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Exclusive measurements of [pi]N --> [pi][pi]N Kermani, Mohammad Arjomand 1997

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E X C L U S I V E M E A S U R E M E N T S OF T T A - » • ir%N By Mohammad Arjomand Kermani B .Sc , The University of British Columbia, 1991 M . S c , The University of British Columbia, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS AND ASTRONOMY We accept this thesis as conforming to^i£_j^ujix4-steiTda,rd T H E UNIVERSITY OF BRITISH COLUMBIA August 1997 © Mohammad Arjomand Kermani, 1997 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. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or 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 of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, Canada V 6 T 1Z1 Date: Abstract The pion induced pion production reactions —> 7 r ± 7 r + n were studied at projectile incident energies of 223, 243, 264, 284, and 305 MeV. The Canadian High Acceptance Orbit Spectrometer (CHAOS) was used to detected the charged particles, which origi-nated from the interaction of the incident pion beam with a cryogenic liquid hydrogen target. The experimental results are presented in the form of single, double and triple differential cross sections. Total cross sections obtained by integrating the differential quantities are also reported. The experimental data, namely the ir~p —> 7r~7r + n double differential cross sections, were used as input to the Chew-Low extrapolation procedure which was utilized to determine on-shell 7r + 7r _ elastic scattering cross sections in the near threshold region. The Chew-Low results (the extrapolated irir cross sections) were then used in a dispersion analysis (Roy equations) to obtain the -KIT isospin zero S-wave scattering length. We find a° = 0.209 ± 0.011/x - 1. In addition, the invariant mass distri-butions from the 7r~) channel were fitted to determine the model parameters for the extended model of Oset and Vicente-Vacas. We find that the model parameters obtained from fitting the ( 7r + 7r _ ) data do not describe the invariant mass distributions in the (7r +7r +) channel. ii Table of Contents Abs t r ac t i i L i s t of Tables v i L i s t of Figures ix Acknowledgments x v i i 1 Physics Mot iva t ions &c Goals 1 1.1 Isospin Symmetry 1 1.2 Chiral Symmetry 3 1.3 Effective Theory 6 1.4 Probes of the -KTI Interaction 7 1.5 Review of Previous Experiments 9 2 The Exper iment 13 2.1 The Cyclotron & Beam Line 13 2.2 The Experimental Apparatus 16 2.2.1 The Magnet 18 2.2.2 WC1 & WC2 19 2.2.3 WC3 20 2.2.4 WC4 21 2.2.5 Counter Telescopes 23 2.2.6 First Level Trigger 24 iii 2.2.7 Second Level Trigger 26 2.3 The Cryogenic Target 29 2.4 Beam Counting 31 2.5 Beam Samples 33 3 Event Reconstruction 36 3.1 Chamber Calibrations 37 3.2 Track Sorting 41 3.3 Momentum Reconstruction 42 3.4 Interaction Vertex & Scattering Angle 44 3.5 Particle Identification 46 4 Data Analysis 51 4.1 Primary Backgrounds 51 4.2 Particle Identification Process 52 4.3 Missing Mass Calculation 57 4.4 Data Reduction 58 4.5 Empty Target Data 64 4.6 The Binning Parameters 64 5 Monte-Carlo Simulations 69 5.1 Statement of the Problem 69 5.2 Event Generation & Tracking 71 5.3 Analysis of G E A N T Data 75 5.4 Remarks on Phase-Space Extrapolation 84 6 Cross Sections 87 6.1 Acquisition Live Time 87 iv 6.2 Beam Calculations 88 6.3 Wire Chamber Efficiencies 95 6.4 Target Thickness Calculations 100 6.5 PID Efficiency 102 6.6 Normalization Checks & Elastic Cross Sections 104 6.7 The Cross Sections I l l 6.8 Systematic Errors 121 7 Interpretation 123 7.1 Chew-Low Analysis 123 7.2 Roy Equation Analysis 129 7.3 The (TT + TT +) Channel 134 7.4 Remarks 137 7.5 Another Consistency Check 137 7.6 Model Dependent Analysis 140 7.7 Model Calculations & O P E Dominance 154 8 Concluding Remarks 156 Bibliography 159 A Relevant Kinematic Parameters 163 A . l The Lowest |t| 165 B Tables of ir~p —> 7 r + 7 r _ n Cross Sections 167 C Tables of n+p —» 7 r + 7 r + n Cross Sections 184 D Explicit Functional Forms for Gjf 201 v Lis t of Tables 6.1 The pion decay fraction for different incident pion energies 92 6.2 Table showing the incident beam rate, singles fraction, and fraction of beam on the target for all energies and reaction channels studied in this work 94 6.3 Table of particle identification efficiencies for the (TT+ IT~) channel 104 6.4 Table showing the measured total cross sections. The first error bar is the statistical uncertainty, and the second reflects the systematic error. . . . 118 7.5 Table of 7T7T cross sections (mb) 127 7.6 A summary of predictions and experimental results for the isospin zero S-wave scattering lengths 132 7.7 Table of 7 r + 7r - elastic phase shifts 139 7.8 Table showing the results of the global fit to the ( 7 r + 7 r - ) data 151 B.9 Table of double differential cross sections for the (7r + 7T~) channel at 223 MeV 168 B.10 Table of double differential cross sections for the (ir+channel at 243 MeV : 169 B . l l Table of double differential cross sections for the (ir+ir~) channel at 264 MeV 171 B.12 Table of double differential cross sections for the (TT+ 7r~) channel at 284 MeV 174 vi B.13 Table of double differential cross sections for the (n+ ir ) channel at 305 MeV 175 B.14 for the (TI+ TT") channel at 223 MeV 176 B.15 % for the (%+ ir~) channel at 223 MeV • 176 B.16 for the (TT+TT -) channel at 223 MeV 177 B.17 ^ j f - for the (TT+TT") channel at 243 MeV 177 B.18 ^ for the (TT+ T T ) channel at 243 MeV 178 B.19 ^ for the (TT+TT) channel at 243 MeV 178 B.20 ^ g - for the (TT+ TT~) channel at 264 MeV 179 B.21 <f for the (TT+ T T ) channel at 264 MeV 179 B.22 ^ for the (TT+ channel at 264 MeV 180 B.23 for the (TT+ TT~) channel at 284 MeV 180 B.24 § for the (TT+ ? r ) channel at 284 MeV 181 B.25 ^ for the (TT + TT") channel at 284 MeV 181 B.26 for the ( 7r + 7r _ ) channel at 284 MeV 182 B.27 § for the (TT+ TT - ) channel at 305 MeV 182 B. 28 ^ for the (TT+ T T ) channel at 305 MeV 183 C. 29 Table of double differential cross sections for the (TT+ 7r + ) channel at 223 MeV 185 C.30 Table of double differential cross sections for the (7T+7T+) channel at 243 MeV 186 C.31 Table of double differential cross sections for the (TT+ 7r + ) channel at 264 MeV 188 C.32 Table of double differential cross sections for the (7r + 7r + ) channel at 284 MeV 190 vii C.33 Table of double differential cross sections for the (TT+ 7r + ) channel at 305 MeV 192 C.34 ^ g - for the (TT+ TT +) channel at 223 MeV 193 C.35 ^ for the (TT+ TT+) channel at 223 MeV 193 C.36 ^ for the (TT + TT +) channel at 223 MeV 194 C.37 for the (TT+ TT +) channel at 243 MeV 194 C.38 ^ for the (TT+TI+) channel at 243 MeV 195 C.39 ^ for the (TT + TT +) channel at 243 MeV 195 C.40 ^ j j - for the (TT+ TT+) channel at 264 MeV 196 C.41 % for the (TT+ TT +) channel at 264 MeV 196 C.42 ^ for the (TT + TT +) channel at 264 MeV 197 C.43 for the (TT+ TT +) channel at 284 MeV 197 C.44 % for the (TT+ TT +) channel at 284 MeV 198 C.45 ^ for the (TT + TT+) channel at 284 MeV 198 C.46 ^ g - for the (ir+ir+) channel at 284 MeV 199 C.47 ^ for the (TT+ TT+) channel at 305 MeV 199 C.48 ^ for the (TT+ TT +) channel at 305 MeV 200 viii Lis t of Figures 1.1 The Feynman diagram for the One Pion Exchange mechanism is shown. In addition to O P E other diagrams contribute as well 8 2.2 The M i l pion channel 15 2.3 The CHAOS spectrometer. A corner post and the top pole tip have been removed for a better view. In addition, a quadrant of the detector has also been removed 17 2.4 Illustration of the top view of the experimental setup. The beam counting scintillators (Si and S2) and the veto counter are shown. The cryogenic target is represented by the solid circle located at the center of the spec-trometer 18 2.5 A n illustration of WC3 cell geometry. 20 2.6 WC4 cell geometry. Here A , R, and G denote the anode, resistive and field shaping wires, respectively. 22 2.7 A schematic of the first level trigger is shown. For simplicity only one C F T block is included. . . . '. 25 2.8 A simplified schematic of the first stage of the 2LT. Here M L U represents the LeCroy 2372 Memory Lookup Unit, A L U is the LeCroy 2375 Arith-metic Logic Unit, and Stack denotes the LeCroy 2375 Data Stack. In addition, MLU21 is a 21 bit Memory Lookup Unit developed at T R I U M F . 27 2.9 Top and side views of the cryogenic liquid hydrogen target 30 2.10 A n illustration (not to scale) depicting the beam counters Si and S^. . . 31 ix 2.11 Full schematic of the CHAOS trigger system 35 3.12 Illustration of the CHAOS coordinate system. Units are mm 36 3.13 A typical drift time spectrum for WC4. Short drift times correspond to large T D C values 38 3.14 The tree structure employed in the track sorting algorithm. The numbers correspond to the appropriate chamber (layer) 41 3.15 Illustration of a reconstructed (7r,27r) event. The track trajectories in the region of the target (inside WC1 radius) were approximated by circles. The C F T blocks removed at the entrance and exit of the beam are not shown. The incident beam is from the left and is registered by the WC1/2 hits located at negative x-values 45 3.16 Example of the reconstructed interaction vertex for data acquired with a coincidence 1LT and liberal 2LT cuts. They are dominated by irp elastic events 46 3.17 Cerenkov calibration data acquired during the commissioning phase (using the incident beam directly). The electron pulse heights are larger because they produce an electro-magnetic shower in the lead glass. For the pions, the light output is only due to Cerenkov radiation 49 4.18 Flowchart of the particle identification process 54 4.19 A typical scatter plot of pulse height in AEi versus momentum. Pion and proton bands are well separated. These data were acquired with 280 MeV positive polarity pions. The momentum spread is due to the variation of the pion and proton momenta with angle 55 x 4.20 Scatter plot of pulse height in the Cerenkov counters versus momentum. The elastic events have been eliminated. Negative momenta represent negatively charged particles 56 4.21 Scatter plot of pulse height in AEi versus momentum. Events identified as electrons by the Cerenkov as well as n+7r~ events are shown. The ir~p events have already been removed with the momentum sum cut. Negative momenta represent negatively charged particles 56 4.22 Scatter plot of the reconstructed interaction vertex for full (top) and empty target (bottom). The polygon illustrates the vertex cut. Data were ac-quired with liberal 2LT cuts 59 4.23 Plots showing the vertical coordinate of the projection of the incident beam for all events (top) and (TT,2TT) events (bottom) 61 4.24 Momentum sum spectra for np elastic and (TT, 2TT) reactions. The distri-butions were generated using two and three-body phase-space 62 4.25 Missing mass distributions for all events except irp elastic 63 4.26 Flowchart describing the various steps of the analysis 65 4.27 Missing mass spectra for full (top) and empty target (bottom). For each reaction channel the spectra have been normalized to reflect the same number of incident pions 66 5.28 The flowchart describing the acceptance determination process. The "an-alyzer" is the same software used to reconstruct the experimental events. 72 5.29 Spectrum of the out-of-plane angles for Monte-Carlo events, which the simulation predicted would be detected with CHAOS. The solid lines de-note the out-of-plane Monte-Carlo window. The fraction of events within ±7° is 0.95 75 xi 5.30 Missing mass distributions for the Monte-Carlo (bottom) and experimental data (top) 77 5.31 Typical phase space distributions of m ^ , t and cosO in 47r (left) and in the CHAOS acceptance (right) for the (-K+ 7r~) channel at 264 MeV. . . . 79 5.32 Sample phase space distributions for m^n, t and cosd in Air (left) and in CHAOS (right) 80 5.33 Distribution of (TT, 2TT) events generated with F O W L at T^=264 MeV. Plots labeled (A) represent the distributions in Air sr. The out-of-plane angle of the events in histograms denoted by (B) lie within ± 7°. Additional in-plane angular restrictions (accounting for the missing C F T blocks) were placed on the events appearing in the distributions labeled (C) 81 5.34 Scatter plots of m^, t and cosO in 3-dimensions versus those obtained from the reconstruction process, which ignores the third (z) dimension. . 83 5.35 The phase space distribution for a; 85 5.36 The acceptance-corrected distribution measured in this experiment for the (n+ ir~) channel at 305 MeV is represented by the solid points. The line shows the results of Jones at al. [13], obtained at 300 MeV in a Air sr bubble chamber experiment. The Jones data were published in arbitrary units. Thus they have been normalized (multiplied by a constant) to reflect the units of the data from this experiment 85 6.37 Sample-gated spectra for in-plane (top) and vertical (bottom) projections of the incident beam. The solid lines represent the dimensions of the target vessel. From the distributions shown, the fraction of beam on the target was determined to be 0.90 (0.948 in-plane and 0.95 vertical) 90 xii 6.38 The singles fraction as a function of the incident beam rate for selected values of (3 94 6.39 Typical plots of wire chamber efficiency as a function of angle. The dashed lines denote the one sigma deviation and the solid line is the mean. . . . 97 6.40 Illustration of the effective target thickness calculation 101 6.41 Missing mass plot for misidentified TT+TT~ events at 7^=305 MeV . . . . 103 6.42 Missing mass plot for 7 r + e + events at 1^=280 MeV. Regardless of the PID, the pion mass was used in calculating the energy of the final state particles. 105 6.43 The correlation between the scattering angles of the two tracks for np events. A l l events in which both tracks passed the interaction vertex cut are included. As expected, the non-elastic background is minimal 106 6.44 Absolute differential cross sections for ir~p elastic scattering 109 6.45 Absolute differential cross sections for 7r +p elastic scattering 110 6.46 distributions for ir~p —»•n+K'n 112 6.47 t distributions for ir~p —> 7r + 7r _ n 113 6.48 cosd distributions for ir~p —> ir+n^n. For clarity the horizontal error bars are not shown 114 6.49 m2^ distributions for 7r +p —> 7r + 7r + n 115 6.50 t distributions for ir+p —> 7 r + 7 r + n 116 6.51 cos9 distributions for 7r +p —> 7i+7i+n. For clarity the horizontal error bars are not shown 117 xiii 6.52 7r+p —• TT+TY+n total cross sections from: this work (solid circles), Refer-ence [16] (open squares), Reference [17] (solid stars), and Reference [50] (solid triangles). A l l previous experimental results shown here were ob-tained from a comprehensive list presented in Reference [21]. The solid line is the results of an amplitude analysis performed by Burkhardt and Lowe [41] 119 6.53 7r~p —> ir+ir~n total cross sections from: this work (solid circles), Refer-ence [14] (open squares), Reference [42] (solid stars), References [43, 44, 45] (open diamonds), Reference [46] (solid squares), Reference [47] (solid di-amonds), Reference [48] (open circles), Reference [49] (open triangles), and Reference [13] (solid triangles). Results of previous experiments were obtained from a comprehensive list presented in Reference [21] 120 7.54 Feynman diagram of the O P E mechanism 124 7.55 Typical Chew-Low extrapolation curves for F ' ( t , m ^ ) as a function of t. The solid point at t=+l is that deduced from the extrapolation (the TTTT cross section). The solid points denote the experimental data used in the linear fit. The experimental points denoted by crosses were not included in the fit 128 7.56 7 r + 7 r - cross sections obtained in this experiment as a function of m ^ . The solid points denote the energy-averaged values from the Chew-Low analysis (Table 7.5), and the open points are from Reference [53]. The solid line represents the cross sections obtained in Reference [54], using the Roy equations, with only higher momenta data included as input. The dashed lines indicate the calculated errors in the Roy equation analysis. 131 xiv 7.57 Profiles of the reduced y 2 - The top plot represents the results obtained by excluding the lowest bin and the bottom graph shows the profile determined by including the smallest m%n bin 133 7.58 Typical plots of the Chew-Low function, F', for the (7r + 7r + ) channel. . . 136 7.59 Feynman diagrams for processes included in the OT model. The next to leading order diagrams (bottom) contribute to the constants di 141 7.60 Diagrams for ix'nNN P-wave coupling through p exchange with interme-diate A and nucleon states. For simplicity, not all possible permutations are shown 144 7.61 Three point Feynman diagrams with nucleon and A intermediate states. Not all permutations are shown 145 7.62 Feynman diagrams for N* —> N(Tnr)S-wave- Not all permutations are shown. 146 7.63 Additional Feynman diagrams added by Sossi et al. to describe the N* —> N(TT7r)P^wave mechanism. Not all permutations are shown 147 7.64 Measured differential cross sections (solid points) and extended OV model fits (solid lines) for the (7r + TX~) channel. The dashed lines represent three-body phase-space. The model predictions are absolute, and the phase-space curve has been normalized to the experimental total cross sections. 149 7.65 Illustration of the experimental total cross sections and predictions of the extended OV model for the (7r + 7r~) channel 150 7.66 Measured differential cross sections (solid points) and predictions of the extended OV model (solid lines) for the ( 7r + 7r + ) channel. Three-body phase-space is represented by the dashed lines. The phase space curve has been normalized to the experimental total cross sections. The model predictions are absolute 152 xv 7.67 The experimental total cross sections and predictions of the extended O V model for the (TT+ TT+) channel are shown xvi Acknowledgments There are many people whom I would like to thank, but I do not think that I will be able to list them all. First and foremost I would like to thank my research supervisor Dr. Greg Smith who has taught me everything I know, and perhaps I would not have gotten this far if it was not for his continuing optimism. Thanks Greg you have really made this an experience. I would like to thank all of the members of the CHAOS collaboration (past and present) in particular Gertjan, Larry, Martin, Faustino, Nevio, Paolo, Pierre, Pino, Rinaldo, Rudi, and Tony. Thanks guys I will miss working with you, but I am sure I will still get invited to the parties. I would also like to thank Dr. Oleg Patarakin who helped me with the Chew-Low and Roy equation analysis and Dr. Richard Johnson for his help with the model-dependent analysis. I would like to thank my parents for giving me the opportunity to come to Canada and embark on the long journey that has brought me here today. Last but not least I would like to thank Sheila who has supported me through the years and has brought stability to my life. xvii Chapter 1 Physics Motivations &; Goals The field theory of the strong interactions, Quantum Chromo Dynamics (QCD), describes the interaction between quarks which make up the hadrons. The strong interactions are mediated via the exchange of gluons, which are the gauge bosons of Q C D . The Q C D coupling constant, as varies depending on the energy scale. To first order as may be written as [1] 12TT Ois = 7vT i 1-1) ( 3 3 - 2 r a / ) i n $ where Q2 is the absolute momentum transfer squared, n/ is the number of quark flavors involved in the interaction, and A is the Q C D scale parameter thought to be ~ 500 MeV. At large Q2, it is possible to perform calculations in terms of perturbations in the coupling constant. In the low energy region (E < 1 GeV), however, the coupling constant is large and perturbations in as are no longer valid. As a result, effective field theories incorporating the symmetries of Q C D are employed. 1.1 Isospin Symmetry Examining the particle spectrum reveals that bound states appear in isospin multiplets. For example, the proton and neutron form an isospin doublet and pions are in an isospin triplet. It was first thought that isospin symmetry was exact, and that differences in the masses of the multiplet members were due to electro-magnetic effects. This explanation is incorrect, since such a scenario would make the proton heavier than the neutron. 1 Chapter 1. Physics Motivations &, Goals 2 The Q C D Lagrangian density is C = \tr (GTGp,) + Q(iYD, - M)q, (1.2) where i=\ z Here aj, are the eight color gauge fields (gluons), A' are the Gell-Mann matrices, q is the column vector of quarks, and M is the quark mass matrix. If we consider SU(2) (only the up and down quarks) and assume that the quark masses are equal, Equation 1.2 is invariant under the set of unitary transformations generated by 2x2 matrices. These can be represented by q -> exp(i^^-)q, (1.3) where are the Pauli spin matrices, and a is a constant but arbitrary vector. In the infinitesimal limit, the transformation is represented by q —» q + ia • ^q. By Noether's theorem, the SU(2) invariance leads to the conservation of 3 vector currents given by V; = qiJ-q. (1.4) Defining V+ = 1 + iV2, and computing its divergence yields d^V + = d^u'fd) = i{mu - md)ud. (1.5) It is clear that for the case of equal quark masses, isospin is an exact symmetry of the Lagrangian. In practice, the quark masses differ and hence isospin is only an approximate symmetry of the strong interactions. Chapter 1. Physics Motivations & Goals 3 1.2 Chiral Symmetry A closer examination of the particle table also reveals that pions are the lightest hadrons. In addition, there seems to be a large mass gap between between the pion and the kaons which are the next heaviest hadrons. Consider the Q C D Lagrangian in the theoretical limit where quark masses are zero (chiral limit). Given the definition for left and right-handed quark states, qL>R = \(l±^)q. (1.6) there are no terms in the Lagrangian which include both qL and qR, and consequently the left and the right-handed quark states are decoupled. Each of the column vectors qL and qR may now be transformed with the set of unitary 2x2 matrices, and the Lagrangian will remain invariant. Thus in the theoretical limit mu = m,d = 0, the theory is invariant under SUL(2) x SUR{2). Once again Noether's theorem states that there exist conserved currents given by ft/AV, (1-7) (1.8) Writing the above in terms of q yields the three axial vector currents 4 = (1-9) Taking the divergence of the A+, defined as A\\ + iA2, results in + = fyfaVVd) = i(™<u + md)u-f5d. (1.10) The above equation shows that the axial vector current is conserved if and only if the quark masses are both zero. Chapter 1. Physics Motivations & Goals 4 Once again consider the chiral limit, mu = m,d = 0. Here both the vector and axial vector currents are conserved. For the vector current, the conserved charge is given by Q+ = J d*xV0+ = J d3xv)d. (1.11) Q+ is the isospin raising operator which causes transitions between the members of the isospin multiplets and leaves the vacuum state unchanged. The chiral isospin raising operator is defined as Q5+ — J d3xu^/y5d. It acts differently on the right and left-handed quarks and carries negative parity. Consequently, if chiral symmetry were realized, one would expect to find chiral multiplets whose members have equal mass and opposite parity. This is not observed in nature. The proton for example, does not have a coun-terpart of equal mass and opposite parity. Although the Lagrangian was invariant under chiral transformations, the eigenstates of the Hamiltonian do not have this symmetry. In other words, chiral symmetry is spontaneously broken. By Goldstone's theorem, there must then exist three massless bosons (Goldstone bosons) with zero spin, isospin 1, and negative parity. The lightest mesons, (ir~, 7r°, 7r + ) , have the required quantum numbers, but they are not massless. This is because chiral symmetry is explicitly broken by the mass term in the Lagrangian. In SU(3) the spontaneous breakdown of chiral symmetry results in additional Goldstone modes, namely the kaons and the rj. To examine how massive quarks lead to a non-zero pion mass, consider the expec-tation value of the axial vector current A+ between the 7r + and the vacuum states. By Lorentz invariance, the matrix element up to a constant factor is given by the pion four-momentum p^. Hence < ir+{p)\u(x)YY°d{x)\0 >= -ip»V2F„eip'x, (1.12) where denotes the pion decay constant. Computing the matrix element of the diver-gence of between the same two states and using equation 1.10 yields Chapter 1. Physics Motivations & Goals 5 < ir+(p)\dl>(u(x)'fyid(x))\0 > = i(mu + md) < n+(p)\u(x)j5 d(x)\0 > (1.13) = \f2F„Mleiv'x. Let Gv denote the pseudoscalar density matrix element appearing on the right hand side of Equation 1.13. < ir+(p)\u(x)-f5d(x)\0 >= -iV2Gneip'x. (1.14) Combining equations 1.13 and 1.14 provides a relation between the quark masses and the pion mass, Ml = + (1.15) Equation 1.15 shows that in the chiral limit the pion is massless and explains the non-zero Goldstone boson mass. It is interesting to note that the quark masses are related to the pion mass squared. As such, a small pion mass is indicative of light quarks. This implies that the symmetry breaking mass term in the Lagrangian may be treated as a perturbation to the chirally symmetric part. In the Hamiltonian formalism this may be written as HQCD = H0 + H ' , (1.16) where H0 respects chiral symmetry, and H' includes the mass terms which are not in-variant under chiral transformations. This is different from the standard perturbation theory approach where the Hamiltonian is written as, UQCD = Hfree + Hint- Here Hfree and Hint represent the free and interaction parts of the Hamiltonian, respectively. Recall that performing an expansion in terms of the coupling constant (ie. in powers of Hint) is not fruitful at low energies. An expansion in powers of H1 on the other hand, treats the quark masses as the perturbation and preserves quark-gluon interactions within H0. Chapter 1. Physics Motivations & Goals 6 1.3 Effective Theory In QCD, pions are the Goldstone bosons arising from the spontaneous breakdown of SUL(2) x SUR(2) symmetry At low energies, the exchange of Goldstone bosons produces singularities (poles). In addition, the strength of interactions between the Goldstone modes decreases as their momenta are reduced [2]. This is in contrast to quark-gluon interactions, which become stronger at low energies. Thus in the low-energy domain, the interactions between Goldstone bosons may be treated as a perturbation in powers of momenta. It was pointed out by Weinberg [3] that this approach could be the basis of an effective field theory referred to as Chiral Perturbation Theory (ChPT). In ChPT, the quark and gluon fields of the Q C D Lagrangian are replaced by a set of scalar pion fields organized in a 2 x 2 matrix denoted by U(x). The effective Lagrangian is constructed from U(x) and its derivatives and has the same symmetry properties as the underlying theory (QCD). As such only even powers of the derivatives of U(x) are allowed. Given that the effective Lagrangian contains all of the terms allowed by the symmetries of Q C D , the two theories are mathematically equivalent [4], In the framework of Chiral Perturbation Theory, it is possible to compute observables of pion-pion interactions to various orders in powers of momenta, TIT: elastic scattering is the simplest form of interactions between the Goldstone bosons, and consequently is one of the most fundamental processes in Chiral Perturbation Theory. In particular, 7T7T scattering lengths provide a measure of chiral symmetry breaking in Q C D . This is because, in the chiral limit these quantities are identically zero. The most recent calculated values of the S-wave scattering lengths for isospin zero and two, in inverse pion mass units are [5] a£ = 0.217 Chapter 1. Physics Motivations & Goals 7 al = -0.041 Experimental determination of the above scattering lengths will be a direct test of C h P T and hence Q C D at low energies. 1.4 Probes of the irir Interaction 7T7T scattering plays a crucial role in testing the predictions of Chiral Perturbation Theory. However since colliding pion beams are not attainable, indirect methods must be used to study this process. To date all of the experimental data on ITIT interactions have been obtained from the study of processes which result in the production of pions in the final state. In this section a brief overview of the processes used to determine S-wave pion-pion scattering lengths is presented. The best and the least ambiguous probe of TTTT scattering is the study of 7r+7r - atoms. There is currently a proposal for measuring the decay rates of 7r+7r_ bound states to 7r°7r° [6]. Although this would be a beautiful and elegant method, the experimental difficulties are substantial. Another reaction used to study TI-K scattering is the so called Ke4 decay, K+ -> TT+n-e+u (1.17) In Ke4 decay, both final state pions are on the mass shell. In addition, the pions are the only strongly interacting particles in the final state, and the decay process is well understood in terms of the standard electro-weak theory. Consequently the extraction of 7T7T observables from Ke4 data is less prone to theoretical difficulties. Furthermore, the 7T7T invariant mass distribution is peaked near the -KIT threshold, and as such, this reaction is a good probe of threshold irir interactions. However the branching ratio for Ke4 decay is ~ 4 x 10~5. Chapter 1. Physics Motivations & Goals 8 N N I - |- Others 7T 1 (OPE) Figure 1.1: The Feynman diagram for the One Pion Exchange mechanism is shown. In addition to O P E other diagrams contribute as well. The most recent measurements of Ke4 decay were performed by Rosselet et al. at the C E R N proton synchrotron [7]. In this experiment 30,000 decays were analyzed. The S-wave isospin zero 7TTT scattering length from Rosselet et al. is a° = 0.28 ± 0.05 / i - 1 [7]. It was Weinberg who first suggested that the reaction -KN —> nirN may be used as a probe of 7TTT scattering. Amongst the processes contributing to the reaction irN —> -KTTN is the One Pion Exchange mechanism (OPE). In the O P E graph (Figure 1.1), the off-shell pion interacts with the physical pion. Although the O P E mechanism may be used to study 7T7T interactions, other processes such as resonance exchanges also contribute to the reaction amplitude. This presents theoretical difficulties as well as ambiguities in extracting -KTX scattering observables from TTN —> 7r7riV data. Chapter 1. Physics Motivations & Goals 9 In this thesis, the differential and total cross section measurements for the reactions i&p -> 7 r ± 7 r + n at incident pion energies of 223, 243, 263, 285, and 305 MeV will be presented. The 7 r + 7 r _ n data will be interpreted using two different techniques. The first is a model independent approach which employs the Chew-Low extrapolation process [8] and Roy equation analysis techniques [9] to extract the isospin zero S-wave scattering length. The other is based on a model of the nN -> imN reaction first developed by Oset et al. [10] and later extended by Sossi et al. [11, 12]. The model-dependent analysis will be used to determine coupling constants for isobar exchanges which contribute to the non-OPE background. 1.5 Rev iew of Previous Exper iments There are five different experimentally accessible channels of elementary pion induced pion production. These are %+p —>• 7r + 7r + n ( 7r + 7r + ) Channel (1-18) 7r~p —»• 7 r - 7 r + n (TT+ 7T") Channel Ti+p —> 7r+7r°p (7r + 7T°) Channel rc~p —• 7r~7r°p (7T_ 7T°) Channel -K~p —>• 7r°7r°?i (7T° 7T°) Channel In the past, the above reactions have been studied over a wide range of incident pion energies. However, only the near threshold measurements ( T W < 400MeV) are of interest in the current work. Over the past decade there has been a flurry of activity in performing inclusive and exclusive studies of pion-induced pion production near threshold. In 1974 Jones et al. measured angular distributions in the (7r + TT~) and (n~ n°) chan-nels at an incident pion momentum of 415 MeV/c [13]. They employed a 180 liter Chapter 1. Physics Motivations & Goals 10 hydrogen bubble chamber at Saturne. However, even with 140,000 pictures taken, this experiment suffered from poor statistics. Only 881 (ir+ and 140 (7r~7r°) events were detected. Using an isobar model along with data acquired at higher energies, they find —0.06p~l < a° < 0.03/U"1, which is consistent with zero. In the late 1980's and early 1990 's the Omicron group at C E R N measured total cross sections and angular distributions for the (TT+TT~), ( 7 r _ 7 r ° ) , and (TT+n+) channels at in-cident pion momenta between 295 and 450 MeV/c [14][15][16]. These experiments were performed at the C E R N Synchro-Cyclotron and employed a large solid angle magnetic spectrometer and a thin gas target. For each reaction the two charged particles in the final state were detected. In order to determine total cross sections, the angular distri-butions were extrapolated to the regions of phase-space not covered by the experimental apparatus. They assumed that all kinematic variables except the dipion invariant mass in the (TT + 7T _) channel are distributed according to phase-space. The experimental total cross sections were analyzed in the context of the Olsson and Turner model [18]. This model assumes that O P E is the dominant mechanism close to threshold, and it param-eterizes the threshold (TT,2T:) amplitude and the 7T7T scattering lengths in terms of the chiral symmetry breaking parameter £. As such the threshold amplitudes may be fitted to determine £ and the scattering lengths. In the framework of Chiral Perturbation The-ory £ is zero. The Omicron results for £ and the scattering lengths (in units of inverse pion mass) are: £ = -0.5 ± 0.8, o°2 = -0.05 ± 0.02, a°0 = 0.15 ± 0.03 for the (TT+TT0) channel, £ = 0.1±°0;57, a20 = -0.03 ± 0.02, a°Q = 0.18 ± 0.04 in the (TT+TT-) channel, and £ = 1.56 ± 0.26, al = -0.08 ± 0.01 for the (TT+TT+) channel. These data have been criticized for the limited phase-space coverage as well as the extrapolation process used to determine total cross sections. In 1993 at T R I U M F Sevior et al. performed an inclusive measurement in the (TT+ TT+) channel at incident pion energies of 180, 184, 190, and 200 MeV [17]. They used a novel Chapter 1. Physics Motivations & Goals 11 method in which a set of 5 thin plastic scintillators were employed as active targets in which the final state pions were detected. An array of neutron detectors placed down stream were used to detect the final state neutron. A magnet placed between the target and the neutron counters was used to deflect the incident beam away from the detectors. The measured cross sections of Sevior et al. are in disagreement with those obtained by the Omicron group. The Sevior data were analyzed using the Olsson and Turner model, and the resulting TTTT scattering length is a°2 = (-0.040 ± 0.001 ± 0.003)/TT. To date, these represent the experimental data acquired closest to the pion production threshold energy of 172.38 MeV. At PSI, the Erlangen group performed exclusive measurements of the (TX+ TT~) channel at pion incident energies of 247, 284 , and 330 MeV with a liquid hydrogen target [19]. They employed a magnetic spectrometer, plastic scintillators and wire chambers in order to cover large regions of phase space, including regions out-of-plane. The published results were in the form of pion-pion angular correlations, and triple differential cross sections at 1301 MeV total center-of-mass energy. No total cross sections were published. Measurements of the (ir° 7r°) channel at pion incident energies ranging from 5 MeV above threshold to 293 MeV were performed by Lowe et al. at Brookhaven [20]. They used the Crystal Box detector to acquire kinematically complete data over a large region of phase space. In this experiment the four final state photons originating from TT° decay were detected, and the TT°TT0 events were isolated by reconstructing the missing mass of the final state neutron. The total cross section data were interpreted in the context of the Olsson and Turner model. The value of £ from this experiment is —0.98 ± 0.52. The corresponding S-wave TTTT scattering lengths are eft = (0.207 ± 0 .028) / / - 1 , a° = (-0.022 i O . 0 1 1 ) / / - 1 . In 1993, the Virginia group measured total cross sections as well as angular distri-butions for the ( 7 T + 7 r ° ) channel at incident pion energies of 190, 200, 220, 240, and 260 Chapter 1. Physics Motivations & Goals 12 MeV [21]. They performed an analysis in which all of the existing nN —• TXTXN were fitted to extract partial amplitudes in the framework of the Olsson and Turner model. Their results indicate that £ = -0.25 ± 0.10 and o°2 = -0.041 ± 0.001//-1. The above constitutes a large body of threshold irnN data, but the only published differential cross sections are at a single incident pion energy [19]. The published angular distributions from other works are often in arbitrary units [13, 14]. As such, it is im-possible to interpret these data in a model independent approach such as the Chew-Low extrapolation technique. Chapter 2 The Experiment The experiment was performed in two stages at the M i l pion beam channel at T R I U M F . The TT~P —> ir+ir~n reaction was studied in the first stage, and in the second part data were acquired for the ir+p —> 7 r + 7r + n channel. A liquid hydrogen target was employed in both phases, and projectile kinetic energies of 223, 243, 264, 284, and 305 MeV were studied. The CHAOS magnetic spectrometer [23] was used to detect the charged particles in the final state. The details of the experiment and the experimental apparatus will be discussed in this chapter. 2.1 The Cyclotron &i Beam Line The T R I U M F facility houses the largest cyclotron in the world, which accelerates H~ ions to a maximum energy of 520 MeV. A low value of the average magnetic field is required in order to ensure that the loosely bound extra electron is not stripped off. The low field necessitates a large accelerating radius, which for a 520 MeV proton is 7.8 meters. To extract the proton beam, the two electrons bound to the H~ ion are stripped off by passing the ions through a thin foil of carbon. The positively charged protons will then curve in the opposite direction and exit the machine at a precise point. The cyclotron resonant cavity operates at a frequency of 23 MHz, producing a proton burst every 43 ns. A typical 500 MeV extracted beam current is ~ 150 \xA. One of the four proton beams exiting the cyclotron is directed at a pion production target. Interaction of the proton beam with the target results in the production of 13 Chapter 2. The Experiment 14 positive, negative, and neutral pions via A decay. The positive to negative pion ratio is about 9:1. The neutral pions decay in the production target, and the interaction of the resulting gamma rays with the target and surrounding materials produces electron-positron pairs, which contaminate the pion beam. Additional contamination is caused by pion decay in the secondary beam channels leading to the experimental area. A diagram of the M i l channel is shown in Figure 2.2. The channel consists of three parts: the front end, middle section, and rear section. The goal of the front part is to separate the pions from the primary proton beam. Particles exiting the production target, T l , are passed through the quadrupole magnet, 1AQ9, which deflects the lighter charged particles of the selected polarity from the proton beam and focuses the beam (pions more than protons) in the vertical direction. The septum magnet, SI, further reduces the proton contamination by increasing the angular separation between pions and protons1. The middle section consists of the dipole B l and quadrupoles Q l and Q2. B l is the primary momentum selector; it bends the pion beam by 60° and selects particles whose momenta lie within ~ ±2 .5% of the central momentum setting of the channel. The quadrupoles Q l and Q2 provide a dispersed double-focus at the mid-plane. The momentum dispersion at the mid-plane focus is 18 mm per %Ap/p [22]. If the channel is set to select positive pions, the proton to pion ratio at the mid-plane is about 10:1. Absorbers placed at the mid-plane differentially degrade the protons and allow the dipole B2 to deflect these particles from the pion beam. The rear section consists of the dipole B2 and quadrupoles Q3 through Q6. These quadrupoles provide a doubly-achromatic double-focus in the experimental target posi-tion. In addition, B2 removes particles whose energy has been degraded by the mid-plane absorber. The sextupoles SX1 and SX2 correct for aberrations introduced by the fringe field 1 There is no proton contamination when the channel polarity is set to select negative pions. Chapter 2. The Experiment Figure 2.2: The M i l pion channel. Chapter 2. The Experiment 16 of B l , and SX3 and SX4 do the same for B2. The sextupole SX6, located near the mid-plane focus, corrects for chromatic aberrations. The momentum bite and the pion flux are adjusted via slits positioned at the mid-plane focus of the channel. Slit settings used in this experiment provided a momentum resolution (Ap/p) of 1% and 5% for positive and negative incident pions, respectively. The momentum distribution of the channel is uniform with the centroid, p, and standard deviation, Ap. 2.2 The Experimental Apparatus The experiment employed the Canadian High Acceptance Orbit Spectrometer (CHAOS), a cylindrical magnetic spectrometer designed for pion physics studies [23, 24]. CHAOS consists of a dipole magnet, four cylindrical wire chambers ( W C l , WC2, WC3, WC4), and an array of plastic scintillators and lead glass Cerenkov counters (see Figure 2.3). A top view of the experimental setup is shown in Figure 2.4. Here the target is represented by the solid circle located at the center of the spectrometer. Figure 2.3: The CHAOS spectrometer. A corner post and the top pole tip have been removed for a better view. In addition, a quadrant of the detector has also been removed. Chapter 2. The Experiment 18 Figure 2.4: Illustration of the top view of the experimental setup. The beam count-ing scintillators (Si and S2) and the veto counter are shown. The cryogenic target is represented by the solid circle located at the center of the spectrometer. 2.2.1 The Magnet The CHAOS magnet has a pole diameter of 95 cm and is equipped with return yokes located at the four corners. It is capable of producing magnetic fields of up to 1.6T, and a 12 cm diameter bore hole along its symmetry axis allows for the insertion of cryogenic targets. The magnitude of the nominal field is measured with an N M R probe located on the bottom pole face at a radius of about 22 cm. The magnet assembly is fitted with rails and a stand which allow it to move perpen-dicular to the beam line and rotate about its symmetry axis. This translation is required because the CHAOS magnetic field causes the incident pion beam to deflect prior to Chapter 2. The Experiment 19 hitting the target. Consequently the magnet assembly must be translated with respect to the beam line, to ensure that the beam strikes the target. The rotation allows for the beam to enter and exit the spectrometer at convenient locations relative to the return yokes. During the experiment, the nominal field setting at 284 MeV incident pion energy was 0.5 T. In order to keep the incident beam trajectory fixed, the ratio of the spectrometer field to the incident momentum was kept constant at all incident beam energies. 2.2.2 WC1 & WC2 The two inner CHAOS chambers ( W C l and WC2) are multi wire proportional chambers, located at radii of 114.59 and 229.18 mm, respectively [23]. Each chamber has a half gap of 2 mm and a vertical active height of 70 mm. W C l consists of 720 anode wires with a pitch of 1 mm; WC2 also has 720 wires but the wire pitch is 2 mm [23]. The angular pitch of both chambers is 0.5°, and the anode wires are 12 pm diameter gold plated tungsten. Both chambers are instrumented with 360 cathode strips inclined at 30° with respect to the anodes. The anode wires provide in-plane position information, and the cathodes are used to obtain out-of-plane spatial information [23]. In each chamber the anode wires are instrumented with 16-channel pre-amplifiers (four bipolar Fujitsu MB43458PF quad chips), LeCroy 2735 P C amplifier/discriminator cards, and the LeCroy Proportional Chamber Operating System (PCOS III). The cathode strips are connected to 8-channel pre-amplifiers, inverter/amplifier cards, and LeCroy FASTBUS Analog to Digital Converters (ADC's) [23]. A gas mixture of 80% C F 4 and 20% isobutane is used. W C l and WC2 are capable of operating at incident pion fluxes of up to ~ 3 MHz without loss of performance. This is crucial for all CHAOS experiments since the incoming beam hits are needed to determine the trajectory of the incident beam, which is in turn required to obtain track scattering Chapter 2. The Experiment 20 W C 3 cell structure 4.0 mm 3.0 mm Signal Signal H.V. S ignal Signal 3.75 mm O © <— 75 mm > O Anode Cathode Anode Signal Signal H.V. S ignal S ignal 2.0 mm 1.0 mm Figure 2.5: An illustration of WC3 cell geometry. angles. The nominal operating voltages for W C l and WC2 are 2450 V and 1950 V , respectively. 2.2.3 W C 3 WC3 is a single plane cylindrical drift chamber located at a radius of 343.77 mm. The chamber was designed to operate in a region of high magnetic field, and it employs a "rectangular" cell geometry [25, 26]. Each cell consists of alternating anodes and cathodes (see Figure 2.5), separated by 7.5 mm. In addition, alternating high voltage and signal cathode strips are also employed. The high voltage strips are meant to provide a uniform electric field distribution, and the signal strips are used to resolve the left-right ambiguity [25]. The latter is acheived by examining the difference between the pulse heights of the appropriate signal strips in a given cell. Since the chamber operates in a region of high magnetic field, the drift electron trajectories are curved. This complicates the left-right ambiguity resolution. However it is possible to resolve left from right by examining the quantity, Chapter 2. The Experiment 21 where Pa and Pb are the pulse heights on a pair of diagonal signal strips in a given cell. The correct combination of strips depends on the direction of the magnetic field. In total, there are 144 sense wires and 576 cathode strips. The anode and cathode wires are 50 pm and 100 pm diameter gold plated tungsten, respectively. The chamber half gap is 3.75 mm, and the vertical active area is 90 mm. This chamber does not provide any vertical position information. The anode wires are instrumented with 8-channel pre-amplifiers (composed of two Fujitsu MB43458PF chips) and LeCroy 2735DC amplifier/discriminator cards, and the readout cathode strips are equipped with 8-channel pre-amplifiers, inverters, and FAST-BUS ADC's . The output of the 2735DC cards is split and fed to the LeCroy 4290 drift chamber system as well as the PCOS III system. The PCOS signal is intended for use in the second level trigger. To avoid long cable lengths, the 4290 system is operated in common stop mode. The chamber is operated with a gas mixture of 50% argon and 50% ethane. To prevent damage caused by the large incident flux, regions of WC3 located at the entrance and exit of the beam into the spectrometer were deadened in groups of four adjacent cells by removing the anode bias to that group. The nominal operating voltages for WC3 are: 2250 V on the anodes, -600 V on the cathode wires, and -300 V on the cathode strips. 2.2.4 WC4 WC4 is a vector drift chamber with a trapezoidal cell geometry (see Figure 2.6). Each cell contains fourteen concentric cylindrical wire planes separated by 5.0 mm. The middle eight wire planes (A) are anodes that provide in-plane spatial information, and layers 2 and 13 are resistive wires (R) used to obtain vertical position information. The other four planes (1, 3, 12, and 14) are field shaping wires (G). There are a total of 100 equally Chapter 2. The Experiment 22 Back Wall Upper/lower plate boundry Figure 2.6: WC4 cell geometry. Here A, R, and G denote the anode, resistive and field shaping wires, respectively. sized cells, and the radius to the first resistive wire is 613 mm. To resolve the left-right ambiguity, the anode wires are alternately staggered by 250 in the direction perpendicular to the radial line bisecting the cell. The boundaries between adjacent cells are defined by cathode strips glued on thin wafers of Rohacell (a low density material, p ~ 50 mg/cc, similar to Styrofoam). These strips are kept at a negative voltage and together with the anodes provide the electric field. The other two boundaries of the cell are formed by two additional cathode planes. Since the cell geometry is trapezoidal, to obtain a uniform electric field, the voltage on the cathode strips at the narrow end of the cell must be less than that at the wide end. This is accomplished by means of a chain of resistors, which drop the voltage by 55 V across successive strips. Chapter 2. The Experiment 23 The anode wires are instrumented with 8-channel pre-amplifiers, LeCroy 2735DC amplifier/discriminators, and the LeCroy 4290 drift chamber system. Signals from the resistive wires are pre-amplified and fed into LeCroy FASTBUS ADC's . As for WC3, those cells located in the path of the incoming pion beam were deadened by removing the bias voltage from the cathode strips. This has the effect of deadening two cells at once since the cathode strips form the common boundary between two adjacent cells. The chamber gas is a mixture of 50% argon and 50% ethane. Typical chamber operating voltages are -5200 V on the cathode strips and -2500 V on the cathode planes. There is no bias voltage on the anodes. Since this chamber has eight anode planes, it is capable of providing a vector along the direction of the particle trajectory, which is essential for track sorting. 2.2.5 Counter Telescopes The outermost layer of detectors in the spectrometer is a ring of counter telescopes referred to as the CHAOS Fast Trigger (CFT) counters. Each telescope is made up of three layers. The first, A E X , is a NE110 scintillator 3.5 mm thick with an area of 25 x 25 cm 2; the second layer consists of two adjacent NE110 scintillators (AE2R and AE2L), each 13 mm thick with a cross sectional area of 13x25 cm 2 . The third layer C is made up of three adjacent lead glass Cerenkov counters, each 12 cm thick with a frontal area of 9.2x25 cm 2 . The index of refraction of the lead glass is ~ 1.7. In total there are 20 such counter telescopes, each covering an angular arc of 18°. A detailed discussion of the CHAOS first level trigger is presented in Reference [27]. The A E X , A E 2 and Cerenkov counters are viewed by photo-multiplier tubes which are shielded from the CHAOS magnetic field. The AEX and AE2 counters are equipped with Phillips TDC's and LeCroy Fast Encoding and Readout ADC's (FERA); the Cerenkov Chapter 2. The Experiment 24 detectors are also instrumented with the F E R A system. During the course of the exper-iment, the blocks positioned at the entrance and exit of the beam were removed. The information from the C F T blocks is used in the CHAOS first level trigger, which r forms its decision based on event multiplicity. The word multiplicity refers to the number of C F T blocks that fired in a given event. In the analysis stage, the pulse heights from the C F T blocks are combined with the momentum information to provide particle identification. 2.2.6 First Level Trigger The CHAOS first level trigger (1LT) provides the gate for the ADC's , the PCOS, and the stop signal for the 4290 drift chamber system. In other words, it initiates the digitization of all of the chamber information. The 1LT decision is formed based on the multiplicity information provided by the C F T counters. During the course of the experiment, it was required that at least two C F T blocks be activated for an event to pass the 1LT. The signal for a given block was formed by the coincidence of the AE\ and at least one of the AE2 counters in that block. This may be written as 1LT = BEAM *Bi*Bj , % ± j (2.20) Bk = AElk.{AE2Lk + AE2Rk), (2.21) where B E A M denotes the logic signal from the beam counting scintillators (see Sec-tion 2.4), and the subscripts i,j and k are the C F T block numbers, ranging from 0 to 19. A simplified schematic of the 1LT is shown in Figure 2.7, and its operation may be summarized as follows. The signals from the C F T blocks are fed into programmable dis-criminators which produce an E C L logic signal, if the input is above a certain threshold. This particular type of discriminator also acts as a fan-out which allows for the analog Chapter 2. The Experiment 25 AE2r/ \ aE1 4n Start .(Loteh)] Stop END BUSY 2LT NO RUN START RUN END Stort |(Ldch)| Stop Scalers - Logical OR of PLU outputs -f OF FRST LEVEL TRIGGERS PCOS. FERA, AND FASTBUS GATES * Gate to 2nd Level Trigger § of 1st LT Accepted by J11 Beam Siqnol DRIFT CHAMBER UNCOMMON STOP LE TDC START Camac and Visual Scaler Gate Manual] Start/ |Stop (Lotch)J WC Stop Figure 2.7: A schematic of the first level trigger is shown. For simplicity only one C F T block is included. input to be delayed and directed to Fast Encoding and Readout (FERA) ADC's for digi-tization. The E C L output of the discriminator is delayed to compensate for the different cable lengths and is fed into a programmable lookup unit (PLU), which outputs differ-ent logic combinations of its input. During the experiment this unit was programmed to output the logic signal given by Equation 2.21. The output of the P L U is fed to a majority logic unit (MALU) , which makes the multiplicity decision. The M A L U output is the first level trigger accept signal. Given that the system is not busy processing a previous event, the 1LT accept signal initiates the second level trigger and provides the Chapter 2. The Experiment 26 gates for the PCOS, F E R A , and FASTBUS systems. From the time that the 1LT starts making its decision until the event is either recorded or rejected by one of the two hardware triggers, the entire trigger system is blocked so as to ensure that only a single event is processed in each cycle. In other words, once an event passes the first level trigger, the system is disarmed until either the " E N D - B U S Y " (a signal indicating that the computer has finished recording the event) or the "2LT-NO" (the second level trigger reject signal) re-arms the trigger. 2.2.7 Second Level Trigger The CHAOS Second Level Trigger (2LT) is a fully programmable system which uses the PCOS information from W C l , WC2, and WC3 to make decisions based on track polarity, momentum, and distance of closest approach to the target. In addition, it is capable of rejecting events in which the incident pion decays prior to reaching WC2. A detailed description of the 2LT is provided in References [28, 29]. The trigger system consists of three stages, and the operation of the first two can be summarized as follows. In the first stage (see Figure 2.8), the hits from the PCOS system are stored in LeCroy 2375 data stacks, and the angular coordinate of each hit is determined. Next, all possible combinations of the hits in W C l , WC2, and WC3 are examined until a group is found such that the angular difference between the hit in WC2 and each of the hits in W C l and WC3 is less than ±32°; such a group forms a track. Since the CHAOS magnetic field is uniform up to the radius of WC3, the trajectory of a charged particle (up to the location of the WC3 hit) may be approximated by a circle. As such, given three points along with the magnitude and direction of the magnetic field, it is possible to determine the particle's momentum, polarity, and distance of closest approach to the center of CHAOS. If the track satisfies the user-defined requirements for these quantities, it is passed to the second stage which simply stores its momentum Chapter 2. The Experiment 27 PC0S3 PC0S1, PC0S2 Get Oj MLU A Get S, and 0, sort WC1, WC2 MLU A WC3 Stack B' WC2 Stack C WCl Stack B -32° * 8,' * 32° MLU J -32° =£ 6,' £ 32° MLU I momentum, polarity and target cuts I MLU21 K To the second stage Figure 2.8: A simplified schematic of the first stage of the 2LT. Here M L U represents the LeCroy 2372 Memory Lookup Unit, A L U is the LeCroy 2375 Arithmetic Logic Unit, and Stack denotes the LeCroy 2375 Data Stack. In addition, MLU21 is a 21 bit Memory Lookup Unit developed at T R I U M F . Chapter 2. The Experiment 28 and directs the first stage to find the next track. The above process continues until all possible combinations of hits in W C l , WC2 and WC3 are examined. If more than two tracks passed the requirements of the first stage, the event is ac-cepted, and those events in which less than two tracks passed the first stage are rejected. In the cases where two tracks passed the first stage, the sum of their momenta is deter-mined and the event is accepted if this sum is less than a predefined upper limit. The third stage of the trigger is known as the muon rejection part. It requires the existence of hits in predefined regions of W C l and WC2, which correspond to those illuminated by the incident beam (thus termed "incident beam hits"). This allows for the rejection of events in which the incident pion decays prior to arriving at WC2. In a large fraction of incident pion decays, the momentum of the resulting muon is substantially different from that of the original pion. Thus the trajectory of the muon in the CHAOS field will also differ, and the particle will not produce hits in the predefined regions of W C l and WC2. In addition to rejecting muons, this stage eliminates events in which an incident beam hit was not recorded due to chamber inefficiency. Such events could not be reconstructed in later analysis, because incident beam hits are required to determine the scattering angle. During the experiment one of the main tasks of the 2LT was to reduce the background events coming from irp elastic scattering. For both incident polarity pions, the np cross section is many orders of magnitude larger than that of (TX, 2TT). Thus it was imperative to reduce this background. This was achieved by choosing an appropriate upper limit for the 2LT momentum sum. Since the sum of the momenta of the two pions is much smaller than that of the pion and proton, placing a liberal upper limit on this quantity removed a substantial fraction of np elastic events. The 2LT does not use the drift time information in WC3; thus its momentum res-olution is relatively poor. To determine the momentum sum and distance of closest Chapter 2. The Experiment 29 approach cuts, simulations were carried out using the F O W L [30] phase space generation program to generate events for (7r,27r) and irp reactions. For each event, the generated particle momenta (from FOWL) were used to simulate tracks originating from the center of CHAOS, and the location of the corresponding hits in W C l , WC2, and WC3 were digitized. The digitization included the resolution of the PCOS data. From the simulated PCOS hits, the momentum and distance of closest approach to the center of CHAOS were calculated by approximating the particle trajectory with a circle. Once a large number of tracks was analyzed, the distributions for the momentum sum and distance of closest approach were used to determine the 2LT limits. The above process was repeated for each incident pion energy and magnetic field setting. Although great care was taken in determining the 2LT cuts, some valid (IT, 2TT) events could still be rejected by the trigger. As such, a software simulation of the 2LT was incorporated into the Monte-Carlo simulation of CHAOS presented in Chapter 5. 2.3 The Cryogenic Target The cryogenic target vessel is cylindrical with a radius of 25.5 mm and a height of 50 mm. A diagram of the target vessel is shown in Figure 2.9. The vessel is constructed from a 0.125 mm thick mylar cylinder, which is held in place by two copper disks that are each 54 mm in diameter. These form the top and bottom of the target cell. A 0.007 mm thick aluminum heat shield is located at a radius of 29.6 mm with respect to the center of the cell. The target cell was kept under vacuum and the outside pressure was supported by a mylar foil wrapped around a honeycomb structure and located at an outer radius of 46.8 mm. The target temperature was determined by measuring the vapor pressure of the liquid hydrogen in the target cell. The graphs presented in Reference [31] were used to relate Chapter 2. The Experiment 54 m m Side View Cel l on ly 50 m m I t 9-5 m m 0.125 m m Mylar copper Figure 2.9: Top and side views of the cryogenic liquid hydrogen target. Chapter 2. The Experiment 31 10 cm Si 20 cm 0.8 o m 1.2 c m 10 cm 1.2 c m 0.8 c m Figure 2.10: A n illustration (not to scale) depicting the beam counters Si and S2. the vapor pressure to the temperature. The nominal operating temperature of the target was 18 K . At this temperature the liquid hydrogen density is 0.074 g/cm3 [31]. 2.4 Beam Coun t ing Incident pions were counted using two plastic scintillators located at the exit point of the beam line and at the entrance to the spectrometer (see Figure 2.4). The first counter, S i , consisted of four 10 cm wide horizontal adjacent strips each 5 cm high and 3.2 mm thick (see Figure 2.10). The signal from each Si strip was directed to a constant fraction discriminator where lower and upper discriminators thresholds were applied. The lower threshold (set at ~ 30 mV) removed noise while the upper was set to about two times higher than the signal pulse height produced by incident pions. This allowed the removal of the protons present in the 7r + beam. The logic signal of S i was produced from the Chapter 2. The Experiment 32 logical O R of the strip signals. The second scintillator, S2, consisted of four 1.6 mm thick vertical adjacent strips each 10 cm high. The two center strips were 0.8 mm wide while the two outer ones had a width of 1.2 cm each. S2 was located ~ 90 cm upstream of the target and was used to provide a better definition of the beam at the entrance to CHAOS. The analog signals from the S2 strips were threshold-discriminated, and the S2 logic signal was formed from the logical OR of the four strips. The incident pions were counted via the coincidence signal between Sx and S2. The Sx signal also provided the reference time with respect to which the wire chamber drift times were measured. Typical incident beam rates were < 1 MHz and < 2.5 MHz for the TT~ and 7r + beams, respectively. The C F T block located at the exit of the incident beam from the spectrometer was removed. It was replaced by a veto counter, V , consisting of two adjacent plastic scintil-lators each 3 mm thick. The veto counter spanned an angular arc of 18° (initiated from the target center), which corresponds to the removed C F T block (see Figure 2.4). The signal which started the first level trigger (STROBE) was formed by S?»S[»S1»S2:V = BEAM • V. (2.22) Here u and I denote the upper and lower threshold signals, respectively. To better define the negative incident pion momenta, a hodoscope consisting of 16 scintillation counter strips, each 6 mm wide, was placed at the intermediate focus of the channel. The momentum dispersion at the mid-plane is 18 mm per %Ap/p [22]. Pions with momenta equal to the central channel momentum setting activate a given hodoscope strip, 7VC. For each incident pion that caused an event, the hodoscope data indicate which strip was activated by the pion. Given the width of the scintillator strip and the momentum dispersion at the mid-plane, the corrected incident momentum is Chapter 2. The Experiment 33 given by P = (1 - (N — iV c)/300) * P c , (2.23) where TV is the activated strip number, and P c is the central momentum of the channel. This reduces the uncertainty in the momentum of a given incident pion to ~ 0.3%. Recall that the uncertainty provided by the slits was 5%. Due to the large number of protons in the 7T + beam, {jp/ir ~ 10), it was not practical to use the hodoscope in the second stage of the experiment. At the incident pion energies studied here, the proton pulse height is about 2 to 3 times greater than that of pions. 2.5 Beam Samples To determine quantities such as the profile of the incident beam on the target, it was necessary to record events ( S A M P L E events) which were not biased by the first and second level trigger requirements. This was achieved by the implementation of the "beam-sample" circuit, which bypassed the first and second level triggers. The logic signal initiating a sample event was given by Si • Si • S 2 • CLOCK (2.24) where CLOCK denotes the logic signal generated by a pulser operating at a rate of a few Hertz. Once the above coincidence is made, the beam-sample circuit issues the gate to all of the read-out systems and directs the computer to record the event. The sample events provided a means of recording unbiased information at regular intervals during the course of the experiment. A complete schematic of the CHAOS trigger system including the 1LT, 2LT, B U S Y , and Beam Sample circuits is presented in Figure 2.11. A circuit similar to the beam-sample setup was used to record a fraction of the rejected 2LT events. This (2LT-sample circuit) was implemented during the second Chapter 2. The Experiment 34 stage of the experiment and was not utilized in the analysis, but is employed in other CHAOS experiments. Chapter 3 Event Reconstruction Prior to discussing the data analysis procedure, it is necessary to describe the recon-struction techniques used to determine the track momenta and scattering angles. The coordinate system in which the reconstruction takes place is referred to as the CHAOS coordinate system and is shown in Figure 3.12. Here angles are measured counter clock-wise with respect to the positive x-axis, and the positive z-axis is directed out of the page. CHAOS coordinate system Figure 3.12: Illustration of the CHAOS coordinate system. Units are mm. 36 Chapter 3. Event Reconstruction 37 3.1 Chamber Calibrations The first step in the reconstruction process was the conversion of wire chamber hit in-formation to spatial coordinates in the CHAOS system. Absolute chamber positions were determined from calibration data acquired during the commissioning phase of the detector and will not be discussed here. A detailed explanation of chamber position determination is presented in References [24, 32]. In the case of the proportional chambers, the coordinates of a hit are given by x = Rccos(8), (3.25) y = Rcsin(6), (3.26) _ - 6) i U 180tan(30°) ' 1 j where 8 is the angular coordinate of the hit, Rc is the radius of the wire plane, and 4> is the polar angle at z=0 of the cathode strip corresponding to the charge centroid. Both angles in Equation 3.27 are measured in degrees, and the charge centroid corresponding to an anode hit is given by ~eS7*~' ( 3 ' 2 8 ) here qi is the A D C value, Ni is the cathode strip number, and n represents the number of strips activated. The PCOS system provides an address directly related to the angular coordinate of each recorded hit. The system operates in what is termed cluster mode. In other words, if a track activates a single wire, the PCOS address is given by the angle of that wire, but if more than one wire fired, the PCOS address corresponds to the average of the angular coordinates of the activated wires. For the drift chambers, the process of determining track coordinates is more complex. Here the position of the hit is obtained from the time it takes the drift electrons (liberated by the charged particle) to reach the sense wire. A typical drift time (TDC) spectrum Chapter 3. Event Reconstruction 38 250 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r—-j 1 1 1 r 0 100 200 300 400 500 . TDC (ns) Figure 3.13: A typical drift time spectrum for WC4. Short drift times correspond to large T D C values. for WC4 is shown in Figure 3.13. Since the TDC's are operated in common stop mode, tracks passing closest to the anode wire correspond to the larger T D C values. The right-hand edge of the spectrum is the T D C offset for this particular wire, which corresponds to drift electrons liberated very close to that wire. To determine the drift time, the T D C offset must be subtracted from the T D C value. Since the cables carrying the signals from the pre-amplifiers to the TDC's have different lengths, the T D C offset for different wires in a given chamber varies. In addition, timing jitters result in T D C spectra that do not have sharp cutoffs. To determine the offsets, T D C spectra were formed for each of the 944 anode wires in WC3 and WC4. Enough tracks were analyzed such that each drift cell was fully illuminated. The T D C offset for a given wire was found by examining each individual spectrum and recording the T D C value at which the number of counts dropped to ~ 60% of the maximum. Here maximum refers to the number of events in the bin that Chapter 3. Event Reconstruction 39 had the highest number of counts in the histogram. The minimum number of events in each spectrum was on the order of 2000. The time-distance (x(t)) relation determines the distance of the hit from the anode, given the drift time. With the exception of the T D C offsets, which are determined for each anode, the x(t) relations for WC3 and WC4 are assumed to be the same for each wire in a given chamber. For chambers located in regions of intense magnetic field, the x{t) relation is nonlinear because the trajectories of the drift electrons are curved. This was not a concern for WC4, since at the field settings used in the experiment (B ~ 0.5T) the magnitude of the CHAOS field in the region of WC4 was small (< 0.1T). WC3 on the other hand, is located in a high field area, and as such the x(t) relation for this chamber is nonlinear. In addition, the drift distance is also a function of the angle, 7, of the track with respect to the drift cell. Hence the x(t) relation was determined for small bins in 7. To determine the WC3 time-distance relations, tracks of different momenta were analyzed in order to obtain a wide range of incident angles. Using the Quintic Spline method discussed in Section 3.3, an analytic form for the trajectory of the particle in the magnetic field was obtained. In this calculation, the position of the WC3 hit was based on an initial estimate of the x(t) relation, for which 7 was set to zero. Once the track trajectory was known, the tangent to the track at WC3 was used to determine a new value of 7, and the Quintic Spline method was applied again. Next, the angle of the WC3 hit, 9d, was calculated by finding the intersection of the track and the circle describing the WC3 anodes in the CHAOS coordinate system. The angular distance corresponding to drift time, td, and incident angle, 7, is given by e{td)j)=\ed-ew\, (3.29) where 9W is the angle of the sense wire. The above procedure was repeated for a large number of tracks, and average values of #(£<j,7) were calculated for each small bin in Chapter 3. Event Reconstruction 40 7. Next, a third order polynomial fit was applied to the drift distance as a function of time for each 7 bin, and a better estimate of the x(t) relation was obtained. The entire process was repeated until the change in the time-distance relation was negligible. A more detailed discussion of the above process is given in Reference [32]. During the design and construction of the chamber, the resolution of a prototype using a similar cell geometry was determined to be 150|iirn (<r) [25]. To achieve this accuracy, the chamber was calibrated with the aid of other wire chambers which provided equal or better spatial resolutions. In CHAOS the resolution of the other chambers ( W C l , WC2 and WC4) does not allow for such extremely accurate calibrations. Thus, the effective resolution of the chamber is 2bQpm (a) [33]. For WC4, the electric field lines are such that the drift electron trajectories are nearly perpendicular to the track. Hence the time distance relation was assumed to be inde-pendent of the incident angle. Otherwise, the x(t) relation for WC4 was determined in a similar manner to WC3. Position information from the other three chambers, ini-tial estimates of the x(t) relation in WC4, and the Quintic Spline method were used to determine the analytic form of a given particle's trajectory. Next, the point of intersec-tion of the track with each anode plane was used to obtain the drift distance. A large number of tracks was analyzed, and the drift distances were averaged over small bins of td, producing a better estimate of the time-distance relation. The new set of x(t) relations was then used to reanalyze the same data, and the process was repeated until the change in the x(t) relation was small. Roughly, the time-distance relation for WC4 is ~ 0.05 mm/ns [24]. From this estimate, the intrinsic resolution of the chamber (for one cell) is ~ 150 pm (a) per wire. However, the effective chamber resolution (averaged over all cells) is ~ 200 pm. Due to the small nominal field setting, the Lorentz angle [26] correction to the x(i) relation was negligible. This is not the case with all CHAOS experiments. Chapter 3. Event Reconstruction 41 3.2 Track Sorting The track sorting algorithm employed in this work uses a "tree pruning" method in which various combinations of chamber hits correspond to different branches of a tree structure. Starting with each of the hits in W C l , all hits within a given angular window in WC2 are selected. This forms the first (lowest) branches of the tree (see Figure 3.14). Next, the WC3 hits within a given angular window of each WC2 hit are selected, and they form the next stage of the tree structure. A similar prescription is applied to the tracks in WC4. Once all of the possibilities are exhausted, each branch of the tree (from start to finish) corresponds to a possible way of grouping hits into tracks. Clearly some of the branches will be incomplete because there might not be any hits within the given angular window. These branches are eliminated (pruned) at each layer. 1 3 3 4 4 1 3 3 4 4 Figure 3.14: The tree structure employed in the track sorting algorithm. The numbers correspond to the appropriate chamber (layer). Once the tree structure is constructed, the algorithm decides on the correct combina-tion of hits (a particular branch) that corresponds to the true trajectory of the scattered particle. This is achieved by repeated applications of the Quintic Spline method (see Section 3.3) to each group of hits. Equation 3.31 provides an analytic form of the track trajectory inside CHAOS. Hence fitting the coordinates of the group of hits in a given Chapter 3. Event Reconstruction 42 branch yields a %2 for that branch. Those combinations with a %2 less than a lower limit are selected as tracks. The efficiency of the track sorting algorithm is part of the overall reconstruction efficiency which was determined from Monte Carlo simulations discussed in Chapter 5. 3.3 M o m e n t u m Reconstruct ion The Quintic Spline method, developed by Wind [34], forms the basis of the momentum reconstruction software. Only in-plane position information was used as input to the reconstruction algorithm, and this was due to two main reasons. First, the combined W C l and WC2 cathode strip efficiency for a single track was only about 50%. Thus the efficiency of obtaining vertical coordinate information for three tracks (the incident beam and two final state pions) is only about 13%. Second, the vertical acceptance of CHAOS is small (~ ±7°), and as such only particles with a small out-of-plane momentum component will be detected. The error in the momentum may be estimated from the relation pcos{\) = 0.3BR, (3.30) where p is the true momentum of the track in MeV/c , A is the out-of-plane angle, B is the field magnitude in Tesla, and R is the radius of curvature (in mm) of the trajectory of the singly charged particle (in a uniform magnetic field) . Thus for A < 7°, the error is less than 1%. Consequently, the in-plane approximation is adequate. This is also confirmed by the Monte-Carlo studies discussed in Chapter 5. The equations of motion describing the trajectory of a particle with charge q, in a nonuniform magnetic field B, may be used to obtain a simple relation between the Chapter 3. Event Reconstruction 43 particle's trajectory and its momentum. This is given by [24] P | £ = - , * ( * , „ ) + , (3.31) where P is the momentum, and (x, y) are the coordinates of arbitrary points along the particle's trajectory. The second order derivative on the left-hand side of Equation 3.31 may be integrated twice to obtain 72„ y = a + bx + ^- I* da fa(P^)dr, (3.32) r Jo Jo dr* where a and b are integration constants, and a and r are dummy variables. Thus, once the the double integral is evaluated numerically, P may be obtained by fitting the coordinates of the track with Equation 3.32. The steps in the momentum reconstruction software may be summarized as follows: • The hits in W C l , WC2, and WC3 are used to obtain the parameters of the circle passing through them. • From the equation of the circle two artificial ("pseudo") hits are generated. One is between W C l and WC2, and the other lies between WC2 and WC3. • A straight line fit is performed on the hits in WC4. • A cubic spline is used to generate pseudo hits between WC3 and WC4. • The analytic forms of the circle, spline and straight line are differentiated to obtain ^ at all track and pseudo hit coordinates. • Given the polarity of the track (from the direction of curvature) and the magnetic field strength at each point, Equation 3.31 provides the corresponding value of P | ^ f at each point. Chapter 3. Event Reconstruction 44 • A cubic spline fit is applied to the values of P ^ . • The polynomial obtained from the cubic spline fit (above) is analytically integrated twice, and the value of the integral at each chamber hit is recorded. The double integral yields a fifth order polynomial, and hence the name Quintic Spline. • Once the double integral is known, an analytic form for the track trajectory is a fifth order polynomial given by Equation 3.32. A least square minimization between the values of y obtained from Equation 3.32 and the chamber hits yields the constants a, b, and js. The pseudo hits are not used in the minimization. • ^ is recalculated from the equation for the track, and the cycle is repeated once more to obtain a stable value of P. Corrections for energy loss were applied through extensive Monte-Carlo studies using the G E A N T simulation package. A discussion of this process is presented in Chapter 5. 3.4 Interaction Vertex & Scattering Angle Once the track momentum and polarity were known, the interaction vertex and the scattering angle in the CHAOS coordinate system were determined. The trajectory of a track inside W C l may be estimated by a circle because the CHAOS field in this region is nearly uniform. The equation of this circle was obtained from the track momentum, the coordinates of the hits in W C l and WC2, and the magnetic field setting. Similarly, the incident beam trajectory was approximated by a circle passing through the incident beam hits. The radius of curvature of the incident beam circle was determined from the known magnetic field strength and the incident beam momentum. The coordinates of the interaction vertex were then determined by computing the intersection point of the incident beam with each of the scattered track circles, (see Figure 3.15). Equations Chapter 3. Event Reconstruction 45 describing the intersection of two circles have two solutions. Thus, for each track there are two possible intersection points; the vertex was chosen to be the one closest to the center of CHAOS. Due to the finite resolution of the detector, there were exceptional cases where the incident and scattered particle trajectories did not cross. Such tracks were eliminated from the analysis, and they contribute to the reconstruction software inefficiency, which was accounted for via Monte Carlo simulations (see Chapter 5). T 7 r=264 MeV + -7T p->7T 7T n 5 0 0 0 - 5 0 0 - 5 0 0 0 5 0 0 Figure 3.15: Illustration of a reconstructed (7r,27r) event. The track trajectories in the region of the target (inside W C l radius) were approximated by circles. The C F T blocks removed at the entrance and exit of the beam are not shown. The incident beam is from the left and is registered by the WC1/2 hits located at negative x-values. The vertex resolution could be obtained by computing the difference between the reconstructed vertices for pairs of outgoing tracks, which were detected in coincidence. The vertex resolution is angle dependent since it is more difficult to determine the inter-section of trajectories that are nearly parallel. The resolution is best near 90° and worst around 0°. On average, the vertex resolution was determined to be ~ 1.6 mm. It was found that tracing the scattered track trajectory by numeric integration of the equations Chapter 3. Event Reconstruction 46 100 60 ? 20 > H -20 -60 -100 -100 -60 -20 20 60 100 X ( m m ) Figure 3.16: Example of the reconstructed interaction vertex for data acquired with a coincidence 1LT and liberal 2LT cuts. They are dominated by 7rp elastic events. of motion did not improve the vertex resolution. Figure 3.16 shows a typical scatter plot of the reconstructed vertex. The liquid hydrogen target and the vessel walls are clearly visible. The scattering angle is defined as 9 = cos~l{ki • ks), (3.33) where ki and ks are unit vectors pointing along the direction of the tangents to the incident and scattered track trajectories at the interaction vertex, respectively. Since the vertex location is used in determining 6, the scattering angle resolution is also angle dependent. On average, the scattering angle resolution is ~ 0.5° [24]. 3.5 Particle Identification The energy deposited by a charged particle (with mass much greater than that of the electron) in a thin counter of thickness Ax is given by the Bethe-Bloch expression (in Chapter 3. Event Reconstruction 47 natural units, where Ti = c = 1) [35]: . ATrNz2e4Zp { In ( 2mf32 ) (3.34) m/32A 1(1 - (32) where (3 and are the velocity and charge of the particle, m is the electron mass, N is Avogadro's number and Z, A , and p are the atomic number, atomic mass, and density of the counter material, respectively. In addition, I represents the medium's effective ionization potential. For a given material, the energy loss only depends on the velocity and charge of the particle. Once the velocity increases past a critical limit ((3 ~ 0.85), the energy deposited is almost constant before increasing logarithmically. The particle is then said to be minimum ionizing. Since the proton is much heavier than the pion, a non-minimum ionizing proton will deposit more energy than a pion with the same momentum. Hence it is possible to separate pions from protons by examining plots of pulse height in the AE\ counters versus momentum. The same plots may also be used to distinguish pions from electrons over a limited range of momenta. In order to apply the same software filters (cuts) to all pulse height spectra in a given layer, it is necessary to make corrections for photo-tube gain differences. A separate correction factor is required for each AE\ counter. These coefficients were determined by examining pion and proton tracks produced in -np elastic scattering and ixd breakup. The breakup data were acquired during the second stage of the experiment. The gain calibration process can be summarized as follows: • Pion and proton tracks were identified by placing kinematic cuts on their momenta and scattering angles. • Using the angle of the track in WC4 and the track momentum, the expected energy loss in a given AEX counter was determined. This value was scaled by a constant so that it was of the same order of magnitude as the digitized pulse heights (measured energy loss). The constant was the same for all tracks. Chapter 3. Event Reconstruction 48 The gain correction factor was determined from a = 1 " 7 5 c ' ( 3 ' 3 5 ) where A is the difference between the scaled energy loss and the A D C value, and ADC represents the digitized pulse height. • A large number of tracks was analyzed and distributions of a were recorded for each AEi counter. • The final gain correction factor for a given block was determined by computing the mean value of a. Once the gain correction factors were determined, they were monitored throughout the analysis procedure. This was achieved by examining pion and proton tracks resulting from TT^P elastic scattering as well as those originating from inelastic reactions taking place in the walls of the target vessel. In the case of the positive incident pions, ixd —• pp data acquired using a liquid deuterium target were also used. A l l of these results were consistent with the lack of any significant changes in the photo-tube gains during the two data acquisition periods. The details of the particle identification process are discussed in Chapter 4. To separate pions from electrons, the lead glass elements of the C F T blocks were used. For electrons, the total light output is due to Cerenkov radiation produced by the initial electron, as well as photons created from the electro-magnetic shower in the lead glass. Hence the gain-matching procedure could not be carried out employing an analytic expression relating the pulse height to the incident momentum. During the commissioning phase of the detector, a single Cerenkov counter (CFT block) was fully illuminated with pions and electrons (directly from the incident beam) over a wide range of incident momenta, and the resulting pulse heights were recorded. For these data, Chapter 3. Event Reconstruction 49 pion-electron identification was independently available via time of flight analysis. This is possible using the incident beam by utilizing a long flight path (the entire length of the pion channel and the experimental area), with cyclotron R F based timing from the pion production target to the C F T block being calibrated. 200 p—i id > u 100 Q < 50 H J I I I I i I L. J i t i i i t I !—I—I L Electrons ~i—i—i—I—i—i—i—r i 1 1 1 1 i 100 200 300 400 Momentum (MeV/c) 500 Figure 3.17: Cerenkov calibration data acquired during the commissioning phase (using the incident beam directly). The electron pulse heights are larger because they produce an electro-magnetic shower in the lead glass. For the pions, the light output is only due to Cerenkov radiation. The calibration data were placed in suitable bins of incident momenta, and for each bin the Cerenkov pulse height spectrum was formed. The A D C values corresponding to the centroids of the distributions for electrons and pions were used to determine an empirical relation between pulse height and momentum (see Figure 3.17). Using the pulse height versus momentum relation, the photo-tube gains were matched in a procedure similar to that used for the AE^s. Initially, electron and pion tracks were Chapter 3. Event Reconstruction 50 selected by placing conservative cuts on plots of the pulse height versus momentum for the AEi counters. The gains were then adjusted to obey the correlation between momentum and pulse height shown in Figure 3.17. Once the initial gain correction was made, the Cerenkov counters along with the AEi's were used to select more pion and electron tracks, and the gains were recalculated. This process was repeated in an iterative fashion until a stable gain correction factor was obtained. Throughout the analysis process, the cuts made on the Cerenkov pulse heights were monitored for changes in photo-tube gains; no significant changes were observed. In later chapters, further tests and checks of the particle identification process are discussed. Chapter 4 Data Analysis The goal of the experiment was to determine many-fold differential cross sections for the n+p —> Ti+ii+n and ir~p —> TT+TT'TI reactions. To achieve this, (7r, 27r) events first had to be isolated from background reactions and the reconstructed data binned in terms of suitable kinematic parameters. The data reduction process leading to this consisted of many separate steps. In the first part of this chapter, the primary background reactions are discussed, and the steps in the data reduction procedure are outlined. In Section 4.4 the details of how various cuts were used to eliminate background events are presented. Unless otherwise stated, all histograms and scatter plots shown in this chapter were obtained with a coincidence first level trigger, and the second level trigger was enabled. The vast majority of the recorded interactions were itp elastic events which were not filtered by the second level trigger. the walls of the target (4.36) (4.37) (4.38) 4.1 Primary Backgrounds In addition to events coming from interactions taking place in vessel, the primary background reactions were -K^p —• ^p, TX~P —> 7T°7l, TT'P —> 771. 51 Chapter 4. Data Analysis 52 For both positive and negative polarity incident pions, the cross section for elastic scat-tering is many orders of magnitude greater than that for pion production. For example, at 280 MeV the total cross section for n+p scattering is 76 mb [36]; in contrast, the pion production cross section is only 48 ph. Furthermore, as the incident energy approaches the pion production threshold (172.38 MeV), there is a quadratic decrease in the (IT, 2TT) cross section. Hence, the largest contribution to the background events came from irp elastic scattering. Although the second level trigger was used to minimize this back-ground, the large cross section combined with the relatively poor momentum resolution of the trigger made it impossible to discard all of the np events in hardware. The other source of background events, present only for the ix~ data, was due to electron-positron pairs created when one of the gamma rays originating either from the decay of the 7r° produced in the single charge exchange (SCX), reaction 4.37, or reac-tion 4.38 converts to an e+e~~ pair inside the detector. The e+e~ pairs have a wide range of momenta, and they do not display any kinematic correlation. They mimic 7 r + 7 r -pairs. To obtain reliable (7r,27r) results, the e+e~ events not only had to be tagged, but procedures had to be developed to measure the pion-electron particle identification efficiency. 4.2 Par t ic le Identification Process One of the most crucial stages of the analysis procedure was particle identification (PID). The PID information was obtained by combining the pulse height information from the C F T counters with the track momenta, in scatter plots of pulse height versus momentum. Pion-proton-electron identification was made by placing suitable two dimensional cuts on these plots. At the energies studied in this experiment, the only charged particles in the final state Chapter 4. Data Analysis 53 were pions, muons, electrons, and protons. Since it was not possible to identify decay muons from the C F T information, the PID strategy was aimed at tagging proton and electron tracks and identifying the rest as pions or muons. The muons were accounted for using missing mass limits, track sorting constraints sensitive to the "kink" in the track produced at the TX —>• pv decay vertex, as well as Monte Carlo studies (discussed later). This strategy proved to be fruitful because prior to the particle identification stage, other cuts were used to greatly reduce the 7ip and e+e~ backgrounds. Figure 4.18 depicts a flowchart showing the various steps in the particle identification process. Here p is the track momentum, AE is the pulse height in the AEi counter and C denotes the combined pulse heights from the three lead glass counters (right-middle-left). In the first stage of this process, pion-proton distinction was made based on cuts on plots of pulse height in AE\ versus momentum. As shown in Figure 4.19 the pion and proton bands are well separated and events inside the polygon are identified as protons. Pion-electron identification on the other hand is more challenging. This is because in plots of AEi A D C versus momentum, the pion and electron bands overlap for momenta greater than ~ 120 MeV/c . Cerenkov information is required to distinguish pions from electrons at higher momenta. Figure 4.20 shows a scatter plot of Cerenkov A D C versus momentum. The pion and electron bands are separated above 120 MeV/c , and electrons may be identified by making two-dimensional cuts on this plot. In making the electron cuts, extreme care was taken such that only those events that were clearly separated from the pion band were identified as electrons. As a check on the pion-electron identification process, particles tagged as electrons by the Cerenkov were examined in plots of pulse height in AEi versus momentum (see Figure 4.21). The electron band is clearly visible and overlaps with the pion band for momenta greater than ~ 120 MeV. The most stringent test of pion-electron identification was made by constructing the missing mass spectrum for events identified as 7r +7r _ and those tagged Chapter 4. Data Analysis 54 P AE C Protons Proton cut on AE vs P ^ ) Electrons P< 120 MeV/c electron cut on AE vs P P> 120 MeV/c electron cut on C vs P Pions Figure 4.18: Flowchart of the particle identification process. Chapter 4. Data Analysis 55 1200 800 M o m e n t u m (MeV/c) Figure 4.19: A typical scatter plot of pulse height in AE\ versus momentum. Pion and proton bands are well separated. These data were acquired with 280 MeV positive polarity pions. The momentum spread is due to the variation of the pion and proton momenta with angle. as e+e~. This is discussed in Section 4.4. Clearly the PID process is not perfectly efficient, and consequently the particle iden-tification efficiency is a parameter that enters into the calculation of differential cross sections. A detailed discussion of the procedure employed for determining the PID effi-ciency is presented in Chapter 6. Chapter 4. Data Analysis 56 3 0 0 u Q : < > o a CD t-t CD o 100 1 1 I 1 , i i i ! i i • . i 1 1 • 1 - v '• •. . ; . " • , Electrons / \ . •.. Pidjis '. '. .Pions - • -^Siwi: 'isffiHi . . . -300 -150 0 150 Momentum (MeV/c) 300 Figure 4.20: Scatter plot of pulse height in the Cerenkov counters versus momentum. The elastic events have been eliminated. Negative momenta represent negatively charged particles. 6 0 0 5 0 0 U 4-00 h Q 3 0 0 < 2 0 0 100 K : Pions is* E lec trons J I I I I I L_ - 2 0 0 - 1 0 0 0 100 Momentum (MeV/c) 2 0 0 Figure 4.21: Scatter plot of pulse height in AEi versus momentum. Events identified as electrons by the Cerenkov as well as TT+7T~ events are shown. The ir~p events have already been removed with the momentum sum cut. Negative momenta represent negatively charged particles. Chapter 4. Data Analysis 57 4.3 M i s s i n g Mass Calcu la t ion Consider the fixed target reaction a + b^c + d + e. (4.39) In the laboratory frame, the energy and momentum conservation relations may be written as Ea + mb = Ec + Ed + Ee, (4.40) Pa = fc+fd+Pe- (4.41) where E denotes a particle's total energy, raj, is the target mass, and p is the three-momentum vector. For the pion production reactions studied in this work, the neutron in the final state was not detected. However, the magnitude and direction of the momentum for each of the final state pions were measured, and the incident beam momentum was known. Using the PID information and the measured momenta, the total energy of the detected final state particles was determined. From the energy and momentum conservation equations, it is possible to determine the missing energy, AE, and the missing momentum Ap. In the case where particle e was not detected, the missing mass is given by A M = ^AE2 - (Ap)2, (4.42) where AE = Ea + m b - E c - E d (4.43) Ap = Pa-Pc-Pd (4.44) For (7r,27r) events, the spectrum of AM has a peak centered about the neutron mass. Determination of the missing mass thus provides a method of separating the (n, 2ir) events Chapter 4. Data Analysis 58 from the background, but if any of the final state particles are misidentified, this quantity is no longer useful. Since the strength of the background reactions is large compared to the foreground, even a small fraction of misidentified particles will be problematic. As such, prior to calculating the missing mass spectrum, other restrictions were used to reduce the background events. 4.4 D a t a Reduc t ion Events coming from the interaction of the incident pion beam with the target vessel could be eliminated by placing a cut on the location of the interaction vertex. Figure 4.22 shows typical scatter plots of the reconstructed vertex in the x-y plane. The top figure refers to data acquired with the liquid hydrogen target, and the data shown in the bottom plot were recorded after the LH2 was pumped out of the target vessel. In both plots the walls of the target vessel are clearly visible. Events located inside the polygon enclosed by the solid lines are those selected for further analysis. Since the efficiency of obtaining out-of-plane coordinate information for all three tracks (the incident beam and the two final state particles) is poor, the interaction ver-tex was reconstructed in the x-y plane only. Thus, no limitations were placed on the out-of-plane location of the vertex. Figure 4.23 shows the out-of-plane coordinate for the projection of the incident beam on the target. This was computed by finding the intersection of the incident beam track with a plane containing the center of CHAOS and oriented perpendicular to the beam direction in the x-y plane. The top figure shows the spectrum for all events except Tip elastic data. The copper disks (see Figure 2.9) appear as two distinct peaks above and below the main peak representing interactions in the LH2 target. In the bottom plot, only events which were identified as (IT, 2TI) are shown. It is clear that any contributions from the disks have been removed. These peaks Chapter 4. Data Analysis 59 100 r-60 7 m) 20 -(m -(m - 20 '--60 1 -100 -I 1 1 1 1 I ' 1 1 ' I ' ' 1 ' I 1 1 1 1 Full Target J I I I I I I L_ - + -7T p->7T 7T D. T f f = 284 MeV 100 -60 -20 20 X ( m m ) 60 100 100 60 h £ 20 p-V -20 -60 h -100 —| 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 Empty Target _j i i I i i i i — L _i i I i i — 77 p ->TT 7T n 264 MeV -100 -60 -20 20 X ( m m ) 60 100 Figure 4.22: Scatter plot of the reconstructed interaction vertex for full (top) and empty target (bottom). The polygon illustrates the vertex cut. Data were acquired with liberal 2LT cuts. Chapter 4. Data Analysis 60 are mainly due to electron-positron pairs, which are produced when one of the gammas originating from Reactions 4.37 and 4.38 converts in the copper disks. Since copper is denser and has a higher atomic number than hydrogen, the gamma rays produced in the disks are more likely to convert than those produced in the liquid hydrogen target. The sum of the magnitudes of the momenta of the two charged particles in the final state may be used to eliminate the 7rp elastic events. This is because the momentum of the pion and proton combined is greater than that of the two pions from (IT, 2ir). Figure 4.24 shows the phase space distribution for the momentum sum of the two charged particles in the final state for elastic scattering and (TT, 2TT). This plot does not reflect the strength of the irp interaction relative to (7r,27r). It is clear that placing an upper limit on the momentum sum will remove the majority of irp elastic events. Since only events with two pions in the final state are of interest, events having a proton in the final state were not considered for further analysis. This removed any remaining elastic events and eliminated pion-proton final states produced by TT+p -> T T 0 ^ . (4.45) The above is especially important for the 7r + data, since the total cross section for the 7r +p —> 7Y°7i+p reaction is nearly twice that of ir+p —> 7r+n+n. The vertex was also useful in reducing the e+e~ background, because the gammas are more likely to convert in the walls of the target vessel or the faces of the pole tips. Those which convert inside the LH2 target may be eliminated using the PID information. Past this stage, the only tracks selected for further analysis were those in which both particles were identified as pions. As a test of the pion-electron identification process, consider the plot in Figure 4.25. Here the solid line shows a typical missing mass spectrum formed by all events (except 7rp elastic). In calculating this missing mass spectrum, the pion mass was used when Chapter 4. Data Analysis 61 -80 -60 -40 -20 0 20 40 60 80 Z P r o j e c t i o n ( m m ) 300 I I I I L ~ J " T " I I I | 1 1 1 I | I T I I | I I I I | I I ! I J" I I I I | 1 I I I -80 -60 -40 -20 0 20 40 60 80 Z P r o j e c t i o n ( m m ) Figure 4.23: Plots showing the vertical coordinate of the projection of the incident beam for all events (top) and (7r,27r) events (bottom). Chapter 4. Data Analysis 62 800 100 Tw=285 MeV 300 500 700 S|p| ( M e V / c ) 900 800 600 2 400 O O 200 i i i i i i i i i i i i i i i i i i i i i i i i i i i I • T7r=243 MeV 0 100 200 300 400 500 600 700 800 E|p| ( M e V / c ) Figure 4.24: Momentum sum spectra for irp elastic and (it, 2-K) reactions. The distribu-tions were generated using two and three-body phase-space. Chapter 4. Data Analysis 63 4000 3000 w -p 3 2000 O O 1000 - \ _J I I L_ _l I I 1_ T w = 284 MeV 7T p->7T 7T n 700 TT n & e e + -7T 7T n 1 r"—I 1 r 1 1 r 800 900 1000 Missing Mass (MeV) 1100 Figure 4.25: Missing mass distributions for all events except irp elastic. computing the energies for all of the detected final state particles (all tracks were assumed to be pions). There is a peak centered at the neutron mass, and this is superimposed on top of a broad background. The peak is due to the (7r,27r) events, and the background is caused by the electron-positron pairs. The dashed line corresponds to events which were tagged as e+e~ by the PID system. It describes the background perfectly and demonstrates the success of the pion-electron identification. The peak labeled TT+TT~ corresponds to those events which were tagged as 7r + 7r~. This corresponds to a clean signal, which is virtually free of background. The mean and standard deviation of the 7 r + 7 r - peak shown in Figure 4.25 are 946 and 4.2 MeV, respectively. Placing a window around this peak isolates the (TT, 2-K) events from the small fraction Chapter 4. Data Analysis 64 that were misidentified. The missing mass cut also removes events in which one or both of the final state pions decayed before arriving at WC4. This is because the momentum and scattering angle measured for such tracks do not equal those of the original pion, and hence they will lie outside the missing mass window. Monte-Carlo simulations are required to fully account for pion decay. A flowchart summarizing the different stages of the analysis process (discussed above) is shown in Figure 4.26. This figure indicates the order in which the various cuts were applied to the data. 4.5 Empty Target Data During the course of the experiment, empty target data were acquired in order to ensure that the (it, 2it) events coming from the windows of the target vessel and surviving the cuts were negligible. The empty target data were analyzed using the process outlined in Figure 4.26. A sample missing mass spectrum for the full and empty target data is shown in Figure 4.27. For both reaction channels, the full and empty spectra have been normalized to reflect the same number of incident pions. It is clear that the empty target background is negligible. Although the total cross sections for the two reaction channels are substantially different, the ratio of background to foreground counts should be the same. This is indeed the case; for both channels the background counts are 2.3% of the foreground. 4.6 The Binning Parameters Once all of the (it, 2TT) events were fully reconstructed, the data were binned in terms of the following kinematic variables: the dipion invariant mass squared, ra2^, the four-momentum transfer to the nucleon squared, t, and the pion-pion scattering angle in the Chapter 4. Data Analysis 65 Electrons M— Vertex Cut rep Elastic — ZpCut Target Vessel Figure 4.26: Flowchart describing the various steps of the analysis. Chapter 4. Data Analysis > CD CO <N -i-> cd t ft T—|—I—I—I—I—I—!—I—I—I—|—I—I—I—R-J—I—I—I—I—|—r—i—I—R—i—I—I—I—R-I O OB U crj E-< 3 fa u cd I ft -L fa _j i I i * ' • "" I > CD cd on • i-H g 00 o m o o o O in o o HI O CM O o > CD to CO CM crj t ft r | i i i I | I I i i | r i T i | i u crj H 2 fa 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -t-> d) M f-n crj J. ft 4 H i • • 1 1 1 1 1 1 • 1 1 1 1 1 1 1 1 1 . i . . . 111 o o o o o o o CO o o o o o o o o o o o o o o o ^ • C M O O O l O ^ C M T f C N O O O t O ^ - C N > CD ui cd • I-H 03 J-H +3 o 0) O H CO CP a P i a o P i .2 o3 CD o a3 CP (-1 o CO -s « W> .2 -^-2. CD . g P i ° o3 s-j CD o B ^ P i ^ cp 03 CO p3 ^ (-1 -1-3 CP CP o co q=l CP (-H o hO cp .9 'co ^ CO fo S—I o P i P i CD CD r-CP i-l 2 3? • I-H 03 fa rPl Chapter 4. Data Analysis 67 dipion center of mass frame, 9. For the ir+ir+n channel, 9 is defined as the angle between the incident 7r + and one of the outgoing pions (chosen at random), and in the case of the 7r+TT~n data, 9 denotes the angle between the two negative pions. In this work both t and m2^ will be presented in units of pion mass squared. At a given incident pion energy, the kinematics of TTN —• TTTTN (ignoring polarization) are completely described by ra2^, t, 9, and x, where x is the angle between the dipion and the plane defined by the incident pion and the scattered nucleon. Relatively low statistics and the limited coverage of CHAOS in the out-of-plane angle (limited coverage in x) make the determination of four-fold differential cross sections impractical. Thus the data were binned in terms of the other three parameters. These particular kinematic variables were chosen since they are well suited for use in the Chew-Low extrapolation process, a model-independent analysis used to extract ITIT scattering cross sections from (7r, 2TT) data, as described in Chapter 7. The pion production reactions studied in this work are: where fci, fc2, and kz denote the four-momenta. The expressions denning m2^, t, and cos9 may be written as ^{h) +p(kp) -»• TT±(k2) + TT + ( /C 3 ) + n(kn) (4.46) ml '7T7T = 2p2 + 2k2-h (4.47) cos 9 t = ml„ + p2 - 2ki • k2 - 2fci • k: Oi ' 3 , (4.48) (4.49) y/AB' here p is the pion mass and 9i, A, and B are given by 9 7 -(fci • h - h • k2) 4 (4.50) A = (4.51) Chapter 4. Data Analysis 68 o m47r7T-2m2,(p2 + t) + (p2-t)2 B = 4 ^ • ( 4 5 2 ) A derivation of the above equations is presented in Appendix A. The (7r, 2TT) events were placed in a 10 x 10 x 10 lattice of m2n, t, and cosd. In units of pion mass (p), the bin widths for m2^ ranged from 0.3-0.5p2. The larger width bins were used for the 305 MeV data. The bin widths for t ranged from 0.5-lp2; smaller widths were chosen for those t values close to the physical limit of t ~ 0. This is because the determination of TTTT scattering lengths requires the extrapolation of the double differential cross sections to t = +p2. The cosQ bins had a constant width of 0.1. Prior to the calculation of differential cross sections, extensive Monte-Carlo simula-tions were performed in order to correct for factors such as the geometrical acceptance of CHAOS, pion decay, energy loss, second level trigger efficiency, and reconstruction software inefficiencies. This process is discussed in the next chapter. Chapter 5 Monte-Carlo Simulations In order to produce many-fold differential cross sections, the event lattice must be cor-rected for the following factors: geometrical acceptance of CHAOS, final state pion decay, energy loss, 2LT inefficiency, and reconstruction software inefficiencies. The combined correction factor corresponding to all of these effects is referred to as the acceptance cor-rection. To determine the acceptance, extensive Monte-Carlo simulations of CHAOS were performed using the C E R N detector simulation package G E A N T [37]. The Monte-Carlo simulation process and results are discussed in this chapter. 5.1 Statement of the Problem At a given incident beam energy, the triple differential cross section for irN —> irirN can be expressed as = ' N(m2^,t,cos9) dm2nndtdcos9 A(m2T7T, t, cos9) Ntgt e A m ^ At Acos9' 1 ' ' where N is the number of reconstructed events in a given bin of m2^, t and cos9, is the number of incident pions on target, and Ntgt denotes the number of target particles per unit area, e represents the overall detection efficiency, which includes the wire chamber efficiencies, computer live-time, and PID efficiency. In addition, A is the acceptance correction factor (weight) corresponding to a given bin, and A m J T , At and Acos9 are the bin widths in TO2^, t and cos9, respectively. Due to the limited geometrical acceptance of CHAOS and factors such as pion decay, the acceptance correction factor A is different 69 Chapter 5. Monte-Carlo Simulations 70 from unity. The geometric acceptance of CHAOS is about 10% of An. Thus for an interaction for which detection of two charged particles in the final state is required, the solid angle coverage is only about 1% of Air. As such, by far the largest contribution to the weight is due to the limited geometrical acceptance. The contribution from pion decay is much smaller. The survival probability for a 100 MeV/c (32 MeV) pion in CHAOS (path of ~ 1 m) is about 0.84. This is a lower detection limit (worst case). Depending on the location of the decay (inside CHAOS) and the angle of the resulting muon, the reconstructed momentum for the track may be nearly the same as that of the initial pion. Recall that at a given center-of-mass energy, the kinematics of TTN —> TTITN can be fully described in terms of the following four independent kinematic parameters: ra2^, t, 9 and x. In other words, in this four-dimensional space all possible combinations of momenta and scattering angles of the final state particles correspond to unique points. Since the out-of-plane acceptance of CHAOS is limited, and the number of experimental events was relatively low, the formation of the event lattice based on four variables was impractical. Consequently, the acceptance correction factor was determined in terms of the other three. Thus, in the process of acceptance calculation, it is implicit that the fourth variable, x, is distributed according to phase-space. The determination of the acceptance weighting factors is represented in the flowchart shown in Figure 5.28, and may be summarized as follows. First three-body phase-space was used to generate (IT, 27r) events into An solid angle, and each event was placed in an appropriate bin of m 2^., t, and cos9 in the lattice denoted by G. Next, the C E R N detector simulation package, G E A N T , was employed to simulate CHAOS, and those generated events which fell within the detector's acceptance were tracked and digitized. The digitized tracks were then analyzed in the same manner as the experimental data, and each fully reconstructed event was placed in a suitable bin of the lattice R. The acceptance correction factor, A, for a given bin was determined from the ratio of the reconstructed events to the generated Chapter 5. Monte-Carlo Simulations 71 ones in that bin. It is important to note that the procedure outlined above does not make any assump-tions about the distributions of ra^, t, or cos9. Each of these parameters is treated explicitly, and hence the acceptance correction factor is independent of the distribution used to generate the events in 4ir. Phase space merely provides a convenient and standard method of generating multi-particle final states which conserve energy and momentum. The only assumption is that x is distributed according to phase-space. In the following sections, each of the individual steps shown in Figure 5.28 is discussed in detail. 5.2 Event Generation & Tracking The effort to create the CHAOS simulation software has spanned a period of five years. It began when the detector was at the design stage, and today the software is capable of simulating all components of the detector. Throughout the development process, the goal has been to make the simulation as realistic as possible. To this end, all of the materials used in the construction of the wire chambers as well as the chamber gases are incorporated into the simulation package. In addition, AEX) AE2, Cerenkov, and Veto counters as well as the magnetic field map obtained from field measurements made (in three dimensions) with a Hall probe during the construction of CHAOS were all included. Furthermore, the cryogenic target including all of the components which make up the target vessel are included in the code. Event generation and tracking can be summarized as follows. Incident pions were generated such that their momenta were distributed according to a Gaussian distribution with the mean at the central beam momentum and the width given by the momentum spread of the incident pion beam (typically 1%). The pions were then tracked from Chapter 5. Monte-Carlo Simulations 72 I G(mln,t,cosO) \ 2LT Simulation I GEANT 3-Body Phase \ S p a c e / A(nL,t, c o s 0 ) : R(mlK,t,cosO) GirrlA, c o s Q) Figure 5.28: The flowchart describing the acceptance determination process. The "ana-lyzer" is the same software used to reconstruct the experimental events. Chapter 5. Monte-Carlo Simulations 73 the exit of the beam pipe, through CHAOS to the LH2 target (located at the center of the detector). The dimensions of the beam spot on the target were matched to those observed during the experiment. At a random location inside the target, the pions were forced to undergo an interaction for which the distribution of the final state particles was given by three-body phase space for nN —• irirN. The phase-space generation subroutine, G E N B O D [37], was used to determine the momenta and directions of the secondary particles at the interaction vertex. The interaction point in the target was chosen at random along the incident beam trajectory. The final state pions were then tracked through CHAOS and the hit information in the chambers and C F T counters was recorded. The energy deposited in the AEi and AE2 counters was used to simulate the first level trigger decision. As with the experimental data, tracks which passed through the veto counter did not produce a trigger. It should be noted that energy loss, multiple scattering, and decay were all considered in the tracking of the final state pions. Since hadronic interactions generated by G E A N T are thought to be unreliable in the energy region studied in the current experiment, these processes were not included in the acceptance calculation. The CHAOS chambers are constructed from low density material, and as such, hadronic interactions are more likely to take place in the plastic scintillators and lead glass counters. This does not affect the reconstructed momentum and scattering angle, but such events may lead to a lower particle identification efficiency (discussed in Chapter 6). The first level trigger required at least two C F T blocks to be activated, and the logic signal for each block was formed from the coincidence between the AEX and AE2 counters in that block, just as in the actual experiment. The C F T blocks that were removed during the experiment (those located at the entrance and exit of the beam) were also removed in the Monte-Carlo simulations. If the first level trigger requirements were met, the hit information was digitized so as to mimic the experimental situation. Chapter 5. Monte-Carlo Simulations 74 For the drift chambers, the hit coordinates were converted to drift times. In the case of the proportional chambers, the angular coordinates of the hits were used to determine the activated wires. The finite resolution of the chambers was included in the digitization process. For WC3 and WC4 this was accomplished by adding a Gaussian distributed random number (noise) to the original hit position. The a of the distribution was set to the chamber resolution (250 for WC3 and 200 pm for WC4). In the case of W C l and WC2, the resolution is determined by the wire spacing. The simulated drift time and wire numbers were then put in a binary format, identical to that produced by the readout electronics. The digitized information was recorded in a data file that had the same format (YBOS) as the one produced by the data acquisition system. As such, identical analysis software could be used to analyze both the experimental and simulated data. In addition to the digitized information, the exact hit coordinates, PID, polarity, and momenta were also recorded in the output data file of the simulations. This additional information was useful for testing the digitization and other algorithms. The computing time required to track and digitize simulated events is substantial. Typically, the tracking and digitization rate on a D E C 3000 (model 400) with an Alpha-166 MHz processor and a single user is about 0.5 events per second. To reduce the computing time required for sufficient statistics, events with trajectories that were clearly outside the acceptance of CHAOS were not tracked. Only particles whose out-of-plane angle was within a ±15° window (Monte-Carlo window) were tracked by G E A N T . This range is about twice the out-of-plane angular coverage of CHAOS, and the difference is due to the fact that the interaction vertices of the generated events were distributed over the target, which has a finite size. Consequently simulated events with relatively large out-of-plane angles may be reconstructed. To determine safe limits for the Monte-Carlo window, limited simulations with no out-of-plane restrictions were performed, and the resulting data were analyzed. The limits used for the bulk of the simulations were Chapter 5. Monte-Carlo Simulations 75 300 250 200 OT § 150 o 100 50 0 -60 -40 -20 0 20 40 60 Out of P lane Angle (Deg.) Figure 5.29: Spectrum of the out-of-plane angles for Monte-Carlo events, which the simu-lation predicted would be detected with CHAOS. The solid lines denote the out-of-plane Monte-Carlo window. The fraction of events within ±7° is 0.95. then chosen by examining the spectrum of the out-of-plane angles for those events that were successfully reconstructed (see Figure 5.29). Here the vertical solid lines represent the out-of-plane angular window, and over 99% of all the counts in this histogram are included within the limits shown. Those tracks which do not fall in this window will never be detected by CHAOS, and hence tracking such events is futile. At all incident pion energies, each simulated event (regardless of whether it was tracked or not) was placed in the appropriate bin of a 10 x 10 x 10 lattice, G, of m ^ , t and cosO. This represents the underlying distribution in An solid angle. 5.3 Analysis of G E A N T Data To determine the number of fully reconstructed events in a given bin of G, the simulated events were analyzed in exactly the same manner as the experimental data. In addition, Chapter 5. Monte-Carlo Simulations 76 a simulation of the 2LT was included in the analysis software. This used the digitized (simulated) PCOS data to make decisions based on the momentum sum of the final state particles and the distance of closest approach to the center of CHAOS. In the course of the experiment, a small number of the nearly 4500 wire chamber pre-amplifier channels stopped functioning, and time constraints did not allow for the immediate replacement of these units. These "dead" channels were accounted for by deactivating the wire numbers corresponding to the malfunctioning channels during the analysis of the Monte-Carlo data. In addition, chamber hits in the regions of WC3 and WC4 that were deadened at the entrance and exit of the beam were excluded from the analysis. The deactivated channels account for those events which were either rejected by the 2LT or not reconstructed due to missing chamber hits. Once the Monte-Carlo data were analyzed, the reconstructed events were binned in a 10 x 10 x 10 lattice of m 2 ^ , t and cos6, denoted by R. As a test of the Monte-Carlo simulations, the missing mass spectra for the simulated and experimental data are shown in Figure 5.30. The excellent agreement between the mean and width of the two distributions is an indication that the simulation of the detector has been successful. Once the G E A N T data were analyzed, the acceptance correction factor, A, was de-termined from the ratio of the reconstructed and generated events in each bin: At 2 ± n\ R(ml^,t,COS9) tr-rA\ Assuming Poisson statistics, the uncertainty in the weighting factor corresponding to a given bin is the square root of the number of reconstructed events in that bin 1 . For each pion incident energy and polarity, the total number of events in R was on the order of 105; depending on the incident energy, this is about one to two orders of magnitude 'The statistical uncertainty in G is negligible compared to that in R Chapter 5. Monte-Carlo Simulations 77 - + 7 T p->7\ 7 T n at ^=284 MeV 3000 | — i — 1 — 1 — 1 — i — 1 — ' — 1 — 1 — i — 1 — 1 — 1 — 1 — i — 1 — • — 1 — r ~ 2500 h 2000 173 1500 o o 1000 500 1400 1200 1000 w 800 -+J fl o 600 o 400 200 0 E x p e r i m e n t a l D a t a /j, = 947 MeV a - 4.8 MeV H 1 h H 1 1 h t - / I | V Monte—Carlo /j, = 946.1 MeV a = 4.6 MeV fl i i i i i i i i i_ 800 850 900 950 Missing Mass (MeV) 1000 Figure 5.30: Missing mass distributions for the Monte-Carlo (bottom) and experimental data (top). Chapter 5. Monte-Carlo Simulations 78 greater than the number of experimental events. Figure 5.31 shows typical spectra for m^, t and cos8 in the (ir+n~) channel. The plots labeled CHAOS were obtained from the analysis of Monte-Carlo data and thus include the limitations of the detector geometry. Those labeled 47r represent the corre-sponding distributions in 4n, as determined from three-body phase-space. Figure 5.32 shows similar distributions for the (ir+ ir+) channel. Both figures indicate that CHAOS covers a wide range in m 2 ^,* and cosd, but some features of these distributions require fur-ther explanation. Evidently, the accepted cos9 distributions for the (TT+ 7T~) and (TT+ 7r+) channels are significantly different. This is to be expected, since for the (n+ 7r + ) case, 9 is the angle (in the dipion rest frame) between the incident 7r + and one of the outgoing positive pions (chosen at random), whereas in the (7r+ IT~) channel, 9 is the angle between the incident pion and the outgoing n~. Hence the distribution in the (7r+7r+) channel should be symmetric about zero, but this is not true of the (ir+n~) case. Comparing the right and left plots in Figures 5.31 and 5.32 reveals that there is approximately two orders of magnitude difference on the vertical axes. This is due to the limited solid angle coverage of CHAOS. Recall that the detector covers about 10% of 47r sr. Thus the probability of detecting two charged particles is ~0.01. A closer examination of these plots also shows that there is a significant "dip" in the distributions of and cos9. This feature can also be attributed to the geometrical acceptance of the detector by means of the following study. The F O W L phase-space generation program was used to generate (7r+ 7r+) events at 264 MeV. The plots labeled A in Figure 5.33 show the distributions for and cos9 with no limitations on the in-plane or out-of-plane angles of the two final state pions. This is simply the phase-space distribution of these variables in 47r sr. Now, if the out-of-plane angles of the two pions are forced to be within ± 7°, the resulting distributions are those labeled B. It is clear that the "dip" in the cos9 distribution is well reproduced, but the distribution still looks Chapter 5. Monte-Carlo Simulations 79 T f f=264 MeV - + 7T p ^ T T 7T n ( ^ ) 4.5 5.0 5.5 6.0 6.5 7.0 ( CHAOS ) 4.0 4.5 5.0 5.5 6.0 6.5 7.0 4.2 '4.0 w +> a 51 3.8 o 3.61 -1.0 -0.6 -0.2 0.2 0.6 1.0 -1.0 -0.6 -0.2 0.2 0.6 1.0 Cos9 C o s d Figure 5.31: Typical phase space distributions of m ^ , t and cos9 in ATT (left) and in the CHAOS acceptance (right) for the (TT+ 7r~) channel at 264 MeV. Chapter 5. Monte-Carlo Simulations 80 T>264 MeV + + + 7T p->7T 7T 11 (CHAOS) 4.0 4.5 5.0 5.5 6.0 6.5 7.0 4.0 4.5 5.0 5.5 6.0 6.5 7.0 3.5 3 o .—' 2.5 in +J 2 fl 0 0 1.5 U 1 0.5 -1.0 -0.6 -0.2 0.2 0.6 1.0 Cos0 -1.0 -0.6 -0.2 0.2 0.6 1.0 Cos0 Figure 5.32: Sample phase space distributions for m ^ , t and cosd in An (left) and in CHAOS (right). Chapter 5. Monte-Carlo Simulations 81 R o 50000 40000 30000 b 20000 — 1 10000 h 25000 r w -P tf o u DO 0 o u 400 I 1 1 1 1 I (C) _ l • _ . I i i i . L_ 1.0 -0.5 0.0 0.5 cost? 1.0 Figure 5.33: Distribution of (TI^TT) events generated with F O W L at 7^=264 MeV. Plots labeled (A) represent the distributions in Air sr. The out-of-plane angle of the events in histograms denoted by (B) lie within ± 7°. Additional in-plane angular restrictions (accounting for the missing C F T blocks) were placed on the events appearing in the distributions labeled (C). Chapter 5. Monte-Carlo Simulations 82 significantly different than those shown in Figures 5.31 and 5.32. This is because there are no restrictions on the in-plane angles of the scattered pions. Recall that the C F T blocks located at the entrance and exit of the beam were removed, and hence pions scattered into these regions will not be detected. Implementing this in-plane restriction as well results in the distributions labeled C in Figure 5.33. The above process does not include tracking or digitization; it is only meant to qualitatively associate gross features of the m2n, t, and cosO distributions with specific geometrical constraints. To ensure that the in-plane momentum reconstruction was satisfactory, the values of m2^, t and cosO obtained from the reconstructed Monte-Carlo data were compared to those determined from the three dimensional information provided by the simula-tion code. Figure 5.34 shows a scatter plot of the binning parameters computed in 3-dimensions versus that obtained from the 2-dimensional reconstruction process. The average spread in all three parameters is comparable to the bin sizes chosen. It is possible to use the results of the Monte-Carlo simulations to make estimates of the reconstruction efficiency and the average survival probability of a (TT^TT) pion. To determine the former, G E A N T data were analyzed and only those events in which none of the scattered pions decayed were selected. The ratio of the number of events to those that were successfully reconstructed provides the lower limit on the reconstruction efficiency. This is a lower limit because the track sorting algorithm requires the presence of hits in all four chambers. However if a track passes through the deadened regions of WC3 or WC4, the event will not be reconstructed. This contributes to the reconstruction inefficiency. At 284 MeV incident energy, this lower limit for the reconstruction efficiency was determined to be ~0.94 per track. A similar analysis was employed to determine the average survival probability for final state (n, 2n) pions. At an incident energy of 284 MeV, the average survival probability was determined to be ~ 0.80. It should be noted that the reconstruction efficiency and the average survival probability determined Chapter 5. Monte-Carlo Simulations 83 Figure 5.34: Scatter plots of m ^ , t and cos9 in 3-dimensions versus those obtained from the reconstruction process, which ignores the third (z) dimension. Chapter 5. Monte-Carlo Simulations 84 from the above procedure are not independent of the underlying distribution of events. The trajectory and decay probability of a final state pion (inside CHAOS) depend on its momentum and scattering angle. The efficiencies provided are meant to give the reader a sense of the extent to which different factors contribute to the detector acceptance. 5.4 Remarks on Phase-Space Extrapolation The main criticism of the Ornicron data [16] has been that their extrapolation to regions of phase space not covered by their detector assumed that all kinematic parameters, except the dipion invariant mass, were distributed according to phase space. The same is not true of the CHAOS data. The CHAOS detector has extensive coverage in m2^, t and cos6; as such no assumptions are made regarding the distribution of these parameters. Since the out-of-plane acceptance is limited, only x is assumed to be distributed according to phase space. The phase space distribution for x is shown in Figure 5.35. Each of the other three parameters were explicitly treated, and hence the acceptance determination process does not assume a constant matrix element (phase-space distribution) for irN —> TTTTN. Due to the lack of previously measured differential cross sections in this energy re-gion, it is impossible to state with 100% certainty that x is distributed according to phase-space. However, some existing experimental results may be used to support this assumption. The out-of-plane behavior of the n~p —> -K^-n'n reaction was studied in Reference [19]. They observe increasing deviations from phase space as the bombarding energy is raised. For our incident pion energies, these deviations are small, and range from only ~2% at 223 MeV to - 15% at 305 MeV. An independent check of the acceptance corrections was made by comparing the corrected mL. distribution at 305 MeV to that measured in the 47r sr bubble chamber Chapter 5. Monte-Carlo Simulations 85 400 i i i i i i i ' ' i i i i i i i i i i i i i i i i i i i i i i ' i i ' 300 o u 200 100 0 20 40 60 80 100 120 140 160 180 x (deg . ) Figure 5.35: The phase space distribution for x. 500 TT p ^ T T 7r n, Tw = 305 MeV i 1 1 1 1 i 1 1 1 1 i Jones et al. (Normalized) m e (^ 2) Figure 5.36: The acceptance-corrected m%n distribution measured in this experiment for the ( 7 r + 7T~) channel at 305 MeV is represented by the solid points. The line shows the results of Jones at al. [13], obtained at 300 MeV in a An sr bubble chamber experiment. The Jones data were published in arbitrary units. Thus they have been normalized (multiplied by a constant) to reflect the units of the data from this experiment. Chapter 5. Monte-Carlo Simulations 86 experiment of Jones et al. [13]. This is shown in Figure 5.36. There is excellent agreement between the shapes of the two distributions, lending further credibility to the assumption that x behaves like phase space. Unfortunately, there are no such previous studies of the (7r + 7r + ) channel. As a final check on the acceptance determination process, total cross sections were calculated by integrating the measured angular distributions. It will be shown in Chap-ter 6 that the CHAOS total cross sections for both the (7r + 7r~) and (TI+ TT+) channels are in good agreement with previously published total cross section data acquired using 4-7T sr detectors. Chapter 6 Cross Sections Once the acceptance corrections were applied, the appropriate efficiency factors were determined in order to obtain the differential cross sections. Later, these constants were checked by measuring absolute differential cross sections for np elastic scattering and comparing them to the partial wave analysis predictions. The aim was to produce many-fold differential cross sections with systematic uncertainties < 10%. The triple differential cross section is given by = N'(ml„,t,cos6) dm%„dtdcos6 Ni Ntgt e e ^ A m ^ A t AcosO' { ' ; where N' denotes the acceptance-corrected event lattice, Ni is the number of incident pions on the target, and Ntgt is the number of target nuclei per unit area. In addition, e denotes the combined wire chamber and data acquisition efficiencies, and e^d is the particle identification efficiency. 6.1 Acqu i s i t i on L ive T i m e While the data acquisition computer is recording the readout information or the second level trigger is making its decision, the "BUSY" circuit (see Figure 2.11) prevents the processing of other events. Interactions that occur during this period (dead-time) were not recorded, and hence the differential cross sections had to be corrected for the resulting acquisition efficiency. This quantity may be computed directly from the scalers recorded 87 Chapter 6. Cross Sections 88 during the experiment and is given by ec = (6.56) where Nt is the number of events accepted by the first level trigger, and Npass represents those events which were allowed to pass through the B U S Y circuit to the second level trigger. The first level trigger live-time is 100% because it is capable of operating at more than 6 MHz, and although typical incident beam rates varied between 0.5 to 2 MHz, the 1LT rates were typically only ~ 9 to 270 kHz. 6.2 Beam Calculations Recall that the BEAM scaler (defined by the coincidence between Si and S 2) was used to count the number of incident particles. However due to several factors, this quantity by no means provided the number of incident pions on the target. First, the incident beam had a small but non-zero contamination of muons and electrons, which originate primarily from pion decay at the production target. In addition, a fraction of incident particles decay between the beam counting scintillators and the target, and some of the incident particles physically miss the target. As such, the number of incident pions on the target is given by Ni = BEAM *U*(1- fd) * ftgt * fs, (6.57) where ftgt is the fraction of the incident beam hitting the target, fd is the fraction of incident pions which decayed but produced hits in the beam-counting scintillators, /„• is the pion fraction in the beam, and fs is the probability of having a single pion in a beam burst. In 1992, a detailed study of the composition of the M i l pion beam at various incident Chapter 6. Cross Sections 89 energies was performed by Pavan et al. [38]. In a phase-restricted beam tune, they mea-sured the time of flight of particles from the T l production target to plastic scintillators located at the exit of the beam from the channel. This allowed them to measure the IT, p, and e fraction of the beam for both positive and negative incident pions at various incident energies. At the energies studied in this experiment, the percentage of pions in the 7r + beam was 99.5% and it varied by less than 0.25% over the five energies studied in this work. For the TT~ beam, /„. was between 0.97 to 0.98. The Pavan results are in agreement with another measurement performed by Smith et al. four years earlier [39]. The particle time-of-flight spectra recorded for the current experiment were not useful in pion fraction determination. This is because they were improperly timed relative to the S\ counter. Hence pion fractions from Reference [38] were used, and the associated uncertainty in /„. was estimated to be ±0.01. This is reasonable because at most there is a 0.01 change in the pion fraction over the incident energies studied here. In addi-tion, the overall normalization of the (IT, 2TT) cross sections was confirmed by computing differential cross sections for ir^p elastic scattering. Next, the fraction of the beam incident upon the target was determined. This was accomplished by examining the spectra for B E A M - S A M P L E events of the projection of the incident beam on the target using the incoming beam hits in W C l and WC2. Both the in-plane (rproj) and vertical (zproj) projections were considered in this calculation. Figure 6.37 shows the SAMPLE-gated spectra for the in-plane and vertical projections of the beam on a plane located at the center of CHAOS and oriented perpendicular to the tangent of the incident beam at the origin. In these plots, the solid lines represent the dimensions of the target vessel. The ratio of the number of events in the target region over the total number of counts in each histogram yields the fraction of the incident beam on the target. The overall fraction of the beam impinging upon the target is simply the product of the fractions in the horizontal and vertical directions. This product ranged Chapter 6. Cross Sections 90 rr p -> 7T 7T n a t = 3 0 0 M e V 8 0 0 I ' ' 1 > i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 I - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0 in—plane p r o j e c t i o n ( m m ) 3 5 0 I i i i i i i M i | i i i i | i i i i | i i i i | i i i i | i i i i | i i i i - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0 V e r t i c a l p r o j e c t i o n (mm) Figure 6.37: Sample-gated spectra for in-plane (top) and vertical (bottom) projections of the incident beam. The solid lines represent the dimensions of the target vessel. From the distributions shown, the fraction of beam on the target was determined to be 0.90 (0.948 in-plane and 0.95 vertical). Chapter 6. Cross Sections 91 from 0.8 to 0.9 for both the (ir+ 7r + ) and ( 7 r + 7 r _ ) data. For the data represented in Figure 6.37, the overall fraction of beam on the target is 0.9. The variation of ftgt is not surprising considering that the size and location of the beam spot on the target did not remain precisely fixed throughout the two running periods. In addition, the dimensions of the beam "spot" were comparable to that of the target. The error in the fraction of beam on target was estimated by examining the variation in this quantity for consecutive runs that were acquired with identical M i l channel settings. Using this technique the uncertainty in the fraction of beam on the target is estimated to be ±0.02. The decay fraction, fd, was determined from the Monte-Carlo studies discussed in Chapter 5. Recall that the second beam counting scintillator, S2, was 10 cm high and 4 cm wide. It was located ~ 90 cm upstream of the target and was separated from Si by approximately 150 cm. Si was 10 cm wide and 20 cm high, and it was placed ~ 5 cm downstream of the beam exit (about 240 cm upstream of the target). The simulated incident pion tracks were traced from the exit of the beam pipe to the target. Next, the number of incident tracks (initially all pions) which activated both Si and S2 (as a 7T or a p) but decayed to a muon somewhere between Si and the target were recorded. The decay fraction ranged from 0.036 to 0.045 depending on the incident pion energy (see Table 6.1). This fraction is entered as an efficiency in Equation 6.57 because in practice pions and muons can not be distinguished in the beam counting scintillators. The estimated uncertainty in the decay fraction is ±0.5%. The error estimate reflects the variation of the decay correction obtained from different simulations for the same incident energy pions. It is possible for two pions within the same beam bucket to reach the target at the same time, and this probability increases with the incident beam rate. Here "the same time" means within one ~ 5 ns R F bucket, in which case only a single count is registered by the Si • S2 coincidence regardless of how many pions were actually present. The Chapter 6. Cross Sections 92 Incident Energy (MeV) Decay Fraction (%) ±0 .5% 223 45 243 4.4 264 4.0 284 3.9 305 3.6 Table 6.1: The pion decay fraction for different incident pion energies. probability of both pions interacting in the target is negligible. However if one of them interacts, the event will not be recorded because the second pion will almost always produce a hit in the veto counter. Consequently, the measured yield was corrected to correspond to those beam bursts in which only a single pion was present. The singles fraction, fs, is thus Number of beam buckets with a single pion . (6.58) Total number of BEAM counts Ignoring decay, the total rate of pions exiting the beam pipe, R, is R = u(P(l) + 2P{2) + ...), (6.59) where P(n) is the probability of having n pions in a single beam burst, and v is the cyclotron R F frequency, which is fixed at 23.058 MHz. The latter represents the rate at which the proton beam bursts arrive at the T l production target. Assuming that the distribution for the number of pions, n, in a single R F bucket is described by Poisson statistics, P(n) is given by e~x\n P{n) = (6-6°) where A is the average number of pions per beam burst. In the case where all of the beam is intercepted by the in-beam scintillators and impinges upon the target, A is simply Chapter 6. Cross Sections 93 where Rm is the measured beam rate. The expression for fs may then be written as P ( l ) _ Ae~A 1 - P(0) ~ T ^ e -In practice only a fraction of the total beam is incident upon both the in-beam counters and the target. Thus Equation 6.62 must be modified to reflect this. Letting (5 denote the fraction of the total beam which hits both the in-beam counters and the target, fs may be written as ^ = i _ e - / ? A - ( 6 - 6 3 ) A derivation of the above equation along with a detailed discussion is presented in Ref-erence [38]. Due to its large size, it is reasonable to assume that the veto counter intercepted nearly all of the incident beam. Rate studies conducted during the experiment compared the veto counter rate to that of the BEAM scaler. These showed that the fraction of the incident beam intercepted by the in-beam counters was nearly 100%, and since the fraction of the beam on the target varied from 0.8 to 0.9, /8 lies within the same range. Figure 6.38 shows the singles fraction as a function of the incident beam rate for various values of (3. It is clear that at incident beam rates below 1.5 MHz the doubles correction is small and does not vary significantly for different values of f3. The singles fraction and the fraction of the beam on the target at all incident pion energies and polarities studied in this experiment are listed in Table 6.2. As the value of /5 changed by 0.1 over the course of the experiment, it is reasonable to assume that (5 is known to within ±0.05. Based on Figure 6.38 and the error in /?, the estimated uncertainty in the singles fraction is 0.005. Chapter 6. Cross Sections 94 Beam Rate (KHz) Figure 6.38: The singles fraction as a function of the incident beam rate for selected values of (5. Incident Polarity Incident Energy (MeV) Beam rate (MHz) ftgt ± 0.02 fs ± 0.005 + 223 1.9 0.83 0.952 + 243 1.8 0.77 0.952 + 264 1.5 0.80 0.961 + 284 1.1 0.79 0.971 + 305 1.5 0.86 0.963 - 223 1.5 0.80 0.961 - 243 1.2 0.80 0.968 - 264 0.7 0.86 0.983 - 284 0.6 0.87 0.985 - 305 0.17 0.90 0.996 Table 6.2: Table showing the incident beam rate, singles fraction, and fraction of beam on the target for all energies and reaction channels studied in this work. Chapter 6. Cross Sections 95 6.3 W i r e Chamber Efficiencies In order to group chamber hits into tracks, the track sorting algorithm requires the pres-ence of hits in three of the four chambers. In this work, in order to obtain unambiguous and reliable wire chamber efficiencies, hits in all four chambers were required. Thus, the combined chamber efficiency for a single track is given by where denotes the chamber efficiency of the ith chamber. WC4 has eight layers of sense wires, and the efficiency of a single layer is ~ 0.96 [24]. Since a minimum of only three layers is required to form a track, it is reasonable to take e4 « 1. To fully reconstruct the event, the incoming beam hits are also needed. This is because together with the known beam momentum and CHAOS field setting, they define the incident beam trajectory, which is required for the determination of the scattering angle and the interaction vertex. Hence the combined chamber efficiency for fully reconstructing a reaction with two charged particles in the final state is The combined chamber efficiency is the largest source of systematic uncertainty. The chamber efficiencies for the inner three chambers were computed using Tip elastic scattering data recorded during the experiment. During these running periods the second level trigger was disabled, and two C F T blocks were required in the 1LT. The process of efficiency determination may be summarized as follows: • Proton tracks were identified in the usual way from scatter plots of pulse height in AEi versus track momentum (see Figure 4.19). • Given the incident beam momentum and the proton scattering angle, two body kinematics were used to compute the momentum for a Tip recoil proton at that 616263^4) (6.64) (6.65) Chapter 6. Cross Sections 96 angle. If the selected proton momentum was consistent with irp kinematics, the event was selected for further analysis; otherwise another event with a proton track was selected. • Given the proton momentum and scattering angle as well as the incident beam mo-mentum, the expected momentum and angle of the scattered pion were computed using the kinematics of 7rp elastic scattering. • The expected pion momentum and angle were used to solve the equations of motion describing the expected pion trajectory in the CHAOS field. This was used to generate pseudo hits in W C l , WC2, WC3, and WC4. These hits provided an estimate for the expected coordinates of the scattered pion. • If an actual hit was found within a narrow angular window around the pseudo hit in a given chamber, then that chamber was considered to have detected the pion, and a histogram of such events binned in 10° wide intervals was incremented. • For an angular bin centered about 6, the efficiency of the ith chamber is given by £<W = | | , (6-66) where and N% are the number of physical and pseudo hits in chamber % at a given angle, 9, respectively. To avoid underestimating the chamber efficiency (due to scattered pion decay), only tracks for which the actual WC4 hits were consistent with the pseudo hits were used to determine the chamber efficiencies. Figure 6.39 shows typical chamber efficiencies as a function of angle for data acquired with both polarity incident pions. The solid line denotes the mean of the data points. Chapter 6. Cross Sections 97 •tt", T f f = 2 4 0 M e V 01 1,2 11' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ' 11 1.1 1.0 0.9 0.8 0.7 0.6 1.2 1.1 1.0 £• 0.9 0.8 -0.7 : 0.6 1.2 1.1 -1.0 -0.9 t-0.8 0.7 0.6 0.5 i i i i | i i i i | i i i i | i i i i | i i i i | i i i i i i i i | i i I I | i i i i | i i i i | i i i i | i i i i * • • • • i • * • • i • • ' • 1 * * • * 1 0 60 120 180 240 300 360 6 (Deg.) t t + , T f f = 2 2 0 M e V 0 ^ 1.20 1.08 F-0.97 -0.85 h 0.73 0.62 10 01 1.2 1.1 1.0 0.9 0.8 0.7 0.6 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 1 I I I | I 1 I I | 1 T I I | I ! ! V P I I 1 1 I I I I I I I I I | I I I I | I I I I | I I I I | I I I I | I M I I I 1 I | I I I I | I I 1 I | I I I I | I I I I | I I I I 1 • ' • ' 1 ' • 1 ' 1 • • ' • 1 • ' ' • 1 0 60 120 180 240 300 360 8 (Deg.) Figure 6.39: Typical plots of wire chamber efficiency as a function of angle. The dashed lines denote the one sigma deviation and the solid line is the mean. Chapter 6. Cross Sections 98 The one sigma deviation is represented by the dashed lines. The efficiencies were averaged over all angles to determine an overall efficiency for the entire chamber. It is noted that the efficiency of WC3 during the second running period (7r + beam) was significantly (~ 10%) lower than during the first. The reason for this is not clear; however, all of the TT+ data consistently showed the same trend. One possible explanation may be a change in the chamber gas, but perhaps the most likely scenario is that the lower threshold settings on the LeCroy 2735 cards were set higher in order to reduce the electronic noise present on these amplifier/discriminator channels. The chamber efficiency for protons was determined in exactly the same manner, except here -np pion tracks were chosen, and two body kinematics were used to determine the expected trajectory of the proton. Since 7tp elastic data were used, the plots show a limited angular range. This reflects the fact that not all of the kinematically available protons could be detected in coincidence with the scattered pion. The uncertainty in the wire chamber efficiencies (for pion tracks) was estimated from the standard deviations of the efficiencies shown in Figure 6.39. Typically this was ± 0.02. As a consistency check, the efficiencies of W C l and WC2 in the incident beam re-gion were also obtained by examining the B E A M - S A M P L E events. Since these events correspond to straight-through beam trajectories, the efficiency for a single hit is given by £ , = ^ % ^ , (6.67) Aj where Nj(in, out) is the number of events for which both the incident and exit beam hits were present, Nj represents the number of S A M P L E events for which the exit beam hit was present, and j denotes the chamber in question. If a chamber was perfectly efficient, nearly all S A M P L E events would produce 2 hits in that chamber (1 in and 1 out). The only exception would be those events which decayed after passing through the incident Chapter 6. Cross Sections 99 beam region of a given chamber. The diameters of W C l and WC2 are approximately 23 and 46 cm, respectively. The survival probability of a 280 MeV pion over 46 cm is 0.98, and it increases to 0.99 for a flight path of 23 cm. The S A M P L E events provide an independent way of determining W C l and WC2 efficiencies. The wire chamber efficiencies obtained using the S A M P L E events were con-sistent with those determined by selecting np elastic tracks. For example, for the n+ data acquired at 285 MeV, the efficiency of W C l obtained using the S A M P L E and n+p elastic events was 0.93 and 0.94, respectively. The results for WC2 were 0.92 (SAMPLE) and 0.94 (n+p). In determining the cross sections, it was assumed that the wire chamber efficiency for the np and (n, 2n) pions was the same within the given uncertainty; that is, the efficiency was assumed to be independent of the pion energy. However, it is important to elaborate on this point. For example, in W C l the efficiencies for detecting np pions and protons were determined to be 0.94 and 0.98, respectively. At Tn = 280 MeV, the kinetic energy of a typical np proton (9P = 45°) is about 90 MeV, and the energy of the corresponding pion is ~ 210 MeV. For a given material, the energy lost by a 90 MeV proton (dE) in a given thickness (dx) is approximately 4 times greater than that for a 210 MeV pion. The resulting change in efficiency is 0.04. At 285 MeV, the kinetic energy of a typical (n, 27r) pion is about 51 MeV, and the dE/dx for a 51 MeV pion is about 1.5 times greater than that of a 210 MeV pion. From the above it is reasonable to interpolate the efficiency for a 51 MeV pion to be ~ 0.96. Making the conservative estimate that the uncertainty in this quantity is also 0.02, the efficiencies for typical np and (n,2n) pions agree. The above is meant to provide an estimate of the change in chamber efficiency with momentum. It is true that the energy loss and thus the wire chamber efficiency for a Chapter 6. Cross Sections 100 non-minimum ionizing particle depend on the particle's momentum. However, the deter-mination of the efficiencies for a wide range of pion momenta requires irp elastic data at many different incident pion energies. Given the time and experimental constraints, this was not practical. The crude estimate (above) indicates that the change in efficiency is not substantial, and since the incident beam pions are minimum ionizing, the incoming beam hits are not affected anyway. Overall, it is our opinion that approximating the relation between wire chamber efficiency and pion momentum (without sufficient experi-mental data) is not practical. Instead we use the efficiencies obtained from the irp elastic data, and include an extra error of 0.01 in each wire chamber efficiency (for the scattered pions only). The uncertainty due to the momentum dependence of the efficiency and that resulting from the variation of the efficiency with angle are independent quantities. In order to determine the overall error, these were added in quadrature. 6.4 Target Thickness Calculat ions To determine the number of target protons per unit area, the effective target thickness is required. Since the target vessel was cylindrical, the profile of the incident beam on the target was needed for this calculation. Consider the target cylinder as shown in Figure 6.40. Here R denotes the target radius and r is the distance from the center. Assuming the incident beam trajectory over the target region is a straight line, the target thickness d is a function of r, where Given that the distribution of the beam on the target is f(r), the effective target thickness, t, is d(r) = 2VR2 - r2. (6.68) t = (6.69) Chapter 6. Cross Sections 101 Beam direction Figure 6.40: Illustration of the effective target thickness calculation. In practice, an explicit measure of f(r) is provided by the SAMPLE-gated r_proj histogram. The number of target particles per unit area is given by = ( 6 . 7 0 ) where p is the target density in grams per cubic centimeter, NA is Avogadro's number, A is the atomic weight of molecular hydrogen (2.0159), and t is the effective target thickness in centimeters. The nominal target temperature was 18 K , and this corresponds to a liquid hydrogen target density of 0.074 g/crn? [31]. The temperature was constant to within 0.3 K; the resulting relative error in the target density is 0.4%. Assuming that the profile of the beam on target has a negligible uncertainty, the error in t is dominated by the "bowing" of the target windows. At worst, this is estimated to be about 2% of the diameter of the target cell [40]. As such, the relative uncertainty in the number of target nucleons per unit area is estimated to be 2%. Depending on the beam profile, the effective target thickness ranged from ~ 4.40 to 4.55 cm. For a typical case, where t = 4.45 ± 0.09, the number of target nucleons per Chapter 6. Cross Sections 102 unit area is (1.97±0.04) x 10 2 3. It should be noted that this parameter was computed in the same manner for both up and (ix, 2TX) reactions. Thus, the computation of absolute cross sections for np elastic scattering (see Section 6.6) provides an independent check of this as well as other normalization factors. c 6.5 P I D Efficiency The particle identification efficiency was determined from the probability, P(inT,xy), of misidentifying an actual pion-pion final state as xy. This quantity is given by where N(TTTT, xy) denotes the number of events in which the actual pion-pion final state was identified as xy. Recall that the particle type assigned to a given track is either a pion, electron or proton (note that muons were accounted for via Monte-Carlo simulations discussed in Chapter 5). Thus, the particle identification efficiency, e^, is given by tpid = 1 - (P(7T7r, ee) + P(nn, 7rp) + P(mr, ep) + P(inr, ire) + P(mr,pp)). (6.72) The number of actual pion-pion events that were identified as xy was determined by computing the missing mass spectrum for events tagged as xy. In this calculation the pion mass was used to compute the energy of the particles in the final state, regardless of their PID assignment. The misidentified (IT, 2TT) events produced a peak in the missing mass spectrum centered at the neutron mass, and the remaining events resulted in a broad background. The number of mis-tagged events was obtained by integrating the area under this peak, but above the background. Figure 6.41 shows the missing mass spectrum for the 7re events. It is important to note that this plot shows the worst case scenario since it represents data acquired at the highest (305 MeV) incident pion energy. P(nii, xy) N(ITIT, xy) (6.71) N(TTTT, 7T7r) + N(irir, xy)' / Chapter 6. Cross Sections 103 350-300-250-m 200-PI o u 150 -100-50 -o -860 910 960 Missing Mass (MeV) Figure 6.41: Missing mass plot for misidentified 7T+7T events at 7^=305 MeV In the ( 7 r + 7 r - ) channel, the biggest contribution to the PID inefficiency comes from (7r, 27r) events, which were misidentified as ire. Nearly all of the np elastic background was removed by the momentum sum cut, and it is clear from Figure 4.25 that there is no peak in the missing mass spectrum for the e+e~ events. Hence P(ir-n,ee) and P(iriT,iTp) were negligible. The PID efficiencies at various incident beam energies are listed in Table 6.3. The efficiency decreases as the incident pion energy is increased. This is because the final state pions become more energetic. Since higher energy pions also emit Cerenkov light, the probability of misidentifying a pion as an electron increases. This problem is compounded by the diminishing separation between pion and electron bands in the AEi versus momentum plots. The uncertainty in the PID efficiency reflects the error associated with the area under the missing mass peak corresponding to mis-tagged events. The increase in the uncertainty at low energies is due to lower statistics (decreasing cross section). For the (ir+ 7r + ) channel, both the pp and up events were eliminated with the Chapter 6. Cross Sections 104 Incident Energy (MeV) PID efficiency (%) E R R (%) 223 93 6 243 91 6 264 90 5 284 88 2 305 84 2 Table 6.3: Table of particle identification efficiencies for the n ) channel. momentum sum cut along with the PID information obtained from plots of AEi pulse height versus momentum. For positive polarity pions, the n+p S C X reaction is forbidden by charge conservation. Thus the e+e~ background is minimal and is mainly due to ir+n S C X taking place in the windows of the target vessel. Furthermore, since the polarity of the final state pions is the same, all negatively charged tracks were eliminated from the analysis. This removed all e+e~ events. The e + e + background is negligible since both 7 rays (from SCX) must convert in the target region, and the two positrons must be detected while both electron tracks are missed. Figure 6.42 shows a typical missing mass spectrum (computed using the pion mass for all particles in the final state ) for events identified as 7 r + e + . It is clear that any enhancement around the neutron missing mass is only barely significant. The above factors combined allow for a much improved particle identification efficiency. Using the same technique as for the (ir+ TT~) data, the PID efficiency ranged from 0.97 to 1.00. 6.6 Normal i za t ion Checks &; Elas t ic Cross Sections TT^P elastic scattering data acquired during the experiment were used to provide an independent check on the normalization of the (7r, 27r) cross sections. In the it'p channel, elastic data were acquired at 223, 243 and 264 MeV, and during the second running Chapter 6. Cross Sections 105 2 0 0 In Events tagged as Tr+e+ ISO 1 a o 100 V-50 \-0 850 900 950 Missing Mass (MeV) 1000 Figure 6.42: Missing mass plot for 7 r + e + events at 7^=280 MeV. Regardless of the PID, the pion mass was used in calculating the energy of the final state particles. period, ir+p elastic scattering events were recorded at all five incident pion energies. During the elastic runs, the second level trigger was removed from the data acquisition electronics, and two C F T blocks were required in the 1LT. Given that the particle identification efficiency, e^, for distinguishing pions from protons is 100%, the differential cross section for Tip elastic scattering at a given pion scattering angle, 6, is given by Here N(9) and dVteff(9) are the number of scattered pions and the effective solid angle for a given bin of 9, respectively. Apart from dQeff, the normalization factors that appear in the denominator of Equation 6.73 are the same as those in Equation 6.55. Thus, comparison of the experimental elastic cross sections to the partial wave analysis (PWA) [36] predictions provides an independent check on the overall normalization of the (IT, 27r) cross sections. To determine the elastic differential cross sections, irp elastic events were isolated da _ N(9) (6.73) dfl NiNtgtedileff(9Y Chapter 6. Cross Sections 106 Figure 6.43: The correlation between the scattering angles of the two tracks for np events. A l l events in which both tracks passed the interaction vertex cut are included. As expected, the non-elastic background is minimal. from background events. First, two-track events that passed the interaction vertex and lower momentum sum cuts were selected, and scatter plots of the scattering angle of the first track versus the second track were made (see Figure 6.43). Next, the elastic events were selected by placing suitable two dimensional cuts around the bands appearing in Figure 6.43. Since elastic scattering has a two-body final state, the first and second track angles for np events are perfectly and uniquely correlated. Figure 6.43 also indicates that the non-elastic background is minimal. The requirement that the first and second track angle be perfectly correlated removes any background or decay events. Note that electron-positron events are no longer a problem here. On average the np elastic cross section is many (~ 3) orders of magnitude larger than that for (n,2n). Thus the signal-to-noise ratio is improved significantly. In Chapter 6. Cross Sections 107 addition, the kinematical constraints embodied in the lower limit momentum sum and the scattering angle cuts further suppress the e+e~ events. Once the elastic events were isolated, they were histogrammed in 5° wide bins of pion scattering angle. To ensure that the efficiency for pion-proton identification equalled unity, the cross sections were computed with and without the use of the pulse height information from the AEi counters. In the case of the latter, the following strategy was used. For the n~p channel, the negatively charged particle was tagged as a pion, and in the n+p channel, the following algorithm was employed. There were two possible scenarios: either track 1 was a pion and track 2 a proton, or vice versa. To determine the correct PID, the missing energy was computed for both scenarios, and the one with the lowest missing energy was chosen (aside from energy loss, the missing energy for np events should be zero). There is a small angular region where the pion and proton tracks have the same momentum, but here the scattering angle of both particles is also the same. Hence the correct scattering angle bin is incremented. For both reaction channels the computed cross sections with and without the AEi information were consistent. This is to be expected since at the incident energies studied here, the scattered pions are minimum ionizing and thus can be easily separated from protons in plots of AEi pulse height versus momentum. The effective solid angle was computed through Monte-Carlo simulation of CHAOS. Due to factors such as pion decay and missing wire chamber channels, the effective solid angle and the geometric one are not the same. The procedure for the calculation of dQ,eff was similar to that of the (n,2n) acceptance discussed in Chapter 5. Here two-body np final states were generated and tracked through CHAOS. The digitized data were analyzed using the same reconstruction software employed to treat the experimental events. The solid angle for a 5° wide bin of pion scattering angle centered at 6^ is given Chapter 6. Cross Sections 108 b y 1 P(d„) r^+2-5 dfie// = 7 r - ) ^ ( / sinBdB, (6.74) where T(67r) and P(#tt) are the total and reconstructed number of simulated irp events in a given bin of pion scattering angle, respectively. Figures 6.44 and 6.45 illustrate the experimental absolute differential cross section for ir~p and ir+p reactions, respectively. The solid line denotes the predictions of the P W A (SAID-SM95) [36], and the dashed lines reflect a ± 10% deviation. The statistical errors are small compared to those associated with the combined chamber efficiency ewc. The uncertainty in ewc was derived from the observed variation of the chamber efficiency with angle as discussed in Section 6.3. Typically this was about 0.02 for each chamber. The error bars shown in Figures 6.44 and 6.45 include both statistical and systematic errors, but are dominated by the uncertainty in ewc. There is good agreement between the experimental cross sections and the P W A pre-dictions. From these data it is clear that the overall normalization of the (irp) cross sections is accurate to at least ± 10%. Since the pion and proton tracks were detected in coincidence, the cross sections were only measured over a limited range of pion angles. Recall that the in-plane acceptance of CHAOS is nearly 360°; thus, to ensure symme-try the cross sections were measured in both halves of the detector. These are denoted by the solid and open points. In the ir~p channel, the missing solid points around 90° are due to dead wires in WC4. The largest pion angle for the ir+p and n~p data is different, and this is due to the following. Back angle pions correspond to forward going protons; for the 7r~p data, the proton bends in the opposite direction to the outgoing pion beam, and as such it will be detected (it misses the dead regions of the drift chambers and the removed C F T block). In the case of the ir+p data, the recoil proton bends in the same 1Here a factor of -K appears instead of 2iv because pions that scattered into the two halves of the detector were placed in separate angular bins. Chapter 6. Cross Sections 109 7r"p-*7T~p at Tw=223 MeV 4 w \ 6 3 v ' \ b T J 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 \ \ \ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 — P W A : \\\ • \\\ • ' \\\ • : x\\ • 1 \\\ -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i J ^ J ^ i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4.0 3.5 TT"p->7fp at T>243 MeV u 3.0 w \ fi d 2.5 -—• 2.0 G 1.5 \ b 1.0 20 40 60 80 100 120 140 160 180 0Lab (deg.) 0.5 0.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 \\ | I I I I | I I 1 1 | 1 1 l l | 1 1 1 1 m — P W A : \\\ -i \ \ \ : : \\\ : \\\ ; r x\ * i i i i 1 i i i i 1 i i i t 1 i i i i 1 i i i i 1 i i i i 1 i i i i 1 i i t i u w \ G 3.0 2.5 2.0 1.5 1.0 7T p^TT p at Tff=264 MeV 20 40 60 80 100 120 140 160 180 0Lab (deg.) b 0.5 0.0 1^ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Ax P W A : : \ \ \ • : \ V : X V • : \ \ \ -: \ \ \ • - 4 — — - ^ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 M I . M I I I M . I . I M ' 20 40 60 80 100 120 140 160 180 0Lab (deg.) Figure 6.44: Absolute differential cross sections for 7r p elastic scattering. Chapter 6. Cross Sections 110 + + 7T p^TT p Elastic Cross sections + 7T p -» 7T p at T„=243 MeV 50 I j i i i i | i r i i | i I I I | i i I I | i i rr | T T i i ) i P W A • ' ' 20 40 60 80 100 120 140 160 180 + + 35 n 25 !» w 15 a V 10 \ 7T p -» 7T p at Tjr=284 MeV ' 1 1 1 1 . P W A i k : - \ 1 \ • V • . . . . 1 . . . . 1 , . . . 1. • * t - - — ' " . . . 1 . . . . 1 . . . . 1 . . . . 1 . . . 50 7r p -> 7T p at T,=223 MeV 30 2 20 \ £ 10 11 1 1 I 1 1 11 11 1 I 1 I i I I i | i i i i | i i i i | i i i i | i I I i P W A • \ \ \\ • \ -: \ '. . , , I l l . ... 1.... 1 .... 1.... 1 ,., .' 20 40 60 80 100 120 140 160 180 + + 40 ^ 3 5 ? 3 0 ^ 25 A 20 b 7T p -» n p at T,T=264 MeV TTTTJTTTTJTTTTTT i i i r j r r i i | r i i i | i i i i •[ T T i T — P W A j "\\\ f \ : X • X 1 • » • * * * • • ^ * * • ' 1 ' . . , i . . > . i . . , . i i i . . i . . . .-u m \ a 30 25 20 15 20 40 60 80 100 120 140 160 180 / p -» 7r+p at T)r=305 MeV a 10 \ b _ ' ' i * * * 1 1 i i . P W A : \ '•\ '-• \ '-11111111111111111 . . i — i — i — i — • 20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180 Figure 6.45: Absolute differential cross sections for n+p elastic scattering. Chapter 6. Cross Sections 111 direction as the outgoing beam, and hence it exits through the missing C F T block. This is also the reason why the pion scattering angle range varied for the ir+p cross sections. Note that the use of different spectrometer field settings can have a large impact on the extent of the angular range. 6.7 T h e Cross Sections Tables of the double and single differential cross sections measured in this experiment for both the ( 7 r + 7 r + ) and (TT+TT~) channel are listed in Appendices B and C. The two-fold differential cross sections were obtained by integrating out the the TTTT scattering angle (cosO) dependence. Due to the large number of pages required to accommodate all of the data, the triple differential cross sections are not included in this thesis. Instead, the data files are available on the T R I U M F computing cluster and may be obtained electronically by contacting the author (kermani@triumf.ca) or the head of the CHAOS collaboration (smith@triumf.ca). Figures 6.46, 6.47 and 6.48 show the measured differential cross sections for the (TT+ TT~) channel. The same distributions for the ( 7 r + 7 r + ) channel are shown in Figures 6.49, 6.50 and 6.51. The distributions shown were determined by integrating the triple differential cross sections over two of the parameters. The errors shown are statistical and were determined using the following relation: where o, 6, and c are the binning parameters, and da, db, and dc are the bin widths. In addition, 5 represents the statistical uncertainty. It is interesting to note that the shapes of the single distributions for the two channels differ significantly; however, the shapes alone do not provide great insight into the reaction mechanism. (6.75) (6.76) Chapter 6. Cross Sections 112 80 Distributions For 7T p ^7T 7T +n ^—.* 6 TJ \ b TJ 60 \ 40 20 l 1 I I J I I I r ) 1 1 r— T = 223 MeV i i—i i m^ 2 (M2) 43 150 120 ~ 90 S 60 30 0 300 TJ \ b TJ —1 1 1— I | r . I i | 243 MeV f-H \ : 4 -. 1 . . . . -1 . 6 7 ™-J (M2) b TJ 250 200 150 100 50 0 —i 1 1 1 1 1 1 1 1 1 1 1 1 1 p Tff=284 MeV T 1 1 r— 6 TJ \ b TJ -1 250 200 150 100 50 0 400 300 200 100 1 ' 1 ' 1 ' 1 ' ' 1 1 1 1 1 : Tn= 264 MeV | 1 1 1 1 >-i-> t - H h ~ " ~ H ^' ^ 1 1 1 1 1 1 1 1 al i la 1 • • • ™J ill) b TJ —i 1 1 r 1 r—i 1 1 1 1 t—i 1 y '• T =305 MeV * -\ + r-#H ^ P • i i I i i i i | i i i • I m n n 2 ( M 2 ) Figure 6.46: ra2^ distributions for 7r p —>• 7r+7r n. Chapter 6. Cross Sections 113 t Dis t r ibu t ions For 7T p^TT 7T +n 3. b 30 25 20 15 10 5 0 T =243 MeV -8 -2 t (M Z ) CM -p b 70 60 50 40 30 20 10 0 T7r=284MeV -8 -6 t (M 2) N £2 13 \ b T3 =1 -P 20 15 10 5 T„=223 MeV -8 -6 -2 t (fl) t (//) 120 100 80 60 40 b 20 0 -10 -8 -6 t (fl2) 1 1 1 1 ) 1 l 1 ' i i i . t - i - i " -: T„=305 MeV • , i , , . . i . , , . i , . i . » Figure 6.47: t distributions for n p —> TT+-K n. Chapter 6. Cross Sections 114 f T w i T w 10' = 223 MeV * T w = 264 MeV • T„ = = 243 MeV • T,, = 284 MeV - - + 7T p ^ T T 7T II 305 MeV t= 2 OS 10 o o b T3 10' - i — i — i — i — i — i — i — i — i — i — i — i — i — i — r -1.0 s - i 1 1 1 1 1 r • • • i • i i I i i i i I i i i i I -0 .6 -0.2 0.2 c o s O . a a Q • s i 0.6 i _ i i i i i L_ 1.0 7T7T Figure 6.48: cos9 distributions for 7r p —> 7r+7r n. For clarity the horizontal error bars are not shown. Chapter 6. Cross Sections 115 m 7T7T 3. \ TJ \ b TJ 5. 5. TJ \ b TJ 15 10 40 35 30 25 20 15 10 5 0 Distributions for + + + 7T p^TT 7T n . 1 1 1 1 1 1 1 1 • JL Tw= - i i > i i i i i =243 MeV 4 5 6 7 8 . . I 1 1 I ' 1 ' 1 ' ' ' ' 1 ' ' ' ' . T>284 MeV \ 1— \ + : ; ^ : : . . . , i . i • i -. . . i , , . - ^ - i . . . . 3. \ 5. £ TJ \ b TJ =1 \ rQ 3. a TJ \ b TJ 10 8 6 4 2 0 30 25 20 15 10 5 0 1 1 1 1 1 1 1 1 1 1 1 1 1 , , • . • . Tn=223 MeV ; + n -i—i—i—i—|—i—i—i—i—(—I I i—i—|—i—i—i—r-T =264 MeV l-l-H ''It Figure 6.49: distributions for n+p —> 7r + 7r + n . Chapter 6. Cross Sections 116 <N \ - P TJ \ b TJ \ 3. - P TJ \ b TJ t Distributions for + + + 7T p^TT 7T n -8 -6 t (//) -8 -6 -2 0 N \ -p TJ \ b TJ \ -Q 5 -p TJ \ b TJ 3. X TJ \ b TJ 8 -6 -4 -2 0 t (//) -10 -8 -6 -4 -2 0 t (/x2) t G O Figure 6.50: £ distributions for TT+P —• 7r + 7r + n . Chapter 6. Cross Sections 117 223 MeV i T w = 264 MeV § = 305 MeV 50 - 2 40 in o o b T3 30 -20 h 10 S T w = 243 MeV ^ T , = 284 MeV + + + 7T p ^ T T 7T n T 1 1 1 1 1 1 1 r - i 1 1 R n 1 1 1 1 1 1 1 r 0 -1.0 i I] i i i _ i i i _ • i -0.6 -0.2 0.2 c o s O 0.6 1.0 Figure 6.51: cosO distributions for ir+p —> -K+ir+n. For clarity the horizontal error bars are not shown. Chapter 6. Cross Sections 118 Reaction Channel Incident Energy Incident Momentum 5 a stat sys (MeV) (MeV/c) (pb) (lt+ 7T +) 223 335 ± 3.4 5.0 0.3 0.48 ("7T+ 7T +) 243 357 ± 3.6 14.9 0.7 1.44 ( 7 T + 7 T + ) 264 378 ± 3.8 32.7 0.9 3.13 ( 7 T + 7 T + ) 284 400 ± 4.0 49.7 1.4 4.80 (7T+ 7T +) 305 422 ± 4.2 56.8 1.4 5.44 (7T+ 7T _) 223 336 ± 2.9 35.6 2.1 3.98 ("7T+ 7T~) 243 357 ± 3.4 106.2 5.8 11.3 (7T+ 7T~) 264 377 ± 4.0 198.0 4.6 21.0 (7T+ 7T _) 284 400 ± 4.6 366.5 5.0 34.2 ("7T+ 7T~) 305 433 ± 4.7 620.0 17 58.0 Table 6.4: Table showing the measured total cross sections. The first error bar is the statistical uncertainty, and the second reflects the systematic error. As an independent check on the differential cross sections, the distributions were integrated to produce the total cross sections listed in Table 6.4. Recall that x was assumed to be distributed according to phase-space. Figures 6.52 and 6.53 show the experimental total cross sections as a function of the incident beam momentum, along with the results from previous experiments. The error bars shown on the graph (for our results) are systematic (see Section 6.8). The statistical errors are small compared to the systematic. Recall that for the (7r + TT~) data the hodoscope was used to better define the momentum of the incident beam. As such, the incident ir~ momenta listed in Table 6.4 were obtained from the mean of the beam momentum distribution, which was computed using the distribution of activated hodoscope strips corresponding to events that were identified as (7r,27r). The horizontal error bars reflect the momentum spread determined from the hodoscope distributions. The momenta listed represent the incident pion momentum at the center of the target. Since the hodoscope was not used for the 7r + data, the 1% uncertainty in the central beam momentum is reflected in the total cross Chapter 6. Cross Sections 119 section plot for the (TT+ channel. 250 300 350 4 0 0 4 5 0 500 ( MeV/c) Figure 6.52: ir+p —> 7 r + 7 r + n total cross sections from: this work (solid circles), Refer-ence [16] (open squares), Reference [17] (solid stars), and Reference [50] (solid triangles). A l l previous experimental results shown here were obtained from a comprehensive list presented in Reference [21]. The solid line is the results of an amplitude analysis per-formed by Burkhardt and Lowe [41]. In the ( 7 r + 7 r _ ) channel, the measured total cross sections are consistent with results of previous experiments. The (TT+ 7r + ) data on the other hand, disagree with the Omicron results [16] below 400 MeV/c; however, the Sevior data [17] also suggest that the Omicron cross sections are too high. The solid curve in Figure 6.52 is the predicted cross section Chapter 6. Cross Sections 120 250 300 350 4 0 0 4 5 0 5 0 0 ( MeV/c) Figure 6.53: ir~p —> Ti+TT~n total cross sections from: this work (solid circles), Ref-erence [14] (open squares), Reference [42] (solid stars), References [43, 44, 45] (open diamonds), Reference [46] (solid squares), Reference [47] (solid diamonds), Reference [48] (open circles), Reference [49] (open triangles), and Reference [13] (solid triangles). Re-sults of previous experiments were obtained from a comprehensive list presented in Ref-erence [21]. Chapter 6. Cross Sections 121 obtained from an amplitude fit for all isospin channels of the irN —> irixN reaction [41]. It indicates that the CHAOS results are consistent with the total cross sections measured by Sevior et al. [17] as well as the bulk of previously published cross sections. In the next chapter the ( 7r + 7r _ ) differential cross sections are interpreted through both the Chew-Low technique and a model-dependent analysis. 6.8 Systematic Er rors Throughout this section Equation 6.77 was used to determine the uncertainty, 5f, of an arbitrary function f(xi,x2, ...,xn) where 5xi represents the uncertainty associated with a^. The overall normalization constant required to obtain differential cross sections from the acceptance corrected event lattice is where e« is the acquisition live-time. The uncertainty associated with the live-time is negligible, and thus the relative error in K is given by (6.79) Based on the discussion presented in Section 6.2, the error in Ni is dominated by the uncertainty in fn and ftgt- Considering the ( 7r + 7r _ ) channel at 265 MeV where = 0.98 ± 0 . 0 1 and ftgt = 0.86 ±0.02, the resulting relative uncertainty in Ni is 2.5%. Recall that ewc is given by n Of (6.77) K — t-WC ^ -It tpid Ntgt Ni (6.78) •wc (6.80) Chapter 6. Cross Sections 122 where the primed quantities are used to denote the efficiency of W C l and WC2 associated with the incoming beam hits. Although the central values of the primed and non-primed quantities are the same, the additional uncertainty, introduced due to the momentum dependence, results in a slightly higher error in the chamber efficiencies associated with the scattered tracks. Using e'x = 0.96 ± 0.02, e'2 = 0.94 ± 0.02, ix = 0.97, e2 = 0.96, e3, and 8ex = 5e2 = 8ea = 0.022, the relative error in the overall chamber efficiency is 8.7%. This is by far the largest contribution to the systematic error of the data. Using Epid = 0.90±0.05 and a 2% relative error in Ntgt, the systematic error associated with all of the (IT, 2TT) cross sections at this energy is 10.6%. The systematic errors presented in Table 6.4 were determined in the above manner. Chapter 7 Interpretation The measured double differential cross sections in the (IT+ir~) channel were analyzed in two different ways. In the first model-independent approach, the Chew-Low extrapolation technique [8] and Roy equation [9] analysis were used to determine the isospin zero S-wave 7T7T scattering length, 0%. The second was a model-dependent analysis based on an isobar model, first developed by Oset et al. [10] and later extended by Sossi et al. [11, 12]. Throughout this chapter, pion mass, p, will be set to unity unless explicitly written to avoid confusion. In addition, t is the square of the four-momentum transfer to the nucleon, and m2^ is the dipion invariant mass squared. 7.1 Chew-Low Analysis The extraction of TTTT scattering observables from TTN —• inrN data is based on isolating the contribution of the one-pion exchange (OPE) mechanism (see Figure 7.54) from background processes which involve resonance and isobar exchanges. The interest in the O P E mechanism stems from the fact that the intermediate (off-shell) pion interacts with the on-shell incident particle, giving rise to pion-pion interactions. Therefore this mechanism provides an indirect means of studying %% interactions. However, separating the O P E contribution from background processes is a non-trivial task, and often the 7T7T scattering observables obtained from TTN —> TT-KN data are determined through a model-dependent analysis. However, if experimental irN itirN data are acquired in suitable kinematic regions, it is possible to obtain KIT scattering cross sections in a 123 Chapter 7. Interpretation 124 (p-q) \ " q' \ \ \ \ \ Figure 7.54: Feynman diagram of the O P E mechanism. "model-independent" manner. This approach is referred to as Chew-Low analysis. The t-dependence of the cross section for n~p —»• 7 r + 7 r _ n is given by [51] t / l " 2 + X, (7.81) where ]x is the pion mass. The first term in Equation 7.81 represents the O P E mechanism and X stands for all other processes. The denominator of the first term is simply due to the pion propagator, and the i-dependence of the numerator is not specific to OPE. Rather, it is a consequence of the pseudo scalar nature of the pion coupling at the nucleon vertex. To see this, consider the TTNN vertex in the Feynman diagram shown in Figure 7.54. The coupling of the intermediate pion at the nN vertex has the form ~u(q)j5u(p), (7.82) where u represents the nucleon spinor, and p and q are the four-momenta of the initial and final state nucleons, respectively. Taking the square of the above matrix element and summing over spins yields Chapter 7. Interpretation 125 ~ - J M „ T r ( 7 ' V ) + 4 M 2 , (7.83) ~ -Ap • q + AM2, 2t where M represents the nucleon mass. The t-dependence of the numerator is of critical importance since it indicates that the O P E contribution vanishes at t = 0. In other words, the O P E process is peripheral. From Equation 7.81, it is also clear that as t—» p2 (the pion pole) the first term diverges, and if the second term remains finite, it is possible to isolate the O P E contribution from the rest. In addition, at large values of \t\, the O P E contribution diminishes. As such, experimental TTN —» ix-nN data close to the physical limit of t ~ 0 are required, and the CHAOS data satisfy this condition. The exact prescription for determining mr scattering cross sections via extrapolation to the pion pole (t = p2) was developed by Chew and Low in their 1959 paper [8]. The 7T7T scattering cross section at a given center of mass energy, m^, is expressed in terms of the Chew-Low function, F(m^t), defined as F(m^,t) - — • QTg-r, (7-84) where f2 = 0.08 is the TTN coupling constant, a J ^ 2 is the experimental TTN —> ITIIN dou-ble differential cross section, and pv is the incident pion momentum in the laboratory frame. The function F extrapolated (in t) to the pion pole yields the on-shell TTTT scat-tering cross section. The Chew-Low method is based on the hypothesis that the O P E mechanism is domi-nant in the region of extrapolation; however, there are no reliable models which determine the t region in which O P E is dominant. As such, we employed the pseudo-peripheral-approximation (PPA) [52], which extrapolates an auxiliary function F' — F/\t\ to the pion pole. This approach is based on the fact that in the case of O P E dominance, F Chapter 7. Interpretation 126 vanishes at zero, a statement which is consistent with Equation 7.81. In the PPA, the O P E dominance also implies that F'(m™-,£) is linear in t (since F vanishes at t = 0). To select t regions consistent with the hypothesis of O P E dominance, it is sufficient to restrict the analysis to these regions. In the past the P P A and Chew-Low extrapolation techniques have been used to interpret -nN —» TTTTN data acquired at high incident mo-menta (see for example Reference [52]), but due to the lack of sufficient experimental data, Reference [53] presents the only other experimental results of Chew-Low analy-sis near threshold. However, the near threshold region (m\^ > 4) is most sensitive to the scattering lengths, and the determination of 7T7T cross sections in this region would represent an important step towards understanding pion-pion interactions. To determine the ir+ir~ scattering cross sections, the measured double-differential cross sections for 7r _p —• ix+-K~n were used to calculate F^m^^t) for each bin in m\v. Next, suitable regions in t were chosen such that F' could be described by a linear function of t, consistent with the O P E dominance hypothesis. The data points in the selected interval were fit with a straight line, and the parameters of the fit were used to determine the on-shell 7r+7r~ scattering cross section by evaluating F' at the pion pole. The most crucial part of the Chew-Low technique was the selection of t intervals consistent with the O P E dominance hypothesis. It is important to elaborate on this point. The central values of the t bin in units of inverse pion mass squared (p~l) were -0.3, -0.6, -1.1, -1.9, -2.6, -3.5 , -4.5, -5.5, -6.5, and -7.5. It should be noted that at a given incident pion energy and m ^ , the kinematically allowed region in t is limited. This is the reason for the lack of counts in certain t-bins. In the following discussion these will be referred to as "zero bins". The exact procedure for choosing the t region is summarized in the following paragraph. At each incident energy the Chew-Low function was formed for each bin in , and the zero bins were discarded. Recall that the O P E signal is expected to vanish at large Chapter 7. Interpretation 127 Kinetic Energy (MeV) ml^p2) 223 243 264 284 Weighted Average 4.15 8.5 ± 1.1 5.0 ± 1 . 1 1.5 ± 0.7 -0.1 ± 1.1 3.1 ± 0.5 4.45 - (6.7 ± 2.0) 3.5 ± 0.7 2.2 ± 0.8 3.2 ± 0.5 4.75 - 23.8± 3.0 9.3 ± 1.1 5.1 ± 1.0 8.0 ± 0.7 5.05 - (7.6 ± 4.1) (3.0 ± 1.5) 8.4 ± 1.0 6.8 ± 0.8 5.35 - - (7.5 ± 5.4) 7.5 ± 5.4 5.65 - - (5.7 ± 3.6) 5.7 ± 3.6 Table 7.5: Table of 7T7T cross sections (mb). values of \t\ as well as at t = 0. We selected the t region such that 0.6 < |t|. There is no reliable model which describes the non-OPE background. As such we chose to place the lower |t| cutoff at a bin which was filled in most of the cases. This provided a consistent method of choosing the lower \t\ limit. The points in the selected t interval were fitted to a straight line and extrapolated to the pion pole. If the reduced x2 f ° r the fit was less than 2.5, the m2^ bin was selected for further analysis. Otherwise it was discarded. It should be noted that the \ 2 for the discarded bins was much greater (typically ~ 8). We believe these m2^ bins were not consistent with the P P A and the O P E dominance hypothesis. From Equation 7.81, the O P E drops to ~ 50% of its maximum at | i | ~ 6. For the m2n bins which passed the \ 2 test, the selected t region was studied. If the removal of the point at largest and/or smallest t changed the value of the extrapolated cross section by more than the associated error, the t point(s) in question were removed. In all cases the remaining points provided a stable value of onn. Typical extrapolation plots are shown in Figure 7.55. The resulting (7T7T) cross sections are listed in Table 7.5. Entries enclosed in paren-theses correspond to those cases where point(s) were removed from the fit. Typically, the fits were performed in the region 1.0 < \t\ < 7.5 and 1.0 < \t\ < 6.0, depending on Chapter 7. Interpretation 128 25 r. (mb 20 -15 -~ i 10 : fa 5 -50 ^ (mb; 40 -30 -J , 20 -fa 10 -50 i 40 -30 ' : " i • 1111111 i o o fa 20 < (qui) 15 -10 ' : 5 -o t. Tw=223 MeV =4.15 M T =243 MeV H • I • • • • I • 11111 • • • t • • • • I • • • • I • • • • I • • • • I T „ = 2 8 4 MeV 2 A AC  2 m„„ =4.45 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 t ( / / ) 15 (qui) 10 " B 5 fa 15 (mb) 10 ~ i 5 fa 15 (mb) 10 -i-> 5 ~ i 0 fa 20 15 J , 10 J , 5 0 ' I " " ! ' " '"P" T „ = 2 4 3 MeV Z . 2 =4.15 fj. l | l l l l | l l l l | l l l l | l l l l | l l l l | l l l l | l l M | l T „ = 2 6 4 MeV mn 7 T 2=4.45 /x2 — I M n n | n n | i T =264 MeV T „ = 2 8 4 MeV 2 A -in:  2 m„„ =4.75 fx 4 - 4 -• • - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 t ( ^ 2 ) Figure 7.55: Typical Chew-Low extrapolation curves for F'(t,m^) as a function of t. The solid point at t=+l is that deduced from the extrapolation (the nn cross section). The solid points denote the experimental data used in the linear fit. The experimental points denoted by crosses were not included in the fit. Chapter 7. Interpretation 129 the incident pion energy. The errors shown in Table 7.5 are statistical only and do not reflect the systematic (~ 10%) uncertainty in the total cross sections. Although the majority of the cross section values at different projectile energies agree within the error bars, the smallest m\v bin shows a considerable spread; this may reflect the growing importance of other mechanisms (besides OPE) in this region. We have made no attempt to remove potential background contributions, as currently a reliable model for such a procedure does not exist. Instead, Section 7.2 presents the results of the Roy equation analysis with and without the lowest mvlT point. Table 7.5 reports only fits which passed the x2 test-Applying exactly the same formalism to the 305 MeV data was not successful. This was because the extrapolated cross sections for all of the bins were consistently negative. We speculate this is due to the dominance of background processes over O P E at this energy. Since a negative value for the cross section is clearly not physical, we did not pursue the application of the Chew-Low method at this energy. 7.2 Roy Equation Analysis Once the 7 r + 7 r _ cross sections were determined, dispersion constraints embodied in the Roy equations were used to check whether the extrapolated cross sections were consistent with results obtained at much higher incident momenta. The work presented here closely follows that of Patarakin et al. [54], and the Roy equation analysis was performed in collaboration with the authors of Reference [54]. Analyticity, unitarity, crossing and Bose symmetry properties of the partial scattering amplitudes for ITTT scattering were studied by Roy [9] and Basdevant et al. [55, 56]. The results of their work are a set of dispersion relations (the Roy equations), which provide a means of determining the real part of the scattering amplitude over a wide range of Chapter 7. Interpretation 130 center of mass energies ( - 4 < m 2 ^ < 60 ) given the magnitude of the amplitude in the physical domain (m^ > 4) [54]. As such, it is possible to obtain the TTTT scattering amplitude in the near threshold region given the phase shifts at higher incident momenta. Considering only S and P waves, the Roy equations for charged pions may be written as [54] Refl(s) = Af(s) + £ E f1 G}/(s,s')Imff (s')ds' + (7.85) l'=0 I'=0J4 where sfs is the center of mass energy, / is the isospin index, I is the orbital angular momentum index, and G\j! are functions of the center of mass energy. The expressions denning GJ'f are presented in Appendix D. In addition, </>((s) represents the contribution of I > 2 waves (see Appendix D), and the subtraction constants, Af, are related to the scattering lengths in the following manner: Ag(s) = a°0+S-^(2a00-5al), (7.86) Ag(s) = al-S-^(2a°0-5a20), (7.87) A}(s) , = ^ ( 2 a ° - 5 a 0 2 ) , (7.88) where a° and al are the S-wave 7T7T scattering lengths for isospin zero and two, respectively. To check the extrapolated 7r+7r~ cross sections, phase shifts determined from a partial wave analysis of data at higher incident momenta (obtained in Reference [54]) were used to determine the real and imaginary parts of the partial amplitudes by using the unitarity constraints, Ref!(s) = U-L^ain2S{{8)t (7.89) ImfUa) = 1^-^(1-co825{(s)). (7.90) Next, Equation 7.85 was employed to determine the real part of the partial amplitudes over the energy region of the CHAOS data. The 7T7T scattering lengths used for this Chapter 7. Interpretation 131 18 16 14 12 f 10 8 6 4 2 0 4.0 1 1 1 1 | 1 I 1 1 ^ I ; L 1 : 1 l—•—i 1 r-«H 1 , I , I I I -4.5 5.0 2 5.5 6.0 Figure 7.56: TT+/K~ cross sections obtained in this experiment as a function of m2^. The solid points denote the energy-averaged aww values from the Chew-Low analysis (Table 7.5), and the open points are from Reference [53]. The solid line represents the cross sections obtained in Reference [54], using the Roy equations, with only higher momenta data included as input. The dashed lines indicate the calculated errors in the Roy equation analysis. analysis were those obtained in Reference [54] (a$ = 0.225// - 1, a 2 = —O.Oip'1). Given the partial amplitudes, the inr scattering cross sections near threshold were computed using ann(mb) = 4TT (^j • (\S\2 + 3 |P | 2 ) , (7.91) where S and P are given by S = 1< + \ A 2 0 , (7.92) P = A\. Here Aj = exp(i5{)sin(8{) are the irn partial amplitudes in the elastic region. Figure 7.56 shows the CHAOS cross sections along with those calculated on the basis of phase shifts determined in Reference [54]. Clearly the bulk of the extrapolated cross sections lie well Chapter 7. Interpretation 132 Method Reference Global amplitude analysis of 7rA" —» -KTTN data Global Colour Symmetry Model Threshold miN total cross sections K E 4 decay KE4 & Roy HT^N Phase-shifts & Roy Olsson & Turner Model Schwinger ChPT (one-loop) ChPT (two-loops) Generalized ChPT Weinberg 0.205 < a°0 < .270 0.188 ± 0.016 0.197 ±0 .010 0.209 ± 0.011 0.21 ± 0.07 0.28 ± 0.05 0.26 ± 0.05 0.16 0.10 0.20 ± 0 . 0 1 0.217 0.263 0.192 [57] [58] [59] [5] [60] [61] [62] [7] [7, 63] [54] [64] [41] This work Table 7.6: A summary of predictions and experimental results for the isospin zero S-wave scattering lengths. within the band predicted from the Roy Equations, and they are in agreement with the results of Reference [53]. To determine a^, TTTT scattering cross sections in the near threshold region were com-puted (via Roy equation analysis) as a function of al at values of m2^ matching those of the current experiment. A x2 based on the difference between the extrapolated cross sections (without the lowest m2^ bin) and those computed from the Roy equations was determined as a function of OQ . We found that x2 depends only weakly on the value of al, as one would expect, since the reaction ir~p —> n+ir'n deals predominantly with the isospin 0 channel. However, x2 develops a very sharp minimum as a function of a°0, as shown in Fig. 7.57. With a20 fixed at -OMp'1, we obtain a° = 0.209 ± C O l l / i " 1 . The quoted uncertainty corresponds to the change in al which causes the total x2 to increase by unity from its minimum value. The scattering length obtained by including the lowest m2^ point is a°0 = 0.205 ± 0.014/x -1. This agrees well with the results of Reference [54], Chapter 7. Interpretation 133 Figure 7.57: Profiles of the reduced x2- The top plot represents the results obtained by excluding the lowest TO2^ bin and the bottom graph shows the profile determined by including the smallest m 2 ^ bin. Chapter 7. Interpretation 134 which quotes a central value of 0.225// - 1 and a range 0.205/x - 1 < a°Q < 0.27/i - 1 . Val-ues of a[J obtained from various experimental approaches as well as several theoretical predictions are listed in Table 7.6. The CHAOS results are in good agreement with the predictions of Chiral Perturbation Theory [59, 5]. 7.3 The (7T+7T+) Channe l In principle the Chew-Low technique along with Roy Equation analysis may also be applied to the (-7T+ 7r+) data in order to determine the isospin two S-wave scattering length. However this analysis was not possible. In the context of the P P A , F' is linear in the region consistent with the O P E dominance hypothesis, and for the ( 7 r + 7 r ~ ) data, this linearity condition was used to select a suitable t region over which the fit was performed. The existence of linear regions was taken as support for the hypothesis of the PPA. Figure 7.58 shows typical plots of the Chew-Low extrapolation function, F', for the (TT+ 7 r + ) channel. At the incident energies studied in this experiment, the absolute minimum value of \t\ attainable experimentally is slightly greater than zero (See Appendix A). This is the reason for the sharp drop in F' near \t\ = 0. The same behavior is seen in the phase-space distribution of t (see Figures 5.31 and 5.32). As the dependence of F' on t has no distinguishable linear region, it was not possible to apply the same formalism to this channel in our energy range. It is interesting to compare the pion-pion elastic scattering cross sections for the 7r + 7r + and 7r+7r - channels. The 7r+7r~ channel is predominantly isospin zero while only isospin two contributes to the 7r+7r+ channel. It is thus possible to write the threshold cross sections for the two reaction channels in terms of the S-wave isospin zero and two scattering lengths [65]. a++ = 47r(a02)2, (7.93) Chapter 7. Interpretation 135 r t = 47r(a°)2. (7.94) Here the small contribution of the isospin two S-wave scattering length in the 7T+7T channel has been neglected. From the above equations, the ratio of the cross sections is H I ) 2 <795) Using a° = 0.209^ _ 1 and a2, = —0.041/i - 1 [5], it is clear that the 7r + 7r + elastic cross section at threshold is a factor of ~ 25 smaller than that for the ir+iT~ channel. Although charge conservation forbids N* exchange in the n+p —> 7 r + 7 r + n channel, mechanisms with A intermediate states contribute to the non-OPE background in both channels. At the energies studied in this experiment, the total cross section for n~p —> 7r + 7r~n is about 10 times greater than that for n+p —> 7r + 7r + n (see Figures 6.53 and 6.52), whereas the pion-pion scattering cross section for T T + / K + is about 25 times smaller than for ir+ir~. Associating the O P E component of the total (TT, 2TT) cross section with the strength of the pion-pion interaction, and assuming that the non-OPE background in the ir+p —>• 7T+/K+n channel relative to the ir'p —> 7 r + 7 r _ n channel is suppressed by about the same order as the TTN —> 7T7rAf total cross section, it follows that O P E ( + - ) OPE(++) B K G ( + - ) BKG(++) ' 1 j where B K G is the non-OPE background for the two reaction channels. It is thus not surprising that the application of the Chew-Low formalism to the ir+ir+n data is prob-lematic, since one is trying to observe a signal which is ~ 2.5 times smaller than in the 7r~p —• n+/n~n reaction. Chapter 7. Interpretation 136 + + + 7T p->7T 7T n 10 -Q 8 L =4.25 pL 6 h 4 h 1 — 1 —r 2 T n=264 MeV \ * • L _i i i ] i i_ - 6 -4 t (p.) i 20 mW 7 I 2=4.45 /x 2 h T7 r=284 MeV 10 5 h - i — i — i — i — i (• —1 1 1 1-_l I I L_ J i i i i L - 8 - 6 -2 t ( / / ) Figure 7.58: Typical plots of the Chew-Low function, F', for the (7r + TT+) channel. Chapter 7. Interpretation 137 7.4 Remarks As previously mentioned, the Chew-Low analysis procedure is based on the assumption that the O P E mechanism is dominant in the region of extrapolation. To date, the mech-anism of the TTN —>• TITIN reaction is not well understood, and knowledge of background processes is limited. As such it is impossible to prove that the non-OPE backgrounds do not contribute to the final TTTT cross section. The linearity of F' does not guarantee the absence of non-OPE signals. This is a valid criticism of this approach and in fact applies to all TTN —» irnN data. It is however reassuring that the results of the current Chew-Low analysis are consistent with the phase shifts obtained from the high incident energy data. This work is not meant to prove the validity of the Chew-Low technique as a tool for studying pion-pion interactions. It merely provides an approach for the interpretation of the experimental TTN —>• TTTTN data, and suggests that methods previously used to treat the high energy (IT, 2-K) data may also be employed in the threshold region. The experimental results presented in this thesis represent the most extensive measurements of differential cross sections and angular distributions of the TTN —>• ITIYN reaction in the threshold region. These data should provide our theoretical colleagues with useful infor-mation, and perhaps improve the current understanding of the processes and mechanisms involved in pion-induced pion production. 7.5 Ano the r Consistency Check Given the 7r+7r~ elastic cross sections as a function of the center of mass energy, m^, it is possible to extract the isospin zero S-wave phase shift and scattering length using a very simple formalism. This method relies on some reasonable approximations regarding the behavior of the ixix phase shifts near threshold. The results presented here are not meant to replace those obtained from the more rigorous Roy equation analysis. The goal Chapter 7. Interpretation 138 is simply to provide another consistency check on the extrapolated -K-K cross sections. From Equations 7.91 and 7.92, the TT+-K~ total cross section may be written as where s is the square of the center of mass energy of the two pions [m2^), k is the wave number, and 8f represents the appropriate phase shift (I and I denote the isospin and angular momentum indices, respectively). Reference [54] presents a series of fits performed to the previously published experimental phase shifts (S and P waves of isospin zero and two) over a wide range of center of mass energies. These results indicate that at the center of mass energies studied in this work, the isospin one P-wave phase shift is nearly zero (8\ < 1°) and cos(8° — 8$) « 1 [54]. Hence, Equation 7.97 may be approximated by From Equation 7.98, it is possible to compute the isospin zero S-wave phase shifts, given the isospin two S-wave phase shifts and the 7r+7r~ total cross sections. The extrapolated 7r +7r~ cross sections from this work, along with the appropriate values of 8% (from Ref-erence [54]) were used to determine 8® as a function of s. The phase shifts provide yet another consistency check on the -K-K cross sections presented in this work. The results of this analysis are shown in Table 7.7. The isospin zero S-wave phase shifts from the CHAOS data are in agreement with the results of Reference [54], which were obtained from phase shift analysis of high energy TT-KN data, although our results are consistently higher by about 1 standard deviation. It is now possible to utilize the calculated phase shifts to determine the isospin zero S-wave scattering length. For the range of s presented in this work, the relation between cw(s) (7.97) 47T 9fc2 Asin2(8°Q) + sin2(8l) + Asin(8°Q)sin(8l) (7.98) Chapter 7. Interpretation 139 S2 from Reference [54] (deg) 8% from Reference [54] (deg) 8$ from this work (deg) 4.15 -1.1 ± 0.3 1.0 ± 1.5 2.4 ± 0.2 4.45 -1.5 ± 0.3 3.1 ± 1.5 4.0 ± 0.3 4.75 -2.0 ± 0.3 5.2 ± 1.5 7.5 ± 0.3 5.05 -2.4 ± 0.4 6.2 ± 1.5 8.4 ± 0.5 5.35 -1.1 ± 0.5 7.2 ± 1.5 10.0± 3.0 5.65 -1.1 ± 0.5 8.2 ± 1.5 9.8 ± 2.5 Table 7.7: Table of 7r+7r elastic phase shifts, the scattering length and the phase shifts may be written as [7] A 2 2 sin(26) = 2( ^ )1/2(a°Qp + 6^-), (7.99) S jJL where 8 — 8° — 8{, q is the center of mass momentum, p is the pion mass, and b is defined as [7]: b = b°0-a\, (7.100) where 6° is the S-wave slope parameter, and a\ is the P-wave scattering length. Assuming 5\ ~ 0 for the energy range considered here, it is possible to compute O,Q and b by fitting the phase shifts listed in Table 7.7. We obtain a°0 = 0 .25±0 .05^- 1 and b = O ^ O i O . l S ^ r 1 . The value of a[] is consistent with the results of the Roy equation analysis. In addition, b agrees well with the value obtained from the analysis of Ke4 data [7], where b is reported to be 0.11 ± 0 . 1 6 J U - 1 . The above analysis raises a very interesting point which is worth noting. In the near threshold region, the 7r + 7r + elastic scattering is purely isospin 2. It is thus possible to extract 8% directly from the experimental total cross sections. Furthermore, if the 7T+TT+ and 7 r + 7 r - cross sections are combined, both 8Q and 8$ may be obtained from a given experimental data set. Chapter 7. Interpretation 140 7.6 M o d e l Dependent Analys is From Bose symmetry, at threshold the final state pions (from all channels of nN —>• TT-KN) must have isospin 0 or 2. Thus, there are only two independent threshold amplitudes, denoted by A2IiI^. Here / and J w represent the isospin of the TXN and TTTT systems, respectively. For the experimentally accessible channels the isospin decomposition of the TIN 7nrN amplitudes at threshold are [66] 7r + 7r + n) = 7T +7T 0p) = A(ir~p - > 7r°7r u n) = A(-K~p —> TT~7T°p) = A(-K+p A(TT+P A(ir~p _2_ 7E A32, •1 I A 3 2 , A ^ A 2 A - l A* (7.101) (7.102) (7.103) (7.104) (7.105) x/iO ^ Based on the effective Lagrangian formulation [67, 68, 69] of P C A C and current algebra, Olsson and Turner (OT) developed a model which enabled the extraction of nn scattering lengths from the near threshold (TN < 200 MeV) itN —> mrN total cross sections [18]. In this model, the threshold amplitudes for the (n+n+) and the TT~) channel were parameterized in terms of the chiral symmetry breaking parameter £, which in turn was related to the S-wave TT-K scattering lengths. The OT model pre-dates Q C D , and with £ = 0, it is equivalent to the tree level (lowest order) Chiral Perturbation Theory calculation. The explicit form of the OT relations with £ = 0 are [66] Re{A32) = -2Vl0n 9TXN M * + efc Re(A10) = 4TT 9-KN M On + dD (7.106) (7.107) Chapter 7. Interpretation 141 \ 1 C o n t a c t P o l e \ / ' \ / / \ / / \ / / \ / \ / H i g h e r O r d e r Figure 7.59: Feynman diagrams for processes included in the OT model. The next to leading order diagrams (bottom) contribute to the constants di. where gVN is the pion-nucleon coupling constant, and M is the nucleon mass. The con-stants di are of order p and represent the contributions of next to leading order diagrams (see for example Figure 7.59). The magnitude of the corrections di is significant; they are approximately 40% and 50% of the leading terms for A 1 0 and A 3 2 , respectively [62]. The above relations are based on the dominance of the pole and contact terms (see Figure 7.59) in the threshold region. In the past, the OT model has been used to ob-tain 7T7T scattering lengths from the near-threshold irN —• TTTTN data (see for example Reference [17]). In the current work two important factors make the use of the OT formalism im-practical. First, it has been suggested that in order to obtain reliable scattering lengths, better estimates of the constants di were required [70]. In addition, regardless of the validity of the OT model as a whole, the application of this formalism to the energy Chapter 7. Interpretation 142 region studied in the current experiment is invalid. The reason for this is as follows. To determine the partial amplitudes, the -K-KN amplitude at threshold is needed. The modulus of the threshold -K-KN amplitude is determined via the extrapolation of the experimental total cross sections to threshold. The exact relation between the modulus of the ITTTN amplitude and the experimental total cross sections is given by [62] where T^, Tth, and a are the laboratory incident energy, the -inrN threshold energy, and the TT-KN total cross section, respectively. The constant S is a symmetry factor (S = 1/2 if the two final state pions are identical), and C is given by [62] 3(2 + a)(2 + 3a) 1 2 8 T T 2 ^ ( l + 2a) n ' a = £ , (7.110) where M is the nucleon mass. Since the determination of \A\ requires extrapolation of the experimental total cross section to threshold; the use of higher energy (T^ > 200 MeV) data may lead to unreliable values of the threshold amplitude. In addition, the energy dependence of the amplitude is not known. In 1996, Bernard, Kaiser and Meissner [62] provided the chiral expansion of the irirN threshold amplitude to second order in the pion mass. In the framework of Heavy Baryon Chiral Perturbation Theory, they provide relations between the threshold amplitude and the S-wave TTTI scattering lengths, and a2,. Their work may be thought of as an improve-ment on the OT results, and requires the experimental miN amplitudes at threshold. For the reasons given above, this formalism may not be used to treat the experimental results presented in this work. The model of the n~p —• ii+ii~n reaction developed by Oset and Vicente-Vacas (OV) added diagrams involving nucleon, A , and iV*(1440) intermediate states to the OT Chapter 7. Interpretation 143 model [10]. These additional mechanisms may be summarized as • TITTNN coupling through P-wave p-exchange, with nucleon and A intermediate states (Figure 7.60). • Three-point diagrams with nucleon and A intermediate states (Figure 7.61). • Two point diagrams with N* intermediate states and NN*TTIT S-wave coupling through e (an isoscalar resonance state also referred to as the a meson) exchange (Figure 7.62). OV assumed that the Lagrangian density for NN*TTTT was given by £yvw,7r7r = —C$N*(t> • <^ 0JV + Hermitian Conjugate (7.111) where ipN*, 4> and ipN are the N*, pion and nucleon fields, respectively [10]. In addition C is a parameter of the model. In their original work O V determined C by estimating the fraction of the width for N* —> nnN which goes into the Ne channel [10]. They estimated C to be 0.91 ± 0.20// - 1. Subsequently, Sossi et al. [12, 71] added P-wave NN*^ coupling (Figure 7.63) and replaced the TTTT scattering amplitudes in the original OV model with those obtained by Donoghue et al. to one-loop order in Chiral Perturbation Theory [72]. The TTTT amplitudes in the original OV model were based on the Weinberg 7T7T Lagrangian, which included the chiral symmetry breaking parameter1 £. The Donoghue amplitudes involve two parameters, d\ and d2, which are renormalization constants that must be determined from experimental data. In the framework of chiral perturbation theory, £ is zero. Chapter 7. Interpretation 144 Figure 7.60: Diagrams for TITINN P-wave coupling through p exchange with intermediate A and nucleon states. For simplicity, not all possible permutations are shown. Chapter 7. Interpretation 145 \ \ / \ / / / / \ / \ \ A / / \ \ / \ / / / \ \ / \ A A / / / / Figure 7.61: Three point Feynman diagrams with nucleon and A intermediate states. Not all permutations are shown. Chapter 7. Interpretation 146 Figure 7.62: Feynman diagrams for N* -» N(inr)S-wave. Not all permutations are shown. Chapter 7. Interpretation 147 Figure 7.63: Additional Feynman diagrams added by Sossi et al. to describe the TV* —> N(inr)p_wave mechanism. Not all permutations are shown. Chapter 7. Interpretation 148 Reviewing the relevant literature brings up several points which are worth noting: • In the original model, the strength of the N* —> N(-KIT)s-wave process was not accurately determined [12]. A more accurate determination of this quantity requires good knowledge of C and the N*Nir coupling constant, gN*Nn-• The addition of the N* —> N(7nr)p-wave mechanism requires the determination of the N*Air coupling constant, <?JV*ATT-In this work, the extended OV model [12, 71] was used to compute the m2^ distri-butions for the ( 7 r + 7 r ~ ) channel at all five incident energies. The C E R N minimization package, MINUIT, was employed to perform a global fit (at all incident energies simulta-neously) of the experimental mln distributions by varying &i, d2, C, gN'Nn, and pjvAw Initial studies indicated that the x2 °f the global fit did not show a strong dependence on d2 or g^Ntx- Consequently, d2 was fixed at the value given in Reference [72] (0.013), and g^'NiT was set to 0.02/i - 1 [10, 12]. The results of the fit are listed in Table 7.8, and the fits to the u?^ distributions and the predicted total cross sections along with their experimental counterparts are shown in Figures 7.64 and 7.65, respectively. The reduced x2 f ° r the global fit was 10.8. The uncertainty in each parameter was determined from the change in that parameter that caused the \ 2 to increase by 1. These errors were negligible (on the order of 10~5), however, they do not represent the systematic uncertainties present in the model. Given the large reduced x2, the systematic errors may be substantial. The value for C obtained in the current work is very different from those obtained by Oset et al. [10] and Sossi et al. [71]. However, it should be noted that the previously published values of C are not consistent. In their 1993 work, Sossi et al. state that inclusion of the Donoghue amplitudes has a dramatic effect on C, and that the value of C obtained with the Weinberg amplitudes is -1.97// - 1 [71]. In Reference [71], the value of Chapter 7. Interpretation 149 + -7T p-*7T 7T n M o d e l Phase—Space J2 a. b x i 3. s X l \ b x) 120 100 80 60 40 20 0 T„=243 MeV J X \ ^ : / / -/ / (A \ \ J a. b a. 6 x( \ b x) J3 a. 6 XJ \ b x) 60 50 40 30 20 10 0 200 150 T7r=223 MeV : : At -m7m2(At2) 400 300 200 100 0 T„=305 MeV - / \ \ -/ / \ - V -./ A~ \ \ 1 / \ \ . .' . . 1 . . . . 1 \ \ : . V ) Figure 7.64: Measured differential cross sections (solid points) and extended OV model fits (solid lines) for the (TI+ channel. The dashed lines represent three-body phase-space. The model predictions are absolute, and the phase-space curve has been normalized to the experimental total cross sections. Chapter 7. Interpretation 150 + -320 340 360 380 400 420 440 Pn (MeV/c) Figure 7.65: Illustration of the experimental total cross sections and predictions of the extended O V model for the (7r+ 7r _ ) channel. Chapter 7. Interpretation 151 Parameter CHAOS (p~l) Previous Results (p *) Reference C -1.35 ± 0.001 -0.91 ± 0.2 [10] -2.07 ± 0.04 [12] -3.04 [71] 9N'An 2.0 ± 0.01 1.35 ± 0.225 [74] 0.017 ± 0.005 -0.007 ± 0.011 [71] Table 7.8: Table showing the results of the global fit to the TT ) data. C obtained with the Donoghue amplitudes is -3.04^ - 1 . References [10] and [71] also state that their value of C is consistent with the N* —> TTTTN branching ratio. The total width for N* —>• 7nrN is (350 ± lOOMeV), and the fraction of the width for which the final state pions are in a relative S-state is (7.5±2.5%) [73]. Due to the large uncertainties associated with these parameters, the N* decay width does not provide a stringent constraint on C. The difference between the values of C (and to some degree, the uncertainties) ob-tained in Reference [71] and that determined in this work may be attributed to the extra constraints imposed by the differential cross sections. In the past, the extended OV model has predominantly been used to fit the experimental total cross sections [71, 74]. In general, the parameters determined by fitting the experimental total cross sections may not be well suited towards describing the m 2^. distributions. The invariant mass dis-tributions contain new information concerning the reaction mechanism and place extra constraints on the model. In order to test the predictive power of the extended OV formalism, the model re-sults for the ( 7 r + 7 r + ) channel using the parameters determined from the (n+ data are shown in Figure 7.66. The predicted total cross sections are shown in Figure 7.67. There is significant disagreement between the experimental results and the model predictions Chapter 7. Interpretation 152 4. T J b X i Xi 3. X l \ b 13 + + + TT p ^ T T 7T n M o d e l Phase—Space 3. a x l b s a XI b XI a. b x l T ^ = 2 2 3 M e V : : + Figure 7.66: Measured differential cross sections (solid points) and predictions of the extended OV model (solid lines) for the ( 7 r + 7 r + ) channel. Three-body phase-space is represented by the dashed lines. The phase space curve has been normalized to the experimental total cross sections. The model predictions are absolute. Chapter 7. Interpretation 153 10' 10' 10^  + + + 7T p - ^ 7 T TT I I — i — i — i — | — i — i — i — i — | — r -$ This work Model - 1 — i — i — i — i — i — i — i — I — i — i — r J I I I I I I I I I I L. I 1 ' _ l I I I J I I I L. 320 340 360 380 400 420 440 P w (MeV/c) Figure 7.67: The experimental total cross sections and predictions of the extended OV model for the (7r + 7T+) channel are shown. Chapter 7. Interpretation 154 in the (TT+ 7r + ) channel. Figures 7.66 and 7.67 indicate that the model parameters re-quired to describe the m 2 ^ distributions in the (TT+ 7r + ) channel are not consistent with those obtained by fitting the (7r + 7r~) distributions. The (7r + 7T+) data seem to be better described by three body phase-space. The extended OV model has been criticized in the work of Olsson, Meissner, Bernard, and Kaiser [66], where they state that the model is inconsistent because " . . . the 7T7T and irirN amplitudes should be treated at the same order in the chiral expansion." They also state that the mesonic low energy constants (d\ and o72) determined in Refer-ence [72] were obtained from " . . . data over an energy range which clearly exceeds the validity of the one-loop calculation." Prior to a more extensive fit and in order to obtain reliable values of C, d\ and gN*An, more theoretical input into the model is required. This clearly falls outside the scope of this thesis. 7.7 Model Calculations &: O P E Dominance A qualitative study of the contribution of the O P E mechanism to the (n, 2TT) cross section was performed using the extended OV model. In this study the O P E contribution was mitigated by setting the constants d\ and d2 to zero [75], which leaves only the effects of the Weinberg Lagrangian included. Other model parameters were fixed at the values obtained by fitting the experimental distributions for the ( 7 r + 7 r _ ) channel. The model was employed in order to produce m 2 ^ distributions for both reaction channels and at all pion incident energies studied in this experiment. The predicted distributions Chapter 7. Interpretation 155 with the full calculation (including OPE) were then compared to those obtained with reduced OPE. The results indicate that in the (7r +7r~) channel there is about a factor of 2 to 3 (depending on the incident energy) reduction in the total cross section if the O P E contribution is reduced. These results support the O P E dominance hypothesis for the n~p —>• 7r +7r _ra reaction. However a strong caveat is in order. The Chew-Low extrap-olation process utilized the experimental two-fold differential cross sections. However, currently the model does not predict double differential cross sections. As such the dis-tribution of the background in t is not known. Consequently this approach only provides a qualitative estimate of the strength of the O P E relative background terms. In the (7r + 7r + ) channel, the model calculations suggest that the O P E and background mechanisms interfere destructively. Here the predicted cross sections with the O P E term are approximately 2 times smaller than those with reduced OPE. Given the significant disagreements between the model calculations and the experimental distributions in the (7r + 7r + ) channel, interpretation of the results of this study for the 7r+p —• ir+-K+n reaction is not clear. Chapter 8 Concluding Remarks We have performed an exclusive study of the elementary pion induced pion production reactions —> 7 r + 7 r _ n and it+p —*• 7 r + 7 r + n at incident pion energies of 223, 243, 264, 284, and 305 MeV. A cryogenic liquid hydrogen target was employed, and the CHAOS magnetic spectrometer was used to detect the two outgoing charged pions in coincidence. Between 2,000 and 12,000 (ir, 2-K) events were recorded at each energy. The experimental distributions were corrected for detector acceptance and experimental efficiencies in order to produce single, double and triple differential cross sections for both reaction channels. The overall normalization of the measured distributions was confirmed by comparing our measured absolute differential cross sections for irp elastic scattering to those obtained from phase-shift predictions [36]. One of the main goals of this experiment was to determine the S-wave -KIT scattering lengths. To this end, the ( 7 r + 7 r ~ ) experimental double differential cross sections were used to obtain on-shell pion-pion scattering cross sections via the model-independent Chew-Low technique. The Chew-Low results along with 7 r + 7 r - phase shifts, obtained from high energy TTN —> irnN experiments, were combined in a dispersion analysis (Roy Equations) to determine the isospin zero S-wave scattering lengths. Our results indicate that 0% = 0.209 ± 0.011, which is in good agreement with the predictions of Chiral Perturbation Theory [59, 5]. The ambiguity of the O P E dominance for the (TT+ TT+) data prevented the use of the Chew-Low technique in this reaction channel. In addition to the above, the extended Oset and Vicente-Vacas (OV) model [10,11,12] 156 Chapter 8. Concluding Remarks 157 was used to fit the experimental m2^ distributions in the (ir+ channel. In this analysis, the parameters of the model, namely C, < ? J V * A T T , and di were varied in order to best describe the measured distributions. The resulting parameters do not agree with previous values determined by fitting total cross sections (see Table 7.8). The model parameters obtained from the (ir+ -K") data were then used to predict the m2^ distributions for the TT+P —» 7r + 7r + n reaction. There is significant disagreement between the model predictions and the experimental data in this channel, highlighting the need for more theoretical effort in this area. Pion-pion interactions are one of the most fundamental strong interactions, and as such they are of crucial importance to our understanding of the manifestation of Q C D in the low energy domain. However the experimental study of these processes is not a triv-ial task, and all experimental -KIT scattering data have been obtained via indirect means. One such method has involved the study of elementary pion induced pion production reactions. Although numerous experiments have been performed in this area (mostly total cross sections and studies in the GeV region), the prescription for determining pion-pion scattering parameters from irN —> mrN data has been plagued by theoretical uncertainty and ambiguity. Thus, often the resulting TTTT observables are extracted in a model-dependent manner, or they are subject to underlying assumptions regarding the -KN —>• TTTTN reaction mechanism. In their recent work, Bernard, Kaiser, and Meissner (BKM) [62] provided the chiral expansion of the irN —> HKN threshold amplitudes in the framework of heavy baryon chiral perturbation theory. In this formalism they use the experimental total cross sections near threshold to extract the S-wave scattering lengths. However they caution that resonance exchanges make the accurate determination of a° difficult. Although B K M have improved the theoretical situation near threshold, the procedure for extracting TT-K scattering observables from -KN —»• 7T7rA" data above thresh-old still remains uncertain. The largest obstacle to extracting TTTT scattering observables Chapter 8. Concluding Remarks 158 from TTN —» ITTIN data is the lack of a reliable model for describing the non-OPE back-ground. Clearly a large theoretical effort is required in order to utilize the existing •KN —> TTTTN data to determine accurate pion-pion scattering observables. It is not yet clear whether additional experimental studies of irN —> TTTTN interactions, without sig-nificant concomitant theoretical developments, will significantly increase our knowledge of 7T7T scattering. Despite the theoretical uncertainties present in relating the large body of existing (7r,27r) cross sections to 7T7T scattering observables, it is the experimental data presented in this thesis that should be most emphasized. The cross sections measured in this work are unique and represent the world's most complete data set on pion induced pion production near threshold. We have measured many-fold differential cross sections near threshold, and we believe that these data are of great importance in furthering the un-derstanding of the reaction mechanism involved in (7r,27r). 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Halliday, Introductory Nuclear Physics, John Wiley & Sons, New York, 1955. Bibliography 162 [66] M . Olsson et al, HEP-PH-9503237 (1995). [67] S. Weinberg, Phys. Rev. Lett. 18, 188 (1967). [68] S. Weinberg, Phys. Rev. Lett. 18, 507 (1967). [69] S. Weinberg, Phys. Rev. 166, 1568 (1968). [70] A . A . Bolokhov, Chiral Dynamics: Theory and Experiment, Proceedings of the Workshop held at MIT, Cambridge, M A , USA, 25-29 July 1994. Aron M . Bernstein and Barry H . Holstein eds., P115. [71] V . Sossi et al, Phys. Lett. B298, 287 (1993). [72] J.F. Donoghue et al, Phys. Rev. D38, 2195 (1988). [73] Particle Data Book, Phys. Rev. D54, 575 (1996). [74] N .B . Fazel, M.Sc. Thesis, University of British Columbia, Unpublished, 1992. [75] R.R. Johnson, Private Communication, 1997. Appendix A Relevant Kinematic Parameters For simplicity, in the following discussion the symbol, s, is used to denote the dipion invariant mass squared. Consider the following reaction: ^ ( fc i ) +P - -K±{k2) + n+{h) + n, (A.112) where hi, k2, and fc3 are the four-momenta. In addition let k[, k'2, and k'3 represent the corresponding four-momenta in the dipion rest frame. The expression for s is given by s = (k2 + k3)2 = (k'2 + k'3)2 (A.113) = 2fi2 + 2k2-k3 = 2/i 2 + 2E2E3 - 2p2ps cos(0), where p, E, and 4> a r e the magnitude of the three-momentum, the total energy and the opening angle between the two pions, respectively. Using the same notation t may be written as t = (k2 + k3-h)2 (A.114) = [k!2 + k's — k\)2 = s + n2 - 2ki • k2 - 2ki • h = s + n2 - 2EXE2 + 2plP2 cos(^i) - 2E1E3 + 2plPs cos(62), where di is the scattering angle of the ith particle. 163 Appendix A. Relevant Kinematic Parameters 164 Similarly cos$ is given by c o s ^ ^ - T 1 , (A.115) P1P2 where pj denotes the three-momentum of the ith particle in the dipion rest frame. In the center of mass of the two scattered pions, k'2 = ( £ , P ' 2 ) , (A.116) K = (E,-p'2), K = ( E L p i ) . Hence ^ • ( ^ - ^ ) = 2 p ' 1 - p 2 . (A.117) In addition, from Lorentz invariance we have, k[-(k'3-k'2)=k1-(k3-k2). (A.118) From the above it is clear that (kl-k,-kl-k2) = 2p'1-p'2. (A.119) Similarly s = {k'2 + Kf (A. 120) = AE2. Using the standard relation between momentum and energy we get p , 2 = ( s - _ V ) ( A m ) p\ may also be determined in terms of the invariant quantities, t = (k'2 + k'z-k\)2 (A. 122) = AE2 - AEE[ + p2. Appendix A. Relevant Kinematic Parameters 165 Hence p2 = E{-p2 (A. 123) \2 {t-p2-sf - - 2 4s s 2 - 2 S ( ^ + //2) + ( t - / / 2 ) 2 4s Combining the above relations yields, cos9 = ^ hfhM. (A124) A . l The Lowest |t| It is interesting to look at the lowest value of |t| allowed by kinematics. Consider the reaction ir(kl) +N(k2) ^ X{k3) +N(k4) (A.125) where X represents the dipion system. The four-momentum transfer squared is given by t = (kl-k3)2. (A.126) In the center of mass frame, the four-momenta are kl = (El)P), (A. 127) k2 = (E2,-p), k3 = (E3,p'), fc4 = {E4,-p'). From the above, the expression for t is t = \p2 + M2X - 2 £ i £ 3 l + 2pp'cos{(t))) (A. 128) Appendix A. Relevant Kinematic Parameters 166 where <$> is the scattering angle of the dipion. Note that the term enclosed in brackets is negative, and the minimum of \t\ occurs at <f> = 0. At a given center of mass energy S', the expressions for Ei and E3 are given by The lowest possible (kinematically allowed) value of \t\ (\tmin\) is reached at Mx = 2p. For the range of energies considered in this experiment, tmin varies from — Q.Qp2 to —0.3/i2. The 7T7T configuration corresponding to \tmin\ is when the pions are produced along the direction of the incident beam with zero relative momentum (same momenta and zero opening angle). E, = (A. 129) (A. 130) \ A p p e n d i x B Tables of 7i~p —»• ir+ir~n Cross Sections Tables of differential cross sections for the (TT+ -K ) channel are listed in this section. The overall systematic errors for the 223, 243, 264 285, and 305 MeV are 11.2%, 10.6%, 9.3%, and 9.4%, respectively. TT-p -»• ir+7r-n at T w = 223 MeV. (fj2) t(n2) E R R (pb/v2) 4.15 -5.500 4.36 3.87 4.15 -4.500 1.11 1.04 4.15 -3.500 1.29 0.31 4.15 -2.625 2.10 0.35 4.15 -1.875 2.65 0.35 4.15 -1.125 3.00 0.21 4.15 -0.625 0.03 0.01 4.45 -4.500 1.27 0.52 4.45 -3.500 2.12 0.41 4.45 -2.625 6.00 0.80 4.45 -1.875 7.68 0.54 4.45 -1.125 5.05 0.38 4.75 -5.500 7.16 4.92 4.75 -4.500 5.34 0.94 4.75 -3.500 10.87 0.84 4.75 -2.625 14.63 0.99 4.75 -1.875 18.47 1.33 4.75 -1.125 1.01 0.19 167 Appendix B. Tables of n~p —> TT+TI n Cross Sections 168 %-p -> 7r+7r-n at T w = 223 MeV. ™IAV2) t(p2) zS^rA^/^) E R R (pb/p2) 5.05 -5.500 0.87 0.60 5.05 -4.500 3.24 1.01 5.05 -3.500 17.79 1.92 5.05 -2.625 18.89 2.50 5.05 -1.875 4.89 0.82 Table B.9: Table of double differential cross sections for the (TT+ TT~) channel at 223 MeV. ir~p -»• TT+TT-W at Tv = 243 MeV. t(p2) E R R (pb/p2) 4.15 -5.500 0.64 0.36 4.15 -4.500 1.20 0.34 4.15 -3.500 1.25 0.28 4.15 -2.625 1.63 0.36 4.15 -1.875 1.08 0.30 4.15 -1.125 2.09 0.29 4.15 -0.625 0.30 0.14 4.45 -6.500 0.46 0.46 4.45 -5.500 0.38 0.22 4.45 -4.500 6.15 4.57 4.45 -3.500 4.48 1.43 4.45 -2.625 3.75 0.82 4.45 -1.875 4.93 0.79 4.45 -1.125 5.17 0.46 4.45 -0.625 0.01 0.01 4.75 -7.500 16.33 16.77 4.75 -6.500 11.08 3.50 4.75 -5.500 4.24 1.38 4.75 -4.500 5.19 1.34 4.75 -3.500 10.93 2.38 4.75 -2.625 11.40 1.52 4.75 -1.875 15.57 1.20 4.75 -1.125 6.26 0.65 Appendix B. Tables of 7r p —> TT+TI n Cross Sections 169 7r-p -»• vr+TT-n at TV = 243 MeV. ™l* ( M 2 ) t{p2) E R R (pb/p2) 5.05 -6.500 17.02 3.25 5.05 -5.500 15.24 1.56 5.05 -4.500 15.10 1.49 5.05 -3.500 16.15 1.56 5.05 -2.625 16.36 1.41 5.05 -1.875 21.51 1.44 5.05 -1.125 1.79 0.41 5.35 -6.500 2.65 1.18 5.35 -5.500 20.87 2.87 5.35 -4.500 30.50 1.93 5.35 -3.500 28.32 1.66 5.35 -2.625 29.91 1.89 5.35 -1.875 13.26 1.88 5.65 -5.500 1.08 0.83 5.65 -4.500 11.25 2.20 5.65 -3.500 24.07 2.33 5.65 -2.625 10.48 1.81 5.65 -1.875 0.04 0.02 Table B.10: Table of double differential cross sections for the (n+ IT ) channel at 243 MeV. TT'P -»• vr+x-n at Tn = 264 MeV. < r ( ^ 2 ) *(/*2) aSnzWf) E R R ( ^ / / , 2 ) 4.15 -7.500 0.51 0.20 4.15 -6.500 0.75 0.21 4.15 -5.500 0.49 0.14 4.15 -4.500 1.00 0.22 4.15 -3.500 0.48 0.15 4.15 -2.625 0.44 0.17 4.15 -1.875 1.16 0.35 4.15 -1.125 0.87 0.32 4.15 -0.625 0.69 0.21 Appendix B. Tables of IT pir+7i n Cross Sections 170 7f-p -> 7r+7T-n at Tn = 264 MeV. <Af) t(p2) o$^iW»2) ERRQufr/M 2) 4.45 -7.500 0.50 0.30 4.45 -6.500 1.16 0.45 4.45 -5.500 0.86 0.32 4.45 -4.500 1.47 0.43 4.45 -3.500 1.74 0.50 4.45 -2.625 1.09 0.36 4.45 -1.875 0.91 0.37 4.45 -1.125 2.74 0.36 4.45 -0.625 1.17 0.37 4.75 -7.500 5.72 2.42 4.75 -6.500 2.86 1.20 4.75 -5.500 1.70 0.39 4.75 -4.500 3.79 1.48 4.75 -3.500 5.66 1.20 4.75 -2.625 4.86 1.17 4.75 -1.875 8.01 1.13 4.75 -1.125 5.96 0.58 4.75 -0.625 0.03 0.03 5.05 -7.500 10.09 1.42 5.05 -6.500 13.85 3.65 5.05 -5.500 15.89 6.48 5.05 -4.500 18.61 7.56 5.05 -3.500 15.04 2.97 5.05 -2.625 11.18 1.56 5.05 -1.875 11.39 1.12 5.05 -1.125 8.72 0.77 5.35 -7.500 7.96 1.18 5.35 -6.500 20.96 1.66 5.35 -5.500 26.79 3.21 5.35 -4.500 15.96 2.61 5.35 -3.500 21.56 2.20 5.35 -2.625 27.10 2.21 5.35 -1.875 23.34 1.41 5.35 -1.125 5.21 0.71 5.65 -7.500 9.16 3.19 5.65 -6.500 20.47 1.67 Appendix B. Tables ofnp^> 7r+7r n Cross Sections 171 -K-p - > Tr+TT-n at T T = 264 MeV. ™ L (M2) t(p2) E R R (pb/p2) 5.65 -5.500 32.51 1.75 5.65 -4.500 37.86 2.24 5.65 -3.500 41.08 2.16 5.65 -2.625 36.35 1.82 5.65 -1.875 23.12 1.74 5.65 -1.125 0.15 0.06 5.95 -7.500 0.15 0.09 5.95 -6.500 5.98 1.55 5.95 -5.500 28.50 1.97 5.95 -4.500 43.38 1.85 5.95 -3.500 39.84 1.71 5.95 -2.625 31.33 2.22 5.95 -1.875 4.72 0.83 6.25 -6.500 0.04 0.04 6.25 -5.500 3.46 0.87 6.25 -4.500 14.86 1.63 6.25 -3.500 19.43 1.86 6.25 -2.625 4.22 0.89 Table B . l l : Table of double differential cross sections for the (7r+ 7r ) channel at 264 MeV. 7T-p -»• ir+TT-n at % = 284 MeV. <Af) Hp2) ^^(pb/p2) E R R (pb/p2) 4.15 -7.500 0.87 0.19 4.15 -6.500 0.91 0.18 4.15 -5.500 0.48 0.12 4.15 -4.500 0.63 0.14 4.15 -3.500 0.54 0.13 4.15 -2.625 0.31 0.12 4.15 -1.875 0.54 0.19 4.15 -1.125 0.27 0.11 4.15 -0.625 0.37 0.14 Appendix B. Tables of n~p —> 7r+7r n Cross Sections 172 TX'P - > Tr+Tf -n at T w = 284 MeV. ^ ( M 2 ) t(»2) eS^iWv2) ERRQu&// i 2 ) 4.45 -7.500 1.24 0.34 4.45 -6.500 1.32 0.34 4.45 -5.500 2.27 0.58 4.45 -4.500 0.97 0.27 4.45 -3.500 1.01 0.31 4.45 -2.625 1.26 0.35 4.45 -1.875 1.21 0.43 4.45 -1.125 1.93 0.33 4.45 -0.625 0.61 0.20 4.45 -0.250 0.00 0.00 4.75 -7.500 3.72 0.80 4.75 -6.500 1.54 0.45 4.75 -5.500 2.45 0.47 4.75 -4.500 2.92 0.70 4.75 -3.500 2.76 0.62 4.75 -2.625 2.45 0.69 4.75 -1.875 4.30 0.70 4.75 -1.125 3.92 0.40 4.75 -0.625 0.94 0.43 5.05 -7.500 6.87 2.17 5.05 -6.500 3.84 0.48 5.05 -5.500 8.47 3.28 5.05 -4.500 11.78 4.18 5.05 -3.500 6.89 1.08 5.05 -2.625 8.47 1.20 5.05 -1.875 8.89 1.04 5.05 -1.125 7.22 0.58 5.35 -7.500 25.36 3.26 5.35 -6.500 32.74 7.10 5.35 -5.500 18.56 6.53 5.35 -4.500 14.37 4.49 5.35 -3.500 18.56 2.39 5.35 -2.625 16.96 1.68 5.35 -1.875 17.20 1.19 5.35 -1.125 8.61 0.73 5.65 -7.500 30.78 1.64 5.65 -6.500 32.67 2.66 Appendix B. Tables ofnp-> -K+TT n Cross Sections 173 7T.P ->• TT+Tr-n at T w = 284 MeV. <A^) t(p2) o^r-gj (fJ'b/p2) E R R (pb/p2) 5.65 -5.500 29.73 3.39 5.65 -4.500 26.88 2.75 5.65 -3.500 23.50 1.82 5.65 -2.625 29.38 1.90 5.65 -1.875 23.90 1.20 5.65 -1.125 5.91 0.69 5.95 -7.500 28.79 1.51 5.95 -6.500 43.91 1.83 5.95 -5.500 50.60 2.61 5.95 -4.500 43.21 2.58 5.95 -3.500 40.18 1.99 5.95 -2.625 38.24 1.63 5.95 -1.875 23.35 1.27 5.95 -1.125 0.51 0.14 6.25 -7.500 15.95 1.58 6.25 -6.500 37.84 1.61 6.25 -5.500 51.89 1.74 6.25 -4.500 53.60 1.80 6.25 -3.500 49.74 1.60 6.25 -2.625 37.51 1.62 6.25 -1.875 13.78 1.41 6.55 -7.500 0.49 0.18 6.55 -6.500 17.37 1.66 6.55 -5.500 42.54 1.92 6.55 -4.500 49.83 1.73 6.55 -3.500 43.68 1.70 6.55 -2.625 21.86 1.97 6.55 -1.875 0.19 0.05 6.85 -6.500 0.31 0.10 6.85 -5.500 9.73 1.29 6.85 -4.500 18.53 1.50 6.85 -3.500 11.62 1.13 6.85 -2.625 0.51 0.11 Appendix B. Tables of 7r p —> 7r + 7r n Cross Sections 174 7T"p -> Tr+TT-n at T w = 284 MeV. < r f c 2 ) *(/*2) a&iWv2) E R R (pb/p2) Table B.12: Table of double differential cross sections for the (ir+ 7r~) channel at 284 MeV. 7i-p -> Tr+TT-n at T w = 305 MeV. t(p2) & ( ^ 2 ) E R R (pb/p2) 4.25 -9.500 1.28 0.52 4.25 -8.500 2.16 0.51 4.25 -7.500 1.45 0.40 4.25 -6.500 1.62 0.39 4.25 -5.500 1.15 0.32 4.25 -4.500 1.08 0.26 4.25 -3.500 1.31 0.31 4.25 -2.500 0.67 0.26 4.25 -1.500 0.94 0.31 4.25 -0.500 0.55 0.24 4.75 -9.500 7.55 2.03 4.75 -8.500 3.70 1.03 4.75 -7.500 3.88 1.61 4.75 -6.500 4.15 1.26 4.75 -5.500 1.60 0.44 4.75 -4.500 1.60 0.67 4.75 -3.500 2.75 0.82 4.75 -2.500 3.98 1.05 4.75 -1.500 1.41 0.47 4.75 -0.500 1.19 0.23 5.25 -9.500 11.33 1.77 5.25 -8.500 12.46 3.10 5.25 -7.500 8.54 1.47 5.25 -6.500 31.47 30.69 5.25 -5.500 8.73 2.43 5.25 -4.500 6.94 2.70 5.25 -3.500 4.69 1.31 5.25 -2.500 4.35 0.99 5.25 -1.500 8.67 0.91 Appendix B. Tables ofirp-* -K+TI n Cross Sections 175 7T~p -»• Ti+Ti-n at T T = 305 MeV. t(p2) a & ( ^ / M 2 ) E R R (fib/p2) 5.25 -0.500 1.16 0.28 5.75 -9.500 33.38 2.52 5.75 -8.500 25.59 2.02 5.75 -7.500 22.32 2.67 5.75 -6.500 21.16 3.60 5.75 -5.500 20.39 4.12 5.75 -4.500 28.79 5.73 5.75 -3.500 24.65 2.79 5.75 -2.500 20.00 1.79 5.75 -1.500 14.22 1.07 6.25 -9.500 27.90 4.41 6.25 -8.500 51.99 2.98 6.25 -7.500 45.58 2.27 6.25 -6.500 37.47 2.29 6.25 -5.500 34.18 2.61 6.25 -4.500 40.45 2.99 6.25 -3.500 34.72 2.26 6.25 -2.500 35.06 1.75 6.25 -1.500 15.31 1.32 6.75 -8.500 12.99 2.72 6.75 -7.500 58.08 3.32 6.75 -6.500 77.14 3.07 6.75 -5.500 64.70 2.46 6.75 -4.500 56.08 2.19 6.75 -3.500 52.39 2.12 6.75 -2.500 36.10 1.95 6.75 -1.500 0.44 0.14 7.25 -7.500 0.24 0.10 7.25 -6.500 19.26 2.20 7.25 -5.500 59.36 3.85 7.25 -4.500 61.27 3.42 7.25 -3.500 29.19 2.10 7.25 -2.500 1.65 0.24 Table B.13: Table of double differential cross sections for the IT ) channel at 305 MeV. Appendix B. Tables of TT p —• TT+TT n Cross Sections 176 TT-p -»• Tr+Tr-n at T^ = 223 MeV. (^) agrW/*2) E R R (^6/pl) 4.15 12.59 4.04 4.45 17.43 1.02 4.75 48.96 5.23 5.05 39.74 3.00 5.35 0.081 0.03 Table B.14: -M- for the (TT+TT-) channel at 223 MeV TT-p ->• TT+TT-w at Tv = 223 MeV. t(p2) f Qz6//i 2) E R R ( ^ ) = -5.50 3.72 1.89 -4.50 3.29 0.54 -3.50 9.64 0.65 -2.625 12.49 0.85 -1.875 10.11 0.51 -1.125 2.72 0.14 -0.625 0.009 0.00 Table B.15: f for the (TT+TT") channel at 223 MeV ir~p -»• TT+TT—7i at T^ = 223 MeV. cos9 A W E R R (pb) -0.90 19.16 3.22 -0.70 19.82 2.08 -0.50 18.06 1.64 -0.30 18.97 4.19 -0.10 16.53 1.22 0.10 16.59 1.90 Appendix B. Tables of' -n p —> 7r +7r n Cross Sections 177 ir~p -> Tr+TT-n at T w = 223 MeV. ~~c^s9 A {jib) ERR(Atb) 0.30 13.56 0.96 0.50 13.57 1.17 0.70 27.55 8.56 0.90 14.41 2.30 Table B.16: ^ for the (TT+ TT") channel at 223 MeV n'p -)• TT+TT-n at T w = 243 MeV. •£t WW E R R (pb/p2) 4.15 7.37 0.93 4.45 21.85 4.91 4.75 72.69 17.470 5.05 93.27 4.48 5.35 114.71 4.49 5.65 44.28 3.58 5.95 0.02 0.01 Table B.17: for the (TT+ T T ) channel at 243 MeV Appendix B. Tables of IT p —> ir+ix n Cross Sections 178 Ti-p -> Ti+iT-n at Tn = 243 MeV. t(p2) f W ) ERR (pb/p2) -7.50 4.90 5.03 -6.50 9.54 1.49 -5.50 12.74 1.10 -4.50 20.82 1.74 -3.50 25.56 1.29 -2.62 22.06 1.04 -1.88 16.92 0.84 -1.12 4.59 0.28 -0.62 0.10 0.04 Table B.18: g for the (TT+ TT") channel at 243 MeV TT-p ir+7t-n at Tv = 243 MeV. cose jgsbib) E R R (fib) -0.90 67.90 23.50 -0.70 77.10 7.32 -0.50 52.76 4.07 -0.30 63.71 10.79 -0.10 41.67 3.06 0.10 40.81 3.83 0.30 45.97 3.17 0.50 50.33 7.68 0.70 49.43 3.95 0.90 41.61 3.81 Table B.19: ^ for the (TT+ T T ) channel at 243 MeV Appendix B. Tables ofnp^ 7T+7T n Cross Sections 179 Tt'p -»• 7r + 7r -n at Tn = 264 MeV. E R R (pb/p2) 4.15 8.98 3.91 4.45 11.08 1.83 4.75 35.07 3.68 5.05 97.46 11.22 5.35 135.77 5.53 5.65 185.80 5.41 5.95 144.89 3.98 6.25 40.97 2.71 6.55 0.17 0.05 Table B.20: for the (TT+ TT") channel at 264 MeV 7r-p -> Tr+TT-n at Tv = 264 MeV. t(M 2 ) § (MS) E R R (pb/p2) -7.50 10.23 1.33 -6.50 19.82 1.44 -5.50 33.06 2.33 -4.50 41.10 2.64 -3.50 43.48 1.54 -2.62 34.97 1.27 -1.88 21.79 0.87 -1.12 7.10 0.39 -0.62 0.57 0.13 Table B.21: ^ for the (it+ T T ) channel at 264 MeV Appendix B. Tables of'IT p —> -K+TT n Cross Sections 180 TT'P -> ir+Tr-n at % = 264 MeV. cosd E R R [pb2) -0.90 109.83 4.66 -0.70 122.48 7.40 -0.50 113.25 6.36 -0.30 110.66 8.68 -0.10 105.06 10.70 0.10 96.31 11.27 0.30 96.30 7.18 0.50 85.48 4.38 0.70 79.03 4.07 0.90 71.87 4.66 Table B.22: ^ for the (TT + T T ) channel at 264 MeV TT'P -> TT+Tr-n at T w = 284 MeV. Wf) E R R (pb/p2) 4.15 7.60 0.75 4.45 13.59 1.34 4.75 34.35 2.64 5.05 75.44 6.34 5.35 163.93 11.61 5.65 205.04 6.11 5.95 262.80 5.29 6.25 247.81 4.07 6.55 170.47 3.82 6.85 40.58 2.28 Table B.23: for the (n+%') channel at 284 MeV Appendix B. Tables of'IT p —> 7r + 7r n Cross Sections 181 TT-p -> 7r +7r-n at 7 ; = 284 MeV. t(p2) E R R (pb/p2) -7.50 34.23 1.46 -6.50 51.74 2.45 -5.50 65.03 2.70 -4.50 66.85 2.34 -3.50 59.56 1.39 -2.62 47.09 1.26 -1.88 28.01 0.86 -1.12 8.51 0.38 -0.62 0.57 0.15 Table B.24: % for the (TT+TT") channel at 284 MeV TT'P - + TT+vr-n at % = 284 MeV. cosO E R R (fib) -0.90 238.73 5.74 -0.70 223.48 7.33 -0.50 211.90 8.80 -0.30 183.62 8.95 -0.10 192.78 13.75 0.10 160.96 8.15 0.30 163.67 8.00 0.50 161.36 6.35 0.70 157.23 4.66 0.90 138.94 4.18 Table B.25: ^ for the (TT+ vr") channel at 284 MeV Appendix B. Tables of-np^- 7r+7r n Cross Sections 182 it~p -> TT+Tr-n at % = 305 MeV. ml* (^2) •& E R R (pb/p2) 4.25 14.60 1.76 4.75 37.85 4.89 5.25 111.39 31.27 5.75 224.89 10.09 6.25 322.67 8.03 6.75 357.92 6.85 7.25 170.96 5.99 Table B.26: for the (TT+ TT") channel at 284 MeV TX'P -»• Tr+TT-n at = 305 MeV. t(p2) f WP2) E R R (pb/p2) -9.50 40.72 2.89 -8.50 54.45 2.80 -7.50 70.04 2.66 -6.50 96.14 15.62 -5.50 95.06 3.57 -4.50 98.12 4.06 -3.50 74.86 2.47 -2.50 50.91 1.75 -1.50 20.49 1.01 -0.50 1.45 0.22 Table B.27: ^ for the (TT+ TT - ) channel at 305 MeV •K~P -»• 7i+7i~n at Tn = 305 MeV. cosfl E R R ( ^ ) : -0.90 480.86 15.32 Appendix B. Tables ofirp—* 7r+7r n Cross Sections ii-p - » 7 r + 7 r - 7 i at Tn = 305 MeV. cosd E R R (pb) -0.70 396.02 17.08 -0.50 392.18 78.43 -0.30 272.58 13.88 -0.10 289.75 16.32 0.10 246.48 12.74 0.30 259.86 12.70 0.50 261.57 12.26 0.70 240.71 11.00 0.90 260.84 10.19 Table B.28: ^ for the (TT+ TT") channel at 305 MeV A p p e n d i x C Tables of ir+p —> 7r + 7r + n Cross Sections Differential cross sections for the (T\+ 7 r + ) channel are presented in the following tables. The systematic errors in the listed cross sections at 223, 243, 264, 285, and 305 MeV are 9.6%, 9.7%, 9.6%, 9.7%, and 9.6%, respectively. TT+P -»• 7 r + 7 r + n at TV = 223 MeV. t(p2) a f e Wp2) E R R (pb/p2) 4.15 -4.5 0.01 0.01 4.15 -3.5 0.28 0.09 4.15 -2.625 1.11 0.22 4.15 -1.875 1.72 0.20 4.15 -1.125 1.82 0.24 4.45 -3.5 0.40 0.19 4.45 -2.625 1.61 0.42 4.45 -1.875 3.08 0.30 4.45 -1.125 0.84 0.13 4.75 -4.5 0.61 0.18 4.75 -3.5 2.04 0.31 4.75 -2.625 3.38 0.32 4.75 -1.875 1.82 0.17 4.75 -1.125 0.13 0.05 5.05 -4.5 0.13 0.13 5.05 -3.5 0.37 0.08 5.05 -2.625 0.75 0.13 Appendix C. Tables of 7t+p —> 7 r + 7 r + n Cross Sections 185 TT+P n+ir+n at T, = 223 MeV. " ^ r f c 2 ) HV2) i££wtW\?) E R R (//&/^2) 5.05 -1.875 0.89 0.84 Table C.29: Table of double differential cross sections for the ( 7r + 7r + ) channel at 223 MeV. TT+P -»• 7 r + 7 r + n at T T = 2 4 3 MeV. ml, (p2) t(p2) E R R (pb/p2) 4.15 -7.500 0.91 0.96 4.15 -6.500 1.10 1.13 4.15 -5.500 0.23 0.13 4.15 -4.500 0.39 0.10 4.15 -3.500 0.94 0.17 4.15 -2.625 1.40 0.22 4.15 -1.875 2.60 0.23 4.15 -1.125 2.02 0.27 4.45 -5.500 0.14 0.06 4.45 -4.500 0.47 0.18 4.45 -3.500 0.72 0.19 4.45 -2.625 1.65 0.39 4.45 -1.875 3.76 0.40 4.45 -1.125 3.56 0.28 4.75 -6.500 0.41 0.24 4.75 -5.500 0.00 0.00 4.75 -4.500 0.88 0.61 4.75 -3.500 2.19 0.76 4.75 -2.625 2.98 0.43 4.75 -1.875 6.23 0.41 4.75 -1.125 2.48 0.38 5.05 -6.500 0.11 0.11 5.05 -5.500 0.67 0.16 5.05 -4.500 0.80 0.16 5.05 -3.500 1.94 0.25 5.05 -2.625 4.35 0.35 Appendix C. Tables of 7i+p —> n+n+n Cross Sections 1 8 6 TT+P -> TT+vr+w at = 2 4 3 MeV. < ( ^ 2 ) t(p2) E R R (pb/p2) 5 . 0 5 - 1 . 8 7 5 5 . 4 0 0 . 3 7 5 . 0 5 - 1 . 1 2 5 0 . 1 9 0 . 0 6 5 . 3 5 - 6 . 5 0 0 0 . 0 1 0 . 0 1 5 . 3 5 - 5 . 5 0 0 0 . 7 1 0 . 2 1 5 . 3 5 - 4 . 5 0 0 1.80 0 . 2 1 5 . 3 5 - 3 . 5 0 0 2 . 6 0 0 . 2 0 5 . 3 5 - 2 . 6 2 5 4 . 0 9 0 . 3 3 5 . 3 5 - 1 . 8 7 5 0 . 8 5 0 . 1 3 5 . 6 5 - 5 . 5 0 0 0 . 0 2 0 . 0 1 5 . 6 5 - 4 . 5 0 0 0 . 4 0 0 . 1 4 5 . 6 5 - 3 . 5 0 0 0 . 7 8 0 . 1 4 5 . 6 5 - 2 . 6 2 5 0 . 5 4 0 . 1 9 5 . 6 5 - 1 . 8 7 5 0 . 0 1 0 . 0 1 Table C . 3 0 : Table of double differential cross sections for the ( 7r + 7r + ) channel at 2 4 3 MeV. TT+P _> TT+yr+n at T-n- = 2 6 4 MeV. t(p2) E R R (pb/p2) 4 . 1 5 - 7 . 5 0 0 0 . 0 6 0 . 0 3 4 . 1 5 - 6 . 5 0 0 0 . 2 1 0 . 0 8 4 . 1 5 - 5 . 5 0 0 0 . 2 5 0 . 0 6 4 . 1 5 - 4 . 5 0 0 0 . 7 2 0 . 1 0 4 . 1 5 - 3 . 5 0 0 1 .19 0 . 1 2 4 . 1 5 - 2 . 6 2 5 2 . 0 6 0 . 2 4 4 . 1 5 - 1 . 8 7 5 2 . 5 8 0 . 2 1 4 . 1 5 - 1 . 1 2 5 2 . 8 4 0 . 3 5 4 . 1 5 - 0 . 6 2 5 1.69 1.81 4 . 4 5 - 7 . 5 0 0 0 . 0 5 0 . 0 4 4 . 4 5 - 6 . 5 0 0 0 . 0 6 0 . 0 3 4 . 4 5 - 5 . 5 0 0 0 . 1 3 0 . 0 3 4 . 4 5 - 4 . 5 0 0 0 . 6 7 0 . 3 6 4 . 4 5 - 3 . 5 0 0 0 . 7 2 0 . 1 4 Appendix C. Tables of n+p —> 7 r + 7 r + n Cross Sections 187 TT+p ^ TT+TT+W at T w = 264 MeV. mlAf) t(p2) sSzaiWp2) E R R [pbj p2) 4.45 -2.625 2.06 0.32 4.45 -1.875 4.25 0.40 4.45 -1.125 5.41 0.35 4.45 -0.625 0.25 0.26 4.75 -7.500 0.04 0.04 4.75 -6.500 7.89 11.05 4.75 -5.500 0.10 0.03 4.75 -4.500 0.24 0.08 4.75 -3.500 1.21 0.30 4.75 -2.625 4.80 0.64 4.75 -1.875 6.86 0.56 4.75 -1.125 6.85 0.48 4.75 -0.625 0.01 0.01 5.05 -7.500 0.42 0.14 5.05 -6.500 0.78 0.41 5.05 -5.500 1.03 0.59 5.05 -4.500 3.75 1.94 5.05 -3.500 3.49 1.18 5.05 -2.625 6.51 0.63 5.05 -1.875 10.91 0.57 5.05 -1.125 3.69 0.36 5.35 -7.500 0.34 0.12 5.35 -6.500 0.96 0.16 5.35 -5.500 1.46 0.34 5.35 -4.500 2.51 0.48 5.35 -3.500 4.42 0.49 5.35 -2.625 8.27 0.58 5.35 -1.875 7.53 0.35 5.35 -1.125 0.46 0.08 5.65 -6.500 0.44 0.10 5.65 -5.500 1.77 0.17 5.65 -4.500 2.94 0.26 5.65 -3.500 5.58 0.35 5.65 -2.625 6.17 0.32 5.65 -1.875 2.19 0.18 5.65 -1.125 0.06 0.05 Appendix C. Tables of ir+p —> 7r + 7r + n Cross Sections 188 TT+P 7r + 7r + r i at = 264 MeV. t(p2) E R R (pb/p2) 5.95 -7.500 0.01 0.01 5.95 -6.500 0.42 0.20 5.95 -5.500 0.57 0.10 5.95 -4.500 2.04 0.16 5.95 -3.500 2.82 0.17 5.95 -2.625 2.15 0.22 5.95 -1.875 0.54 0.22 6.25 -6.500 0.02 0.01 6.25 -5.500 0.39 0.16 6.25 -4.500 0.50 0.15 6.25 -3.500 0.59 0.14 6.25 -2.625 0.43 0.15 Table C.31: Table of double differential cross sections for the (n+TT+) channel at 264 MeV. TT+P -> Ti+7i+n at TTT = 284 MeV. t(p2) E R R (pb/p2) 4.15 -7.500 0.27 0.07 4.15 -6.500 0.38 0.08 4.15 -5.500 0.74 0.12 4.15 -4.500 1.03 0.12 4.15 -3.500 1.24 0.14 4.15 -2.625 1.80 0.20 4.15 -1.875 2.86 0.26 4.15 -1.125 1.76 0.21 4.15 -0.625 1.67 1.26 4.45 -7.500 0.12 0.06 4.45 -6.500 0.16 0.06 4.45 -5.500 0.77 0.34 4.45 -4.500 0.58 0.11 4.45 -3.500 1.46 0.20 4.45 -2.625 2.56 0.31 4.45 -1.875 4.23 0.47 Appendix C. Tables ofn+p -> 7 r + 7 r + n Cross Sections 189 n+p -> 7r + 7r + n at = 284 MeV. <A\?) t (p2) -^(pb/p2) E R R (pb/p2) 4.45 -1.125 7.83 0.56 4.45 -0.625 0.38 0.22 4.75 -7.500 0.08 0.04 4.75 -6.500 0.05 0.03 4.75 -5.500 0.29 0.08 4.75 -4.500 0.33 0.07 4.75 -3.500 1.06 0.22 4.75 -2.625 3.23 0.58 4.75 -1.875 6.62 0.63 4.75 -1.125 8.18 0.44 4.75 -0.625 0.21 0.10 5.05 -7.500 0.75 0.64 5.05 -6.500 0.16 0.09 5.05 -5.500 0.41 0.12 5.05 -4.500 0.62 0.20 5.05 -3.500 1.70 0.39 5.05 -2.625 5.42 0.70 5.05 -1.875 8.63 0.68 5.05 -1.125 8.07 0.81 5.35 -7.500 1.47 0.53 5.35 -6.500 2.12 1.49 5.35 -5.500 3.10 3.35 5.35 -4.500 1.31 0.68 5.35 -3.500 4.38 0.85 5.35 -2.625 7.37 0.75 5.35 -1.875 12.70 0.72 5.35 -1.125 3.56 0.32 5.65 -7.500 1.27 0.20 5.65 -6.500 1.17 0.31 5.65 -5.500 1.17 0.44 5.65 -4.500 4.25 0.76 5.65 -3.500 6.85 0.69 5.65 -2.625 8.85 0.68 5.65 -1.875 9.29 0.48 5.65 -1.125 0.63 0.19 5.95 -7.500 0.83 0.14 Appendix C Tables of n+p —> 7 r + 7 r + n Cross Sections 190 7T+p TT+vr+n at = 284 MeV. ml, (p2) t(p2) E R R (pb/p2) 5.95 -6.500 1.57 0.19 5.95 -5.500 2.37 0.31 5.95 -4.500 4.25 0.47 5.95 -3.500 5.46 0.42 5.95 -2.625 8.20 0.45 5.95 -1.875 3.50 0.26 5.95 -1.125 0.05 0.02 6.25 -7.500 0.27 0.10 6.25 -6.500 0.88 0.13 6.25 -5.500 2.17 0.19 6.25 -4.500 3.64 0.25 6.25 -3.500 5.91 0.30 6.25 -2.625 4.02 0.27 6.25 -1.875 0.38 0.08 6.55 -7.500 0.10 0.06 6.55 -6.500 0.29 0.10 6.55 -5.500 0.68 0.11 6.55 -4.500 1.42 0.14 6.55 -3.500 1.84 0.16 6.55 -2.625 0.61 0.17 6.55 -1.875 0.02 0.01 6.85 -6.500 0.01 0.01 6.85 -5.500 0.34 0.11 6.85 -4.500 0.34 0.10 6.85 -3.500 0.28 0.08 6.85 -2.625 0.07 0.02 Table C.32: Table of double differential cross sections for the (ir+ TT+) channel at 284 MeV. 7T+p -> ir+it+n at T, = 305 MeV. mlAv2) H^2) J ^ W ) ERR (pb/p2) 4.25 -9.500 0.49 0.20 Appendix C. Tables of n+p —> 7 r + 7r + n Cross Sections 191 T T > TT+TT+W at T„. = 305 MeV. mL (M2) t (p2) ISSHWP2) E R R (^/// 2 ) 4.25 -8.500 0.21 0.07 4.25 -7.500 0.29 0.08 4.25 -6.500 0.48 0.10 4.25 -5.500 0.93 0.14 4.25 -4.500 1.31 0.14 4.25 -3.500 1.36 0.16 4.25 -2.500 2.46 0.21 4.25 -1.500 3.05 0.36 4.25 -0.500 1.34 0.22 4.75 -9.500 0.06 0.06 4.75 -8.500 0.03 0.02 4.75 -7.500 1.91 1.76 4.75 -6.500 0.17 0.06 4.75 -5.500 0.32 0.07 4.75 -4.500 0.78 0.18 4.75 -3.500 1.45 0.25 4.75 -2.500 2.14 0.27 4.75 -1.500 6.53 0.39 4.75 -0.500 1.94 0.19 5.25 -9.500 0.06 0.06 5.25 -8.500 0.02 0.02 5.25 -7.500 0.31 0.11 5.25 -6.500 0.26 0.10 5.25 -5.500 0.23 0.06 5.25 -4.500 0.85 0.23 5.25 -3.500 1.83 0.46 5.25 -2.500 5.22 0.50 5.25 -1.500 9.90 0.43 5.25 -0.500 0.45 0.06 5.75 -9.500 0.66 0.14 5.75 -8.500 0.57 0.18 5.75 -7.500 0.72 0.21 5.75 -6.500 0.39 0.18 5.75 -5.500 1.07 0.43 5.75 -4.500 3.38 1.05 5.75 -3.500 4.28 0.57 5.75 -2.500 8.30 0.55 5.75 -1.500 6.58 0.27 Appendix C. Tables of n+p —> 7r + 7r + n Cross Sections 192 TT+P 7l+^+n at % = 305 MeV. < ( M 2 ) t(p2) E R R (pb/p2) 6.25 -9.500 0.16 0.08 6.25 -8.500 0.65 0.11 6.25 -7.500 1.29 0.18 6.25 -6.500 1.27 0.23 6.25 -5.500 1.41 0.29 6.25 -4.500 3.18 0.38 6.25 -3.500 5.61 0.41 6.25 -2.500 7.33 0.32 6.25 -1.500 1.76 0.15 6.75 -8.500 0.17 0.08 6.75 -7.500 0.45 0.09 6.75 -6.500 1.27 0.14 6.75 -5.500 2.38 0.19 6.75 -4.500 3.93 0.24 6.75 -3.500 4.64 0.24 6.75 -2.500 2.80 0.20 6.75 -1.500 0.15 0.08 7.25 -7.500 0.02 0.02 7.25 -6.500 0.12 0.05 7.25 -5.500 0.40 0.08 7.25 -4.500 0.92 0.11 7.25 -3.500 0.69 0.10 7.25 -2.500 0.12 0.03 Table C.33: Table of double differential cross sections for the (n+ 7r + ) channel at 305 MeV. Appendix C. Tables ofn+p —> -n+Ti+n Cross Sections 193 n+p -»• Tr+TT+n at T, = 223 MeV. sgrW/*2) E R R (pb/p2) 4.15 3.28 0.28 4.45 4.56 0.44 4.75 6.64 0.45 5.05 1.81 0.66 5.35 CX26 0.10 Table C.34: for the (TT+ TT+) channel at 223 MeV 7T+J9 TT+TT+U at T, = 223 MeV. t(p2) ^(pb/p2) E R R (pb/p2) -5.50 0.03 0.03 -4.50 0.25 0.07 -3.50 0.94 0.11 -2.625 2.09 0.18 -1.875 2.25 0.28 -1.125 0.63 0.07 Table C.35: g for the (TT+ TT+) channel at 223 MeV Appendix C. Tables ofix+p —>• -K+n+n Cross Sections 194 7i+p -»• 7 r + 7 r + n at T w = 223 MeV. cosd (/*&) E R R (pb) -0.90 2.08 0.99 -0.70 2.24 0.33 -0.50 2.37 0.38 -0.30 3.02 0.37 -0.10 2.80 0.34 0.10 3.33 0.45 0.30 3.33 0.42 0.50 2.67 0.34 0.70 1.83 0.28 0.90 1.17 0.21 Table C.36: ^ for the (TT+ TT+) channel at 223 MeV TT+P -> 7i+7r+n at = 243 MeV. <AS) ifirWl?) ERR (MS) 4.15 8.08 1.54 4.45 8.06 0.54 4.75 12.25 1.14 5.05 10.97 0.52 5.35 8.83 0.44 5.65 1.62 0.25 Table C.37: -M- for the (TT+ TT+) channel at 243 MeV Appendix C. Tables of n+p —> 7 r + 7 r + n Cross Sections 195 TT+P -»• vr+TT+n at = 243 MeV. t(p2) £ W ) E R R (pb/p2) -7.50 0.27 0.29 -6.50 0.49 0.35 -5.50 0.53 0.09 -4.50 1.42 0.21 -3.50 2.75 0.26 -2.625 4.50 0.24 -1.875 5.66 0.22 -1.125 2.48 0.16 Table C.38: ^ for the (TT+ TT+) channel at 243 MeV TT+P _> vr+vr+n at T,,- = 2 4 3 MeV. cosQ E R R (pb) -0.90 6.58 1.53 -0.70 7.00 0.57 -0.50 7.53 0.56 -0.30 6.81 0.51 -0.10 8.78 1.30 0.10 8.22 0.90 0.30 8.12 0.64 0.50 8.64 1.77 0.70 7.47 0.64 0.90 5.57 0.65 Table C.39: ^ for the (TT+ TT+) channel at 243 MeV n+p -+ Tr+ir+n at T, = 264 MeV. miA»z) agrW/x 2) E R R (pb/pF) 4.15 8.85 0.67 Appendix C. Tables of ix+p —>• n+'K+n Cross Sections 196 TT+P _ , TT+TT+TI at T T = 264 MeV. m2^ (p2) a & r WS) E R R (pb/p2) 4.45 10.49 0.61 4.75 23.47 11.08 5.05 25.42 2.49 5.35 21.89 0.95 5.65 17.04 0.56 5.95 7.88 0.40 6.25 1.81 0.29 6.55 0.04 0.01 Table C.40: for the (ir+ TT+) channel at 264 MeV TT+P —• TT+TI+TI at Tn = 264 MeV. t(p2) § (MS) E R R (pb/p2) -7.50 0.28 0.06 -6.50 3.23 3.32 -5.50 1.71 0.22 -4.50 4.02 0.62 -3.50 6.01 0.42 -2.625 9.74 0.36 -1.885 10.46 0.31 -1.125 5.79 0.23 -0.625 0.58 0.55 Table C.41: g for the (TT+ TT +) channel at 264 MeV Appendix C. Tables of n+p —• n+n+n Cross Sections 197 n+p -»• TT+vr+n at T w = 264 MeV. cos9 E R R (pb) -0.90 11.63 0.87 -0.70 15.02 0.90 -0.50 17.78 1.08 -0.30 31.60 16.67 -0.10 19.93 2.79 0.10 19.32 1.61 0.30 18.56 1.27 0.50 15.84 1.17 0.70 15.68 1.10 0.90 9.96 0.65 Table C.42: ^ for the (n+ n+) channel at 264 MeV 7T+p _> TT+yr+n at T w = 284 MeV. ™L ( M 2 ) E R R (pb/p2) 4.15 9.46 0.54 4.45 14.23 0.73 4.75 15.52 0.77 5.05 20.79 1.25 5.35 31.19 3.95 5.65 29.06 1.34 5.95 23.42 0.84 6.25 16.17 0.51 6.55 4.81 0.29 6.85 1.03 0.17 Table C.43: for the (n+ n+) channel at 284 MeV Appendix C. Tables of n+p —>• 7 r + 7 r + n Cross Sections 198 TT+P -»• TT+TT+W at T T = 284 MeV. t(p2) %(pb/p2) E R R (pb/p2) -7.50 1.55 0.26 -6.50 2.04 0.46 -5.50 3.61 1.03 -4.50 5.34 0.36 -3.50 9.05 0.40 -2.625 12.64 0.45 -1.875 14.47 0.42 -1.125 9.02 0.35 -0.625 0.68 0.38 Table C.44: ^ for the (TT+ TT+) channel at 284 MeV TT+P - > 7 r + 7 r + n at T-k = 284 MeV. cosO A W E R R (pb/pz) -0.90 17.42 1.23 -0.70 24.44 1.23 -0.50 27.05 1.56 -0.30 31.73 5.26 -0.10 30.46 2.60 0.10 25.27 1.49 0.30 28.87 1.65 0.50 23.46 1.11 0.70 22.50 1.10 0.90 17.32 1.01 Table C.45: ^ for the (?r+ TT+) channel at 284 MeV Appendix C. Tables of n+p —> 7 r + 7 r + n Cross Sections 199 7T+P TT+vr+r! a t T, = 305 MeV. •& Wf) E R R (pb/p2) 4.25 11.91 0.59 4.75 15.55 1.87 5.25 19.38 0.87 5.75 21.70 1.46 6.25 22.72 0.79 6.75 15.79 0.48 7.25 2.28 0.18 7.75 0.03 0.01 Table C.46: for the (TT+ TT+) channel at 284 MeV 7T+p _ , Tr+Tr+n at % = 305 MeV. t(p2) f Wp2) E R R (pb/p2) -9.50 0.72 0.14 -8.50 0.82 0.12 -7.50 2.49 0.89 -6.50 1.98 0.18 -5.50 3.37 0.29 -4.50 7.19 0.60 -3.50 9.94 0.46 -2.50 14.18 0.45 -1.50 13.98 0.38 -0.50 1.86 0.15 Table C.47: <g for the (vr+ vr+) channel at 305 MeV Appendix C. Tables of Tr+p —> •K+ir+n Cross Sections 2 0 0 TT+P _ , TT+yr+n at T T = 3 0 5 MeV. cos9 E R R (fib) - 0 . 9 0 1 8 . 0 1 1 .10 - 0 . 7 0 2 6 . 9 9 1.41 - 0 . 5 0 3 0 . 6 0 1.69 - 0 . 3 0 3 2 . 7 8 4 . 6 2 - 0 . 1 0 3 0 . 6 7 1 .57 0 . 1 0 3 5 . 5 7 2 . 9 6 0 . 3 0 3 2 . 6 8 1.75 0 . 5 0 2 9 . 3 0 1.45 0 . 7 0 2 8 . 8 0 1.56 0 . 9 0 1 8 . 9 8 1 .19 Table C.48: ^ for the (TT+ TT+) channel at 3 0 5 MeV Appendix D Explicit Functional Forms for Gj The expressions defining the functions, GII'f(s, s'), of Equation 7.85 are presented in this appendix. These were taken from the work of Patarakin et al. [54]. "-^0,0 1 7T s - 4 (s'-s)(s'-A) + s' + s - 4 - 1 2(g - 4) 3s '(s '-4) (D.131) ^ 0 , 0 — 7T 10 / s' , s' + s-A 3s' \ s - 4 In- - 1 -5(s - 4) 3s'(s' - 4) (D.132) ^o.o — 7T s'-Aj Is - 4 s' + s - 4 3 ( 8 - 4 ) ' s'(s' - 4) , (D.133) 1 7T 4 3 ( s - 4 ) s' + s - 4 1 -2 ( 3 - 4 ) 3s'(s' - 4) (D.134) r2>° 1 7T 10 3(5 - 4) '1 s' N 2 + s' - 4 7^1 s' + s - 4 5 ( s - 4 ) 18s '(s ' -4) (D.135) 201 Appendix D. Explicit Functional Forms for Gjf 202 ^1,1 1 7T •s' - 4 + s' — s s - 4 1 + 2s' s'-A 1 s 2 + s' - 4 . , s' + s - 4 ' In • 1 ^2,0 — — 7T 3s' I s - 4 s' + s-A 7' r<2>° — ^2,0 — — 7T 3s7 s - 4 s' + s - A s' - 1 -1 + 3(3 - 4) " 2s'(s' - 4)_ ( 3 - 4 ) " 3s'(s' -4 )_ : 5 ( s - 4 ) ' 6s'(s' - 4 ) J ' (D.136) (D.137) (D.138) 1,1 I 71 2s' V + s' - 4 s' + s-A -In \s-A In addition, <p{ are given by [54] 4>» = i -5(s - 4) 3s'(s' - 4) (13 ± 5 ) x 10~5(s2 - 16), bl(s) = (13 ± 6 ) x 10~ 5 s(s -4) , b\(s) = (3.0 ± 1 . 5 ) x 10~ 5 s (s -4) . (D.139) (D.140) (D.141) (D.142) 

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