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Derivation of amplitudes for [pi] - 2[pi] reaction channels using the oset-vicente-vacas model, and calculation… Fazel, Neil Behzad 1992

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DERIVATION OF AMPLITUDES FOR n  — 2n REACTION CHANNELSUSING THE OSET—VICENTE-VACAS MODEL, AND CALCULATIONOF THE CHIRAL SYMMETRY BREAKING PARAMETERNeil Behzad FazelB. Sc. (Applied Physics) Sharif University of Technology, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJune 1992© Neil Behzad FazelIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department ofPhysics  The University of British ColumbiaVancouver, CanadaDate June 19, 1 992DE-6 (2/88)AbstractThe Oset and Vicente-Vacas model for nN --> nnN reaction has been used to derivetheoretical 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 obtainagreement with the world data for the n-p --> n-n+ and n-p --->n0n0n channels. Thex2 analyses of the world data near threshold for all the channels strongly support theWeinberg's value of e = 0 for the chiral symmetry breaking parameter. The value ofC, 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.iiTable of ContentsAbstract 	 iiList of Tables 	 vList of Figures 	 viiAcknowledgements 	 viii1 Introduction 	 11.1 Overview  	 11.2 Chiral Symmetry and PCAC 	  31.3 Olsson and Turner's Work 	  42 7r — 27r Reaction Channels	92.1 Introduction  	 92.2 x2 Analysis of the World Data 	  112.2.1 The 7r-p —* 7r07r0n Channel 	  112.2.2 The 7r- p ---* 7r-7r+n Channel 	  192.2.3 The 7r+p -- 7r+7r+n Channel 	  212.2.4 The 7r-p --4 7r-7°p Channel 	  262.2.5 The 7r+p —* 7r+70p Channel 	  293 Discussion and Conclusion 	 33A Derivation of the Amplitudes		 41111A.1 The Oset and Vicente-Vacas Model 	  41A.2 Expansion of the Lagrangians 	  43A.3 Recipe for Obtaining the Amplitudes 	  46A.4 Calculation of Some of the Amplitudes for the Three Channels 	  48A.5 Taking Care of Spin 	  56B Amplitudes for the 7r- p --* 71070n reaction 	 59C Amplitudes for the 7r- p 	 7r-ep reaction 	 71D Amplitudes for the 7r+ p —* 7r+7°p reaction 	 83E World Data for 7r — 27r Channels 	 95Bibliography 	 100ivList of Tables	2.1 Results of the x2 analysis for the 7r07r0 channel    182.2 Errors in the x2 analysis of the 7r°7r0 channel 	  182.3 Results of the x2 analysis for the 7r-7r+ channel. 	  202.4 Errors in the x2 analysis of the 7r-7r+ channel. 	  202.5 Results of the x2 analysis for the 7r+7r+ channel, excluding the OMI-CRON data. 	  24E.1 Batusov's 7r-7r+ data [BA65]. 	  95E.2 Bjork's 7r-7r+ data [BJ80] 	  96E.3 OMICRON's 7-7+ data [0M89a]. 	  96E.4 Kravtsov's 7r+7+ data [KR78]. 	  97E.5 OMICRON's 7r+7+ data [0M90]	 97E.6 Sevior's 7r+7r+ data [SE91]. 	  97E.7 Lowe's 7r07r0 data [L091]. 	  98E.8 OMICRON's 7r-7r0 data [0M8913]. 	  99E.9 World data for 7+7r° at lower energies 	  99List of Figures2.1 The diagrams which do not vanish at threshold in the rr0 channel. . . 142.2 Comparison of the model's predictions for the 7r07r0 channel, with andwithout the N* 2-point diagrams (C = —2.13 it' and = 0.3) 	  142.3 7r07r0 two point diagrams 	  152.4 7rr0 three point diagrams 	  152.5 7rr0 three point diagrams with one delta intermediate state. 	  162.6 71-r0 three point diagrams with two delta intermediate states 	  162.7 7r07r0 two point diagrams with N* and delta intermediate states. 	  162.8 The surface plot of x2 for the 7rr0 channel 	  182.9 Feynman diagrams for the 7r-7r+ channel. 	  192.10 The surface plot of x2 for the 7r-7r+ channel 	  212.11 Feyn.man diagrams for the 7+71-+ channel. 	  222.12 x2 analysis of the 7r+7r+ channel. 	  232.13 World data for 7r+7r+ channel, and the theoretical prediction of the modelfor = 0.15 	  252.14 Feynman diagrams for the 7r-7r0 channel. 	  262.15 The contribution of different groups of diagrams to the total cross sectionfor the 7r-7r0 channel with = 0. 	  272.16 (a) x2 analysis of the 7r-7r0 channel, (b) world data for the 7r-7r0 channel,and model's prediction for — 1  9	   282.17 comparison of the cross sections given by the model for different valuesof the chiral symmetry breaking parameter	  29vi2.18 Feynman diagrams for the 7r+7r° channel. 	2.19 X2 analysis of the 7r+7r° channel, and comparison of the model's predic-30tion for different values of 4" 	 323.1 Comparison of model's prediction for the 7r+7r+ channel with 6 = 0 and 	 353.2 The contour plots of X2 for the 7r+7r- and 7r°7r° channels 	 373.3 World data for the 7r+7r-, 7r+7r+, 7r°7r° and 7r-7r0 channels, and model'sprediction for C = —1.97 pc', 6 =--- 0 	 383.4 World data for the 7r+7r0 channel, and model's prediction for 6 =--- 0. . . . 39viiAcknowledgementsI would like to thank Dr. Richard Johnson for supervising this work.Thanks are also due to Eulogio Oset and Manolo Vicente-Vacas for making theirMonte Carlo code containing the amplitude calculations for the 7r- p -4 7r-7r+n and7r+p --+ 7r+7r+n channels available to us; as well as to Vesna Sossi, for her assistance,and Martin Sevior, for his comments.viiito my parentsixChapter 1IntroductionThe knowledge of 7r — 7 scattering amplitudes at zero relative momentum provides uswith clues on the nature of chiral symmetry breaking. Experimentally this involves thestudy of Ke4 decay and pion-induced pion production (7r —27r) reactions. In the presentchapter I will first give a broad perspective of the theoretical environment that makesthe 7rN --4 7r7rN reactions experimentally significant, and will then review some of itsaspects in greater detail.1.1 OverviewA good starting point for the discussion of 7r — 27r reactions would be chiral symme-try. This refers to the symmetry which would exist if the up, down, and one mightadd, strange quarks were massless; then the Quantum ChromoDynamic Lagrangianwould consist of two parts, one for the right-handed quark fields and the other for theleft-handed ones —without any coupling between them— i.e. handedness of the fieldswould 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 thecase of chiral symmetry this conserved quantity turns out to be the axial current. Themethodology based on this symmetry, provides us with the only rigorous approach tolow energy QCD. However in reality chiral symmetry is broken spontaneously, as wellas explicitly. The difference between these two I will make clear later, but the con-sequence of chiral symmetry being explicitly broken is that the axial current will no1Chapter 1. Introduction 	 2longer be conserved. Still it would be approximately conserved if the symmetry wereonly slightly broken; this leads to the idea of Partially Conserved Axial Current, orPCAC.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 el-ements for emission and absorption of soft pions and determine the pion scatteringlengths. Weinberg used PCAC and current algebra for calculations of 7 — 7 scatteringlengths [WE66]; later he changed his approach and used phenomenological Lagrangiansto reproduce his previous results for 7r —7r scattering [WE67]. However, his results werenot entirely consistent with the work done by Schwinger [SC67] in which, using a non-operator method and eliminating all reference to current algebra, he had constructeda 7r — 7 interaction Lagrangian which differed from Weinberg's by a term represent-ing chiral symmetry breaking. Later it was shown by Olssen and Turner [0T68] thatthe Lagrangians of Weinberg and Schwinger were cases of the most general 7 — 7r La-grangian derived in accordance with current algebra and PCAC; this Lagrangian hadone free parameter, , to be determined experimentally, and the Lagrangians of Wein-berg and Schwinger corresponded to = 0 and = 1 cases, respectively (Weinberg'slater work in this area [WE68] has further implications for both Schwinger's and Olssonand Turner's results; this will be discussed in the final chapter). Weinberg, and Olssonand Turner also noted the similarity between near-threshold chiral symmetry breakingproperties of the 7rN --> 77N reaction channels and the low-energy 7r — 7r scattering,which means that experimentally, the cross section measurements of 7 — 27r reactionsnear threshold can be used to study the nature of chiral symmetry breaking.Chapter 1. Introduction	 31.2 Chiral Symmetry and PCACIn what follows, the quark will be considered as a doublet containing the up and downquarks. This is a good approximation for pion-nucleon interactions, in light of the factthat the strange quark which is the lightest of the heavy quarks, is still about 200 MeVmore massive than the u and d quarks.It is known that strong interactions conserve helicity. For massless quarks, thehelicity 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 aframe of reference moving at a speed greater than that of light will reverse the directionof the particle's momentun, required to change the helicity). These two facts togethersuggest that, at least ideally, the worlds of the righthanded and lefthanded quarks areindependent and do not interact. As a result, in the QCD Lagrangian for the freequarks, the fields representing left-handed and right-handed quarks will decouple togiverquark =_  'IT L jooL +71 TIR ipAR (1.1)The left- and right-handed parts are each invariant under an SU(2) transformationand the Lagrangian as a whole is said to be invariant under an SU(2)L 0 SU(2)Rtransformation. This invariance leads to the idea of chiral symmetry, which is, however,spontaneously broken. A symmetry is spontaneously broken when the system hasdegenerate ground states not displaying the underlying symmetry of its Lagrangian.It has been shown that whenever a symmetry is spontaneously broken, there mustexist massless bosons, called Goldstone bosons, whose interactions with the particlesof 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 coursemassive, and the reason is that chiral symmetry, in addition to being spontaneouslyChapter 1. Introduction	 4broken, is also explicitly broken, since in the real world, quarks, however relatively lightsome of them may be, are still massive. The consequence of this is that the Goldstonebosons acquire mass. This is why pions are massive. Their low mass in the hadronspectrum 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 limitwhich is the basis of several theorems concerning pionic amplitudes [EW88]. Naivelyspeaking, 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 atransformation implies the existence of a conserved current. In the case of the invarianceof QCD Lagrangian under chiral transformation, the conserved current is the axialcurrent1/) (x)-y 574 0(x) (1.2)where is the quark doublet and 7-i the Pauli isospin matrices. Under exact chiralsymmetry, the conservation of current would make the divergence of the axial currentvanish; since chiral symmetry, as pointed out, is an approximate symmetry, the axialcurrent is rather partially conserved and one has49„A'it —fu20i(x) (1.3)here fir, ft and 02(x) are the pion decay constant, pion mass and the pion field, respec-tively. Eq. (1.3) is the PCAC relation which is obtained in a natural way within theframework of chiral symmetry [EW88].1.3 Olsson and Turner's WorkAs mentioned earlier, Weinberg and Schwinger approached the problem of deriving7r —7r Lagrangians differently and their results were not similar; Weinberg's LagrangianChapter 1. Introduction 	 5was2	,C7r7r	 G2 	  ( Y-1±) 	 [2a, c-6." (9,2 -04+ it,242] 	 (1.4)8m2 gA(m is the nucleon mass), whereas Schwinger obtained	firir	 (f0/)22[akt(T (9,2- +1/22-0421	 (1.5)Using a more familiar notation, and noting the equality —2Gm • 322-1- =- L3 = 	 theseequations become, 2einber g :=	 1  [	 oil	 021	71-7	 4f 2 	 21 	 H2	r S chwinger   02 op7 	02]171" 4./7 	 4the 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, Olssonand Turner [0T68] showed that the most general Lagrangian derived in accordancewith current algebra and PCAC is"Crir = (G/2rn)2(gv/gA)2 x 	 (1.8)[2(1 — 2gof,)(040)2 — 2g0f42(51-`0)2 -1(3gof, — 2h0f — 1)1'2(02)21with go and ho as free parameters. They also defined the parameter later known asthe chiral symmetry breaking parameter, as a combination of go, ho and f, in the forme = 2f,r(go 2h0f,) 	 (1.9)the choice of go = 	 results, using Eq. (1.9), in ho 	 , and Eq. (1.8) reduces to1firr 	 [020402 _1 (1 102(02)214f72. 	 2	 2(1.10)(1.6)(1.7)so by choosing go, and using e as the new parameter, one obtains a family of Lagrangianswhich depend on the chiral symmetry breaking parameter as their single parameter;Chapter 1. Introduction	 6as will be shown in the conclusion to this thesis, there are further constraints whiche has to obey, but for now it is clear that the choice of e = 0 in Eq. (1.10) gives theWeinberg Lagrangian, whereas e = 1 gives Schwinger's (the sign difference in the firstterm of Eq. (1.10) as compared to Eqs. (1.6) and (1.7) is due to Olsson and Turnerusing a different metric).Relating the experimental results from it — 27r scattering to the it — it scatteringinvolves 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 La-grangian .C„ (Eq. 1.10), as well as three 7r —N interaction Lagrangians, ENNiririrand ENN„, generated using Weinberg's covariant-derivative formalism, and calculatedthreshold amplitudes for two of the it — 27r reaction channels, namely 7r-p 7r-7r+nand 7r+p 7r+7r+n channels'. Although their results do not adequately reproduce theexperimental data even at pion lab kinetic energies 20-30 MeV above threshold, theydid derive equations which expressed the threshold amplitudes in terms of only the chi-ral symmetry breaking parameter e, and which are still widely used by experimentaliststoday. According to their calculations [0T69]ath(7-7+) =82(—)2(-1.36 + 0.6e) (1.11)ath(7-1-7r+) (L-2)2(1.51 + 0.6e) (1.12)ath(7ro 	 ) (-3-2)2(2.11 — 0.3e)fir(1.13)where fir is the pion decay constant. They had earlier related the isospin 0 and 2 (swave) 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-pointdiagrams (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	 7the equations(1.14)(1.15) a2 	 0- 22a0 — 5a2 = 6Lwhere L = —a-.	 Solving Eqs. 	 (1.11)-(1.15) for the threshold amplitudes and the8irgscattering lengths, one gets for the 7r-7r+, 7r+7r+ and 7r°70 scattering lengths(_+) 	= 	 [-0.120ath(7r-7+) + (82)2ao 	0.038]m1a2") = [-0.048ath(7r-7r+) — (82)2 0.123]M1fir(1.16)a(++)o = [-0.120ath(7r+7r+) + (-8-2--) 2 0.382]/71-1(++)a2 = [-0.048ath(7r+7r+) + (82-Tr) 2 0.0151M7T1 (1.17)(00)ao =(82)2[0.240ath(7r°7°) + 	 o.3o5inc1(oo)a2 = [0.096ath(7r°7°) — (--2) 2 0.260]77C1 (1.18)frIn 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 canobtain the scattering lengths by first deriving the value of , from a x2 analysis of theexperimental data versus the theoretical cross sections, and then using Eqs. (1.14)-(1.15), which explicitly giveao --= —L(148 — 50a2 = — —L4 ( + 2)(1.19)(1.20)Chapter I. Introduction	 8hence the importance of knowledge of the chiral symmetry breaking parameter.As pointed out earlier, the above model does not reproduce the data for all chan-nels to the desired accuracy. It is the objective of this work to show that anotherpresently available model, namely that of Oset and Vicente-Vacas, will provide a morecomplete theoretical description of the 7r — 27r reactions.Chapter 27r — 27r Reaction Channels2.1 IntroductionIn order to test the predictions of chiral symmetry and soft pion theory, one has toresort to indirect means. This is because the particles of the theory have half lives inthe region of 26 nanoseconds, making experiments involving direct scattering of pionsunfeasable. One such indirect method is by studying the Ke 4 decay (ITthis reaction has small branching ratio, but involves pions as the only strongly inter-acting particles in the outgoing channel, hence facilitating the data analysis.The other reaction, and the one we are concerned with here, is the pion-inducedpion production reaction, or 7 — 27r scattering. For pion energies < 1 Gev, this is themajor inelastic reaction between a pion and a nucleon. The threshold measurements ofthe 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 out-going pions are created at rest in the centre of momentum system, which, as a resultof 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 amenableto experimental investigation. Since strong interactions are isospin invariant, the am-plitudes for the different charge channels can be written in terms of isospin amplitudesA21,1„ where 1,,„ is the isospin of the two outgoing pions, which, when combined withthe isospin of the outgoing nucleon, gives I, the total isospin; the isospin channels are9Chapter 2. 7r — 27r Reaction Channels	 10(/, /„) 	 (,2), (, 1), (1,1), (12--, 0) (see [MM84]) and the charge channels, along withthe isospin decomposition of their amplitudes, and their threshold energies, are1. 7r-7r+ channelA(7r-p 	 7r-r+n)rrh = 172.4 MeV2. 7r+7r+ channelA(7r+p 	 7r+7r+n)Titrh 172.4 MeV3 — A 	 A1A.10 	 31 - 5-"--1113. 7°7r0 channelA(7r-p 	 7r°70n)rrh = 160.5 MeV4. 7r-7r0 channelA(7-p 7r -70p) =rrh = 164.8 MeV5. 7r+7r0 channelA(7r+p 7+°p) = —rrh = 164.8 MeV1-5- A 32 + 32 A 1 010 32 3 12 A31 + \5A3—A + 	  11jA31The wave function of a system consisting of two pions (i.e. two bosons) will besymmetric under interchange of the pions. The total wave function of a two-pion statemay be written as zb(total) 0(space)a(spin)x(isospin); thus its symmetry is givenby (-1)(-1)s+1(-1)/4.1. Since s 0 for the pion, and at threshold 0, in orderto maintain the symmetry of the total wave function, I must be even. As a result, atthreshold the odd isospin amplitude Lrir =- 1 will vanish, leaving only the isospin 0 and2 amplitudes; for this reason these reactions have high isospin selectivity.Chapter 2. 7r — 27r Reaction Channels	 112.2 x2 Analysis of the World DataAs previously mentioned, the purpose of this work is to use the Oset and Vicente-Vacas model for 7rN 7r7rN reactions [0V85] to study the world data for 7F — 27rreactions channels. A description of the model, and the details regarding the derivationof theoretical amplitudes from it, are gived in Appendix A. The model has two freeparameters; the chiral symmetry breaking parameter, and C, the factor related tothe NN*(7r7r),,_,,,„ coupling; these are to be determined from a comparison of thetheoretical cross sections with experimental data and to this end, the amplitudes havebeen integrated into a Monte Carlo code, which has as its output the theoretical valuesof cross section predicted by the model. In the following sections, these values are usedwith the world cross section data for the five 7r — 27r reaction channels, in a x2 analysesinvolving and C as the parameters. The analysis was done according to the relation1 E  —x 2 = 	olh(c,e)] 2 (2.1)N 	 1 E, [	 So-fxin which N is the number of experimental data used in the analysis, n is the numberof parameters varied during the analysis and o and cr:h are the experimental andtheoretical cross sections, resp.; (Sur is the experimental error and c is a factor relatedto C by the relation C = —c x 1.52 it'. In order to calculate the error in the x2-analysisdue to the theoretical error, the following relation was used2	 v, icrs8(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 ChannelThe first reaction channel examined is the 7r07r0 channel. This is the only one in whichall the outgoing particles are neutral, thus excluding any electromagnetic effects in theChapter 2. 7r - 27r Reaction Channels 	 12outgoing 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 frompole and contact term diagrams, as well as the two-point diagrams with N* intermedi-ate state and NN*(7r7r),_„„, coupling (Fig. 2.1); these are the only diagrams which donot vanish at threshold, and at energies as much as 60 MeV above the threshold, theamplitude from these 4 diagrams makes up 95% of the cross section with the rest of thediagrams included. The existence of the N* 2-point diagrams is an important featureof the Oset-Vicente-Vacas model; for the two channels in which the N* -÷ N(77)s-wavemechanism exists, i.e. the 7070 and 7r-7r+ channels, this mechanism is necessary toobtain agreement with experimental cross sections. A comparison of the results, withand without this mechanism, is shown in Fig. 2.2; the inclusion of N* mechanism givesagreement 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. 1The diagrams in Fig. 2.3 are two-point diagrams with nucleon intermediate stateand 7rN s-wave amplitude. It should be noted that the phenomenological Lagrangianused in the Oset-Vicente model for NN7r7r coupling differs from the one used by Ols-son and Turner, and for this reason, this model would give values for the cross sectionwhich are not necessarily equal to theirs, even when the same Feynman processes areconsidered.Three-point diagrams with nucleon and N* in the intermediate states are shownin Fig. 2.4. The diagrams in which the N* would be formed after the emission of oneor two pions, were omitted on kinematic grounds; at 277,=300 MeV the total CM energyis 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 followingparagraphs.Chapter 2. 7 — 27r Reaction Channels 	 13nucleon) were to be formed after the emission of the 70, less than 1313-135=1178 MeVwould be available for it; if this happened before the arrival of the incoming pion, evenless 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 availablefor the formation of the N* at T,- = 300 MeV; since N* has a mass of (1440 + 40) MeVwith a width at resonance of (200 + 80) MeV [EW88], the formation of such diagramsis not energetically favoured for pion lab kinetic energies < 300 MeV, in which we'reinterested (stated more formally, the propagator factor is too small).The remaining diagrams contribute to a lesser degree, however becoming impor-tant at higher energies, especially for the 7+7r° and 7r-70 channels (in the case of 7°70channel, there are some cancellations among these diagrams, making them less impor-tant at lower energies). Using similar arguments as in the previous paragraph, amongthe diagrams in Fig. 2.7, the first and last one are expected to be important in ourenergy region.Chapter 2. 7r — 27r Reaction Channels	 14( 1)1 7T' 	 `... 	 ...•\ 	 a 	 ,..,71A 	 . 	 -,.. 	 7r- 	 ..,7T- 	 li 	 `... ifI	4,..a 	  ....-I	, 	 TIP --	. 	 S..	IT 	-S	.. 	 ........-,.it 	 S.. 	 ...." 	 ---•••. 	 ./..... ----1 	( 3 )(2)n 	 p(4), 	 •s'rr 	 4/\ 	 /71-w\\ /N'Figure 2.1: The diagrams which do not vanish at threshold in the 7rr0 channel.10'_o10._ 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 	 1 1 1 1 1 1 1 1 1- p__,7107t.ox 	 [L091]	  with Ni 2—pointdiagrams_ without Nrdiagrams1111111111111111111111111milmiliiiiiiiiiiiiipm111111111111111160 	 180 	 200 	 220 	 240 	 260 	 280 	 300pion lab kinetic energy (MeV)Figure 2.2: Comparison of the model's predictions for the 7070 channel, with andwithout the N* 2-point diagrams (C = —2.13 itt-1 and = 0.3).\ Tr' TIN \\: 	 — 	 % 	 -I 	 `1Chapter 2. 7r — 27r Reaction Channels 	 15	(5 ) 	 (6)\ 	 \ 	 0 	 0,	o /\ 	 _ 	 \7T 	 7T ' 	 Iii 	 x IT ir /	ki 7 	 lk 	 0 / 	 I 	 t	\ 	/	\ 	 \ /	\ 	 / 	 /\ 	 \P 	 n 	 n	 P 	 P 	 nFigure 2.3: 7r07r0 two point diagrams	(9) 	 (10)\ 	 /  . 	 0' 	 0' 	 Ilit 7V0// 71- i 	 7ri 	 1 _\ 	 _	14 71- 	 1 	 /	 / 	 I 	 1 7\	 I 	 / 	 I 	 / 	 I	\ 	 I 	 / 	 1 	 i 	 I P 	 n 	 n	 n 	 P 	 P 	 p 	 n	(11)/ 	(12)I 	0/0	nio /	 o /	i 	 I 	 _ 	 7T/0	7T 	 7T4	/ 	 i it 	 / 	\ 	 _	/ 	 1 	 / 	 /I 	/ 	 /I 	 / 	 \	 / 	 /i 	 ‘77777774 	 1 P 	 P 	 n 	 n 	 P 	 nr 	 n 	 nFigure 2.4: 7r07r0 three point diagrams.(24)	/	 I	o 	 0 /	71: 4 	 I 	 _ 	 71: 4	1 	 '17	 /	I 	 I 	 /	■ 	 i=---, (22)\ 0/ 	 0/\ 	 _	 'Tr } 	71-4)1 7T 	 /\ 	 / 	 /N---lk•-,•0-7,-,-,0/ PChapter 2. 7r — 27r Reaction Channels 	 16	(13)	 (14) 	 (15)\ 	 7T0 /o / 	 0 / 	 I 	 o I 	 Io / 	 0/_ 71- i\ 	 II 	 Tr 4 	 I 	 _ 	 rt- 4 	 7T 4 	 I - 	 7 47T	/ 	 I 	 171: 	 / 	 1 	 17T	 /\ 	 / 	 / 	 / 	 I 	 / 	 / 	 I	V 	 / 	 /	 I 	 V	 I	1 P 	 A° 	 n 	 n 	 P 	 P 	 A° 	 n 	 P 	 A° 	 n 	 n	(16)	 (17) 	 (18)al 	 a! 	 Io / 	 0 / 	 I 	 \ 	 / a	 0/7T 4 	 71 - 4 	 I 	 _ 	 71: 4 	 71: 4 	 I 	 _ 	 \ 	 - 	 4 71: 	 71: 41 	 I	 411: 	 1 	 / 	 47 	 47T	 1	 // 	 I 	 I 	 / 	 / 	 I 	 \ 	 / 	 // 	 r	1	r 	,	 k 	 \ 	 I	V P 	 P 	 A+ 	 n 	 P 	 A' 	 P 	 n 	 P 	 n 	 A°	 nFigure 2.5: 7r07r0 three point diagrams with one delta intermediate state.	(20) 	 (21)01 	 0! 	 I 	 0! 	 I 	 0/71:4	 7 4 	 I n- 	 7 4 	 I _ 	 7 #/ 	 I 	 4 	 I 	 v n 	 /I 	 I 	 t 	 I 	 II	 r 	 I 	 /	 I 	 i■ P 	 A° 	 A° 	 n 	 p 	 A.' 	 A° 	 nFigure 2.6: 7r07r0 three point diagrams with two delta intermediate states.(25)o I 	 I 	 I7r 4 	 74 	 I/ 	 I 	 4 7T-1 	 / 	 I	h-7-m-,,-A	 I P 	 NI 	 A. 	 n	(26) 	 (27)	o 1/ 	 I 	 \ 	 0! 	 o71:4	 704 	 I 	 _ 	 \ 	 _ 	 71: 4 7 4I 	 I 	 471 	 7 	 / 	 // 	 / 	 I 	 1 	 I 	 //----10-0,0•th 	 1,-,-,--,- 77.4 	 V P 	 4* 	 Nr 	 n 	 P 	 N° 	 A° 	 nFigure 2.7: 7r070 two point diagrams with N* and delta intermediate states.Chapter 2. 7r — 27r Reaction Channels	 17Analysis of the Data for the 7r07r0n ChannelFor 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 283.99MeV which is the highest energy for which experimental result is available from Lowe'smeasurement, they yield a cross section which around 86% of the cross section with allFeynman processes included. For this reason, this channel provides the least model-dependent estimation of the chiral symmetry breaking parameter. In the case of thechannels without the N* mechanism, diagrams other than the pole and contact be-come important, particularly due to the fact that experimental data very close to thethreshold are not available (see sections on 7r-7r0 and 7r+7r° channels). Of the 5 chan-nels investigated, the 7rr0 is the one for which the most reliable near-threshold datais available [L091]. Measurements of this reaction involve the detection of the 4 -y'screated as a result of decay of the 2 pions in the outgoing channel; due to the absence ofCoulomb interaction between the outgoing particles, measurements closer to thresholdcan be performed. The threshold pion kinetic energy for this reaction is 160.5 MeV andin order to minimize the effect of the model-dependent parts of the Lagrangian, onlyLowe'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(alldiagrams) > 95%.The results of the x2 analysis, as well as the error in x2 caused by the theoreticaluncertainty in the cross sections given by the Monte Carlo code, are given in Tables 2.1and 2.2. Figure 2.8 shows the surface plot of the x2 VS. e and c. It can be seen that asharp, well-defined minimum for x2 does not exist; in fact, by varying c, one can mini-mize 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	 182.0c \ e -1.25 -1.0 -0.75 -0.50 -0.25 0 0.25 0.50 0.75 1.0 1.250.6 7.26 8.34 9.18 10.09 11.34 12.45 13.64 14.73 15.96 17.22 18.490.8 4.46 4.90 5.78 6.40 7.25 8.31 9.32 10.27 11.19 12.53 13.771.0 2.96 3.05 3.39 3.83 4.17 5.00 5.67 6.56 7.23 8.32 9.131.2 3.18 2.77 2.73 2.56 2.58 2.80 3.19 3.65 4.14 4.84 5.461.4 5.79 4.59 3.78 3.18 2.68 2.49 2.39 2.46 2.49 2.68 3.031.6 11.16 8.83 7.90 6.16 4.85 4.12 3.41 2.88 2.54 2.17 2.061.8 18.73 16.59 14.02 11.83 9.80 8.28 6.69 5.52 4.46 3.68 3.132.0 30.99 27.10 23.99 21.37 18.58 15.07 13.29 11.10 9.34 7.61 6.17Table 2.1: Results of the x2 analysis for the 7rr0 channelc \ e -1.25 -1.0 -0.75 -0.50 -0.25 0 0.25 0.50 0.75 1.0 1.250.6 0.19 0.20 0.21 0.22 0.22 0.22 0.23 0.23 0.23 0.22 0.220.8 0.16 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.22 0.22 0.221.0 0.17 0.16 0.16 0.16 0.16 0.16 0.17 0.18 0.20 0.20 0.211.2 0.19 0.16 0.16 0.16 0.15 0.16 0.15 0.16 0.15 0.16 0.181.4 0.35 0.29 0.25 0.20 0.17 0.16 0.15 0.14 0.15 0.14 0.141.6 0.55 0.45 0.43 0.36 0.30 0.27 0.22 0.20 0.18 0.15 0.141.8 0.78 0.72 0.66 0.56 0.50 0.44 0.38 0.33 0.29 0.25 0.222.0 1.19 1.06 0.94 0.91 0.79 0.66 0.62 0.54 0.47 0.42 0.36Table 2.2: Errors in the x2 analysis of the 7r07r0 channel110101).00 	IOb. AO.4'Af dp 0.500.751.01-251.81 61.41.2	$:10 Po-0.50-0.250.251.0 -0.750.8 -1.2531x 20.6Figure 2.8: The surface plot of x2 for the 71-07r0 channel.7r+7TChapter 2. 7r — 27r Reaction Channels	 192.2.2 The 7r-p 	 7r-r+n ChannelThe 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 exper-iment [0V85][S090][S092]. The processes contributing to this channel are shown inFig. 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 contributionto the cross section.—	 TT 7T	 _- —	E,o- 	r  71--/\TVTV,'A++ 	 P+	7T7 	17Tr	A\ TV _	 TV,' 	 ' TVA' 	 A 	 A+ 	n 	 P	 A+ n	\ 7C—	 TV,,P 	 A° 	 A- 	p	 AA	 n\7T 	7V>', 	7v *,/	/ 	 ‘,, 7T— 	TV / 	 TV/ — 	 %, 	 7v-.-,,, 17T/	TV / 	 TV// 	 ‘. 7T	-.- / 	 — /■■ 	 ...---..., 	 r	,_______,	/ 	 ,P 	 Ao 	 NI 	 n 	 P 	 N1* 	 A 	 n P A" N. 	 nin+'P 	 N. P 	 A— 	 1\1 P 	 N. 	 A+TV7_—TV 4'	 7T 4 	 \1 	 a 71-A+ 	 n71_44,A' 	 A'TT,' ,r 7v— 7T-1TV ,P 	 NI. 	 A+- 	 Th 	 7T+,'\TTFigure 2.9: Feynman diagrams for the 7r-7r+ channel.Chapter 2. 7r - 27r Reaction Channels 	 20Analysis of the Data for the 7r-7r+n ChannelFor 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 surfaceplot of x2 is shown. For this channel too, the analysis does not result in a well-definedminimum for x2. However for both channels containing the N* mechanism, e = 0yields a value for C consistent with C = (-2.25 ± 0.75) it', obtained from the recentcalculation based on the branching ratio of N* N(77)s-wave [S092] .2c \ -1.25 -1.0 -0.75 -0.50 -0.25 0 0.25 0.50 0.75 1.0 1.250.6 9.58 13.98 18.06 23.78 30.14 37.03 42.99 50.17 55.82 64.00 70.180.8 5.01 5.94 9.11 12.28 17.49 21.52 28.33 34.18 41.21 48.45 54.861.0 7.74 5.36 4.68 5.37 8.46 11.47 15.59 20.93 26.63 32.90 39.121.2 19.82 13.54 8.55 5.70 5.04 5.12 7.43 10.70 13.66 18.86 25.121.4 47.36 31.95 22.50 14.77 10.24 5.71 5.07 5.12 6.49 9.17 12.951.6 86.87 63.81 48.87 35.08 24.88 16.80 11.52 7.16 5.02 5.08 5.731.8 138.18 120.99 89.77 69.21 51.94 39.28 26.13 20.37 11.67 9.00 5.742.0 228.01 176.25 151.61 127.35 95.93 76.46 55.72 43.30 32.20 21.59 15.35Table 2.3: Results of the x2 analysis for the 7r-7r+ channel.c \ -1.25 -1.0 -0.75 -0.50 -0.25 0 0.25 0.50 0.75 1.0 1.250.6 0.51 0.52 0.55 0.59 0.66 0.64 0.61 0.63 0.61 0.59 0.560.8 0.38 0.43 0.49 0.47 0.52 0.61 0.59 0.66 0.67 0.65 0.621.0 0.50 0.31 0.33 0.40 0.50 0.51 0.53 0.54 0.59 0.59 0.671.2 1.14 0.79 0.56 0.32 0.33 0.38 0.46 0.50 0.53 0.58 0.581.4 2.22 1.66 1.21 0.89 0.64 0.37 0.31 0.35 0.42 0.48 0.501.6 3.26 2.66 2.21 1.77 1.33 0.98 0.71 0.44 0.26 0.33 0.371.8 4.67 4.63 3.42 2.97 2.35 1.95 1.38 1.20 0.72 0.58 0.322.0 7.28 5.49 5.32 4.62 3.60 3.18 2.49 2.08 1.72 1.18 1.00Table 2.4: Errors in the x2 analysis of the 7r-7r+ channel.In the next chapter it will be shown that the possible values that e can take on is2This value supercedes the the original value from branching ratio calculation which Oset Vicente-Vacas had obtained [0V85].2282XChapter 2. 7r — 27r Reaction Channels 	 21constrained by further theoretical arguments as well as the consideration of the resultsof the X2 analyses of all channels; this will then be used to determine C.Figure 2.10: The surface plot of X2 for the 7r-7r+ channel.2.2.3 The 7r+p —* 7r+7r+n ChannelThe third reaction channel to be investigated is the 7r+7r+ channel. The diagrams forthis reaction are shown in Fig. 2.11. All the amplitudes, except those corresponding tothe 3-point diagrams with N* and A intermediate states (which have not been includedin 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 intermediatestate and NN*77(6) coupling (N* has charge states similar to nucleon), thus at lowenergies (where diagrams with A intermediate states do not contribute significantly) the« • 	 IIT 	 %,4IChapter 2. 7r — 27r Reaction Channels 	 22amplitudes from this model can be used in Olsson and Turner's formula (Eq. 1.17) toobtain the scattering length (provided the effect of using different N1\177-7 Lagrangians isnot 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 withall the diagrams.liT71.1.37T.'7T 4' ir 7r,4/•*'VT. 	 1/7T 	 ?To/•Tr/,n 4'p 	 n  ,IT.,,re-,' 	 7T+ i 	 \ + 	 7T+,/' \ 7I*	IT 	 I11.. i 	 A re 	 7r7 %% 	 ,I 1	...	 _I' 	 , P 	 A*+ 	 p 	 n 	 P 	 n 	 A+ 	 n 	 P 	 A° 	 p 	 n•	/V*/ 	 \ ir+55 	 I p 	 n	 A- 	 n•	7T 	 71-7\ Tr+p 	 A++	n	7T 4	 % 	 7r+1 	 IT 1 	 n' I 	 \ .„...	/I 	 \ lv. 	 I 	 ,' 	 k"i	r 	' 	vj 	I 	 1 	 '1 P 	 A° 	 A' 	 n 	 p 	 A. 	 A- 	 nn*/ \An+ 	 777T+ 	 7Ty 	 7T+„P	P 	 A°	Nt 	 n	P 	 NtFigure 2.11: Feynman diagrams for the 7r+7r+ channel.Analysis of the Data for the 7r+7r+n ChannelThe experimentalexperi ental 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 poleand contact diagrams make up ,-- 61% of the total cross section. There are discrepanciesDE o0s 0_-7T+p —>ir+-rr+n 1i--_-- 0 with OMICRON data -x without OMICRON data0x 0 -0o 0 	 Ino 0 0 	 oT<o 0 D 	 ..._-x 	 3-< -xxxxxxs -Sx 	 -zx_ ---->r<±  --Chapter 2. 7r — 2r Reaction Channels 	 23in the data in the energy region 190-255 MeV; the model favours the measurements bySevior et al. and Kravtsov et al. over those by the OMICRON group. For this reasonthe 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 inclusionI „ 	 I I -, , 	' 1 ' 	 " i ' ' " I—1.0 	 —0.5 	 0.0	 0.5 	 1.0eFigure 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 valueof lie close to zero or be negative. Also the lower-energy OMICRON data is largerthan both Sevior's and Kravtsov's data, and the isospin 2 scattering length obtainedby the OMICRON group is inconsistent with both Ke4 and chiral perturbation theoryvalues (see [SE91]). Consequently it is believed that the exclusion of their data (and20152X1050Chapter 2. 7 - 27r Reaction Channels 	 24using the cross sections at the remaining 5 energies) yields a more accurate estimateof the chiral symmetry breaking parameter.' The results of the x2 analysis of datafrom [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-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3x2 11.89 10.83 9.48 8.41 6.91 5.62 4.57 3.548(X2) 0.19 0.18 0.18 0.16 0.16 0.15 0.15 0.13-0.2 -0.1 -0.05 0 0.05 0.1 0.2 0.3x2 2.60 2.01 1.66 1.39 1.22 1.04 1.04 1.628(X2) 0.10 0.12 0.12 0.11 0.11 0.10 0.11 0.160.4 0.5 0.6 0.7 0.8 0.9 1.0X2 2.33 3.54 4.97 7.64 9.81 12.69 17.116(X2) 0.17 0.23 0.30 0.43 0.49 0.57 0.76Table 2.5: Results of the x2 analysis for the 7+7+ channel, excluding the OMICRONdata.for = 0.15. There is good agreement with Sevior's and Kravtsov's data, as well aswith 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 allchannels internally consistent below 284 MeV.1O -1Chapter 2. 71- - 27r Reaction Channels	 25I	 I 	 I 	 I 	 I 	 I 	 I 	 I 	 I	 I	 I 	 I 	 I 	 I 	 I 	 I 	 I	 I 	 I 	 I 	 I 	 I 	 I 	 I"10-3 	 I i 	 IIIIIIIIIIIIIIIIIIIIII	150 	 200 	 250 	 300 	 350	400pion lab kinetic energy (MeV)Figure 2.13: World data for 7F+71-+ channel, and the theoretical prediction of the modelfor e = 0.15.Chapter 2. 71- — 27r Reaction Channels 	 262.2.4 The 7r-p 	 7r-70p ChannelThe next reaction studied is the 7r-7r0 channel. The Feynman diagrams contributingare shown in Fig. 2.14 and the amplitudes are given in Appendix C. Diagramatically,(1)ifit 	 ITit(2)7T:s='s \71/4(3) '‘JT••••7T:''••(4)4/ 	 • 	 _71-/\\,,•iy(5)_\IT	 ,	 7%,ro,,7T041(14)- 	 Ait] • 	 -/ itP‘s 	 -VTn(6)in-,P0,Irl••P 	 P(7),, 	 7T0\TT 	 i,P7v;•,fP P	0, 	 i 	 (8)	7T, 	 it, IT-1P7T, ,P, ,	 _\71-n 	 P(9) 	,it°,4/P7T-•4Ip 	 n 	 n(10) 	 _• 7T4'‘,7T• /P	 N' 	 P(12)7T: 	 \ 	 -/ 	 • 7T,/ •p7T74'/P7T, -1p ' p 	 n(13)	Tv-4/ 	 % 	 _	/ 	 k 71-p07T„,•P• _• .7N. 	 n(1 1)	 -IT/fr /I'p.7v° ,P\Tv -A° 	 P(15)AnP_71P 	 A°(16)\--4/• 7T-\•P7Ti,vP\IVA++ 	 P(17) 	 a- 	 71-1/P_7T1P/P 	 A++ 	 P(18)_kit•prt-n	 A0(19)p n(20)„- 	 np0p A0 	 n	_ 	 (21)	.71- 01 	 IV• •/p-71-piT:41n 	 p(22)%.7r--7T/A°(23) A7T/P-P 	 A° 	 A(24)7T?P7e/P	A 	 A++(27)	7T4-, 	 -i n"P0PTT /4A* 	 A 	 P (28)" 	 ■ 	 -7TP\TvA++ 	 A++(25)PIr.:,P 	 A 	 A(26)	T  4 	 -P7T,T'P\ 7v-A° 	 N'(29) 	 04171-P-7T1•P 	 N'(30) 	 _• IT ,• 7v-	 ;,•Pa,P\N(31)IT,,P-7,%P0//N' 	 A++ 	 P(32)S no,kitIP N P P P P A° 	 N' P P A* 	 N' 	 PFigure 2.14: Feynman diagrams for the 7r-7r0 channel.the notable difference between this channel and the 7r-7r+ and 707r0 ones is the absenceof two-point diagrams with N* intermediate state and E coupling. Figure 2.15 showsb 10'3=10' -=-071-- —>7V 7T p1--Chapter 2. 7r — 27r Reaction Channels	 27that diagrams involving A's contribute significantly. In fact, the pole and contact termdiagrams 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 diagramscombined 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[0M89b]egaacriddiagramsoole,gontact and2—point diagramspole,contact,2—point,and 3—pointwith n or Nintermediate stateuole,gontact 2—point,3—point ■qnd1—delta diagramsall diagrams1111111111111111111111111111111111160	200 	 240 	 280	320 	 360pion lab kinetic energy (Mev)Figure 2.15: The contribution of different groups of diagrams to the total cross sectionfor the 7r-70 channel with e = 0.Analysis of the Data for the 7r-7r°p ChannelThe experimental data available for this channel is scarce, with few reliable measure-ments at low energies, partly as a result of the large background, mostly elastic scat-tering, which exists for this reaction (at around T7, = 190 MeV, where 7r-p 7r-p-_---_-_---3.0 _---2.5 —-_2.0 —_x' __1.5 —__--1.0 —-_Chapter 2. 7r — 27r Reaction Channels	 28reaction 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 olderdata which is for T, > 275 MeV, were discarded (see Ref. [0M8913] for references tothese). However due to the large error bars in the OMICRON measurements at lowerenergies, 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 fromdiagrams other than pole and contact term is very large (--, 85%).The result of the analysis is shown in Fig. 2.16. x2 is minimized somewhere111111111111111111111111111111 1111111111 1i11i IT - p ->71" -7r° p	 (a)_0.5 11111111111111111111111111111111111111111111—1 	 0 	 1 	 2 	 31 	 1 	 it 	 II 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 110-2 	 i „,,i,„,„,i,,‘, i150 	 200 	 250 	 300	350pion 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'scontribution is significant (compared to other diagrams) only at low energies, a is notvery sensitive to the variations in 6 at most of the energies used in the analysis, thusmaking x2 insensitive to 6. Indeed, for 6 = 0 one gets x2 es, 1, compared to x2 es, 0.6Chapter 2. 7F — 2r Reaction Channels	 29obtained for e = 1.8. As can be seen from Fig 2.17, e E [-1, 3] gives decent fits toOMICRON's data.In order to do a more accurate and model-independent estimate of e, more11111111 11111111 11111111111111111111111111111111111111111111111111111111111111 IIIillimmilimuly111111111mmililin11111111111111tylimminimin1180 200 220 240 260 280 300 320 340pion lab kinetic energy (MeV)Figure 2.17: comparison of the cross sections given by the model for different values ofthe chiral symmetry breaking parameter.measurements with better statistics near threshold are required, so that the effect ofdiagrams other than the pole and contact term will less dominate.2.2.5 The 7r+p --÷ 7+71-°p ChannelThe 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 thisreaction is the most model dependent among all channels, since already at — 5 MeVup 	 35% of the cross section.(1)171+7T 	 7T(2)ITIT+ -.	 +	 •/7T+	 :• 'o,,	P	 P	 P	 P	(3)	 (4)	 (5)0	0 ,	 ,	 ,	 .	7- ,	 .	 + 7+ ,	 7r+ /	 ‘s	 ...	 7T/• •asTr	 ,a•	 •	 •‘.7T	 ,...	 \Tv	 Tr.o• 1II	 ,	 ,	 ,	,	P	 P	 P	 P	 n	 P	 P	 P	 P(6)(7) ,	 (70).	 1	 7T ,	 I	 (9)	7T i	 iv ,	 7+;	 7T /	 ■ 	 +A	 ‘,I 7r.	 f	 t 7T	 //	 \ Tv+1	I	 I	 ,	 ■ 0/TV /(9)vv	 no/	 \7/	P	 P	 n	 p	 P	 n	 n	 p	 P	 n	 P	 P	.(10)	 (11),	 .,,	if	.74,	 a,	 *	 ,/	V	 4	 /	 %I	 It	 l	 '	 N	Tr	712'	 7Ti	 * •	 '	r	%	 +s1	 kir	/'	 /	,/	 rP	L\	 P	 P	 P	 A++	 P	 P	 P	 A°	 P	 P	 n	(13)	 (14)	 (15)	—1	 /V	 I	 +	 71-/	 it;	 I	o	 I+	 Tri	 ni	 '	 *	 7Ti	i	 i	 kn	 ,	 VT	 r	 r	 417r	 i• +.,s‘P	 n	 A,	 p	 P	 A+	 n	 P	 p.	 p° '	 n	 p	 P,	A°(17.)...	(18)	 (19)	.714'	 7Ti	'+	 7T+,/‘, n-	 4,71-	 ,	0	 0	7TI	 71-/	 ,■ 	 +	 TV,/	 71-i	,	 ,	 A	 /	 /	 /	 /	/	 I	 /	 1	 /	 /r	 .	rr./ 	 .,	 7V4'	r	 Ir 7T1	/	 I	 'I	o	 / 	 /	 /1	 .	P	 A++	 A++	 P	 P	 A++	 A+	 P	 P	 A+	 A++	 P	 P	 A+	(21)	 *	 (22)	o	,	 .71-/	 77	 I 7TNTk71"P	 L0A+	 P	 P	 A	 A	 P(23)	 (24)	 (25)Tr 4	 IT	r	 Tr	 "4/P	 N	 A++	 PP	 A++	 N.	 P	 .	 A	 P	 N(27)	 (28)	 (29)71-7T I"	 I	 71"7r+/	 VT	 /	 VT■7T •P	 A°	 P	 N	 PA	 A°P	 N	 P	 P.7r / 	 \ 7T+/A° 	 PChapter 2. 7r — 27r Reaction Channels	 30above the threshold energy of 164.8 MeV, diagrams other than pole and contact makeFigure 2.18: Feynman diagrams for the 7r+7r0 channelChapter 2. 71 — 27r Reaction Channels 	 31Analysis of the Data for the 7+7°p ChannelFew measurements have been done for this channel [BA75][AR72] [DE66][BA63] and theexisting data points are far from the threshold energy of 164.8 MeV, the lowest energymeasurement being at T, = 230 MeV where pole and contact diagrams together give across section which is only — 20% of the total cross section. This, as well as the factthat both energies and cross sections have large error bars, prevent the determinationof using x 2 analysis. Figure 2.19 shows the x2 analysis of this channel using all theworld data; in can be seen that x2 does not have a minimum in the interval [-2,2]; thiscan be misleading, since the comparison of the cross sections for different values of (inthe lower part of the figure) shows that only at energies sufficiently close to thresholdwill the model become sensitive enough to distinguish between different values of inthat interval; in fact, excluding the measurement by Barnes et al. which was doneusing hydrogen bubble chamber at CERN in 1963, the model's prediction goes throughthe error bars for any value of c between -2 and 2. As a result, until more precisemeasurements at lower energies are done, we cannot use this channel to unambiguouslyderive a value for the chiral symmetry breaking parameter. However, it should be notedthat = 0 still provides a good fit to the data..95.852 	 .75 —X.65-- 	 --r102, =---_----=_---1 	 1 	 1 	 I 	 I 	 I 	 I 	 I 	 I 	 I 	 I 	 I 	 I	 I 	 I 	 I 	 I 	 I 	 I 	 I 	 I 	 I 	 I1 	 1 	 1 	 1+ 	 + o7T p-71- 'TT p   1          -II	1 	 1 	 1 	 I 	 1 	 1 	 1 	 1 	 I 	 1 	 1 	 1 	 1	 1 	 1 	 1 	 1 	 1 	 1 	 I 	 1 	 1	i 	 1	 1-2 	 -1	 o 	 2e11111111 	 1111111 	 1 	 111 	 11 	 11 	 11 	 1 	 11 	 1111 	 11 	 11 	 11 	 11 	 111Jill 	 1111 	 11 	 1 	 1111 	 11 	 I ll 	 11.55---x CEIA75]0 [AR72jEl [BA63]0-2--1111111111111111 	 11 	 111111 	 111111 	 i 	 1 	 1 	 11111 	 1 	 1 	 111111 	 III[	 1111111 	 11 	 11111111 	 11180 	 20010-1220 	 240 	 260 	 280 	 300 	 320Chapter 2. 7r — 27r Reaction Channels	 32pion kinetic energy (MeV)Figure 2.19: x2 analysis of the 7r+7r° channel, and comparison of the model's predictionfor different values of .Chapter 3Discussion and ConclusionIn the analysis of the previous chapter it was shown that, given the present status ofthe world data for the five 7 - 27r reaction channels, only the 7r+7r+ channel unambigu-ously yields a value for the chiral symmetry breaking parameter e. This channel givese = 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 thresh-old are needed. Significantly, however, for the channels in which the N* ---* N(77)8—..vemechanism makes a significant contribution to the cross section, i.e. the 7r-7r+ and 7r07r0channels, it was seen that the x2 analysis of the world data does not give unique valuesfor both C and e; together they can vary over a large range of values to minimize thex2• However, e can only take on certain values, as presently will be shown, and thisgreatly restricts the range of values that C can have.Weinberg has provided us with a way of directly relating the chiral symmetrybreaking parameter, to the transformation properties of the term in the Lagrangianwhich breaks the symmetry. He uses a nonlinear method in which the Lagrangian is socontructed as to be invariant under chiral transformations expressed in terms of isospinmatrices and the pion field. He then assumes that Girs,B, the symmetry breaking termin G„, transforms according to the (N/2,N/2) representation of SU(2) x SU(2) andobtains the following for the term in r,s,B which contributes to the 27r ---> 27r process[WE68]C3 - _22 {1 - [N(N + 2) + 2] 4f1 2 } °2	(3.1)33Chapter 3. Discussion and Conclusion 	 34N = 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 hadderived, for the scattering lengths= -3L[N(N + 2) + 2] 	 (3.2)52a0 - 5a2 = 6L 	 (3.3)where L = -a--• explicitly these give8 7r f 7Lao = -4 [N(N + 2) + 4]La2 = —10 [N(N + 2) - 8]from a comparison of these equations and Eqs. (1.19)-(1.20), one obtainse = -2[3 - N(N + 2)]5the importance of this relation is that it gives the chiral symmetry breaking parameterin terms of the tensor rank N, which only takes on positive integer values; then thecorresponding values for e will beN=1 -+ = 0N = 2 --- e = -2N = 3 --- (3.7)therefore consistency between Weinberg's formulation of the problem and that of Olssonand Turner, requires that e < 0.1 Therefore concentrating on the zero and negativevalues for e, the OMICRON's measurement for the 7r-7r0 channel seems to rule out2a0 + a2(3.4)(3.5)(3.6)'note that Eq. (3.6) does not give = 1 (Schwinger's value) for the chiral symmetry breakingparameter.40010-4 	 I 	 1 	 I 	 1 	 1 1 	 1 	 1 	 1 	 1 	 1 	 I 	 1 	 1 	 1 	 ' 1150 	 200 	 250 	 300 	 3501 	 1 	 1 	 1 	 )	 1 	 II 	 1 	 1 	 II	 1 	 I I 	 1 	 1 	 1 	 1 	 1 	 1Chapter 3. Discussion and Conclusion	 35< —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 energyexperimental data for the 7r+7r+ channel adequately.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 reproducesthe few data available finely, except the last data point, which however seems to besomewhat anomalous.This brings us back to the two channels in which the value of depends on theChapter 3. Discussion and Conclusion 	 36value of C. Figure 3.2 shows the contour plots of the results of the X2 analyses of theprevious chapter for the 7r-7r+ and 7r071-0 channels (see Figs. 2.10 and 2.8). A 100 x 100grid has been used to interpolate the values listed in Tables 2.3 and 2.1; only thecontours which, according to the errors listed in Tables 2.4 and 2.2, reasonably canrepresent the minima for the X2 have been shown. The analysis was done over therange 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 probablybeyond, C can be chosen to minimize the x2. However if, with regard to the discussionof 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 	 4.4 (contour 1) and the corresponding error from Table 2.4 is6(x2) 	 0.4, so 4.4 < x2nii„ < 5.2, i.e. contours labeled 1-9 which give 1.20 < c < 1.38or 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 obtainedfrom 7r-7r+ and 7070 channels and its uncertainty is then obtained using E = Ec'/(8')2Ei/(sc,)2and 6E =Ei/(s)2, where i = 1, 2 and Sci and 6Z are the uncertainties in c, and Z, resp.c, This procedure results in = 1.30 + 0.07, which gives for C the followingC = (-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 fromX2 analysis of the 7r-7r+ channel alone and C = (-2.25 + 0.75) ,a-1 from branchingratio calculations [S092].In Figs. 3.3 and 3.4 the world data at lower energies for all the reaction channelsare shown, along with what the model gives using C = -1.97 p' and 6 = 0.7T p -47 -n2.0111= p-> oo1.8	n1- 2.10E+00-- 	 2- 2.20E+00= 	 3- 2.30E+00= 	 4. 2.40E+00-- 	 5- 2.50E+00-- 	 6- 2,60E+002= 2.70E+00-- 	 8- 2.80E+00-- 	 9- 2.90E+001=Chapter 3. Discussion and Conclusion	 372.01.81.60.81.01.41.2x 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=0.6 I) 	 I 	 II 	 I-1.5 	 -1.0	-0.5 	 0.0	0.5	1.0	 1.50.6 I 	 I 	 1 	 1 	 I 	 I-1.5 	 -1.0II 	J 	I 	 III 	 I 	 I 	 I 	 I-0.5	 0.0 	 0.5	1.0	 1.5Figure 3.2: The contour plots of x2 for the 7+71-- and r°71-° channels.1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 II 	 1 	 1 	 1 	 1 	 11 	 1 	 I 	 1 	 II 	 II 	 II 	 1 	 1 	 I 	 1 	 1 	 1 	 1 	 I'	 1 	 1 	 1, 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1o [KR78][0M90][SE91]- model with =01 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1 	 1150 200 250 300pion lab kineticII350energy (MeV)Chapter 3. Discussion and Conclusion 	 38200 250 300 350 400pion lab kinetic energy (MeV)103 	1111 1111111 111111111111111111111111111111 1111111111111111111 11111111110 	160 180 200 220 240 260 280 300pion lab kinetic energy (MeV)10I Iit I Him150 	 200 	 250 	 300 	 350 	 400pion lab kinetic energy (MeV)Figure 3.3: World data for the 7r+7r-,+ +7r 7r 	 o 	 -7r 7r aria 7r 7ro channels, and model'sprediction for C = —1.97 	 = 0.Chapter 3. Discussion and Conclusion	 391o3 	 _ 111111111.1111111111111111111111111111111111111111111111111111111111111111111111,-10-2 	 mriiiiiiiiiiiiiirmiliumiiiiiiiiiitimiiimiiiirrminirmill160 180 200 220 240 260 280 300 320pion 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. Veryclearly, 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'sdata, 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 theregion 190-250 MeV have been underestimated, not the least of which is the fact thatthe 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 asChapter 3. Discussion and Conclusion 	 40it does at lower energies. Considering that below this energy the model fits the datafinely, 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 athigher energies, if anything, should be more accurate than the other channels. Futuremeasurements for 7°70 will settle this matter.For the other two channels too, the model's cross sections for = 0 is in agreementwith the available data. The r+7° channel at present only allows a consistency checkon 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 existenceof the N* ----* N7r7r(E) mechanism in the 7rN —> 7r7rN reaction, as well as establish thevalue of = 0 for the chiral symmetry breaking parameter.Appendix ADerivation of the AmplitudesIn this part an exposition of the Oset-Vicente-Vacas model is given and the modelis used to derive the amplitudes for representative diagrams of the 7r — 27r reactionchannels. A complete list of amplitudes for the 7070, 7-70 and 7+7° channels can befound in Appendices B-D.A.1 The Oset and Vicente-Vacas ModelThe interaction Lagrangian used by Olsson and Turner [0T68], includes terms whichlead to diagrams for pole and contact terms, as well as to two-point diagrams for the7rN s-wave amplitude. To this, Oset and Vicenta-Vacas [0V85] have added piecesto account for 1) 771-NN coupling through p-wave p-exchange with nucleon or A inthe 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 andNN*(77r).„0„„ coupling through the exchange of the scalar meson E. The contributionof the diagrams containing the p meson has been found to be negligible [S090]. Insteaddiagrams 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-Vicente-Vacas model. The components of the interaction Lagrangian, in the nonrelativisticapproximation, are [0V85][S090]:LNN7	 LIPt Cr2(1910)70	(A.1)41Appendix A. Derivation of the Amplitudes	 42f„ = 	 f  1  [02(ait 0)2 1(1 1 0/22(02)2],a 4 )7.	2' 	 2f  1  of crioi0)1527,0£NNirirr -_=.CNN„„ =--- —470t[—Ai0.0 + —A2r(0 x at (15)}0tt	 ga2,CNA7r -- = f'CbtAS1(aie)TtAON + h.c.it.f6,,,At cf 	 (,1 A \r7-7,/,rNAA -,itfNN.7r7r 	 —COtN.O.00N h.c.ffN.N7r =--- 	 + h.c.,CN*A7r =- gN*A,-01:6, s")- ( a ) TtA oN* + h.c.(A.2)(A.3)(A.4)(A.5)(A.6)(A.7)(A.8)(A.9)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 thePauli 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 scatteringlengths, resp.; f,,= 93 MeV used here); 12 is the pion mass and 	 -p—rn, where g„ is theNN7r 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 valuesAi=0.0075 and A2=0.053.In (A.5) and (A.6), Op is the A particle field, S and T transition spin and isospinmatrices, resp. (see Sec. A.2), and Sp and TA are ordinary spin and isospin matricesfor the A; f* and fzx are related to the NAr and NAA coupling constants, resp. withnumerical 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* particleand 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 in0- = -(01 + i02)	 0+ = -(01 - i02)	 00 = 037_ = 2andAppendix A. Derivation of the Amplitudes 	 43[0V85], using the branching ratio for N* 	 N(771, s—wave (^-' 5 — 20% [PDG90]) andthe Lagrangian in (A.7), to be C (-0.91 +0.20)/4". The recent calculation by Sossiet al. [S092] gives the more accurate value of C = (-2.25 + 0.75)y'; this containsthe large uncertainties in the N*N€ and en-R- coupling, as well as the uncertainty inthe mass of E, the s-wave two-pion resonance. The calculation based on x2 analysis ofthe 7r-7r+ data [S092] has given C (-2.07 + 0.04)1-1, while the x2 analysis of the7r-7r+ and 707r0 channels in the present work gives C -= (-1.97± 0.11)-1. The N*A7rcoupling 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 LagrangiansUsing the relationswhich 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 annihilationand creation operators for pion states of charge IC (tz=-1,+1,0) and 4-momentum q; thepion states have the normalization< 7r,(q)17r,,(q)>=-- 8„,83(q — q') 	 (A.14)Appendix A. Derivation of the Amplitudes	 447-.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 asf.tht ( 	 (1, _4_ 	 a 	 a do )71,-r 0-i \ 	 1 I 72 7- 2 	 73-z r3iIL—f 0-i(N/rfaio+ +	 + mai00)71)	(A.15)IL£NNirir ir•CNNIrir,CNN*fN*Nir14.n [(20_0+ + 0(2)).(25(r0+ + (ö0)2))2)1 	 1-2 (1 -2 OiL2(402-0- + 40-0+4 + 0O)]f1Otcri(\&+ai0+ +	 + mai00) x,Lt 4f,?.(20-0+ + O)OAl 	 A2-4701-(20- 0+ +	 +	 00 - C.60)1-1V-2-ird-(0+ - 1.5+0o) + iTo(-0+ --C4*(2.0-0+ + 0DON + h.c.t-ON*cri(v 4T+ai0+ + 12-7_ap0- + mai00)0 + h.c.I-t(A.16)(A.17)(A.18)(A.19)(A.20)where ,-- is used to indicate that the effect of normal ordering in perturbation theoryexpansion has already been considered in writing the Lagrangian.s. In order to putthe other Lagrangians in a convenient form, the spin-isospin transition matrices aretransformed from Cartesian to the spherical coordinates, defined by,CNN7r= 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 asVOA = -1.1+10+ + 11-10- + to0oTtA0A	 _t-41-1.0+	 igooTcbA	-11)„.4_10+ +11A,-10- +116„o0oAppendix A. Derivation of the Amplitudes	 45using these, the expressions for .CAT6,7, £1T0,1,5(-17+1ao, + tiazo_ +toa000tN +ofAst(-niaio+ + 	 + 73io0)1PN	(A.22)CNAZ1 	 141- 	 —116+1 ai 0+ + 	 ai(b_ + t,Oai 00)1PA 	 (A .23)fN*pir becomesfN*Air = gN*A7r{ oAsi( 	 +1ai+ tai000k. +	ot.As?(-ttiaio+ + 	 + gai000N• 	 (A.24)	I is the transition isospin operator for objects with isospins 1 and 	 and tt2its Hermitian conjugate. Their matrix element between isospin states 11,rn > andj, m > is obtained by making use of the Wigner-Eckart theorem, according to whichthe matrix element of a tensor operator te between angular momentum eigen.statesmay be written as<j,mITj,m >=< j,mJ, 1C; rni ><	>	(A.25)here < j, mIJ,Kji% > 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 natureof 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 valuewhen sandwiched between two isospin states can be obtained using the Wigner-Eckarttheorem. t and t both are objects of angular momentum 1, and here a normalizationhas been used according to which the reduced matrix elements of -7", tt and t havethe values —f2-, 1 and   resp.; so for future reference< j, 	 \12- < j, mil, K ; j' , 	 > 	 (A.26)Appendix A. Derivation of the Amplitudes	 46< j, 	 >=< j, 71211, 	 >.<	 ,k1:11	 >	 2	  < J,"111,K;j',rn' >(A.27)(A.28)A.3 Recipe for Obtaining the AmplitudesInstead 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 as00	Sfi < f ISii > = < f {exp[— i 	 dt 1-11(t)]} >	—i(27048(Pf —	 (A.29)where Ii > and If > are the initial and final states, resp., T is the time ordering opera-tor 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 di-agrams. Each such amplitude is calculated by multiplying together the contributionsfrom all fundamental processes occuring at vertices of a given diagram. The contri-bution of each vertex is calculated by finding the part of the interaction Lagrangianwhich can create the process at that vertex; to this end, the interaction Lagrangiansgiven in this appendix are to be used, in conjunction with Eqs. (A.12) and (A.13). Theresulting expression will then be sandwiched between the in- and out-states for thatvertex.Depending on the vertex, the in-state can be the intial or an intermediate stateand the out-state can be the final or an intermediate state. The field operators have thetask of creating and annihilating the particles at each vertex, and in the cases where thepart of Lagrangian used contains derivatives, the exponentials in Eqs. (A.12)-(A.13)Appendix A. Derivation of the Amplitudes 	 47will contribute +iq or ±ie. To each intermediate state in a diagram, corresponds apropagator factor E, 2E7, where Ei is the total energy of the initial state and ET, thetotal 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 multipliedwith the propagators, giving that diagrams contribution to the total amplitude.Appendix A. Derivation of the Amplitudes 	 48A.4 Calculation of Some of the Amplitudes for the Three ChannelsIn the following section some of the diagrams for each channel are explicitly calculatedwith the rest of the amplitudes being given in Appendices B,C and D. It should benoted that in order to obtain the amplitudes given in the FORTRAN code at the endof 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 72n(p3)— NN71- vertex: N/It,cr.pusing 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 1,2so overall it becomes,-f 	 -pip5 - P5p6 - p6p1 + 21-e [12v2- —cr.p	.a	 p02 p2 it2	Appendix A. Derivation of the Amplitudes 	 49• 71-p	 7r-70pI TV-(P6)	11- (PI) 	 ilk 	 71-°(p5)70(p)	P(P2) 	 P(P3)— NI\17 vertex: Icr.pusing '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 — p2so overall it becomesf 1 P1	P5P6 P6P1 -1g1-12tt 	0- 10 	0222Appendix A. Derivation of the Amplitudes• r	 71-+ 70pTT*(P5)7.(1D1) 7e(Pe)7t0 (p)P(P2) 	 P(P3) 50— NI\TR- vertex: icr.pusing 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 _p2so overall it becomes6),tt2} _iT1,P f 1 cr ,P1P5 + P5P6 — p6p1 + 6/12 P02 — p2 — itt 2(1)/ 0/ \7T lID6)/Appendix A. Derivation of the Amplitudes	512. Contact diagramsThere's only the contribution from NN7r7r7r vertex(2) 	 (3)7T-(P6)//	 \\\ 7 (p,) 	 / 07 , 0D0 	 \PIv \\ 7T 	 ) 	r 0\ 	 / 	 \ 	 /1 7 (P),\ 	 / 	 , \ / 	 ,\ 	 / „...7" 	 \ 	 / -.-\	 /...-' 	 \ 	 /___\Z-	\.4-PODO 	 n(p3) 	 P(P2) 	 P(100 	 P(P2) 	 P(P3)• 7r-p 	 7r07r0n (first diagram above)using —/44-0tcri(A/2-7_aic5_)0(2)0 in (A.17), givesf 1—iTLc =• 7r-p 	 7r-70p (second diagram)using —/4-4-1-f7rOto-i(Thaic0).20_0+0 in (A.17), gives1 f 1- = --2 —tt —g cr D5• r+p 	 7r+7r°p (third diagram)using ---*,01-cri(7-0,9i00).20+0 in (A.17), gives1 f 1—iT1'c = -2--y cr.p6Appendix A. Derivation of the Amplitudes 	 523. Two-point diagram with N* intermediate state7T(p5) 70(p6)7-(P1)	 ,/P(P2)	 n(p3)• N*Nr vertex: —1-12-0-.piusing 	 in (A.20)• NN*71-7 vertex: -i.2Cusing —00I-NOOON* in (A.19)• propagator: ri_iEn = 	 m2*+0,*where r* is the width of the N*, to be obtained frome(E) F* = r*(E) _ q3(M*) r(M*)with E = E — — 	 M*, r(m*) = 235 MeV andq(E) [(E2 + /12 m2)2/4E2 /111/2(q is the momentum of the pion from N* N7 decay)so overall one getscr.p_iTa2,N* _ 2v 2 I- C 	 1 N/Ts — M*(A.30)Appendix A. Derivation of the Amplitudes 	534. Two-point diagram with nucleon intermediate state7- (PI) 	 7° (P5) 7.° (P 6)P(N) 	 n 	 n(p3)• NThr vertex: --\121(7.Piusing (A.15)• NThrir vertex; —47r(i2-1-)using _4 tq5 in (A.18)• propagator: Ernso overall it becomes_iTa2,s _ _ 8 \/--,7r/2 [t NC9 — mAppendix A. Derivation of the Amplitudes 	 545. Three-point diagram with nucleon and/or N* intermediate statesusing (A.15)\ 	 I 	 /\ 	 I.Trot.., \ 	 /\ 	 _ 	 A 	 \ H5)7T (Pi) 	 i 	 o\ 	 / 	 / 7C (Pe)\ 	 I 	 /\ 	 i 	 / P(Pa)	 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—mE,—En	 vi-p2--rq/2m-mso overall, taking into account the identical particle nature of the outgoingpions, one gets1 	 cr.136/7.Pscr.Pi = 	 f )3 	p \is — m \Ts — — pg / 2m — m + (5 <-+ 6)for diagrams with both nucleon and N* intermediate states, the result is similarwith appropriate changes to account for NN*7 coupling constant, addition of theN* width and replacement of nucleon with N* mass in the propagator.• second propagator:	Appendix A. Derivation of the Amplitudes	 556. Three-point diagram with A and nucleon (or N*) intermediate stateo„7 MO 7-e(p6)i\_	7 (Pi)	 //\ 	 / 	 /\ 	  / 	 /	P(P2) A° 	 n 	 n(p3)• first NA7 vertex. —11i--St•using r--:,otAV't iaio_oN in (A.22)• second NA7 vertex: .MS.p5using 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 usingF(E) =2	1 	 1 1 (9 M' 3	—2	 Eq347r ttwhere E = — — 	 mA and q is as in Eq. (A.30)• second propagator: 	EE -so overall one getsf—iTa3'° = 	(f 2— 1 	 c•.p6S.p5St.pi--3 ti 	 V=s — 	 \fi — — pU2m, — m + (5 4* 6)In the case of three-point diagrams with A and N* intermediate states, the cal-culation is similar, with changes to the coupling constants, addition of N* widthin the propagator and replacement of the nucleon with the N* mass.Appendix A. Derivation of the Amplitudes	 567. Three-point diagram with two A intermediate states\ 	 71-°(P5)/ 	 Tr°(p.),\ 	 A/	\ 71--(1D1)	 I/ 	 /\ 	 / 	 /\ 	, 	/i 	 I	 i 	p(p2) Aa 	 A° 	 n(p)• first NAT- vertex:using L:01:s,f laicLON in (A.22)• Ni vertex: —1°-1SA.p5p. 2using ofstai000 in (A.23)• second NA7 vertex: Li' HS.136using	 Sit oai0o0tN in (A.22)• first propagator: 	E1—En = 1:5—rnpd±r• second propagator: 		Ei—En	 V75-192--pU2m—mp-qirso overall one gets—iT°'° =	1 fA	 \ 23-\/-F(5	 6)1 	 S.p6SA.p5St.p1 — mA 	 —p — pOm — TriAA.5 Taking Care of SpinThe amplitudes from the Feynman diagrams contain combinations of spin and transi-tion spin matrices. Generally one has to deal with objects of the form(1) cr • A(2) cr •A a • B a- • CAppendix A. Derivation of the Amplitudes	 57(3) cr•A S • B St•C , S • A St-13 a•C(4) S•A Sp •B St -Cwhich can be reduced to expressions having the general form Ia + a•b, in which I isthe 2 x 2 unit matrix, a is a scalar and b is a 3-component vector; in fact the totalamplitude T has just such a formT =la+ cr•b	 (A.31)Since cross section is proportional to the square of the amplitude, it is sufficient tocalculate ITV.For an unpolarized target, one has to average over the spin states of the proton inthe 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 willhave the general form2(<1 I+ <I 1)(Ia + cr•b)-(1 1> +1 1>) (A.32)here I 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 loweringoperators, one will obtain, after some algebra1T12 = 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 bothdown, 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 anddown polarization, resp.Appendix A. Derivation of the Amplitudes 	 58The form cr•A o-•B o--C can be reduced to the form Ia o--b by writing it usingimplicit summation notation as ii.2B3Ck aaok and then usingCriaj = 6ij 	 jEijkak 	 (A.35)or directly by usingo-•Acr•B = A•B io--(A x B) 	 (A.36)either way the result will beo--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 become1 	 1o-•A S • B St•C = —2(B C)cr-A + —3(C • A)o--B — —3(A • B)o-•C — —iA • (B x C) (A.38)3 	 31 	 1S • A St •B o--C = —2(A • B)o-•C + —3(C • A)o-13 — —3(B • C)o...A. — —iA • (B x C) (A.39)3 	 3using the relationSS = E S I ms >< m-siS = 3Mswhere Ms is the spin of the A intermediate state.Finally the form S-A St -BSt•C becomes—3 Eiikuk(A.40)1 	 1S-A St-B St-C = —2(C • A)o--B — —6(A • B)o--C — —6(B • C)o-•A 4- —5 iA • (B x C) (A.41)3 	 6using the relationsisstk 	 E siim's >< misls,„jims ><ms,mis5. 	 12 	 1, 	 ,ka.— 	 — —096 	 - 	 6 2- 	 3 2 	 - 	 6 -(A.42)1 	 Ai= 	,- 	/„/ 	 vs 	 — 772 /2 (B.5)Appendix BAmplitudes for the 7r-p —> 7r071-0n reactionThe amplitudes in this part correspond to diagrams in chapter (2.2) with same thenumber. All quantities are centre of momentum values; pi is the 4-momentum ofthe incident pion, p5 and /36 are the 4-momenta of the identical outgoing pions, andp = p5+ p6 — pi the 4-momentum of the off-shell exchange pion.1----- f 1— o-.p 	f72. 	 p02 p2 y 2[-Pi.P5 - p5•p6 - p6.p1 + -1C(12)	(B.1)\12- f 1 cr.p1 	 (B.2)iT2 , N* L=- —2v2—Ccr.pi	— M* (B.3)=-2v 2 —C o- x  1   (B.4) \Cs 137 — — - * - (p + p 5 + P6)2 2m* +59Appendix B. Amplitudes for the 7r- p —> 71-071-0n reaction	 60A2 0 	 ■= 4V-2-7rf-cr.p5 	1	 0/-t 	 m — —p1-2— (pi + p 5)2 12m itt 2 U51+136)+ (5 4-* 6)	 (B.6)f=	 27—cr.pi x1(B.7)— m — — - 192 - (P1 p5+p6)2/2m=1 	 A2 0	 0\—iTc2/8 	 /Pi +Poit 	 1/4 — m, —P2 — 134/2m 122+ (5 4-* 6)	 (B.8)1 	 1—iTa3'N = 	 (f) 3 Cr.P60-.135Cr.131m v-s- — p — pU2m — m(B.9)+ (5 4-* 6)—i Tb3 N	 (1) 3 cr.picr.p6a.p5 x1P7 — - m - (p1+p5)2/2rn x1Nti;—P7-14—PS---rn—	+ Ps + p6)2/2m+ (5 4-* 6)	 (B.10)_ 	 f 3—iTc3'N 	 —v 2 (— cr.p6a.pia.p5 x) 1- - - m - (Pi P 5)2 I 2n1 x1— — m — pU2m+ (5 4-* 6)	 (B.11)Appendix B. Amplitudes for the 7r- p	 7r07r0n reaction	 61--iT3'N'N* =	 (2I-21 le- Cr.1360..P50-•Pi Xitt	 itM* -i2-ir*Ncs -p — pg/2m — m+ (5 4-)' 6) 	 (B.12)-iTa34 =-_ -12-f If * )2cr.p6S.p5St.p,3 IttrnA + Ncs - - m- pg/2m+ (5 4-* 6) 	 (B.13)-i7163'° •\/ f f* 2= 3 - (=-) S.p6St.plcr.p5 xft 	 itt1    \Cs — m —	 — (131 + p5)2/2m1   \Cs —p — P.U2ma, — rnA ++ (5 	 6) (B.14)—iTc34 f f*) 2- —3 -,11 	 cr.p6S.p1St.p5 x   1   — — (P1 + p5)2/2m6, rnA 	 x  1    + (5 4-* 6) 	 (B.15)Appendix B. Amplitudes for the 7r- p	 71-0en reaction	 62= —	(f)2S.p1St.p60-.p5 x3 ii1\Cs — m — A — A — (pi + p5)2/2in1N/75 — A — A — A — (pi + p5 + p6)2/2mA — mA ++ (5 f-3 6) 	 (B.16)2\12-f ( f*) 2— Te3 A	 cr.piS.p6St.p5 x3 P	 1-11—p—p — (P1 +p5)2/2mA — m6, + iF x1—m —A —A 	 — (P1+ p5 + p6)2/2m+ (5	 6) (B.17)—iT3'°2/ f f*) 2S.p6St.p5cr.p1 x3 P P    \ — m \Cs — — Pg/2mA — TrIA+ (5 	 6) (B.18)—iTa°41 f6,3fL)2S.p6SA.p5St.p1 x— m6, + iF\is —p — 	 — pg/2niA ++ (5 4-4 6) (B.19)Appendix B. Amplitudes for the 7r- p	 71-0en reaction	 63—iTbA'A 1	 f (t) 2S S	 Sip3,/j-	 /.1 	 •Pl 	 .P6 	 5 X1— 137 131:5) — (P1 P5)2/2MA — MA +1— p — P.(5) — P06 — (pi + p5 + p6)2/2m, — mA ++ (5 4-* ) 	 (B.20)—iTc°4 _ (f )23	Q.p5 xft  1   \Ts — — p — (P1+ P5)2/2m — rnA + iF x1  N1-9 — — PU2rnA TnA+ (5 	 6) (B.21)_ f (f * \- - gN*Air cr.P6S.P5St.Pi x3 ft 	 ft\Ts — m, + iF,v7s —p — m* — pg/2M*+ (5 4-* 6) (B.22)_ji,b3,N.,A _ 	 f (f*_ 3 —tt —it) gN*Air S.p6St.picr.p5 x  1   ,V7s — M* — p?— 13? — (Pi + p5)212M* 	 x1  \Cs — p — pU2rnA — m ++ (5 	 ) (B.23)Appendix B. Amplitudes for the 7r- p	 71-0en reaction	 64_iT3,N*A (f*- —) gN*Air ( .P6S•PiSt -P5 x3 1-11— — (Pi + P5)2 /2mA x1— — M* — pU2M* dm'	+ (5 4-4 6) 	 (B.24)- iT3'N* 	 (.9 gN*AirS•P1St •P60. •P53 it	 it1M*	PSI – (P1+ P5)2 / 2m* + 	 x1– –p 	 p? – (p1 + 5 + P6)2 /21716, TiL6, ++ (5 	 6) 	 (B.25)—iT3'N* 	2 f (f*gN*Air cr•pis•p6st.p, x3 [I 	 tt1Nt; – – – (131 +P5)2/277/6, — mA1– m* 	 – – P2 – (p1 +P5 + p6)2/2M*+ - jr*+ (5 	 6) 	 (B.26)2\/ f ( --f*—iTPN*'° —	 ) gN*AirS-P6St -P5cr .Pi x3 	 ttv:s — it4-* +	—p — pU2mA — 	 ir+ (5 4-+ 6)	 (B.27)Appendix B. Amplitudes for the 7r-p -4 r'en reaction	 65What follows is the portion of the Monte Carlo code in which the amplitudes , withthe spin taken care of, are given.SUBROUTINE BOBA(SRAJD)IMPLICIT COMPLEX (U)LOGICAL LOSOLDIMENSION ID(48),X1D(48)COMMON /MOMEN/P1(4),P5(4),P6(4),P5MOD,P6MOD,P5MO2,P6MO2,P1MO2COMMON/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,FNDCOMMON/MANIP/TPI,EPILAB,ETOT,PPILAB,S2,S,EPICM,ENUCM,PPICM• ,XL,EREM,PLIM,AT,BT,FI5,FI6,CO5,C06,XMAX,XMINCOMMON/CASO/ICASO,LOSOL,IPOL,JPOL,MPOLC SOME COUP. CONSTANTS FN,FD,FDD,FNS INCLUDE /MASS PIONC UNITS ARE FMIF(LOSOL)THENUY.(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=1,4P(I1)=P5(I1)-1-P6(I1)-P1(I1)ENDDOP2=P(4)**2-P(1)**2-P(2)**2-P(3)**2FORM--,(XLAM**2-XMU2)/(XLAM**2-P2)DO 395 1=1,48395 XID(I)=FLOAT(ID(I))DO 1=1,3U1P(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))*FORMU1C(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	 66• ((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(01J2D(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))-1-U3NAFAC2*(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)-FU3 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. XMNSDN3A 2=SRA-P 6(4)-P 66/2. /XMNS1J3DNAFAC1=-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./Appendix B. Amplitudes for the 71--p 	 7r° 71-'n reaction	 67• (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. /XMDD3B2=SRA-P6(4)-P66/2./XMDU3DBFAC1=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./XMNSD3NB2=SRA-P 1(4)-P 6(4)-(P 11-FP 66+2.*P16)/ 2./XMNSU3DNBFAC1=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./XMDD3C2=SRA-P1(4)-P6(4)-(P11-FP66+ 2.*P16)/2./XMDU3DCFAC1=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./XMNSD3NC2=SRA-P6(4)-P66/ 2./XMNSU3DNCFAC1=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./XMDU3DD 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)Appendix B. Amplitudes for the 7r- p 	 7071-0n reaction	 68U3DD(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./XMNSD3ND2=SRA-P 6(4)-P1(4)-(P11-I-P66+2.*P16)/2 ./XMNSU3DNDFAC1=-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./XMDD3E2 -,SRA- P1(4)-P6 (4)-(P11-I-P 66+2 .*P16)/ 2./XMDU3DEFAC1.----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./XMDD3F2=SRA-P6(4)-P66/2./XMDU3DFFAC1=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-Appendix B. Amplitudes for the 7r-p --* 7r07r0n reaction 	 69F U3DNFFAC2*(2./3.*P56*P1(I)+1./3.*P15.P 6(I)-1./3.*P16*P5(0)DDA3= SRADDA4=SRA-P5(4)- P55/ 2. /XMDDDA5=SRA-P6(4)-P66/2./XMD1.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./XMDDDB3=SRA-P 1(4)-P6(4)-(P11-1-P66+ 2.*P16)/2./XMDDDB4=SRA-P 1(4)-P5(4)-P6(4)-F 	 (P11-f-P55 -FP66-1-2.*P15+2.*P16+ 2.*P56)/2./XMDUDDBFAC1=-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./XMDDDC2=SRA-P1(4)-P6(4)-(P111-P66-1-2.*P 16)/2. /XMDDDC3=SRA-P5(4)-P55/2./XMDDDC4=SRA-P6(4)- P66/2. /XMDUDDCFAC1=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./ XMNSU2NB(I)=-CNS*2.*SQ2*FNS/(DNS1- X MNS-FUY*GDELS(DNS1)/2.)*P1(I)UTOT(I)=XID(21)*(U1P(I)-FUlC(I))-I-F 	 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-Appendix B. Amplitudes for the 71-773	 7r071-0n reaction	 70• 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 0UJAC=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)RETURNELSEUTOT(1)=0.UTOT(2)=0.UTOT(3)=0.UTOT(4)=SQRT(SIGMA(PPILAB,TPL-1))RETURNENDIFENDAppendix CAmplitudes for the 7r-p 7r-7r°p reactionThe amplitudes in this part correspond to diagrams in chapter (2.3) with same thenumber. All quantities are centre of momentum values; pi is the 4-momentum of theincident 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 	 1cr.p 	/IR	 P02 /32[-PITS + P5•P6 P6•P1 + 2141211 f 1— 	 •iTl'e = 	 0 1352 tt fl.f 	 1 	 A2 01LL07—cr-Pi- 	2 (P5 - /4)vS 147r—f a-.p5	- m - p - p - (pi + p5)2/27n2A1	 A2 0 	 0(P1 +	A2 0 	 01 	 r p2 (pi + PO]4L0.5V:s — m — /32 _ pU2m IttTa2 , STc2 ,S(C.1)(C.2)(C.3)(C.4)(C.5) (C.6)cr•Pscr-Pscr.Pi \Cs — m	 732 Pg/ 2m — m71Appendix C. Amplitudes for the 7r- p —> 71--'ir'p reaction	 72if. P0 •135 .Pi (C.7)N/75 m	—130m m— iT3'N 	 2 ( —je ) 3 cr.p6cr.picr.p5 xIt )1-\18- — p — p — m — (Pi + p5)2/2m1(C.8)—iT3'N'N* = 2— —1-tcr.P517.P6cr.p1 x (C.9)— M*	 \is — — p4/2m — m_iTb3,N,N*	f (1) 2= —2— —It 	 ftcr.P60-.P50..p1 x1	1 (C.10)V7s — M* 	 — p(5) — 2m — m—iT34 = if (f*) 2 cr.3 ia itt 	 p5S•po 	 x    (C.11)  Ncs — rnA + ir \Fs — — m — p4/2m,2a.p6L (f* 	 ct— 	 ,3 .picr.p5 xItt1  m 13? 13(5) — (Pi + P5)2 / 2m x1(C.12)0 	 , 	 1 .-1-12 /2MV7S — p5 — p5/ A — MA -r Appendix C. Amplitudes for the 7r-p —> 7r-70p reaction	 73Ittf (f )2— cr.p5S.piSt.p6 x1AFS –p —p - mA (pi p6)2/2rnAi iF x1(C.13)= L — S.piSt.p6cr.p5 x1-11\Cs — — — — (P1+ p5)2/2m x\[s — — — p6P — (P1+ P5 + p6)2/2771A - rnA +(ttf )23 1,1S•p6St•p5cr.pi x- m Nfi- - - W12 rnA - mA + ir—iTP°2 f f*) 2 S.p5St.p6o-.p1 x3 It¼— m — 132— pg/2mA — m6, +(C.14)(C.15)(C.16 )=	(L-*) 2 cr.p6S.p5St.p1 x3 it 	 Itt     (C.17) \cs - rnA +	 - p(5) — — pg/2m_1 fn, ((f* 261\is — 77/A + iFN[s — p — m, — pU2rnAS•P6Sp •P5S 1-.131 X1 ( C.1 9 )Appendix C. Amplitudes for the 7r-p —> 71-ep reaction 74 (11 2 	 t3 	 ft 	 0- P6S•PiS .p5 x1V.; — — _ mA (131 p 5)2 / 2rn + ir x  1 (C. 18) Ncs — p — m — pU2m  2 f6, f*) 2S.p5SA•p6St.p1 x3 ti     (C.20) N/79 — rnA + iF	 — p — rnA 134/21np = x(f)2S.piSA•p6St.p5f2   1— — pC81 — (pi + p5)2/2mA rnA	 2iir1 (C.21) \Cs —p —p — p — (Pi + P5 + P6)2/2MA 2 fA f*) 2=— n•psaA Tin •p5 x3 it 	 it   1   \Cs —p — p — ( 131+ p5)2/2mA — mA1(C.22) — — 0/2mA — rnA + iF —iTb3'N*'° =f (f * gN.A7 S.PeSt.Picr.Ps xIL 	 ttAppendix C. Amplitudes for the 7r-p	 7r-ep reaction	 75—iTA'A =I.	 23f (L-) S.p1SA.p5St.p6 x2 ,tt1  —Pi — Ps — 11)1 p5) 2 /2MA — MA +1 (C.23)  — — PS — P2 — (131 + Ps + p6)2/2mA — f*2=	 O.P5Op•Pi3.-136 Xitt ( itt)     1    N[s — P — P6P — (P1 + 136)2/2mA —1 (C.24)  0	 2	rn 	1 -P6 — P6/2MA	 A 	-1-1  - iT3'N*'°1j (f*_—	 gN*Air cr.P5S.P6St.Pi x3 la 	 Itt1 	 1(C.25)  — mA + iF —p — m* — p4/2M* ir*1  — pg/2mA — rti„n,1 (C.26)— m* — pi) —p — (Pi + p5)2/2M*+-iT3 'N*'° (f;) g N*6,7 Cr -P5S-P1St.P6 XIL 1• xNtr; — P7 — P6°— rn•A — (p1 + p6)2/27716,1(C.27)N[5 — — — pS/2m* +Appendix C. Amplitudes for the ir-p	 7-71-°p reaction	 76—iT,31'N* gN*A7S.P1St .P6ff •P5 X1  —,n0 	 _j_ 	 , 1 •-r, X— F5 — F6 	 Fs p6)2, c,771A Trip 1 (C.28)— m* 	 — — (p1 + p5)2 /2M* + —iT3'N*4f (f*—	 ) gN*A7S.P6St .135°..P1—	 x3 tt -- It1(C.29) v7s — p(5) — pg/2mA — rrtA 	 — M*2 (f*gN*A,S.p5St .p6cr.pi x3—i21.f3'N*4   (C.31) 0 	 2	 •1/7-9 — Ps — P6/2mA — rnA + -1 zr 	 — M* lir*	2 	 2-iTpN*'°2 (f*=	 gN*Airff-P6S.P5St.Pi x3 bt1 	 1(C.31)  — rnA 	 — p — m* — pg/2M*---273'N*142gN*Aircr.P6S.P1St .p51o	 1V.75 — pi — p05 — rizA — (131 + p5) 2 /2772p 1 (C.32)\Cs — p — Mt — p0m*+ Appendix C. Amplitudes for the 7r-p	 7r-71-°p reaction	 77What follows is the portion of the Monte Carlo code in which the amplitudes , withthe spin taken care of, are given.SUBROUTINE BOBA(SRA,ID)IMPLICIT COMPLEX (U)LOGICAL LOSOLDIMENSION ID(48),XID(48)COMMON /MOMEN/P1(4),P5(4),P 6(4),P5MOD,P6MOD,P5MO2,P6MO2,P 1MO2COMMON/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,FNDCOMMON/MANIP/TPI,EPILAB,ETOT,PPILAB,S2,S,EPICM,ENUCM,PPICM• ,XL,EREM,PLIM5,PLIM6,AT,BT,F15,FI6,CO5,C06,XMAX,XMINCOMMON/CASO/ICASO,LOSOL,IPOL,JPOL,MPOLC SOME COUP. CONSTANTS FN,FD,FDD,FNS INCLUDE /MASS PIONC UNITS ARE FMIF(LOSOL)THENUY.(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=1,4P(I1)=P5(I1)+P6(I1)-P1(I1)ENDDOP2=P(4)**2-P(1)**2-P(2)**2-P(3)**2FORM=(XLAM**2-XMU2)/(XLAM**2-P2)DO 395 1=1,48395 XID(I)=FLOAT(ID(I))DO 1=1,3U1P(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)*FORMU1C(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 reaction	 78• (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./XMNSU3DNAFAC=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./XMDU3DBFAC=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./XMNSU3DNBFAC=FNS*FD*FND/Appendix C. Amplitudes for the 71--p 	 7 -7°p reaction	 79• (D3NB-XMNS-1-UY*GDELS(D3NB)/ 2.)/• (D3B-XMD-FUY*GDEL(D3B)/2.)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./XMDU3DCFAC=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./XMNSU3DNCFAC=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 ./XMDU3DDFAC=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./XMNSU3DNDFAC=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./XMDU3DEFAC=-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./XMDU3DFFAC=-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))Appendix C. Amplitudes for the r- p 	 7r-R-°p reaction	 80U3DNFFAC=-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./XMNSU3DNGFAC=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 /XMD113DHFAC=-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. /XMNSU3DNHFAC=-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./XMDU3DDAFAC=-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./XMDU3DDBFAC=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./XMDDD C2= SRA- P1(4)- P5(4)-P6 (4)-(P11-1-P 55 -f-P66 2.*(P15-FP16 P56))/ 2. /X MDU3DDCFAC=-FDD*FD**2/• (1JDC1-XMD-I-TiY*G1JEL(DDC1)/2.)/Appendix C. Amplitudes for the 7r-p —> 71--7°p reaction	 81• (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./XMDDDD2=SRA-P5(4 )-P55/2./XMDU3DDDFAC=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./XMDDDE2=SRA-P 1(4)-P 5(4)- P6(4)-(P11 4FP554-P664-2.*(P15+ P16-1-P56))/ 2./XMDU3DDEFAC=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. /XMDDDF2=SRA-P6(4)-P66/2./XMDU3DDFFAC=-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 	 20)*U3DD B(I)+XID (21)*U3DD C(I)4-F XID(22)*U 3DD D (I)+XID (23)*U3DDE(I)-FX ID (24)*U3DDF(I)4F• 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)ENDDOUJAC=UY*XMIXT(P1,P 5,P6)U3A(4)=U JAC*U3A FA CU3NA( 4 )= UJA C*U3NAFA CU3B(4)=-UJAC*U3BFACU3NB(4)=-UJA C*U3NBFACU3C(4)=UJAC*U3CFACU3DA(4)=(-1./3.)*UJAC*U3DAFACU3DB(4)=(-1./3.)*UJA C*U3DBFA CAppendix C. Amplitudes for the 7r-p —÷ 7r-7r°p reaction 	 82U3DC(4)=(1./3.)*UJAC*U3DCFACU3DD(4)=(1 ./3.)*UJAC*U3DDFACU3DE(4)=(1.13.)*UJAC*U3DEFACU3DF(4)=(-1./3.)*UJAC*U3DFFACU3D G(4)=(1 ./3.)*UJAC*U3D GFAC113DH(4)=(- 1./3. )*UJAC*U3DHFACU3DNA(1)= (-1. /3.)*UJA C*U3DNAFACU3DNB(4)=(-1./3.)*UJAC*U3DNBFACU3DNC(4)=(1./3.)*UJAC*U3DNCFACU3DND(4)=(1./3.)*UJAC*U3DNDFACU3DNE(4)= (1./3.)*UJAC*U3DNEFACU3DNF( 4)=(-1. /3. )*UJAC*U3DNFFAU3DNG(4)=-- (1./3.)*UJAC*U3DNGFACU3DNH(4)= (-1./3.)*UJAC*U3DNHFACU3DD A( 4)=-5./ 6.*UJAC*UDDAFACU3DDB(4)=5./6.*UJAC*UDDBFACU3DD C(4 )=-5./6.*UJAC*UDDCFA CU3DD D (4).5 ./6.*UJAC*UD DDFA CU3DDE(4)=5./6.*UJAC*UDDEFACU3DDF( 4)=-5./ 6.*UJAC*UDDFFACITTOT(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( 	 (11)*U3DDB( 4)-PCID ( 12)*U3DDC( 4)-F• 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-• 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)RETURNELSEUTOT(1)=0.UT OT(2 )=0.UTO T( 3)=0.UTOT(4)=SQRT(SIGMA(PP ILAB,TPI,-1))RETURNENDIFENDx(D.1)(D.2)pz 	 /121 fr	 2,— 136•P1	 J1—	 — p - P2 - (P1 + p6)2/2m xPn] (D.3)01 	 A2(0Pi +P)	 (D.4)rrt 	 P? 	 PSI — (Pi + P5)212m 1121rn —	 — PU2rnPC5))] (D.5)J= 	 cr.p.t7 	 p0211.P5	 135•P6= 47—cr.p62Ai. 	 A2Ii	 111 f 1—iT1'c	 — —2 —tt 	 a-.P6—iTa2's = 47-f-cr.p62A1	 A2[ 	 —03(1)tt2—iTb2's 	 —871a-.p5/-1Appendix DAmplitudes for the 7r+p —> 7+70p reactionThe amplitudes in this part correspond to diagrams in chapter (2.4) with same thenumber. All quantities are centre of momentum values; p1 is the 4-momentum of theincident 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.83x(t) 2 S . p5S t .picr.p6 IL 	1Appendix D. Amplitudes for the 7r+p	 7+70p reaction	 842 () 3 cr.pio-.p50-.p6 x1Nrs- - rn p(1) P(6) — (pi + p6)2/2m,1V,--.14-1,2-152—m—(pi+ P5 + P6)2/2M—2 (—f) 3 cr.p1cr.p6cr.p5 x1N[s m 	 — — (P1 + p5)2/2m x1N/7-3 137 115)	 —	 (131 + P5 + p6)2 /2m3—iT3'N = 2 (1) cr.p6r.p1cr.p5 x1— m — 137 — — (Pi + p5)2 /2ni x10"f3- — P5 — rrt Ps2 /2m(D.6)(D.7)(D.8)-zT3,°	f* ) 2 cr.p6S.p5St.p, x(D. 9)0 	 2 	 1— p5 — m — p5/2m— mA + -2 21— m — — — (131 p6)2/2m1 (D.10)0 	 2 	 1 T-1Ni; P6 — P6/27nA m 	 -A—iTc3/ 	2 f f*) 2 S.p6St.p1a.p5 x3 it1\Cs — m — — — (Pi + p5)2/2mx1\Cs —IA —Pil2mA—mA+—iTe34(—f*) 2 S.piSt.p6cr.p5 x3 it 	 /-11m — —p — (Pi + p5)2/2m x1\fi. — —p — — (pi p5+p6)2/2mA — rriA— iT3'° — 	 (112	 S St3fL	 ,tt 	 Pi .P5 .P6 x1\Cs — p — pcs' — (Pi + p6)2/2m A — mA 	 x1\Cs—m-14-14—A— (Pi +P5 + P6)2/2mAppendix D. Amplitudes for the 7r+p	 7r+ep reaction	 85- z77,342=	 S.piSt.p5a.p6 x1— — — (pr +136)2/2m x1'Nfs —P —P —P — (Pi + Ps +136)2/27nA — mA(D. 11)(D.12)(D.13)(D. 14)= 	(—f*) 2 cr.piS.p6St.p5 x3 tt   1   —p — p —m — (Pi + P5)2/21-nA 	 x1 (D. 15 ) \Cs — m — 	 (col +5 + p6)2 12m Appendix D. Amplitudes for the 7r+p --* 7r+ep reaction	 86f (j2cr.p6S.p1St.p5 x1(D.16)0Vs- —Pi0 —P5 — rnA — (Pi + P5)2/2mA 10Ar9 — P5 — rn	 Ps2 /2711—iT" 3f6, /f* \2 —ft )2 	 '3•P5°A•P6'°'•P1 X1 	 1(D.17)\Cs — 	 Vs- —A— p4/2mA — mA— T' (f*)2— 	 S.p6SA•p5St.pi xitt1 	 1(D.18)— 77/A 	 —p — nip — pg/2777,p 12-ir2fa, f* s.p5sA.piSt .136 xit 	 it1o o	T\Cs —Pi —P6 — (131 P6) 2/2m — mA 	 A 1 (D.19)0 	 2	 1— m TA iTd6"° fA f*= 	 —) 2 0.pi .p5St.p6 x(la	 [I1— — — (pi + p6)2/2rnA — mA + lir21 1fi p? — p — p — (pi + ps + p6)2/2mA — n-tA + iF (D.20)Appendix D. Amplitudes for the 7r+p	 7r+ep reaction 	 873 It \ )2 S •Ps S •P iSt•P 51  .	  X0 	 0 	 1 T1N4-5 - Pi P5 - (pi P5) 2 /2MA — rnA 1 ( D.21 )	2 	 1	1fi P5° P 	 .-1-15/2MA —iTA'A1 jr6, f* 2S•P1Sp•P6St.135 X1\Cs — 	 i 	— (p1 + p5)2/2m — mA r1 VT5 — 	 (p1 p5 p6)2/2mA _ mA 	 (D•22)(_iTa3,N*,° = I fl gN*An-cr.p5s.p6st.pi xit 	 ,tt   (D.23) 0- — Pc5) — M* — PU2M* -jr* — m + lir—iTb3'N*'° = —(f*--) gN*Air S.P5St 	 r .P6 Xbt    1    .\15 - m* — — — (pi + p6)2/2m* + 	 x1(D.24)  0 	 ,2 /2VT5 Ps --- P6/ MA — TrIA L (f*— —)gA,A,s.pist.p5cr.p6xft 	 ft1— — p(-0) — (pi + p6)2/2m* + 	 x1 \fi — — — — (pi + p5 + p6)2/2mA rnA + 	 (D.25)Appendix D. Amplitudes for the 7r+p 	 7+71-°p reaction'= __2 f (f*)d	 — gN*A, S.p6Sa-t.pi.p5 x3/1	 it88 1   Ars — m* — p — p — (Pi + P5)2 / 2m* + 	 x1(D.26) 2 	 1p°5 — P5/2mA — 	T 2 f (f*—)= 	 gN*A7rs.p1st.p6ff•p5 x3 it	 it   1\Ts — m* 	 — — (Pi + p5)2 / 2M* + 	 x1(D.27)  \fi — — p — p — (pi + p5 +136)2/2mA — mA +—iT3'N*42 f (f*)gN*A- cr-PIS-P5St•P6 x3 itt1N/7-9 — — 132 — (pi +136)2/2mA — rrtz, +1N[s — m* 	 — — — (p1 + P5 + p6)2/2M*+ lir* (D.28)2 (f*)gN*Air €7•131S•P6St.p5 x1	  X0 	 0 	 12 /e) 	 1 .T-1— P1 P5 — 411 + p5) 	 — rrlp1(D.29)v— m* — —p — — (pi + P5 + 136)2/2M* +—iTh3'N*41 f (f= 	 *—) g 	 CinSDSt 10 X3 ti 	 N*A7 .r-6 	 .1-5   1    — p — p(5) — (131 p5)2/2m A — 	• 	   1 (D .30)  \Cs —p — m* — pU2M* Appendix D. Amplitudes for the 71--Ep 	 7r+7°p reaction	 89What follows is the portion of the Monte Carlo code in which the amplitudes , withthe spin taken care of, are given.SUBROUTINE BOBA(SRA,ID)IMPLICIT COMPLEX (U)LOGICAL LOSOLDIMENSION ID(48),XID(48)COMMON /MOMEN/P1(4),P5(4),P6(4),P5MOD,P6MOD,P5MO2,P6MO2,P1MO2COMMON/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,FNDCOMMON/MANIP/TPLEPILAB,ETOT,PPILAB,S2,S,EPICM,ENUCM,PPICM• ,XL,EREM,PLIM5,PLIM6,AT,BT,FI5,FI6,CO5,C06,XMAX,XMINCOMMON/CASO/ICASO,LOSOL,IPOL,JPOL,MPOLC SOME COUP. CONSTANTS FN,FD,FDD,FNS INCLUDE /MASS PIONC UNITS ARE FMIF(LOSOL)THENUY=(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=1,4P(I1)=P5(I1)+P6(I1)-P1(I1)ENDDOP2=P(4)**2-P(1)**2-P(2)**2-P(3)**2FORM=(XLAM**2-XMU2)/(XLAM**2-P2)DO 395 1=1,48395 XID(I)=FLOAT(ID(I))DO 1=1,3U1P(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))*FORMU1C(I)= - .5*FN/FP2*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	 90• (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./ XMNSU3DNAFAC=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./XMDU3DBFAC=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 ./XMNSU3DNBFAC=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./XMDU3D C FA C =FN*FD**2/• (SRA-P1 (4)-P6(4)-X MN-(P11+ P66+ 2*P 16)/2./XMN)/Appendix D. Amplitudes for the 7+ p	 71-+R0p reaction	 91• (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./XMNSU3DNCFAC=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 MDU3DDFAC=-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 MNSU3DNDFAC=-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./XMDU3DEFAC=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./XMNSU3DNEFAC=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./XMDU3DFFAC=-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 ./XMNSU3DNFFAC=-2./3.*FNS*FD*FND/• (D3F-XMD-FUY*GDEL(D3F)/2.)/• (DN3F-XMNSA-UY*GDELS(DN3F)/2.)U3DNF(I)=U3DNFFAC*Appendix D. Amplitudes for the 7r+p 	 71-1-71-°p reaction 	 92• (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 DU3DGFAC=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./XMNSU3DNGFAC=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. /XMDU3DHFAC=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./XMNSU3DNHFAC=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 /XMDU3DDAFAC=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 MDU3DDBFAC=-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. /XMDDDC2=SRA-P6(4)- P66/2. /XMDU3DD 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))Appendix D. Amplitudes for the r+p —4 7+70p reaction	 93DDD1=SRA-P1(4 )-P6(4)-(P11-1-P661-2.*P16)/2./XMDDDD2=SRA-P1(4 )-P5(4)-P 6(4)-(P11-I-P 55+P66+ 2.*(P15+P 1 6-I-P56))/2./X MDU3DDDFAC=-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./XMDDDE2=SRA-P 5(4)-P55/ /XMDU3DDEFAC=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./XMDDDF2=SRA-P1(4)-P5(4)-P6(4 )-(P 11-FP 551-P66+2 .*(P 15 -I-P56 -I-P16))/2./XMDU3DDFFAC=-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 	 (14)*U3DDE(I)+X ID (15)*U3DDF(I)-1-• XID(23)*U3DNA(I)+XID(24)*U3DNB(I)-1-XID(25)*U3DNC(I)-F• XID(26)*U3DND(I)+XID(27)*U3DNE(I)+XID(28)*U3DNF(I)-1-• XID(29)*U3DNG(I)-1-XID(30)*U3DNH(I)ENDDOUJAC=UY*XMIXT(P1,P 5,P6)U3A(4)=UJAC*U3A FACU3B(4)=- UJAC*U3BFACU3C(4)=UJAC*U3CFACU3DA(4)=(1./3.)*UJAC*U3DAFACU3DB(4)=( 1. /3.)*UJAC*U3DBFACU3DC(4)=(-1./3.)*UJAC*U3DCFACU3DD(4)=(- 1./3.)*UJAC*U3DDFACU3DE(4)=(1. /3 .)*UJAC*U3DEFACU3DF(4)=(-1./3.)*UJAC*U3DFFACU3DG(4)=(1./3.)*UJAC*U3DGFACU3DH(4)=(- 1. /3.)*UJAC*U3DHFACAppendix D. Amplitudes for the r+p -4 7r+7r°p reaction	 94U3DNA(4)=(1./3.)*UJAC*U3DNAFACU3DNB(4)=(1 ./3.)*UJAC*U3DNBFACU3DNC( 4) =(-1./ 3 .)*UJA C*U3DNCFACU3DND(4)=(-1 )*UJAC*U3DNDFA CU3DNE(4)=(1./3.)*UJAC*U3DNEFACU3DNF(4)=(-1./3.)*UJAC*U3DNFFACU3DNG(4)=(1./3.)*UJAC*U3DNGFACU3DN1T(4)=(-1./3.)*UJAC*U3DNHFACU3DDA(4)=5./6.*UJAC*UDDAFACU3DDB(4)=-5./6.*UJAC*UDDBFACU3DDC(4)=-5./6.*UJAC*UDDCFACU3DDD(4).5./6.*UJAC*UDDDFACU3DDE(4).5./6.*UJAC*UDDEFACU3DDF(4).-5./6.*UJAC*UDDFFACUTOT(4)=X ID (22)*U 3A(4)-I- XID(1)*( U3B(4)-1- U3 C(4))-1-F 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)-1-F 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)RETURNELSEUT 0 T(1).0.ITTOT(2)=0.UTOT(3)=0.UT 0 T(4)=S QRT(SIGMA(PPILAB,TPL-1))RETURNENDIFENDAppendix EWorld Data for 7F — 271 ChannelsThe world data at lower energies are given below. Where the original data was givenin terms of the incident pion's lab momentum and rms beam momentum spread (poand Apo), the pion lab kinetic energy and its error, 77, and A T„ , were calculated usingT„ = /p2 + ,a2 — it and AT, = poApol \ IA + ta2• The errors in cross section includeboth statistical and systematic errors, unless separate entries given, in which case thetotal error has been calculated using Ao-— total = NAACrstat)2 + (Lass)2 •• 7r— p --* 7r-7-EnT, (MeV) (3-(1b)210+7 15+3222+5 27+5233+7 53+13246+6 125+28264+12 160+60288+12 380+90Table E.1: Batusov's 7-7± data [BA65].95Appendix E. World Data for 7r — 27r Channels	 96T,(MeV) a(jib)203 13.8+1.5230 60.3+3.2255 166+6279 374+15292 546+31331 1160+52357 1880+77Table E.2: Bjork's 7r-7r+ data [BJ80].Po ± Apo(MeV/c)TR.(MeV)a(ub)A a stat(sub)AO. sy st(fub)Acrtotai(sub)295+9 186.8+8.1 5.1 1.1 0.5 1.2315+10 205.0+9.1 20 2.4 1.8 3.0334+10 222.4+9.2 51 10 6 12354+10 240.9+9.3 118 15 13 20375+11 260.5+10.3 211 27 24 36394+10 278.4+9.4 327 18 37 41413+10 296.4+9.5 477 17 53 56432+11 314.4+10.5 785 55 88 104450+12 331.6+11.5 1052 42 118 125Table E.3: OMICRON's 7r-7r+ data [OM89a].Appendix E. World Data for 7 - 27r Channels 	 97• 7r+p --+ 7+7+nT(MeV) a(1ub)226 9.4±2.3250 25.0±5.3312 57±10357 100±21Table E.4: Kravtsov s 7r+7r+ data [KR78].PO(MeV/c)Tir(MeV)a(,ub)Acrstat(1ib)Ausyst(ub)A atotal(4ub)297 188.6 1.8 0.2 0.3 0.4317 206.8 8.0 1.3 1.2 1.8338 226.1 21.7 3.0 3.3 4.5358 244.7 27.4 3.2 4.1 5.2378 263.4 39.0 4.4 5.5 7.0398 282.2 45.1 5.2 8.9 10.3418 301.1 65.0 4.7 12.7 13.5439 321.1 74.0 5.3 14.4 15.3459 340.2 83.0 7.3 16.2 17.8480 360.3 94.0 8.0 18.4 20.1Table E.5: OMICRON's 7+ 7+ data [0M90].T,(MeV)a(pb)180 0.11±0.03184 0.28±0.05190 0.60±0.10200 1.46±0.22Table E.6: Sevior's 7r+7r+ data [SE91].Appendix E. World Data for 7r - 27r Channels	 98• 7 p _ ), 70 70 nBeam momentum(MeV/c)T11-(MeV)a(,ub)Aristat(ab)Aasyst(ub)Aatotal(gb)272.5 166.59 0.382 0.096 0.023 0.099275.5 169.45 0.59 0.14 0.04 0.14279.7 173.02 1.18 0.22 0.07 0.23283.9 176.78 2.06 0.35 0.12 0.37285.7 178.40 2.31 0.65 0.14 0.66286.9 179.48 3.33 0.64 0.20 0.67291.0 183.40 3.81 0.81 0.23 0.84292.6 184.61 8.1 1.3 0.5 1.4297.7 189.22 8.5 1.0 0.5 1.1304.7 195.58 17.1 1.9 1.0 2.2313.8 203.87 21.9 2.0 1.3 2.4322.5 211.84 30.3 3.0 1.8 3.5330.5 219.15 59.8 6.4 3.6 7.3339.4 227.41 75.2 7.3 4.5 8.6349.4 236.68 98.1 9.3 5.9 11.0359.1 245.68 118 11 7 13389.6 274.28 388 46 23 52399.9 283.99 479 49 29 57Table E.7: Lowe's 7r07r0 data [L091].Appendix E. World Data for 7r — 27r Channels	 99• 7r-p	 7r-7r°pPo ± Apo(MeV/c)T„(MeV)a(sub)A Cr stat(pb)A a syst(jib)AO-total(ab)295+9 186.8+8.1 0.75 0.3 0.3 0.4315+10 204.9+9.1 2.2 0.6 0.4 0.7334+10 222.4+9.2 8.5 1.4 0.8 1.6354+10 240.9+9.3 20 3 4 5375+11 260.5+10.3 27 4 4 6394+10 278.4+9.4 50 4 12 13413+11 296.4+10.4 73 4 14 15432+11 314.4+10.5 119 8 18 20450+12 331.6+11.5 157 9 36 37Table E.8: OMICRON's 7r-7r° data [0M89b].• 7r +p 	 71- + pT,(MeV) cr(pb) source230+13 18+192 [BA75]275+15 48iT [BA75]294+4 120 + 50 [AR72]300 110 + 40 [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 onPhys. 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) 536.100Bibliography	 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.

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