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Pion induced pion production on deuterium Sossi, Vesna 1990

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PION I N D U C E D PION P R O D U C T I O N O N D E U T E R I U M By Vesna Sossi Laurea in Physics, University of Trieste, Trieste, Italy  A THESIS THE  S U B M I T T E D IN P A R T I A L F U L F I L L M E N T REQUIREMENTS DOCTOR  FOR THE DEGREE  OF  OF  PHILOSOPHY  in THE  FACULTY OF GRADUATE  STUDIES  PHYSICS  We accept this thesis as conforming to the required standard  THE  UNIVERSITY  O F BRITISH  COLUMBIA  November 1990 © Vesna Sossi, 1990  OF  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department o r by his or her representatives. /  It is understood  that  copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  Abstract  This thesis describes measurements of the pion induced pion production reaction 7 r d +  —• 7 r 7 r ~ p +  p performed with a 280 M e V incident 7r beam at T R I U M F . The data are +  compared with an improved version of the Oset and Vicente-Vacas theoretical model [12]. The goal of the experiment and of the analysis was to provide a larger body of data for the free reaction and to test the validity of theoretical models. In the process, the ability to determine the values of the coupling constants C ,  , gx*&  T  within such  a model framework would be explored. The knowledge of the precise value of these coupling constants would constrain A * decay branching ratios and other pion induced :  reaction mechanisms like Double Charge Exchange. A previous experiment [23] had indicated that the pion induced pion production on deuterium is essentially a quasifree process with the reaction occurring on the neutron leaving the proton merely a spectator. The main difference with respect to the free reaction is the effect of Fermi motion of the neutron. Although we were interested in studying the free reaction (7r~p —• 7r 7r~n), we chose a deuterium target so that the +  experiment could be run with a 7r beam, since the 7r~ beam flux is about 6 times lower +  than the flux of the positive pion beam at 280 M e V , the energy at which our experiment was performed. Such a flux would have required a much longer running time for the experiment in order to achieve the same statistical accuracy. The quasifree nature of the process was also confirmed in our experiment. This experiment involved a coincidence measurement of the quasifree process and as such provided four-fold differential cross section spectra of the reaction thus allowing  for a microscopic comparison between data and theoretical models. In the theoretical description we incorporated additional amplitudes for the N*—• N(mr) .  p wave  diagrams  required to describe the reaction cross section at T = 280 MeV. We also added the T  Fermi motion of the nucleon to the model to account for the deuterium environment. The 'free' parameters of the model are the largely unknown coupling constants listed above. We fixed C to be -2.08 by requiring the energy dependence of the model to be that of the measurement of [22] and compared the energy and angular distributions of the model to our data for several values of the f& and g^-^r coupling constants ranging between 0 and 2 (where the units are 4/5 fssu)  and between 1.08 and 1.53  respectively. We found reasonable sensitivity of the model to the f& variation, but only limited sensitivity to the value of the gs*& coupling constant. Overall we achieved a r  very good agreement between data and the theoretical predictions for f& values smaller than 0.5 and ^ v»^ values closer to its lower limit. Improved statistical accuracy of the ;  r  data would however be needed to better constrain the values of the coupling constants. On the basis of our results we feel that this model is a useful tool for planning future experiments and that a more extensive ( r , 2n) experimental program, where differential cross sections are measured for differing isospin channels, would provide a further, more stringent test on the model allowing for a more precise determination of the coupling constants.  iii  Table of Contents  Abstract  ii  List of Tables  viii  List of Figures  x  1 Introduction  1  1.1  Pion Induced Pion Production (7r,27r)  2  1.1.1  Theoretical Environment  3  1.1.2  Experimental Efforts and Justification for Our Experiment  2 Experiment  ...  7 9  2.1  M i l Channel  9  2.2  Apparatus  11  2.2.1  Beam Counters  11  2.2.2  QQD Spectrometer  lo  2.2.3  CARUZ  16  2.2.4  Targets  IS  2.3  Data Acquisition  IS  2.4  Electronics  19  2.4.1  QQD  19  2.4.2  CARUZ . .  23  2.4.3  Coincidence and Beam Electronics  24  iv  3  Apparatus Calibration  26  3.1  26  3.2  3.3  4  3.1.1  Wire Chamber Calibration  27  3.1.2  Target Traceback Reconstruction  29  3.1.3  Delta Coefficients  29  3.1.4  Angular Acceptance  31  CARUZ  33  3.2.1  Absolute Energy and Time Calibration  33  3.2.2  Particle Identification  35  Number of Target Scattering Centers Determination  35  3.4  Determination of the Number of Incident Pions  38  3.5  Background Removal  39  D a t a Analysis  44  4.1  Apparatus Settings  44  4.2  Differential Cross Sections  45  4.2.1  45  4.3  5  QQD  Weight Factors  Histograms  48  4.3.1  T  4.3.2  T^- spectra  4.3.3  6 + spectra  7r+  spectra  49 52 53  Theoretical Model  57  5.1  Modified Weinberg Lagrangians and Related Diagrams  58  5.2  Isobar Intermediate State Lagrangians and Related Diagrams  61  5.2.1  Diagrams with one and two A's in the Intermediate State . . . .  63  5.2.2  Diagrams with N* in the Intermediate State  68  v  6  5.3  Improvement of the Oset Vicente-Vacas Model  69  5.4  Cross Section Sensitivity to Various Parameters  73  C o m p a r i s o n between the T h e o r e t i c a l M o d e l a n d O u r D a t a  "6.1  6.2  7  8  A  75  Introduction of the Fermi Motion in the Model  76  6.1.1  Neutron Momentum Probability Distribution  76  6.1.2  Insertion of the Neutron Momentum into the Model  79  MonteCarlo Simulation of Experimental Conditions  83  6.2.1  Event Vertex Selection and Definition of the Coordinate System  85  6.2.2  Pion Direction  87  6.2.3  Energy Loss and Decay Probability Calculation  88  6.2.4  Pion Decay Vertex Determination and Event Acceptance  6.2.5  Input and Output Spectra  . . . .  93 94  Results  98  7.1  C determination  98  7.2  QN*A*  and  101  7.3  x Analysis and Total Cross Section Estimate  114  7.4  Comparison to other data  117  f&  Variation  2  Discussion and Conclusion  118  8.1  Discussion  118  8.2  Conclusion  119  E x a m p l e o f A m p l i t u d e C a l c u l a t i o n and \T\ for a P o l a r i z e d Target 2  A.l  121  Amplitude Calculation  121  A.1.1  122  Vertex Factors Calculation  vi  A.1.2  Propagator Calculation  126  A . 2 Square of the Coherent Sum of the Amplitudes  B  126  A.2.1  Unpolarized Target  127  A . 2.2  Polarized Target  128  Amplitudes Related to the New Diagrams and Model Cross Section Calculation  131  B. l  Amplitudes Related to the Added Diagrams  131  B.2  Model Cross Section Calculation  133  B. 2.1  Event Determination  133  B.2.2  Grid Elements and Output Format  134  Bibliography  135  vii  List of Tables  4.1  Apparatus settings. The runs with the Q Q D at 80° were performed in 1987, the rest in 1988.  In the abbreviation the number refers to the  Q Q D angle, the first letter to the central Q Q D energy setting, Low, if 50 M e V , High, if 80 M e V , and the second letter to the C A R U Z angle. Low, if - 5 0 ° , High , if - 1 0 0 ° 4.2  45  Uncertanties in the variables defining the four-fold differential cross section expressed in %  4.3  49  Multiplicative correction factors for the T^- spectra that account for r  +  loss for each setting 5.1  56  Total cross section sensitivity to the coupling constants variation. The coupling constant f& is expressed in units (4/5 f)  6.1  74  Material layers from target center to S i . The horizontal lines separate various elements of the apparatus: target, air, Tl-(-wrapping, air, W C l l , air, W C 1 2 , air  6.2  89  Total pion decay probability between target and S i for different distance intervals. Note that not always all the layers were considered (see text). The materials that were not skipped are listed in table 6.3  91  6.3  Materials considered in the pion energy loss calculation  92  7.1  Calculated cross section with different C values  99  VI11  7.2  Parameter sets used in the theoretical calculations and \  2  divided by  the number of points for each configuration. The last column shows the average reduced \ 7.3  2  value (see text)  Calculated total cross section values for three parameter sets  ix  115 117  List of Figures  1.1  Successive mechanism for D C X  2  2.1  M i l channel  10  2.2  Apparatus layout  12  2.3  Beam profile from the in-beam PCOS. The Full Width Half Maximum during the experiment ranged between approximately 1.6 and 2.2 cm. .  14  2.4  QQD related electronic circuit diagram  20  2.5  CARUZ related electronic circuit diagram. PD denotes a passive divider. 21  2.6  Coincidence and beam monitoring related electronic circuit diagram. . .  3.1  Solid angle 6 dependence. The solid circles are the values obtained with  22  the W C l as front end QQD W C , while the open boxes are the points obtained with PCOS. The solid lines are the fits to the points 3.2  32  ToF vs. ENER spectra, a) All the data. Protons, pions and positrons are clearly separated, b) Pion band after positron removal and box cut c) Proton band after positron removal and box cut  3.3  AE vs. ENER plot. Pions and protons are clearly separated except for the ones stopping in S i that populate the straight line  3.4  36 37  Target frame reconstructed from the in-beam PCOS and the front end QQD W C particle trajectory backtrace  40  3.5  Background rejection in the QQD arm  41  3.6  Particle momentum as a function of the ToF on the QQD side  43  x  4.1  Measured four-fold differential cross sections for the Q Q D 50° angular setting. Each spectrum refers to an apparatus setting. The error bars include the statistical and systematic uncertainties  4.2  50  Measured four-fold differential cross sections for the Q Q D 80° angular setting. Each spectrum refers to an apparatus setting. The error bars include the statistical and systematic uncertainties  4.3  51  T - and 6^+ differential distributions for Q Q D 50° angular setting. The T  error bars include the statistical error and the systematic error 4.4  54  T - and 0^+ differential distributions for Q Q D 80° angular setting. The T  error bars include the statistical error and the systematic error 5.1  55  Diagrams described by Weinberg Lagrangians: a) pion pole term, b) contact term, c) two point diagrams, d) three point diagrams  5.2  Feynman diagram for the xNxN  59  amplitude through the p exchange in  the t-channel  61  5.3  Two point diagrams with nN s wave amplitude  62  5.4  Two point diagrams with TTTTNN coupling through p-wave p exchange. .  62  5.5  Three point diagrams with only nucleons in the intermediate state. . . .  63  5.6  Two point diagrams with a A in the intermediate state.  The A " A "  coupling occurs through a p exchange  64  5.7  Three point diagrams with a A and a nucleon in the intermediate state.  65  5.8  Three point diagrams with two A's in the intermediate state  66  5.9  Cross section calculated with different mechanisms  67  5.10 Two point diagrams with A**in the intermediate state and coupling  N*N{Tnr) -wavt t  6S  5.11 Diagrams with the A * —• A (7T7r) _ T  7  p  xi  decay  70  5.12 Diagrams describing the reaction mechanisms that contribute to the pion induced pion production at T = 280 MeV T  ;  6.1  Hulthen wave function as a function of momentum (q)  6.2  Neutron momentum probability distribution.The distribution is not nor-  72 77  malized to 1  78  6.3  Calculated 7^+ distribution and data for two geometrical configurations.  80  6.4  Neutron momentum projection along the x axes. The projections on the y and z axes follow the same distribution  81  6.5  Invariant mass distribution including the nucleon Fermi motion  82  6.6  Schematic structure of the program that simulate the experimental conditions  6.7  86  Distribution of the differential pion decay probability (dP/dx) as a function of the distance from the target center (cm). Discontinuities of the slope occur at the interface of different materials due to an abrupt pion energy loss variation  93  6.8  Fit to the theoretical T * distribution points  95  6.9  Example of Montecarlo input and output spectra  97  7.1  x distribution as a function of C. The solid line connects the estimated  v  2  points  100  7.2  theoretical distributions compared to data  102  7.3  7~+ theoretical distributions compared to data  103  7.4  T - and 8^+ theoretical distributions compared to data  104  7.5  TV- and 8^+ theoretical distributions compared to data  105  7.6  T + theoretical distributions compared to data  106  7.7  T + theoretical distributions compared to data  r  r  n  JT  xii  107  7.8  T„- and 0^+ theoretical distributions compared to data  7.9  T„- and 8^+ theoretical distributions compared to data  108 :  109  7.10 T + theoretical distributions compared to data  110  7.11 T + theoretical distributions compared to data  Ill  n  7.12 T„- and 9^+ theoretical distributions compared to data 7.13 7^- and # + theoretical distributions compared to data pi  112 113  7.14 Average reduced \' value as a function of f& variation. The three curves 2  correspond to three different g^^ values. The lines connect the points. 116 r  A.l 2 point diagram  122  xm  Chapter 1  Introduction  T h e p i o n has a peculiar position i n the hadron family. Its mass is almost one order of m a g n i t u d e smaller t h a n the typical hadronic mass scale of 1 G e V . Consequently the p i o n defines the length scale of the nuclear physics d o m a i n (sr 1.4fm) a n d the range of the strong nucleon-nucleon interaction. Current strong interaction theories consider the smallness of the pion mass to be of fundamental importance. T h e y relate it to a basic quark symmetry, the chiral symmetry, that arises from the quark helicity conservation i n the limit of zero quark mass.  In this limit the pion mass is zero.  S t u d y of pion  properties is thus relevant on a double level: on the nuclear physics level, where the inner structure of the hadrons can be neglected and on a more fundamental quark level, where it can test Q u a n t u m C h r o m o D y n a m i c s ( Q C D ) theories. Several pion-nucleon, pion-nucleus and pion-pion interactions must be understood i n order to achieve a comprehensive description of the d y n a m i c properties of the pion when interacting w i t h either a free nucleon or a nucleon i n a nuclear environment. These interactions include pion elastic scattering, pion absorption and p r o d u c t i o n , Single and D o u b l e Charge Exchange ( S C X , D C X ) . P i o n induced pion p r o d u c t i o n is the dominant 7rN inelastic reaction channel for pion energies below 1 G e V a n d above p i o n production  —• 7 r 7 r n , 160 M e V for TT~P —• ~°~°n  threshold (173 M e V on a free nucleon for a n d 165 M e V for  ±  +  —» 7r 7r°n). T h i s feature together w i t h the fact that the reaction ±  also involves a irir interaction vertex and a high isospin selectivity t h r o u g h the selection of i n i t i a l a n d final pion charge states makes it an attractive choice for experimental and  1  Chapter  1.  2  Introduction  TT  TT  Figure 1.1: Successive mechanism for D C X theoretical work.  1.1  Pion Induced Pion Production  (7r,2TT).  Many different interesting questions can be addressed using this reaction as an investigative tool. They can be separated into two main groups. The first, concerning the reaction mechanism itself, involves topics like the pion-pion scattering length with underlying chiral symmetry connections, nature of the intermediate states involved and strength of the coupling constants. The second one addresses the possible nuclear effects on the reaction mechanism. The reaction occurring on a single free nucleon (free process) is the most suitable tool to provide answers to the first category of questions. One additional interesting aspect of the free (7r,27r) is its connection to the successive mechanism for D C X shown in fig. 1.1 where the left side of the diagram is essentially a (7r,27r) reaction ([1]). The importance of D C X reactions is the fact that by their very nature they involve two nucleons and can thus be used to study nucleon-nucleon  Chapter  1.  3  Introduction  correlation functions provided the reaction mechanism is thoroughly understood. One source of uncertainty is the A ^ A vertex coupling constant (see chapter 5 ). The basic  (ir, 2n) reaction is likely a more suitable environment for its determination. Answers to the second type of questions require the reaction to occur in complex nuclei. Here the attention is focused on problems such as excitation of a double A system, pion condensation precursor phenomena and pion propagation through nuclear matter. Again, knowledge of the  (7r,27r)  reaction is relevant to DCX cross section  measurements since it is technically difficult to separate inclusive DCX from the (ir, 2ir) reaction above the pion production threshold in nuclei with A > 3. The (7r,27r) cross section has to be subtracted from a DCX cross section measurement to achieve a correct DCX cross section estimate. A substantial body of theoretical work addressing both types of questions has been done in the last 25 years [2], [3], [4], [5], [6], [7], [8], [9]. This theoretical work has stimulated the experimental activity that has been accomplished so far. Only the free process will be discussed here, since it was the reaction which was the subject of our experiment. 1.1.1  Theoretical Environment  Several models have been developed to describe the free process. They are all based on effective Lagrangians derived from Current Algebra (see following paragraphs) for describing the reaction mechanism involving only pions and nucleons. However, the models differ in the way the isobars and mesons are handled in the intermediate states. These models can be divided into two major groups. Arndt and followers [10], [11] perform an isobar-model partial-wave analysis of the final state particles, while the approach originated by Oset and Vicente-Vacas describes the reaction in terms of Feynman diagrams [12]. Our model which follows the latter approach is extensively  Chapter  1.  4  Introduction  described in chapter 5. Chiral Symmetry  In the limit of zero mass, quarks are eigenstates of the helicity operator and are associated with one of the two possible eigenvalues. The QCD interactions are the same for left- and right-handed quarks and since they do not flip helicity, the left-handed and right-handed quark worlds are completely separate, but still governed by exactly the same laws. There is thus a separate flavour SU(3) invariance in each world, that can be described as: SU(3)  x  L  SU(3)  R  This statement is the description of chiral symmetry. Since in nuclear physics it is a very good approximation to consider only the lighter u and d quarks we can restrict ourselves to the SU(2) group. According to Noether's theorem [13], whenever a system exhibits a global symmetry property, there is a set of conserved currents that can be derived by subjecting the Lagrangian describing the states of the system to a symmetry transformation. In our case the application of a SU(2) chiral rotation provides a conserved vector (J£) and a conserved axial (A%) current. W?)  = q(x)r ?q(T) T  and  A»{x) =  q(x)ris fq(x) T  where 7^ are the usual Dirac matrices. 7 is proportional to the product of the s  r  a  are the isospin operators and q is the pion field. Conservation of these currents can also be expressed in terms of the so called current algebra commutation relations [14]. The chiral limit of zero quark mass would require the existence of mass degenerate states with opposite parity (for a particle of spin-parity 0 or 1~ there is a mass degenerate +  Chapter  1.  5  Introduction  partner of spin-parity 0" or  Since nature does not exhibit such symmetry this  is a manifestation of the chiral symmetry being spontaneously broken. This means that although the QCD Lagrangian is still chirally invariant there exists a continuous family of ground state solutions which are related to each other by the symmetry but which are not individually invariant under the symmetry. Whenever such a symmetry is spontaneously broken, the massless Goldstone boson appears. In this case it can be identified with a massless pion field (soft pion) [17]. To endow the pion with mass the chiral symmetry must be explicitly broken thus leading to a non conserved axial current. Effective Lagrangians Derived from Soft Pion Theory and P C A C - Weinberg Lagrangians A convenient way of describing a system is to make use of effective Lagrangians [17]. [16]. An effective Lagrangian is obtained by writing the most general Lagrangian that obeys the required symmetry laws, but then considering only those terms that are relevant to the particular system under examination. In our case the effective Lagrangians are derived from the most general pion nucleon Lagrangian that is consistent with the current algebra commutation relations given by soft pion theory. The chiral symmetry requirement together with a finite nucleon mass generates the massless pion field. The chiral symmetry is then explicitly broken by adding a pion mass term to the chirally symmetric Lagrangian. This mass term can be written as a power series in  \ f~ 4> 2  7  as  shown by Weinberg [18]: ,  =  _ l  m  ^  2  [ l  - l(N(N  + 2) + 2)]}/"V + ....]  where f is the pion decay constant (93 MeV). <p is the pion field. LK transforms under r  a chiral rotation as a tensor operator of rank N. The rank of the tensor, N , is related to  Chapter  6  Introduction  1.  the isoscalar (a ) and isotensor (a ) 0  2  7T7T  scattering length and proportional to the more  popular chiral symmetry breaking parameter f: 2a  0  + a  = %L[N{N + 2) + 2]  2  with  2a - 5a = 6X 0  2  and e H [ 3 - J V ( J V + 2)]  where I =  m /(ST:f ). r  2  The P C AC theory [20], which states that the pion mass term corresponds to the following divergence of the axial current:  predicts £ to be zero. These effective Lagrangians describe only tree level reaction diagrams (no loops the terms tree level diagram and loop diagram refer to topological properties of the diagrams) and differ by the value they assume for the £ parameter [19], [21]. Experiments (see 1.1.2) seem to confirm the P C A C prediction favoring the Weinberg lagrangians that are consistent with P C A C . Chiral Perturbation Theory It is worthwhile to mention the chiral perturbation theory approach to the chiral symmetry breaking, since in this framework the pion-pion scattering lengths (which can be measured from (7r,27r) data) could shed some light on QCD questions. In this approach the pion mass arises from the introduction of the quark mass in the QCD chiral lagrangian in a perturbative way [17], [15]. By introducing a chiral transformation the Lagrangian is expanded in a power series of the pion four momentum (q). where  Chapter  1.  7  Introduction  the lowest order term coincides with the Lagrangians derived from soft pion theory. At sufficiently low energies the higher order terms can be regarded as corrections to the lowest order term parametrized by constants. The inclusion of higher order terms requires a considerable theoretical effort and presently a calculation up to the order q  4  has been done. According to [17] the different theoretical approaches predict different values for the coefficient of this term. Experimental results are needed to discriminate amongst them. 1.1.2  Experimental Efforts and Justification for Our Experiment  A complete list of pion induced pion production experiments on a single nucleon can be found in [10] and [11] (except for the experiment described in [26], [27] and [25]). Two of the experiments will be mentioned here since they are directly comparable to our model and our data. The first one is the 7r~p —* 7r 7r~n double differential and total cross section mea+  surement of Manley et al. [22], [23].  It was a single arm experiment in which the  outgoing 7r was detected for seven different energies of the incident pion: 203, 230, +  256, 280, 292, 331 and 356 MeV. The detection of the outgoing n was a clear signature +  of the reaction, since DCX is not possible on a single nucleon. The energy range 203280 MeV is the range of applicability of our model. Thus these measurements provide a useful test of the validity of the model. The second one is the double differential and total cross section single arm measurement of the Tt d +  —+  TT 7t~pp and ir~d —> 7T 7r~nn reaction of Lichtenstadt et al. +  +  [24]. In the interpretation of this experiment, the authors considered a model based on the reaction being a quasifree process with free process phenomenological amplitudes deduced from the results of the first experiment described above. In comparing such a model with their data they were able to conclude that pion production on deuterium  Chapter  1.  Introduction  8  was consistent with a quasifree process on a single nucleon. More specific tests of the various theoretical models were, however, hampered by the lack of adequate detailed experimental information concerning the differential cross section. The intent of our experiment was to obtain information of this kind by detecting the two outgoing pions in coincidence using a double arm system. Taking advantage of the conclusions of the earlier results we decided to investigate the free process on a deuterium target: in this way an incident ?r beam could be employed rather than +  a 7T~ beam, whose intensity is 6 times lower (see next chapter). This allowed us to perform the experiment with good statistics within a time frame consistent with beam scheduling constraints.  Chapter 2 Experiment  This experiment was performed at TRIUMF (Tri-University Meson Facility) in two separate time periods, July 1987 and June 1988. The TRIUMF cyclotron provides two independent unpolarized or polarized primary proton beams whose energy can be varied from 183 to 520 MeV with 150  and 600  nA maximum intensity respectively. Two pion production targets each serving several pion channels are situated in the proton channel (Bl) of the meson hall area. Our experiment was performed in pion channel M i l , that provides positive and negative pions in the energy range 60 to 300 MeV. The beam intensity distribution is described in [27]. The maximum 7r beam intensity reaches 100 MHz for a momentum acceptance +  of 5% and a pion kinetic energy of 210 MeV. The maximum TT~ beam intensity is about 10 MHz for the same momentum acceptance and pion kinetic energy 170 MeV.  2.1  M i l Channel  The M i l channel (fig. 2.1) [27] originates at the production target T l where the pions produced at 2.89° are deflected with a quadrupole and a septum magnet into the rest of the channel situated at 19° relative to the proton beam. The momentum dispersion at the midplane is I8mm/(%AP/P)  and its magnitude can be controDed by setting the  midplane slit widths appropriately. The beam divergence at the center of the target position in the experimental area is ±0.67° in the horizontal and ±3.2° in the vertical plane. This divergence is largely independent of the midplane slit width. 9  Chapter  2.  10  Experiment  Figure 2.1: M i l channel.  Chapter  2.  Experiment  11  The muon and electron components of the beam at the lower energies can be identified and consequently separated by measuring the time of flight (ToF) of the particles over the length of the channel (15 m) with respect to the time of a proton beam burst. At the energy of our experiment the muon and electron contaminations of the raw beam 'are negligible. The proton contamination of the beam when the channel is operating in the TT mode is reduced by a differential absorber wheel located in the midplane. The +  absorber degrades more the energy of the protons than the pions causing a difference in their momenta. The protons are then.deflected out of the beam by the downstream magnets. During the experiment we used a 0.56^/cm thick CE 2  2  absorber which re-  duced the proton contamination to 2% at the exit of the channel. The absorber wheel position, the width of the midplane slits and the strength of the magnetic fields in the magnets were controlled directly from the counting room by means of program control separate from that of the operator controlled primary beam. During the experiment the slit width ranged between 30 and 35 mm. 2.2  Apparatus  The apparatus used in the experiment is shown in fig. 2.2. It is essentially identical to the apparatus described in [28] apart from some minor changes which were implemented in the second time period of the experiment. It consisted of several beam counters, a total absorption range telescope, CARUZ, used for 7r detection and the Quadrupole +  Quadrupole Dipole (QQD) spectrometer, which identified the negative pions. 2.2.1  Beam Counters  Several counters were used to measure the beam intensity and beam stability. Such a severe monitoring of the absolute measurement of the beam intensity was needed  Chapter  2.  12  Experiment  E x p e r i m e n t a l Setup  Figure 2.2: Apparatus layout.  Chapter  2.  Experiment  13  because no beam counter was included in the event definition. We used a 5x5 cm  2  multiwire proportional counter ( M W P C ) , called P C O S for the absolute measure of the beam flux. P C O S was placed 10 cm upstream of the target. The M W P C has 64 wires in each of the x- and y-plane with 0.7 mm spacing between them. Each wire is read separately by the P C O S III readout system of LeCroy . 1  This system is particularly  suitable for high flux measurements, since only the particular wire which produces the signal remains inactive for 100 ns following particle detection. The rest of the chamber stays active. The beam intensity was calculated from the product $ = Nt, • v where  was the  number of beam bursts per second and v was the average number of pions per beam burst.  The logic outputs of the M W P C were directed into two separate electronic  paths: one to count Nb and the other to obtain the distribution of the average number of pions per beam burst. The first path consisted of a logical ' O R ' between all the wires, while the v distribution was obtained by recording the number of wires firing for each accepted event. The efficiency of the chamber depended both on the intrinsic efficiency of the chamber and on the dead time of the P C O S readout system. A complete description of the chamber and of the algorithms used to determine its efficiency are described in detail in reference [29]. The efficiency was found to vary linearly with the beam intensity from 97% at 2 MHz to 80% at 20 M H z with an uncertainty of about 2.5%. Typical beam intensities during the experiment ranged from about 10 to about 20 M H z . P C O S was also used to center the beam on the target and to constantly monitor the beam profile during the experiment. Fig. 2.3 presents a typical beam profile. The beam monitoring was completed by an in-beam plastic (NE102) ' L e C r o y Corporation Research System Division, New York 'Nuclear Enterprises  2  scintillation counter,  Chapter  2.  Experiment  14  200  '  1  1  •)•  p  100  o  10 wire  •  \ \ I  Jr  o  -10  1  f\  -  3  i  1  i  V  30  50  70  number  Figure 2.3: Beam profile from the in-beam PCOS. The Full Width Half Maximum during the experiment ranged between approximately 1.6 and 2.2 cm. B l , and two downstream plastic scintillators. BI, a thin 6 cm diameter circular counter fitted into a fast base capable of handling fluxes up to 10 MHz, was used mostly for timing purposes during calibration work (see chapter 3). B l also provided an absolute measure of the beam flux [30] in the runs with lower beam intensity thus providing an independent check of the PCOS performance. A segmented hodoscope consisting of four 10x10 cm plastic scintillators arranged 2  to form a square (HODO) was placed downstream from the target to monitor the beam stability and rate. Each scintillator intercepted approximately a quarter of the beam  Chapter  2.  Experiment  15  thus reducing the counting load on each photomultiplier. The ratio between the sum of the counts in the upper two vs. the sum of the counts in the lower two and the ratio between the sum of the counts in the right pair vs. the sum of the counts in the left pair provided an additional measure of the vertical and horizontal beam stability. A logical 'OR of the four scintillators provided an independent monitor of the beam intensity. 1  A final measure of the incident beam intensity was provided by a coincidence between three small aligned counters (TT 1 • 7r2 • 7r3) viewing the downstream side of the HODO, at the centre of which a 6 mm thick piece of aluminum was attached. The number of counts in (7rl • 7r2 • 7r3) was then proportional to the beam flux since the scattering cross section from the aluminum sheet is a constant for a constant geometrical configuration of the apparatus and constant pion beam energy. The ratios between all the flux measurements agreed to about 2% throughout the whole experiment, a value which is within the uncertainty of the PCOS efficiency estimate. 2.2.2  Q Q D Spectrometer  The QQD [31] consists of a quadrupole-quadrupole-dipole spectrometer containing two MYVPC's at its entrance and two MWPC's together with two plastic scintillators at its exit.  It accepts particles in the momentum range of 77 MeV/c to 195 MeY/c  (pion energy range from 20 to 100 MeV) for 2 central momentum magnet settings, corresponding to a momentum acceptance ±20%. The quadrupole magnets focus the incoming particles thus increasing the spectrometer solid angle (about IS msr), while the dipole magnet provides the dispersion, bending the selected particles 73° to the left (defined by an observer looking downstream the beam direction). During the experiment the horizontally focusing quadrupole was inoperative due to a short circuit in its power coils. Although this did not greatly affect the spectrometer solid angle (at the most 10 %), it moved the focal plane further downstream from the back wire chambers  Chapter  2.  Experiment  16  thus decreasing the intrinsic resolution of the spectrometer. P a r t i c l e identification was obtained f r o m knowledge of the m o m e n t u m together w i t h the T o F measurement.  The  m o m e n t u m was determined by particle trajectory reconstruction i n the Q Q D using the M P W C position i n f o r m a t i o n , while the T o F was measured over the 2.38 m path length o f the Q Q D . T h i s p a t h is long enough to provide a good separation between pions and electrons throughout the whole Q Q D m o m e n t u m range. D u r i n g the first experimental time period the M W P C ' s described i n [28] were employed. T h e vertical (y-plane) information was provided by cathode wires parallel to the horizontal anode wires and the horizontal (x-plane) i n f o r m a t i o n was given by another set of cathode wires oriented vertically, and placed on the opposite side of the anode wire plane.  T h e coordinate system was chosen i n such a way that the z axis  coincided w i t h the direction of particle m o t i o n , y was vertical (positive i n the up direction) and x perpendicular to the other two (directed to f o r m a conventional right-hand coordinate system).  T h e cathode wires were connected to printed circuit delay lines  that were i n t u r n connected to time-to-digital c o n v e r t e r s ( T D C ' s ) , f r o m w h i c h the particle position information was extracted as described i n the following chapter. In the second experimental time period the first Q Q D front end wire chamber was replaced w i t h a 5x5 cm  2  P C O S wire chamber having 64 wires per plane operating i n the same  way as the in-beam P C O S described earlier. T h i s allowed higher rates i n the front end Q Q D chamber.  2.2.3  CARUZ  T h e total absorption range telescope C A R U Z and its calibration are extensively described i n [28] a n d [32]. T h i s detector subtends a solid angle of 190 msr w i t h respect to a pointlike target placed about 1 m away. It consists of a t h i n scintillation counter T I , two M P W C ( W C l l and W C 1 2 ) , and a stack of 5 plastic scintillators (NE102) to  Chapter  2.  Experiment  17  which an additional one, 5 cm thick, was added in the second series of runs to increase the energy range of the detected particles. The particles were identified by a ToF vs. ENERgy algorithm, where the time of flight was measured between T l and the first scintillator of the stack, S i , over a 76.8 cm pathlength and E N E R is the sum of the particle energy losses in each scintillator. To guarantee that the total energy of a particle was measured we required it to stop in the stack by using the last scintillator as a veto counter.  The total thickness of  the scintillators was 10 g/cm in the first experimental period and 15.1 g / c m in the 2  second.  2  The energy ranges of the detected particles in the two periods were 8 - 60  M e V and 8 -70 M e V for pions and 16 - 140 MeV and 16 - 150 M e V for protons. The lower energy threshold was determined by the amount of material the particles had to traverse before reaching the first scintillator of the stack. For the first part of the experiment W C l l was a standard T R I U M F chamber having etched strips on an aluminized mylar foil acting as a cathode plane whereas WC12 was identical to the exit Q Q D wire chambers. In principle position information from these wire chambers could be used to define the angular coordinates of the particles and to enable extrapolation of the particle trajectories back to the production point together with some information about the n to u inflight decay.  Even if a detected muon  originating from a decay of a pion belonging to the reaction would constitute a valid count, it has to be distinguished from a pion, since the muon has a different kinetic energy from the decaying pion and this would introduce a distortion in the pion energy spectra.  Since in the second experimental period the wire chambers were not used,  we preferred to ignore the information supplied by the W C in the first experimental period in order to apply the same analyzing techniques to both sets of data.  The  information described above were obtained in the following way. The particle angle was obtained from the position calculated from the time information supplied by the  Chapter  2.  18  Experiment  T D C ' s connected to the two sides of S i (see following chapter), the event vertex was determined by reconstruction of the spectrometer trajectories back to the target using the information from the front wire chamber of the Q Q D (see following chapter). The muon distortion of the 7r energy spectra was modelled by MonteCarlo simulations +  (chapter 6).  2.2.4  Targets  Several targets were used during the Q Q D calibration procedure.  Two strip targets  consisting of 3 mm wide nichrome strips were used to check trajectory traceback using the Q Q D front wire chambers. One of the targets had strips oriented horizontally with 12 mm spacing and the other was oriented vertically with 10 mm spacing strips. A thin 1 2  C target with area! density \2Amg/cm  2  was used for the Q Q D solid angle acceptance  measurement (see chapter 3). For the experiment itself we used a cylindrical 5.08 cm high liquid D target having 2  a radius of 2.54 cm and density 0.1644y/cm at 23.0 K . This cryogenic target was 3  protected by a liquid nitrogen heat shield at 77.0 K . In this condition its density could change at most 4%.  The target frame as well as the vacuum chamber foils and the  vessel containing the LD  2  were a source of background events when measuring the the  pion induced pion production occurring on deuterium. A run without LD  2  in the target  (empty target run) was performed to determine the position of the background events with trajectory backtracing so as to be able to put appropriate software cuts on the target backtrace region.  2.3  Data Acquisition  The data acquisition system used in the experiment was identical to the one described in [29].  We used a PDP-11/34 computer with a R S X operating system on which we  Chapter  2.  19  Experiment  ran a modified version of STAR [34] used to both collect the data from the CAMAC crates and to perform some online analysis for diagnostic purposes. A prompt, called LAM (look-at-me), to the acquisition system was provided by the output of an event coincidence unit (see 2.4.3.). The relevant event information, time and pulse heights in a digital form obtained from TDC's, ADC's (Analog to Digital Converter) and scalers as well as a C212 CAMAC module bit pattern value were then 3  read from the CAMAC system and stored in a buffer that was periodically emptied to tape. An online analysis was performed on these events before new events were accepted. As the scalers were not inhibited during the online event analysis, a dead time correction had to be applied to our data. However this correction was generally only about 5% - 10%, since the event rate during the experiment was typically 3.5 per second (see section 3.4). 2.4  Electronics  Since this was a two-arm experiment the electronics associated with the apparatus could be separated into three distinct parts: electronics related to the QQD, to the CARUZ, and circuitry related to the beam monitoring and coincidence requirement between the two arms. Figures 2.4,2.5,2.6 show the three electronics circuit diagrams (same as [29] except for the modification implemented in the second experimental time period). 2.4.1  QQD  The QQD provided only timing information from the wire chamber and scintillator signals. These were later translated to the particle ToF and its trajectory. The STOP for the time offlightwas provided by the E l scintillator. Outputs from the two sides 3  E G k G Ortec  Chapter  2.  20  Experiment  irr  1, I — v  SPEC7  <3 B>  fAN OVT  >DJSCRIKiNA70R  £>• £>,  >CONST  nuc DISC  >71MINC  riLTXF  B r—v  A N O D E S  A M P  K E A K TIKES  | CC [ CATE CENERATOF  ) AND  l  •  SCALER  C Ctt  O ADC  Figure 2.4: QQD related electronic circuit diagram.  Chapter  2.  Experiment  21  Figure 2.5: CARUZ related electronic circuit diagram. PD denotes a passive divider.  Chapter  2.  Experiment  22  •roc  STAJ5T  surrs  KIM D R A T R KA  KB  HC  HC *2  Figure 2.6: Coincidence and beam monitoring related electronic circuit diagram.  Chapter 2. Experiment  23  of the scintillator were therefore first directed into Constant Fraction Discriminators (Ortec 934) (CFD's) to reduce timing uncertainty. The outputs of the CFD's were routed to a Mean Timer (LeCroy 624) (MT) to generate a signal which was independent of the particle position in the scintillator. Although only E l was used as a ToF stop, the signal from E2 was fed through the same electronic path to provide some redundant information. The E l and E2 signals were fed to a coincidence unit (SPECT), one output of which was fed into a scaler while a second output was used to construct further coincidences. In the first half of the experiment there were 4 anode and 24 cathode delay line signals (see following chapter). The latter were first discriminated to reject background noise and to produce logic pulses and ultimately fed into TDC modules. The anode pulses were part of the fast logic in defining the QQD event. After being discriminated the following coincidence signal was generated  (W1A + W3A) • (W4A + W5A) = ANODES (where WiA is the i  th  MPWC anode signal) to guarantee that at least one of the front  end and one of the back end of the QQD wire chambers fired. An accepted ('good') QQD event was then identified by the coincidence SPECT • ANODES. In the second part of the experiment the signal from the first wire chamber anode was replaced by the PCOS fast 'OR' output. The coincidence logic remained the same. 2.4.2  CARUZ  Both time and energy information were extracted from the signals generated by the CARUZ detectors. The signals coming from each side of T l and Si were passively split and directed to TDC's and ADC's. The pulses directed to the TDC's werefirstfed into CFD's and MT's to yield pulses whose timing was independent of the pulse amplitude and particle position in the scintillator. One very narrow output pulse from the T l MT  Chapter  2.  Experiment  24  (about 10 ns wide) was brought into coincidence with the broader LAM output. One output of this coincidence provided the START for all the TDC's. A second output from the T l MT was fed into the coincidence 2T • 51 that defined an accepted ('good') CARUZ event, while a third output was directed to a scaler. The signals from each side of Si werefirstsent to a Timing Filter Amplifier (TFA) to compensate for their distortions due to the large size of the scintillator bar (95 x 20 x .7 cm ). The left and right outputs were then fed to a CFD and then to a MT. 3  One output of the mean timer provided the input for the T l • 51 coincidence, while the other one was fed to a TDC module. The signal coming from the right side of the scintillator bar was also directed to a TDC. The timing of this signal was recorded to provide particle position information (see following chapter). The scintillators S2-S6 were used only for energy information: the signals were first attenuated and then connected to ADC inputs. Since S6 was used as a software veto counter, its discriminated signal was also fed to the C212 pattern unit. 2.4.3  C o i n c i d e n c e a n d B e a m Electronics  An overall valid event corresponded to a coincidence between the two arms: EVENT  = SPECT  • ANODES  • (Tl • 51).  Some additional circuitry was therefore needed to provide this coincidence signal as well as to allow for the dialog with the PDP-11. The key coincidence was the LAM, a four-fold coincidence. The first input to the LAM was a logic 'OR' between an EVENT and a BEAM SAMPLE. These were also fed to the C212 pattern unit. The BEAM SAMPLE event defined as BI- PCOS • GG (where GG is a LeCroy Gate Generator 222) was recorded once a second to provide a constant monitor of the beam structure. This was particularly necessary during calibration work, where the energy of the incident pion beam was  Chapter  2.  25  Experiment  lower so that the electron and muon contaminations of the beam were not negligible. The second input to the LAM, provided by the computer  BUSY  signal produced  by a CAMAC NIM driver, inhibited further data acquisition during acquisition and reading of the event by the computer. The third input was a RUN IN PROGRESS signal. This was a coincidence between a signal from a beam rate monitor, a wire chamber high voltage trip monitor, a manual start/stop and a NIM driver output that provided a start or stop from the computer when a run was started or ended. In this way data acquisition was inhibited when the cyclotron duty factor was too low and when any of the MPWC's was not working. The last input to the LAM came from the NIM output of the LAM itself in order to provide a fast inhibit of the LAM while waiting for the BUSY signal from the computer. The output of the LAM coincidence provided the strobe for the C212 pattern unit and gated the ADC's and the PCOS readout system, while the start for all the TDC's was provided by the coincidence between the LAM and T l . During the single arm calibration runs the role of T l was provided by Bl. In the double-arm configuration T l was timed with respect to Bl to avoid any timing shift between the two arms. The beam monitoring circuitry, constructed of discriminators and scalers, was connected to B l , to each of the scintillators forming HODO and to (7rl • TT2 • 7r3). Bl was also used to monitor the proton component of the beam: its signal was fed in parallel to high and low threshold discriminators, whose outputs were then connected to scalers. The proton contamination was measured by the ratio of these two rates.  Chapter 3  Apparatus Calibration  In order to be able to convert the pulse time and height characteristics to absolute particle time and energy characteristics it was necessary to calibrate the instruments. A correct calibration also makes more stringent offline software tests and cuts possible and meaningful. The calibration performed in this experiment follows closely the calibration procedure described in [29] and can be divided into a QQD and a CARUZ part. In this chapter the common methods will be only briefly summarized while the ones unique to this experiment will be described in detail. The determination of the number of target scattering centers and of the incident pions as well as background removal are described at the end of the chapter. 3.1  QQD  As mentioned in the previous chapter the spectrometer identifies particles by their ToF and momentum, the latter calculated from the position information supplied by the wire chambers. The calibration procedure can be subdivided into four parts: wire chamber calibration, target traceback reconstruction, delta coefficient determination and angular acceptance measurement. Since the pulse timings are recorded in the form of TDC channel numbers the linearity of the response of the modules had also to be checked. This was accomplished by applying a stop signal with a series of known delays with respect to the start signal to each module. The response was then fitted with a linear function where the slope represented the desired conversion factor. It was found 26  Chapter  3. Apparatus  Calibration  27  to be linear throughout the modules dynamic range.  3.1.1  W i r e Chamber Calibration  The W C calibration determines the coefficients that convert the W C time information to particle position.  The time information was provided by pulses induced on the  cathode wires by the electron sheath around the anode wires formed when electrons produced during the ionization of the gas by the charged particle traversing the chamber were collected at the anodes. The wire chamber cathode wires were connected to printed circuit delay lines. The signals from their outputs (2 per x- and 2 per y-plane of each chamber) were first amplified and then fed into T D C modules. The particle position was calculated as the calibrated difference between the left and right side of each delay line: Xi  where A', refers to the i  th  = a, • (tdc  lT  — tdcu)  -f 6,  position coordinate, a, is the conversion factor from time  to position and t, is the offset used to position the center of the chamber. Since the cathode wires in the y plane were parallel to the horizontal anode wires the y plane spectrum had a picket fence structure. It was straightforward to find o, knowing the distance between the wires, that in our case was 2 mm. The first two wire chambers had the same delay line in both planes; therefore the time-to-position conversion factors for the x plane were assumed to be the same as for the corresponding y planes. The offsets for the front end chambers were fixed by centering the position distribution on the middle of the chamber. In the two back Q Q D wire chambers the x plane is segmented in three parts each containing 203 wires 1 mm apart. The x plane conversion factors were found by identifying the events that produced a signal in the middle and left or middle and right segments simultaneously and requiring their position to correspond  Chapter  3.  Apparatus  Calibration  28  to the segment borders. The offsets for these chambers was also obtained from this information. The particle x- and y-position measured by the PCOS chamber, that replaced W C l in the second part of the experiment was determined using a lookup table which associated the wire readout channel numbers with the wire positions in the chamber. One test was applied to the raw TDC information before a wire chamber position r  determination was considered valid. We defined the variables WCSUMi  where  tdc,  r  and  tdc i t  = (tdc  ir  WCSUM,  as follows:  + tdci,)  were the values of the TDC's connected to the left and right  (respectively up and down for the y direction) delay line outputs for each cathode plane of the wire chamber. WCSUM is proportional to twice the electron drift time to the anode wires. Its value was required to fall in the appropriate TDC channel interval (a typical drift time for those chambers is about 40 nsec). Once the position information was obtained, we performed one more test before further processing the event. We constructed the variable: DIFY45  = WCY4  - 0.65  •  WCY5  where W C Y i are the y positions in the respective wire chambers. The DIFY45 value must be very close to zero , since we were considering in-plane trajectories only (the factor 0.65 accounts for the defocussing effect of the exit fringe field of the magnet). Any value differing significantly from 0 indicated some other particle such as a muon originating from a decayed pion. This cut ( — 12mm < be very effective for muon rejection.  DIFY45  <  16mm) proved to  Chapter  3.1.2  3.  Apparatus  Calibration  Target Traceback  29  Reconstruction  The particle position information from the front QQD WC's was used to reconstruct the event vertex. It was not possible to use a simple geometrical reconstruction of the particle trajectory from the WC1 and WC3 positions, since the two chambers were separated by a quadrupole magnet. We therefore used the strip targets with strips parallel to the QQD x and y axes respectively to obtain the proper target traceback. The X and Y coordinates of the event vertex, XO and YO had to coincide with the location of the target strips. The form of the dependence assumed is: X  0  where  X{  and  Yi  = ai •  + a  3  •X  3  and  Y  0  = bi •  Yi +  b •Y 3  3  are the x and y positions in W C l and WC3. The ratio  was determined from the slope of the lines in the plot X  3  ai/a  3  (bi/b ) 3  vs. X\ (Y vs. Y\). The 3  magnitude of the coefficients was obtained by requiring the proper separation between the reconstructed strips. The determination of the particle polar (THO) and azimuthal (PHO) angle was easily accomplished by knowing the event vertex and the W C l coordinates since there was no magnet between the target and W C l . 3.1.3  D e l t a Coefficients  The position information from both the back and front wire chambers was needed to calculate the particle momentum, since the back wire chambers were not positioned on the focal plane. Only three points are actually necessary to calculate the particle trajectory. Since the QQD is equipped with four wire chambers, we could calculate the particle momentum using two independent sets of coordinates obtained from either W C l , WC3 and WC4 or W C l , WC3 and WC5. In order to check overall consistency we required agreement between the two values. The conversion factors that allow calculation of the momentum from the particle trajectory were found by routing particles  Chapter  3.  Apparatus  Calibration  30  of known momentum through the QQD. In what follows the particle momentum is expressed in terms of 6's, the percentage deviation of the momentum from the central momentum setting of the spectrometer. The relation between 6 and the positions (JV,) in the back WC's is:  X = A + B-6 + C-6 4  4  2 4  (1)  (same relation for WC5). A,B,C are polynomial functions of coordinates in the two front end chambers, typically up to the second or third order. A regression technique contained in the software package QQDMP [34] was used to optimize the polynomial coefficients ( 6 coefficients) to achieve the best momentum resolution. In order to obtain monoenergetic pions we placed the QQD at 50° with respect to the beam, inverted its polarity and performed a series of T T elastic scattering runs +  using the thin  1 2  C target to select elastically scattered pions with known momentum.  We varied the pion beam energy to span the QQD momentum acceptance (±20% of its central momentum setting) in nine steps and histogrammed those events, for which all four wire chambers fired, as a function of energy. With QQDMP we forced the peaks containing the events originating from the elastic scattering to fall on the required location in the energy spectra. QQDMP optimized the 6 coefficients to achieve the desired peak location and the minimum peak width. During the optimization procedure cuts were applied to DIFY45 (described earlier) and to DDIF (defined as the the difference between the momenta calculated from each set of coordinates  6 — 6 : only values of b  4  \DDIF\  {DDIF  =  < 1.2 were retained: in principle DDIF should be a 6  function, the value 1.2 corresponds to rejecting the tails of the peak distribution)). As was the case for DIFY45, any DDIF value far from zero indicates either an in-flight 7r to \i decay or the passage of some other particle. For 82 MeV pions we obtained a 2.6 MeV resolution at 6 = 0%. This number is a convolution of the intrinsic spectrometer  Chapter 3. Apparatus Calibration  31  resolution, the beam width and the straggling effect occurring in the in-beam counters. During data taking (1) was solved for 6 to provide momenta as a function of position.  3.1.4  A n g u l a r Acceptance  The spectrometer solid angle is not constant over all of its momentum acceptance range. We determined its functional 6 dependence from the same data used to calculate the 6 coefficients. For each beam energy step we determined the solid angle ( A f i in msr ) from:  AQ = N  peak  • 10 • J(6) • g(6) • 4>~ • N$ • o£ 30  l  c  (2)  where  Np k ea  is the number of events in the fitted elastic peak  J(6) is the Jacobian converting from lab to c m . frame <j> is the number of incident pions (see section 2.4)  Nt t  is the number of scattering centers in the target (see section 2.3)  g  a\ic is the c. m. cross section summarized by Rozon in [29]  g(B) = ^ f f f r l L )  ^  6 t is tg  WC jj t  nf=i  the angle between the normal to the target and the beam direction is the overall Q Q D W C efficiency calculated as w i t h  * ^ J' ^ * /  T  h  e  subscripts refer to different W C and JV,- is the  number of times each W C fired expressed in percentages of the total number of events on a run by run bases.  (n — dec) is a weighting factor correcting for pion in-flight decay (see following chapter). The solid angle distributions were slightly different in the first and second part of the experiment because of the different front end wire chambers used in each time period. Their respective 6 dependences are shown in fig. 3.1.  The distributions were fitted  Chapter  3.  Apparatus  Calibration  32  Figure 3.1: Solid angle 6 dependence. The solid circles are the values obtained with the WC1 as front end QQD W C , while the open boxes are the points obtained with PCOS. The solid lines are the fits to the points.  Chapter  3. Apparatus  Calibration  33  with a fifth and fourth order polynomial respectively and the corresponding functional dependence was introduced as a weighting factor in the event by event analysis to normalize the solid angle to 18 msr, its nominal value. The errors on the spectra were obtained by summing in quadrature all errors in the terms present in the solid angle calculation. The major contribution to the error was given by the uncertainty in the elastic scattering cross section (up to 9%). 3.2  CARUZ  A complete calibration of the CARUZ arm includes the determination of the scintillator response as well as TDC and ADC conversion factors. The TDC's were calibrated as described earlier. To obtain the calibration of an ADC it was fed a signal that produced a nearly full-scale value and then the signal was progressively attenuated by known steps. The A D C response was then fitted with a linear function to obtain pedestals and conversion slopes. 3.2.1  Absolute Energy and Time Calibration  A common calibration procedure for all the bars had to be performed, since we measured the total energy released by each particle in the whole stack. We selected 150 MeV electrons from the pion beam to obtain such an absolute energy calibration. Since the rate of energy loss of such electrons is essentially constant, the energy loss in the stack was proportional to the traversed NE102 scintillator thickness (the scintillator stack was thin enough not to induce significant electromagnetic showers). We directed the electron beam through the center of the bars and scaled the calibrated ADC values so that the mean value of the electron ADC distribution corresponded to half of the scintillator thickness in mm. The sum (left plus right) of the scaled A D C values thus  Chapter  3.  Apparatus  Calibration  34  amounted to the total telescope scintillator thickness. This procedure assured the conversion from A D C value to energy to be independent of the detailed characteristics of the individual A D C ' s and scintillator/base units. A t this stage the pion kinetic energy could be obtained from the electron calibration of the A D C ' s by using the the analytic expression determined in the first C A R U Z calibration [32]. A check of this procedure was provided by comparing the A D C values resulting when pions of known energy (50 and 65 M e V ) were incident on the stack with those calculated from the calibration. The agreement between the experimental calibrated A D C values and the ones calculated from the original calibration curves was within 0.4%. We also confirmed the original C A R U Z energy resolution to be 2 M e V for 50 M e V pions. A similar procedure was used to convert proton pulse response to energy. We also used the 150 M e V electron beam to determine the attenuation of the light in the scintillator. The amount of light collected for each particle depends on the position of the particle in the bar, since the light is exponentially attenuated while traversing the scintillator. We obtained an attenuation curve for each bar by firing the electrons through the stack at various distances from the center of the bars spanning their whole horizontal range. The summed (left plus right side ) response of each scintillator was plotted against the particle position for each bar. Using these curves we normalized each scintillator response to its central region value. The ToF was calculated between T l and S i . T l provided the start signal to all the T D C ' s thus defining time zero. A linear expression of the form: time = b • (a + T D C , ) related time to the T D C channel value (TDC,).  b is the conversion factor from channel  number to time (typically 50 psec/ch), while a determines the T D C channel offset with respect to time zero. The offset of the T D C connected to the S i meantimer ( S l M T )  Chapter  3.  Apparatus  Calibration  35  was fixed so that the S i T D C value corresponded to the electron travel time from T l to S i (their velocity is ~ c). The offset of the T D C connected to the right output of S i (SIR) was determined by requiring its difference with respect to S 1 M T to be zero (SIR - S l M T = 0), when the electrons were sent through the center of the bars. This time difference was used to determine the particle position in the scintillator (S1POS) and consequently the angle at which the particle was emitted. The timing resolution of S l P O S was about 250 psec corresponding to a position uncertainty of about 5 cm. 3.2.2  Particle Identification  The particles were identified by plotting their ToF vs. total energy released in the stack ( E N E R ) as shown in fig 3.2 . Pions were separated from protons by applying a software box cut around the pion band. The counts in the lower left corner of the spectra were identified as positrons and so were removed by an additional cut. Protons and pions are clearly separated also in AE vs. E N E R spectra (fig 3.3) (where AE is the energy released in S i ) except for those that stop in S i since in that case AE is equal to E N E R for both pions and protons. The first method was preferred, because it provided a cleaner pion separation throughout the whole energy range. Muons originating from pion in-flight decay could not be identified with these techniques however, since their mass is too close to the pion mass to cause a significantly different ToF. A detailed description of the TT decay problem is given in chapter 6. 3.3  N u m b e r of Target Scattering Centers D e t e r m i n a t i o n  The Number of Target Scattering Centers (NTSC) was another variable present in the cross section. There are two methods which could be used to determine it. The first one is purely geometrical, using the known geometry of the target, while in the second  Chapter  3. Apparatus  36  Calibration  ?0'  u  u  v  m c  10 TT  o  ^  i .•  s  TT XK  ?oc  200  yx  ENER (ADC Channel)  ENER (ADC Chonnel)  b)  o)  •-T  i  . . . .  i  . ~ ^ S ^ r — n - .  i >  ENER (ADC Channel)  Figure 3.2: ToF vs. E N E R spectra, a) A l l the data. Protons, pions and positrons are clearly separated, b) Pion band after positron removal and box cut c) Proton band after positron removal and box cut.  Chapter  3.  Apparatus  Calibration  37  ENER (ADC Channel)  Figure 3.3: AE vs. E N E R plot. Pions and protons are clearly separated except for the ones stopping in S i that populate the straight Une. one N T S C was calculated from data characterizing a 7 r d elastic scattering run where +  the nuclear cross section is well known [36]. In the first method N T S C is given by the formula:  NTSC = 2 •  gmole  where  t is the target 'effective' areal thickness (in g/cm ) and N^ is Avogadro's number 2  v  (the 'effective' area! thickness was calculated as a convolution of the beam profile and the target geometry). The factor 2 accounts for the presence of two deuterium atoms in a deuterium molecule. The N T S C found with this method was (2.36 ± .11) • 1 0 c m " . 23  2  The error originates from the uncertainty in the target density (see chapter 2) and to the uncertainty in the 'effective' areal thickness (about ± 2%) due to the varying beam profile width during the experiment (see fig. 2.3). The elastic scattering run was performed at pion kinetic energy of 84 M e V and at a Q Q D angle of 50° with respect to the beam. The Q Q D polarity was inverted to  Chapter  3. Apparatus  Calibration  38  detect the outgoing 7r . The central momentum was set to correspond to 80 MeV, +  since a first, approximate, estimate of the pion total energy loss in reaching the back end of the Q Q D was 4 M e V . During the off-line analysis a more accurate energy loss calculation showed that pions originating from elastic scattering from deuterium in the target entered the spectrometer with a kinetic energy of 77 M e V . We solved formula (2) for N T S C using the Q Q D solid angle corresponding to the 6 value - 4 % and the differential cross section value (1.82 ± 0 . 1 ) mb/sr. The N T S C obtained in this way was (2.09±0.2)- 10 cm~ where the error is mostly due to the Q Q D solid angle uncertainty. 23  2  The two values agree within the error limits. The value obtained from the elastic scattering run was used in the subsequent analysis since it was calculated in a way consistent with the Q Q D solid angle determination: any systematic error would therefore cancel since the product of the solid angle and the N T S C is the relevant quantity in the cross section calculation.  3.4  Determination of the Number of Incident Pions  The Number of Incident Pions (NIP) was calculated from the beam intensity measured by the P C O S . Two additional corrections, however, had to be applied besides the ones described in chapter 2: a factor accounting for the proton, electron and p. contamination of the incident beam and a factor correcting for the dead time of the data acquisition system. The proton contamination was determined to be about 2% by the high-low discriminator threshold method described in the previous chapter. The beam contamination was measured online using the beam sample spectra that show the ToF distribution of particles down the channel (see chapter 2). The pion fraction was typically 97% consistent with the proton background. This added a 3 % uncertainty to the beam intensity measurement leading to a total uncertainty of 4 % .  Chapter  3.  Apparatus  39  Calibration  The fractional event loss due to the data acquisition dead time was provided by the ratio between the number of E V E N T S written on tape and the number of E V E N T S recorded by the C A M A C scalers. Its value usually ranged between 5 % and 10% with a peak of about 25% for the detector configuration fully covering the allowed region for r r p elastic scattering (see following section). +  3.5  Background Removal  The background could be divided into two main parts: events originating from a (TT, 2T) reaction on a nucleon other than deuterium and events originating from reactions of a completely different type (such as random coincidence between two elastically scattered pions). To eliminate the first a software cut on the event vertex region reconstructed from the front end Q Q D wire chambers limited the reaction volume to a region within the target walls. The position of the target frames and walls was determined in an empty target run as shown in fig. 3.4. The dominant reactions that simulate a charge exchange (SCX) (n n +  —• ir p c  (7r,27r)  reaction on deuterium are single  ) and 7 r p quasielastic scattering. The first step +  to eliminate such backgrounds was the offline requirement that all four Q Q D wire chambers fired for each event. The impressive background rejection obtained with this condition is shown in fig. 3.5. The S C X reaction can generate background by producing a TT° that decays into two 7 s. Then one of the 7's produces an (e ,e~) pair so that the e" is detected by the 1  +  Q Q D while the proton triggers the C A R U Z . The Q Q D flight path is long enough to separate pions from electrons by ToF, so an appropriate cut on this quantity easily eliminated this background.  Chapter  3.  Apparatus  Calibration  40  a 0 Q  Of CX X  I  -r* 1  40  1  1  20 -  1  1  -k-  b ,.  0 - 20  -  -40 -40  -20  0  X_Beam  20  40  (mm)  Figure 3.4: Target frame reconstructed from the in-beam P C O S and the front end Q Q D W C particle trajectory backtrace. For this run the spectrometer was situated at 50°. X_Beam is the x coordinate in the reference frame where the z axis coincides with the beam direction, while X . Q Q D is the x coordinate in the reference frame, where the z direction coincides with the particle direction in the Q Q D .  Chapter 3. Apparatus Calibration  (TDC  41  Channels)  Figure 3.5: Number of counts vs.ToF in the Q Q D arm. Upper histogram: number of events with the hardware trigger A = (WCl + WC3)-{WC4 + WCb). Lower histogram: events with all four Q Q D W C firing. No other tests are imposed on the events.  Chapter  3.  Apparatus  Calibration  42  The quasielastic scattering 7r p constituted a high background (about a factor 200) +  for those runs where the detector settings were consistent with ?r p kinematics. That +  occurred for the two configurations C A R U Z at —50° and Q Q D at 50° with respect to the beam and C A R U Z at —50° and Q Q D at 80° (see next chapter). The outgoing proton is detected by the C A R U Z while the coincident 7 r can simulate a 7r~ in the Q Q D in +  two ways. In the first case the emitted 7r hits the front end W C ' s and then scatters +  from the magnet yokes towards the back end of the Q Q D . In the second mechanism the emitted 7 r only hits the front W C ' s while the back W C ' s and the scintillators +  detect uncorrelated particles in random coincidence with these 7 r ' s .  These could,  +  for example, be residual protons delivered by the beam line. The 7r can thus be +  erroneously identified as a 7r~ when the timing between the firing of the front and back Q Q D detectors simulates a 7r~ defining coincidence. Most of this background was removed by requiring a signal from all four wire chambers and imposing the D D I F and D I F Y 4 5 cuts. The residual, ranging from 8% to 20% of the acquired events depending on the Q Q D central momentum setting, was removed in the following fashion. Using the momentum information,the corresponding ToF of a pion through the Q Q D was calculated. The particle was then identified as a pion only if its measured ToF fell within a narrow gate around the calculated value . The effects of these requirements are shown in fig. 3.6.  Chapter  3.  Apparatus  700  Calibration  800  E1SUM.C0RR  43  I  7—I—I  goo  700  600  E1SUM_C0R  a)  b)  [ "I—i—i—r—|—i—I—I—p—j—r  700  800  E1SUW.C0R  900  i  l  1000  c)  Figure 3.6: Particle momentum as a function of the ToF on the Q Q D side. Abscissa units are T D C channels, a) Remaining spectrum after the software cut on the T D C channel value that separates electrons from pions. b) Events that satisfy the following requirements: 'good' C A R U Z , all four Q Q D W C firing. D D I F , D I F Y 4 5 , event coming from target region, c) Events after the very tight cut on the ToF (see text).  Chapter 4  Data Analysis  Once the apparatus was properly calibrated, the  (7r,27r)  events were unambiguously  separated from background. Four measured variables expressed in the laboratory reference frame identified each event: the kinetic energy of the the respective azimuthal angles  7r  +  and n~  (T +,T^-) n  and  The events were histogrammed to obtain  T T and 7r" kinetic energy distributions as well as T T angular distributions. The total +  +  number of collected (7r, 27r) events is 4168. We used the M O L L I histogramming package for the off-line analysis [37], after converting the data format from the one used by S T A R to write data on tape to the one readable by M O L L I .  4.1  Apparatus  Settings  During the experiment geometrical and energy settings of the apparatus were changed several times to cover as much reaction phase space as possible while still collecting a sufficient number of events per setting to keep the average statistical error below 10%. The apparatus settings are listed in table 4.1.  To obtain an estimate of the amount of  phase space covered we assumed the reaction to be a three-body process and calculated the percentage of the in-plane phase space covered for each Q Q D angular setting using the F O W L MonteCarlo software package [38]. We constrained the outgoing rr" angle within the limits of the Q Q D angular acceptance and found the ratio between the number of events falling within the apparatus detection region and the total number of 44  Chapter  4.  Data  Analysis  45  QQQDC) T QD(MeV)  OcARUzC)  Abbreviation  80  50  -50  80-LL  80  -50  80.HL  50  -100  80XH  80  -100  80-HH  50  -50  50XL  80  -50  50.HL  50  -100  50XH  80  -100  50-HH  Q  50  Table 4.1: Apparatus settings. The runs with the Q Q D at 80° were performed in 1987. the rest in 19SS. In the abbreviation the number refers to the Q Q D angle, the first letter to the central Q Q D energy setting, Low, if 50 M e V , High, if 80 M e V , and the second letter to the C A R U Z angle, Low, if - 5 0 ° , High , if - 1 0 0 ° . events. We estimated the in-plane phase space acceptance to be 85% when the Q Q D was positioned at 80° and 75% for the other Q Q D angular setting.  4.2  Differential Cross Sections  Energy and angle dependent weight factors, as presented below, were applied to the events before they were histogrammed in the four differential spectra. They corrected for the non uniform apparatus acceptance and for pion loss due to its decay.  4.2.1  Weight Factors  Each negative pion was weighted with the factor:  Chapter  4.  Data  Analysis  46  W, = 18./f  (6)  where f(6) is the polynomial fit to the Q Q D solid angle 6 dependence. In this way the solid angle was normalized to its central (6 = 0%) value throughout its momentum acceptance. Most of the muons originating from the pion decay escaped from the Q Q D path either because they were emitted at an angle with respect to the pion trajectory or because they were produced with a different momentum. The remaining muons were distinguished from pions with the D D I F and DIFY45 cuts. A weight factor was therefore introduced to account for pion loss as follows:  ^  = ( P(7S)) ex  _1  =  e x  P(^)  where m is the pion mass, p its momentum, d the distance from the target to the T  T  last W C (2.38 m), c the speed of light and r the pion mean lifetime in the pion rest 0  frame (26 nsec). The correction factors were more difficult to determine on the C A R U Z side for two reasons. The C A R U Z does not distinguish pions from muons and the energy threshold for detecting particles is low enough to make the detection sensitive to the production point of the particle in the target. The C A R U Z energy detection range includes pions whose range is less than the target thickness (which corresponds to the range of about a 12 M e V pion). The detection probability of the low energy pions thus depends on location within the target where the particle was produced ('extended target effect'). Both the extended target and pion decay cause a distortion of the integrated yield as well as of the shape of the  spectra. The distortion in the yield occurs because the  pions with a range less than the target thickness are detected only if they are produced close to the target exit, while the muons originating from pion decay can be emitted outside the detector angular acceptance. The extended target affects the shape of the  Chapter  4.  Data  Analysis  47  target affects the shape of the whole energy distribution, since the measured kinetic energy of even those pions having a range larger than the target thickness depends on the amount of target material the pion traversed before exiting the target.  The  detected muons also introduce a shape alteration, since they are produced with an energy different from that of the pion from which they originate. There is no simple way to unravel the true spectra from the contributions of these different mechanisms. If the particle is truly a pion, its energy at the production point can be calculated by knowing the interaction vertex reconstructed from the Q Q D W C . However, if the particle is a muon, this cannot be accomplished, since there is no unique correspondence between the measured muon energy and the energy of the pion from which it originated, at the point where the pion was produced. Our main goal was to find a reliable means of comparing the experimental data with theoretical distributions. To achieve this and still treat the pion and muon contamination on a reliable basis, we constructed a F O R T R A N program utilizing MonteCarlo techniques ('MonteCarlo Program'). This program folds the extended target effect and the pion decay as well as the experimental geometric and energy cuts into a theoretical pion energy spectrum which describes the pion kinetic energy distribution at the point where they were produced. The spectra generated by the program output were then compared to the experimental data. Both the MonteCarlo program and the comparison are extensively described in chapter 6. The only weighting factor applied to the positive pions was a correction for the position dependence of the C A R U Z solid angle. The C A R U Z height is constant over its horizontal range thus decreasing the subtended solid angle towards the ends of the system. Each positive pion count was multiplied by the ratio between the vertical angle subtended at the center of the C A R U Z and the vertical angle subtended at the position where the TT was detected: +  Chapter  4. Data  Analysis  48  arctan(^) Mr  V 968 /  _  where 107.5 and 968 are the half height of S i and its distance from the center of the target expressed in mm and  BQARVZ  is the  7r  +  angle with respect to the center of the  CARUZ. Histograms  4.3  It is not completely accurate to regard the measured experimental spectra as representative of the expected four-fold differential cross section, since the binning intervals used in an experimental spectrum are always finite due both to a finite apparatus acceptance and because of the need to achieve enough events per bin to have a statistically significant measurement. This last requirement dictated the binning sizes chosen for our spectra. Therefore, the experimental four-fold differential cross sections presented in this work are averages of the actual four-fold differential cross sections over the appropriate binning intervals. The four-fold differential cross section is expressed as: rfV 1  dKv  cM+dn^-dT^+dT TX  TX  TX  - ~ e• N • N b  tgt  TX  s  ' A f t + • AQ r  TX  TX  • A71+ • A T TX  TX  where t is the Q Q D wire chamber efficiency, typically about 50 %, Nb and N  tgt  are the number of particles in the beam and the number of target scattering  centers as discussed in the previous chapter, is the number of weighted events within the four-fold interval specified by Aft .+ •  dN  7  ev  A f t . - • A 7 L + • A T . - . Except for A f t . - , the size of the other unidimensional bins TX  TX  TX  r  TX  1  defining the four-fold interval was varied from one variable histogram to another to  Chapter  4. Data  Analysis  49  variable  error (%) 5 10 4  N  tgt  5.5  Table 4.2: Uncertanties in the variables denning the four-fold differential cross section expressed in %. maintain a reasonable statistics in each four-fold differential interval. The values are described in the following sections. 4.3.1  T+ K  spectra  We produced a four-fold differential cross section spectrum as a function of T + for each apparatus setting. The size of the binning intervals is listed below: A f i i = IS msr, Q Q D solid angle. Aft  2  = 190 msr, C A R U Z solid angle covering the angular region (—50° ± 30°) or  ( — 100° ± 30°) with respect to the beam according to the C A R U Z angular setting. A T ^ - = 40 M e V or 30 M e V according to the Q Q D central energy setting covering the (80 ± 20) M e V or (60 ± 15) M e V interval respectively. AT  7 r +  = 4 MeV  The 7 r kinetic energy spectra are shown in fig. 4.1 and fig. 4.2. +  The kinetic energy  of the positive pion is calculated at the exit of the target. The errors were obtained by summing in quadrature the percentage statistical error calculated on the number of events for each bin and the uncertainties listed in table 4.2.  The uncertainty in the  Chapter  4. Data  0  TT —  Analysis  50  =50°±5°  • data points  T  (MeV)  Figure 4.1: Measured four-fold differential cross sections for the Q Q D 50° angular setting. Each spectrum refers to an apparatus setting. The error bars include the statistical and systematic uncertainties.  Chapter  4.  ©  Data  TT —  51  Analysis  • data points  =80°±5°  1.0  CM  2.  " 0)  T  =60x20 M e V  6  «-100°i:28  i  1  1  1  1  r  T . =80±20 MeV 6  e  0.5  =-50°±28°  1. -  c + fe  3TT  o.o i  E-  •!!E 20  OBZD  60  80  fe 3. -  T  = 4 5 ± 1 5 MeV  €  =-ioc°±2e°  b  20  40  60  80  20  40  60  80  3. -  fe  G  0.  ' +  G  40  2.  2.  - t  1  T5 o.  iS  x 2 -  5  1. -  i i  1  20  40  -  DD 60 ;  T  -  L  80  (MeV) 7T  +  Figure 4.2: Measured four-fold differential cross sections for the Q Q D S0° angular setting. Each spectrum refers to an apparatus setting. The error bars include the statistical and systematic uncertainties.  Chapter  4. Data  Analysis  52  C A R U Z solid angle was calculated as the difference between the solid angle subtended by the C A R U Z with respect to the extreme points of the target and the solid angle subtended with respect to the center of the target. This last value, 190 msr, was used in the analysis. The uncertainty associated with the other variables were discussed in the previous chapter. The uncertainty for N  tgt  of the n  presented here is only due to the error  d elastic scattering differential cross section (see previous chapter), since the  +  error in the solid angle is included separately. Comments on data and comparison to theory are described in chapter 6. 4.3.2  Tr  spectra  The binnings intervals for these spectra are as follows: Aft] = IS msr, Q Q D solid angle. Aft2 = 190 msr, C A R U Z solid angle covering the angular region (—50° ± 30°) or ( — 100° ± 30°) with respect to the beam according to the C A R U Z angular setting. A T . - = 3 MeV AT  7 r +  = 75 M e V  To obtain these spectra we histogrammed together data from the two Q Q D energy settings for each Q Q D angular position and averaged over the whole 7 r kinetic energy +  range. Since we only accepted an event when we had a pion in each arm, loss of positive pions yields an underestimation of the negative pion yield. The shape of 7r~ energy distributions was not greatly affected, since to a good approximation negative pions from the same energy bin have the corresponding 7r s evenly spread over the allowed +,  energy range. Therefore the probability for loss of counts is approximately uniform over the whole T^- spectrum. These effects were taken into account in a quantitative way, however, by calculating  Chapter 4. Data Analysis  53  a multiplicative correction factor using our 'MonteCarlo program'. The theoretical TT  +  distributions for each setting were used as input to the program. The ratios between the number of initial events (events subtended by the curve, see chapter 6) and the number of events in the program output spectrum that contained pions and muons provided an estimate of the number of lost events and were used as the appropriate multiplicative correction factor on a setting by setting basis. They varied according to the shape of the initial energy distribution as expected, since the pion energy loss and its decay probability are a function of its energy (see chapter 6). The values of the ratios are listed in table 4.3 and the resulting spectra are shown in fig. 4.3 and 4.4. The uncertainties in the correction factors were estimated by comparing the ratios obtained with different input spectra that were produced with the theoretical model for different values of the coupling constants (see chapter 7). They varied between 3% and 6% and were neglected in the total error calculation since they would contribute to the total uncertainty of the data at the most by 1% (compared to a typical value of 15% - 20%). 4.3.3  6 + spectra n  Here we used the following binning: Aftj = 18 msr, QQD solid angle. = 21 msr A T ^ - = 40 MeV or 30 MeV according to the QQD central energy setting covering the (80 db 20) MeV or (60 ± 15) MeV interval respectively. AT„.+ = 75 MeV These spectra were obtained by histogramming together data relative to both CARUZ angular settings for each QQD energy and angular setting. The number of  Chapter  6  4. Data  Analysis  54  =50°±5°  •  data  points  TI-  ~\  i  1  1  r  T  1  '  r  T"  —I  '—  * . . - + 5 0 " i 5" 8. =-100 ±28*  6, =-50°i28°  0<T  0<T  ,  4  4  20  —l  37  .  ,  1  53  T..  i_  70  (MeV)  87  <75 MeV  103  120  <75 MeV  'A DO  20  37  53  T  70  87  (MeV)  103  120  T..=80i20 MeV  T..=45*6 MeV  0<T  0<T  8 . . - + 5 0 ' ! 5°  K>  <75 MeV  <75 MeV  -D  33  57  80  103  127  150  10  33  57  80  103  127  150  Figure 4.3: T ~ and 6^+ differential distributions for Q Q D 50° angular setting. The error bars include the statistical error and the systematic error. n  Chapter 4. Data Analysis  6  =80°±5  55  •  C  data  points - i —  25  9 8  20  # <  1  — i — ' —  -+80°*  5"  =-100•±28•  0<T ,<75 MeV  0<T  <75 MeV  15 K)  20  37  120  53 70 T . (MeV)  20  37  B a t e•Pod  53 T  70 87 (MeV)  103  120  15 T„_=45±15 MeV  T „ _ = 8 0 ± 2 0 MeV  12  e . . - + 8 C ' ± 5°  • . . - + 8 0 ° * 5" 0<T <75 MeV  0<T  <75 MeV  9 x  !55  55.  55 'Do  ° ° P nDa 103  u  c b  TJ  K)  33  57  80  Ot)  127  _ i — . — i — .  150  10  33  57  80  103  pti  127  150  e  Figure 4 . 4 : T^- and 8^+ differential distributions for Q Q D 8 0 ° angular setting. The error bars include the statistical error and the systematic error.  Chapter  4.  Data  56  Analysis  Setting  Correction factor  80.LL  1.57  80.HL  1.27  80.LH  1.32  80.HH  1.3  50.LL  1.18  50.HL  1.13  50XH  1.09  50.HH  1.25  Table 4.3: Multiplicative correction factors for the T _ spectra that account for 7r loss for each setting.  +  events was multiplied by the correcting factors described above, since we can assume that 7r *s of different energies give an approximately uniform contribution over the +  whole angular span. The data are presented in fig. 4.3 and 4.4; the angle 6 is the absolute value of the angle of the outgoing 7T with respect to the beam direction. +  Chapter 5  Theoretical  Model  The theoretical model developed to interpret the experimental results is based on the model of Oset and Vicente-Vacas [12] that describes the free pion-induced pion production process: ir~p  —> 7 r 7 r ~ n . +  They describe the reaction mechanism with Feynman  diagrams involving nucleons, pions and intermediate state isobars. The amplitudes that involve only nucleons and pions are derived from the Weinberg effective Lagrangians to which they added terms corresponding to intermediate state isobars. Although they compare their model to data up to center of mass energies of 1350 M e V , their model is incomplete already at the energy our experiment was performed (1300 M e V c m . ) , since at that energy more reaction mechanisms become relevant. We therefore added nine new diagrams to their model. In implementing their model with these additional diagrams we also required different coupling constants for the additional interaction vertices. Some of these coupling constants are experimentally well determined, while others are permitted a wide range of allowed values. B y comparing the model to data we accomplished two goals. We tested the model itself and we narrowed the range of values of some of the poorly known coupling constants within the model framework. The first task was to establish the model sensitivity to various coupling constants.  A starting value for the parameter to which  the cross section is most sensitive (the one which determines the isobar (1/2,1/2) A"*, N*—• N(Tr7c) _ t  wavt  contribution) was found by requiring the energy dependence of the  total cross section to be that of earlier data [22]. We then compared the model to our 57  Chapter  5.  Theoretical  58  Model  four-fold differential cross sections to test it on a more detailed level and to investigate the importance of those coupling constants that do not significantly change the magnitude of the total cross section. To perform a correct comparison between model and data we accounted for the presence of the second nucleon in the deuteron by introducing nucleon Fermi motion and then applying the model in a quasifree sense using free process amplitudes. The complete model was finally compared to the other existing (7r,27r)  data. The model and its development are presented in this chapter while the  comparison between data and theory is covered in the next chapter. The results and comparison to other data are described in chapter 7. The description of the model can be divided into three parts: the modified Weinberg Lagrangians, isobar intermediate state Lagrangians and completion of the model to make it a suitable description of the reaction occurring at the energy of our experiment. The first two sections of this chapter are essentially a description of the Oset-Vicente-Vacas model and follow [12], except that we fixed the chiral symmetry breaking parameter, £, to zero, which is consistent with the soft pion theory. The last section of this chapter (model improvements) and what follows in the next chapters (Fermi motion and Montecarlo simulation of the extended target effect) target effect constitute a new, original effort to compare data to a possible theoretical description.  5.1  Modified Weinberg Lagrangians and Related Diagrams  The Weinberg Lagrangians (WL) used to describe the pion induced pion production are the following:  (5.1)  5.  Chapter  Theoretical  59  Model  t  *7T~  TT -* - •f  o)  b)  d)  c)  Figure 5.1: Diagrams described by Weinberg Lagrangians: a) pion pole term, b) contact term, c) two point diagrams, d) three point diagrams. I A  Lrr  A > r r  =  = -ij.md^f  LNNrr  (5.2)  - ^ ^ y ^ T ^ d r f W  = —fat  -  (l/2)m^ ]  \Y + it^iP'  (5.3)  4  ~ Pi)*} W T  *  (-) 5  4  where ip and d> are the nucleon and pion fields, respectively, g„ = 13.5, is the xNN coupling constant, m the nucleon mass,  the pion decay constant (in [12] / =87 r  M e V , we used 93 M e V ) and A'v = 1.85, the anomalous magnetic moment factor of the nucleon. The diagrams that can be constructed with the amplitudes derived from the above Lagrangians are shown in fig. 5.1. They are separated into one, two and  Chapter  5.  Theoretical  Model  60  three point diagrams. The diagram 5.1.a) is particularly interesting, since it involves the  (TT,7T)  interaction vertex, and can be related to the pion scattering length. In this  case the reaction occurs via a nucleon emitting a virtual pion that is subsequently scattered by the incoming pion thus producing two outgoing pions. This diagram is called the pion pole term, while the diagram 5.1.b) is known as the contact term. The two and three point diagrams involve two and three interaction vertices, respectively. The W L description of the two point diagram is however not complete, since the two point diagrams also contain an isoscalar part of the irN scattering amplitude that can not be derived from the W L . This part becomes relevant at low energy, where the isovector part contained in the W L vanishes. The vector part y part of equation (5.4) M  is therefore substituted by: = -47T \^4' 4> • M  Lxxrr  +  + %4< T(<p X d <p)i] +  (5.5)  t  where Aj and A are related to the s-wave 7rN scattering lengths [38]. The expression 2  (5.5) is presented in a non-relativistic approach that is used throughout the model except when relativistic corrections are non negligible. The above substitution requires the equivalence 1/4/  2  = 47rA /^ that agrees to within 3%. 2  2  The isovector part of  the Lagrangian (5.4) is believed to come from the p exchange in the t-channel. The Lagrangian Lssw  where G  SSfi  describing an intermediate p formation (fig 5.2) is the following:  and Gji  Sf)  are the pNN vector and tensor couplings, / / 4 : r = 3.0 deter2  mined from experiment [39] and q = (pi — pj) . v  v  The zero component of this expression  can be separated from the spatial component. The zero component is then equivalent to the isovector part of (5.5) while the spatial component, that accounts for the p-wave part of the p exchange, can substitute the remaining piece of (5.4), provided that  Chapter  5.  Theoretical  Model  61  Pi  p N  Pi  Figure 5.2: Feynman diagram for the t-channel.  TTNTTN  amplitude through the p exchange in the  Considering the experimental values for the constants involved [39] the above relationships do not hold exactly, but since the p exchange contribution to the reaction cross section is very small, the Oset-Vicente-Vacas model still replaces (5.4) with (5.5) and the spatial part of (5.6) thus introducing the isoscalar part in the irirNN  Lagrangian.  The two point diagrams of fig. 5.1 are consequently separated into two groups: equation (5.5) leads to the isoscalar part of the amplitude and to the s-wave part of the p exchange, fig. 5.3, while the spatial part of (5.6) describes the p-wave part of the p exchange (fig. 5.4).  A posteriori, they omitted the p diagrams from the model, since  the contribution of these was negligible at all energies at which the model applies. The three point diagrams described by the unmodified W L are shown in fig. 5.5.  5.2  Isobar Intermediate State Lagrangians a n d R e l a t e d D i a g r a m s  The total cross section calculated from the Lagrangians described above is significantly lower than the experimental one [22] throughout the measured energy range. It differs by a factor of 3 at threshold and about a factor 5 to 6 at T , « 3 0 0 M eV. The W L do  Chapter  5.  Theoretical  62  Model  n  - TT x7T  7T  T.  o)  b)  TT  d)  c)  Figure 5.3: Two point diagrams with nN s wave amplitude.  n  TT  *  •* TT  TT  w  n  ' TT  * . TT  o)  n  TT  TT  ^TT  b)  Figure 5.4: Two point diagrams with  c)  KTTNN  Y*J\T^ TT  d)  coupling through p-wave p exchange.  Chapter  5.  Theoretical  63  Model  n '  m  n  7T  TT  P  a)  P  b)  Figure 5.5: Three point diagrams with only nucleons i n the intermediate state. not include the isobar formation in the intermediate state. The other new feature of the Oset and Vicente-Vacas model is the introduction of diagrams with one and two A ' s i n the intermediate state as well as the N* (1/2,1/2) resonance intermediate state. The relative contribution of these diagrams to the cross section is very dependent on the energy of the incoming pion, the A diagrams contributing more significantly at higher energy (TV as 300 M e V ) , while the N* diagrams together with the pion pole and the contact term are dominant at energies close to the reaction threshold energy. 5.2.1  D i a g r a m s w i t h one a n d two A ' s i n the Intermediate State  The diagrams with one and two A ' s in the intermediate state are shown in fig. 5.6, 5.7 and 5.8.  The corresponding phenomenological Lagrangians and coupling constants  are given by:  Chapter 5. Theoretical Model  64  «r- n  •» n  ' Tf  , rr  7T  b)  o)  c)  d)  Figure 5.6: Two point diagrams with a A in the intermediate state. coupling occurs through a p exchange. y/C~ ^e S (d 4> )T r!^  IA-AP  =  L,AA  = ^vtS 4d,<f> )T rP  p  1jk  A  ]  x  x  where f* /An = 0.36 and C 2  x  x  k  +  The AKKN  h.c.  A  p  2 assuming the same scaling between A A7r and A A p T  T  as in the case A A 7r and NNp. c, is the p meson polarization vector and S and T are r  T  the transition spin and isospin raising operators connecting states of spin (isospin) 1/2 with 3/2. S  and T ^ are the spin and isospin matrices for the A ' s . The value for the  A  coupling constant / A related to the A ^ A vertex, is taken from quark model predictions to be f&/f = 4/5 (where f is the pion-nucleon-nucleon coupling constant determined from / = pg /2m r  = 1, with p. the pion mass), since it has never been determined  experimentally. The contribution of the diagrams containing a p meson was found to be negligible and those diagrams were therefore omitted. The contribution of the other A diagrams is shown in fig. 5.9. They do not contribute to the reaction cross section  65  Chapter 5. Theoretical Model  n  TT  <  TT  V  if,  TT  TT  > TT  ' TT , TT  o) TT  n  D  b) >JT  V  n  TT TT  P  V  n  A"  TT  TT  d)  c) TT  ^ f)  9)  h)  Figure 5.7: Three point diagrams with a A and a nucleon in the intermediate state.  Chapter  5.  Theoretical  Model  'TT  IA  ' TT  V 'TT  *-  , 0  'TT TT,  b)  o)  *TT  c)  d)  . TT  e)  0  Figure 5.8: Three point diagrams with two A ' s in the intermediate state.  Chapter 5. Theoretical Model  67  T  Figure 5.9: Cross section for the reaction n~p — • n ir~n calculated with different mechanism. Lower dashed curve: contribution of diagrams from fig. 5.3, 5.4 and 5.5. Upper dashed curve: addition of the A diagrams. Full line: full calculation with C=-.91 / x . Circles: data points from [22]. +  -1  Chapter 5. Theoretical Model  68  f  TT  7T~ N  Tf  o)  »TT  b)  Figure 5.10: Two point diagrams with JV~*in the intermediate state and couphng.  N*N(Tnr),_  wave  close to the threshold energy, but they do become more and more important at higher energies. 5.2.2  D i a g r a m s w i t h N* i n the Intermediate State  Oset and Vicente-Vacas introduced only those diagrams containing N* that do not vanish at threshold. This is the case when the two outgoing pions are emitted in the relative s state couples to spin and isospin J=I=0. This pion relative state is referred to an 'effective' c o r e meson with a mass parameter between 500 and 600 M e V [20]. The diagrams are shown in fig. 5.10 and are formally identical to the two point diagrams 3.a) and 3.c). But while the diagrams 3.a) and 3.c) mutually cancel at threshold, since both contain the nucleon as propagator, this is not the case i n the diagrams of fig. 5.10, since the N* has a gaussian mass distribution. The Lagrangian describing the N*Nir part is identical to (5.1), which in nonrelativistic approximation is:  Chapter  5. Theoretical  Model  69  with / /4rr = 0.02. In the part related to the N*Ne vertex there are large uncertainties 2  in the vertex coupling constant, in the c mass and in the CTTTT coupling. A l l of these uncertainties are included in one single coupling constant C, by calculating directly the A *A*7rTf r  vertex. C was calculated from the available experimental information about  the A *€ fraction of the A *width for the Nnir decay using the following Lagrangian: r  T  The fraction of the A * —• N(Trn) _ T  a  wave  [40] is not well known, having values ranging  from 5% to 209c (leading to C = -(0.91 ± 0.20)/J  _1  according to [12]). This value for  C was later changed (see next section). Although the inclusion of all these terms increases the calculated cross section by 80%. the model still fails to reproduce the experimental cross sections. The full model calculation is shown in fig. 5.9 with C = —0.91/i . It reproduces fairly well the cross -1  section close to threshold energy, while is still about 50% lower than the experimental results at higher energy.  5.3  Improvement of the Oset Vicente-Vacas Model  The energy at which our experiment was performed is high enough that the A *A (7r7r) _ T  T  p  u ; a v e  decay fraction cannot be neglected. We thus need to add to the the-  oretical description of the reaction also diagrams that have an A * in the intermediate r  state and the two emitted pions in the relative p wave state. A l l such possible diagrams are shown in fig. 5.11. The first four containing a p meson were neglected, since we know that the p-wave part of the p contribution is very small. The next two are formally equivalent to the diagrams of fig. 5.5 and the last eight to those offig.5.7.  Chapter  5.  Theoretical  Model  fi  n  N"  p  r  P  \AA'"-  o)  *'  T'  V  c)  y- r  \T.  • -  *  L  \ T  ' f  v r  d)  Figure 5.11: Diagrams with the  N*—*  N(7nr)j . >  wave  decay.  Chapter  5.  Theoretical  Model  71  The contribution of diagram 5.11.c) was also neglected, since in this case a real pion is emitted before the formation of the N* leaving the system at an energy (1160 MeV) too far from the N* mass peak to favour its formation. For the same reason only the ••first nucleon propagator in the diagram 5.11.b) was replaced by the N* propagator. For the remaining ones we can use the same amplitudes associated with the diagrams of fig.5.5 and fig. 5.7. provided that the appropriate vertex coupling constants are introduced and that the nucleon mass is replaced by the N* mass in the propagators where necessary. A detailed example of the calculation of an amplitude from a Lagrangian is presented in appendix A and the amplitudes related to the added diagrams are listed explicitly in appendix B . The first part of this set of diagrams does not require new coupling constants but a new vertex N*ATT is introduced in the diagrams that involve A  and A * . The coupling constant :  gj^*^  r  w  a s calculated in an analogous way to C ,  since it has never been measured. The A *ATT decay branching ratio ranges from 1 0 % T  to 2 0 % of the total A *decay. The corresponding range for <J/V*A* is 1.305 ± 0.225 [41]. T  The contribution of these new diagrams to the total cross section calculated at T  r  = 280 M e V with <7A**A»=1-305 is approximately 15%. The model now contains the description of all the reaction mechanisms that we consider relevant at the energy at which our experiment was performed (see fig. 5.12). We also believe that this energy constitutes the upper energy limit for the applicability of the model in this isospin reaction channel, since at higher energies the contribution of more massive A * isobars 7  might become non negligible. While calculating gx*A >C was also recalculated and the new value range is now r  (—2.28 ± 0.76)/i [41]. This result is extremely important because with this range of -1  values the theoretically calculated cross sections now overlap the measured values. Furthermore the -model was also implemented with the cross section calculation  Chapter 5. Theoretical Model  72  IT'*'  "1  b)  f  *  J  t ^ v  r  i  V  .  -  <-  it  X  c)  is:'  d)  r  *  v  \  / V  Figure 5.12: Diagrams describing the reaction mechanisms that contribute to the pion induced pion production at T = 280 M e V . a) diagrams derived from the Weinberg Lagrangians, 6) one A intermediate state diagrams, c) two A intermediate states diagrams, d) N*(irir)luiave diagrams, c) N^irn)^^ diagrams. r  Chapter 5. Theoretical Model  73  for the reaction occurring on a polarized nucleon (see appendix A) and can be easily applied to other isospin reaction channels. The amplitudes related to the 7r p +  —•  Tr+Tr+n  cross section are already calculated and the insertion of the amplitudes relative to the :  ^emaining isospin channels can be done directly in any extension of this work. 5.4  Cross Section Sensitivity to Various Parameters  The 'free" parameters of the model are C, gN*&* and the / A coupling constants. These have not been well determined experimentally . They are related to reaction mechanisms that have a different energy dependence, therefore the variation of each one will differently influence the cross section at different energies. We studied the model cross section sensitivity to these parameters in the energy range 203-280 M e V by calculating the cross section change for a particular parameter variation. The cross section was calculated for the lower and upper limits of the C and  <7N*ATT  coupling constants sepa-  rately, maintaining the other parameters at a fixed value. There are several theoretical predictions for / A - In what follows we will express them in units 4/5 f (where f is the TTNN coupling constant described previously). The quark model value, which is the most frequently used, gives / A = 1 [42], SU(6) calculations predict / A = 1.064 [43]. the Goldberg-Treiman relation gives either 0.966 or 1.076 according to the value used for the pion decay constant  [14], [44] gives 1.4 using an Adler-Weisberger relation and  [45] gives 1.14 using a quark model. A l l of these values are very close to 1 except for the value from [44], but the analysis of [10] seems to favour smaller values. We arbitrarily chose / A ' = 1/2/Ato test the cross section sensitivity to the variation of this parameter. The calculation was done with a fortran program that evaluates the coherent sum of the amplitudes from all the diagrams (see appendix B) for three body phase space  Chapter  5.  T„(MeV)  Theoretical  Model  74  A<T(C = - 1 . 5 2 -» - 3 . 0 4 )  A c r ( < 7 * , = 1 . 0 8 - ^ 1.52)  A < r ( / = . 5 - * 1.)  9N*A* = 1.5  C = -2.0  C =  U  h  9N*A* = 1.5  = i.  N  A  = 1.  A  -2.0  203  120 %  not detectable  not detectable  230  100 %  not detectable  not detectable  255  100 %  1 %  not detectable  280  100 %  5 %  4 %  Table 5.1: Total cross section sensitivity to the coupling constants variation. The coupling constant / is expressed in units ( 4 / 5 f). A  events randomly generated with a Montecarlo technique ('Theoretical Montecarlo Program'). The coupling constants appear in the program as multiplicative factors for the corresponding amplitudes and therefore are very easy to vary. The square of the sum of the amplitudes is then multiplied by the phase space factor and properly normalized to the required number of events. The results are presented in table 5.1. The cross section is clearly most sensitive to the variation of the C parameter throughout this energy range, showing the very significant contribution of the diagrams of fig. 5.10. The diagrams involving two A ' s and the ones of fig. 5.11 do not contribute to the cross section until T  v  « 2 5 0 M e V and even at T  T  = 2 8 0 M e V the sensitivity  of the total cross section to the variation of their coupling constants is modest. Since the C variation influences the cross section magnitude in this whole energy range, we use the data of [22] to find that value of C that best reproduces the total cross section energy dependence. We then use that C value as a starting point i n the estimate of the whole set of parameters using the more detailed information provided by our data.  Chapter 6 Comparison between the Theoretical Model and Our Data  Two effects were addressed before comparison of the theoretical model to our data was carried out, the nucleon Fermi motion in the deuterium, and the experimental effects on the  spectra. The original model described the reaction occurring on a single  free nucleon, while we used a deuterium target. Although the mechanisms for pioninduced pion production in deuterium may still involve only one nucleon, the nucleon in this case is not at rest in the laboratory system, since there is some Fermi motion. We therefore implemented neutron Fermi motion in the model, transforming it in a quasifree calculation with free process amplitudes. The fact that the model actually describes the isospin symmetric reaction rather than the one we measured (7r~p —• 7r~7r n +  as compared to 7 r d —• 7r 7r~p p), did not require any amplitude modifications, +  +  since the Langrangians describing the reaction are invariant under isospin rotation. The second correction was of experimental nature. Since the C A R U Z was not able to distinguish pions from muons, a distortion in the yield and shape of the  energy  spectra resulted, as described in chapter 4. A further contribution to such distortion arose from the finite size of the target (extended target effect). Before we could make a valid comparison between model and data we had to either correct the experimental spectra or include the experimental conditions in the theoretical predictions that calculate the pion energy distributions at their production point to take into account the experimental conditions for the events. We chose this second approach and wrote a fortran program based on MonteCarlo techniques ('Experimental MonteCarlo Program'), 75  Chapter 6. Comparison between the Theoretical Model and Our Data to introduce the experimentally-based corrections into the theoretical distributions. The parametrized theoretical T„ spectra were used as input to this MonteCarlo program (see section 6.2.5) and its output was then compared to the experimental spectra. Introduction of the Fermi Motion in the Model  6.1  The Fermi motion in the deuterium is related to its bound nature through the Heisenberg uncertainty principle. The neutron momentum probability distribution was inferred from the Hulthen wave function [47] and inserted in the Theoretical MonteCarlo Program (TMP). The nucleon thus contributes as an off-shell particle with its energy equal to its mass (minus the binding energy of the deuteron) and its momentum selected according to its probability distribution. 6.1.1  Neutron Momentum Probability Distribution  The Hulthen wave function consists of two terms. Thefirst,which is the dominant, longranged one, is related to the actual Fermi motion, while the second, short-ranged one, is due to the correlation potential between the two nucleons. We are therefore slightly imprecise when we only mention Fermi motion in calculating the nucleon momentum. The Hulthen form used in our work is:  where q is the neutron momentum, a=  y/triN  • Bi = 45.7 MeV is the factor related to the Fermi motion,  is the nucleon  mass and Bi = 2.225 MeV is the binding energy of the deuteron, b=237.3 MeV, arises from a coarse parametrization of the nucleon correlation potential, Ni = ^!*°)*^ arises from the normalization requirement /<Pg\tpd(9)\ = 12  We normalized the wave function value to its value at q = 0 (fig. 6.1):  77 Chapter 6. Comparison between the Theoretical Model and Our Data  50  100 150 200 Momentum (MeV/c)  250  300  Figure 6.1: Hulthen wave function as a function of momentum (q). R =  MO)  (<7 + a )(<7 + *> ) 2  2  2  2  and took the product of the momentum squared and the square of the wave function as the neutron momentum probability distribution:  The resulting, unnormalized, probability distribution is shown in fig. 6.2.  78 Chapter 6. Comparison between the Theoretical Model and Our Data  Figure 6.2: Neutron momentum probability distribution.The distribution is not normalized to 1.  79 Chapter 6. Comparison between the Theoretical Model and Our Data 6.1.2  Insertion of the Neutron Momentum into the  Model  The T M P calculates the reaction amplitudes for each event in the pion-nucleon centerof-mass system in terms of the kinematic variables expressed in this system.  The  variable that determines the kinematic limits of the outgoing particles is the invariant mass of the system (S), which is constant in the case of a free nucleon. This is not the case, however, when Fermi motion is present, since each target particle can have a different momentum. The consequence of such Fermi motion is a broadening and smearing of the kinematic ranges allowed for the outgoing particles. This effect becomes particularly evident in the kinematic region close to the free reaction phase space limits. Two such regions fall in the range of our detectors and in both cases the effect of Fermi motion is significant as shown in fig. 6.3 (the two theoretical curves are calculated with the same set of parameters C=-2.08, ^ . ^ , = 1 . 5 3 / A = 1 , see chapter 7) (Data in these regions were previously [48] attributed to a possible four body contribution to the reaction, since they are not allowed according to pure three body phase space). The momentum selection and the calculation of the invariant mass were performed in the laboratory system, since in this system the nucleon momentum distribution is isotropic. The random nature of the nucleon momentum was introduced by selecting its magnitude according to the probability distribution (1) and then randomly selecting its projection on the x, y and z axes. (fig. 6.4). The momentum selection was done on an event-by-event basis, thus facilitating the insertion of this coding in the original theory program. The momentum probability distribution (1) is not a monotonically increasing function, therefore is is not possible to establish a one-to-one correspondence between a given probability and a momentum. To obtain a monotonically increasing function  Chapter  6.  Comparison  between the Theoretical Model and  Our  Data  80  Figure 6.3: Calculated T +(MeV) distributions (nb/sr MeV ) and data for two geometrical configurations: a) 9 - = 50°, T - = (45 ± 15)AfeV, $ + = - 5 0 ° ± 30°, b) 6^- = 80°, T - = (80 ± 20) M e V , 0^+ = - 1 0 0 ° ± 30°. Solid curve: no Fermi motion. Dashed curve: Fermi motion included. Left: theoretical curves. Right: theoretical curves corrected for experimental conditions (see last paragraph and ch. 7). 2  V  t  v  2  v  6.  Chapter  Comparison  between the Theoretical Model and Our  400-1  1  1  1  Data  81  L  267<  -300  -200 -100 0 100 200 300 x momentum component distribution (MeV/c)  Figure 6.4: Neutron momentum projection along the x axes. The projections on the y and 2 axes follow the same distribution. that allows the inverse function, we calculated the cumulative momentum probability distribution as a function of momentum for the nucleon momentum range 0-200 M e V / c . Using the inverse function (momentum as a function of cumulative probability) we could randomly select a number between 0 and the integral value at q = 200 M e V / c and find the corresponding momentum value for the neutron momentum. In order to obtain the appropriate kinematic limits for the outgoing particle we calculated S for each event using the following expression: 5  where m  T  and  ml  =  +  m-v  +  2{m E , N  T  -  P^PK,,)  -  p%  are the pion and nucleon mass, E„, is the total energy of the  incident pion calculated in the laboratory system, p , is the pion momentum in the v  laboratory system and p  N  and pyv are the nucleon momentum and its z component in 1(  the laboratory system. The S distribution obtained from 50000 momentum selections with the above procedure is shown in fig. 6.5. Its peak is at the zero nucleon momentum  82 Chapter 6. Comparison between the Theoretical Model and Our Data  Figure 6.5: Invariant mass distribution including the nucleon Fermi motion.  83 Chapter 6. Comparison between the Theoretical Model and Our Data value (1300 M e V ) and it is slightly asymmetric with a longer tail i n the low energy region due to the off-shell nature of the nucleon. Since we needed the theoretical distributions to be expressed in the laboratory system, we added a rotation to the existing Lorentz boost that transformed the variables of the outgoing particles from the center-of-mass system to the laboratory system (see appendix B ) . Due to the isotropic nature of the nucleon momentum the direction of motion i n the laboratory system did not necessarily coincide with the beam direction. The rotation angles (9 and <j>) were determined from the polar and azimuthal angles of the momentum of the pion-nucleon system i n the laboratory system described in section 6.2.1. and obviously varied for each event. The transformation is described in matrix form as follows: pi  =  R  • L  •  p. . c m  where pi is the momentum energy four vector expressed in laboratory system, R is the rotation, L is the Lorentz transformation and p . . is same four-vector expressed i n the c TO  center-of-mass system. 6.2  M o n t e C a r l o S i m u l a t i o n of E x p e r i m e n t a l C o n d i t i o n s  The estimate of the pion decay fraction on the C A R U Z side of the system was a non trivial task. The pion decay probability is an exponential function of its momentum. The different material thicknesses the pion traverses before reaching S i cause an abrupt change i n the pion energy loss and consequently momentum, thus introducing discontinuous slope changes of the pion decay probability distribution on their path from the production point to S i . Furthermore the decay probability depends on several other factors: the energy of the pion when it is produced, the geometry and nature of the materials the pion traverses, the production point in the target and the angle at which  84 Chapter 6. Comparison between the Theoretical Model and Our Data  the pion is emitted. We did not attempt to unravel these effects with an analytic approach, but instead wrote the 'Experimental MonteCarlo Program' that simulated all these experimental conditions. The extended target was reproduced by randomly -selecting the event vertex within the target boundaries along the beam direction. The beam shape was simulated by selecting randomly within a gaussian distribution the out-of-plane vertex coordinates while maintaining as a boundary condition the target physical limits. Once the vertex coordinates were chosen, the pion direction was selected randomly within a defined angular interval (see next paragraphs). Its trajectory was then divided into small steps (Ax) and the pion energy loss as well as its decay probability  (AE/Ax  and  Pi(x  + Ax))  were calculated over each step separately to  obtain the decay probability distribution as a function of the pion position along its path. The size of Ax was allowed to vary according to the density of the traversed material density and was determined by the requirement of AE/Ax  being fairly constant  over each step in order to maintain the accuracy of the decay probability distribution reasonably uniform. On the basis of this probability distribution the MonteCarlo program decides if and where the pion decays. If the pion decays the polar and azimuthal angles 0 and  <f> of the resulting muon are selected randomly in the pion reference frame, since in that frame the decay is isotropic. The muon energy and angle are then transformed into the laboratory system using a Lorentz transformation. At this stage the program checks to determine whether the muon is emitted in such a direction that it would hit the detector. If so, it tracks the muon to Si and records its energy there after calculating its energy loss along its path. The overall requirement for an event to be accepted is the reproduction of the experimental trigger: (Tl • 51 • 56) (note:this was (Tl-51-55) in thefirstrun period) either in the form of pion or as a combination of pion  Chapter 6. Comparison between the Theoretical Model and Our Data subsequent muon. The program records the energy at Si of all those events that satisfy this trigger condition and calculates the particle energy at the target exit using the empirically determined curves described in chapter 3. This method of determining '•'the energy of the particle at target exit was preferred to simply taking the recorded value of its energy at that point, in order to reproduce as close as possible the procedure used for the experimental data. The events were then histogrammed according to their kinetic energy at the exit of the target in three separate histograms: events that were pions at S i , events that were muons at S i and a histogram that contained all events. This last histogram was the one compared to the experimental data.  A schematic  flow structure for the Montecarlo program is presented in fig. 6.6 and a more detailed description of each step of the program is given in what follows.  6.2.1  Event Vertex Selection and Definition of the Coordinate System  The deuterium target had a cylindrical shape with a radius of 2.52 cm and a height of 5.04 cm as described in chapter 2. The center of the target defined the origin of the coordinate system used throughout this calculation. The z axis was defined by the beam direction. The y coordinate is perpendicular to the two, positive in the up direction. The x axis then, is in the detector plane, forming a right handed Cartesian coordinate system with the other axes. The polar angle 6 of an outgoing particle was defined as the angle between the direction of the particle and the beam direction, while its azimuthal angle 4> is the angle between the y axis and the projection of the particle trajectory onto the xy plane. Since the beam had a finite width the events were not produced only on the z axis, but rather their vertices were distributed within the target according to the beam shape distribution. The beam profile had a gaussian shape both in the y- and x-direction with  86 Chapter 6. Comparison between the Theoretical Model and Our Data  calculculation of AE/Ax Pj(x-t-Ax)  n  Tr r e a c h e s S1  stops  end of n t r a v e l J_  Irondom selection 0 < R < 1  fj. a c c e p t e d  porticle lost  Figure 6.6: Schematic structure of the program that simulate the experimental conditions.  Chapter  6.  Comparison  between  the  Theoretical  Model  and  Our  Data  87  a Full W i d t h Half Maximum ( F W H M ) of about 2.0 cm as estimated from P C O S beam profile (see chapter 2). The x- and y-coordinate of the event vertex were thus randomly chosen according to a normal gaussian distribution with the F W H M scaled to 2.0 cm. :  T h e beam divergence i n the target region was neglected, since at the edges of the target the beam F W H M increases by only 3 mm in the y direction and 0.6 mm in the x direction. We assumed a uniform probability distribution for the occurrence of the reaction along the beam direction in the target and therefore selected the z coordinate randomly between the target limits in the z direction. These limits varied according to the transverse distance of the vertex from the z axis ranging between ± 2 . 5 4 cm on the beam axes to 0 at the extremes of the target in the x direction. 6.2.2  Pion Direction  The theoretical four-fold differential cross section spectra (see appendix B) contain only those events that fall within the kinematic region covered by the detectors. The counts in the theoretical T^+ energy spectra therefore correspond only to those events characterized by a 7r emitted into the solid angle subtended by the C A R U Z with the +  corresponding TX~ detected by the Q Q D . The same restriction on the emission angles could not be applied a priori to pions of the T^+ spectra that are used as input to the Montecarlo, since this would have led to an underestimate of the number of detected muons. Since the muons arising from pion decay are distributed within a cone around the direction of the original pion, some of the muons originating from pions emitted towards the C A R U Z would miss the detector thus introducing a particle loss. But there is a compensating particle gain, since the detector detects some of those muons that originate from those pions that would originally have missed the detector. To take this second contribution into  88 Chapter 6. Comparison between the Theoretical Model and Our Data account and still use the theoretical spectra as input data we introduced the following procedure.  We checked that the 9 and <j> cross section dependence was reasonably  flat over the solid angle subtended by an artificially enlarged CARUZ. We 'spread' the -«vents of the theoretical energy distribution over the new solid angle and then corrected the number of pions and muons hitting the actual C A R U Z with a weighting factor given by the ratio of the enlarged solid angle with respect to the actual geometric solid angle subtended by the C A R U Z pj^-  AQ i g en ar  ed  &&CARUZ  The variation of the number of detected muons as a function of the range of the pion emission angles was studied by gradually increasing the angular limits for the initial pions. At sufficiently large angles the yield of detected muons multiplied by the above weighting factor became constant. This indicated that the pion angular range was wide enough to account for all the muon contributions. We chose the smallest of these values in order that the assumption of a flat angular dependence for the cross section to be reasonable. The weighting factor corresponding to the selected $ and 9 pion emission angular range intervals was 4.357 (corresponding to At? = 70° and A<f> = 40°) for the C A R U Z set at 100° and 4.237 (corresponding to A9 = 70° and A<f> = 50°) for the C A R U Z at50° setting. 6.2.3  Energy Loss and Decay Probability Calculation  On its way through the system from inside the target the pion traverses several different materials. These are listed in table 6.1. The energy loss per unit path length for each material was calculated using the Bethe-Bloch equation for heavy charged particles [47]:  dE _ DZ  z> (1  dx  P\2  mtdPmtd  A  n  m  t  d  2m /? c e7  I  2  2  2  *  g2 P  2  2  C \ Z ) med  89 Chapter 6. Comparison between the Theoretical Model and Our Data  material  width (cm)  p (g/cm )  deuterium  2.54  0.17  mylar  0.013355  1.39  aluminum  0.002  2.7  vacuum  2.525  0  aluminum  0.00762  2.7  vacuum  2.533  0  kapton  0.0127  1.43  air  14.4  0.00129  scintillator(NE102)  0.129  1.03  air  11.75  0.00129  mylar  0.0127  1.39  3  gas(30% argon+30% isobutane) 4.975  0.00205  mylar  0.0127  1.39  air  17.3  0.00129  mylar  0.0127  1.39  gas(30% argon+30% isobutane)  4.975  0.00205  mylar  0.0127  1.39  air  35.05  0.00129  Table 6.1: Material layers from target center to S i . The horizontal lines separate various elements of the apparatus: target, air, Tl+wrapping, air, W C l l , air, WC12, air.  Chapter  6.  Comparison  between  the  Theoretical  Model  and  Our  Data  90  where D = 0.3071 M e V c m / g , z is the particle charge, /9c its speed (with the associated 2  "r), Zmed is the charge number of the medium,  p d mt  is its mass density and  A  m  t  d  its  atomic mass number, I is the ionization potential, 6 is a density correction applicable at high energy and C represents an atomic shell correction. The values of the last two parameters are negligibly small, so we set them to zero. The pion decay probability in a spatial interval depends on its energy and on the length of the interval. This length can be arbitrarily long when the pion travels in the vacuum, since in that case its energy is constant. This is not the case when the pion traverses material where it undergoes energy loss.  Since its energy is then a  function of its position the decay probability is calculated more accurately for shorter path intervals. We therefore divided the distance from the particle production point to S i into small intervals, Ax,-, and calculated the decay probability for each step  (Pj{x  + A x , ) ) thus obtaining the decay probability distribution as a function of position.  The decay probability at x + A x , is thus the product of the survival probability up to that point (P„(0 — x)) and the decay probability in that particular space interval P (Ax,): d  P (x d  + A x , ) = P (o - x) • P ( A x , ) d  t  where p.(o-*) =  n;  = 1  exp(-^)  and P (Ax ) d  t  =  1 -  exp(-=^)  where the symbols are defined in chapter 4. The choice of the step size was constrained by the requirement of a reasonable program execution speed on one hand and the  91 Chapter 6. Comparison between the Theoretical Model and Our Data  A x (cm)  all layers? (yes/no)  Pt  10  0.002  yes  0.4061  10  0.01  no, skipped only A l  0.4123  10  0.05  yes  1.0  10  0.05  no  0.4049  10  0.5 for air, 0.05 for rest  no  0.40866  20  0.002  yes  0.2128  20  0.5 for air, 0.05 for rest  no  0.2127  V  (MeV)  Table 6.2: Total pion decay probability between target and S i for different distance intervals. Note that not always all the layers were considered (see text). The materials that were not skipped are listed in table 6.3. accuracy of the decay probability distribution on the other. A good test of the accuracy was provided by the value of the total decay probability over the distance from the production point to S i , defined as: p _ W0-Sl)/Ar  I  r  ~  2-1=1  p r  d,  We calculated P of a 10 M e V pion produced at the target center and emitted towards t  the center of the C A R U Z for several step sizes and compared their values (table 6.2). When A i was large compared to the dimensions of the materials the pion traversed, its energy loss was highly overestimated thus introducing a false decay probability distribution. Some of the thinnest layers were therefore neglected. The best compromise between accuracy and computing speed was found for A x = 0.5 cm in the air and A T = 0.05 cm in the materials listed in table 6.3. Fig. 6.7 shows the decay probability  Chapter 6. Comparison between the Theoretical Model and Our Data92  Materials considered in pion  dE/dx calculation  deuterium (target) vacuum inside the target air: target - T l scintillator ( T l -f wrapping) air: T l - W C l l gas: W C l l air: W C l l - WC12 gas: WC12 air: WC12 - S i Table 6.3: Materials considered in the pion energy loss calculation.  93 Chapter 6. Comparison between the Theoretical Model and Our Data  .000006 b  Figure 6.7: Distribution of the differential pion decay probability (dP/dx) as a function of the distance from the target center (cm). Discontinuities of the slope occur at the interface of different materials due to an abrupt pion energy loss variation. distribution for Ax = 0.002 cm for a 10 M e V pion. It is interesting to observe the discontinuities of the slope occurring at the borders of different materials. 6.2.4  P i o n Decay Vertex Determination and Event Acceptance  The decay probability distribution was calculated for each pion separately, since each one underwent a different energy loss depending on the energy at which it was produced and on the amount of material traversed. This in turn depended on the production point in the target and on the pion emission angle. For each pion we calculated the total decay probability P , and recorded its energy t  at each step as well as the cumulative decay probability P  u  defined as:  To decide if a pion decayed we randomly selected a number between 0 and 1 and  Chapter  6.  Comparison  between  the  Theoretical  Model  and  Our  94  Data  compared it to P . The pion was assumed not to decay if the selected number was larger t  than the total decay probability. Otherwise, we determined the closest P value to the ti  randomly selected number and assigned the decay vertex location to the corresponding position and the decaying pion energy to the energy associated with that position. To verify that an event satisfied the experimental trigger T l • 51 • 56(55) in the form of either a muon, a pion, or the sequential combination of the two, we first checked, if the particles trajectories intercepted the scintillator planes. If so, we calculated the particle energy at S i and tracked its fate to S6. The event was considered valid if the energy at S i was positive and the particle did not hit S6 (S5) (see figure 6.6). We determined the nature of an accepted particle by its identity at S i where we also recorded its energy as the difference between the energy at which the original pion was produced and the energy lost between the production point and S i . The calculated energy at S i was then related to the particle energy at T l using the same curves that were used for the experimental data (see chapter 3). This last procedure is not an exact reproduction of the experiment, since the particle energy was actually determined in the C A R U Z by the amount of light released in the scintillator stack. We estimated that any further simulation would be inefficient, introducing only higher-order corrections into the spectra at the expense of a much longer computing time. 6.2.5  Input and Output Spectra  In the theoretical four-fold differential  {d o/dT^+dT^-dSl^+dQ^-) A  T^+  spectra the  event distributions were histogrammed in 7 M e V energy bins (see appendix B ) . We fitted the distributions with fourth order polynomials to obtain an analytic relation between energy and yield (see fig. 6.8) that was used as input for the experimental Montecarlo. To obtain a better yield versus energy resolution we scaled the theoretical  95 Chapter 6. Comparison between the Theoretical Model and Our Data  Figure 6.8: Fit to the theoretical T + distribution points. Squares represent the histogrammed theoretical distribution for this particular apparatus configuration.  96 Chapter 6. Comparison between the Theoretical Model and Our Data cross section yields from nb/(MeV sr ) 2  2  to arbitrary units before the fitting procedure  (usually spanning a range between 0 and 200). The experimental Montecarlo program divides the energy axis into 1 M e V bins and associates to each of them the number of events determined by the integer of the corresponding yield value. Since the experimental program uses Montecarlo techniques we increased the statistical significance of the results by multiplying the initial number of events in each bin by a factor of four and subsequently weighted each event in the final spectra with a factor 1/4.  The  program output consisted of three spectra: pion, muon and (pion + muon) kinetic energy distribution. A typical example of the input and output spectra is presented in fig. 6.9. The left top picture shows the pion kinetic energy distribution spectrum used as input to the program and the right bottom picture compares this distribution (solid line) to the program output spectrum that contains pions and muons. The particle loss appears constant over the whole energy region confirming the existence of a feeding mechanism from higher to lower energy channels. The stronger muon contribution to the lower part of the energy spectrum tends to reduce the effect of the particle loss in this region. The muon percentage (about 15%) is consistent with the values of the pion decay probability for these energies. It is also possible to observe the high energy cutoff of the spectrum due to the vetoing of the last scintillator of the stack.  This  cutoff is present also in one of the spectra of fig. 6.3, though it seems not to account completely for the suppression of the events in the higher energy region. Some possible explanations are presented in the last chapter.  97 Chapter 6. Comparison between the Theoretical Model and Our Data  CO  CO  T (MeV)  J  (MeV)  Figure 6.9: Example of Montecarlo input and output spectra. Left top: input spectrum, parametrized theoretical T + distribution at pion production point. Right top: output spectrum, only pions. Left bottom: output spectrum, only muons. Right bottom: output spectrum, pions + muons. Spikes are due to statistics. n  Chapter 7  Results  Once the theoretical model was completed with reaction diagrams, Fermi motion, and with the experimental conditions included, we tested the model on our data. As already mentioned in chapter 5 the theory allows three 'free' parameters: C ,  and f . A  We  constrained C with the energy dependence of the total cross section from the data of [22]. Two degrees of freedom where thus left to be determined by the comparing the model with our data. We calculated theoretical distributions for fifteen sets of parameters and compared them to data using the standard \  2  technique.  The \  2  distribution favours values of gs*^ close to its lower limit and f& values smaller than 1. The latter is consistent with the conclusion of [10]. The results show that the model reproduces the experimental distributions surprisingly well and is quite sensitive to variations in the parameters. The current analysis is not sufficient to set more stringent limits on  f&  and  gN*& . v  However, we have de-  veloped necessary tools for a more extensive program investigating  (7r,27r)  reactions  occurring in different isospin channels. Such a program would lead to a more accurate determination of the coupling constants by allowing all three parameters to be varied simultaneously. 7.1  C determination  The energy dependence of the total cross section is a significant constraint on the coupling constant C , since the magnitude of the total cross section is strongly sensitive 98  Chapter  7.  99  Results  T (MeV)  <C=-1.8  °*C=-1.9  <TC=-2.0  °'C=-2.03  T  203  14.1 ± 1.5  4.2 ± .2  15.0 ± .2  16.3 ± . 3  16.0 ± . 3  230  60.8 ± 3 . 2  58.6 ± .8  59.7 ± .8  66.6 ± 1 .  67.4 ± 1 . 2  255  168.7 ± 6 . 4  147.7 ± 2 .  153.4 ± 2.3  158.2 ± 2.3  164.4 ± 2.6  280  384. ± 1 6 .  300. ± 5.  314. ± 5 .  331. ± 6 .  334.7 ± 5 .  T (MeV)  <7C=-2.05  <7C=-2.08  <7  C=-2.09  <7C=-2.1  CC=-2 15  203  16.4 ± . 3  16.7 ± . 3  16.5 ± . 2  17.0 ± . 3  17.4 ± . 2  230  67.4 ± 1.2  67.2 ± .9  69.5 ± . 9  71.4 ± 1 . 1  70.7 ± . 9  255  168.7 ± 2 . 1  171. ± 2 . 9  168.6 ± 2 .  175.2 ± 4 .  180. ± 3 .  280  343. ± 5.  349 ± 6.  353. ± 5.  367. ± 6.  358. ± 5.  Table 7.1: Calculated cross section with different C values. Errors from Montecarlo statistical error. o from [22]. t  to the variation of this parameter in the energy range from threshold up to the energy of our experiment. We varied C keeping the gN*&r and / A values fixed at 1.5 and 1, respectively. These two values were chosen arbitrarily, since the total cross section magnitude is not influenced by their variation up to about 270 M e V and even then the sensitivity is small (see table 5.1). We calculated the cross section for nine values of C ranging from -2.15 to -1.8 for four energy points: 203, 230, 255 and 280 M e V . The results are shown in table 7.1. Subsequently we calculated the x for each value of C according to: 2  X  where z  m  -2-,=i  and x are the measured and calculated values of the total cross section c  Chapter  7. Results  100  50  x  u i  -2.20  1  -2.08  1  -1.96  1  -1.84  C value  1  h  -1.72  -1.60  Figure 7.1: \ distribution as a function of C. The solid line connects the estimated points. 2  respectively and o  m  is the experimental error associated with each measurement.  The  X distribution is shown in fig. 7.1. We fitted the four points in the region of the 2  minimum with a second order polynomial to obtain an analytic expression around the x minimum. Such a procedure allowed us to determine the value of the coupling 7  constant within one standard deviation. To accomplish this we used the \ distribution 2  property: X (a + 2  Aa)  = x (a) + 1 2  for  Aa  =  a  a  that is the \ changes by 1 when the value of a parameter changes by one standard 7  Chapter  7.  101  Results  deviation [49]. When the minimum reduced \ value is larger than 1, it is common 2  procedure to determine the errors of the parameters by scaling their value obtained with the method described above by the square root of the minimum reduced \ value. 7  We thus obtained the following range for the C coupling constant: -2.105 < C < -2.029  7.2  <7,v*A,.  and /  A  Variation  We fixed C at -2.08 and then evaluated the model for three values of g^'Ar and five values of f& (a total of 15 combinations, see table 7.2.). We varied g^A-r from its lower to its upper limit with one intermediate step at  gs<*An =  1.31 and for each of them we  tested five different values o f / A : 0, 0.1, 0.5, 1, and 2. As already described in chapter 5 there is no direct measurement currently available for this last coupling constant i n spite of its importance (see chapter 1). The vast majority of the theoretical predictions give values around one, while [10] favours smaller values. We therefore concentrated our attention on values equal to or smaller than one. However we also tested one larger / A value to confirm the validity of the accepted range of values. In the figures 7.2-7.13 we show the theoretical distributions obtained with these parameter sets compared to the T +, T -, and 0 + T  T  r  experimental spectra.  The T + distributions were obtained by including the experimental effects i n the theoretical distributions (see chapter 6). The curves associated with the values of / A equal to or smaller than one agree with the data remarkably well, while the curves associated with the value / A = 2 overestimate the data i n almost all the spectra. A quantitative comparison was performed by calculating the \ value for each parameter 2  set as explained in the next section.  Chapter  7.  102  Results  Figure 7.2: theoretical distributions compared to data. Q Q D at 50°. <7N*A*= 1-05. Short-dashed curve: / =0; solid curve: / =0.1; dotted curve: / =0.5; long-dashed curve / = l ; dashed-dotted curve / = 2 . A  A  A  A  A  Chapter  103  Results  7.  Figure 7.3: T + theoretical distributions compared to data. Q Q D at 80°. gx*&T= 108. Short-dashed curve: / A = 0 ; solid curve: / A = 0 . 1 ; dotted curve: / =0.5; long-dashed curve / A = 1 ; dashed-dotted curve / A = 2 . n  A  Chapter  7.  6  Results  104  =50°±5  •  C  T  '  data  points T  I '  • -•»•* 5*  1  1  1 —  •..-+50** 5*  w—  3h  0<T  <75 MeV  0 < 7 . <75 MeV  3  2 I1 20  53  T  70  87 i  T . . » 8 0 i 2 0 MeV  3  103  (MeV)  1  1  120 1  —  T..=45416 MeV •..-+50 t 5° #  r  0<T  <75 MeV  0<T .<75 MeV  2h 1  r  K)  103  33  127  150  150  10  Figure 7.4: T - and 0^+ theoretical distributions compared to data. Q Q D at 50°. 0 N * A * = 1-08. Short-dashed curve: / A = 0 ; solid curve: / a =0.1; dotted curve: / =0.5; long-dashed curve f —\\ dashed-dotted curve f&=2. v  A  A  Chapter  6 2.5  Tt-  T  105  Results  7.  • data points  =80°±5° 1  1  1  ,-  T  -i  <  1  '  r  T  —i •..••HO'i  20 0<T  3. r  <75 MeV  0<T  1  —  5*  <75 MeV  1.5 2. r  1.0  0.5 0.0  20  37  T  53  BB&-  70  103  (MeV)  120  20  37  53  70  87  103  120  T„_ (MeV)  15 T . . = 8 0 i 2 0 MeV  T . . = 4 5 i 1 S MeV  0<T  0<T  12  •80*1 <75 MeV  1 r  3 r c  5»  <7S MeV  10  150  150  10  b  e  e  Figure 7.5: T - and 0^+ theoretical distributions compared to data. Q Q D at 80°. gN***= 1-08. Short-dashed curve: / A = 0 ; solid curve: / A = 0 . 1 ; dotted curve: / =0.5; long-dashed curve / A = 1 ; dashed-dotted curve / A = 2 . T  A  Chapter  7.  Results  106  Figure 7.6: theoretical distributions compared to data. Q Q D at 50°. gx*A = 1.31. Short-dashed curve: /a=0; solid curve: / A = 0 . 1 ; dotted curve: /a=0.5; long-dashed curve / A = 1 ; dashed-dotted curve f =2. T  A  Chapter  7.  Results  € )  7 I  107  _=80°±5°  • data points  T  (MeV)  Figure 7.7: 7^+ theoretical distributions compared to data. Q Q D at 80°. gn-* =- 1.31. Short-dashed curve: / A = 0 ; solid curve: / A = 0 . 1 ; dotted curve: / A = 0 . 5 ; long-dashed curve / A = 1 ; dashed-dotted curve / A = 2 . AT  Chapter  7.  6  108  Results  • data points  =50°±5* 71-  -r  T •,.-•50**  •..-+50**  5*  5*  4 h  3 r  0<T  0<T  <75 MeV  <75 MeV  3 r  53  20  T  70 87 (MeV)  120 — i  3  r  T _ « 8 O ± 2 0 MeV  T  •..-•50*t  •,.-+50*t  0<T  10  r  5*  3  <7S MeV  150  33  r  10  —  =45*MeV 5'  0<T ,<75 MeV  33  57  tso  80  Figure 7.8: T , - and 6^+ theoretical distributions compared to data. Q Q D at 50°. 9N*AT 1.31. Short-dashed curve: / =0; solid curve: / =0.1; dotted curve: / =0.5; long-dashed curve / = l ; dashed-dotted curve / = 2 . A  =  A  A  A  A  Chapter  6  109  Results  7.  • data points  =80°±5' TT —  2.5  -i—•  T" •,.-+«0»t  r  T • . - + B 0 i 5"  5*  o  >  2.0  e..=-50 ±28• , ,  0<T  <75 MeV  ( X T . <75 MeV  1.5  2. r  1.0  1. r  0.5 0.0  <  20  15 12  37 '  53 70 T._ (MeV)  1  1—  r-  "T  120 '  T - ™»  T  20 — i  '  T . . = 4 5 i t t MeV  T . . = 8 O i 2 0 MeV ' -  e._-+60*i 5*  -  0 < T < 7 5 MeV t>  • . . - + 8 0 * * 5*  3h  0<T  <75 MeV  52 X  6  r  TV  3 0  >  10  „i  33  57  I  80  .  r n i-l n B 3 — i  103  127  150  O-B-  10  150  e  Figure 7.9: T - and 8^+ theoretical distributions compared to data. QQD at 80°. 9N*A*= 1-31. Short-dashed curve: fA=0; solid curve: / =0.1; dotted curve: / =0.5; long-dashed curve / A = 1 ; dashed-dotted curve / A = 2 . m  A  A  Chapter  110  Results  7.  © _=50°±5°  • data points  7t  0  20  40  60  80  0  20  40  60  80  Figure 7.10: T + theoretical distributions compared to data. Q Q D at 50°. 0 N * A * - 1-53. Short-dashed curve: / =0; solid curve: / A = 0 . 1 ; dotted curve: / =0.5; long-dashed curve / A = 1 ; dashed-dotted curve / =2. A  A  A  Chapter  7.  111  Results  0  =80°±5°  • data points  71 —  •*  Figure 7.11: T + theoretical distributions compared to data. Q Q D at 80°. gx*Ar= 1-53. Short-dashed curve: / = 0 ; solid curve: / =0.1; dotted curve: / =0.5; long-dashed curve / = l ; dashed-dotted curve / = 2 . A  A  A  A  A  Chapter  7.  6  Results  112  =80°±5°  • data points  TT —  25 20  1  f-  —  '  —  1  —  e..=-looks' 0<T  <75 MeV  1.5 1.0 0.5 CC  20  120  o. i  103  120  15 T . . = B C ± 2 C MeV 12  T . . = 4 5 i l S MeV 0<T  <75 MeV 0<T  9h  80  103  127  150  <75 MeV  10  6  150 e  Figure 7.12: T - and 9^+ theoretical distributions compared to data. Q Q D at 50°. 1-53. Short-dashed curve: / A = 0 ; solid curve: / a =0.1; dotted curve: / =0.5; long-dashed curve / A = 1 ; dashed-dotted curve / = 2 . K  0/VA»r=  A  A  Chapter  113  Results  7.  6  =50°±5°  • data points •..-+50** 5* W>**28* 0<T  <75 MeV  0<T <75  MeV  120  4  '  1  1  i  >  1  . —-  i  —  i  —  i  i  —  —  T._=B0±20 MeV .  3  0<T„<75 MeV  2  1  0  D —  »  -  —  0 i  —  .  —  33  i  —  57  ,  '  SO  J  103  —DX3.  127  .  150  e  Figure 7.13: T , - and 0^+ theoretical distributions compared to data. Q Q D at 80°. gN*&*= 1-53. Short-dashed curve: / A = 0 ; solid curve: / A = 0 . 1 ; dotted curve: / A = 0 . 5 ; long-dashed curve /A==1; dashed-dotted curve / A = 2 .  Chapter 7. Results 7.3  x  114  Analysis and Total Cross Section Estimate  2  Our data are histogrammed as a function of three different variables (T^*, T ~, 6^+). r  These spectra provide independent information even though produced from the same data base. For each parameter set we therefore estimated the x value relative to each 2  of the three distributions, divided it by the number of points (since the number of points is significantly larger than the number of fitted parameters, we approximated the degrees of freedom with the number of points) contained in each distribution and averaged the three reduced x values: 2  (xi /AV +  where A j , ^T -» Ne„+ T  +  T  +  w+  a r e  X?v/AV + •xl.+ /JV».  +  )/3  =  xl  the number of points in each distribution as summarized  in table 7.2. In this way the same weight was applied to all the histograms.  Table  7.2 also shows the average x value obtained for each parameter set. These values are 2  histogrammed in fig. 7.14, which thus provides a summary of the results and shows the sensitivity of the model to variations in the parameters. We cannot find a clear x  2  distribution minimum for any particular value of the coupling constants, but we can definitely rule out f& values larger than 1.0 for any value of g^An- The smallest value of gN'Ar seems to be favoured except for / preferred  ^A'-A*  A  values very close to 0, where there are no  values.  Since there is no strongly preferred set of parameters we cannot rigorously determine the coupling constants values. We can still achieve an estimate of the total cross section by calculating its value according to the model for a number of sets of parameters that provide the best agreement between the differential cross sections of the model and our data. We selected three sets of parameters that represent the lower and the upper limit and an in-between value for the cross section and performed the total cross section calculation. The range of the theoretical cross section values for our experiment as well  Chapter 7. Results  9N' T A  115  A-  U  98 points  73 points  64 points  1.08  0  2.63  4.13  2.02  2.92  1.08  0.1  2.94  4.51  1.83  3.09  1.08  0.5  2.93  4.16  1.96  3.02  1.08  1  3.14  4.82  2.41  3.46  1.08  2  5.11  8.71  5.96  6.59  1.31  0  2.85  4.27  2.03  3.08  1.31  0.1  2.78  4.49  1.96  3.07  1.31  0.5  3.08  5.32  2.22  3.54  1.31  1  3.33  6.01  2.87  4.07  1.31  2  5.11  10.78  7.31  8.25  1.53  0  2.94  4.07  2.07  3.05  1.53  0.1  3.37  4.80  2.10  3.42  1.53  0.5  2.93  5.89  2.49  3.77  1.53  1  3.99  8.23  3.51  5.24  1.53  2  8.27  12.00  11.23  10.5  Table 7.2: Parameter sets used in the theoretical calculations and \ divided by the number of points for each configuration. The last column shows the average reduced X value (see text). 2  2  Chapter  7.  116  Results  Figure 7.14: Average reduced \' value as a function of / variation. The three curves correspond to three different g * * values. The lines connect the points. 2  N  A  A  Chapter 7. Results  c  0N»Air  117  / A  w  quasifree calc.  free calc.  quasifree calc.  -2.048  0  1.08  316 ± 4  319 ± 4  156 ± 2  -2.080  0.5  1.08  317 ± 4  319 ± 4  160 ± 3  -2.086  0.8  1.53  340 ± 4  336 ± 4  172 ± 3  Table 7.3: Calculated total cross section values for three parameter sets. First column: total cross section for our experiment according to this model. Second column: values to be compared to [22]. Third column: values to be compared to [23]. Errors are statistical errors from the Theoretical Montecarlo Program. as for the experimental conditions of [22] and [23] (see next section) is presented in table 7.3.  7.4  C o m p a r i s o n to other data  We used the same sets of coupling constant to calculate total cross sections for the free reaction at the incident pion energy 280 M e V and for the quasifree reaction at T„ = 256 M e V . The results in table 7.3 can be compared to the experimental values (384 ± 16)pb of [22] and (160 ± I0)ub of [23]. In the case of the free reaction even the largest value expected from the model underestimates the measured cross section by more than two standard deviations, while the quasifree reaction cross section is reproduced remarkably well.  Chapter 8  Discussion and  8.1  Conclusion  Discussion  There are three important observations arising from the comparisons described in the previous chapter. The model reproduces the experimental total cross sections up to about T =2bO  M e V very well with the lower values of the coupling constants being  T  favoured. This is also the case when the model is compared with our differential cross section data at T,=280 M e V , except that at this energy it underestimates the free total cross section measurement [22] by about 15%. Thus,if the model is complete, the total cross section measurement is inconsistent with our data. O n the other hand the model may not completely account for some mechanisms that contribute to the total cross section at higher energy in a region of phase space that was not covered by our experiment. Possible contributions of this type could arise from mechanisms involving heavier N* isobars in the intermediate state. A further test of the model could be provided by a more comprehensive experiment that would measure pion induced pion production in a greater variety of isospin channels, since the various diagrams in the theory contribute differently to the cross section in different isospin channels. Such an extended experiment would also facilitate determination of the coupling constants, since different isospin channels provide different limits for the coupling constants; for example a larger f& value decreases the 7r p —* T r 7 r n cross section, while it increases +  the Tr"p —> 7r 7r~n cross section. +  118  +  +  Chapter  8.  Discussion  and  Conclusion  119  Comparison of the theoretical curves to the data still indicates some discrepancies as shown in figures 7.2-7.13. One of them could be explained by experimental problems. The theoretical cross section systematically overestimates the data i n the high energy part of the T + spectra obtained in the second series of runs, when one additional v  scintillator bar was added to the detector to increase its energy range (see chapter 2).  This overestimation is present in spite of the correction to the theory for the  altered experimental conditions, while the same spectra for the first experimental period ( Q Q D set at 80°) show a much better agreement between data and theory in that energy region. This suggests the possibility of an energy miscalibration of the last scintillator bar. However, in the comparison of data to theory it was apparent that the statistical accuracy was inadequate to extract better limits on the coupling constants. Significantly more work on calibrations was therefore deemed unprofitable. A similar comparison was performed by the authors of [24], where a partially different model was employed [50]. They do not account for the e induced contribution and treat the off-shell pion in the pion-pole term differently from ours. In a preliminary analysis they concluded that their model underestimates the in-plane reaction cross section, which seems in disagreement with our results. However, since the two theoretical models are different, the two conclusions are not necessarily contradictory. 8.2  Conclusion  We extended the existing [12] model to be applicable to pion production in a deuteron in the energy range of interest by incorporating additional diagrams together with allowance for Fermi motion of the contributing nucleon. We demonstrated that the model reproduces both our data very well and the total cross section measurements at lower energy.  Chapter  8.  Discussion  and  Conclusion  120  W i t h i n the framework of the model we were able to determine the C parameter for the  N*—»  of the „[10].  N(irir),_  ATTA  wave  to be between  -2.105  and  -2.029  coupling constant to be smaller than  1.0  and demonstrated a preference  in agreement with the analysis of  We did not detect significant sensitivity of the cross section to the A *Arr coupling r  constant in agreement with [24]. However our data show a preference for the lower values within the allowed range. The results of our experiment and analysis contribute to the understanding of the overall mechanism of the free pion induced pion production reaction, even though we realize that they are not quite adequate for providing significantly improved limits on the coupling constants,  gN*&*  and  / A -  We believe that a more stringent test of  the model and consequently a more accurate determination of such parameters would result by comparing the theoretical distributions to a more extensive set of precise differential cross section measurements of the free process where there is no broadening of the angular and energy range due to Fermi motion. A natural candidate for further investigation is the reaction  ir p +  —> 7 r 7 r n . +  +  Due to isospin conservation constraints  there are only a few intermediate states permitted in this reaction. It is thus easier to isolate the contributions of each reaction mechanism. Currently studies are being performed to understand how the / A coupling constant and the pion scattering length could be extracted from the measurement of the cross section of this reaction [51]. Since the cross section for this process is about one order of magnitude smaller than that of 7r"p —+  7r"7r n +  the energy and the geometrical setting of the possible experiment would  have to be carefully planned in order to guarantee the maximum information from the data. We also feel that more extensive measurements of the in- and out- of the reaction plane cross section are needed for a more complete test of the model. A more extensive set of data would also allow a simultaneous parameter variation, making their determination even more significant.  Appendix A  Example of Amplitude Calculation and \T\ for a Polarized Target 2  In the first part of this appendix we present an example of a detailed amplitude calculation. In the second part we show the square of the coherent sum of the amplitudes for a polarized and unpolarized nucleon target. A.l  Amplitude Calculation  The cross section calculation for any rection has a term related to the dynamics of its mechanism and a part accounting for the kinematic constraints. The first part is usually described by the T matrix that connects the initial and the final state through the possible intermediate states weighted with their strength and energy dependence. If the T matrix is graphically expressed in terms of Feynman  diagrams, then the  strength of a mechanism is described by the vertex coupling constants and the energy dependence by the intermediate state propagators and the vertex factors. The T matrix must, of course, obey all the conservation rules relative to the particular interaction, that is energy, isospin and and parity in our case, since we are considering a strong interaction. The kinematic constraints are given by the overall momentum and energy conservation requirement and the probability for a particular kinematic state is decribed via the phase space factor. The total cross section is obtained with the integration of the T matrix weighted by the phase space factor over the geometrical and momentum space. The phase space factor used in our cross section calculation is described in [12] as well 121  Appendix  A.  Example  of Amplitude  Calculation  and |T|  2  for a Polarized  Target  122  Figure A . l : 2 point diagram. as a detailed listing of the amplitudes related to the (Tr,27r) interaction mechanisms. Here we present the derivation of the amplitude related to the diagram 5.3.a) as an example (fig. A . l ) . According to the Feynman  rules a diagram can be translated in an amplitude fol-  lowing the convention that any diagram external line corresponds to a free particle field, any internal line to a propagator, while the vertex factor arises from the Lagrangian terms that contain the interacting fields. We follow the convention of [20]. A.1.1  V e r t e x Factors C a l c u l a t i o n  The diagram of fig. 1 is a two point diagram, containing thus two vertices and a nucleon in the intermediate state. The first vertex involves two nucleons and a pion, while the  Appendix  A.  Example  of Amplitude  Calculation  and  \T\  2  for a Polarized  Target  123  second one two pions and two nucleons; therefore, we derive the related vertex factors from: LNN* = {flu)ip+Oi{di<t>) rxp  (1)  and LsNr,  = -4rr  \^+<p  • <prP + %ip T(<p  x d <p)rp]  +  t  (2)  where  <p represents the pion field that is a triplet in isospin space and a pseudoscalar, its derivative is thus a pseudovector. Oi are the spin matrices, projections of the spin operator, that is a pseudovector. T represents the isospin operator matrices that form a vector in the isospin space. Both Lagrangians are thus scalars and isoscalars as requested by the invariance under Lorentz transformations and under rotations in the isospin space. Here they are expressed in a nonrelativistic form, since the energies involved are smaller then the nucleon mass. The isospin matrices r j , T , T can be combined in a particularly convenient form 2  3  to form isospin lowering and raising operators: T±  =  §(71  ±  T ) 2  where T.\p>=\n>  r |n>=|p> +  r \p >= r_|n >= 0 +  The components of the isovector 4> can be expressed in cartesian isospin coordinates as: ^ = (^1,^2,^3)  Appendix A.  Example  of Amplitude  Calculation  and  |T| for a Polarized 2  Target  124  or:  In terms of second quantization language #  +  creates a 7r and destroys a T T " , while +  creates a 7r~and destroys a 7r . They can be written in the form of creation and +  annihilation operators in the following way: <£+ cc (a.(q)exp(-iqx)  a^iq)exp(iqx))  +  (3.1)  <j>. o c ( a ( g ) e x p ( - i g : r ) -f ai(t7)exp(tgx))  (3.2)  +  where a_ and a+ annihilate a 7r~and a 7r with four momentum q respectively, while +  at and a.£ are the related creation operators. Equation (1) can be thus developed: LNNT  = (///^"Wnd.^i +  T di<t> + r di<t> )ip 2  2  3  3  or, substituting coordinates: IAA>  =  (f/v)iP cn(\/2(T-d <t> +  l  +  +  T d,(f>.) +  +  T d,> )^ 3  3  Since we have a proton in the initial state and a neutron in the intermediate state only the term with the isospin lowering operator T _ will give a non zero constribution to the Lagrangian that is thus reduced to: INN*  =  ifI>)v^t/>tf,T_6\<?!> ^ +  Remembering (3), we can use <f> to destroy the incoming n~, that has a four momentum +  pi. Since we are using a nonrelativistic approach, we only need the spatial part of the ^  +  exponent (-p^x): d,(a_(p)exp(t - (-Pji))) = t p ,  The vertex is proportional to the i L interaction term:  Appendix  A.  Example  TTNN  of Amplitude  Calculation  and  vertex factor = - (f/u)V%  The factor for the vertex  \T\  for a Polarized  2  Target  125  (4)  tr •  is derived in a similar way from Lagrangian (2).  TTXNN  ..The first part of (2) contains a <p term, that can be expanded i n : 2  <t> = 2  +$ + #  or in terms of charged states:  <t> = < M - + < M + + 4>l  (5)  2  In this vertex we are creating a 7r and a T T " . We can thus use 4> and 4>- as creation +  +  operators for pions of the appropriate charge, and since we have two such terms in (5) this part contributes with a factor 2. In the second addend of (2) only the part containing r survives since the nucleon 0  isospin does not change from the intermediate to the final state contributing with a factor -1, that is the To\n > eigenvalue. This leaves only the k component of the vector product (<p x d (p) that is: t  4> d <t>2 — 1  fadt&i  t  or, in terms of charged states: i (<t>+d <f>t  Using (3) and the operators  <f>  +  - <f>-d 4>+) t  and <^>_ as above this term leads to:  (-1) i i (rf - pi)  =  p° 6  p°  s  The factor relative to the irnNN vertex is thus: xrrNN  vertex factor =  (-t)47r  + ^ ( p g - pf))  (6)  Appendix  A.1.2  A.  Example  of Amplitude  Calculation  and \T\ for a Polarized 2  Target  126  Propagator Calculation  Since we are using a non relativistic approach the propagator is calculated as Ei  -  E  n  where Ei is the total energy of the initial state and E  n  is the total energy of the  intermediate state. In our example, the propagator has a particularly simple form: the initial energy is the invariant mass of the system, defined as y/S, while the energy in the intermediate state is the nucleon mass (m):  Putting together (4),(6) and (7) we obtain the full expression for the amplitude - i T : - i T = - (f/u) y/2*.p ±^ l7  47T (a*- + %(  P6o  -p  S o  ))  where all the variable are expressed in the 7r nucleon center of mass system. This particular amplitude contains only a dependence on the particle momentum and nucleon spin, but in the general case the amplitude has also an energy dependent part. The general form of an amplitude related to the reaction diagrams obtained in the same way as the sample calculation can be thus written as: T  = a + b-<r  where a is the part of the amplitude containing the energy dependent terms, while b contains the momentum dependent terms. A.2  S q u a r e o f the C o h e r e n t S u m o f the A m p l i t u d e s  The cross section of a reaction is proportional to the square of the transition matrix T . Since in our case all of the diagrams contribute to the reaction and we are not able  Appendix  A.  Example  of Amplitude  Calculation  and  \T\  for a Polarized  2  Target  127  to distinguish amongst them in the cross section calculation, we have to construct the square of the coherent sum of the amplitudes relative to each diagram. The form in which the amplitudes are written is particularly convenient, since it allows for an easy •T  summing of the various diagram contributions.  where t refers to a particular diagram and j to the momentum and spin cartesian components. The square of the amplitude T is evaluated between initial and final state as: \T\ = T,Zj  <i\T+\fxf\T\i>  2  where i and f denote the initial and final state respectively. A.2.1  U n p o l a r i z e d Target  In an unpolarized case the initial state is not completely determined, therefore we have to average over the two possible states of the undetermined spin variable. In what follows a state with spin oriented up w.r.t. the direction of o$ axes will be indicated with an uparrow (f), while in the case of a state with spin down we use a downarrow (J.).  It is useful to express the spin matrices in terms of the spin raising and lowering  operators a  +  and a_ (analogous to the isospin raising and lowering operators of the  previous section):  0\  = cr + +  c7  o~  2  =  i(cr~  —  er =  a ) +  0  In this coordinates we can express the scalar product 6 • tr as follows: b • tr  <r b~ + c~b +  —  +  +  c7 6 0  0  <?3  Appendix  A.  Example  of Amplitude  where b = fcj + t & , fc = &i +  —  2  Calculation  and \T\ for a Polarized 2  Target  128  *&2 and bo = 63.  Now we can start evaluating |T| : 2  |T| = 2  § [<T  |a* +  6* • *\ T X T  § [<T  |a* +fc*• <r|  § [<1  |a* + fc* • <r| T > < T  § [<1  |a* +fc*• «r|  | > < T  |a + * • «r| T>] +  |a + 6 • <r\  |a +  6 • cr| 1>]  1>] + +  1><! |a +fc• a\ 1>] =  acting with the spin operator expressed in spherical coordinates and listing only the terms w-ith a non zero contribution we obtain:  = § [2|a| + 2|6 | + (6-)"6 + (6*) 6"]= 2  3  2  +  +  = 1 [2|a| + 2|6 | + (K - «62)(6i +162) + (&* + t&SX&i - ^2)] = 2  3  2  = i[2|a| + 2|fe| + 2|6| + 2|6|] 2  3  2  1  2  2  2  that finally gives: |T| = |a| + |fc| 2  A.2.2  2  2  Polarized Target  In the case described above we can immediatelly observe that there is no interference amongst terms that have a momentum dependence and the ones that have a pure energy  Appendix A. Example of Amplitude Calculation and \T\ for a Polarized Ta 2  dependence. This is not the case when we have a polarized nucleon target as shown in what follows, thus we expect to possibly obtain different information about the reaction mechanisms in the two cases. The only (TT, 27r) experiment that used a polarized proton -.target was performed at T^- = 17.2 GeV, that is far beyond our energy [52]. Sofar there are no experimental data about a (7r,27r) reaction on a polarized nucleon at our energy and also no theoretical studies. The implementation of the model with these amplitudes provides a useful investigation tool for future experimental plans. Initial State Polarized " d o w n " In this case the initial state is completely defined and we can evaluate the square of the T matrix as:  m  2  = £/ < i | T i / x / | T U > +  following exactly the same procedure as above we find:  |T| = M + |b| 2  2  c  2  where C = [a*h  + bla + i(b\b 2  blh)]  I n i t i a l State P o l a r i z e d " u p " Here we have:  m  2  = Z  f  <T|r |/x/|T|T> +  tha leads to: |T| = |a| + 2  2  |b| + C 2  Appendix A. Example of Amplitude Calculation and \T\ for a Polarized T 2  In identifying the polarization direction w.r.t. the laboratory reference frame particular care has to be taken in associating the third component of the above vectors to the direction of the polarization axes.  Appendix B  Amplitudes Related to the New Diagrams and Model Cross Section Calculation  B.l  Amplitudes Related to the Added Diagrams  In calculating these amplitudes we followed the convention of [12] for the A and A ' r  mass distribution. The A * mass ( M * ) is taken to be 1440 M e V and its width (P") T  at any energy is appropriately scaled to the width value at the peak energy (235±115 M e V [40]). The amplitudes related to the diagrams of fig. 5.11 are thus:  _ 3jV* zT  =  _(//  A I  ) / / ) a ^ o r • p <r • p <r . (  / 1  2 v  6  s  P l  \/S-p° ,-m-pl/2m y/S-M'+(i/2)r* i  N/S-m +(i/2)r  N/5-pg-p|/2Af-Af*+(i/2)r*  A  -iT ' ~ 3  b  N  S  = -(f/ti)(r/p.)(g . )y/2S' N Ar  P6  +  .v Pl  •p  5  t  I  pj/2m+m-p°-(p  l  +  p ) /2A/--A/*+(./2)r* J  5  -Hf-**' = -(f/n)(r/li)(9N*A*)y/2<r-p  VS-pl-pl/2m -m*+{i/2)r  S.p,S+.p  s  131  A  6  Appendix B. Amplitudes Related to the New Diagrams and Model Cross Section i  P]l2m-pl-u> -{p R  l  i  + P 6 ) / 2 m + ( , / 2 ) r >/S-p°-M*+(t/2)r*-pf/2M« s  A  - t 7 j - * = -V/n)(r/M9N*A*) $S.p N  3  S+ .p <r - p 6  l  5  >r  PJ/2m-pS+m-M-+(t/2)r«-(p  + p )'/JM« Pj/Sm-pJ-pO-WH-tp, + P  l  s  = -(f/u)(r/u)( N^)f<r-  - i l * * *  S • p  Pl  9  pj/2m-p°-w«-(p  l  6  5  J  A  N  P6) /2m +(./2)r J  A  s  + p ) /2M»+(,/2)l>-Af-  5  + p ) /2A^+(</2)r--A/'  6  3  S -p <r. +  6  ;  +  5  + p ) /2m +(,/2)r pJ/Sm-pg-pg+m-tp, + p  -if}- * = -(f/u)(f*/u)(g ^)s/2S'p N  5+ . p  5  s  Pi  t  \/S-A/*+(i/2)r* x / S - p ° - m - p ? / 2 m + (i/2)r A  A  _  j T  3,A-  =  - ( f / W / M g ^ y f a . p ,  pJ/2m-p°-w -(p K  x/S-A/« + («/2)P>  where w/j = m — m  A  .  a  S.  P  s  S+.p  6  + p ) / 2 m + (i/2)r p j / 2 m - p ° - p ° + m - ( p + p 6  J  A  l  >/5-pg-m -pJ/2m +(i/2)r A  A  6  J  Appendix B. Amplitudes Related to the New Diagrams and Model Cross Sectio B.2  M o d e l Cross Section Calculation  The program used to calculate the theoretical cross sections proceeds with a Montecarlo selection of an event and an analysis on an event by event basis. It is structured in three main parts. In the first one it calculates the kinematic variables for each event and in the second one it calculates the cross section for that particular event. After completing the calculation for the required number of events it outputs the results in a form that simulates the experimental apparatus acceptance. The kinematic as well as the cross section calculations are performed in the particle center-of-mass system, while the final distributions can be given either in that or in the laboratory system. The conversion from the center of mass system to the laboratory system occurs via a simple Lorentz boost in the case of the free process, since the direction of motion is always along the z axes. To account for the possible x and y components of the nucleon momentum in the case when Fermi motion is present, the coordinate system had to be chaged with an additional rotation. B.2.1  Event Determination  The final state of the reaction allows for nine degrees of freedom, since there are three particles in the final state. Five of them ( # „ . - ,  C ^ - i  IPITT' ^7r -  +  IPITC+)  a r e  chosen  randomly by the program between their physical limits: the angular ranges cover the whole space, the lower limit for the momentum absolute value is 0, while the upper limit is calculated for each particle from the invariant mass of the system assuming the other two particles to be produced at rest. This limit, of course, varies for each event when the Fermi motion is present, while it is constant in the case of the free process. The other four variables are fixed by the energy and momentum conservation requirement. For each set of five randomly selected variables the program checks if the energy and momentum conservation equations allow a solution for the other four variables. If so, it  Appendix  B.  Amplitudes  Related  to the New  Diagrams  and Model  Cross  Section  CalculationlM  calculates the square of the coherent sum of the amplitudes for the coordinates of that events, multiplies it by the appropriate phase space factor and stores the cross section value i n a predefined grid element. B.2.2  G r i d Elements and Output Format  Since the program is able to calculate all the variables of all three particles, we can construct any differential cross section we need by binning the cross section in a grid of arbitrary (up to nine) dimensions. The size of the grid element is determined by the size of the interval that we choose for each variable. In principle the binning size should be as small as possible to conserve the concept of a differential cross section, while in practice we have to compromise between a reasonable computing time and accuracy. This is not a major drawback if the theoretical distributions are to be compared to experimental data.In that case they must have a binning size that is comparable to that used in the histograms of the experimental data. Our data are presented in terms of kinetic energy distribution histograms of the two pions and of the azimuthal angle distribution of the ir . +  Consequently we pre-  defined a grid which element size was determined by a four dimensional bin A f t ^ - H  •  A f i ^ - • AT + • A T ^ - , where the energy intervals were 7 M e V and the angular intervals 7J>  were essentially defined by  A(cos(0))  —  0.1, since we required the two particles to be  coplanar. 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