DERIVATION OF AMPLITUDES FOR n — 2n REACTION CHANNELS USING THE OSET—VICENTE-VACAS MODEL, AND CALCULATION OF THE CHIRAL SYMMETRY BREAKING PARAMETER Neil Behzad Fazel B. Sc. (Applied Physics) Sharif University of Technology, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1992 © Neil Behzad Fazel 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. (Signature) Department of Physic The University of British Columbia Vancouver, Canada Date June 19, 1 DE-6 (2/88) 992 Abstract The Oset and Vicente-Vacas model for n N --> nnNreactionhsbuderiv theoretical values of the amplitudes and cross sections for the n - 2n reaction channels. It is shown that the N* --> N(nn)s—wave mechanism is required in order to obtain agreement with the world data for the n-p --> n -n+ and n-p --->n0n0n channels. The x2 analyses of the world data near threshold for all the channels strongly support the Weinberg's value of e = 0 for the chiral symmetry breaking parameter. The value of C, the factor related to N*Nnn coupling through the exchange of the scalar meson e , was estimated to be C = (-1.97 + 0.11) p-1. ii Table of Contents Abstract ii List of Tables v List of Figures vii Acknowledgements viii 1 Introduction 1 1.1 Overview 1 1.2 Chiral Symmetry and PCAC 3 1.3 Olsson and Turner's Work 4 2 7r — 27r Reaction Channels 9 2.1 Introduction 2.2 x2 Analysis of the World Data 9 11 2.2.1 The 7r-p —* 7r07r0n Channel 11 2.2.2 The 7r- p ---* 7r-7r+n Channel 19 2.2.3 The 7r+p -- 7r+7r+n Channel 21 2.2.4 The 7r-p --4 7r-7°p Channel 26 2.2.5 The 7r+p —* 7r+70p Channel 29 3 Discussion and Conclusion 33 A Derivation of the Amplitudes 41 111 A.1 The Oset and Vicente-Vacas Model 41 A.2 Expansion of the Lagrangians 43 A.3 Recipe for Obtaining the Amplitudes 46 A.4 Calculation of Some of the Amplitudes for the Three Channels 48 A.5 Taking Care of Spin 56 B Amplitudes for the 7r- p --* 71070n reaction C Amplitudes for the 7r- p 7r-ep reaction D Amplitudes for the 7r+ p —* 7r+7°p reaction E World Data for 7r — 27r Channels Bibliography 59 71 83 95 100 iv List of Tables 2.1 Results of the x2 analysis for the 7r07r0 channel 18 2.2 Errors in the x2 analysis of the 7r°7r0 channel 18 2.3 Results of the x2 analysis for the 7r-7r+ channel. 20 2.4 Errors in the x2 analysis of the 7r-7r+ channel. 20 2.5 Results of the x2 analysis for the 7r+7r+ channel, excluding the OMICRON data. 24 E.1 Batusov's 7r-7r+ data [BA65]. 95 E.2 Bjork's 7r-7r+ data [BJ80] 96 E.3 OMICRON's 7-7+ data [0M89a]. 96 E.4 Kravtsov's 7r+7+ data [KR78]. 97 E.5 OMICRON's 7r+7+ data [0M90] 97 E.6 Sevior's 7r+7r+ data [SE91]. 97 E.7 Lowe's 7r07r0 data [L091]. 98 E.8 OMICRON's 7r-7r0 data [0M8913]. 99 E.9 World data for 7+7r° at lower energies 99 List of Figures 2.1 The diagrams which do not vanish at threshold in the rr0 channel. . . 14 2.2 Comparison of the model's predictions for the 7r07r0 channel, with and without the N* 2-point diagrams (C = —2.13 it' and = 0.3) 14 2.3 7r07r0 two point diagrams 15 2.4 7rr0 three point diagrams 15 2.5 7rr0 three point diagrams with one delta intermediate state. 16 2.6 71-r0 three point diagrams with two delta intermediate states 16 2.7 7r07r0 two point diagrams with N* and delta intermediate states. 16 2.8 The surface plot of x2 for the 7rr0 channel 18 2.9 Feynman diagrams for the 7r-7r+ channel. 19 2.10 The surface plot of x2 for the 7r-7r+ channel 21 2.11 Feyn.man diagrams for the 7+71-+ channel. 22 2.12 x2 analysis of the 7r+7r+ channel. 23 2.13 World data for 7r+7r+ channel, and the theoretical prediction of the model for = 0.15 25 2.14 Feynman diagrams for the 7r-7r0 channel. 26 2.15 The contribution of different groups of diagrams to the total cross section for the 7r-7r0 channel with = 0. 27 2.16 (a) x2 analysis of the 7r-7r0 channel, (b) world data for the 7r-7r0 channel, and model's prediction for — 1 9 28 2.17 comparison of the cross sections given by the model for different values of the chiral symmetry breaking parameter vi 29 2.18 Feynman diagrams for the 7r+7r° channel. 30 2.19 X2 analysis of the 7r+7r° channel, and comparison of the model's prediction for different values of 4" 3.1 32 Comparison of model's prediction for the 7r+7r+ channel with 6 = 0 and 3.2 The contour plots of X2 for the 7r+7r- and 7r°7r° channels 3.3 World data for the 7r+7r-, 7r+7r+, 7r°7r° and 7r-7r0 channels, and model's prediction for C = —1.97 pc', 3.4 37 6 =--- 0 World data for the 7r+7r0 channel, and model's prediction for vii 35 38 6 =--- 0. . . . 39 Acknowledgements I would like to thank Dr. Richard Johnson for supervising this work. Thanks are also due to Eulogio Oset and Manolo Vicente-Vacas for making their Monte Carlo code containing the amplitude calculations for the 7r- p -4 7r-7r+n and 7r+p --+ 7r+7r+n channels available to us; as well as to Vesna Sossi, for her assistance, and Martin Sevior, for his comments. viii to my parents ix Chapter 1 Introduction The knowledge of 7r — 7 scattering amplitudes at zero relative momentum provides us with clues on the nature of chiral symmetry breaking. Experimentally this involves the study of Ke4 decay and pion-induced pion production (7r —27r) reactions. In the present chapter I will first give a broad perspective of the theoretical environment that makes the 7rN --4 7r7rN reactions experimentally significant, and will then review some of its aspects in greater detail. 1.1 Overview A good starting point for the discussion of 7r — 27r reactions would be chiral symmetry. This refers to the symmetry which would exist if the up, down, and one might add, strange quarks were massless; then the Quantum ChromoDynamic Lagrangian would consist of two parts, one for the right-handed quark fields and the other for the left-handed ones —without any coupling between them— i.e. handedness of the fields would exist in a symmetrical way; this is where the chiral symmetry gets its name. Symmetries in physics are accompanied by the conservation of some quantity; in the case of chiral symmetry this conserved quantity turns out to be the axial current. The methodology based on this symmetry, provides us with the only rigorous approach to low energy QCD. However in reality chiral symmetry is broken spontaneously, as well as explicitly. The difference between these two I will make clear later, but the consequence of chiral symmetry being explicitly broken is that the axial current will no 1 Chapter 1. Introduction 2 longer be conserved. Still it would be approximately conserved if the symmetry were only slightly broken; this leads to the idea of Partially Conserved Axial Current, or PCAC. In the sixties theorists were using the commutation relations between the currents, known as the current algebra, and the PCAC assumption, to calculate the matrix elements for emission and absorption of soft pions and determine the pion scattering lengths. Weinberg used PCAC and current algebra for calculations of 7 — 7 scattering lengths [WE66]; later he changed his approach and used phenomenological Lagrangians to reproduce his previous results for 7r —7r scattering [WE67]. However, his results were not entirely consistent with the work done by Schwinger [SC67] in which, using a nonoperator method and eliminating all reference to current algebra, he had constructed a 7r — 7 interaction Lagrangian which differed from Weinberg's by a term representing chiral symmetry breaking. Later it was shown by Olssen and Turner [0T68] that the Lagrangians of Weinberg and Schwinger were cases of the most general 7 — 7r Lagrangian derived in accordance with current algebra and PCAC; this Lagrangian had one free parameter, , to be determined experimentally, and the Lagrangians of Weinberg and Schwinger corresponded to = 0 and = 1 cases, respectively (Weinberg's later work in this area [WE68] has further implications for both Schwinger's and Olsson and Turner's results; this will be discussed in the final chapter). Weinberg, and Olsson and Turner also noted the similarity between near-threshold chiral symmetry breaking properties of the 7rN --> 77N reaction channels and the low-energy 7r — 7r scattering, which means that experimentally, the cross section measurements of 7 — 27r reactions near threshold can be used to study the nature of chiral symmetry breaking. 3 Chapter 1. Introduction 1.2 Chiral Symmetry and PCAC In what follows, the quark will be considered as a doublet containing the up and down quarks. This is a good approximation for pion-nucleon interactions, in light of the fact that the strange quark which is the lightest of the heavy quarks, is still about 200 MeV more massive than the u and d quarks. It is known that strong interactions conserve helicity. For massless quarks, the helicity of the quarks emerges as a quantity that is the same in all frames of reference (this is because for massless particles moving at the speed of light, only a boost to a frame of reference moving at a speed greater than that of light will reverse the direction of the particle's momentun, required to change the helicity). These two facts together suggest that, at least ideally, the worlds of the righthanded and lefthanded quarks are independent and do not interact. As a result, in the QCD Lagrangian for the free quarks, the fields representing left-handed and right-handed quarks will decouple to give rquark =__ 'IT L jooL +71 I T R ipAR (1.1) The left- and right-handed parts are each invariant under an SU(2) transformation and the Lagrangian as a whole is said to be invariant under an SU(2)L 0 SU(2)R transformation. This invariance leads to the idea of chiral symmetry, which is, however, spontaneously broken. A symmetry is spontaneously broken when the system has degenerate ground states not displaying the underlying symmetry of its Lagrangian. It has been shown that whenever a symmetry is spontaneously broken, there must exist massless bosons, called Goldstone bosons, whose interactions with the particles of theory cause the system to make transitions between the degenerate ground states. In this case the bosons are 7r's, K's and the 77 particle. These particles are of course massive, and the reason is that chiral symmetry, in addition to being spontaneously 4 Chapter 1. Introduction broken, is also explicitly broken, since in the real world, quarks, however relatively light some of them may be, are still massive. The consequence of this is that the Goldstone bosons acquire mass. This is why pions are massive. Their low mass in the hadron spectrum reflects the extent to which the chiral symmetry is explicitly broken. A massless pion is called a soft pion, and the limit ft —> 0 is the soft pion limit which is the basis of several theorems concerning pionic amplitudes [EW88]. Naively speaking, these low-energy theorems may have an accuracy of around 2%, however see [GL83]. It follows from Noether's theorem that the invariance of a Lagrangian under a transformation implies the existence of a conserved current. In the case of the invariance of QCD Lagrangian under chiral transformation, the conserved current is the axial current 1/) (x)-y 574 0(x) (1.2) where is the quark doublet and 7-i the Pauli isospin matrices. Under exact chiral symmetry, the conservation of current would make the divergence of the axial current vanish; since chiral symmetry, as pointed out, is an approximate symmetry, the axial current is rather partially conserved and one has 49„A'it —fu20i(x) (1.3) here fir, ft and 02(x) are the pion decay constant, pion mass and the pion field, respectively. Eq. (1.3) is the PCAC relation which is obtained in a natural way within the framework of chiral symmetry [EW88]. 1.3 Olsson and Turner's Work As mentioned earlier, Weinberg and Schwinger approached the problem of deriving 7r —7r Lagrangians differently and their results were not similar; Weinberg's Lagrangian 5 Chapter 1. Introduction was 2 ( Y-1±) G2 ,C7r7r 8m2 gA 04+ it,242] [2a, c-6." (9,2 - (1.4) (m is the nucleon mass), whereas Schwinger obtained firir 0421 (f0/)22[akt(T (9,2- +1/22- (1.5) 2Gm • 322-1- =- L3 = Using a more familiar notation, and noting the equality — these equations become einber g := 71-7 1 [ 4f 2 r S chwinger1 7 171" ,2 oil 021 (1.6) 02] (1.7) 2 02 op H2 4 4./7 the second term in the above Lagrangians, known as the symmetry-breaking term, differs by a factor of two in these equations; in order to resolve this ambiguity, Olsson and Turner [0T68] showed that the most general Lagrangian derived in accordance with current algebra and PCAC is "Crir (1.8) = (G/2rn)2(gv/gA)2 x [2(1 — 2gof,)(040)2 — 2g0f42(51-`0)2 -1(3gof, — 2h0f — 1)1'2(02)21 with go and ho as free parameters. They also defined the parameter later known as the chiral symmetry breaking parameter, as a combination of go, ho and f, in the form e = 2f,r(go 2h0f,) the choice of go = results, using Eq. (1.9), in ho firr 1 4f72. 1 (1 [020402 _ 2 (1.9) , and Eq. (1.8) reduces to 102(02)21 2 (1.10) so by choosing go, and using e as the new parameter, one obtains a family of Lagrangians which depend on the chiral symmetry breaking parameter as their single parameter; Chapter 1. Introduction 6 as will be shown in the conclusion to this thesis, there are further constraints which e has to obey, but for now it is clear that the choice of e = 0 in Eq. (1.10) gives the Weinberg Lagrangian, whereas e = 1 gives Schwinger's (the sign difference in the first term of Eq. (1.10) as compared to Eqs. (1.6) and (1.7) is due to Olsson and Turner using a different metric). Relating the experimental results from it — 27r scattering to the it — it scattering involves coming up with a theoretical model to describe the 7rN 7r7rN reaction. Olsson and Turner [0T68] used a model which included the it — it interaction Lagrangian .C„ (Eq. 1.10), as well as three 7r —N interaction Lagrangians, and ENN„, ENNiririr generated using Weinberg's covariant-derivative formalism, and calculated threshold amplitudes for two of the it — 27r reaction channels, namely 7r-p 7r-7r+n and 7r+p 7r+7r+n channels'. Although their results do not adequately reproduce the experimental data even at pion lab kinetic energies 20-30 MeV above threshold, they did derive equations which expressed the threshold amplitudes in terms of only the chiral symmetry breaking parameter e, and which are still widely used by experimentalists today. According to their calculations [0T69] ath(7-7+) ath(7-1-7r+) ath(7ro ) = 82 (—)2(-1.36 + 0.6e) 2 (L- )2(1.51 + 0.6e) 2 (-3- )2(2.11 — 0.3e) (1.11) (1.12) (1.13) fir where fir is the pion decay constant. They had earlier related the isospin 0 and 2 (s wave) scattering lengths to the chiral symmetry breaking parameter [0T68] through 'It should be noted that in their derivation they have included the contribution from the two-point diagrams (eg. Fig. 2.3), as well as the diagrams which contribute more significantly near threshold, i.e. the diagrams representing the pole and contact term contributions (Fig. 2.1). Chapter 1. Introduction 7 the equations (1.14) 0- 2 a2 2a0 — 5a2 = 6L where L = —a-. 8irg Solving Eqs. (1.15) (1.11)-(1.15) for the threshold amplitudes and the scattering lengths, one gets for the 7r-7r+, 7r+7r+ and 7r°70 scattering lengths (82)2 (_+) ao = [-0.120ath(7r-7+) + 0.038]m1 (82)2 fir a2") = [-0.048ath(7r-7r+) — a(++) o = [-0.120ath(7r+7r+) + (-8-2--) 2 0.382]/71-1 (++) a2 = [-0.048ath(7r+7r+) + ( -Tr) 2 0.0151M7T1 (00) 0.123]M1 82 (82)2 ao = [0.240ath(7r°7°) + (oo) a2 = [0.096ath(7r°7°) — -- ( 2) 2 fr (1.16) (1.17) o.3o5inc1 0.260]77C1 (1.18) In Olsson and Turner's formulation, scattering lengths can be obtained using Eqs. (1.16)-(1.18), having the threshold amplitudes for the reaction. Alternatively, one can obtain the scattering lengths by first deriving the value of , from a x2 analysis of the experimental data versus the theoretical cross sections, and then using Eqs. (1.14)(1.15), which explicitly give L ao --= — (14 — 50 8 (1.19) a2 = —— 4 ( + 2) (1.20) L Chapter I. Introduction 8 hence the importance of knowledge of the chiral symmetry breaking parameter. As pointed out earlier, the above model does not reproduce the data for all channels to the desired accuracy. It is the objective of this work to show that another presently available model, namely that of Oset and Vicente-Vacas, will provide a more complete theoretical description of the 7r — 27r reactions. Chapter 2 7r — 27r Reaction Channels 2.1 Introduction In order to test the predictions of chiral symmetry and soft pion theory, one has to resort to indirect means. This is because the particles of the theory have half lives in the region of 26 nanoseconds, making experiments involving direct scattering of pions unfeasable. One such indirect method is by studying the Ke 4 decay (IT this reaction has small branching ratio, but involves pions as the only strongly interacting particles in the outgoing channel, hence facilitating the data analysis. The other reaction, and the one we are concerned with here, is the pion-induced pion production reaction, or 7 — 27r scattering. For pion energies < 1 Gev, this is the major inelastic reaction between a pion and a nucleon. The threshold measurements of the 7r —27r reactions provide information on 7r —7r scattering at zero relative momentum (threshold being defined as the pion lab kinetic energy, 7,„ at which both of the outgoing pions are created at rest in the centre of momentum system, which, as a result of the conservation of momentum, requires the outgoing nucleon to be at rest as well). Of the reactions in the 7rN 7r7rN family, there are five channels which are amenable to experimental investigation. Since strong interactions are isospin invariant, the amplitudes for the different charge channels can be written in terms of isospin amplitudes A21,1„ where 1,,„ is the isospin of the two outgoing pions, which, when combined with the isospin of the outgoing nucleon, gives I, the total isospin; the isospin channels are 9 Chapter 2. 7r — 27r Reaction Channels (/, /„) 10 (,2), (, 1), (1,1), (12--, 0) (see [MM84]) and the charge channels, along with the isospin decomposition of their amplitudes, and their threshold energies, are 1. 7r-7r+ channel A(7r-p 7r-r+n) 3 — 1A.10 A A 31 - 5-"--111 rrh = 172.4 MeV 2. 7r+7r+ channel A(7r+p Titrh 7r+7r+n) 172.4 MeV 3. 7°7r0 channel A(7r-p 7r°70n) 1-5- A 32 + 32 A 1 0 rrh = 160.5 MeV 4. 7r-7r0 channel A(7-p 7r -70p) = —A + 10 32 11 3 1 2 A31 + \5A 3 rrh = 164.8 MeV 5. 7r+7r0 channel A(7r+p 7+°p) = — jA31 rrh = 164.8 MeV The wave function of a system consisting of two pions (i.e. two bosons) will be symmetric under interchange of the pions. The total wave function of a two-pion state may be written as zb(total) 0(space)a(spin)x(isospin); thus its symmetry is given by (-1)(-1)s+1(-1)/4.1. Since s 0 for the pion, and at threshold 0, in order to maintain the symmetry of the total wave function, I must be even. As a result, at threshold the odd isospin amplitude Lrir =- 1 will vanish, leaving only the isospin 0 and 2 amplitudes; for this reason these reactions have high isospin selectivity. Chapter 2. 7r — 27r Reaction Channels 11 2.2 x2 Analysis of the World Data As previously mentioned, the purpose of this work is to use the Oset and VicenteVacas model for 7rN 7r7rN reactions [0V85] to study the world data for 7F — 27r reactions channels. A description of the model, and the details regarding the derivation of theoretical amplitudes from it, are gived in Appendix A. The model has two free parameters; the chiral symmetry breaking parameter, and C, the factor related to the NN*(7r7r),,_,,,„ coupling; these are to be determined from a comparison of the theoretical cross sections with experimental data and to this end, the amplitudes have been integrated into a Monte Carlo code, which has as its output the theoretical values of cross section predicted by the model. In the following sections, these values are used with the world cross section data for the five 7r — 27r reaction channels, in a x2 analyses involving and C as the parameters. The analysis was done according to the relation x 2 = N E 1 1 E, [ — olh(c,e)] 2 So-fx (2.1) in which N is the number of experimental data used in the analysis, n is the number of parameters varied during the analysis and o and cr:h are the experimental and theoretical cross sections, resp.; (Sur is the experimental error and c is a factor related to C by the relation C = —c x 1.52 it'. In order to calculate the error in the x2-analysis due to the theoretical error, the following relation was used 2 v, icrs 8(X2) = N _ n —1 IT/ 1 (2.2) where 6c4h is the theoretical error as calculated by the Monte Carlo code. 2.2.1 The 7r- p 7r07r0n Channel The first reaction channel examined is the 7r07r0 channel. This is the only one in which all the outgoing particles are neutral, thus excluding any electromagnetic effects in the Chapter 2. 7r - 27r Reaction Channels 12 outgoing channel. The Feynman diagrams for the reaction are shown in Figs. 2.1, 2.3-2.7, and the corresponding amplitudes are given in Appendix B. There are 27 diagrams, with the largest contribution at low energies coming from pole and contact term diagrams, as well as the two-point diagrams with N* intermediate state and NN*(7r7r),_„„, coupling (Fig. 2.1); these are the only diagrams which do not vanish at threshold, and at energies as much as 60 MeV above the threshold, the amplitude from these 4 diagrams makes up 95% of the cross section with the rest of the diagrams included. The existence of the N* 2-point diagrams is an important feature of the Oset-Vicente-Vacas model; for the two channels in which the N* -÷ N(77)s-wave mechanism exists, i.e. the 7070 and 7r-7r+ channels, this mechanism is necessary to obtain agreement with experimental cross sections. A comparison of the results, with and without this mechanism, is shown in Fig. 2.2; the inclusion of N* mechanism gives agreement with Lowe's data [L091] for pion lab kinetic energies (T,) up to 212 MeV, and at higher energies it makes a significant contribution to the cross section. 1 The diagrams in Fig. 2.3 are two-point diagrams with nucleon intermediate state and 7rN s-wave amplitude. It should be noted that the phenomenological Lagrangian used in the Oset-Vicente model for NN7r7r coupling differs from the one used by Olsson and Turner, and for this reason, this model would give values for the cross section which are not necessarily equal to theirs, even when the same Feynman processes are considered. Three-point diagrams with nucleon and N* in the intermediate states are shown in Fig. 2.4. The diagrams in which the N* would be formed after the emission of one or two pions, were omitted on kinematic grounds; at 277,=300 MeV the total CM energy is around 1313 MeV (using N/Ts = [2mT, + (ft + m)2]1/2) and if an N* (instead of a 'this is in spite of the large energy denominators in the N* propagators [0V85]; see following paragraphs. Chapter 2. 7 — 27r Reaction Channels 13 nucleon) were to be formed after the emission of the 70, less than 1313-135=1178 MeV would be available for it; if this happened before the arrival of the incoming pion, even less energy would be available, as in that case 137, its total CM energy, is also deducted; using 137 , s+2\rs -m2 gives A - 329 MeV, leaving only 1178 — 329 = 849 MeV available for the formation of the N* at T,- = 300 MeV; since N* has a mass of (1440 + 40) MeV with a width at resonance of (200 + 80) MeV [EW88], the formation of such diagrams is not energetically favoured for pion lab kinetic energies < 300 MeV, in which we're interested (stated more formally, the propagator factor is too small). The remaining diagrams contribute to a lesser degree, however becoming important at higher energies, especially for the 7+7r° and 7r-70 channels (in the case of 7°70 channel, there are some cancellations among these diagrams, making them less important at lower energies). Using similar arguments as in the previous paragraph, among the diagrams in Fig. 2.7, the first and last one are expected to be important in our energy region. Chapter 2. 7r — 27r Reaction Channels 14 (2) ( 1) `... 1 7T' A 7T- li . \ a -,.. `... I 4 ,.. ....- a I . S.. -S itIT 1 n 71 .., 7r- , S.. .. ...." --./..... ---- •••. ,.., ...• ifTIP -........-,. p (4) (3) , • \ 4 s'rr 71-w / \ \ / / N' Figure 2.1: The diagrams which do not vanish at threshold in the 7rr0 channel. _ 1 1 1 1 Ii 1 1 1 I 1 1 1 I 1 I I 1 1 1 1 1 1 1 1 1 1 .1 I 1 1 1 I 1 1 1 I I 1 1 1 1 1 1 1 I I 1 1 1 1 1 1 1 I 1 111111111 - p__,71 0 7t. o 10' _o x [L091] with Ni 2—point diagrams _ without Nr diagrams 10. 1111111111111111111111111milmiliiiiiiiiiiiiipm111111111111111 160 180 200 220 240 260 280 300 pion lab kinetic energy (MeV) Figure 2.2: Comparison of the model's predictions for the 7070 channel, with and without the N* 2-point diagrams (C = —2.13 itt-1 and = 0.3). Chapter 2. 7r — 27r Reaction Channels (5 15 (6) ) \ \ \ _ \7T ki 7 \ \ lk 0 n P Iii / / n \ Tr' TIN \ \: — % I `1 o/ x IT ir / I t \ / \ 0, 7T ' 0 / \ \ / P n P - Figure 2.3: 7r07r0 two point diagrams (9) (10) \ \ / . lit _ 1 14 71- \ \ / I n I 1 _ / I I / 17 I / I P 0' 7ri 0' 71- i 7V0/ / / n 1 n / P n \ o / ni / / _ \ /I P nr 7T o/ 4 / / 1 ‘77777774 n n (12) i P p P (11) I / 0/0 7T i I _ 7T/ 0 / i it / / 1 / I I i P n Figure 2.4: 7r07r0 three point diagrams. n Chapter 2. 7r — 27r Reaction Channels 16 \ \ 7T0 /o 71- / _ 7T / / A° P I Tr 4 I / / / V 0 / i II \ n Io 71 - 4 I / I n A° P n I / A+ n \ I I 71: 4 1 _ \ 47 / / , k n P - 4 n / a 71: 71: 1 47T \ \ I A' P 0/ 7 4 / (18) 0 / / 71: 4 411: I P P _ r 1 r / V A° P (17) a! 4 1 I - 1 17T / I I 1 / I P I 7T 4 / I / n o I o / rt- 4 I _ 171: (16) al 7T (15) (14) (13) 0/ 4 / / / I V n P A° n Figure 2.5: 7r07r0 three point diagrams with one delta intermediate state. (20) (21) 01 71:4 / I 7 4 7 4 I 4 I t r I _ v n I I / I A° I 0! I n - I I P I 0! A° n I / i■ A.' p 0/ 7# A° n Figure 2.6: 7r07r0 three point diagrams with two delta intermediate states. (22) (24) \ 0/ \ _ 'Tr } )1 7T \ 1 0/ 71-4 / o 71: 4 / I I ■ N---lk•-,•0-7,-,-,0/ 0/ 71: 4 / / I _ '17 / / I i=---, P (25) (26) I o I 7r 4 74 1 / / I I h-7-m-,,-A P NI A. I o 1/ 71:4 704 I 4 7T- I (27) I I I / / \ \ I _ 471 I Figure 2.7: P 7r070 4* Nr o 7 4 / / / 7 1 /----10-0,0•th n 0! 71: 4 _ I V 1,-,-,--,,-77.4 n P N° A° two point diagrams with N* and delta intermediate states. n Chapter 2. 7r — 27r Reaction Channels 17 Analysis of the Data for the 7r07r0n Channel For the 7r°7r0 channel, at sufficiently low energies, the pole, contact and two-point N* diagrams are almost exclusively responsible for the cross section. In fact, at 283.99 MeV which is the highest energy for which experimental result is available from Lowe's measurement, they yield a cross section which around 86% of the cross section with all Feynman processes included. For this reason, this channel provides the least modeldependent estimation of the chiral symmetry breaking parameter. In the case of the channels without the N* mechanism, diagrams other than the pole and contact become important, particularly due to the fact that experimental data very close to the threshold are not available (see sections on 7r-7r0 and 7r+7r° channels). Of the 5 channels investigated, the 7rr0 is the one for which the most reliable near-threshold data is available [L091]. Measurements of this reaction involve the detection of the 4 -y's created as a result of decay of the 2 pions in the outgoing channel; due to the absence of Coulomb interaction between the outgoing particles, measurements closer to threshold can be performed. The threshold pion kinetic energy for this reaction is 160.5 MeV and in order to minimize the effect of the model-dependent parts of the Lagrangian, only Lowe's results up to 77„. = 219.15 MeV (i.e. cross section measurements at 13 energies) have been included in the analysis. At this energy (3-(pole,contact,2-point N*)/a(all diagrams) > 95%. The results of the x2 analysis, as well as the error in x2 caused by the theoretical uncertainty in the cross sections given by the Monte Carlo code, are given in Tables 2.1 and 2.2. Figure 2.8 shows the surface plot of the x2 VS. e and c. It can be seen that a sharp, well-defined minimum for x2 does not exist; in fact, by varying c, one can minimize x2 by using any value for e in the interval [-1.25,1.25] and probably beyond. To = 0 there corresponds C '--' —2/./-1, which is in agreement with previous calculations (see Appendix A). Chapter 2. 7r - 27r Reaction Channels c\e 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 -1.25 7.26 4.46 2.96 3.18 5.79 11.16 18.73 30.99 -1.0 8.34 4.90 3.05 2.77 4.59 8.83 16.59 27.10 -0.75 9.18 5.78 3.39 2.73 3.78 7.90 14.02 23.99 -0.50 10.09 6.40 3.83 2.56 3.18 6.16 11.83 21.37 18 -0.25 11.34 7.25 4.17 2.58 2.68 4.85 9.80 18.58 0 12.45 8.31 5.00 2.80 2.49 4.12 8.28 15.07 0.25 13.64 9.32 5.67 3.19 2.39 3.41 6.69 13.29 0.50 14.73 10.27 6.56 3.65 2.46 2.88 5.52 11.10 0.75 15.96 11.19 7.23 4.14 2.49 2.54 4.46 9.34 1.0 17.22 12.53 8.32 4.84 2.68 2.17 3.68 7.61 1.25 18.49 13.77 9.13 5.46 3.03 2.06 3.13 6.17 Table 2.1: Results of the x2 analysis for the 7rr0 channel c\e 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 -1.25 0.19 0.16 0.17 0.19 0.35 0.55 0.78 1.19 -1.0 0.20 0.16 0.16 0.16 0.29 0.45 0.72 1.06 -0.75 0.21 0.17 0.16 0.16 0.25 0.43 0.66 0.94 -0.50 0.22 0.18 0.16 0.16 0.20 0.36 0.56 0.91 Table 2.2: Errors in the x2 0 0.22 0.20 0.16 0.16 0.16 0.27 0.44 0.66 2 2.0 1.8 00 16 1.4 1.2 1.0 0.25 0.23 0.21 0.17 0.15 0.15 0.22 0.38 0.62 0.50 0.23 0.22 0.18 0.16 0.14 0.20 0.33 0.54 0.75 0.23 0.22 0.20 0.15 0.15 0.18 0.29 0.47 1.0 0.22 0.22 0.20 0.16 0.14 0.15 0.25 0.42 1.25 0.22 0.22 0.21 0.18 0.14 0.14 0.22 0.36 analysis of the 7r07r0 channel 110101). Ob. 31 x -0.25 0.22 0.19 0.16 0.15 0.17 0.30 0.50 0.79 I AO Af dp 0.500.751.01-25 .4' o $:10 P -0.250.25 -0.50 -0.75 0.8 0.6 -1.25 Figure 2.8: The surface plot of x2 for the 71- 07r0 channel. \ - Chapter 2. 7r — 27r Reaction Channels 2.2.2 The 7r-p 19 7r-r+n Channel The Oset-Vicente-Vacas model was first used to obtain cross sections for the 7r-7r+ channel [0V850090], and the results have been in very good agreement with experiment [0V85][S090][S092]. The processes contributing to this channel are shown in Fig. 2.9. There are 33 diagrams, of which the first four do not vanish at threshold. Similar to the previous channel, the N* mechanism makes an important contribution to the cross section. 7 — - r — TT 7T _ E ,or 7T + TV7 71--/ \TV A++ + r \ TV _ A' A A° TV,' P ■■ Ao ...---..., NI n P A in+ P N. ' 17T ' TV P A+ _ 71_44, n A' A' , ‘,, 7T— N1* n TV, AA 7v *,/ / 7V>' , \7T n A+ p a 71- A+ A A- \ 1 P 7T7 TT,' 7T-1 , — 7C— r 7v P — 7T 4 TV 4' TV,' n n %, 7v-.-,,, -.- / TV / 17T / r ,_______, / n — TV/ TV / P A" N. TV , P \TT P A— 1\1 Figure 2.9: Feynman diagrams for the 7r-7r+ channel. P —/ TV/ / ‘. 7T , NI. - A+ 7T+,' Th N. A+ Chapter 2. 7r - 27r Reaction Channels 20 Analysis of the Data for the 7r-7r+n Channel For the 7r-p 7r-7r+n reaction, experimental data from P3A65MBJ80] and [0M89a] have been used. The threshold is at 172.4 MeV and measurements up to 230 MeV (i.e. 7 cross section measurements) were included in the analysis. At this energy, the pole, contact and N* diagrams give 85% of the cross section. The results of the x2 analysis are given in Tables 2.3-2.4. In Fig. 2.10 the surface plot of x2 is shown. For this channel too, the analysis does not result in a well-defined minimum for x2. However for both channels containing the N* mechanism, e = 0 yields a value for C consistent with C = (-2.25 ± 0.75) it', obtained from the recent calculation based on the branching ratio of N* N(77)s-wave [S092] c\ 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 -1.25 9.58 5.01 7.74 19.82 47.36 86.87 138.18 228.01 -1.0 13.98 5.94 5.36 13.54 31.95 63.81 120.99 176.25 -0.75 18.06 9.11 4.68 8.55 22.50 48.87 89.77 151.61 -0.50 23.78 12.28 5.37 5.70 14.77 35.08 69.21 127.35 -0.25 30.14 17.49 8.46 5.04 10.24 24.88 51.94 95.93 0 37.03 21.52 11.47 5.12 5.71 16.80 39.28 76.46 0.25 42.99 28.33 15.59 7.43 5.07 11.52 26.13 55.72 0.50 50.17 34.18 20.93 10.70 5.12 7.16 20.37 43.30 0.75 55.82 41.21 26.63 13.66 6.49 5.02 11.67 32.20 .2 1.0 64.00 48.45 32.90 18.86 9.17 5.08 9.00 21.59 1.25 70.18 54.86 39.12 25.12 12.95 5.73 5.74 15.35 Table 2.3: Results of the x2 analysis for the 7r-7r+ channel. c\ 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 -1.25 0.51 0.38 0.50 1.14 2.22 3.26 4.67 7.28 -1.0 0.52 0.43 0.31 0.79 1.66 2.66 4.63 5.49 -0.75 0.55 0.49 0.33 0.56 1.21 2.21 3.42 5.32 -0.50 0.59 0.47 0.40 0.32 0.89 1.77 2.97 4.62 Table 2.4: Errors in the -0.25 0.66 0.52 0.50 0.33 0.64 1.33 2.35 3.60 x2 0 0.64 0.61 0.51 0.38 0.37 0.98 1.95 3.18 0.25 0.61 0.59 0.53 0.46 0.31 0.71 1.38 2.49 0.50 0.63 0.66 0.54 0.50 0.35 0.44 1.20 2.08 0.75 0.61 0.67 0.59 0.53 0.42 0.26 0.72 1.72 1.0 0.59 0.65 0.59 0.58 0.48 0.33 0.58 1.18 1.25 0.56 0.62 0.67 0.58 0.50 0.37 0.32 1.00 analysis of the 7r-7r+ channel. In the next chapter it will be shown that the possible values that e can take on is 2This value supercedes the the original value from branching ratio calculation which Oset VicenteVacas had obtained [0V85]. Chapter 2. 7r — 27r Reaction Channels 21 constrained by further theoretical arguments as well as the consideration of the results of the X2 analyses of all channels; this will then be used to determine C. 228 X 2 Figure 2.10: The surface plot of X2 for the 7r-7r+ channel. 2.2.3 The 7r+p —* 7r+7r+n Channel The third reaction channel to be investigated is the 7r+7r+ channel. The diagrams for this reaction are shown in Fig. 2.11. All the amplitudes, except those corresponding to the 3-point diagrams with N* and A intermediate states (which have not been included in the present analysis) have been calculated in [0V91]. For the 7r+p .--4 7r+7r+n channel, charge conservation does not permit two-point diagrams with N* in the intermediate state and NN*77(6) coupling (N* has charge states similar to nucleon), thus at low energies (where diagrams with A intermediate states do not contribute significantly) the Chapter 2. 7r — 27r Reaction Channels amplitudes from this model can be used in Olsson and Turner's formula (Eq. 1.17) to obtain the scattering length (provided the effect of using different N1\177-7 Lagrangians is not significant). For this channel, the contribution to the cross section at T„ = 200 MeV (highest energy in Sevior's measurement), of three-point diagrams with only nucleon, nucleon-IX and two A intermediate states, is about 28% of the total cross section with all the diagrams. liT * 71.1. 37T.' 7T 4' • ir • Tr/ 7r,4/ n 4' p 'VT. n 7T+ i ' \ 7I* % P p Tr+ \ + p 7T A++ • n 11.. i A+ 4 % \ lv. P « • P n A' vj I n A % _I p i n p 1 n' I I p A- '1 A° n A- n 7T+ n*/ \An+ P n k" I %,4 I \ ir+ I \ .„... ,' 1 A. /V*/ 7r7 55 A° IT • re , P 7r+1 ' A° IT I 1 ... ' n P 71-7 n 7T+,/ IT ,I A*+ ?To/ 1/7T , ,IT.,, re-,' 7T \ /I r 22 77 Nt n 7Ty 7T+ „ P Nt Figure 2.11: Feynman diagrams for the 7r+7r+ channel. Analysis of the Data for the 7r+7r+n Channel The experimental experimental data used for the analysis of the 7r+p 7r+7+p reaction is from [KR78], [0M90] and [SEE] up to T, 226 MeV at which the amplitudes from the pole and contact diagrams make up ,-- 61% of the total cross section. There are discrepancies Chapter 2. 7r — 2r Reaction Channels 23 in the data in the energy region 190-255 MeV; the model favours the measurements by Sevior et al. and Kravtsov et al. over those by the OMICRON group. For this reason the x2 analysis was done with and without the OMICRON data. As can be seen in Fig. 2.12, excluding the OMICRON data results in a smaller value for x2, and the inclusion 20 7T + + + p —>ir -rr n 1 i 15 X 0 _ - 2 with OMICRON data x without OMICRON data D E o 0 10 s ± ->r< 0 0 x 0 0 z 5 o 0 x o 0 0 o 0 D x I —1.0 „ I I -, , —0.5 In . ._ T< 3-< x 0 o x x xxx x 1 ' 0.0 ' S s _ _ - " i ' ' " I 1.0 0.5 e Figure 2.12: x2 analysis of the 7r+7r+ channel. of the OMICRON data results in a widening out of the minimum region away from zero; as will be shown in the Discussion, it is expected on theoretical grounds that the value of lie close to zero or be negative. Also the lower-energy OMICRON data is larger than both Sevior's and Kravtsov's data, and the isospin 2 scattering length obtained by the OMICRON group is inconsistent with both Ke4 and chiral perturbation theory values (see [SE91]). Consequently it is believed that the exclusion of their data (and Chapter 2. 7 - 27r Reaction Channels 24 using the cross sections at the remaining 5 energies) yields a more accurate estimate of the chiral symmetry breaking parameter.' The results of the x2 analysis of data from [KR78] and [SE91] are given in Table 2.2.3. It follows from the x2 analysis that = 0.15 + 0.1. In Fig. 2.13 the world data has been compared with the theoretical cross sections x2 8(X2) x2 8(X2) X2 6(X2) -1.0 11.89 0.19 -0.2 2.60 0.10 0.4 2.33 0.17 -0.9 10.83 0.18 -0.1 2.01 0.12 0.5 3.54 0.23 -0.8 9.48 0.18 -0.05 1.66 0.12 0.6 4.97 0.30 -0.7 8.41 0.16 0 1.39 0.11 0.7 7.64 0.43 -0.6 6.91 0.16 0.05 1.22 0.11 0.8 9.81 0.49 -0.5 5.62 0.15 0.1 1.04 0.10 0.9 12.69 0.57 -0.4 4.57 0.15 0.2 1.04 0.11 1.0 17.11 0.76 -0.3 3.54 0.13 0.3 1.62 0.16 Table 2.5: Results of the x2 analysis for the 7+7+ channel, excluding the OMICRON data. for = 0.15. There is good agreement with Sevior's and Kravtsov's data, as well as with OMICRON's higher-energy results. 'Also in the isospin amplitude analysis of Burkhardt and Lowe [BL91], data in this region for the -+ -+ channel from all experiments were excluded, in order to make all the available data from all channels internally consistent below 284 MeV. Chapter 2. 71- - 27r Reaction Channels I I I I I I I I I 25 I I I I I I I I I I I I I I I 1O -1 "10-3 I 150 i IIIIIIIIIIIIIIIIIIIIII 200 300 350 400 250 pion lab kinetic energy (MeV) Figure 2.13: World data for for e = 0.15. 7F+71-+ channel, and the theoretical prediction of the model Chapter 2. 71- — 27r Reaction Channels 2.2.4 The 7r-p 26 7r-70p Channel The next reaction studied is the 7r-7r0 channel. The Feynman diagrams contributing are shown in Fig. 2.14 and the amplitudes are given in Appendix C. Diagramatically, (2) (1) if it 7T:s IT it = (4) (3) 's \7 1/4 P (6) n P P , , \71- _ P (9) it°, ,4 / P N. • (1 1) _ n P A° • p \Tv p An n _ (19) / P N' P A0 P p A°(23) A 7T/ A++ A++ (25) (12) \ 7v- (16) - 7T , -1 A° / P P P n N' P 0 - (29) 4171- 7T1 • P N P P k 71- (14) 0 7T„, _ P P 7T 41 P P (18) _ _ 7T1 \IV p - A it] • it / 0 • (17) a - 71-1/ A0 n p A P IV P 71- • A A++ n p A P (22) -/ P P p (21) • A++ kit • / p 0 (24) (26) - N' • • 7v- p iT:41 P %.7r- A* 7e/ A 4 , (13) A++ 7Ti, 7T? P 7T, n % (30) _ IT , ;, A P (27) 7T,T' i n" P P , \ a • P p P v A° P (8) 7T74' P n P P TT \Tv A° p ' 01 „- 7T4-, Ir.:, P it, IT1 p P 0, i (20) rt- P P / \--4/ • 7T\ • .71- 7T/ , 7% _ 7T4' ‘,7T • .7v° , _ 71 P n (10) 7T-• P (15) ,ro,, _ \IT ,f n Tv-4/ • .7 7T: \ • 7T / fr/I • ,/ ' • 7v;• i \TT p 4 IT/- iy P (7) ,7T0 P ,, P I _ \\,, 7T, 0, in- Irl ‘s VT , • • n (5) 4/ • 71-/ '‘JT 7T:'' • • • ••• P P (31) , A° (28) 0 - TT / 4 N P - P N' P N' 0 / 7, % IT, / P ■ " A++ (32) S kit I A* N' 7T - P no, P Figure 2.14: Feynman diagrams for the 7r-7r0 channel. the notable difference between this channel and the 7r-7r+ and 707r0 ones is the absence of two-point diagrams with N* intermediate state and E coupling. Figure 2.15 shows Chapter 2. 7r — 27r Reaction Channels 27 that diagrams involving A's contribute significantly. In fact, the pole and contact term diagrams alone cannot reproduce the experimental cross sections even at low energies; already at around 20 MeV above the threshold 17, of 164.8 MeV, all the other diagrams combined give a cross section which is as much as 30% of the total cross section. 1 ii 1 ii 1 1 1 1 1 1 1 1 1 ii 1 1 1 1 1 1 Ii 1 1 1 1 1 1 1 ii ii 1 ii 1 ii 1 1 1 1 1 1 1 -0 71-- —>7V 7T p 1-- b 10'3= [0M89b] egaacriddiagrams oole,gontact and 2—point diagrams pole,contact,2—point, and 3—point with n or N intermediate state uole,gontact 2—point, 3—point ■qnd 1—delta diagrams all diagrams 10 -= ' 160 1111111111111111111111111111111111 200 240 280 320 360 pion lab kinetic energy (Mev) Figure 2.15: The contribution of different groups of diagrams to the total cross section for the 7r-70 channel with e = 0. Analysis of the Data for the 7r-7r°p Channel The experimental data available for this channel is scarce, with few reliable measurements at low energies, partly as a result of the large background, mostly elastic scattering, which exists for this reaction (at around T7, = 190 MeV, where 7r-p 7r-p Chapter 2. 7r — 27r Reaction Channels 28 reaction peaks, the signal to noise ratio is as low as 10-5). For the analysis in this section, only the data in [0M89b] were used, and the older data which is for T, > 275 MeV, were discarded (see Ref. [0M8913] for references to these). However due to the large error bars in the OMICRON measurements at lower energies, the complete set of data (a total of 9 cross sections) was used in the analysis, even though at T, = 331.6 MeV (which is their highest energy) the contribution from diagrams other than pole and contact term is very large (--, 85%). The result of the analysis is shown in Fig. 2.16. 3.0 _ - 111111111111111111111111111111 1111111111 1 -1 -i 1 2.5 — i IT - p ->71" -7r° p 1 is minimized somewhere it II 1 1 1 1 1 1 1 1 (a) - _ 2.0 — _ x' _ _ 1.5 — _ _ 1.0 — _ _ 0.5 1 x2 _ _ _ 11111111111111111111111111111111111111111111 —1 0 1 2 3 i 10-2 150 „,,i,„,„,i,,‘, 200 250 300 i 350 pion lab kinetic energy (MeV) Figure 2.16: (a) x2 analysis of the 7r-7r0 channel, (b) world data for the 7r-7r0 channel, and model's prediction for 6= 1.9. between 6= 1.2 and 6 = 2.2. However, it is noted that since the pole term diagram's contribution is significant (compared to other diagrams) only at low energies, a is not very sensitive to the variations in making x2 insensitive to 6 at most of the energies used in the analysis, thus 6. Indeed, for 6 = 0 one gets x2 es, 1, compared to x2 es, 0.6 Chapter 2. 7F — 2r Reaction Channels 29 obtained for e = 1.8. As can be seen from Fig 2.17, e E [-1, 3] gives decent fits to OMICRON's data. In order to do a more accurate and model-independent estimate of e, more 11111111 11111111 11111111111111111111111111111111111111111111111111111111111111 III illimmilimuly111111111mmililin11111111111111tylimminimin1 180 200 220 240 260 280 300 320 340 pion lab kinetic energy (MeV) Figure 2.17: comparison of the cross sections given by the model for different values of the chiral symmetry breaking parameter. measurements with better statistics near threshold are required, so that the effect of diagrams other than the pole and contact term will less dominate. 2.2.5 The 7r+p --÷ 7+71-°p Channel The last channel studied is the 7+7° channel. The Feynman diagrams are shown in Fig. 2.18 and the corresponding amplitudes are given in Appendix D. The analysis of this reaction is the most model dependent among all channels, since already at — 5 MeV Chapter 2. 7r — 27r Reaction Channels 30 above the threshold energy of 164.8 MeV, diagrams other than pole and contact make up 35% of the cross section. (1) (2) IT+ - 171 + 7T IT 7T P P P P (4) (3) 0 7r+ / ‘s . . 7T/ 7-0•, , , . •asTr + 7+,a• , , • •‘.7T ,... , n P P P P P (6)(7) o , (70). 7T•i iv A , 7+; 7T / 1 ■ + ‘,I 7r. t17T 1 f , II , I , I , , n p n n p P P P (9).74, .(10) no/ \ * if,/ 7/a, vv .,, Tr , V 4 / /' %I It / P P A++ P L\ P P (13) (14) —1 i /V i I kn + 71-/ it; , IVT + P n s‘ P A, (17.) 714' , A++ p 7Ti . . / , P (21) o rr./ ., 7V4' Ir 7T1 r / 1 o P L0A+ P (23) Tr 4 IT P P A++ A++ N. (27) 7T I" A° P P A+ (18) 7T+,/ / n '+ ‘,A n- I P A++ A+ P * (22) , I 7T . 71-/ / 77 I 'I / ./ P P A A (24) (5) • + \Tv /7T+ P N A++ A P P (9) 0/ 7T , I TV / // \ Tv+ , ■ n P P P (11) 712' i 7T * • '% + s1 kir r l / N ' P P P A° (15) o • I Tri r ni r ' * 417r p. p° ' n p (19) ,■ + 4,71- 71-/ / r 1 P A+ (25) Tr N . P 7r+ I VT N P o ' :• Tr. P r (28) 71/ P P 0 7TI / / . . P . TV,/ ,/ . / A++ P ,r P Figure 2.18: Feynman diagrams for the 7r+7r0 channel n 7Ti+. , i P, A° 0 . 71-i 7r / \ 7T+ / P A+ A° P / "4/ Tk71" A P (29) 71" / VT ■ P N , N 7T • P A° Chapter 2. 71 — 27r Reaction Channels 31 Analysis of the Data for the 7+7°p Channel Few measurements have been done for this channel [BA75][AR72] [DE66][BA63] and the existing data points are far from the threshold energy of 164.8 MeV, the lowest energy measurement being at T, = 230 MeV where pole and contact diagrams together give a cross section which is only — 20% of the total cross section. This, as well as the fact that both energies and cross sections have large error bars, prevent the determination of using x 2 analysis. Figure 2.19 shows the x2 analysis of this channel using all the world data; in can be seen that x2 does not have a minimum in the interval [-2,2]; this can be misleading, since the comparison of the cross sections for different values of (in the lower part of the figure) shows that only at energies sufficiently close to threshold will the model become sensitive enough to distinguish between different values of in that interval; in fact, excluding the measurement by Barnes et al. which was done using hydrogen bubble chamber at CERN in 1963, the model's prediction goes through the error bars for any value of c between -2 and 2. As a result, until more precise measurements at lower energies are done, we cannot use this channel to unambiguously derive a value for the chiral symmetry breaking parameter. However, it should be noted that = 0 still provides a good fit to the data. Chapter 2. 7r — 27r Reaction Channels .95 1 1 1 I I I I 32 I I I I I I I I I I I I I I I I1 1 1 1 + +o 7T p-71- 'TT p .85 2 1 .75 — X - .65 - - -r .55 II 1 1 1 I 1 1 -2 1 -1 1 I 1 1 1 1 1 1 1 1 o 1 I 1 1 1 i 1 1 - 2 e 11111111 1111111 1 111 11 11 11 1 11 1111 11 11 11 11 111Jill 1111 11 1 1111 11 I ll 11 = -- 102, _--- x CEIA75] 0 [AR72j 0 =_-- El [BA63] -2 10-1 1111111111111111 180 200 11 111111 220 111111 i 1 1 240 11111 1 1 111111 260 III[ 1111111 280 11 11111111 300 11 320 pion kinetic energy (MeV) Figure 2.19: x2 analysis of the 7r+7r° channel, and comparison of the model's prediction for different values of . Chapter 3 Discussion and Conclusion In the analysis of the previous chapter it was shown that, given the present status of the world data for the five 7 - 27r reaction channels, only the 7r+7r+ channel unambiguously yields a value for the chiral symmetry breaking parameter e. This channel gives e = 0.15 ± 0.1, which is consistent with the Weinberg value of e = 0. In the case of the 7r-7r0 and 7r+7r° channels, data with better statistics near threshold are needed. Significantly, however, for the channels in which the N* ---* N(77)8 —..ve mechanism makes a significant contribution to the cross section, i.e. the 7r-7r+ and 7r07r0 channels, it was seen that the x2 analysis of the world data does not give unique values for both C and e; together they can vary over a large range of values to minimize the x2• However, e can only take on certain values, as presently will be shown, and this greatly restricts the range of values that C can have. Weinberg has provided us with a way of directly relating the chiral symmetry breaking parameter, to the transformation properties of the term in the Lagrangian which breaks the symmetry. He uses a nonlinear method in which the Lagrangian is so contructed as to be invariant under chiral transformations expressed in terms of isospin matrices and the pion field. He then assumes that Girs,B, the symmetry breaking term in G„, transforms according to the (N/2,N/2) representation of SU(2) x SU(2) and obtains the following for the term in r,s,B which contributes to the 27r ---> 27r process [WE68] C3 _2 - 2 {1 - [N(N + 2) + 2] 4f1 2 } °2 33 (3.1) Chapter 3. Discussion and Conclusion 34 N = 1 in Eq. (3.1) give the symmetry breaking term in the Weinberg Lagrangian (Eq. 1.6). Weinberg also derived relations, similar to those which Olsson and Turner had derived, for the scattering lengths 2a0 + a2 3 = - L[N(N + 2) + 2] 5 2a0 - 5a2 = 6L (3.2) (3.3) where L = -a--• explicitly these give 8 7r f 7 L ao = - [N(N + 2) + 4] 4 L a2 = — [N(N + 2) - 8] 10 (3.4) (3.5) from a comparison of these equations and Eqs. (1.19)-(1.20), one obtains 2 e = - [3 - N(N + 2)] 5 (3.6) the importance of this relation is that it gives the chiral symmetry breaking parameter in terms of the tensor rank N, which only takes on positive integer values; then the corresponding values for e will be N=1 -+ = 0 N = 2 --- e = -2 N = 3 --(3.7) therefore consistency between Weinberg's formulation of the problem and that of Olsson and Turner, requires that e < 0.1 Therefore concentrating on the zero and negative values for e, the OMICRON's measurement for the 7r-7r0 channel seems to rule out 'note that Eq. (3.6) does not give = 1 (Schwinger's value) for the chiral symmetry breaking parameter. 35 Chapter 3. Discussion and Conclusion < —1. this can be seen from the lower energy data in Fig. 2.17. More significantly, however, as seen from the figure below, = —2 does not reproduce the low energy experimental data for the 7r+7r+ channel adequately. 1 10-4 1 1 I 150 1 1 ) I 200 1 1 II 1 1 1 1 250 1 1 II 1 1 I 1 1 I 300 I 1 1 1 1 1 1 ' 350 1 1 1 1 400 pion lab kinetic energy (MeV) Figure 3.1: Comparison of model's prediction for the 7r+7r+ channel with i= 0 and = —2. Indeed the theoretical curve for = —2 does not go through any of Sevior's error bars. So =- 0 seems to be a good candidate. For the 7r+7r° channel this value reproduces the few data available finely, except the last data point, which however seems to be somewhat anomalous. This brings us back to the two channels in which the value of depends on the 36 Chapter 3. Discussion and Conclusion value of C. Figure 3.2 shows the contour plots of the results of the X2 analyses of the previous chapter for the 7r-7r+ and 7r071-0 channels (see Figs. 2.10 and 2.8). A 100 x 100 grid has been used to interpolate the values listed in Tables 2.3 and 2.1; only the contours which, according to the errors listed in Tables 2.4 and 2.2, reasonably can represent the minima for the X2 have been shown. The analysis was done over the range 0.6 < c < 2.0 (or, since C = --c x 1.52 /1-1, -0.91 ,u-1 < C < -3.04 1/-1) and -1.25 < 6 < 1.25 ; in both channels for any value of 6 in this range, and probably beyond, C can be chosen to minimize the x2. However if, with regard to the discussion of the previous paragraphs, 6 = 0 be taken for the chiral symmetry breaking paremeter, the value of c, and therefore C, can be determined. From the top contour plot in Fig. 3.2, 6 = 0 gives xm2 6(x2) 4.4 (contour 1) and the corresponding error from Table 2.4 is 0.4, so 4.4 < x2nii„ < 5.2, i.e. contours labeled 1-9 which give 1.20 < c < 1.38 or c = 1.29 + 0.09. Similarly from the lower contour plot, 6 = 0 gives xn,2 2.4 (contour 4) and the error from Table 2.2 is 6(x2) 0.2, therefore 2.4 < x < 2.8, which corresponds to contours labeled 4-8, giving 1.20 < c < 1.46 or c = 1.33 ± 0.13. Denoting the first value of c by c1 and the other by c2, the weighted average of c obtained from 7r-7r+ and 7070 channels and its uncertainty is then obtained using E = Ec'/(8')2 Ei/(sc,)2 and 6E =Ei/(s)2, c, where i = 1, 2 and Sci and 6Z are the uncertainties in c, and Z, resp. This procedure results in = 1.30 + 0.07, which gives for C the following C = (-1.97 + 0.11) ,u-1 (3.8) The value of C given in Eq. (3.8) is to be compared with C = (-2.07+ 0.04) 1a-1 from X2 analysis of the 7r-7r+ channel alone and C = (-2.25 + 0.75) ,a-1 from branching ratio calculations [S092]. In Figs. 3.3 and 3.4 the world data at lower energies for all the reaction channels are shown, along with what the model gives using C = -1.97 p' and 6 = 0. 37 Chapter 3. Discussion and Conclusion x 2.0 7T 1.8 p -47 -n 2 / 1= 4.40E+00 -- 2= 4.50E+00 = 3= 4.60E+00 4= 4.70E+00 = -- 5= 4.80E+00 == 6= 4.90E+00 ==_ 7- 5.00E+00 -- 8= 5,10E+00 9- 5.20E+00 -- == 1.6 - === 1.4 1.2 = ^- 1.0 .1= 0.8 0.6 I) -1.5 I II -1.0 I -0.5 0.0 0.5 1.0 1.5 2.0 1.8 p-> 111= oo -- = = -- -- n -- -- 1- 2.10E+00 2- 2.20E+00 3- 2.30E+00 4. 2.40E+00 5- 2.50E+00 6- 2,60E+00 2= 2.70E+00 8- 2.80E+00 9- 2.90E+00 1= 0.6 I -1.5 I 1 1 -1.0 I I II -0.5 J 0.0 I III 0.5 I I I 1.0 I 1.5 Figure 3.2: The contour plots of x2 for the 7+71-- and r°71-° channels. Chapter 3. Discussion and Conclusion 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 II 1 1 1 1 38 1 1 , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 o [KR78] [0M90] [SE91] - model with =0 1 1 I 1 II II II 1 1 I 1 1 1 1 I' 1 1 1 200 250 300 350 400 10I 150 200 1111 1111111 111111111111111111111111111111 1111111111111111111 111111111 300 350 400 pion lab kinetic energy (MeV) pion lab kinetic energy (MeV) 103 Iit I Him 250 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 1 1 1 II 160 180 200 220 240 260 280 300 pion lab kinetic energy (MeV) 7r 7r + Figure 3.3: World data for the 7r+7r-,+ prediction for C = —1.97 = 0. 150 200 250 300 350 pion lab kinetic energy (MeV) 7ro 7r aria 7r-7ro channels, and model's Chapter 3. Discussion and Conclusion 1o3 10-2 _ 39 111111111.1111111111111111111111111111111111111111111111111111111111111111111111 , - mriiiiiiiiiiiiiirmiliumiiiiiiiiiitimiiimiiiirrminirmill 160 180 200 220 240 260 280 300 320 pion lab kinetic energy (MeV) Figure 3.4: World data for the 7r+7r° channel, and model's prediction for e = 0. For the 7r-p ---* 7r-7r+n reaction, the experimental data from [BA65], [BJ80] and [0M89a] are shown; there is good agreement even at T, as high as 350 MeV. Very clearly, e = 0 provides adequate fits to data for this channel at all energies. For the 7+7+ channel, e = 0 gives a very good fit to Sevior's and Kravtsov's data, as it does to OMICRON's at higher energies. As discussed in the analysis of Sec. 2.2.3, there is reason to believe that the error bars in OMICRON's measurement in the region 190-250 MeV have been underestimated, not the least of which is the fact that the results are inconsistent with the data from both of the other groups. For the 7r07r0 channel above 220 MeV the model does not provide as good a fit as Chapter 3. Discussion and Conclusion 40 it does at lower energies. Considering that below this energy the model fits the data finely, and the effect of the phenomenological Lagrangians (i.e. those involving the A) in this channel is still small compared to the pole, contact and 2-point N* diagrams, it is expected that the theoretical predictions of the cross section for this channel at higher energies, if anything, should be more accurate than the other channels. Future measurements for 7°70 will settle this matter. For the other two channels too, the model's cross sections for = 0 is in agreement with the available data. The r+7° channel at present only allows a consistency check on the results obtained from other channels. Upcoming measurements for this channel [EF92] will test the results so far obtained. In conclusion, analysis of the world data provides strong support for existence of the N* ----* N7r7r(E) mechanism in the 7rN —> 7r7rN reaction, as well as establish the value of = 0 for the chiral symmetry breaking parameter. Appendix A Derivation of the Amplitudes In this part an exposition of the Oset-Vicente-Vacas model is given and the model is used to derive the amplitudes for representative diagrams of the 7r — 27r reaction channels. A complete list of amplitudes for the 7070, 7-70 and 7+7° channels can be found in Appendices B-D. A.1 The Oset and Vicente-Vacas Model The interaction Lagrangian used by Olsson and Turner [0T68], includes terms which lead to diagrams for pole and contact terms, as well as to two-point diagrams for the 7rN s-wave amplitude. To this, Oset and Vicenta-Vacas [0V85] have added pieces to account for 1) 771-NN coupling through p-wave p-exchange with nucleon or A in the intermediate state, 2) three-point diagrams with two-nucleon, nucleon-A or A — A intermediate states, and 3) two-point diagrams with N* intermediate states and NN*(77r).„0„„ coupling through the exchange of the scalar meson E. The contribution of the diagrams containing the p meson has been found to be negligible [S090]. Instead diagrams involving the N* N(7r7r) p—wave decay were considered important and so , were added to the original model [S090]; hence an improved version of the Oset-VicenteVacas model. The components of the interaction Lagrangian, in the nonrelativistic approximation, are [0V85][S090]: LNN7 LIPt Cr2(1910)70 41 (A.1) 42 Appendix A. Derivation of the Amplitudes f 1 [02(ait 0)2 1(1 1 0/22(02)2] 2' 2 ,a 4 )7. f 1 of crioi0)1527,0 f„ = £NNirirr -_= Ai A2 .CNN„„ =--- —470t[— tt 0.0 + — ga2 r(0 x at (15)}0 ,CNA7r rNAA ---= f'CbtAS1(aie)TtAON + h.c. it -, fNN.7r7r fN.N7r =---- ,CN*A7r =- .f6,,,At cf it (,1 A (A.2) (A.3) (A.4) (A.5) \r7-7,/, (A.6) —COtN.O.00N h.c. (A.7) f + h.c. (A.8) s")- ( a ) TtA oN* + h.c. (A.9) gN*A,-01:6, where h.c. stands for Hermitian conjugate. In equations (A.1) to (A.4), V, and 0 are the nucleon and pion fields, resp., ai the Pauli spin matrices, T the nucleon isospin operator and f„ the pion decay constant (f, 87, 93, 95 MeV, from the Goldberger-Treiman relation, pion decay and 7rN scattering lengths, resp.; f,,= 93 MeV used here); 12 is the pion mass and -p—rn, where g„ is the NN7r coupling constant (g,=13.5); 6 in f, is the chiral symmetry breaking parameter; in (A.4), Ai and A2 are related to the s-wave 7rN scattering lengths and have the values Ai=0.0075 and A2=0.053. In (A.5) and (A.6), Op is the A particle field, S and T transition spin and isospin matrices, resp. (see Sec. A.2), and Sp and TA are ordinary spin and isospin matrices for the A; f* and fzx are related to the NAr and NAA coupling constants, resp. with numerical values given by f2/47 = 0.36 and f6, = 0.8f. In the Lagrangians involving the N*, IPAT., is the field operator for the N* particle and the value of f, related to the N*N7r coupling constant, is given by f2/47r = 0.02. The constant C in (A.7) is related to the N*Thrir vertex; its value was calculated in 43 Appendix A. Derivation of the Amplitudes [0V85], using the branching ratio for N* N(771, s—wave (^-' 5 — 20% [PDG90]) and the Lagrangian in (A.7), to be C (-0.91 +0.20)/4". The recent calculation by Sossi et al. [S092] gives the more accurate value of C = (-2.25 + 0.75)y'; this contains the large uncertainties in the N*N€ and en-R- coupling, as well as the uncertainty in the mass of E, the s-wave two-pion resonance. The calculation based on x2 analysis of the 7r-7r+ data [S092] has given C (-2.07 + 0.04)1-1, while the x2 analysis of the 7r-7r+ and 707r0 channels in the present work gives C -= (-1.97± 0.11)-1. The N*A7r coupling constant, gN.A has also been estimated to be gN.A, (1.305 + 0.225)y" [S090]; the model is not very sensitive to the variation of this constant. A.2 Expansion of the Lagrangians Using the relations 0- = -(01 + i02) 0+ = -(01 - i02) 00 = 03 and 7_ = 2 which give the charged pion fields in terms of their Cartesian components, £NArir, ,C/r7r) CNN„ and £NN,„ can be expressed in terms of the charged pion field operators 0_, 0+ and 00, which explicitly are = fd3q[a_(q)e-ig.x+ at+(q)eq'xj = Of-- (A.12) Oo = f d3q[ao(q)e-iq.x ato(q)e2q.xl (A.13) the operators a, and at, are, in the language of second quantization, the annihilation and creation operators for pion states of charge IC (tz=-1,+1,0) and 4-momentum q; the pion states have the normalization < 7r,(q)17r,,(q)>=-- 8„,83(q — q') (A.14) 44 Appendix A. Derivation of the Amplitudes 7-.4_ and T_ are the nucleon isospin raising and lowering operators, resp. Equations (A.1)-(A.4) and (A.7)-(A.8) then can be rewritten as ,CNN7r f.tht ( -r 0-i \ IL (1,1_4_ I a do )71, a 2 72 73-z r3i 7- + mai00)71) —f 0-i(N/rfaio+ + IL 1 4.n [(20_0+ + 0(2)).(25(r0+ + (ö0)2) )2) 1 1 (1 2 OiL2(402-0- + 40-0+4 + 0O)] 2 £NNirir ir f1 Otcri(\&+ai0+ + ,Lt 4f,?. (A.16) + mai00) x (20-0+ + O)O •CNNIrir ,CNN* fN*Nir (A.15) Al -4701-(20- (A.17) 0+ + + A2 00 - C.60) 1-1 V-2-ird-(0+ - 1.5+0o) + iTo(-0+ - (A.18) -C4*(2.0-0+ + 0DON + h.c. (A.19) t -ON*cri(v 4T+ai0+ + 12-7_ap0- + mai00)0 + h.c. I-t (A.20) where ,-- is used to indicate that the effect of normal ordering in perturbation theory expansion has already been considered in writing the Lagrangian.s. In order to put the other Lagrangians in a convenient form, the spin-isospin transition matrices are transformed from Cartesian to the spherical coordinates, defined by = T3 (A.21) where the - sign is used for the spherical components; the sums over A in Eqs. (2.5)(2.7) can now be written as VOA = -1.1+10+ + 11-10- + to0o TtA0A _t-41-1.0+ TcbA -11)„.4_10+ +11A,-10- +116„o0o igoo 45 Appendix A. Derivation of the Amplitudes using these, the expressions for .CAT6,7, £1T 0,1,5(-17+1ao, + tiazo_ +toa000tN + ofAst(-niaio+ + 141- CNAZ1 —116+1 ai + 73io0)1PN 0+ + ai(b_ (A.22) + t,Oai 00)1PA (A .23) fN*pir becomes fN*Air = +1ai+ tai000k. + gN*A7r{ oAsi( + gai000N• ot.As?(-ttiaio+ + I is the transition isospin operator for objects with isospins 12 and (A.24) and tt its Hermitian conjugate. Their matrix element between isospin states 11,rn > and j, m > is obtained by making use of the Wigner-Eckart theorem, according to which the matrix element of a tensor operator te between angular momentum eigen.states may be written as <j,mITj,m >=< j,mJ, here < j, mIJ,Kji% 1C; > rni >< (A.25) > is a Clebsch-Gordan coefficient and the symbol < > which is defined by (A.25), denotes a quantity which depends on j, ji, J and the nature of the tensor operator Te and is called the reduced matrix element (see [B086]). tn, is isospin operator for particles of isospin particles, and similar to T its value when sandwiched between two isospin states can be obtained using the Wigner-Eckart theorem. t and t both are objects of angular momentum 1, and here a normalization has been used according to which the reduced matrix elements of -7", tt and t have the values —f2-, 1 and resp.; so for future reference < j, \12- < j, mil, K ; j' , > ( A.26) 46 Appendix A. Derivation of the Amplitudes < j, < >=< j, 71211, ,k1:11 > 2 > . < J,"111,K;j',rn' > (A.27) (A.28) A.3 Recipe for Obtaining the Amplitudes Instead of explicitly deriving the Feynman rules and listing them, as a theorist would, one can give a practical recipe for calculating the amplitude. The S matrix element between the initial and final state is defined as Sfi < f ISii > = < f {exp[ i — —i(27048(Pf — 00 dt 1-11(t)]} > (A.29) where Ii > and If > are the initial and final states, resp., T is the time ordering operator and If the interaction Hamiltonian. Pi and Pf are the initial and final 4-momenta, resp. and T is the total amplitude for the process. The total amplitude is the sum of amplitudes corresponding to the Feynman diagrams. Each such amplitude is calculated by multiplying together the contributions from all fundamental processes occuring at vertices of a given diagram. The contribution of each vertex is calculated by finding the part of the interaction Lagrangian which can create the process at that vertex; to this end, the interaction Lagrangians given in this appendix are to be used, in conjunction with Eqs. (A.12) and (A.13). The resulting expression will then be sandwiched between the in- and out-states for that vertex. Depending on the vertex, the in-state can be the intial or an intermediate state and the out-state can be the final or an intermediate state. The field operators have the task of creating and annihilating the particles at each vertex, and in the cases where the part of Lagrangian used contains derivatives, the exponentials in Eqs. (A.12)-(A.13) Appendix A. Derivation of the Amplitudes 47 will contribute +iq or ±ie. To each intermediate state in a diagram, corresponds a propagator factor E, 2E7, where Ei is the total energy of the initial state and ET, the total energy of the intermediate state; to each vertex a factor of i is then associated (this is the i in the exponent of Eq. A.29). Finally for a given diagram, the amplitudes from all its vertices are multiplied with the propagators, giving that diagrams contribution to the total amplitude. 48 Appendix A. Derivation of the Amplitudes A.4 Calculation of Some of the Amplitudes for the Three Channels In the following section some of the diagrams for each channel are explicitly calculated with the rest of the amplitudes being given in Appendices B,C and D. It should be noted that in order to obtain the amplitudes given in the FORTRAN code at the end of each appendix, Eqs. (A.37)-(A.39) and (A.41) of Sec. A.5 in conjunction with Eqs. (A.26)-(A.28) of Sec. A.2 are to be used. 1. Pole diagrams • Tp 70 70 72 n(p3) — NN71- vertex: N/It,cr.p using iIpto-i,\/7-__8icb_zi) in (A.15) 7r7r vertex: —th; {4[—(pip5 + p5P6 pspi) 122] — 4(1 — using —4-* {(20-0+(9400)2 2a49A0+4) (1 01-t24(0-0+0O)} in (A.16) — propagator: r, iEn p2 1,2 so overall it becomes ,-f -pip5 - P5p6 - p6p1 + 21-e [12 v2- —cr.p p02 p2 it2 .a Appendix A. Derivation of the Amplitudes • 71-p 49 7r-70p I TV-(P6) 11- (PI) 71-°(p5) ilk 70(p) P(P3) P(P2) — NI\17 vertex: Icr.p using '0tairoa-i0o0 in (A.15) — 71-7 vertex: {4[—(pip5 — p5p6 — p6p1) — 4(1 — using same parts of (A.16) as in the 71-0 channel — propagator: E2—En — p2 so overall it becomes f 1 0- 10 P1 tt P5P6 P6P1 -1g1-12 0222 50 Appendix A. Derivation of the Amplitudes • r 71- + 0 7 p TT*(P5) 7.(1D1) 7e(Pe) 0 7t (p) P(P3) P(P2) — NI\TR- vertex: icr.p using Lto-0-05.000 in (A.15) — 7r7r vertex: --4:t {4{(M.P5 +P5P6 — pspi) 112] — 4(1 — using same parts of (A.16) as in the 70e channel — propagator: E,:i_En — p2 _p2 so overall it becomes _iT1,P f 1 cr ,P1P5 + P5P6 — p6p1 + 6/12 P 02 — p 2 — itt 2 6),tt2} 51 Appendix A. Derivation of the Amplitudes 2. Contact diagrams There's only the contribution from NN7r7r7r vertex (3) (2) (1) 7T-(P6)// / 0/ \ 7T lID6) \ 7 \ / • 7r-p \ \ n(p3) PODO (p,) \ \\ 0, 0D0 / / 7 , / / „...7" \Z- P(100 7r07r0n (first diagram above) f1 = 7r-70p (second diagram) using —/4-4-1-f7rOto-i(Thaic0).20_0+0 in (A.17), gives • r+p \ \ using —/44-0tcri(A/2-7_aic5_)0(2)0 in (A.17), gives • 7r-p \ _ /1 r 0 7 (P) , / / -.- , \ / __ /...-' .4\ P(P2) —iTLc v \ \ 7T \PI ) 1f 1 = -- — tt — cr D5 2 g 7r+7r°p (third diagram) using ---*,01-cri(7-0,9i00).20+0 in (A.17), gives —iT1'c = 1f 1 - -y 2 cr.p6 P(P2) P(P3) Appendix A. Derivation of the Amplitudes 52 3. Two-point diagram with N* intermediate state 7T(p5) 70(p6) 7-(P1) P(P2) ,/ n(p3) • N*Nr vertex: —1-12-0-.pi using in in (A.20) • NN*71-7 vertex: -i.2C using —00I-NOOON* in (A.19) • propagator: ri_iEn = where r* m2*+0,* is the width of the N*, to be obtained from F* = with E = E — — r*(E) _ M*, e(E) q3(M*) r(M*) r(m*) = 235 MeV and q(E) [(E2 + /12 m2)2/4E2 /111/2 (q is the momentum of the pion from N* N7 decay) so overall one gets _iTa2,N* _ 2v 2 I- C N/Ts —cr.p M* 1 (A.30) Appendix A. Derivation of the Amplitudes 53 4. Two-point diagram with nucleon intermediate state 7- (PI) 7° (P5) 7.° (P n P(N) 6) n(p3) • NThr vertex: --\121(7.Pi using (A.15) • NThrir vertex; —47r(i2-1-) using _4 tq5 in (A.18) • propagator: Ern so overall it becomes _iTa2,s _ _ 8 \/--,7r /2 [t NC9 — m 54 Appendix A. Derivation of the Amplitudes 5. Three-point diagram with nucleon and/or N* intermediate states using (A.15) \ I I.Trot.., \ \ \ _ \ A i 7T (Pi) / / \ H5) o / \ / I \ P(Pa) / 7C (Pe) / i n n n(p3) • first NNR- vertex: --Nicr.Pi • second NNR- vertex: — icr .p5 • third NNR- vertex: —Icr .p6 • first propagator: E1—En 1:s—m • second propagator: E,—En vi-p2--rq/2m-m so overall, taking into account the identical particle nature of the outgoing pions, one gets = f )3 1 cr.136/7.Pscr.Pi p \is — m \Ts — — pg / 2m — m + (5 <-+ 6) for diagrams with both nucleon and N* intermediate states, the result is similar with appropriate changes to account for NN*7 coupling constant, addition of the N* width and replacement of nucleon with N* mass in the propagator. Appendix A. Derivation of the Amplitudes 55 6. Three-point diagram with A and nucleon (or N*) intermediate state o„ MO 7-e(p6)i \ 7 _ (Pi) 7 // \ / / \ P(P2) / / n(p3) n A° • first NA7 vertex.• —11i--St using r-:,otAV't iaio_oN in (A.22) • second NA7 vertex: .MS.p5 using OASil'opiOo'CON in (A.22) • N NT- vertex: —icr.p6, using (A.15) • first propagator: where = r is the width of the A particle and is obtained using 1 — 2 F(E) = where E = — — 1 1 (9 347r tt 2 M' 3 Eq mA and q is as in Eq. (A.30) • second propagator: EE - so overall one gets 2 —iTa3'° = f (f — 3 ti V=s — 1 c•.p6S.p5St.pi -\fi — — pU2m, — m + (5 4* 6) In the case of three-point diagrams with A and N* intermediate states, the calculation is similar, with changes to the coupling constants, addition of N* width in the propagator and replacement of the nucleon with the N* mass. \ Appendix A. Derivation of the Amplitudes 56 7. Three-point diagram with two A intermediate states \ Tr°(p.), 71-°(P5)/ A 71--(1D1) I \ \ \ , i / / I i A° Aa p(p2) n(p) • first NAT- vertex: using L:01:s,f laicLON in (A.22) • Ni vertex: —1°-1SA.p5 p. 2 using ofstai000 in (A.23) • second NA7 vertex: Li' HS.136 using Sit oai0o0tN in (A.22) • first propagator: E1—En = • second propagator: Ei—En 1 / / / / 1:5—rnpd±r V75-192--pU2m—mp-qir so overall one gets fA —iT°'° = 3-\/ -F(5 \2 1 — mA S.p6SA.p5St.p1 —p — pOm — TriA 6) A.5 Taking Care of Spin The amplitudes from the Feynman diagrams contain combinations of spin and transition spin matrices. Generally one has to deal with objects of the form (1) cr • A (2) cr •A a •B a- • C Appendix A. Derivation of the Amplitudes (3) cr•A S (4) S•A • B St•C , S • A St-13 57 a•C Sp •B St -C which can be reduced to expressions having the general form Ia + a•b, in which I is the 2 x 2 unit matrix, a is a scalar and b is a 3-component vector; in fact the total amplitude T has just such a form T =la+ cr•b (A.31) Since cross section is proportional to the square of the amplitude, it is sufficient to calculate ITV. For an unpolarized target, one has to average over the spin states of the proton in the initial state; and since the polarization of the final state nucleon is not measured, the spin state of the final state nucleon is the sum of the two spin states; so IV will have the general form (<1 I+ <I here I 1)(Ia + cr•b)-(1 1> +1 1>) 2 (A.32) I> and I 1> are the up and down nucleon spin states with respect to the z axis. Writing the components of the Pauli matrices in terms of the raising and lowering operators, one will obtain, after some algebra 1T12 = l ar + 1E012 (A.33) For polarized targets, the initial and final nucleon states will have the same spin; using an expression similar to (A.32), but with initial and final states both up or both down, one gets [S090] 1712 = ial2 + 11312 + c (A.34) where c = a*b3 + b3*a + i(bIb2 — b2*b1) and the plus and minus signs are for the up and down polarization, resp. 58 Appendix A. Derivation of the Amplitudes The form cr•A o-•B o--C can be reduced to the form Ia o--b by writing it using implicit summation notation as ii.2B3Ck aaok and then using Criaj = 6ij (A.35) jEijkak or directly by using o-•Acr•B = A•B io--(A x B) (A.36) either way the result will be o--A o--B a•C = (B • C)o- -A — (A • C)o--B +(A • B)o--C iA (B x C) (A.37) In order to simplify the forms (3) and (4), closure relations will be used [0V85]. The forms a- •AS • BSt•C and S • ASt•B o-•C become 2 1 1 i o-•A S • B St•C = — (B C)cr-A + — (C • A)o--B — — (A • B)o-•C — — A • (B x C) (A.38) 3 3 3 3 i 2 1 1 S • A St •B o--C = — (A • B)o-•C + — (C • A)o-13 — — (B • C)o...A. — — A • (B x C) (A.39) 3 3 3 3 using the relation SS = E S I ms >< m-siS = Ms 3 — Eiikuk 3 (A.40) where Ms is the spin of the A intermediate state. Finally the form S-A St -BSt•C becomes 1 1 5 2 S-A St-B St-C = — (C • A)o--B — — (A • B)o--C — — (B • C)o-•A 4- — iA • (B x C) (A.41) 6 6 3 6 using the relation E sisstk siim's >< misls,„jims >< ms,mis 5. 6 - — 12 , 6 2- 3 2 1 , — —09 ka. 6- (A.42) Appendix B Amplitudes for the 7r-p —> 7r071-0n reaction The amplitudes in this part correspond to diagrams in chapter (2.2) with same the number. All quantities are centre of momentum values; pi is the 4-momentum of the incident pion, p5 and /36 are the 4-momenta of the identical outgoing pions, and p = p5+ p6 — pi the 4-momentum of the off-shell exchange pion. f1 — o-.p f72. ----- 1 p02 p2 y 2 [-Pi.P5 - p5•p6 - p6.p1 + -1C(12) \12- i T2 , N* f 1 cr.p1 L (B.3) — M* -2v 2 —C o- x 1 \Cs 137 = (B.1) (B.2) =- —2v2—Ccr.pi = — - * - (p + p 5 + P6)2 2m* + — 1 /„/ Ai ,- vs — 772 /2 59 (B.4) (B.5) Appendix B. Amplitudes for the 7r- p —> 71-071-0n reaction = 60 A2 0 0■ 4V-2-7rf-cr.p5 1 /-t m — —p1-2— (pi + p 5)2 12m itt 2 U51+136) + (5 4-* 6) f 27—cr.pi x = 1 (B.7) — m — — - 192 - (P1 p5+p6)2/2m —iTc2/8 = (B.6) 1 it + (5 4-* 6) —iTa3'N = A2 0 1/4 — m, —P2 — 134/2m 122 0\ /Pi +Po (B.8) (f) 3 Cr.P60-.135Cr.131 1 m v-s- — p 1 — pU2m — m + (5 4-* 6) —i Tb3 N (B.9) (1) 3 cr.picr.p6a.p5 x 1 P7 — - m - (p1+p5)2/2rn x 1 Nti;—P7-14—PS---rn— + (5 4-* 6) —iTc3'N _ + Ps + p6)2/2m (B.10) f 3 —v 2 (— cr.p6a.pia.p5 x ) 1 - - - m - (Pi P 5)2 I 2n1 x 1 — — m — pU2m + (5 4-* 6) (B.11) Appendix B. Amplitudes for the 7r- p 7r07r0n reaction 61 - -iT3'N'N* = I-21 le(2 Cr.1360..P50-•Pi X itt it M* -i2-ir*Ncs -p — pg/2m — m + (5 4-)' 6) (B.12) -iTa34 =-_ -12-f If * )2cr.p6S.p5St.p, 3 Itt rnA + Ncs - - m- pg/2m + (5 4-* 6) -i7163'° = (B.13) •\/ f f* 2 - (=-) S.p6St.plcr.p5 x 3 ft itt 1 \Cs — m — 1 — (131 + p5)2/2m \Cs —p — P.U2ma, — rnA + 6) + (5 —iTc34 - f f*) 2 — 3 ,11 (B.14) cr.p6S.p1St.p5 x 1 — — (P1 + p5)2/2m6, rnA 1 + (5 4-* 6) x (B.15) Appendix B. Amplitudes for the 7r- p 71-0en reaction 62 (f )2 = — S.p1St.p60-.p5 x 3 ii 1 \Cs — m — A — A — (pi + p5)2/2in 1 N/75 — A — A — A — (pi + p5 + p6)2/2m A — mA + + (5 f-3 6) 2\12-f ( f*) (B.16) 2 — Te3 A 3 P cr.piS.p6St.p5 x 1-1 1 —p—p — (P1 +p5)2/2mA — m6, + iF x 1 —m —A —A 6) + (5 —iT3'° 2/ f f*) 3 P P 2 — (P1+ p5 + p6)2/2m (B.17) S.p6St.p5cr.p1 x \ — m \Cs — — Pg/2mA — TrIA + (5 6) (B.18) 2 —iTa°4 1 f6, S.p6SA.p5St.p1 x 3fL) — m6, + iF\is —p — + (5 4-4 6) — pg/2niA + (B.19) Appendix B. Amplitudes for the 7r- p —iTbA'A 71-0en reaction S 1 f (t) /.1 2S•Pl 3,/j- .P6 63 Sip 5 X 1 — 137 131:5) — (P1 P5)2/2MA — MA + 1 — p — P.(5) — P06 — (pi + p5 + p6)2/2m, — mA + + (5 4-* ) (f —iTc°4 _ 3 (B.20) )2 Q .p5 x ft 1 \Ts — — p — ( P1+ P5)2/2m — rnA + iF x 1 N1-9 — — PU2rnA TnA + (5 6) _ f (f * \ - 3 ft - ft (B.21) gN*Air cr.P6S.P5St.Pi x \Ts — m, + iF,v7s —p — m* — pg/2M* + (5 4-* 6) (B.22) f (f* _ji,b3,N.,A _ _ 3 — tt —it) gN*Air S.p6St.picr.p5 x 1 ,V7s — M* — p?— 13? — (Pi + p5)212M* x 1 \Cs — p — pU2rnA — m + + (5 ) (B.23) 71-0en reaction Appendix B. Amplitudes for the 7r- p _iT3,N*A - 3 1-1 64 (f* —) gN*Air ( .P6S•PiSt -P5 x 1 — — (Pi + P5)2 /2mA x 1 — — M* — pU2M* dm' + (5 4-4 6) - iT3'N* 3 it (.9 gN*AirS•P1St •P60. •P5 it M* – –p + (5 — iT3'N* 1 PSI – (P1+ P5)2 / 2m* + 1 p? – (p1 + x 5 + P6)2 /21716, TiL6, + 6) f (f* 2 (B.24) tt 3 [I (B.25) gN*Air cr•pis•p6st.p, x 1 Nt; – – – (131 +P5)2/277/6, — mA 1 – m* + (5 —iTPN*'° — – – P2 – (p1 +P5 + p6)2/2M*+ - jr* 6) (B.26) 2\/ f ( f* 3 -- ) gN*AirS-P6St -P5cr .Pi x tt v:s — it4-* + + (5 4-+ 6) —p — pU2mA — ir (B.27) Appendix B. Amplitudes for the 7r-p -4 r'en reaction 65 What follows is the portion of the Monte Carlo code in which the amplitudes , with the spin taken care of, are given. SUBROUTINE BOBA(SRAJD) IMPLICIT COMPLEX (U) LOGICAL LOSOL DIMENSION ID(48),X1D(48) COMMON /MOMEN/P1(4),P5(4),P6(4),P5MOD,P6MOD,P5MO2,P6MO2,P1MO2 COMMON/AMPLI/U1P(4),U1C(4),U2A(4),U28(4),U2C(4),U2D(4), • U3A(4),U3B(4),U3C(4),U3DA(4),U3DB(4),U3DC(4),U3DD(4), • U3DE(4),U3DF(4),U3DDA(4),U3DDB(4), • U3DDC(4),U2NA(4),U2NB(4), • UTOT(4),P(4),U3DNA(4),U3DNB(4),U3DNC(4),U3DND(4),U3DNE(4), • U3DNF(4),U3NA(4) COMMON/CONSTO/HB,PI,XMN,XMU,XM112,XMD,XMNS,FN,FP2,XL1,XL2 • ,FD,FDD,WR,CNS,FNS,CHI,XLAM,SQ2,FND COMMON/MANIP/TPI,EPILAB,ETOT,PPILAB,S2,S,EPICM,ENUCM,PPICM • ,XL,EREM,PLIM,AT,BT,FI5,FI6,CO5,C06,XMAX,XMIN COMMON/CASO/ICASO,LOSOL,IPOL,JPOL,MPOL C SOME COUP. CONSTANTS FN,FD,FDD,FNS INCLUDE /MASS PION C UNITS ARE FM IF(LOSOL)THEN UY.(0.,1.) P11=ESC(P1,P1) P15=ESC(P1,P5) P16=ESC(P1,P6) P55=ESC(P5,P5) P56=ESC(P5,P6) P66.--ESC(P6,P6) DO 11=1,4 P(I1)=P5(I1)-1-P6(I1)-P1(I1) ENDDO P2=P(4)**2-P(1)**2-P(2)**2-P(3)**2 FORM--,(XLAM**2-XMU2)/(XLAM**2-P2) DO 395 1=1,48 395 XID(I)=FLOAT(ID(I)) DO 1=1,3 U1P(I)=FN*SQ2/(P2-XMU2)/(4*FP2)*(-4.)*(P5(4)*P1(4)-P15+ • P6(4)*P 5( 4)-P.56-1-P 1( 4)*P6(4 )- P16- .5*CHI*X MU2)*(P5(I) • -FP6(I)-P1(I))*FORM U1C(I)=2.*FN*SQ2/(4.*FP2)*P1(I) U2A(I)=-FN*SQ2/(SRA-XMN)*4.*PI*(2.*XLI/XMU)*P1(I) U2B(I)=FN*SQ2*4.*PI*XL2/XMU2* Appendix B. Amplitudes for the p 70n, reaction • ((P1(4)-1-P 6(4))*P5(I)/(SRA- XMN-P1(4 )-P5(4) • -(P11+ P.55-1-2.*P1.5)/(2.*XMN))-1-(P 1(4)-I-P5(4))*P 6(I)/(SRA-XMN- F P1(4)-P 6(4)-(P11-I-P66-1-2.*P1 6)/(2.*XMN))) U2C(I)=-FN*SQ2 /(SRA-XMN-P1(4)-P5(4)-P6(4)-(P 11-FP55-I-P 66+2.*P15 • -1-2.*P16+ 2.*P56)/(2.*XMN))*4.*P1*(2.*XL1/XMU)*P1(0 1J2D(I)=-FN*SQ 2*4.*PI*(XL2/XMU2)*((P1(4)-F P6(4))*P5(I)/(SRA-P6(4)F XMN-P 661(2.*XMN))-F(P1(4)-I-P5(4))*P6(I)/(SRA-P 5(4)- XM N • -P55/ (2.*XMN))) U3AFAC1=FN**3*S Q2/ ( SRA-XMN)/(SRA- P5(4)-XMN-P55/(2.*XMN)) U3AFAC2=FN**3*SQ2/ (SRA-XMN)/(SRA- P6(4)- XMN-P66/(2.*XMN)) U3A(I)=U3AFAC1*(P56*P1(I)-1-P 15*P6(I)-P 16*P5(I))+ • U3AFAC2*(P56*P1(0-1-P16*P5(I)-P15*P6(I)) U3NAFAC1=FN*FNS**2*SQ2/(SRA-XMNS-1-UY*GDELS(SRA)/2.) • /(SRA-P5(4)- XMN-P55/(2.*XMN)) U3NAFAC2=FN*FNS**2*SQ2/(SRA-XMNS-1-UY*GDELS(SRA)/2.) • /(SRA-P6(4)- XMN-P66/(2.*XMN)) U3NA(I)-= U3NAFAC1*(P56*P1(I)-FP15*P 6(0- P 16*P5(1))-1U3NAFAC2*(P56*P1(I)-1-P 16*P5(I)-P 15*P6(I)) U3BFAC1=FN**3*SQ2/(P 11/(2.*XMN)-P5(4)-(P11+P55-1• 2.*P15)/(2.*XMN))/(P11/(2.*XMN)-P 5(4)-P6(4)-(P11-1-P55 • P66-1- 2.*P15-1-2.*P16-1-2.*P56)/(2.*XMN)) U3BFAC2=FN**3*5Q2/(P 11/(2.*XMN)-P6(4)-(P114-P66-1• 2.*P16)/(2.*XMN))/(P11/(2.*XMN)-P 5(4)-P6(1)- (P11-4-P55 • P66+ 2.*P15+2.*P 16+2.*P56)/(2.*XMN)) U3B(I)=U 3BFA C1*(P 16*P5 (I)-I-P56*P1 (1)-P 15*P6 (I))+ U3BFAC2*(P15*P6(I)-1-P56*P1(I)- P16*P5(I)) U3CFAC1=-FN**3*SQ2/(SRA-P1(4)-P 5(4)-XMN-(P11-1-P55 -FP15)/(2.*XMN))/ (SRA-P5(4)- XMN-P55/(2*XMN)) U3CFAC2=-FN**3*SQ2/(SRA-P1(4)-P 6(4)-XMN-(P11-1-P66 -FP16)/(2.*XMN))/ (SRA-P6(1)- XMN-P66/(2*XMN)) U3C(I)=U3CFAC1*(P16*P5(I)-1-P 15*P6(I)-P 56*P1(0)-F U3 CFAC2*(P15*P6(I)-1- P16*P5(I)- P 56*P 1(I)) U3DAFAC1=-FN*FD**2*S Q 2/ 3. /(SRA-XMD-I-UY*GDEL( SRA)/2.)/ • (SRA-P5(4)- XMN-P55 /2./XMN) U3DAFAC2=-FN*FD**2*SQ2/ 3. /(SRA-XMD-FUY*GDEL( SRA)/2.)/ • (SRA-P 6(4)- XMN-P66 /2./XMN) U3DA(I)= • U3DAFAC1*(2./3.*P15*P6(I)-1- 1. /3.*P16*P5(I)-1. /3.*P56*P 1(I))-1- • U3DAFAC2*(2. /3.*P16*P5(I)-1- 1./3.*P15*P6 (I)-1./3.*P 56*P 1(0) DN3A1=SRA-P5(4)-P 55/2. XMNS DN3A 2=SRA-P 6(4)-P 66/2. /XMNS 1J3DNAFAC1=-FNS*FD*FND*S Q 2/3./ • (SRA-XMD+ U Y*GDEL(SRA)/2.)/(DN3A 1- XMNS-FUY*GDELS(DN3A1)/ 2.) U3DNAFAC2=-FNS*FD*FND*SQ2/3./ 66 Appendix B. Amplitudes for the 71--p • 7r° 71-'n reaction (SRA-XMD-FUY*GDEL(SRA)/2.)/(DN3A 2- XMNS-FU Y*GDELS(DN3A2)/ U3DNA(I)= • U3DNAFAC1*(2./3.*P15*P 6(I)+1./3.*P16*P5(I)-1./3.*P 56*P1(I))+ • U3DNAFAC2*(2./3.*P16*P 5(I)+1./3.*P15*P6(I)-1./3.*P 56*P1(I)) D3B1=SRA-P5(4)-P55 /2. /XMD D3B2=SRA-P6(4)-P66/2./XMD U3DBFAC1=FN*FD**2*SQ2/3./(113B1- XMD -FUY*G DEL(D3B1)12.)/ • (SRA-XMN-P 1(4)-P5(4)- (P11 +P55 +2 .P 15)12./X MN) U3DBFAC2=FN*FD**2*SQ2/3./(D3B2-XMD-FUY*GDEL(D3B2)/2.)/ • ( SRA- XMN-P 1(4)-P 6(4)-(P11+P66 +2.*P16)/ 2./ XMN) U3DB(I)= • U3DBFAC1*(2./3.*P16*P5M+ 1. /3.*P56*P1 (I)-1. /3.*P15*P6(I))-1- • U3DBFAC2*(2./3.*P15*P6(0-1- 1./3.*P56*P1(I)-1./3.*P16*P5(I)) D3NB1.SRA-P1(4)-P5(4)-(P11-1-P 55+2.*P15)/ 2./XMNS D3NB2=SRA-P 1(4)-P 6(4)-(P 11-FP 66+2.*P16)/ 2./XMNS U3DNBFAC1=FNS*FD*FN1D*SQ2/3./(D3B1-XMD+UY*GDEL(D3B1)/ 2.)/ • (D3NB1-XMNS-FUY*GDELS(D3NB1)/2.) U3DNBFAC2=FNS*FD*FND*SQ2/3./(D3B2-XMD-FUY*GDEL(D3B2)/2.)/ • (1J3NB2-XMNS-I-UY*GDELS(D3NB2)/2.) U3DNB(I)= U3DNBFAC1*(2./3.*P16*P 5(I)+1./3.*P56*P1(I)-1./3.*P15*P6(I))+ • U3DNBFAC2*(2./3.*P15*P 6(1)+1. /3.*P56*P1(I)-1./3.*P16*P5(I)) D3C1=SRA-P1 (4)-P5(4)-(P 11-I-P55-1-2.*P15)/2./XMD D3C2=SRA-P1(4)-P6(4)-(P11-FP66+ 2.*P16)/2./XMD U3DCFAC1=FN*FD**2*5Q 2/3./ • (D3C1-XMD -FUY*GDEL(D3C1)/2.)/(SRA-P5(4)-XMN-P55/2./XMN) U3DCFAC2=FN*FD**2*5Q2/3./ • (D3C2-XMD -FUY*GDEL(D3C2)/2.)/(SRA-P6 (4)- XMN-P 66/2./XMN) U3DC(I)= • U3DCFAC1*(2./3.*P15*P 6(I)-1-1./3.*P56*P1(I)-1./3.*P16*P5(I))+ • U3DCFAC2*(2./3.*P16*P .5(I)+1./3.*P56*P1(I)-1./3.*P15*P6(I)) D3NC1=SRA-P 5(4)-P55/ 2./XMNS D3NC2=SRA-P6(4)-P66/ 2./XMNS U3DNCFAC1=FNS*FD*FND*5Q2/3./ • (D3C1-XMD-FUY*GDEL(D3C1)/2.)/(D3NCI-XMNS-FUY*GDELS(D3NC1)/2.) II3DNCFAC2=FNS*FD*FND*SQ2/3./ • (D3C2-XMD +UY*GDEL(D3C2)/2.)/(D3NC2-XMNS-I-UY*GDEL S(D 3N C2)/ 2.) U3DNC(I). • U3DNCFAC1*(2./3.*P15*P6(I)+1./3.*P56*P1(I)-1./3.*P16*P5(I))+ • U3DNCFAC2*(2./3.*P16*P5(I)+ 1. /3.*P56*P 1(I)-1./3.*P15*P6(I)) D3D=SRA-P5(4)-P6(4)-P 1(4)-(P11-1-P55-FP66 +2.*P15+2.*P 16+2.*P56 )/ F 2./XMD U3DD FA C1=-FN*FD**2*S Q 2/3. /(D3D- X MD + Ti Y*GD EL(D 3D )/ 2.)/ • (SRA- XMN- P1(4)-P5 (4)-(P11-FP 55+2.*P 15)12./X MN) U3DDFAC2=-FN*FD**2*SQ 2/3. /(D3D- X MD-FUY*GDEL(D3D)/2.)/ • (SRA- XMN- P1(4)-P6(4)-(P11+P 66-1-2.*P16)/ 2./XMN) 67 Appendix B. Amplitudes for the 7r- p 7071-0n reaction U3DD(I). • U3DDFAC1*(2./3.*P16*P 5(0+1./3.*P15*P6(I)-1./3.*P56*P1(I))-1- • U3DDFAC2*(2./3.*P15*P 6(I)-1-1./3.*P16*P5(I)-1./3.*P56*P1(I)) D3ND1=SRA-P 5(4)-P1(4)- (P11+P55-1-2.*P15)/2./XMNS D3ND2=SRA-P 6(4)-P1(4)-(P11-I-P66+2.*P16)/2 ./XMNS U3DNDFAC1=-FNS*FD*FN1J*SQ2/3./(D3D-XMD+1JY*G1JEL(D3D)/2.)/ • (D3ND1-XMNS+UY*GDELS(D3ND1)) U3DNDFAC2.-FNS*FD*FND*SQ2/3./(D3D-XMD+UY*GDEL(D3D)/2.)/ • (D3ND2-XMNS-I-UY*GDELS(D3ND2)) U3DND(I). • U3DNDFAC1*(2./3.*P16*P5(I)+1./3.*P15*P 6(I)-1./3.*P56*P1(I))+ • U3DNDFAC2*(2./3.*P15*F 6(I)+1./ 3.*P 16*P 5(I)-1./3.*P56*P1(I)) D3E1= SRA- P 1(4)-P5 (4)-(P 11-EP 55+2.*P11)/ 2./XMD D3E2 -,SRA- P1(4)-P6 (4)-(P11-I-P 66+2 .*P16)/ 2./XMD U3DEFAC1.----FN*FD**2*2.*SQ2/3./(D3E1-XMD-4UY*GDEL(D3E1)/2.)/ • • (SRA-XMN- P1(4)-P5(4)-P6(4)-(P 11-1-P55-FP 66+2.*P16+2.*P56+2.*P 11)1 2./XMN) U3DEFAC2.--FN*FD**2*2.*SQ2/3./(D3E2-XM1DA-UY*GDEL(D3E2)/2.)/ • (SRA- XMN- P1(4)-P5(4 )-P6(4)-(P 11-PP55-FP 66-1-2.*P16 -1-2.*P56+2.*P 15)/ • 2./XMN) U3DE(I)= • U3DEFAC1*(2. /3.*P56*P1 (I)-1-1./3.*P1.5*P 6(1)-1. /3.*P16*P5(I))+ • U3DEFAC2*(2./3.*P56*P1(I)+1./3.*P16*P 5(I)-1./3.*P15*P6(I)) DNE3=(SRA-P1(4)-P5(4)-P6(4)-(P 11 +P55-1- P66-1-2.*P15+2.*P16-1-2.*P56)/ F 2./XMNS) IT3DNEFAC1=FNS*FD*FND*2.*SQ2/3./ • (D3E1-XMD-I-ITY*GDEL(D3E1)/2.)/(DNE3-XMNS+UY*GDELS(DNE3)/2.) TI3DNEFAC2--.7.FNS*FD*FND*2.*SQ2/3./ • (D3E2-XMD-I-UY*GDEL(D3E2)/2.)/(DNE3-XMNS-FUY*GDELS(DNE3)/2.) U3DNE(I)=. • U3DNEFAC1*(2./3.*P56*P 1(I)+1./3.*P15*P6(I)-1. /3.*P 16*P5(I))+ • U3DNEFAC2*(2./3.*P56*P 1(I)+1./3.*P16*P5M-1./3.*P15*P6(I)) D3F1=SRA-P5(4)-P.55/2./XMD D3F2=SRA-P6(4)-P66/2./XMD U3DFFAC1=FN*FD**2*2.*1Q2/3./ • (D3F1-XMD+UY*GDEL(D3F1)/2.)/(SRA-XMN) U3DFFAC2=FN*FD**2*2.*SQ2/3./ • (D3F2-XMD-FIJ Y*GDEL(D3F2)/2.)/(SRA-XMN ) IT3DF(I)= • U3DFFAC1*(2./3.*P56*P1(0-1-1./3.*P16*P 5(0-1./3.*P15*P6(I))-1- • U3DFFAC2*(2./3.*P56*P1(I)+1./3.*P15*P6(I)-1./3.*P16*P5(I)) U3DNFFAC1=FNS*FD*FND*2.*SQ 2/3. / • (D3F1-XMD 4-UY*GDEL(D3F1)/2.)/(SRA-XMN S-FUY*GDELS(SRA)/2.) U3DNFFAC2=FNS*FD*FND*2.*5Q 2/3. / • (D3F2- XMD Y*GDEL(D3F2)/2.)/(SRA-XMN S-I-UY*GDELS(SRA)/2.) U3DNF(I)= F U3DNFFAC1*(2./3.*P56*P1(0-1-1./3.*P16*P 5(I)-1./3.*P15*P6(I))-1- 68 Appendix B. Amplitudes for the 7r-p --* 7r07r0n reaction F U3DNFFAC2*(2./3.*P56*P1(I)+1./3.*P15.P 6(I)-1./3.*P16*P5(0) DDA3= SRA DDA4=SRA-P5(4)- P55/ 2. /XMD DDA5=SRA-P6(4)-P66/2./XMD 1.JDDAFAC1=-FDD*FD**2/(3.*SQ2)/(DDA3-XMD+UY*GDEL(DDA3)/2.)/ • (DDA4-XMD+UY*GDEL(DDA4)/2.) UDDAFAC2=-FDD*FD**2/(3.*SQ2)/(DDA3-XMD-FUY*GDEL(DDA3)/2.)/ • (DDA5-XMD-FUY*GDEL(DDA5)/2.) U3DDA(I)= • UDDAFAC1*(-1./6.*P56*P1(I)+2./3.*P16*P5(I)-1./6.*P 15*P6(I))-I- F UDDAFA C2*(-1./6.*P56*P1(I)-1- 2. /3.*P15*P 6(I)-i. /6 16*P5(I)) DDB2=SRA-P 1(4)-P5(4)-(P11-1-P55-1-2.*P15)/2./XMD DDB3=SRA-P 1(4)-P6(4)-(P11-1-P66+ 2.*P16)/2./XMD DDB4=SRA-P 1(4)-P5(4)-P6(4)F (P11-f-P55 -FP66-1-2.*P15+2.*P16+ 2.*P56)/2./XMD UDDBFAC1=-FDD*FD**2/(3.*SQ 2)/(DDB2-XMD-1-1.TY*GDEL(DDB2)/2.)/ • (DDB4-XMD-1-1.JY*GDEL(DDB4)/2.) UDDBFAC2=-FDD*FD**2/(3.*SQ 2)/(DDB3-XMD-FUY*GDEL(DDB3)/2.)/ • (DDB4-XMD+UY*GDEL(DDB4)/2.) U3DDB(I)= • UDDBFAC1*(-1./6.*P16*P5(I)-1-2./3.*P1.5*P6(I)-1./6.*P56*P1(I))-1- • UDD BFA C2*( 1. /6.*P 15*P 6(I)-1-2./3.*P1.6*P5(I)-1./6.*P56*P1(I)) DDC]. =SRA-P1( 4)- P5(4)-(Pl. 1-FP55-1-2.*P 15)/ 2./XMD DDC2=SRA-P1(4)-P6(4)-(P111-P66-1-2.*P 16)/2. /XMD DDC3=SRA-P5(4)-P55/2./XMD DDC4=SRA-P6(4)- P66/2. /XMD UDDCFAC1=FDD*FD**2*2 .*SQ2/3./(DDC1-XMD-FUY*GDEL(DDC1)/2.)/ F (DDC3-XMD-FUY*GDEL(DDC3)/2.) UDDCFAC2=FDD*FD**2*2.*SQ2/3./(DDC2-XMD+UY*GDEL(DDC2)/2.)/ • (DDC4-XMD -FUY*GDEL(DD C4)/2.) U3DDC(I)=. • UDDCFAC1*(-1./6.*P16*P5(I)+2./3.*P56*P1(0-1./6.*P15*P6(I))+ • UDD CFA C2*(-1./6.*P15*P 6(1)+2. /3.*P56*P1(I)-1. /6.*P16*P5(I)) U2NA(I)=-CNS*2.*SQ2*FNS /(SRA-XMNS+UY*GDELS(SRA)/2.)*P1(I) DNS1=SRA-P1(4)-P5(4)-P6(4)-(P11 -I-P55-1-P66+2.*P15-1- 2.* • P16+2.9'56)/2./ XMNS U2NB(I)=-CNS*2.*SQ2*FNS/(DNS1- X MNS-FUY*GDELS(DNS1)/2.)*P1(I) UTOT(I)=XID(21)*(U1P(I)-FUlC(I))-IF XID(17)*U2A(I)+XID(1.8)*U2B(I)-1- • XID(19)*U2 C(I)+XID(20)*U2D(I)-1- • XID(22)*U3A(I)+XID(1)*(U3B(I)+U3C(I))-1- • XID(2)*U3DA(I)-1-XID(3)*U3DB(I)+ • XID(4)*U3DC(I)+XID(5 )*U3DD(I)-I- • XID(6)*U3DE(I)+XID(7)*U3DF(I) UT OT(I)=I.TTOT(I)-F • XID(10)*U3DDA(I)+XID(11 )*U3DDB(I)+ XID(12)*U 3DDC(I)-1- 69 Appendix B. Amplitudes for the 71-773 7r071 0n - • XID(16)*(U2NA(I)+U2NE (04 • XID(23)*U3DNA(I)+XID (24)*U3 DNB (I)+XID (25)*U3 DNC(I)-1- • XID(26)*U3DND(I)+XID(27)*U3DNE(I)+XID(28)*TJ3DNF(I)-F • XID(31)*U3NA(I) ENDD 0 UJAC=UY*XMIXT(P1,P 5,P6) U3A(4)=U JAC*(-U3AFAC1-f-U3AFAC2) U3NA(4)=UJAC*(-U3NAFAC1-FU3NAFAC2) U3B(4)=U JAC*(-U3BFAC1-1-U3BFAC2) U3C(4)=UJAC*(U3CFAC1-U3CFAC2) U3DA(4)=(-1./3.)*UJAC*(-U3DAFAC1-FU3DAFAC2) U3DB(4)=(-1 ./3.)*UJAC*(U3DBFAC1-U3DBFAC2) U3DC(4)=(-1./3.)*UJAC*(U3DCFAC1-U3DCFAC2) U3DD(4)=(-1./3.)*UJAC*(-U3DDFAC1-4U3DDFAC2) U3DE(4)=(-1./3.)*UJAC*(-U3DEFAC1-1-U3DEFAC2) U3DF(4)=(-113.)*UJAC*(-U3DFFAC1-FU3DFFAC2) U3DNA(4)=(-1./3.)*UJAC*(-U3DNAFAC1+U3DNAFAC2) U3DNB(4)=(-1./3.)*UJAC*(U3DNBFAC1-U3DNBFAC2) U3DNC(4)=(-1./3.)*UJAC*(U3DNCFAC1-U3DNCFAC2) U3DND(4)=(-1 ./3.)*UJAC*(-U3DNDFAC1-I-U3DNDFAC2) U3DNE(4)= (-1. /3.)*UJAC*(-U3DNEFAC1-I-U3DNEFAC2) U3DNF(4)=(-1./3.)*UJAC*(-U3DNFFAC1-FU3DNFFAC2) U3DDA(4)=5./6.*UJAC*(-UDDAFAC1-1-UDDAFAC2) U3DDB(4)=5./6.*UJAC*(-UDDBFAC1-1-UDDBFAC2) TJ3DDC(4)=5./6.*UJAC*(UDDCFAC1UDDCFAC2) UTO T(4 )=X ID(22)*U 3A(4)-I-XID (1 )*(U 3B(4) +U3 C(4))+ • XID(2)*U3DA(4)+XID(3)*U3DB(4)+ • XID(4)*U3DC(4)-1-XID(5)*U3DD(4)-1- • XID(6)*U3DE(4)+XID(7)*U3DF(4)+ • XID(10)*U3DDA(4)-1-XID(11)*U3DDB(4)+XID(12)*U3DD C(4)-1- • XID(23)*U3DNA(4)+XID(24)*U3DNB(4)+XID(25)*U3DNC(4)-1- • XID(26)*U3DND(4)+XID(27)*U3DNE(4)-1-XID(28)*U3DNF(4)-F • XID(31)*U3NA(4) RETURN ELSE UTOT(1)=0. UTOT(2)=0. UTOT(3)=0. UTOT(4)=SQRT(SIGMA(PPILAB,TPL-1)) RETURN ENDIF END reaction 70 Appendix C Amplitudes for the 7r-p 7r-7r°p reaction The amplitudes in this part correspond to diagrams in chapter (2.3) with same the number. All quantities are centre of momentum values; pi is the 4-momentum of the incident pion, p5 and p6 are the 4-momenta of the neutral and negative outgoing pions, resp. and p = p5 p6 — pi the 4-momentum of the off-shell exchange pion. fl /IR cr.p P02 1 /32 (C.1) [-PITS + P5•P6 P6•P1 + 214121 — iTl'e = Ta2 , S 1f1 2 tt fl. (C.2) 0 • 135 f 07—cr-Pi 1LL 1 A2 0 2 (P5 - /4) vS- f 1 47r— a-.p5 2A1 (C.3) - m - p - p - (pi + p5)2/27n A2 0 (P1 + 0 (C.4) 1 r Tc2 ,S A2 4L0.5V: s — m — /32 _ pU2m I tt cr•Pscr-Pscr.Pi 71 \Cs — m 0 0 p2 (pi + PO] 732 Pg/ 2m — m (C.5) (C.6) Appendix C. Amplitudes for the 7r- p —> 71--'ir'p reaction if. P0 •135 .Pi — iT3'N N/75 m 72 —130m m (C.7) 2( — je ) 3 cr.p6cr.picr.p5 x It ) 1 -\18- — p — p — m — (Pi + p5)2/2m 1 —iT3'N'N* = 2— — 1-t — _iTb3,N,N* cr.P517.P6cr.p1 x M* \is — — = —2— — It ft (C.9) cr.P60-.P50..p1 x 1 V7s — M* = if (f*) 3 ia itt L— (f* 2 — 2 cr. Ncs — rnA Itt p4/2m — m (1) 2 f 1 —iT34 (C.8) p(5) — 2m — m x p5S•po + ir \Fs — (C.10) — m — p4/2m, (C.11) ct a.p6 ,3 .picr.p5 x 1 m 13? 13(5) — (Pi + P5)2 / 2m x 1 0 2 /2 V7S — p5 — p5/M A — MA -r, 1 .-1-1 (C.12) Appendix C. Amplitudes for the 7r-p —> 7r-70p reaction 73 f (f )2 — cr.p5S.piSt.p6 x Itt 1 AFS –p —p - mA (pi p6)2/2rnAi iF x 1 = L 1-1 (C.13) — S.piSt.p6cr.p5 x 1 \Cs — — — — ( P1+ p5)2/2m x \[s — — p6P — (P1+ P5 + p6)2/2771A - rnA + — (f 3 1,1 tt )2 S•p6St•p5cr.pi x - m Nfi- - - W12 rnA - mA + ir 2 f f*) 2 —iTP° 3 It ( *) 3 it LItt 2 (C.15) S.p5St.p6o-.p1 x ¼— m — 132— pg/2mA — m6, + = (C.14) (C.16 ) cr.p6S.p5St.p1 x \cs - rnA + - p(5) — — pg/2m (C.17) Appendix C. Amplitudes for the 7r-p —> 71-ep reaction (11 t 2 ft 3 74 0- P6S•PiS .p5 x 1 V.; — — _ mA (131 p 5)2 / 2rn + ir x 1 Ncs — p — m — pU2m _1 fn, 6 2 ((f* S•P6Sp •P5S 1-.131 X 1 1 \is — 77/A + iFN[s — p — m, — pU2rnA 2 f6, f*) 3 ( C. 18) ( C.1 9 ) 2 S.p5SA•p6St.p1 x ti N/79 — rnA + iF — p — rnA 134/21np (C.20) 2 = (f) f2 S.piSA•p6St.p5 x 1 rnA — — pC81 — (pi + p5)2/2mA 1 2iir \Cs —p —p — p — ( Pi + P5 + P6)2/2MA (C.21) 2 fA f*) = 3 2 — n•psaA Tin •p5 x it it 1 \Cs —p — p — ( 131+ p5)2/2mA — mA 1 — — 0/2mA — rnA + iF (C.22) 7r-ep reaction Appendix C. Amplitudes for the 7r-p I. —iTA'A = 75 2 3f (L-) S.p1SA.p5St.p6 x 2 ,tt 1 —Pi — Ps — 11)1 p5) 2 /2MA — MA + 1 — — PS — P2 — (131 + Ps + p6)2/2mA — f*2 = itt ( itt) (C.23) O.P5Op•Pi3.-136 X 1 N[s — P — P6P — (P1 + 136)2/2mA — 1 0 2 P6 — P6/2MA - iT3'N*'° _ — 1j (f* 3 la rn A (C.24) 1 - -1-1 gN*Air cr.P5S.P6St.Pi x Itt 1 1 — mA + iF —p — m* — p4/2M* ir* —iTb3'N*'° = f (f * tt IL gN.A7 (C.25) S.PeSt.Picr.Ps x 1 — pg/2mA — rti„n, 1 (C.26) — m* — pi) —p — (Pi + p5)2/2M*+ -iT3 'N *'° (f;) IL g N*6,7 Cr -P5S-P1St.P6 X 1 Ntr; — P7 — P6°— rn•A — (p1 + p6)2/27716, 1 N[5 — — — pS/2m* + • x (C.27) Appendix C. Amplitudes for the ir-p —iT,31'N* 76 7-71-°p reaction gN*A7S.P1St .P6ff •P5 X 1 , 1 •-r, X _j_ Fs p6)2, c,771A Trip ,n0 — F5 — F6 — 1 —— — m* (p1 + (C.28) p5)2 /2M* + f (f* —iT3'N *4 — -- *A7S.P6St ) gN .135°..P1 x 3 tt It — 1 v7s — p(5) — pg/2mA — rrtA 2 (f* —i21.f3'N*4 — M* (C.29) gN*A,S.p5St .p6cr.pi x 3 0 1/7-9 — Ps 2 -iTp N*'° ---273'N* 14 = 2 — P6/2mA — rnA + -1 •zr 2 (f* 3 bt 2 — M* lir* 2 (C.31) gN*Airff-P6S.P5St.Pi x 1 — p — m* — pg/2M* 1 — rnA (C.31) gN*Aircr.P6S.P1St .p5 1 o 0 V.75 — pi — p 5 — rizA — (131 + p5) 2 /2772p 1 \Cs — p — Mt — p0m*+ 1 (C.32) Appendix C. Amplitudes for the 7r-p 7r-71-°p reaction 77 What follows is the portion of the Monte Carlo code in which the amplitudes , with the spin taken care of, are given. SUBROUTINE BOBA(SRA,ID) IMPLICIT COMPLEX (U) LOGICAL LOSOL DIMENSION ID(48),XID(48) COMMON /MOMEN/P1(4),P5(4),P 6(4),P5MOD,P6MOD,P5MO2,P6MO2,P 1MO2 COMMON/AMPLI/U1P(4),U1C(4),U2A(4),U2B(4),U2C(4), • U3A(4),U3NA(4),U3B(4),U3NB(4),U3C(4),U3DA(4),U3DB(4), • U3DC(4),U3DD(4), • U3DE(4),U3DF(4),U3DG(4),U3DH(4),U3DDA(4),U3DDB(4), • U3DDC(4),U3DDD(4),U3DDE(4),U3DDF(4), • UTOT(4),P(4),U3DNA(4),U3DNB(4),U3DNC(4),U3DND(4),U3DNE(4), • U3DNF(4),U3DNG(4),U3DNH(4) COMMON/CONSTO/HB,PI,XMN,XMU,XMU2,XMD,XMNS,FN,FP2,XL1,XL2 • ,FD,FDD,WR,CNS,FNS,CHI,XLAM,SQ2,FND COMMON/MANIP/TPI,EPILAB,ETOT,PPILAB,S2,S,EPICM,ENUCM,PPICM • ,XL,EREM,PLIM5,PLIM6,AT,BT,F15,FI6,CO5,C06,XMAX,XMIN COMMON/CASO/ICASO,LOSOL,IPOL,JPOL,MPOL C SOME COUP. CONSTANTS FN,FD,FDD,FNS INCLUDE /MASS PION C UNITS ARE FM IF(LOSOL)THEN UY.(0.,1.) P11.ESC(P1,P1) P15=ESC(P1,P5) P16=ESC(P1,P6) P55.ESC(P5,P5) P56=ESC(P5,P6) P66=ESC(P6,P6) DO 11=1,4 P(I1)=P5(I1)+P6(I1)-P1(I1) ENDDO P2=P(4)**2-P(1)**2-P(2)**2-P(3)**2 FORM=(XLAM**2-XMU2)/(XLAM**2-P2) DO 395 1=1,48 395 XID(I)=FLOAT(ID(I)) DO 1=1,3 U1P(I)=FN/FP2/(P2-XMU2)*(-P1(4)*P5(4)-1-P15 + P5(4)*P6(4)-P56 + • P1(4)*P6(4)-P16 + .5*CIII*XMU2)*(P5(I)+P6(I)-P1(0)*FORM U1C(I)=-.5*FN/FP2*P5(I) U2A(I)=8.*PI*FN*P1(I)*XL2/XMU2*(P5(4)-P6(4))/(SRA-XMN) U2B(I)=4 .*PI*FN*P5(1)*(2.*XL1/XMU - XL2/XMU2*(P1(4)+P6(4)))/ Appendix C. Amplitudes for the 7r-p • 7-71-°p (SRA-XMN-P1(4)-P5 (4)-(P11+P 55+2.*P15)/ (2.*XMN)) U2C(I)=4.*PI*FN*P5(I)*(2.*XL1 /X MU - XL2/XMU2*(P1(4)+P 6(4)))/ • (SRA-P 6(4)- XMN-P66/(2.*XMN)) TJ3AFAC=2.*FN**3/ • (SRA-XMN)/ • (SRA-XMN-P6(4)-P66/(2.*XMN)) U3A(I)=U3AFAC*(P56*P1(I)-1-P16*P.5(I)-P15*P6(I)) U3NAFAC=2.*FN*FNS**2/ • (SRA-XMNS-F UY*GDELS(SRA)/2.)/ • (SRA-XMN-P6(4)-P66/(2.*XMN)) U3NA(I)=U3NAFAC*(P56*P1(I)-1-P16*P5(I)-P15*P6(I)) U3BFAC=-2.*FN**3/ • (SRA- XMN)/ • (SRA- XMN- P5(4)-P55/(2.*XMN)) U3B(I)=U3BFAC*(P56*P1(0+ P15*P6(0-P16*P5(0) U3NBFAC=-2.*FN*FNS**2/ • (SRA-XMNS+ UY*GDELS(SRA)/2.)/ • (SRA-XMN-P5(4)-P55/(2.*XMN)) U3NB(I)=U3NBFAC*(P56*P1(I)+P15*P6(I)-P16*P5(I)) TJ3CFAC=2.*FN**3/ • (SRA-XMN-P1(4)-P5(4)-(P11-FP 55+2.*P15)/ (2.*XMN))/ • (SRA-XMN-P5(4)-P55/(2.*XMN)) 153C(I)=U3CFAC*( P16*P5(I)-1-P 15*P 6(I)- P56*P1(I)) U3DAFAC=1./3.*FN*FD**2/ • (SRA-XMD -FITY*GDEL(SRA)/2.)/ • (SRA-P6(4)-XMN-P 66/2. /XMN) U3DA(I)=U3DAFAC* • (2./3.*P16*P5(0-1-1./3.*P15*P6(0-1 ./3.*P56*P1(0) DN3A=SRA-P6(4)-P66/2./XMNS U3DNAFAC=1./3.*FNS*FD*FND/ • (SRA-XMD -I-UY*GDEL(SRA)/2.)/ • (DN3A-XMN S-FUY*GDELS(DN3A )/2.) U3DNA(I).= U3DNAFAC* • (2. /3.*P16*P 5(I)+1. /3.*P15*P6(I)-1./3.*P56*P 1(I)) D3B =SRA-P 5(4)-P55/ 2./XMD U3DBFAC=FN*FD**2/ • (SRA-XMN-P1(4)-P5(4)-(P11+ P55-1-2.9'15)/2./XMN)/ • (D3B-XMD-I-UY*GDEL(D3B)/2 .) U3DB(I)=1J3DBFAC* • (213.*P16*P5(I)+1./3 P 56*P1(0-1 ./3.*1.15*P6(I)) D3NB=SRA-P1(4)-P5(4 )-(P11-1-1' 55+2.*P15)/2./XMNS U3DNBFAC=FNS*FD*FND/ reaction 78 Appendix C. Amplitudes for the • (D3NB-XMNS-1-UY*GDELS(D3NB)/ 2.)/ • (D3B-XMD-FUY*GDEL(D3B)/2.) 71--p 7 7°p - reaction U3DNB(I),---U 3D NBFAC* • (2. /3.*P16*P 5(1)+1. /3.*P56*PI(I)-1./ 3.*P15*P6(I)) D3C=SRA-P1(4)-P6(4)-(P 11-FP66-1-2.*P16)/2./XMD U3DCFAC=FN*FD**2/ • (D3C-XMD-I-UY*GDEL(D3C)/2.)/ • (SRA-P6(4)-XMN-P 66/ 2./XMN) U3DC(I)=U3DCFAC* • (2./3.*P16*P5(I)-1-1./3.*P 56*P1(I)-1./3.*P15*P6(0) D3NC=SRA-P 6(4)-P66/2./XMNS U3DNCFAC=FNS*FD*FND/ • (D3C-XMD+UY*GDEL(D3C)/2.)/ • (D3NC-XMNS-FUY*GDELS(D3NC)/2.) U3DNC(I)=U3DNCFAC* • (2./3.*P16*1' 5(I)-1-1./3.*P56*P1(I)-1./3.*P15*P6(I)) DD3=SRA-P 1(4)-P5( 4)-P 6( 4)-(P 11+ P55-I-P 66 -1-2.*(P15-1-P 16+ P56))/2 ./XMD U3DDFAC=FN*FD**2/ • (SRA-XMN-P1(4)-P 5( 4)-(P11-1- P55-1-2.*P15)/2./XMN)/ • (DD3-XMD-FUY*GDEL(DD3)/2.) U3DD(I)=U3DDFAC* • (2./3.*P16*P5(I)+1./3.*P15*P6(I)-1 ./3.*P56*P1(I)) D3ND =SRA-P 5(4)-P1(1)- (P11 55 -1-2.*P15)/2./XMNS U3DNDFAC=FNS*FD*FND/ • (D3ND-XMNS-I-UY*GDELS(D3ND)/2.)/ • (DD3-XMD-f-UY*GDEL(DD3)/2.) U3DND(I)=513DNDFAC* • (2./3.*P16*P 5(I)-1-1./3.*P15*P6(I)-1./3.*P56*P1(0) D3E=SRA-P5(4)-P55 /2./XMD U3DEFAC=-SQ2/3.*FN*FD**2/ • (SRA-XMN)/ • (D3E- XMD-FUY*GDEL(D3E )/2.) U3DE(I)=U3DEFAC* • (2./3.*P56*P1(0-1-1./3.*P 16*P5(I)-1 ./3.*P15*P6(0) U3DNEFA S Q2/3.*FNS*FD*FND / • (SRA-XMNS-f-U Y*GDELS( SRA)/ 2.)/ • (D3E-XMD-FUY*GDEL(D3E)/2.) U3DNE(I)= U3DNEFAC* • (2./ 3.*P56*P 1(0+1. /3.*P16*P5(I)-1./3.*P15*P6(I)) D3F=SRA-P 6(4 )-P66/2./XMD U3DFFAC=-2./3.*FN*FD**2/ • (SRA- XMN)/ • (D3F-XMDA-UY*GDEL(D3F)/2.) U3DF(I)=U3DFFAC* • (2./3.*P56*P1(0-1-1./3.*P15*P6(I)-1 ./3 P16*P5(I)) 79 Appendix C. Amplitudes for the r- p 7r-R-°p reaction U3DNFFAC=-2 /3 *FNS*FD*FND/ • (SRA-XMNS+UY*GDELS(SRA)/2 )/ • (D3F-XMD-F1JY*G DEL(D3F)/2 ) U3DNF(I)=U3DNFFAC* • (2 /3 *P56*P1(I)-1-1 /3 *P15*P6(I)-1 /3 *P16*P5(0) U3DGFAC=2./3.*FN*FD**2/ • (SRA-XMD-4UY*GDEL(SRA)/2.)/ • (SRA-XMN-P5(4)-P 55/2./XMN) 11311G(I)=U3DGFAC* • (2./3.*P15*P6(I)-1-1./3.*P 16*P5(I)-1 ./3.*P56*P1(I)) DN3G=SRA-P5(4)-P55/2./XMNS U3DNGFAC=2./3.*FNS*FD*FND/ • (SRA-XMD-FUY*GDEL(SRA)/2.)/ • (DN3G-XMNS-FUY*GDELS(DN3G)/2.) U3DNG(I)=U3DNGFAC* • (2./3.*P15*P 6(I)+1./3.*P16*P5(0-1./3.*P56*P1(I)) D3H=SRA-P 1(4)-P5(4)- (P11 -1-P55-1-2.*P1.5)/2 /XMD 113DHFAC=-2. /3.*FN*FD**2/ • (D3H-XMD-1-UY*GDEL(D311)/ 2.)/ • (SRA-P5(4)-XMN-P 55/2./XMN) II3DH(I)=U3DHFAC* • (2./3.*P15*P6(I)-1-1./3.*P56*P1(0-1./3.*P16*P5(I)) DN3H=SRA- P5(4)-P55/2. /XMNS U3DNHFAC=-2./3.*FNS*FD*FND/ • (D3H-XMD-FUY*GDEL(D3H)/2.)/ • (DN3H-XMNS-4UY*GDELS(DN3H)/2.) 1.13DNH(I)= U3DNHFAC* • (2./3.*P15*P 6(I)+1./3.*P56*P1(I)-1./3.*P 16*P5(I)) DDA=SRA-P5(4)-P 55/2./XMD U3DDAFAC=-1./6.*FDD*FD**2/ • (SRA- XMD -I-UY*GDEL(SRA)/2.)/ • (DDA-XMDA-UY*GDEL(DDA)/2.) U3DDA(I)=U3DDAFAC* • (-1./6.*P56*P1(I)-1-2./3.*P16*P5(I)-1./6.*P 15*P6(I)) DDB =SRA-P 6( 4)-P 66/ 2./XMD U3DDBFAC=2./3.*FDD*FD**2/ • (SRA-XMD-FUY*GDEL(SRA)/2.)/ • (DDB- XMD-f-UY* GDEL(DDB)/2.) U3DDB(I)=U3DDBFAC* • (-1. /6.*P56*P1(I)+2./3 P15*P6(1)-1./6 P 16*P5(I)) DDC1=SRA-P1(4)-P5(4)-(P11-f-P55+2.*P15)/2./XMD DD C2= SRA- P1(4)- P5(4)-P6 (4)-(P11-1-P 55 -f-P66 2.*(P15-FP16 P56))/ 2. /X MD U3DDCFAC=-FDD*FD**2/ • (1JDC1-XMD-I-TiY*G1JEL(DDC1)/2.)/ 80 Appendix C. Amplitudes for the • 7r-p —> 71--7°p reaction (DDC2-XMD UY*GDEL(DD C2)/2.) U3DDC(I)=U3DDCFAC* • (-1./6.*P16*P5(I)4-2./3.*P15*P6(I)-1./6.*P56*P1(I)) DDD1=SRA-P1(4 )-P5(4)-(P11+ P55+2 .*P15)/2./XMD DDD2=SRA-P5(4 )-P55/2./XMD U3DDDFAC=2./3.*FDD*FD**2/ • (DDD1-XMD-FUY*GDEL(DDD1)/2.)/ • (DDD2-XMD -FUY*GDEL(DDD2)/2.) U3DDD(I)=U3DDDFAC* • (-1./6.*P 16*P5(I)+2./3.*P.56*P1(0-1./6.*P 15*P 6(I)) DDE1=SRA-P 1(4)-P6(1)- (P11 -FP66 -1-2.*P16)/2./XMD DDE2=SRA-P 1(4)-P 5(4)- P6(4)-(P11 4FP554-P664-2.*(P15+ P16-1-P56))/ 2./XMD U3DDEFAC=3./2.*FDD*FD**2/ • (DDE1-XMD4F1JY*GDEL(DDE1)/2.)/ • (DDE2-XMD4FUY*GDEL(DDE2)/2.) U3DDE(I)=U3DDEFAC* • /6.*P15*P6(I)+2./3.*P16*P5(I)-1./6.*P 56*P 1(I)) DDF1=SRA-P1(4)-P6(4)-(P11 4FP664-2.*P 16)/2. /XMD DDF2=SRA-P6(4)-P66/2./XMD U3DDFFAC=-FDD*FD**2/ • (DDF1-XMD+UY*GDEL(DDF1)/2.)/ • (DDF2-XMD-FUY*GDEL(DDF2)/2.) U3DDF(I)=U3DDFFAC* • (-1./6.*P15*P6(1)4-2./3.*P56*P1(I)-1./6.*P 16*P5(I)) UTOT(I)=XID(1)*U1P(I)+XID(2)*U1C(I)+XID(3)*U2A(I)-F • XID (4)*U2 B(I)+XID (5)*U2 C(I)+X ID( 6 )*U3A(I)+ • XID(7)*U3B(I)+XID(8)*U3C(I)4F • XID(9)*U3NA(I)4-XID(10)*U3NB(I)4- • XID(11)*U3DA(I)+XID(12)*U3DB(I)+XID(13 )*U3DC(I)-1- • XID(14)*U 3D D(I)-1-XID (15)*U3DE(I)-F XID(16)*U3DF(I)-F • XID(17)*U3DG(I)-1-XID(18)*U3DH(I) UTOT(I)=UTOT(I)4• XID(19)*U 3DD F XID(22)*U 3DD D (I)+XID (23)*U3DDE(I)-FX ID (24)*U3DDF(I)4F 20)*U3DD B(I)+XID (21)*U3DD C(I)4- • XID( 25 )*U 3D NA(I)+XID (26)*U3DNB(I)+ XID (27)*U 3DNC(I)-F • XID(28)*U3DND(I)+XID(29)*U3DNE(I)+XID(30)*U3DNF(I)+ • XID(31)*U3DNG(I)+XID(32)*U3DNH(I) ENDDO UJAC=UY*XMIXT(P1,P 5,P6) U3A(4)=U JAC*U3A FA C U3NA( 4 )= UJA C*U3NAFA C U3B(4)=-UJAC*U3BFAC U3NB(4)=-UJA C*U3NBFAC U3C(4)=UJAC*U3CFAC U3DA(4)=(-1./3.)*UJAC*U3DAFAC U3DB(4)=(-1./3.)*UJA C*U3DBFA C 81 Appendix C. Amplitudes for the 7r-p —÷ 7r-7r°p reaction U3DC(4)=(1./3.)*UJAC*U3DCFAC U3DD(4)=(1 ./3.)*UJAC*U3DDFAC U3DE(4)=(1.13.)*UJAC*U3DEFAC U3DF(4)=(-1./3.)*UJAC*U3DFFAC U3D G(4)=(1 ./3.)*UJAC*U3D GFAC 113DH(4)=(- 1./3. )*UJAC*U3DHFAC U3DNA(1)= (-1. /3.)*UJA C*U3DNAFAC U3DNB(4)=(-1./3.)*UJAC*U3DNBFAC U3DNC(4)=(1./3.)*UJAC*U3DNCFAC U3DND(4)=(1./3.)*UJAC*U3DNDFAC U3DNE(4)= (1./3.)*UJAC*U3DNEFAC U3DNF( 4)=(-1. /3. )*UJAC*U3DNFFA U3DNG(4)=-- (1./3.)*UJAC*U3DNGFAC U3DNH(4)= (-1./3.)*UJAC*U3DNHFAC U3DD A( 4)=-5./ 6.*UJAC*UDDAFAC U3DDB(4)=5./6.*UJAC*UDDBFAC U3DD C(4 )=-5./6.*UJAC*UDDCFA C U3DD D (4).5 ./6.*UJAC*UD DDFA C U3DDE(4)=5./6.*UJAC*UDDEFAC U3DDF( 4)=-5./ 6.*UJAC*UDDFFAC ITTOT(4)=XID(22)*U3A(4)+XID(1)*(U3B(4)-I-U3C(4))-F • XID(2)*U3D A(4)-1-XID (3 )*U3DB(4)+XID(4)*U 3D C( 4)+ • XID(5)*U3DD(4)+X ID(6)*U3DE(4)+X ID ( 7)*U3DF(4)-1- • XID(8)*U3D G ( 4)-I-XID(9)*U3DH( 4 )-F • XID(10)*U3DDA( • XID(13)*U3DDD(4)-1-XID(14)*U3DDE(4)+XID(15)*U3DDF(4)-1- • XID(23)*U3DNA(4)+X ID (24)*U3DNB(4)+X ID (25)*U3DNC(4)-1- (11)*U3DDB( 4)-PCID ( 12)*U3DDC( 4)-F • XID(26)*U3DND(4)-1-XID(27)*U3DNE(4)+XID(28)*U3DNF(4)+ • XID(29)*U3DNG(4)+XID (30)*U3DNH( • XID(16)*U3NA( 4)+XID(20)*U3NB( 4) RETURN ELSE UTOT(1)=0. UT OT(2 )=0. UTO T( 3)=0. UTOT(4)=SQRT(SIGMA(PP ILAB,TPI,-1)) RETURN ENDIF END 82 Appendix D Amplitudes for the 7r+p —> 7+70p reaction The amplitudes in this part correspond to diagrams in chapter (2.4) with same the number. All quantities are centre of momentum values; p1 is the 4-momentum of the incident pion, p5 and p6 are the 4-momenta of the positive and neutral outgoing pions, resp. and p = p5 p6 — pi the 4-momentum of the off-shell exchange pion. J 1 = .t7 cr.p p02 pz 1 1.P5 2, fr 135•P6 — 136•P1 (D.1) J 1f1 —iT1'c ——— tt —iTa2's = 47-f-cr.p6 1 — [ A2 —03(1) tt2 —8 71a-.p5 /-1 = 47—cr.p6 2Ai. Ii (D.2) a-.P6 2 2A1 —iTb2's x /12 —p - P2 - (P1 + p6)2/2m x Pn] (D.3) 1 rrt P? PSI — (Pi + P5)212m 112 0 0 ( Pi +P) (D.4) 1 rn — — PU2rn A2 11 A2 (D.5) PC5))] 83 Appendix D. Amplitudes for the 7r+p 7+70p reaction 84 2 () 3 cr.pio-.p50-.p6 x 1 Nrs- - rn p(1) P(6) — (pi + p6)2/2m, 1 V,--.14-1,2-152—m—(pi+ P5 + P6)2/2M (D.6) ( f) 3 cr.p1cr.p6cr.p5 x —2 — 1 — — ( P1 + p5)2/2m x N[s m 1 — N/7-3 137 115) —iT3'N = 2 ( (131 + P5 + p6)2 /2m (D.7) 1 )3 cr.p6r.p1cr.p5 x 1 — m — 137 — — (Pi + p5)2 /2ni x 1 0 "f3- — P5 — -zT 3,° (D.8) rrt Ps2 /2m f* ) 2 cr.p6S.p5St.p, x 0 — m —2 p5/2m— mA + -1 21 — p5 2 IL (t) 2 (D. 9) S . p5S t .picr.p6x 1 — m — — — (131 p6)2/2m 1 Ni; 0 2 P6 — P6/27nA m A 1 - T-1 (D.10) Appendix D. Amplitudes for the 7r+p 7r+ep reaction 85 2 - z77,34 S.piSt.p5a.p6 x = 1 — — — (pr +136)2/2m x 1 'Nfs —P —P —P — (Pi + Ps +136)2/27nA — mA 2 f f*) —iTc3/ 2 3 it (D. 11) S.p6St.p1a.p5 x 1 \Cs — m — — — (Pi + p5)2/2mx 1 (D.12) \Cs —IA —Pil2mA—mA+ ( f*) — 3 it /-1 —iTe34 2 S.piSt.p6cr.p5 x 1 m — —p — (Pi + p5)2/2m x 1 \fi. — —p — — (pi p5+p6)2/2m A — rriA — iT3'° — 3fL (112 ,tt (D.13) St.P6 x Pi S .P5 1 \Cs — p — pcs' — (Pi + p6)2/2m A — mA x 1 \Cs—m-14-14—A— (Pi +P5 + P6)2/2m = 3 tt ( f*) — 2 (D. 14) cr.piS.p6St.p5 x 1 —p — p —m — (Pi + P5)2/21-nA 1 \Cs — m — (col +5 + p6)2 12m x (D. 15 ) Appendix D. Amplitudes for the 7r+p --* 7r+ep reaction 86 2 cr.p6S.p1St.p5 x f (j 1 0 0 Vs- —Pi —P5 0 Ar9 — P5 — (D.16) rn 1 1 Vs- —A— p4/2mA (f*)2 — T' Ps2 /2711 )2 '3•P5°A•P6'°'•P1 X — \Cs 1 /f* \2 3f6, — ft —iT" — rnA — (Pi + P5)2/2mA — — mA (D.17) S.p6SA•p5St.pi x itt 1 1 — 77/A fa, f* it it —p — nip — pg/2777,p s.p5sA.piSt .136 x 1 0 = 2 fA f* ( —) 2 [I la (D.18) 2 1 o o 2 A — m A \Cs —Pi —P6 — (131 P6) /2m iTd6"° 12-ir —m A T 1 T (D.19) 0.pi .p5St.p6 x 1 — 1fi — — (pi + p6)2/2rnA — mA + 2lir 1 p? — p — p — (pi + ps + p6)2/2m A — n-tA + iF (D.20) Appendix D. Amplitudes for the 7r+p 7r+ep reaction 87 )2 S •Ps S •P iSt•P 5 3 It \ 1 0 0 2 N45 - Pi P5 - (pi P5) /2MA — rnA 1 . T1 X 1 2 1fi P5° P 5/2MA 1 jr6, f* 2 —iTA'A 1 ( .-1-1 D.21 ) S•P1Sp•P6St.135 X 1 — (p1 + p5)2/2m — mA i ir 1 (p1 p5 p6)2/2m A _ mA \Cs — VT5 — _iTa3,N*,° (D•22) = it I( fl gN*An-cr.p5s.p6st.pi x ,tt 0- — Pc5) — M* — PU2M* -jr* — m + —iTb3'N*'° = — (f* --) gN*Air S.P5St bt lir (D.23) r .P6 X 1 .\15 - VT5 m* — — — (pi + p6)2/2m* + 1 , 0 2 /2 Ps --- P6/ MA — TrIA x (D.24) L—(f* —)gA,A,s.pist.p5cr.p6x ft ft 1 — — p(-0) — (pi + p6)2/2m* + x 1 \fi — — — — (pi + p5 + p6)2/2mA rnA + (D.25) Appendix D. Amplitudes for the 7r+p __2 f 3/1 = ' d (f*) — 7+71-°p reaction 88 gN*A, S.p6Sat.pi.p5 x it 1 x Ars — m* — p — p — (Pi + P5)2 / 2m* + 1 1 2 p°5 — P5/2mA — (D.26) T 2 f (f* = —) gN*A7rs.p1st.p6ff•p5 x it 3 it 1 x — — (Pi + p5)2 / 2M* + \Ts — m* 1 \fi — — p — p — (pi + p5 +136)2/2mA — mA + 2 f (f*) —iT3'N*4 3 itt (D.27) gN*A- cr-PIS-P5St•P6 x 1 N/7-9 — — 132 — (pi +136)2/2mA — rrtz, + 1 N[s — m* 2 (f*) — — — (p1 + P5 + p6)2/2M*+ lir* (D.28) gN*Air €7•131S•P6St.p5 x 1 0 12 /e) — P1 P5 — 411 + p5) 0 — rrlp 1 .T-1 X 1 v— m* — —p — — (pi + —iTh3'N*4 = 1 f (f * 3 ti —) g N*A7CinSDSt .r-6 P5 + 136)2/2M* + 10 .1-5 X 1 — p — p(5) — (131 p5)2/2m A — 1 \Cs —p — m* — pU2M* (D.29) • (D .3 0) Appendix D. Amplitudes for the 71--Ep 7r+7°p reaction 89 What follows is the portion of the Monte Carlo code in which the amplitudes , with the spin taken care of, are given. SUBROUTINE BOBA(SRA,ID) IMPLICIT COMPLEX (U) LOGICAL LOSOL DIMENSION ID(48),XID(48) COMMON /MOMEN/P1(4),P5(4),P6(4),P5MOD,P6MOD,P5MO2,P6MO2,P1MO2 COMMON/AMPLI/U1P(4),U1C(4),U2A(4),U2B(4),U2C(4), • U3A(4),U3B(4),U3C(4),U3DA(4),U3DB(1),U3DC(4),U3DD(4), • U3DE(4),U3DF(4),U3DG(4),U3DH(4),U3DDA(4),U3DDB(4), • U3DDC(4),U3DDD(4),U3DDE(4),U3DDF(4), • UTOT(4),P(4),U3DNA(4),U3DNB(4),U3DNC(4),U3DND(4),U3DNE(4), • U3DNF(4),U3DNG(4),U3DNH(4) COMMON/CONSTO/HB,PI,XMN,XMIT,XMU2,XMD,XMNS,FN,FP2,XL1,XL2 • ,FD,FDD,WR,CNS,FNS,CHI,XLAM,SQ2,FND COMMON/MANIP/TPLEPILAB,ETOT,PPILAB,S2,S,EPICM,ENUCM,PPICM • ,XL,EREM,PLIM5,PLIM6,AT,BT,FI5,FI6,CO5,C06,XMAX,XMIN COMMON/CASO/ICASO,LOSOL,IPOL,JPOL,MPOL C SOME COUP. CONSTANTS FN,FD,FDD,FNS INCLUDE /MASS PION C UNITS ARE FM IF(LOSOL)THEN UY=(0.4.) P11=ESC(P1,P1) P15=ESC(P1,P5) P16=ESC(P1,P6) P55=ESC(P5,P5) P56=ESC(P5,P6) P66=ESC(P6,P6) DO 11=1,4 P(I1)=P5(I1)+P6(I1)-P1(I1) ENDDO P2=P(4)**2-P(1)**2-P(2)**2-P(3)**2 FORM=(XLAM**2-XMU2)/(XLAM**2-P2) DO 395 1=1,48 395 XID(I)=FLOAT(ID(I)) DO 1=1,3 U1P(I)=FN/FP2/(P2-XMU2)*(P1(4)*P5(4)-P15 + P5(4)*P6(4)-P56 F P1(4)*P 6(4)+P16 + .5*CHI*XMU2 )*(P5(I)+P6(I)- P 1(I))*FORM U1C(I)= - .5*FN/F P2*P6(I) U2A(I)=4.*PI*FN*P6(I)*(2.*XL1/XMU + XL2/XMU2*(P1(4)+P5(4)))/ • (SRA-XMN-P1(4)-P6(4)-(P11-I-P66+2.*P16)/(2.*XMN)) U2B(I)=- 8.*PI*PN*P5(I)*XL2/XMU2*(P 1(4)+P6(4))/ Appendix D. Amplitudes for the 71-+p --÷ 7r+ep reaction • (SRA-XMN- P1(4)-P5(4)-(P11+P 554-2.*P15)/(2.*XMN)) U2C(I)=4.*PI*FN*P6(I)*(2.*XL1/XMU + XL2/XMU2*(P1(4)+P5(4)))/ • (SRA-P5(4)-XMN-P55/(2.*XMN)) U3AFAC=2.*FN**3/ • (SRA- XM N- P1(4)-P 6 (4)-(P11 +P 66+ 2.*P16)/ (2.*XMN))/ • (SRA-XMN-P1(4)-P5(4)-P6(4)- F (P11+P55+P66+2.*(P 16+P15+P56))/(2.*XMN)) U3A(I)=U3AFAC*(P15*P6(I)+P56*P1(I)-P16*P5(I)) U3BFAC=-2.*FN**3/ • (SRA-X MN- P 1(4)-P5 (4)-(P 11+P 55 +2.*P 15)/ (2. *)( MN))/ • (SRA-XMN-P1(4)-P5(4)-P6(4)- F (P11-1-P55 +P66+2.*(P15+P16+P56))/(2.*XMN)) U3B(I)=U3BFAC*(P16*P5(I)+P56*P1(I)-P15*P6(I)) U3CFAC=2.*FN**3/ • (SRA-XMN-P1(4)-P5(4)-(P11+P 55+2.*P56)/(2.*XMN))/ • (SRA-XMN-P5(4)-P55/(2.*XMN)) 1T3C(I)=IT3CFAC*(P16*P5(I)+ P15*P6(I)-P56*P1(I)) 1.T3DAFAC=FN*FD**2/ • (SRA-XMD+UY*GDEL(SRA)/2.)/ • (SRA-P5(4)-XMN-P 55/2. /XMN) U3DA(I)=U3DAFAC* • (2./3.*P15*P6(I)+1./3.*P 16*P5(0-1 ./3.*P56*P1(I)) DN3A=SRA- P5(4)-P55/ 2./ XMNS U3DNAFAC=FNS*FD*FND/ • (SRA-XMD+UY*GDEL(SRA)/2.)/ • (DN3A-XMNS+1JY*G1JELS(DN3A)/2.) U3DNA(I)=U3DNAFAC* • (2./3.*P15*P 6(I)+1.13 .*P16*P5(0-1./3 .*P56*P1(I)) D3B.SRA-P 6(4)-P66/ 2./XMD U3DBFAC=FN*FD**2/ • (SRA-XMN-P1(4)-P 6(4)-(P11+ P66+2.*P16)/2./XMN)/ • (D3B-XMD+1JY*GDEL(D3B)/2 .) U3DB(I)=U3DBFAC* • (2./3.*P15*P6(I)+1./3.*P56*P1(I)-1./3.*P16*P5(I)) D3NB=SRA-P1(4)-P6(4)-(P11+P 66+2.*P16)/2 ./XMNS U3DNBFAC=FNS*FD*FND/ • (D3NB-XMNS+1JY*GDELS(D3NB)/ 2.)/ • (D3B-XMD+UY*GDEL(D3B)/2.) U3D NB( I) U 3D NBFA C* • (2./3.*P15*P 6(I)+1./3.*P56*P1(I)-1./3.*P16*P5(I)) D3C=SRA-P1(4 )-P5(4)-P 6(4)-(P11+P 66+P55+ 2.*(P16+P15+P56))/2./XMD U3D C FA C =FN*FD**2/ • (SRA-P1 (4)-P6(4)-X MN-(P11+ P66+ 2*P 16)/2./XMN)/ 90 Appendix D. Amplitudes for the 7+ p • 71-+R0p reaction (D3C-XMDA-IIY*GDEL(D3C)) U3DC(I)=U3DCFAC* • (2./3.*P15*P6(I)+1./3.*P16*P5(I)-1./3.*P56*P1(I)) D3NC=SRA-P1(4)-P6(4)-(P66-FP11+ 2.*P16)/2./XMNS U3DNCFAC=FNS*FD*FND/ • (D3NC-XMNS-FUY*GDELS(D3NC)/2.)/ • (D3C-XMDA-UY*GDEL(D3C)/2.) U3DNC(I)=-U3DNCFAC* • (2./3.*P15*P6(I)-1-1./3.*P16*P5(I)-1./3.*P56*P1(I)) DD3=SRA-P5(4)-P55/2 /X MD U3DDFAC=-2 /3 *FN*FD**2/ • (SRA-XMN-P1(4)-P 5(4)-(P11-FP55+2 *P15)12 /XMN)/ • (DD3-XMD-I-UY*GDEL(DD3)/2 ) U3DD(I)=U3DDFAC* • (2 /3 *P16*P5(I)+1 /3 *P 56*P1(I)-1 /3 *P15*P6(I)) D 3N11 = SRA-P 5( 4)-P 1(4)-(P11-FP55 +2.*P15)/ 2./X MNS U3DNDFAC=-2./3.*FNS*FD*FND/ • (D3ND-XMNS-FUY*GDELS(D3ND)/2.)/ • (DD3-XMD-4UY*GDEL(D1J3)/2.) U3DND(I)=U3DNDFAC* • (2./3.*P16*P 5(I)+1./3.*P56*P1(I)-1./3.*P1 5*P6(I)) D3E=SRA-P1(4)-P5(4)-P6(4)-(P11-FP66-FP 55-1-2.*(P16-FP15-FP56))/2./XMD U3DEFAC=2./3.*FN*FD**2/ • (SRA-XMN-P1(4)-P 5(4)-(P11+ P55+2 P15)/2./XMN)/ • (D3E-XM1D-FUY*GDEL(D3E)/ 2.) U3DE(I)=U3DEFAC* • (2./3.*P16*P5(I)+1./3.*P15*P6(0-1 ./3.*P56*P1(I)) DN3E=SRA-P1(4)-P5(4)-(P11-FP55-1-2.*P15)/2./XMNS U3DNEFAC=2./3.*FNS*FD*FND/ • (DN3E-XMNS-FUY*GDELS(DN3E)/2.)/ • (D3E-XMD-FUY*G1JEL(D3E)/2.) U3DNE(I)= U3DNEFAC* • (2./3 .*P16*P 5(I)+1./3 P15*P6(I)-1./3 P56*P1(I)) D3F=SRA-P1(4)-P6(4)-1-(P11-FP66-1-2.*P 16)/2./XMD U3DFFAC=-2./3.*FN*FD**2/ • (D3F-XMD-FUY*GDEL(D3F)/2.)/ • (SRA-XMN-P1(4)-P 5(4)-P6(4)- F (P11-FP66-FP55+2.*(P 16-FP15-FP56))/2./XMN) U3DF(I)=U3DFFAC* • (2. /3.*P56*P1(I)+1./3.*P 16*P5(I)-1 /3.*P15*P6(I)) DN3F=SRA-P1(4)-P5(4)-P 6(4)-(P11-1-P66-1-P55+2.*(P16-FP 15-FP56))/2 ./XMNS U3DNFFAC=-2./3.*FNS*FD*FND/ • (D3F-XMD-FUY*GDEL(D3F)/2.)/ • (DN3F-XMNSA-UY*GDELS(DN3F)/2.) U3DNF(I)=U3DNFFAC* 91 Appendix D. Amplitudes for the 7r+p • 71-1-71-°p reaction (2. /3.*P56*P 1(I)+1./3.*P16*P5(0-1./3.*P15*P6(I)) D3G=SRA- P1(4 )-P5(4 )-(P11 -4-P55-1-2.*P15)/2./ XM D U3DGFAC=2./3.*FN*FD**2/ • (D3G-XMD -FUY*GDEL(D3G)/2.)/ • (SRA-XMN-P1(4)-P 5(4)-P6(4)- F (P11-1-1'55-1-P66-1-2.*P15+2.*P16-1-2.*P56)/2./XMN) U3DG(I)=U3DGFAC* • (2. /3.*P56.1'1(I)-1-1./3.*P 15*P6(I)-1./3.*P16*P5(I)) DN3G=SRA-P1(4)-P5(4)-P6(4)-(P 11-1-P55+ P66-1-2.*(P15.4-P16-1-P 56))/2./XMNS U3DNGFAC=2./3.*FNS*FD*FND/ • (D3G-XMD-FUY*GDEL(D3G)/2.)/ • (DN3G-XMNS-1-1IY*GDELS(DN3G)/2.) U3DNG(I)=U3DNGFAC* • (2./3.*P56*P 1(I)+1./3.*P15*P6(I)-1./3.*P16*P5(I)) D 3H= SRA-P1(4)-P5(4)- (P11-1-P55-1-2.*P15 )/2. /XMD U3DHFAC=1./3.*FN*FD**2/ • (D3H-XMD-FITY*GDEL(D3H)/ 2.)/ • (SRA-P5(4)-XMN-P 55/2./XMN) U3DH(I)=U3DHFAC* • (2./3.*P15*P6(I)+1./3.*P 56*P1(I)-1 ./3.*P16*P5(I)) DN3H=SRA-P5(4)-P55/2./XMNS U3DNHFAC=1./3.*FNS*FD*FND/ • (D3H-XMD-I-UY*GDEL(D3H)/2.)/ • (DN3H-XMNS-1-1JY*GDELS(DN3H)/2.) U3DNH(I)=U3DNHFAC* • (2. /3.*P 15*P 6(I)-1-1./3.*P36*P1(0-1./3.*P16*P5(I)) DDA=SRA-P6(4)-P 66/2 /XMD U3DDAFAC=3 /2 *FDD*FD**2/ • (SRA-XMD-FUY*GDEL(SRA)/2 )/ • (DDA-XMD+UY*GDEL(DDA)/2 ) U3DDA(I)=U3DDAFAC* • (-1 /6 *P56*P1(I)+2 /3 *P15*P6(I)-1 /6 *P 16*P5(I)) DDB.SRA-P5(4)-P55/2. /X MD U3DDBFAC=-FDD*FD**2/ • (SRA-XMDA-1JY*GDEL(SRA)/2.)/ • (DDB-XMD-FUY*GDEL(DDB)/2.) 1.13DDB(I)=U3DDBFAC* • (-1./6.*P56*P1(I)+2./3.*P16*P5(I)-1./6.*P 15*P6(I)) DDC1=SRA-P1(4)- P6(4)-(P11-I-P66-1-2.*P 16)/2. /XMD DDC2=SRA-P6(4)- P66/2. /XMD U3DD CFA C =FDD*FD**2/ • (DD Cl-XMD +U Y*G DEL(DD C1)/ 2.)/ • (DD C2-XMD 1-UY*G DEL(DD C2)/2.) U3DDC(I)=U3DDCFAC* • (-1. /6.*P1.5*P6(I)+2. /3.*P56*P1(I)-1. /6.*P 16*P5(I)) 92 Appendix D. Amplitudes for the r+p —4 7+70p reaction DDD1=SRA-P1(4 )-P6(4)-(P11-1-P661-2.*P16)/2./XMD DDD2=SRA-P1(4 )-P5(4)-P 6(4)-(P11-I-P 55+P66+ 2.*(P15+P 1 6-I-P56))/2./X MD U3DDDFAC=-FDD*FD**2/ • (DDD1-XMD-I-UY*GDEL(DDD1)/2.)/ • (DDD2-XMD-I-UY*GDEL(DDD2)/2.) U3DDD(I)=-U3DDDFAC* • (-1./6.*P15*P6(I)-1-2./3.*P16*P5(I)-1./6.*P56*P1(I)) DDE1=SRA-P 1(4)-P5(4)-(P111-P55 -1-2.*P15)/2./XMD DDE2=SRA-P 5(4)-P55/ /XMD U3DDEFAC=2./3.*FDD*FD**2/ • (DDE1-XMD+UY*GDEL(DDE1)/2.)/ • (DDE2-XMD+UY*G1JEL(DDE2)/2.) U3DDE(I)=U3DDEFAC* • (-1./6.*P16*P5(I)-1-2./3.*P56*P1(I)-1./6.*P 15*P6(I)) DDF1=SRA-P1(4)-P5(4)-(P11 -FP55-1-2.*P 15)/2./XMD DDF2=SRA-P1(4)-P5(4)-P6(4 )-(P 11-FP 551-P66+2 .*(P 15 -I-P56 -I-P16))/2./XMD U3DDFFAC=-1./6.*FDD*FD**2/ • (DDF1-XMD+TJY*GDEL(DDF1)/2.)/ • (DDF2-XMD+1JY*GDEL(DDF2)/2.) U3DDF(I)=U3DDFFAC* • (-1.16.*P16*P5(I)+2./3.*P15*P6(I)-1./6.*P 56*P 1(I)) 1.TTOT(I)=XID(21)*(U1P(I)+U1C(I))+XID(17)*U2A(I)-1• XID(18)*U2B(I)+XID(19)*U2C(I)-1- F XID(22)*U 3A(I)+ XID(1)*( U3 B (I)-1-U3C(I))+XID( 2)*U3D AM+ • XID(3)*U3DB(I)+XID(4)*U3DC(I)+XID(5)*U3DD(I)+ • XID(6 )*U3DE(I)-1-XID 7)*U3DF(I)-1-XID(8)*U 3D GM+ • XID(9)*U3DH(I) UTOT(I)=UTOT(0+ • XID(10)*U3DDA(0-1-XID(11)*U3DDI3(I)-1-XID(12)*U3DDC(I)+ • XID(13)*U3DDD • XID(23)*U3DNA(I)+XID(24)*U3DNB(I)-1-XID(25)*U3DNC(I)-F (14)*U3DDE(I)+X ID (15)*U3DDF(I)-1- • XID(26)*U3DND(I)+XID(27)*U3DNE(I)+XID(28)*U3DNF(I)-1- • XID(29)*U3DNG(I)-1-XID(30)*U3DNH(I) ENDDO UJAC=UY*XMIXT(P1,P 5,P6) U3A(4)=UJAC*U3A FAC U3B(4)=- UJAC*U3BFAC U3C(4)=UJAC*U3CFAC U3DA(4)=(1./3.)*UJAC*U3DAFAC U3DB(4)=( 1. /3.)*UJAC*U3DBFAC U3DC(4)=(-1./3.)*UJAC*U3DCFAC U3DD(4)=(- 1./3.)*UJAC*U3DDFAC U3DE(4)=(1. /3 .)*UJAC*U3DEFAC U3DF(4)=(-1./3.)*UJAC*U3DFFAC U3DG(4)=(1./3.)*UJAC*U3DGFAC U3DH(4)=(- 1. /3.)*UJAC*U3DHFAC 93 Appendix D. Amplitudes for the r+p -4 7r+7r°p reaction U3DNA(4)=(1./3.)*UJAC*U3DNAFAC U3DNB(4)=(1 ./3.)*UJAC*U3DNBFAC U3DNC( 4) =(-1./ 3 .)*UJA C*U3DNCFAC U3DND(4)=(-1 )*UJAC*U3DNDFA C U3DNE(4)=(1./3.)*UJAC*U3DNEFAC U3DNF(4)=(-1./3.)*UJAC*U3DNFFAC U3DNG(4)=(1./3.)*UJAC*U3DNGFAC U3DN1T(4)=(-1./3.)*UJAC*U3DNHFAC U3DDA(4)=5./6.*UJAC*UDDAFAC U3DDB(4)=-5./6.*UJAC*UDDBFAC U3DDC(4)=-5./6.*UJAC*UDDCFAC U3DDD(4).5./6.*UJAC*UDDDFAC U3DDE(4).5./6.*UJAC*UDDEFAC U3DDF(4).-5./6.*UJAC*UDDFFAC UTOT(4)=X ID (22)*U 3A(4)-I- XID(1)*( U3B(4)-1- U3 C(4))-1F XID(2)*U3DA(4)+XID(3)*U3DB(4) F -I-XID(4)*U3DC(4)+XID(5)*U3DD(4)+XID(6)*U3DE(4)+ F XID(7)*U3DF(4) F +XID(8)*U3D G(4 )+XID(9)*U3DH(4)+XID(10)*U3DD A(4)+ F XID(11)* F U3DDB(4)+XID(12)*U3DDC(4)-1-XID(13)*U3DDD(4)-1F XID(14)*U3DDE(4) • • +XID(15)*U3DDF(4) -FXID (23)*U 3DNA(4)-1- XID (24 )*U3DNB(4)+XID (25 )*U3DNC(4) • -1-XID(26)*U3DND(4)-1-XID(27)*U3DNE(4)+XID(28)*U3DNF(4) • -i-XID(29)*U3DNG(4)-1-XID(30 )*U3DNI1(4) RETURN ELSE UT 0 T(1).0. ITTOT(2)=0. UTOT(3)=0. UT 0 T(4)=S QRT(SIGMA(PPILAB,TPL-1)) RETURN ENDIF END 94 Appendix E World Data for 7F — 271 Channels The world data at lower energies are given below. Where the original data was given in terms of the incident pion's lab momentum and rms beam momentum spread (po and Apo), the pion lab kinetic energy and its error, 77, and A T„ , were calculated using T„ = /p2 + ,a2 — it and AT, = poApol \ IA + ta2• The errors in cross section include both statistical and systematic errors, unless separate entries given, in which case the total error has been calculated using Ao— total = NAACrstat)2 + (Lass)2 • 7r— p --* 7r-7-En T, (MeV) 210+7 222+5 233+7 246+6 264+12 288+12 (3-(1b) 15+3 27+5 53+13 125+28 160+60 380+90 Table E.1: Batusov's 7-7± data [BA65]. 95 • Appendix E. World Data for 7r — 27r Channels T,(MeV) 203 230 255 279 292 331 357 96 a(jib) 13.8+1.5 60.3+3.2 166+6 374+15 546+31 1160+52 1880+77 Table E.2: Bjork's 7r-7r+ data [BJ80]. Po ± Apo (MeV/c) 295+9 315+10 334+10 354+10 375+11 394+10 413+10 432+11 450+12 a (ub) (MeV) 5.1 186.8+8.1 20 205.0+9.1 222.4+9.2 51 240.9+9.3 118 260.5+10.3 211 278.4+9.4 327 296.4+9.5 477 314.4+10.5 785 331.6+11.5 1052 TR. A a stat (sub) 1.1 2.4 10 15 27 18 17 55 42 AO. sy st (fub) 0.5 1.8 6 13 24 37 53 88 118 Acrtotai (sub) 1.2 3.0 12 20 36 41 56 104 125 Table E.3: OMICRON's 7r-7r+ data [OM89a]. Appendix E. World Data for 7 - 97 27r Channels • 7r+p --+ 7+7+n T(MeV) 226 250 312 357 a(1ub) 9.4±2.3 25.0±5.3 57±10 100±21 Table E.4: Kravtsov s 7r+7r+ data [KR78]. Acrstat (,ub) (1ib) 1.8 0.2 8.0 1.3 21.7 3.0 27.4 3.2 39.0 4.4 45.1 5.2 65.0 4.7 74.0 5.3 83.0 7.3 94.0 8.0 a PO Tir (MeV/c) 297 317 338 358 378 398 418 439 459 480 (MeV) 188.6 206.8 226.1 244.7 263.4 282.2 301.1 321.1 340.2 360.3 Ausyst (ub) 0.3 1.2 3.3 4.1 5.5 8.9 12.7 14.4 16.2 18.4 A atotal (4ub) 0.4 1.8 4.5 5.2 7.0 10.3 13.5 15.3 17.8 20.1 Table E.5: OMICRON's 7+ 7+ data [0M90]. T, (MeV) 180 184 190 200 a (pb) 0.11±0.03 0.28±0.05 0.60±0.10 1.46±0.22 Table E.6: Sevior's 7r+7r+ data [SE91]. Appendix E. World Data for 7r - 27r Channels •7 98 p _ ), 70 70 n Beam momentum (MeV/c) 272.5 275.5 279.7 283.9 285.7 286.9 291.0 292.6 297.7 304.7 313.8 322.5 330.5 339.4 349.4 359.1 389.6 399.9 T11- a (MeV) (,ub) 166.59 0.382 169.45 0.59 173.02 1.18 176.78 2.06 178.40 2.31 179.48 3.33 183.40 3.81 184.61 8.1 189.22 8.5 195.58 17.1 203.87 21.9 211.84 30.3 219.15 59.8 227.41 75.2 236.68 98.1 245.68 118 274.28 388 283.99 479 Aristat (ab) 0.096 0.14 0.22 0.35 0.65 0.64 0.81 1.3 1.0 1.9 2.0 3.0 6.4 7.3 9.3 11 46 49 Aasyst Aatotal (ub) 0.023 0.04 0.07 0.12 0.14 0.20 0.23 0.5 0.5 1.0 1.3 1.8 3.6 4.5 5.9 7 23 29 (gb) 0.099 0.14 0.23 0.37 0.66 0.67 0.84 1.4 1.1 2.2 2.4 3.5 7.3 8.6 11.0 13 52 57 Table E.7: Lowe's 7r07r0 data [L091]. 99 Appendix E. World Data for 7r — 27r Channels • 7r-p 7r-7r°p Po ± Apo (MeV/c) 295+9 315+10 334+10 354+10 375+11 394+10 413+11 432+11 450+12 T„ (MeV) 186.8+8.1 204.9+9.1 222.4+9.2 240.9+9.3 260.5+10.3 278.4+9.4 296.4+10.4 314.4+10.5 331.6+11.5 a (sub) 0.75 2.2 8.5 20 27 50 73 119 157 A Cr stat (pb) 0.3 0.6 1.4 3 4 4 4 8 9 A a syst (jib) 0.3 0.4 0.8 4 4 12 14 18 36 AO-total (ab) 0.4 0.7 1.6 5 6 13 15 20 37 Table E.8: OMICRON's 7r-7r° data [0M89b]. • 7r + p 71- + p T,(MeV) 230+13 275+15 294+4 300 cr(pb) 18+192 48iT 120 + 50 110 + 40 source [BA75] [BA75] [AR72] [BA63] Table E.9: World data for 7r+7r° at lower energies. Bibliography [AR72] M. Arman et al., Phys. Rev. Lett. 29, (1972) 962. [BA63] V. Barnes et al., CERN Report 63-27 (1963). [BA65] Yu. A. Batusov et al., Soy. J. Nucl. Phys. 1, (1965) 374. [BA75] Yu. A. Batusov et al., Soy. J. Nucl. Phys. 21, (1975) 162. [BJ801 C. W. Bjork et al. Phys. Rev. Lett. 44, (1980) 62. [BL91] H. Burkhardt and J. Lowe, Phys. Rev. Lett. 67, (1991) 2622. [B086] A. Bohm, Quantum Mechanics: Foundations and Applications, second ed., Springer-Verlag, 1986. [DE66] J. Detoeuf et al., Phys. Rev. Lett. 16, (1966) 860. [EW88] T. Ericson and W. Weise, Pions and Nuclei, Int. Ser. of Monographs on Phys. no. 74, Clarendon Press, Oxford, 1988. [EF92] E. Friedman et al., TRIUMF Exp. no. 655. [GL83] J. Gasser and H. Leutwyler, Phys. Lett. 125B, (1983) 321. [KR78] A. V. Kravtsov et al., Nuclear Physics B134, (1978) 413. [L091] J. Lowe et al., Phys. Rev. C 44, (1991) 956. [MM84] D. M. Manley, Phys. Rev. D 30, (1984) 536. 100 Bibliography 101 [0M89a] OMICRON Collaboration, G. Kernel et al., Phys. Lett. B 216, (1989) 244. [0M89b] OMICRON Collaboration, G. Kernel et al., Phys. Lett. B 225, (1989) 198. [0M90] OMICRON Collaboration, G. Kernel et al., Zeit. fiir Physik C 48, (1990) 201. [0T68] M. G. Olsson and L. Turner, Phys. Rev. Lett. 20, (1968) 1127. [0T69] M. G. Olsson and L. Turner, Phys. Rev. 181, (1969) 2141. [OT77] M. G. Olsson, E.T.Osypowski, L. Turner, Phys. Rev. Lett. 38, (1977) 297; 39, (1977) 52(E). [0V85] E. Oset and M.J. Vicente-Vacas, Nuclear Physics A446 (1985) 584. [OV91] E. Oset and M.J. Vicente-Vacas, private communication. [PDG90] Review of particle properties, Particle Data Group, Phys. Lett. 239B, 1990. [SC67] J. Schwinger, Phys. Lett. 24B, (1967) 473. [SE91] M. E. Sevior et al., Phys. Rev Lett. 66, (1991) 2569. [S090] V. Sossi, Ph.D. thesis, 1990, unpublished. [S092] V. Sossi et al., Nuclear Physics (1992), to be published. [WE66] S. Weinberg, Phys. Rev. Lett. 17, (1966) 616. [WE67] S. Weinberg, Phys. Rev. Lett. 18, (1967) 188. [WE68] S. Weinberg, Phys. Rev. 166, (1968) 1568.
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Derivation of amplitudes for [pi] - 2[pi] reaction...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Derivation of amplitudes for [pi] - 2[pi] reaction channels using the oset-vicente-vacas model, and calculation… Fazel, Neil Behzad 1992
pdf
Page Metadata
Item Metadata
Title | Derivation of amplitudes for [pi] - 2[pi] reaction channels using the oset-vicente-vacas model, and calculation of the chiral symmetry breaking parameter |
Creator |
Fazel, Neil Behzad |
Date Issued | 1992 |
Description | The Oset and Vicente-Vacas model for π N --> π πN reaction has ben used to derive theoretical values of the amplitudes and cross sections for the π - 2π reaction channels. It is shown that the N* --> N(π π)s—wave mechanism is required in order to obtain agreement with the world data for the π-ρ --> π-π+ and π-ρ ---> π0 π0 n channels. The x2 analyses of the world data near threshold for all the channels strongly support the Weinberg's value of Ƹ = 0 for the chiral symmetry breaking parameter. The value of C, the factor related to N*Nπ π coupling through the exchange of the scalar meson ε, was estimated to be C = (-1.97 + 0.11) μ-1. |
Extent | 3004630 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-12-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0086505 |
URI | http://hdl.handle.net/2429/2904 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1992-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-ubc_1992_fall_fazel_neil_behzod.pdf [ 2.87MB ]
- Metadata
- JSON: 831-1.0086505.json
- JSON-LD: 831-1.0086505-ld.json
- RDF/XML (Pretty): 831-1.0086505-rdf.xml
- RDF/JSON: 831-1.0086505-rdf.json
- Turtle: 831-1.0086505-turtle.txt
- N-Triples: 831-1.0086505-rdf-ntriples.txt
- Original Record: 831-1.0086505-source.json
- Full Text
- 831-1.0086505-fulltext.txt
- Citation
- 831-1.0086505.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0086505/manifest