals it could try were only the co and neutron~ KICK did not try to fit events containing more than one neutral because there are too many missing parameters for such hypotheseso If any one of the above three reactions seemed reasonable, KICK made a "fit" which is the process of readjusting the measured parameters in accordance with their errors so that the four conservation laws are satisfied within some allowed limito Us28 29 ing the method of least squares2829 in the process of fitting, one can use all four constraint equations for the nonneutral case but only one constraint equation for the case of one missing neutral, since the three other equations must be used to calculate unmeasured parameters e KICK

22 also calculated the readjusted errors, which are smaller than the original errors~ Usually the readjusted errors associated with a four constraint fit were smaller than those associated with a one constraint fit. The goodness of fit is estimated by the X2 value.o29 The X2 probability distribution should follow the curve exp 2 ) P(X,2) = n/2 ) provided that there is no systematic error, where n is number of constraints. Figure 3 shows the X2 distributions for events identified as reactions (2) and (3)~ These are consistent with theoretical curve (dashed curve). Hypotheses having a X2 probability of more than 2% are considered as giving a good fit and printed out in KICK output sheetso The x2 probability is defined as co X probability = dP(X) KICK also printed out the two-parti.cle invariant masses for every successful fit and punched out one card identifying the event and the hypothesiso Another important thing done by KICK was to calculate the theoretical ionization relative to the beam track, and to give its projected value on a plane parallel to the front window for every successful fito

23 500 500 (450 ) (b) 350',4 350 300 3 00 250 250 2200 ' 200 LL~' ~~~~~.~~U 0 0 0 0 0 z!OO z 100 50 50 I 2 3 4 5 1 2 3 4 5 Figure 3. X2 distribution for unambiguous events.

24 4A4 EVENT IDENTIFICATION Through the fit procedure of KICK, events often fit more than one hypothesis, as we consider that a fit means a X2 probability of more than 2%. To decide the correct hypothesis, physicists checked the bubble density of the tracks to see if they agreed with the bubble density predicted by KICKo Bubble density is defined as the number of bubbles per unit length of track normalized to a minimum ionizing track, As a minimum ionizing track we took a beam track, ioeo. 3~7 BeV/c E- beam. The bubble density of the track is approximately proportional to the inverse square of P, the ratio of speed of particle' to the speed of light.13 The quantity P depends on the momentum and mass of the particle according to Since we were not interested in strange-particle interactions, the negative track was always considered as t- whereas for a positive particle choice is t+ or proton~ Therefore for negative particles we checked to see if the bubble density agreed with the E- prediction, and for positive tracks we checked to see if the density agreed with either the Xc or p predictions~ The bubble density appearing in KICK output was actually the linear bubble density divided by the cosine of dip angle, since one can only see the track projected on the plane parallel to the. front window of the bubble chamber. From our experience we could distinguish a proton from a r+ for those cases in which the predicted bubble density

differed by at least 30 to 40o. Figure 4 shows the relation between linear bubble density and momentum~ Thus, from the appearance of the track we were able to distinguish a A+ from a proton up to around 1700 MeV/c momentum~ Also, sometimes the kinematics allow only one reaction so that this case was automatically decided, unless there was an obvious bubble density disagreement0 This stage still leaves a few events which are ambiguous between the different reactions. The statistics for event idenrtification are shown below (all numbers indicate the percentage of total events)o On the basis of bubble density check, the events which had a single fit for reaction (1) were never rejected. The events fitting to reactions (1) and (2) or (1), (2), and (3) were never resolved except reaction (3) was often rejected. Further study has been made of events which remained ambiguous after this stage. Since reaction (1) is a four-constraint fit and the others are one-constraint fits the events which were ambiguous between (1) and either of the others were thrown into reaction (1)o The reason for this is clear from Table II where we see that practically all of the events that fit (1) and (3) were determined to be (1) on the basis of bubble densityo There were, howevers, ambiguous events between the two one-constraint fits0 These events were considered as reaction (3) for the following reasonso First, if we look at the X2 distributions for these events considered as (3) (Figure 5a), and also considered as (2) (Figure 5b), we can see a definite difference between theme Even for these events which have X2 < 1 for reaction (2), the X2 distribution considered as (3)

*H 4 (\o pion 12., -! 1000 2000 3000 Momentum MeV/c Figure 4~ Relation of bubble density and particle momentum.

27 TABLE II STATISTICS ON EVENIT IDENTIFICATION Percentage of Unambiguous Events (After Bubble Density Check) Type of reaction (i) (2) (3) Rejected Total Single fit 4o6 8o2 13&3 6o9 33o 0 (1)&(2) ooo o0o --- oo o0.0 Dou.ble (1)&(3) 20.35 --- 0.o 0o0 20o4 fit (2)&(3) --- 1355 13o5 OoO 27.0 Triple fit 0o0 0o0 0o0 00O 0O0 Total 24o9 217 26o8 6o9 80.4 Percentage of Remaining Ambiguous Events Type of reaction (l)&(2) (1)&(3) (2)&(3 ) (1),(2)&(3 ) Total (1)&(2) 1o2 --- --- 1.2 Double (1),(3) --- 1.2 ---. 12 fit (2)&(3) --- --- 606 -- 6.6 Triple fit 10a. 0o0 0oo0 3 1o Total 11o4 102 6o6 l3 19o5 is still good (Figure 5c). Second, the momentum transfer from the incident pion to final dipion system for those ambiguous events is shown in Figure 6 and we notice reaction (5) predominates by comparing it with the distribution for good events (Figure 7)0 By comparing the ratio of highand low-momentum transfer we conclude that more than 80% of these remaining ambiguous events actually belong to (3).

130; 130 130 120: (a) 120 (b) 120. (c) ASSUMED TT'+ P —TT+Irr++ ASSUMED TT-+ P- F+ TT;-P ASSUMED TF — -i- TT —r+ 110 110 110 FOR EVENTS X < I FOR 100 100 I00 I00 T- P- Tn+ rT' P 90A1 90 90. 801 80 80 70 70: 70 60: 60L 60 50 50 50 Z 40 z= 40 x4 0 1 z w > 30 W30, 30: LL LI. U. 0, 0 0 C 20 '. 620, 620, 1Ig0r 5 x d1 b0 1 amiuI0, lot - 27 ~~\ I or L~t ( --- I 2 3 4 5 x I 2 3 4 5 x I 2 3 4 5x Figure 5. X2 distribution for ambiguous events between the two one-constraint fits.

A DISTRIBUTION FOR AMBIGUOUS EVENTS (a) ASSUMED + p..1- + ITT, n 60. 40 C) Iz w >20 w cr m z 100 200 /itz 300 (b) ASSUMED rr-+ P-.if+ Tr0+p 20 100 200 9 300 Figure 6. d listribution for ambiguous events.

30 300 (a) -.-+ p -- + —T, n+n 150 I00 c) z 50 w LL 0 I00 n OO 200 Az/M2 300 z ((b) 150 p — r- + - -~+ p 100 50 F00 200 2 300 Figure 7. A2 distribution for unambiguous events.

CHAPTER V RESULTS 5o1 INTRODUCTION In this chapter we will discuss and interpret the results of this experiment~ The things which will be covered are the production and decay properties of the p-meson followed by a study of the fo-meson. We begin with a discussion of some of the general properties of these resonances 5o2 GENERAL PROPERTIES OF RESONANCES 5201 Distribution of Two-Particle Mass Spectrum Figures 8 and 9 show the histograms of the various two-particle mass combinationso As explained in previous chapters all the ambiguous events between reactions (2) and (3) are included in reaction (3)~ The shapes of the pion-nucleon mass spectra will be explained later in this chapter on the basis of the Dalitz plots for the reactions~ We will first discuss the di-pion mass spectra (Figure 9). The dashed curves show Lorentz invariant phase space (see Appendix A for the definition of this), and are normalized to the number of events, excluding those events inside the resonance peaks~ The E -~ mass spectrum shows a peak centered at 765 MeV which we attribute to the p-mesono The n ~+ mass spectrum shows a very strong peak also centered at 765 MeV and another peak centered at 1250 MeV which we attribute to p and f, respectively~ The 31

60 ~~~~~~~~~~60 -(a) (b) 7F+ p — s.-7T + 7t + n it-ip -I F+p+r7T 1693 EVENTS 1054 EVENTS 40 40 20 20 LLJ U~~~~~~~~~~U z LU) Z 1200 1500 1800 2100 2400 1200 1500 1800 2100 2400 > 60 M(r-n) MeV o 60M(P)MeV LU (C) d (d) o it- + p-7r —i+fl- Z 7 7r-i-p-.-it+p-+it O 1693 EVENTS 1054 EVENTS z 40 - 40 20 - 20 1200 1500 1800 2100 2400 1200 1500 1800 2100 2400 M (Ir+n) MeV M(itWp) MeV Figure 8. Effective mass distribution for pion-nucleon system.

I00 100 (a) (b) 7r-+p-.- r+7r+ + n + P -+pr + + 7r~p 1693 EVENTS 1054 EVENTS 80 80 60 60 40 - 40 20 2 - 20 300 600 900 1200 1500 1800 300 600 900 1200 1500 1800 M (r 7) MeV Figure 9. Effective mass distribution for pion-pion system.

34 The o s and f ' have also been discovered by earlier workers]71 5,2,2 The Parameters of Resonances An important characteristic of a resonance peak is its width, The estimation of the width at half maximum varies according to the assumed mathemati.cal form of the resonance. We have estimated the width cf these resonances by adopting a Breit-Wigner formula,38 Nr2 E(M) (MM)2+(r/2 )2 where H is the height at the mass value M, Mo is the va.lue of M at the peak, N is a normalization factor, and r is the full width at half maximumo In interpreting our data we have also given an energy dependence to F according to4l r = Cp3 = C (M7e2 —4m)2 3 where C and C' are constants and p is the momentum of one of the pions in the di-,pion systemo We have used this form since it gives a better fit to our mass histograms. Below is a table of resonance peaks and wird.th obtained in this experiment by trial and error wh.ich best fits t.he above formula for H(M) including the effect of background which fol-.icws phase space~ There seems to be a possible dependence of Mo and r for our data. on momentum transfer (A2 defined later in this secticn ) Figure 10 shows the histograms cf ~- + mass in reaction (3) around the region of p for those events which have A& < 4hk(I. = m~,) (Figure LOa) and these

355 TABLE III THE RESONANCES AND THEIR WIDTHS Name M(MeV) r (MeV) Production Cross Section (mib) p- 765 125j 15 0.33 02 -0o. p~ 765 125 ~ 1_5 1.02 -0.1 f~ o1250 130 + 20 o. 2' with A between 4z2 and 2O.2 (Figure lob). The dashed curves are fitted to the Breit-Wigner formula. Note that the width of the peak is considerably less than that of the combined data (Table III). We obtain about 80 MeV for Figure 10a and almost 90 MeV for the higher a2 events in Figure lOb. The center of the peak of Figure lOb is also shifted by 30 MeV toward higher mass values. One possible explanation for this change of the width and the center value is that lower momentum transfer means the reaction is more peripheral so that the impact parameter of the reaction is bigger. Therefore, the resonance has less opportunity to interact with the nucleon field before or after it decays. Thus, interference from the nucleon field becomes smaller as A2 becomes smaller. Such interference could be expected to increase r and also possibly cause a shift in M. 5.3 INTERPRETATION OF p PEAK AS PION-PION SCATTERING RESONANCE* 5.3.1 Production Mechanism Chew and Low17 have proposed a method to analyze the scattering *Since the data process was delayed by various reasons, we used 1398 events for reaction (3) and 868 events for reaction (2).

56 (a) (b) A2a 2 4,24A2< 2 0# 40 40 30- 30 -)I I SI (0 I II II I z I I I-, I: I( U.S I i I L I w \\ 20-,,. 20 - ' 00 z I I17 I II I I I *I ~ I I I I I %~ I I~ % 10 10 t L~~~~L!~~~~~~~~~~~~~~~~~~~~!. I.. 600 900 600 900 M -m+ (MeV) Figure 10. Effective mass distribution for pion-pion system for different A2 regions.

37 process on a virtual target as a method by which to obtain information about scattering processes on unstable particle targets. In particular they have given an expression for the cross section on this virtual target. The process ir+p -+ i+t+N with which we are dealing, could proceed through a one-particle exchange diagram, illustrated in Figure 13. Therefore the Chew-Low idea should be applicable to this experiment. If the process proceeds through one pion exchange, the scattering amplitude has an isolated pole in the nonphysical region at A2 = _2,where A2 is the four-momentum transfer from the target proton to the recoil nucleon (or incident pion to final di-pion system). It is given by A = (Pi-Po) where Pi and Po are the four-momentum of the initial and outgoing nucleons. The value of A2 is Lorentz invariant and is given in the laboratory system by twice the mass times the kinetic energy of the outgoing nucleon. We take p. (equals the pion mass) as the unit of four-momentum transfer. According to Chew and Low, the pion-nucleon cross section is expressed in terms of the pion-pion cross section as 2dape lim f2 A2 M Tr- 1 1/2 dM2dA A + -1 2T (E+1) P2beam in lab. where f2is pion nucleon coupling constant. Therefore, in the physical region the interaction is characterized by strong enhancement at small values of the A2 and this can be used as a test to see whether the interaction appears to be peripheral and consistent with one pion ex

change. Figure 11 shows the scattergram of A2 versus M2 (Chew-Low plot). Here one can see that most of the events are in the lower part, especially for the resonances. The concentration of the dots around the M2 of 0.68 BeV2 is due to the p's and another around 1.56 BeV2 represents the f~ resonance. There are very few events with momentum transfer of more than 20. Figure 12 shows the distribution of A for those events inside the three resonances. The plot is only for A < 50j2 because only a few events remain after that point. The first few depleted bins for the fo show the phase space limitation for this mass region, i.e., the kinematics impose a lower limit on A2 for a given M~a, which at the fo mass is 2.6 L. In all these distributions, we note that most of the events are confined to less than 20ui of momentum transfer. The evidence given by Figures 11 and 12 is consistent with the reactions being peripheral, and proceeding through one particle exchange. The only known particle that can be exchanged in the p~ case is the i+, while the p- case can have AO and c~, provided that only strong interactions contribute to these reactions. Below are the exchange diagrams for these reactions. Treiman and Yang have suggested a way of testing the one-particle exchange mechanism.32 If the exchange particle is spinless (like the pion) there should be no preference of azimuthal angle around the exchange particle d.irection because a spinless particle cannot transmit any directional information. This azimuthal angle can be calculated in

1120 O~.~. ~ - -+ X- -c x+ P 100 100 80 80 60 G 60 K 40 40 * 1 ~ ~ 2 3.123 Figure 11. Chew-Low plot of 14-momentumn transfer.

4o 1170 < M -T <1t330 60 1-F1-f co zI 40 20 5 10 15 20 25 30 35 40 45 50 55 100 80 690 < M rT- T.860 N 0 i 20 5 10 15 20 25 30 35 40 45 50 55 60 690 < MTT- < 840 z 40 0 r 20 5 l0 15 20 25 30 35 40 45 50 55 Figure 12. Four-momentum transfer distribution for p's and fo.

41 the rest frame of the incident pion by cos(TY) = (PlxP2)'(P3xP4) IPlXP21 IP3XP41 where P1, P2, P3, and P4 are the 3-momenta of the target particle, recoiled nucleon and two outgoing pions. Since we are dealing with strong interactions in which parity is conserved at each vertex, we can fold the distribution about 1800 line. The reason for this is that the reaction must be independent of spatial inversion which changes the direction of all momenta, and therefore changes the sign of the T-Y angle (but not of its cosine). 0 / P P pp p Figure 13. One-particle exchange diagrams for p production. Figure 14 shows scattergrams for this angle, taking four-momentum transfer in the vertical direction. We note that all those are consistent with isotropy except for a slight depopulation at higher values of the T-Y angle. One can conclude that one pion exchange cannot be excluded by this test. In Figure 14b, one might notice, besides the depopulation for higher values of the T-Y angle, that there is depopulation around 00. This depopulation might be caused by those events which exchanged an w. This fact contradicts our assumption of a one-pion exchange model. An

42 A2 A,42 TREIMAN-YANG ANGLE 2 TREIMAN-YANG ANGLE vs A2 vs A2 690 < M -if 7r+< 840 MeV 690 < Mfr-7r~< 840 MeV o00 (a) 0oo 0 (b) 80 80 60 60 40 40 20 1 - 20 -DEGREES 30 60 90 120 150 180 DEGREES 30 60 90 120 150 180.t2. TREIMAN-YANG ANGLE ~~~~~- ~vs 2 1170 < M 7f 7rt< 1330 MeV 100- (C) 80 60 40 20.... *, '.. * * '.:.- ', DEGREES 30 60 90 120 150 180 Figure 14. Treiman-Yang angle distribution for p's and f~.

43 other discrepancy for the one-pion exchange model is that the production ratio of p and po given in Table III is about i to 3 whereas it should be 1 to 2 for one-pion exchange~ The ratio 1 to 2 comes from analysis similar to that given in Appendix Bo Thus one might doubt the one-pion exchange assumption; however, this will be explained in the next section by looking at decay angular distributions. 553.2 Decay Angular Distribution of p Let us now concentrate our attention on the p resonanceso The quantum numbers are T = 1, 1 (jparity, G-parity) as shown by earlier workers 7-9 As was shown before, the production of the p is consistent with one-meson exchange dominating the reactions. Assuming that the onemeson exchange hypothesis is correct at least for the major portion of the events, let us study the decay angular distribution of the p-meson. We define the angle Q between incoming and outgoing t- in rest system of final di-pion (Figure 15). out 'in _ (wX)exch S (o) out Figure 15. Space diagram of pion vertex in rest frame of final di-pion system. As stated before, the only possible particles which could be exchanged in these reactions are either the pion or the 0.o Therefore, the distribution of events should go as cos20 for pion exchange and sin20 for X exchange provided there are no final state interactions or

44 interference from other production mechanisms (see Appendix C)o However, none of the angular distributions shows either of these predicted distributions (Figure 16)o Assuming the peripheral interaction to be characterized by low-momentum transfer, we took events with momentum transfer less than 20p2 in Figure 160 One can notice a' strong forwardbackward asymmetry in the po decay angular distribution, indicating the strong presence of odd powers of cos gO The po does not seem to have such an asymmetryo 5~3~3 Possible Explanation of Asymmetry in p~ We can think of several reasons which might explain the heavy asymmetry in po decay angular distribution. They are (a) interference effects from nucleon isobar formation,33 (b) possible interference from 34 the two-pion decay of cu-meson, and (c) interference between the resonance and possible s-wave background, along with off-energy shell effects due to the fact that the struck pions are "virtualo" As will be seen later from the Dalitz plots (Figure 22) the effects due to isobar formation, if any, are very small. The effect due to the possible two-pion decay of u-meson is hardest to estimate, although it should not be very large due to the fact that this decay violates G-parity and cannot be a strong interaction' The sharp peak of po (Figure 9a) and the shifted peak for those events having different momentum transfer (Figure 10) might be due partly to two-pion decay of the w-mesonO However, this might also be caused by secondary interactions

7C~~ p - r +7 n 7p - 'Tr+Tr+ p 40 690<M ee< 840 20 690(Mvre<840 422 2 t<20,2 P < 20<22 C/) 30 15 z w w 20 LL 10 0 i 0 go ~~~z5 -I. 0 I 1 0 cos eww Figure 16. Distribution for cos Q inside the p-meson region.

with the nucleon field, the effects of which are too hard to estimate at this stageO In any case, we will neglect these effects and proceed to discuss the third reason, which seems to give a good explanation for the cos @ asymmetryo 2 j8 We want to make an empirical fit to the cos 0 distribution by means of a partial wave analysis for pion-pion scatteringo The two pions can be in the T = O, 1, or 2 states of isotopic spin. Because of the generalized Pauli principle, even T states can only be accompanied by even angular momentum, and likewise, for odd T, only odd J is allowed~ The simplest case would be a mixture of a T = O, L = 0 state with a T = 1, L = I state, assuming T = 2 terms and higher angular momentum terms are negligible o The absence of a T = 2 state will be verified later in this chapter~ The p0 asymmetry can arise due to the quantum-mechanical interference of these two states~ The lack of asymmetry in the p- case is then understandable also, since T = 0 is not allowed for the ge~o system The partial wave expression for the cross section which contains only T = 0, L - 0 and T = 1, L = 1 terms is given by do _ =1_| 2_ ei0O,Osin o0 o + 3eillsin b1,1 cos O d(cos O) k2 3 1k sino.0+o+4 sin o600 sin 51,1 cos(boo-1l,1) x cos + 9 sin b1, cosa =A + B cos 0 + C cos2 2

where A = -4 sin25,0 B -= sin 0 o sin 51, 1 cos(b 0O-bl1l) 9k2 - k2 and C =.9 sin2bS1 1 k2 The phase difference between s- and p-wave can be calculated by cos(30no 61 1, = B/2 4 where 60,o and 61,1 are the phase shifts for s- and p-wave, respectively, and k is the pion-pion center-of-mass momentum and we take units such that d1 = 1o It is expected that the phase shifts will depend on the total energy (Mc) which, of course, is simply related to k. The factor 2/3 in front of the s-wave term comes from the following argument. The wave function of the incoming di-pion system can be expressed in terms of its isotopic spin components as *(0T-T+) 0 +41+ V2 30 and the constants are the Clebsh-Gordon coefficients. The functions ' can be expressed in terms of the products of the single pion isospin functions as

48 '0 = J ( 1()it+(2)+n'+(1)r(-(2)-VrO(1)cO (2) I = F: -l(12)Xi,+(l2)+()(2)+2X+ (1, )~ (2) Therefore, the final state wave function can be written in terms of the scattering amplitude (AT) for the three T-states as = Ao (=:-+++n+0-_:oo) + Al (-) =3 2 + 62(-+++-+2 ) This can be written as 'f = ( A Ao + 2 A) ) 3 3 The coefficient of the first term represents the amplitude for observing 1-r+ in an L = even state, the second is for A-A+ in an L = odd state and the third for observing two O~'so Since we observe in this experiment an outgoing -At+ but do not determine its L-value we must use L Ao+4 Al as our amplitude (we take A2 = 0 as mentioned previously). The relative strength of the Ao to Al amplitude is therefore 2/3, as indicated in the phase shift formula. The factor 3 in front of the ei6 sin 51,. term is the (2L+1) factor in the partial wave expansion of the A's in terms of the Legendre polynomialso

Since the p- case does not have as large a asymmetry as the p, we conclude that there is either no T = 2 s-wave or, if any, it is very small or incoherent. For the p case we have fitted the angular distributions to A+B cos Q+C cos2O. Figure 17 shows the cos 0 distributions for various Mtr regions and for A2 less than 20 C2. The dashed curves represent the fitted curves as determined by a least squares program, and are normalized to the proper number of events in each case. Below is the table for the coefficients A, B, and C along with B/2 45 and the x2 probability. The coefficients, A, B, and C are normalized to arbitrary units in Table IV. The term B/2 4A would be the cosine of the difference of the s- and p-wave phase shift angles if the target were a real particle. In Figure 18 we have plotted the results of Table IV. The number in Figure 18 is normalized the same way as in Table IV. Figure 18c shows TABLE IV LIST OF FITTED COEFFICIENTS AND X2 PROBABILITY FOR DIFFERENT DI-PION MASS Mass Region, M A B C B/2 m X2 Probability 590 690 o.87 o.43 1.12 0.24 15 (%) 690 740 1.31 3.47 4.44 0.72 64 740 790 2.91 6.60 7.90 0.73 29 790 840 2.03 2.14 4.88 0.33 44 840 940 2.14 1.06 2.82 0.21 15 940 1040 0.79 0.16 2.71 0.04 3

A=.87 A=2.91 I A=2.14 15 A=.870<Mg<69| 740<M 790 5 840<M/<940 B=.43 B= 6.60 B= 1.06 10 C= 1.12 10 C= 7.09 C=2 82 10. 10 10 5- 5 5-.I cFOes 1ij K; o Cos. S-. 0 cos e I. da for0~~ 0t2 0 0 0 6 9 0 <Mrm 740 79O<M-w'4840 z9404M4 4040 15 A= I. 31 15 A=2.03 15 A=.79 B= 3.47 B=2.14 B=.16 C= 4.44 C=4.88 C= 2.71 10 10 10 5 5 5 -I.0 Cos. -e I. 0 COS i. -I. 0 COS. Figure 17. Distribution of cos G for different ~-+ mass and for A2 < 20k2.

51 (c) f 5 S ^AVE IJTERF TERM L) 5 In II 600 700 ooo O rrr MeV 7r - 600 700 800 900 600 700 800 900 phase- shit diaram. phase-shift diagram.

52 the energy dependence of the phase shifts assuming the p-wave phase shift propagates from 0" to 180~ across the resonance. The s-wave phase shift remains approximately constant and large (75~+35) throughout the region. This value of 75~+350 was obtained from Figure 18b by assuming cos(5o,o-51,1) = B/2 v/iF and taking 51.1 by the Breit-Wigner resonance formula. As one can see from Table IV and Figure 18, the s-wave amplitude remains almost constant, as would be expected. The slight increase of the s-wave amplitude at the p mass might be statistically not significant and we are not able to conclude if this effect is real or not. The p-wave and interference terms definitely show a sudden increase in their coefficients at the resonance region. Figure 18b shows the B/2 4EJ term which also peaks near the resonance. All these effects are consistent with the dependence shown in Figure 18c. Since the target is not a real pion, we are not able to interpret these values of the phase shifts as being quantitatively correct, but qualitative agreement with our form for the partial wave expansion is very encouraging. An attempt to make these results quantitative will be given in the next section. Thus our interpretation seems to be justified that the p is a pwave resonance of pion-pion scattering with the forward-backward asymmetry being caused by interference with the s-wave background. However, there is one point which is bothersome. We notice from Table IV that the ratio of C to A is always less than 3 to 1, whereas if the

53 phase shifts behave according to Figure 18c the ratio should be 81 to 4, or about 20 to 1, near the peak~ The ratio 81 to 4 is obtained from the expression for A and C by setting $o0o = 581 o We will see in the next section that this ratio becomes much closer to its expected value when we extrapolate A, B, C to the value A2 = -o 5o3o4 Extrapolation Into the A2 = -1 Pole One can think of several possible explanations for the discrepancy in the ratio of the s- and p-wave amplitudeo The most plausible are: (a) the estimation of A, B9 C at the peak is an average value over some finite interval of pion-pion mass hence one does not get the maximum peak value for the p-wave; and (b) the off-energy shell effect makes the quantitative results unreliableo The first explanation gives a small effect, of order 10%, but cannot take care of the whole discrepancy. For an investigation of the second explanation we have to extrapolate the s- and p-wave amplitudes into -the nonphysical region, i oeo, to A2 = - Figure 19 is the angular distribution in the various A2 regions for the entire p region (690 < MICI < 840 MeV)o The coefficients A, B, and C for each division are shown in Table V o The coefficients A, B, and C are normalized to the number of events. The value for A from 20 to infinity is not reliable because of a large x2 value (or small probability) and too big 2 interval It is clear that the ratio C/A is rapidly increasing as 2 decreases. In order to use these coefficients in the Chew-Low formula we must convert them into values

(a) (c) A= 1.2 A= 1.9 15 B= 3.85 15 B 2.25 C= 8.15 C= 3.65 5 "' ue 5 10 Ia I I I 1 1 I Le B- 3.03 B= 1.06 i-. o cose I. w -I. cos e. (b) o (d) ins 0 h p< ~? 4 6r 0 2 A i. 6 A -A 2.5 B: 3.03 B: 1.06 10 C: 6.4 1 C= 2. 56 5 5 -I. cs I.S 8 -C. 0 Cos q. Figure 19. Distribution for cos Q for different A2 for the events inside the po-region.

55 TABLE V LIST OF FITTED COEFFICIENTS AND X2 PROBABILITY FOR DIFFERENT A2 A2/pa Region A B C C/A X2 Probability 0.0 2.6 1o20 3.85 8.15 6.80 75 (%) 2.6 - 6.0 160 3.03 6.40 4.00 80 6o0 - 20.0 1.90 2.25 3.65 1.92 22 20.0 - infinity 2.50 -1.06 2.56 1.02 0.1 2 2 per unit A interval by dividing by the difference of A2o In order to convert into values per unit steradian we actually divided by TcdA2 d = -2cd(cos ) = - PbeamPp where c is the p production angle and the values of Pbeam and Po (which is practically constant) are taken in the center-of-mass system of the 36 whole reaction.o As suggested by Selleri we make use of the modified Chew-Low formula (all in pion mass units) + 3 f2 A2 ( >2 k1 dca, + d.a2dwc2d.W A+1 (A2+l)2 2+A2 klL d.2V where (A2-l)/(A2+A2) is a phenomenological form factor which is unity 2 2 at the A2 = 1 pole, f is the pion-nucleon coupling constant, k1 and klL are the momentum of incoming particle in the rest system of di-pion and the laboratory system, respectively. The symbol n' represents the solid angle for the outgoing n- in the rest frame of the di-pion, and w is the effective mass of the di-pion system. One then multiplies the coefficients A, B, and C by the factor of ((ea+12/ea2).((A2+f2)/(A2-l))2 which is the inverse of the factor in the above equation. The free param

56 eter A2 was determined from the overall A2 distribution and is 6~2 Figure 20 is a plot of ((A,B,C)/sr). ((A2+l)2/A2) ((A2+A2)/(A2 1))2 and a possible extrapolation to A = -1 in the nonphysical region. The extrapolated values for A, B, and C are 0~76 ~ 0~4, 5~9 ~ 0~9, and 1401 + l19 mb/sr, and the ratio C/A comes to about 20 which is now in excellent agreement with the predicted value (see Section 5o3o3)~ We can also make a calculation of the s-wave phase shift from B 4 sin 6o,o sin 6l os(l- o~ C 9 sin9 At the center of resonance we take 81,1 = t/2, then B = _ sin2bo = 0~ 42 + 0oll C 9 which gives sin2o0, = 0o95 0.25 This value for bo.o is consistent with the value shown in Figure 18c as determined in Section 50353~ Professor Marc Ross has suggested to us that using this cross section for the p-wave (C term), we can calculate the pion-nucleon coupling constant (f2) by using the measured p width to compute d-(I+t) o We dS' have done this39 and obtained the value 0o103 + 0o02, in good agreement with other determinations of feo Thus all this evidence agrees remarkably well with the assumption that the po is a p-wave resonance of pionpion scattering with s-wave background interference and with the phase

57 15 5 -II - v 5 // - PS TERM 0 5 1t0 t2h 2 Figure 20. Extrapolation of partial wave amplitude to A2 = -1.

58 shift behavior being roughly that shown in Figure 18c. 5o3o5 Discussion of p- Peak Now let us turn our attention to the p- decay angular distribution. In this case we cannot have as good an analysis as we had for the po because of the more meager statistics. Figure 16 does not give too much promise of deciding whether an X or a pion is exchanged, since the distribution must be sin2g or cos2, respectively, for these two possibilitieso We investigate this question further by dividing the events into several A2 regions and fitting each region separately. Figure 21 shows these angular distributions for p- events (690 < Ma < 840 MeV). One notices that Figure 21a and 21b do not have the shape of a sin20 or cos20 distribution~ However, Figure 21c shows a more clear dependence of a cos2g distribution and does not show the strong asymmetry we had for the p, in agreement with the fact that there can be no swave mixture in the p-o As in Figure 19, the dashed curve shows the fitted curve for a p-wave coefficient of 4o85, which was normalized to the number of events in Figure 21co The slight asymmetry here might be an interference due to some T = 2, s-waveo However, we believe the more reasonable possibility is that it is due to AI p isobar formation. In fact the subtraction of possible isobar events (as determined from the Dalitz plot) removes the asymmetryo A justification of this isobar subtraction (the shaded area in Figure 21c) will be given in the next sectiono The method of subtraction is to randomly take out a constant

Number of Events 0 ~~~~31 ~ ~ ~ ~ ~ // (Dn (Dv/ 4~ ff CD \10 ~ ~ ~ ~ ~ IA 'lR) I.A 0 H- 0 l C+ cD 3+ CD A A o 0 A J I-.,.) AI \J- 0 Fl o (DH 00 0 H 1::[

60 density of events along the N* band in the Dalitz plot, and this amounts to nine eventso Thus for those events with momentum transfer less than 4~5i, it appears that a substantial part of the reaction is going through one-pion exchange~ The rather isotropic distribution in Figures 21a and 21b may be caused by w exchange mixed with pion exchange or it may be that at higher A2 the reaction does not go through a peripheral mechanism Now, if the p is a p-wave resonance then in p production via onepion exchange there should exist a 2 to 1 ratio between po and p o If we take our observed p-wave coefficient for the p 9 which is 4.85, and convert this number -through the same procedure as we did for the p, we obtain the value 6~(7 ~ 14 mb/sro Since the p-wave part in the extrapolation (Figure 20) stays constant throughout the A2 region (14ll mb/sr) we can compare the above two numbers as representing their extrapolated valueso This gives good agreement with the ratio 2:19. and resolves the discrepancy noted in Section 5o31ol 5A4 DALITZ PLOTS AND ISOBAR FORMATION We next discuss the question of nucleo-n isobar formation and the possible effects of this on the p and f0 resonances. The best plot to examine for this purpose is the Dalitz plot (see Appendix A for a detailed explanation)o Figure 22 shows the Dalitz plots of both reactions in terms of the pion kinetic energies in the center-of-mass system of the whole reaction~ A concentration of the dots (each dot represents

-" 0'1 rA~8 ' I.:.-......., ~ r ~-'.,:' -.''.~.,' '~:':"''~:i.:'.' t ~ ~~~~ ~. —,,~'.",'' -'"-%... '.":. 'O::.,,?.% t r~ ~~~~. x.',.'~, ~~~~~ ~ ~, '. ',, ~~~~.....,,, ~ ~ ~ ~ ~ ` ~,;..' ~ + `' ~.... ~ '..~~~~~~~~~~~ ~'... %-.,.. '",%. ' Y~ 0.1(D),~i 5,~~~~~~~~-F

62 one event) in a vertical or horizontal band indicates a resonance in the pion-nucleon system,since the pion-nucleon effective mass is expressed by M%,<N) - WTOT2 M2 - 2WTOT(YM2+Ti2) where WTOT, M2 and Trd are total center-of-mass energy, mass and kinetic energy of the second pion in the center of mass of the whole reaction. It is seen from the formula that M2(Qi.N) is a linear function of the kinetic energy of the other piono The pion-pion resonances are recognized by the concentration along 1350 lines, since the pion-pion mass is described by M(A) = N - WTO + 2WTOT(T l+Th2+Mri+Mri2) where MN is the mass of the nucleono In Figure 22 there are obvious concentrations along the 135U lines which indicate p~, f0, and p-o In Figure 22b, there is a slight indication of isobar (1238,3/2,3/2) in the f'op system which appears as a concentration in a vertical band at a E- kinetic energy of about 1000 MeVo This means that in Figure 22a there should be also some observation of this iso'bar, since the branching ratio N*(vr+n)/N*(t"p) is 1/2, regardless of the production mechanism~ However, this band is almost not visible because resonances in the pion-pion system dominate~ Any concentration of dots in a horizontal direction apparently does not exist, The reason for this may be that the ratio for NE (3/2,3/2) production is 4:2:1:0 for the modes

63 itp:ir+n:Trp:on (see Appendix B for proof), assuming the production is due to the exchange of a p mesono Since the Dalitz plot (Figure 22) shows that there is a slight but definite concentration of dots around T,- = 1000 MeV for reaction (2), an attempt was made to estimate the N* production. Figure 23b shows the distribution of to kinetic energy for those events inside the N*(j~p) (920 < To_ < 1020 MeV) region, The lower horizontal line shows the estimated background which was drawn by averaging the nonresonant area (neither N* nor p) in the Dalitz ploto The upper horizontal line shows the average value of N* events outside the p region. The events in between these two lines can be considered as due to N* production. We estimate that this amounts to about 90 events for reaction (2)~ By taking half of the amount we get for (2) we therefore estimate about 45 events for reaction (3)~ This corresponds to about a 0035 mb cross section for all N* production via the reaction c-+p + N*++ -o 5~5 STUDY OF f~-PEAK AND DETERMINATION OF fo-SPIN The resonance around Mar of 1250 MeV has been also found by several other workerslO-1l3 Since there is no such bunmp in reaction (2), this f resonance is thought to be an isotopic singlet (T = O) with even spin (L) (due to the generalized Pauli principle) provided isotopic spin is conserved in the reaction~ The most probable quantum numbers originally assigned were T = O, L = 2, parity +, G-parity +o However, recent 16 work done by Frazer et a10, indicates that the quantum numbers of f

64 (a) r+ — p- -+ rr'++ n 940 <Tr-<1020 MeV 20 I0 100 200 300 400 (Me Trr+ (MeV) (b) 7r-+ p- tr-+7+p 940 < To-< 1020 MeV OTHER EFFECT l I N* (1238) BACK GROUND 100 200 300 400 T.o (MeV) Figure 23. Distribution of r+ and ot kinetic energy for events inside the N*+-region.

65 may be T = 1, L = 1, and parity -, G-parity +o According to this viewpoint the recently discovered B-meson (o-pion resonance40) would be the T7+i member of the f meson. In fact the angular distribution given in Refso 12 and 37 gives a better fit to L = 1 than L = 20 Because of the poorer statistics in this case, as compared to the po case, it is difficult to make the same type of analysiso However, we can draw the following conclusions from the angular distribution of cos @ (defined in the previous section for p)o Figure 24 shows the distribution of cos O in the f region (1170 < Mt < 1330 MeV) for A2 of less than 20t2o Figure 24a shows reaction (3) while Figure 24b shows reaction (2). We see almost no asymmetry for reaction (3) and a heavy asymmetry for reaction (2), which is just the reverse of the phenomena we saw in the p caseo The shaded area in Figure 24 shows the random subtraction of N* events from the Dalitz plot taking a constant density along the vertical N*+ band~ This amounts to 10 events from Figures 24a and 24c and 13 events from Figure 24bo This subtraction seems very reasonable, although the conclusions drawn below are not weakened appreciably if this subtraction is not made~ It is certain, however, that the forward peak in the fO region (Figure 24a) is to a large extent caused by N* production The lack of asymmetry in the cos Q distribution for fo is already good evidence against T = 1, L = 1 for f, since the T = 1, L = 1, p from the same reaction shows a large asymmetry through the p resonance~

(a) (b) 20 20 10 -10 IO UJ w -O. O. O. -1. O. 1. o (c) -. —PURE P a3]-PURE D - - S AND P S AND D 20 0.,, COS e Figure 24. Distribution of cos Q for events inside the fo mass.

67 We now attempt to determine the f~-spin from Figure 24ao Because of the symmetry of the f0 angular distribution we have folded it about cos G = 0 (Figure 24c). The normalized least squares fits to A+B cos 0 +C cos20 (s- and p-wave) and A+B cos2Q+C cos40 (s- and d-wave) are shown. Also shown as a comparison are cos2Q (pure p-wave) and 1-6 cos2Q+9 cos4Q (pure d-wave). The X 2probability for s- and d-wave is about 9% while for s- and p-wave it is less than 0.1%. The hump around cos 0 = 0 and the steepness for large values of cos 0 is good evidence of the necessity of a cos4G term, thereby ruling out L = 1 for f~. The hump around the center is not big enough to fit pure d-wave, but this is not surprising in view of the possible s-wave background and the interference between them. This gives good evidence against L = 1 (and, of course L = 0) for f0O We conclude that L = 2 is the most reasonable value for the fospin although we cannot rule out L > 3. The Treiman-Yang angle test for fo (Figure 14c) for the one-pion exchange model does not give a result as good as that for p's. This might be caused by events preceding other than by one-pion exchange. If the reaction does not go through one-pion exchange, other processes can introduce any combination of f~ polarization states which, therefore, can introduce any combination of YIm terms (m / O) into the angular distribution. This would allow practically any values for A, B, and C. This discrepancy in the Treimn-Yang angle doesnot weaken our conclusion against L = 1 since the angular distribution (Figure 24c) requires higher than second-order terms in cos 0.

68 An independent test was made for the consistency of the quantum numbers for the f particle on the basis of the di-pion mass spectrum (Figure 9). Assuming that f production occurs predominantly through one-pion exchange in the T = 1 and L = 1 channel, one would expect onehalf as many events in an f peak as in the f -peak. If one takes half of the events in the f -peak above the phase space and subtracts them from the same mass region of Figure 9b, a definite dip occurs which falls far below the general appearance of the background. Moreover other processes, such as w exchange, which can contribute to t_ Op production and therefore possibly obscure any f' peak, tend to make the ratio of total numbers of f~ events to f- events closer to unity; whereas the ratio we observe is at least 3 to 1, including background. The evidence, therefore, indicates that this ratio is not consistent with the T = 1, L = 1 assumption. The lack of evidence for f formation also seems to rule out the possibility of T = 2 for f0, since pure T = 2 pion-pion scattering requires a 4.5 times bigger bump in f than f. An experiment which is now in progress in this group (s+ in deuterium) should give definite isotopic spin information on the fo as it will measure the ratio (fO -+ Ao+ao)/(f0 + _i-+J+). This ratio should be 1/2, 0 or 2 for T = 0, 1 or 2, respectively. 5.6 PARTIAL CROSS SECTIONS We close this chapter by presenting cross sections for the different

69 types of reactions in this experiment~ Below is a table showing the cross section of each reaction of interest. The total cross section for pion-proton interactions at our energy is fairly well known and the values we give are normalized to a total cross section 31.5 mb, obtained from the counter experiment by Diddens et alo30 By doing this we do not have to take into account the kayon or muon contamination in the total path length scannedo In this table, the cross sections for neutrals (O prongs) were obtained from another similar experiment31 to fill an empty spot in our table. TABLE VI TYPES OF EVENTS AND CROSS SECTIONS FOR 357 BeV/c x-p INTERACTIONS Type of Event Cross Section (mb) ~- + p + X- + p 4.74 ~ 2.0 t +9 p + x~ 2.51 ~ 0.5 + - m+ (+ + n 396 07 + other 2 prongs 8.32 + 2.0 + x + + + p 1.71 + 0.5 4+ - + T + + tp 0 2+42 0.7 + T + C+ + +1 + n lo.00 005 + other 4 prongs 2.45 + 0.7 + 0 prongs 350 ~ 1.0 + > 6 prong and strange particles 1.04 ~ l1. Total 31.5 mb

APPENDIX A DALITZ PLOT AND LORENTZ INVARIANT PHASE SPACE Consider the three body final state in center-of-mass system (e.g., A-+p ~+ +t++N) and define the variables PT total 4-momentum Mi,TiPi rest mass, kinetic energy and 4-momentum of ith particle ET,Ei energy of total and ith particle Mij effective mass of ith and jth particle Pi 3-momentum of ith particle. If there are no final state interactions or angular momentum preference the final state should be distributed in proportional to the energy density of final state p oM4 d4pld4p2d4p354 (pl+P2+P3-pT) 3 2 2 x r5(Pi+Mi) ~ (A-l) The term 6(p i+MOT) can be expressed in another form 6(E2-E?2), where 02 -2.*2 2 Ei= pi+M2 Therefore p c'<. d4p1d4p2d4p364(pl+p2+p3-pT) | 6 (Ei-Ei ) d3p1ld3pad3p3 (pl+p2+p3-PT )A-2 E10E20E30 By carrying out the integration over d3p3 70

71 P | d3pld3P2 5(E) (A-3) 0 0 0 E1E2E3 where 6(E) represents energy conservation. Now for further calculation we need information from 3-momentum conservation in the PT = 0 reference frame (Figure 25)o Pl P3 P12 Figure 250 Momentum vector for 3-body final state in center-of-mass system. p3 P + P2 - 2plp2 cos Q12 (A-4) Thus for given Ipj and Ip2 2IP3lddP31 = 2IplII-2Id(cos Qi2) (A-5) and since pi2 + Mi2 = (Mi+Ti) - Itilldpih = EidTi (A-6) then

72 c~S I;P1 I P2 I jde1|jdrP2 IdldO23(E) EOaEE20 = IP1 IIP2 d2,1d22dTldT25(E) (A-7) E3 where 21, Q2 are the solid angles for particles 1 and 2o This integration over solid angle becomes a numerical factor times the integral over cos @12 p c~x i2+J dTdTidT2d(cos '12)5(E) (A-8) E3 The integration over dT3 eliminates 5(E) and we get p~ CZ dTldT2 (A-9) Therefore the differential of the density of final states is proportional to the area dTldT2 ioeo,, if we make a scattergram for T1 and T2, it should give a uniform distribution inside the kinemnatically allowed regiono The kinematically allowed region can be calculated by Eqo (A-4) and the energy equation E1 + E2 + E3 = ET by eliminating p3 2pIp2 cos 12 = ET + M1 + M2 + M - 2ET(E1+E2) + 2E1E2 (A-10)

75 The kinematically allowed region is the solution corresponding to -1 < cos 012 < +lo We set cos 012 = 1 and square the equation. Then two solutions of E2 are obtained for each value of Elo These two solutions give the kinematical limits~ Since we have the relation 22 2 Mjj + = ET (A-11) where Pij is the 3-momentum of the ith and jth particle system. We express M j. by 1j = (ET-Mk) 2ETTk (A-12) Therefore a Dalitz plot can be made in terms of Mjo Since the density is uniform, the probability that the two-particle mass becomes Mij is proportional to the length between the corresponding kinematically allowed limitso This probability is invariant under Lorentz transformations and we refer to it as "phase space" in graphs where we plot number of events versus Mij o

APPENDIX B PRODUCTION AND DECAY RATIO OF NUCLEON ISOBAR (123853/2,/2) Assume the reaction '-+p + N* 3 3 +t goes through p meson exchange 22 (Figure 26)~ A AV p NP Figure 260 One-meson exchange diagram for N* production~ We use labels A and A' for the meson vertices and B and B' for the nucleon vertices (see Figure 26)o On the vertices A and A' (0) + pow) (B-l) and for the isotopic spin functions we have It do 1lal > 4 1 ( -2- Tcdh ) (B2) This relation shows that the probability of A going to both A and Av vertices is the same~ For the B and B' vertices the reaction is p + o() + N*+(O) (B-3) 74

so we construct |P=P P$> i | > (B-4a) 43 2 2 3 2 2 2 p = a 3 1> 1 1' a 2 2 2 2 where the coefficients are Clebsh-Gordon coefficients for combining angular momentum 35 Therefore we have a 2 to 1 ratio for production of isotopic spin 3/2 states from p~+p and p-+p. Now, the possible decay modes of N*Is are N*+ 3 1 Tr ++ n 2 2 I0 + p (B-5a) N*bO. > + t~ + n 2 2 ~- + p (B-5b) and the isotopic spin functions for these modes can be written 5 12> = P >p + i +n > (B-6a) 2 2 3 3 = p >+ iOn >. (B-6b) 2 2 3 3 They are independent of production mechanism. Since we cannot see the t~n decay mode, only 1/3 of the isotopic spin state - > is ob2 2 servable. N*+ 3 1 > decays into either -+n or tOp,, and the ratio is 2 2 1 to 2o By combining (B-4) and. (B-6), we get that the ratios N*(top); N*(t+n)v N*(~-p) are 4:2:1o

76 Fo'V N* production by one-particle exchange, one must exchange a doubly charged particle. This is not possible for p meson exchange since the p meson has T = 1. Therefore, we get the ratio of the N*'s should be N*Q(~rp); N*(+ n); N*( -p); N*( -n) - 4:2:1:0 provided the production is due to pure p exchange.

APPENDIX C ANGULAR DISTRIBUTION OF p- DECAY ASSUMING a& EXCHANGE Consider the reaction E- + p + p + p which can exchange either a ~o or.w without violating any conservation law. Assume the quantum number of the p is T = 1 (1-+)o The following momentum diagram can be drawn in the rest system of the p (Figure 27). o0 c in p or exch ~t0 ut.Figure 270 One-particle exchange diagram in rest system of final di-piono Since for T~ exchange the angular distribution is well discussed previously in this report, we will only discuss the X -exchange mechanism. Let ub represent the spin function of the u exchange, where a and b are the spin and z-component of spin (a = 1 for the co). Since we assume the parity in this reaction is conserved, the W-d orbital angular momentum should be 1 (p, cu, t all have parity = -) Thus we can have the wave functions 77

78 r1 o o I 1 ~in + eexch + - (Yl~-Ylel) = 1 ( yl- Yol ) = al,2 + (Y[-1 w -Y0Wl) = P where Y's are spherical harmonics, and pl are the p spin functions. Now if we take the axis of incoming r and exchange w as z-axis, the z-component of orbital angular momentum has to be O0 Therefore only Y1 terms are allowed and we get some general combination of p-1 and p-1 for the p-spin function Thus the Y's for the final state pions arise as Yj(Al+~Bc+ -+l) = B. AYp + BYB1 Parity is conserved on the nucleon vertex so the whole process is independent of the polarity of exchange particles. Therefore coefficient IJA = IBI, thus we get for the density of final state EI(- r) = IAY: + BY-1 12cC sin2 g ei(t+) - e idI where 8 is the phase angle between A and Bo If we experimentally integrate over ~ by not observing the ~ angle, we get I(0) cC sin2e

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