IIJk rlJ ABSTRACT FINAL STATE INTERACTION IN STRANGE PARTICLE PRODUCTION by Justin C. Huang Phenomenological methods have been employed in this thesis in hopes of explaining the experimental results to the reaction 1r: P -> K+ Y+11. The theory of final state interactions as formulated by'Watson has been applied to the consideration of two different assumptions concerning the above reaction. These are: A. Neglecting the K—Y and the K-Tr interactions, what are the implications of global symmetry in the final state 7r-Y interaction? B. ‘What information can one obtain concerning the 11-K inter- action by neglecting the TT—Y and K—Y interactions and consider only the K-n' interaction in the final state? For both parts A and B, Fermi's statistical model was used to determine the matrix elements for the primary interaction. For part A, the secondary interaction was treated using the Pawave effective range formula where the arbitrary parameters were fitted to the pion—nucleon I = 3/2, J = 3/2 resonance data in accordance with global symmetry. The secondary interaction in part B was also handled by the use of effective range formulae where the parameters were fitted to the known experimental full width at half maximum and effective mass for an I = 1/2 resonance. Both a S—wave and Pawave resonance were considered. Justin C. Huang The results for part A indicated that the methods used would BOJL give a very strong test of global symmetry as the results did not differ appreciably from a purely statistical theory calculation. For part B, the results indicated a K‘fi'WVE' ratio of 1.7 with a small amount of fir? reactions for both the S and P—wave cases. This is in good agreement with the experimental results obtained so far. The . . . " ° " O o O . calculations also indicated a Krrz A ,7): ratio of 1.7 for a S—wave resonance and a ratio of 0.7 for a P—wave resonance. Unfortunately, the reaction K$r°z° is difficult to determine experimentally and no ex- perimental results have been obtained on them as yet. r Lin-7;.- _fl .1. FINAL STATE INTERACTION IN STRANGE PARTICLE PRODUCTION By A ‘ ‘ f Justin C. Huang A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1961 .-s_ 3L ACKNOWLEDGEMENT I am most grateful to Dr. J. S. Kovacs for his sponsorship of this problem and for his continued guidance and suggestions throughout the course of this work. The helpful suggestions by Dr. D. Lichtenberg are also deeply appreciated. ii II. III. VI. VII. TABLE OF CONTENTS IntrOduCtion O O O O O O O O O O O O O O O O O 0 Final State Interaction Isobar Model . . . Global Symmetry and the Pion-Hyperon Interaction Global Symmetry A B C D E F. Isobar Model Spectrum . . The Pion-Kaon Interaction . . o Kinematics o o o o o o. o. o. .T‘heT-MatriXoooooool . Phase Space Calculations . . Extension of the t- Matrix using JR 0. I. Q. Q C. O. Q 0. C O- omeT-mtruoooooooooo.‘ . Calculation of Probabilities . . . Extension of the t-Matrix using q . On the Energy Shell Scattering . . O O O O . On the Energy Shell Scattering . . . .ISObarMOdeloooooooooo A B C D. Extension of the t- Matrix using k E F G Summary 0 Bibliographyoooooooooooo.o iii 0.99 0.0.0.09. .0”! 0,9. 9 , 9 , 00.09?! I I o - I .l 0999’. Page 10 ll 11 12 20 31 38 ul 41 1+3 60 63 69 72 tr“??"—-eeme1r Table l. 9. 10. ll. 12. 13. 14. LIST OF TABLES Mass differences between baryons . . . ... . . . . . . . . New isospin assignment of quantum numbers . . . . . . . . 7T—Y charge wave functions in terms of the new isospin wave funCtionS . o o o o o o o o o o o o o o o o o o o o o W@ights w for 1T-Y reaCtionS o o o o o o o o o o o o o o o ‘fl—N charge wave functions in terms of isospin wave, _ funCtionS o o o o o o o o o o o o o o o o o o o o o o o o 'I—Y;K charge wave functions in terms of isospin wave. . funCtionS o o o o o o o o o o o o o o o o o o o o o o o o WGights F2 for'fflfff>—firl(+174-y reaCtions o o o o o o o Branching ratio assuming global symmetry and a k eXtenSion a. o o o o o o o o o o o o o o o o o o o o o o o m Branching ratio assuming global symmetry and q . . . extension for E = 167nfl o o o o o o o o o o o o o o o o o Branching ratio at E=l67mrassuming global symmetry and l i neglecting off the energy shell contributions . . . . . . Tfifeights W for K-” reaCtionS 0' o of o. o. o o. o. o. o. o. o o. O. 0 Parameters a and I" for the K-‘F interaction ._ . . .I . . Branching ratio at E—‘l677hr for the lK-JI final state interaction using a q extension . . . . . . . . . . . . . Branching ratio at E=l67mr for the K—]' final state inter— aCtion USng a JEq eXtenSiOn o o o o o o o o o o o o o o o Branching ratio at E = 16 7’7" for the K-1I final state inter- action neglecting off the energy shell contributions. . . K* and'z'quantum numbers 0 o o o o o o o o o o o o o o o o K*—Z charge wave functions in terms of isospin wave. functions . . . . . . . . . . . . . . . . . . . . . . . . BranChing ratio for the K* iSObar mOdel o o o o o o o o 0 iv Page ll 12 14 15 23 24 26 BO 35 37 45 53 59 6h 65 66 68 fair Figure 1. Schematic diagram of a final state process . 2. Schematic diagram of an isobar process . . . 3. Variation Of Ireal With Wq o o o o o o o o o 14'. Variation Of Iimaginer'y With Wq o o o o o o +'- a 5. Momentum spectrum for the KHZ _ reaction at USing the k eXtenSiOn o o o o o o o o o o o- 6. Variation of. probability with the total cente emergence-0000000000000. 7. Momentum Spectrum for the K’II 2° reaction at. USing the q eXtenSion o o o o o o o o o o o 8. Momentum spectrum for the Kilt-£0 reaction at E neglecting off the energy shell contributions 9. Momentum spectrum at E = 16m, using the isoba 10. CO-OI‘dinate syStemS K and X o. o o o of 0' o o. o— o. o o. o 0- 11. Variation of I with W at E = 16711, for a S-wave K—JT _ interaction using a k extension . . . . . . . . . . . . 12. Variation of I with W at E = 161', for a P—wave oK-W. _ interaction using a k extension . . . .'. .‘. . . . . . 13. Momentum spectrum at E = 1611).” for the K01": reaction for a S-wave K—n' interaction using a k extension 1L». Momentum spectrum at E = 16"“ for the k°1f+Z- reaction for. a P—wave K-” interaction using a k extension 15. Variation of I with W at E = 1610, for a S-waveK-Tr. interaction using a $135 extension . . . . . . . . . . . . 16. Variation of I with W at E = 16“)” for a P—wave ,K—1T, interaction using a q extension . . . . . . . . . . . . 17. Momentum spectrum at E '2 16m" for the K0114}. reaction for. ~ /. LIST OF FIGURES E I‘ E I‘ 16 m” of mass model .1 . 16 M, a S—wave K_‘IT interaction using a Jk'q extension . . . . . Page 10 21 22 29 32 31+ 36 no 41 49 50 51 52 55 56 57 Figure 18. 19. 20. Page Momentum Spectrum at E 2 1611), for the K392" reaction for a P—wave K-‘IT interaction using a m extension . . . 58 * .- Momentum spectrum at E = 161". for the K31 2: reaction for a S—wave K-fl' interaction neglecting off the energy shell contributions . . . . . . . . . . . . . . . 61 Momentum spectrum at E = 16'” for the Ktfl-Z'o reaction for a S—wave K-7T interaction neglecting _off..the . energy shell contributions . . . . . . . . . . . . . . . 62 vi I. Introduction Around 1950, physicists' attention was diverted to a class of newly discovered particles later entitled strange particles. They were first observed from showers of particles occurring when high energy cosmic rays struck a lead plate placed inside a cloud chamber. The first of these particles discovered was the A0 and the Go , sometimes called the big V and the little; V particles since their decay products left a. V shaped track in the cloud chamber. Once the effort was on, many new particles appeared.1 They were called the Kgg, K42) 5,) It) I: 22 Z: 5; and recently, the 35° where the subscripts on the ”5 refer to the mode of decay. After studying their decay products, life- times, masses, parities, spins, and other prOperties, these particles at first baffled physicists since they seem to violate some of the fundamental laws of physics. The name "strange" particles was mostly derived from this fact. One disturbing phenomenon was that the particles seem to violate the law of the reversibility of nature. In particular, particles produced by a strong interaction should decay by a strong interaction if it is energetically possible. But the observed lifetime of these particles was around 10-10 seconds, while they were produced by strong interactions whose characteristic time is approximately 10'23 2 who proposed the idea seconds. This disparity was resolved by Pais of associated production, the concept that strange particles can only decay and be produced by strong interaction two at a time. This idea led to the concept of a new quantum number, strangeness, which was formu- lated independently by Gell-Mann3 and NishijimafL Another puzzling problem wifi' was the mystery of the K3) (3) and 98. These particles had identical prOperties except for their parities. This dilemma was resolved in 1956 by Yang and Lee5 who suggested that parity need not be conserved in weak interactions and proposed experiments to test this hypothesis. Their suggestions were later verified in an experiment performed by Wu et a1.6 We now label the K5) rs) and 95 simply as Ks . There are still many unknowns in the study of these particles. f' we would like to know more about how they interact with other particles F in nature and among themselves. Of great benefit in this study are E symmetry conditions. 'We have already mentioned how the strangeness v quantum number has aided us in the understanding of strange particles. By looking for these constants of the motion, we are able to reduce our unknown quantities considerably. For example, charge independence allows us to describe the pion—nucleon interaction by one coupling constant instead of by three. An enlargement of the concept of charge independence to include all baryons is entitled global symmetry.7 Instead of using four different coupling constants to describe all the pion—baryon inter- actions, global symmetry, if true, would allow us to equate all the coupling constants. (The equality of the coupling constants in the kaon— baryon interactions is called cosmic symmetry. Other symmetry models8 have also been attempted, but to date, there has been little experimental verification on any of these. ‘With the advent of the ultra-high energy accelerators and the develOpment of the bubble chamber, the task of studying the strange particles has become much simpler. Still, experimental techniques have not advanced to the stage where we can study many types of strange 'IIT‘E ‘Partixihe reactions directly. However, indirect effects may in certain cases be observed by looking at processes in which these particles appear together in the final state. Roughly speaking, final state inter- action refers to that part of a process in which the particles after they are produced, do not go off as plane waves, but interact strongly with each other as they leave the production region. For example, in the reaction [(I’.f f3 ___9. y/ ‘+' W’ 4' Z7 the energy spectrum of one of the pions is considerably affected by the interaction of the remaining pion with the hyperon.9 The theory of final state interactions as formulated byWatsonlo has been applied successfully to many elementary particle reactions. It is the purpose of this thesis to apply the theory of final state interactions to the consideration of two different assumptions about the reaction 7/'- + P -——-> K -r Y ‘r 7 These are: l. Neglecting the [(-I’ and the K4] interactions, what are the impli- cations of global symmetry in the final state ”LY interaction? 2. ‘What information can one obtain concerning the ‘9” reaction by neglecting the FY and the K'Y interactions and consider only the (‘77 interaction in the final state? Closely associated with the final state model is the isobar model11 which supposes the formation of a relatively long-lived two particle system which behaves like a single particle. Calculations 'using this model will also be included here. 3 3 i u a; ""V\ At this point, the status of global symmetry is not on too good a footing. Although there has been some experimental evidence in sup- port of it,12 other evidence would have us reject it.13 In particular, as an example of the latter, there is experimental evidence which seems to support an odd relative parity between the A and the Z- . A resonance in the TI-Y system which has been observed9 gives us hope for global symmetry although it need not imply it. Other interpretations can also K. ‘ be applied to it as was done by Ross and ShawlLL who suggested that the ‘ TF)’ resonance is mainly due to a resonance in the I-IV system. A strong resonance in the K-TT system is also of great interest to the understanding of strange particle reactions. There have been 15916917 about the role of the K47 interaction considerable speculations in the study of elementary particles. For example, in the days of the parity—doublet kaon theory, SchWinger 18 proposed a strong KKrrcoupling to account for the backward peaking of the I10 in the reaction 1‘ + P _..., /]° + K° One can equally account for this phenomenon by postulating a K, meson which corresponds to a strong Smwave, isotopic spin 1/2, K'TT resonance state and proposing a strong KKITT coupling. The existence of an isotopic spin 1/2 K-Nresonance with an effective mass of 878 mev and a full width at half maximum of 23 mev has been already observed experi— mentally.19 The spin and parity of this Ki. particle have not yet been established.20’21 Throughout the text of this report, the units of ii = C = ‘m-rr= 1 will be employed unless otherwise indicated. '11- Final State Interaction In the final state interaction formalism of Watson,10 we hypothetic- ally break up our interaction If into two parts, a part 1?, called the secondary interaction which is the interaction between two of the particles in the final state and a part V which is designated as the primary inter- action and formally defined as 152/. This division is arbitrary but meaningful if certain conditions are met. These conditions being 1. The short range of the primary interaction. 2. The isolation of the final state interaction between one pair of particles from the effects due to all other particles. The second condition is satisfied if the relative energy between the isolated pair is small (or near a resonance for the two particle interaction) and the interactions between the pair and other particles in the final state are negligible. The other particles will then quickly escape the secondary interaction region leaving the isolated pair alone. Schematically, we can illustrate the reaction in the following way: \\fi>“ Figure 1. Schematic diagram of a final state process The transition probability per unit time in going from an initial state "i" to a final state "f" is given by Z. w}; z 211’ 3(Er'54) [ILI (l) where E refers to the total energy. ‘We now want to separate the reaction inatrix: Z; into two parts, one involving only the primary interaction and the other only the secondary interaction. let us start with the Schroedinger equations HoXi = EiXi HOXf = Efo (2) where H0 is the unperturbed hamiltonian containing the kinetic energies. Conservation of energy gives us Bi = Ef. Next, we can write for the total hamiltonian H: l-é‘f—V- where ‘V: V+Zf . Since Zfis the final state interaction, it can only Operate on the states of type "f." The Q?) Schroedinger equation for H is 1: .— V? .1 + H50, efigb, <3) L p where "+" refers to a solution with plane waves and outgoing scattered waves and "—" refers to a solution with plane waves and incoming scattered waves. The solution to equation 3 can be written in integral form Q .— I“. + (Ll—H sandy-$15: (4) Ar vdmnxa‘LG is an infinitesimal which specifies the path of integration and depends on the boundary conditions. 5» can also be written as i 4’ (75.; = Q- 7t: (5) where from equation 4, 0 must be given by {2: / + (first,Her/VJ?r 521 / + (a; vhg-Vflefl‘l/ (6a) (6b) ,fi? is usually called the Moller'wave matrix and equation 6b is known tun" F as the Chew-Goldberger solution22 to the integral equation (6a). Our reaction matrix T is just 7-.- VI? (7) We shall now define a situation where “‘0 . Our hamiltonian is now H=fé+2fand ”2‘2” = [at = at <8> where i ‘ -I t 2’.‘ 9‘; x: +(Eé-Horte)u’<¢ a w 74. (9) and + -I + 24)" / +(Q-Fétn16)y'w- (10) Making use of the general operator relation ’ ’ ’ 6 A); < A 5 A( B and the fact that 7fdoes not operate on states 1; , we get 7 = w, VQ+ (12) Next, let us define a situation where U30 . Then H -‘- ”0* V and + I W,“ = a 94° ‘13) where (14) and of = x + (as mam: an O Combining equations 11, 12, and 15, we obtain for 7;; - of "_ " 0+ E‘(4;,V5f)+($f TNT?) (16) For the limiting case when l/ is very strong, the second term is negligible compared to the first. This is because If only operates on a very restricted set of intermediate states of type "f" while the strongly interacting V leads to many virtual states. We can thus write the T—matrix for reactions in which final state interactions are considered important as E: ‘ (i3 Vt”) : (if) w‘fVfl; 74;) Let us now define T: VfZand rewrite equation 17 as 7; (F/ w’fT/ P) (18) where a momentum representation is explicitly indicated. Let us also (17) \I + define If: U11] and from equation 10 w' = /+ (El-'Ho'iéf/t‘ (19a) .. .. -/ 0" wf = I + fT(E;'/-/o+‘:6) (19b) Substituting equation 19b into 18 and noting that [7; t, we have 7;, :. (F/T/fs") + (F/NEi—Hgééf'T/fi‘j (20) Summing over intermediate states, we can rewrite equation 20 as a , " "MP? 77 = (P/T/F’ ) + “(PM )0” as" 7t. 5. - 5' + .-e (21) We can see that we now have separated Tf‘i into two separate parts, T which involves only the primary interaction and t which involves only the secondary interaction. 10 III. Isobar Model The isobar model was formulated in 1957 by Lindenbaum and Stern- heimerll to account for the pion production data in proton—proton and pion-proton reactions. This model assumes the production of an isobar or isobars which do not interact with the other emergent particles. Furthermore, the lifetime of the isobar is long enough to allow it to separate from the other emergent particles so that the decay products of the isobar will have no final state interactions with them. In comparing the isobar model with the final state model presented in section II, the isobar takes the place of the isolated pair so that the two models differ in the production mechanism. Secondly, the lifetime of the isobar is sufficiently long so that the primary interaction V does not interfere with the decay of the isobar. They can be thought of as two separate processes. we can represent this schematically by Figure 2. Schematic diagram of an isobar process The mass spectrum for the isobar was shown by Lindenbaum and Sternheimer to be simply if : chsr F65: m1) 03(7’71) 0/7711 (22) where /:fi$flh9 is just the two body phase space factor depending on the total energy E of the isobar and the mass of the isobar 7711-. gfmz) is the cross section for isobar formation and is equal to the resonance (Bross section for the two particles forming the isobar. 11 IV- Global Symmetry and the Pion—Hyperon Interaction A. Global Symmetry As was mentioned in section I, one might look at global symmetry7 as an enlargement of charge independence. In charge independence, we neglect the mass difference between say a proton and a neutron and attri- bute this mass difference to the electromagnetic field. If the electro- magnetic field could be turned off, then the proton can not be distin- guished from a neutron. 'We say that strong couplings possess a symmetry (charge independence) which is destroyed by the electromagnetic field. Now let us divide strong interactions into two classes, VS very strong coupling and MS moderately strong coupling. we furthermore make a much bolder assumption, neglect the mass splitting of the baryons. To see how bad this assumption is, let us examine the magnitude of the mass differences as presented in Table 1. Table 1. Mass differences between baryons Particles AIN Z,/\ Z) AV E! N “V” 0.13 0.07 0.2a 0.34 we can see from Table 1 that it is very large. In analogy to charge independence, we then say that VS coupling possesses a symmetry (global) which is destroyed by the MS coupling. In other words, the MS coupling causes the mass differences between the baryons. Under Gall-Mann's scheme, we associate pion—baryon couplings to the VS classification and the kaon-baryon couplings to the MS group. Since /1 and.Z;are not isospin doublets, we have to resort to a trick, defining a - /|°—Z° 0 , /l°+Z° .: __ o. c/ : ___. y la— n E ff We now proceed to define a new isospinz3 with the following assignments: Table 2. New isospin assignment of quantum numbers i iZ Particles 1/2 +l/2 P -Z* Z“ 5° —l/2 N yo 2‘ _=_' +1 7+ 1 O 779 -1 fl' The statement of global symmetry can now be stated as the invariance of the interaction under rotations in this new isospin spin. We now have a means of relating the TT-Y scattering matrix to the known pion-nucleon scattering matrix. B. The T—Matrix Let us now return to our problem of the WV interaction in the final state. Making the approximation that the center of mass of our final state is on the hyperon, we can write for E, , the total energy in the ”59 state as E” : “3+ 717”: +./l,}"2+ 777,,1 + my (23) ” Q where E and k are the relative K—Y and ”"7 momenta in the state [F9 . .3 -'I/ Let us now denote the state vectors H?) and [FD by /?:g> and Lt, I52) J A where g and I; are the relative W’Y and K’)’ momenta in the IF) state, etc. Since t is to involve only the my interaction, we can rewrite (FM/‘3 as (aw) = «ii—a”) (at/z) so 13 a .3 The matrix (g/f/Al) can be reduced further in accordance with the new isospin Space defined previously. (i/t/g) : M/z(§/t%/:)+M/z {f/[Vz/z) (25) where WB/Z and Wi/Z are weights in the new isoSpin space. The weights are determined by the usual method for the addition of angular momenta using Clebsch-Gordhn coefficients. Table 3 gives a list of the ”3” charge wave functions in terms of the new isospin wave functions. Table 4 gives a list of the values of W for various reactions. An interesting result is that the fif‘: 1r? 2- reactions are forbidden. Global symmetry allows us now to equate the above t3/2 and tl/Z matrix elements to the correSponding ones for pion—nucleon scattering. we should mention at this point that pure global symmetry has been assumed here so that we can to begin with, neglect all K—interactions. Since the resonance for pion—nucleon scattering occurs in the I = 3/2, J = 3/2 state, we shall neglect all other states. The effective range approximation will now be utilized to calculate (i/fifi A?) . The matrix a/{ /5 can be written in terms of a scattering amplitudezu 3a rm) ; — sol/a m,1(g/zs/,/§) (26) For P-waves only, 'débé) can be written as 25- am a 5;? — wit-(IA?) (27a) MAE“: 5 2 I 3 ——- CocSSmS +,, 5/15 = .2. g C = ’3']? {0’55 "”5“ "‘25) g ‘52} 66) (27 > Table 3. + + 77' Y° 77°Z" 77W" 77")” 7T2 *0 7r 2 7r”:~ ”020 TT-Ycharge wave functions in terms of the new isospin wave functions .— ’ (6312,35 1743’; '3/2 J_ 5 M Sir/l 991/2. 9024,35 SL1: ) '3/2 14 F952“. 512/2 +g¢z)I/Z + E %fi"/z £9927. g¢zVL—g:fi,é JZ‘W" J2: S‘s-v. J53: 94315-4. gs/é + £17195sz - Ji— 94,2, Table Reaction a N ll 15 4. weights W for HEY reactions W5/2 2/3 1/3 2/3 1/3 1/3 -1/3 -1/3 1/3 1/3 1/3 2/3 2/3 2/3 2/3 1/2 1/3 1/3 -1/3 1/3 -1/3 2/3 1/3 1/3 -1/3 2/3 -1/3 1/3 1/3 1/3 1/3 16 ( refers to the Spin, 5 refers to the I = 3/2, J = 3/2 phase shift, and B and C describe the direct and spin-flip Pewave scatterings. To calcu- late the phase shift 5» ,‘we shall make use of the P-wave scattering length formula derived by Brueckner,26 2 kSCOfS =-: //a3 " k //' (28) where Q and f‘ are two adjustable parameters. We shall adjust Q g and r to fit the pion—nucleon data. With a choice of Q = 0.59 1g/7’lnC ? and r: 0.91 fihfi, we obtain the J = 3/2, I '= 3/2 resonance at a labora- I tory kinetic energy of 187 mev. ‘we are assuming here that the mass a difference between the nucleon and the hyperon will not shift the (3/2,3/2) resonance27 as obtained for 7H scattering from that observed for 77’” scattering. Equations (26), (27), and (28) are actually defined only on the energy shell. Since we are interested in including the effects of off the energy shell scatterings also, we shall extend the definitions to include off the energy shell interactions too. The extension shall be made first using the initial momentum k as is presented in equations (26), (27), and (28). Later on, we shall make the extension of the definition of that matrix using the final momentum 35 as the momentum upon which it depends. Referring back to equation 21, we have now only to determine quantities of the type (f? /T//f) . Since we are primarily interested in the final state interaction and assume that the most influence will be due to this, we shall use the simplest approach for obtaining these matrices; namely, assume that phase Space effects predominate. A model 17 by FermiZB assumes that for very strong interactions, the matrix elements for various processes are independent of the momenta involved. The assump- tions for the theory are that l. The incident particles in every collicion coalesce and the energy brought in by them is released within a common interaction region. 2. Interactions involved are sufficiently strong and interaction region survives sufficiently long for all of the possible final states to be equally excited. Of course, the various conservation laws of momentum, energy, charge, isotopic spin, etc. must be taken into consideration as well as restrictions such as the physical indistinguishability of identical particles. These constants of the motion are most simply taken care of by reducing the matrix elements. For example, to account for the conservation of isotOpic spin, we write (for processes involving only 3/2 and 1/2 isotopic spin) (CI/T/CII’) "' 5/; (P773). ”3") + 52 {FA/52”,) (29> Mgr/fl 519/2 i gf/(Wa/FQ/Z szz/é/h/p‘j/Z (30) assuming that the cross terms will cancel out. Since all states are equally excited /(7’/7§/, I Fj/ = /(;/ 7:21 /P9/ 0 . . /(/3ji/T/F:I)/Z = 00"‘7‘ (5,:*5/;2)= CW" F2 (31) Ifor'the case considered here, it is sufficient to take the matrix ealxaments to be just a constant F which depends on the weighting in 18 isospin Space. Since we have P—state scattering in the final state, We must also include a factor insuring the production of mesons in the 2 P—state. 9 The matrices we need are then (PVT/P" = F iii/w? (F'ITIF’) = F KAT/wk ‘32) __—\ where A is an unspecified urrit vector which depends on spin and which we will average over when necessary in the spirit of the statistical model. Wq is just the total meson energy and is equal to V39 7”,} Substituting equations 23, 2Q, 25, 26, and 32 into equation 21, we have 7, : F31? , iii f/gf) xii-E") AU? a; 4/3" (33) f ‘05 47,. Wk (En/fimf—q—mra‘e) : FE _ Ira/3f5 #579,412 5/2. (M a “i a: w: (E- ear-awn ‘3 > but E : «621+ MK2 4- ll)? 4- 777), from conservation of energy. ” 5' 4*WKZ‘W : w? (35) 7,7: = F j _ Fwy, {am/s f c/Qdé , 3' 47p. a}: (w; .. we “[4,.6) (36) Substituting for {(2:5) from equation 27 f , = F435 — Wag fWW/ed“ L65V4327ié‘é’aadk / 3 7” #2"): (“fwd“) flaw-ant) (37> 19 The first integral in equation 37 can be reduced by using the following identity d "“ .4 2 .... A for/z) (3"?) 49k ‘ g ’6 flu? (38) It can then be written as f = 231’ 4172‘ "" ca) £2 (M I 3 3 0 wk: (%-w‘+’}e) (39) where kc is a suitable cut off used to prevent the integral from di- verging. The second integral in equation 37 can be similarly simplified in the following manner face?) gm: 49/: — -/ZZA§‘)-E A’vl' an,é (40) = - iélrkzi-(Z'Afi) But A. is parallel to a: -'- by using 3A5” = A? (41) we have z '(24f) = /T '4 (42) The second integral can now be written as J .) kc 2 f = 1:! 41 L" dé 2 3 3 0 10,5 (Lt/{WA ”6) (”3) Equation 37 is now after changing variables from k to wk java-t0 +46) W” 20 G(5) : (I‘l— kzqs) (la/”63 + 4: (c3/§3)2 (#5) [Kr- #6); + (Hay—)2] where and 0% was chosen to be 6m" . The integral in equation 141+ is shown in Figures 3 and 4 for the real and imaginary parts. They show much resemblance to the resonance curves obtained by Kovacs.‘29 Denoting the real and imaginary parts of the integral by IR and I", we can write 7515’ = F7615; {(2.er ‘ 43-2 IR)+ I. 51/3; Ix} (1+6) and using the relation -‘ —> 2 “ . d 2 fg'g) dfl T‘fdA ' g/S (1,7) the absolute square of the scattering matrix becomes WK— 1K4} The F2 s were determined as was indicated by equations 29, 30, and 31 2 /7F—F’/ " where F3/2 and F were determined by the usual method of Clebsch—Gordnn 1/2 coefficients. Tables 5 and 6 give the 7F” and TFXK charge wave functions in terms of the isospin wave functions and Table 7 lists the values of F2 for the various reactions. C. Calculation of probabilities The probability for a certain reaction is given by the following . 30 expreSSion z , 4-» .1 —- ,3 : Kf/g/ SKe—E;)S(P-Kf) c/fl‘°'°’/Z (49> l l I A l 2 3 4 5' a)? Figure 3. Variation of IReal with Wq. 22 -5 L m G 'l- ’/5 .53 N ‘20 Figure 4. Variation of Iimaginary with W . q ‘25— ’50 _ I l l / 2 5 4 Table 5; 7T+f> 77'77 777° 77‘? + 77 n ‘n° n 23 TH! charge wave functions in terms of isospin wave functions. $1534 — Wad/z ; \/§:9§%—A +—¢Q§ sag-fl = E 8%“ + E 545/2 = E 8”“ “AI 554/1 3 [3'2- ‘3/4/2 —/§L 924/1 Table 6. 77’2‘ /<* 7r’2‘ K° 7735K ° ‘7I*Z°K+ NEW fi’k’ 1°23? 77’Z°/<° *z°/<° 7r°z*K° 7T°Z°K 7r*:'l(* 7737/ ”Oi-Ki 24 flLYLK charge wave functions in terms of isospin wave functions : 9%; Vi ) ' My; 2 J, __ _L_ : V771: (#47533. _l/7—0— S‘éz V: £2,474 = :7? (7672,94 ’72.?¢ ‘34 : V%¢/% +F¢—;;+VEL(’é-3é ' v76¢572fi1 Mafia! 75561-72 : ”$472“2 ”$127 IF“: 493% 4567442 ”Esta—4L F’s/2V; uzgflK/Z gay/z Esau-392,5 mS‘ I — = /o SAW: ‘ fir? 22],; ’73“ @347? £44 u? 5:4 (7L .1. 25 Table 6. (Continued) 77’2°k = f5¢ «74/ £55 K/ 'H 3w 75L g4"; 7’05“" = «7% 3947: W? 35H; 7% (it WJZ—Ko = fly”; f5‘éi Til/'44 /+ 3%;5 ”3+” 15¢,KE‘: 55,44 “¢,-%:’§l(¢ixz 7r‘A ° K" = fizzy/1 7r‘A°K° = ¢%,~3/2 7r‘A°/<* = (7:57 924/, fig 94w, 7r°/I°/<+ = E 992/4 +f§ 94m WW/(o = ——\/3I Sig/2,72 + E 42/42 77§4°k0 :. l/E: 9%Q'Z: "VF§F 96$-q; Table 7. Weights F2 for 7f.+/> 9K+TT+ Yreactions. Reaction 7r'+}>——> (377’. 2° 771/” -—-> K: 16+ Z— TT—+/° ak++71"+/]° 7r’+/> «—> M? ”12* 71'+/° ——> k°+7r*+ 2' ”if/5 —>/(°+ F°+/1° 77% —>/<°+7r°+z° 26 27 where the 1?.5 are the center of mass momenta of each of the N particles, K is a constant which we need not specify since we are only interested in relative probabilities, and the two delta functions are there for con- servation of energy and momentum. Equation 49 is just an extension of equation l to include a group of final states. The momentum delta function can be taken care of by just eliminating one of the integrations over the particle momentum. we then have 5 = “far/2 SKE'Efl 45'” J/i/ (50) where the if: are now independent. Applying equation 50 to our problem, let us take the pion and the kaon to be our independent particles. With the approximation that the center of mass is on the hyperon, we can write I; e /é7r’K//Zs}I/25/E'Evf) 2224; 411‘”; (51) Changing variables to total energy E, 43 = ”4”ka /E,*'/ZS(E—;)zZ(j-,§)fi 722427; d5 <52) From equation 23, we have @QK : 3% (53) I; : /67r1K / /7,;,-//2 02200. aid/72 (54) Of interest is the energy spectrum 3? of the kaon as it should be K affected by the strong ”LY interaction. (”B/dz; = KW? Mir/2 2 £47 7,:a I (55) = Wk pal/2 [69- mer- ":7 fir-MM ' my“) 28 where 24* = 524772;. The spectrum is shown on Figure 5 for the reaction I + — 6 fl' 4' f7 ——> AK 4- 77 '* 2: at a center of mass total energy of 16 pion masses. The curve does not seem to differ appreciably from the pure statistical spectrum. To calculate the branching ratio, let us refer back to equation 52. Rewriting it as K? /6 W f/l'w 249%) 2241285), 0/? 4’5 (56) _ P z 2 - M w a a; allv K ._ 2‘) _- Q9 é) - fi = A LUZ- My 3 K7? (57) °° rum - 2 2 T g - /(77/:/ /z,;.),,/ w; (£1.29-my)\/@$z_mf2){€'”§r'7’i%fi)é - "MM/(”Z1 a???) 77771 where Wd max is given by ’/z w [52.— {Mflmkemd ijéz- {om—727,; 7277,)? _. (59) _. + 2 3"“). 452 7% Table 8 shows the relative probabilities for various reactions as well as the comparison with a purely statistical model. Table 8 indicates that the final state does alter the branching ratio, although not appre- ciably. In comparison with the statistical model, it tends to enhance - + —- the Kon'ozo, [5310/1' , K017 if, and K 77 2° reactions, while suppressing - + o - the [off and K 7T2 reactions. Unfortunately, the reactions K9r°z° and K'rrvo which are enhanced the most are the most difficult to deter- mine experimentally. Table 8 also indicates a weak energy dependence 0. .13.?- ” I.“ A‘ 29 FmL Side M09151 / Cousr, x a’v' A153 4; ()1 o I l I l 4.5' 5-0 535’ 6.0 65 4:0 20K Figure 5. Momentum spectrum for the Kgr'zoreaction at E=l61f7,, using the k extension. 30 Table 8. Branching ratio assuming global symmetry and a k extension. Process (47F: ° (+77 °Z' K8171" K°1rZ+ K °71*Z _ k 0 ”0/1 a K0n°Z° Statistical Model 1.2 1.2 1.8 1.5 1.5 1.4 FWNAI E=15 1.6 SfafE fibafil E=16 E=17 E=18 1.6 1.6 1.5 1.1 1.0 1.0 1.6 1.5 1.5 1.7 1.7 1.7 2.1 1.9 1.9 2.1 1.9 1.9 31 for'the branching ratio. Figure 6 shows how the probabilities vary with the total center of mass energy E. D. Extension of the t-Matrix using the final momentum q. Let us now go to the other extreme of extending the definition of the t-matrix to off the energy shell values using the final momentum q. Equations 26, 27b, 27c, and 28 are identical except that the q's and k's are interchanged. Following the same procedure as before, we have for [IF/‘4 2- F232 -66) 6102-l a /7/;/>”/ ' 5352// ”32% ’V aw, fig—4%ffifé (60) The integral 1 can be solved simply by making a change of variables W = Wq - Wk. Then pové I z .. g (‘ng‘/)—2W;A0+cuz ”z" w -I :— --(5wi+/7.5') + (44324)]? '6 (rad — (Quiet/7,3") +(k/g’=—/)[jn /%)] Ekquation 60 can now be written as / 7/9/51 2 22. (61a) (61b) (61c) F? /-' 26m5(:_{3_/__121%' +(%2I%5)2 (62) 3ng 32 (32% my 30 r- [693* 6.30 F 25’ — L' : 5: A7720 20 _ + 2, K72. A42" CLIL I5 - X N M 3 K) /O __ 5 _ 04 ~ ' ' 1 ' / /5— /6 /7 /8 Figuir‘e 6. Variation of probability with the total center of mass energy. 33 Using equations 55 and 58, we can similarly calculate the energy spectrum of the kaon and the branching ratio. The results for E = 16777, are shown in Figure 7 and Table 9. In each case, a comparison with the statistical model is again made. The momentum spectrum now appears sharper while the branching ratio appears similar to the previous case except for the (0713* reaction. E. On the energy shell scattering Let us now restrict ourselves to final state scatterings that are only on the energy shell. In other words, (z/t/A_>=O if /§/?€ [IE/ . This has the effect of removing the principal part of the integral in say equation M. We have left only the -;_ ”SQ—hi) part of the integration. “t —_ ‘1‘ 1 ' ‘ja/L/ 7m ”The *%f “if a” WWW (63) This reduces simply to E" : Fj'i [if]; .4. t: M/3/3_ 6(8) (64) “’2 /7F}'/ ‘ 52f "”4/JZ‘WS'” <69 The results are shown on Figure 8 and Table 10 for E = 16771”. The momentum spectrum is almost identical to that of the statistical model while the branching ratio is now changed considerably. More will be said about these on the energy shell calculations in section V_F. A l \ CONST- x ole/ex; (N l \ 40 Figure 7. / $7lA7lIS/VCAI.\ \ 31L / / \ / \ PM 3m. \’ \ l l l 4.5 5.0 55' 6-0 wK Momentum Spectrum for the Kim-2° reaction at E = 16M, using the q extension. 6.5' I'm—- 5W If 1 ‘1' - won- - ii *0 Table 9. Branching ratio assuming for E 2 16th Reaction «W K on"? global symmetry and a q extension 1.3 1.5 1.15 1.0 1.3 2.0 2.0 F.S. = Final state S.M. — Statistical model 1.8 1.5 1.5 Lu 3O — 25— / / ’— \ \ / / \ 20— / 5”“ 5757‘s \\ 3150;13me / \> \\1 / \ / \\ 03 / \ Q / \ V [5'1— ‘D / \ Z \ S \ 8 \ \ e l / 5'— \ \ \ \ \ 4'0 45‘ 5.0 515' so 6 Cdk FiguI‘e 8. Momentum Spectrum for the K1‘7’Zoreaction at E = 16777;; neglecting off the energy shell contributions. 56' Table 10. Branching ratio at E = 16 my assuming global symmetry and 37 neglecting off the energy shell contributions. Reaction Kir’z" k°7r‘Z+ MHZ- Kano/‘0 /< °7T°Z o F.S. 2.1 1.0 1.7 3.5 1.9 2.1 2.1 Final state Statistical model S.M. 1.2 1.2 1.8 1.5 1.5 1.4 38 F. Isobar Model As a final consideration, let us consider the isobar model. The reaction can now be considered as 77 fl—r p ——> Y "‘ K From equation 22, we can write 34% —. "37:70:; ((4711) NEW (66) where Wk is the total energy of the kaon. From two body kinematics, we can easily Show that (67) d‘ ._ — ’7'? E m E 35" — W I) H l I)??? (68) F(E)mx)) 7418 two AOL/y FAA“. SfAca {inc-for. CAN [>8 written as31 PM = 6?:4/5—‘2Wmkflz-éfTI)‘(Eimf-7"E)(E‘-new?) (69) using equation 67, this can be simplifed to Z Name) = 30.? (E'wk) Vwkz'm <70) 2 . Since Virgis proportional to /Z;/J , we can substitute the V33 values for pion-nucleon scattering in place of 4M1) . Equation 68 can now be written as 06/ : CdNS‘} W?) wk (E‘ldk) “MAB; Mk2. /‘ w (71) K VEZ'ZEWK +714} 39 The results are plotted in Figure 9 for E = 167%. It is much sharper than the final state calculations. However, care must be exercised when using the momentum spectrum from an isobar model. Recent experi- ments32 indicate that the 1r decaying from the 1“ state can interact with the K such that its momentum spectrum will be affected. 4O CONST x ewe/w. gt :5 U1 0‘ O 3‘, 035 20/: Figure 9. Momentum Spectrum at E = 1611]" using the isobar model. 41 V. The TI-K Interaction A. Kinematics. We Shall now neglect the 7f-Y and the XX interaction and consider only the 7T—K interaction in the final state. The kinematics will now become more complicated since we can no longer make the approximation that the center of mass is on the hyperon. Borrowing from the isobar model, let us consider the K-7T system as a particle X. The equiva- lent mass of this particle X can then be written as - / Z 7 / 2 z Mx ' 83K + 777k '7‘ fik + mzr (72) where K is the center of mass momentum of the kaon (or the pion) in the two body 7T-K system. The total energy in the center of mass system for the three bodies can then be written as where l? is the center of mass momentum of the hyperon. Since we are studying the K—77 interaction, it is necessary to eXpress our hamilton- ian in terms of the relative K-7r momentum. In order to do this, let us first consider only the K-TT system and make the approximation that the kaoncentric system is an inertial system with respect to the system X. We can then make a Lorentz transformation from the kaocentric system to the system X. N l K X Figure 10. Co-ordinate systems K and X. 42 8,, X 1‘53 507 4.5x?! _ 'A‘Xfl Pf "El-KIT (74> fin e " BK : X(6’II - 5V€7rz+ 77771) (75) I if... )r : -———‘—" anal = ”‘— Wh _ z (76) ere / (3 f3 /¢;47%; -+ 74k equation 75 can now be written as [D _ ‘ 777k arr ' ( ) XK / 17W?- 4' 771:" i" 2 mk/fik "’7'711' 77 It is also useful to know the inverse relation of 6m in terms of 5/. . The inverse Lorentz transformation will give us - _, 1 2. 1:.- M a+e.ft+m) (78) It is more convenient here to expresslg'd-Z'in terms of quantities in the X system instead of equation 76. : __ wk __. — XK [8 W1 (79) 777 M ' 5“ V5k2+7’7/:[/ + /&kz+m; (80) 777K go H Z 1 xx “flk Our hamiltonian can now be expressed in terms of fin, the relative K_7]’ momentum. 43 B. The T—Matrix. Proceeding as before where ‘3 and -fi are now the relative K4” . . . . . _, f/E final and initial momenta, we can reduce the matrix @/ by charge independence to _l _. —‘ A) -‘ ’9 (WW ; vééQ/QA/A +m4hK2/ti/ (81) . . l9 . . Since experimental data to date seem to indicate an I = 1/2 resonance for the K—TT system, we shall neglect the I = 3/2 state. A table of the W s are shown in Table 11. The K—TT isospin wave functions are identical to the ones shown in Table 5 with the K+ in place of P and 77 replaced by K°. Effective range formulae were then used to calcu- late (filth M) for both 8 and P—wave resonances. The scattering length 2 formulas used were km‘ 25 = 9k; — 3’5 79* SWMS (82) Z kscofsf = 5': — k ice» P—WVES r 7-— P (83) The parameters 05) f3) Q15) and '7’ were determined by fitting them to the experimentally observed total energy (in the L” system) of 878 mev and the full width at half maximum of 23 mev. Table 12 lists the 25 values that were used. The scattering amplitude can be written as fifiifi) - 9E3 700* $va (84) 6 (2:79) 366) 2*? a We g k2 (85) - | I2 where G6) : 0&8 MS + I' MS (86) Table 11. Weights W for K—7T reactions Reaction Kit-fi- ;’ Kt» 7T— + _ K+7r g." /<°+7r° an w1/2 2m 42/3 1/3 1/3 2/3 3/2 1/3 +J2‘/3 2/3 15/3 2/3 1/3 45 Table 12. Parameters Q and I" for the K—‘IT interaction. Angular Momentum State Q r s 0.1u2 3.65 x 10'2 P 0.031 0.788 0— 0M /‘ 15V Umis 07° W/WC 1&6 As before, equations 82, 83, 84, 85, and 86 are only defined on the energy shell. In sections D and B, we shall extend the definition using the initial momentum k and the geometric mean momentum \/k . Fermi's statistical model will again be used to determine the primary interaction matrices, {/1777 7/5) . The matrices are (F/r/F’) = F (F/T/F’) : F1? for S waves (87) for P waves 1 .3 where A is again an unspecified unit vector which we will average over when necessary in the spirit of the statistical model. I C. Phase Space Calculations. Let us write equation 50 as 2 III. —I g .: Kf/ZDTPH/ 5(5'55) 0%: (ll/J (88) V -' a where f: and 5 are the center of mass momenta of the kaon and hyperon. Expanding further, we have 2 2 z: = de/lyl/ «sec/@545 o’Qr/fdl; (89) Let us now assume that f; is spherically symmetric and fix our 5, coordinate axis along the direction of I? . Then - 8M [/7 aggres- Pic/P ac 9 P2000 5-" J W/ 'Jr)’ 05kkk (90) ‘1 \l where 6K is the angle between I; and I; -ri 47 Changing variables to the total energy E, 5:: 877‘kf/ 7:.-,/ alas—e); axe (924695. fife/P a5 PP =87r Kf/W ZCJL-Cgtael’ {'0’}; dF (91) In; Using conservation of energy and momentum relations, we get 3059K = E " wk " 375’ P (92) a}; K Pr where Wk and NY are the total energies of the kaon and hyperon. The momentum spectrum of the hyperon from equations 91 and 92 is “1w. Jog? : (af/O Wf/Ff’lzé’w)’, ”Owl: did“ 367,1/a‘075751‘tfi 7 (”law where Wk max and Wk min are given by31 We = 575%) 1/ E"£-«+>‘—-[( -wr)‘-¢i7[e4$ MW] MIN 2 [(F-af- 27‘] (9n) 2 [(EZ-WV)L+ mkl " 7’7”: "" I; J The final probability is then given by Wow: [E = BUZK 3(‘5 WV) My UIWY (95) Mr ,, '/z where _ [52' «fimk‘m’hzflk' (mi-”’1'- 74,):7 2 307m" ‘ ¢El + My (96) 48 D. Extension of the t—Matrix using the initial momentum. An extension of the definition of the t—matrix using the initial momentum k will now be made. By procedures similar to those of section IV we obtain for the square of the matrix element /T ,/2— = Com?‘ F“Z /— W'/zI a [/ZI‘L FF T R + 7 (97) where WC 5) J}, = IFS»! [044% of i. cfldk 7” 5- E ”+4'é (98) ,r 1;! = [mAG/mpy [amt of we 6Y§j a’ldk 1535””, 1,6 (99) Mr E" refers to the total energy in the intermediate state (corresponding to momentum k) and WC is a suitable cut off (We = 20 Mfr). IR and Ii for S and P wave resonances at E = 16711" are shown in Figures 11 and 12. The momentum spectra of the hyperon are shown on Figures 13 and 14 for the reaction 77f —-—> /<°+ 7f++ Z- . Table 13 gives the branching ratio. As can be seen from Figures ll and 12, the functions IR are different for S and P—wave resonances. However, the Ii curves are quite similar. Since the cross sections depend primarily on Ii, our momentum spectra and branching ratios are almost the same. Thus the above method will not differentiate between S and P—wave resonances. However, there now appears to be a considerable difference when the final state results are compared with the results for a statistical model. Figure 11. | l l l 2 3 1 ,5 Ida, Variation of I with W' at E = léflfltfor a S—wave K—fl' interaction using a k extension. '/6 I l / 2 3 4 5' a) 9' Figure 12. Variation of I with W at E = 1674, for a P-wave K—IT interaction using a k extension. mL 51 25' t— 20- a, l GOA/s7: x rid/d4)y 6 l o 1 l I l g 85 8.7 8-9 9.! 9.3 9. 5' 20,, Figure 13. Momentum spectrum at E = 1611, for the Kofifil’reaction for a S—wave K—‘fl' interaction using a k extension. 52 I4» IO " O) l CONSZ x ctr/dad, O\ I O l L J l l 85’ 8-7 8"? 9/ 9.3 96" ‘07 Figure 14. Momentum specteum at E = 16m" for the KWZ reaction for a P-wave K-IT interaction using a k extension. 53 Table 13. Branching ratio at E = 161$ for the K—fl' final state interaction using a q extension. Reaction s.'M.' F.'s.-s F.'s.-1> Kfir’z" 1.2 2.5 2.8 o . Knoz" 1.0 2.3 2.4 * - . 1? vi”:— 1.2 1.9 2.1 o 4 — . k1r Z 1.5 2A 2.8 o - f - /< 71' 2: 1.5 1.0 1.0 S.M = Statistical model F.S.-S = Final state S—wave F.S.—P = Final state P—wave For example, the statistical model gives a K0fl*Z-/K°Ir'zf ratio of one, while the final state P—wave case gives 2.8 and the final state S—wave case gives 2.“. E. Extension of the t—Matrix using the geometric mean momentum. By using the geometric mean value of the momentum V85 instead of the initial value k, we get I = REAZ/Dflzftfjlfifl "hr K H dcdk (loo) E'E + 4'6 WC. I = ImAG/c-VAR)’ Fflkf 0/" 4&5)— . Jack E- E”+4'g- (101) 1.0/HERE A = F . wk ; V ”‘3"? (102) The phase shift 3 is now given by /— _ r _ J. _ K7? “*5: ‘ '25‘ W“ as 7P” S “m“ (103) 3/2 I kg [(7 ) COfS : ——3 - __ 7Q}. flit/IVES ( ? P 0f I} (101+) instead of by equations 82 and 83. The IR and Ii curves at E = 1677), are shown on Figures 15 and 16. There now appears to be a considerable difference between the S and P—wave curves for both I R and 11' Figures 17 and 18 give the momentum spectra and Table 14 the branching ratio. The peak of the S-wave spectrum seems to be shifted considerably from the peak expected from the purely statistical model. In the P—wave case, l I l l /'O /-5' 2.0 2-5' 3.0 302 Figure 5. Variation of I with W at E = 160), for a S-wave K-ff inter- action using a J}; extgnsion. I l 1 l J / 2 3 ‘7- 5’ 6 Figure 16. Variation of I with W at E = 167,711 for a P-wave Kw interaction using a 3' extension. 57 26'— L71 I 0mm” x 06709.0}, 5 I O — l I 8.7 5.9 9.! 93 9-5' Figure 17. Momentum spectrum at E = 161), for the k'VZ' reaction for a S—wave K-IT interaction using a J}? extension. eat 58 4. 3 3X 3 ‘0 x k g 2 / O l l l l l 8.5 8- 7 8.9 9- / 9,3 9. 5" 2L)” 4 .— Figure 18. Momentum spectrum at E = 16:71,, for the KOTT 2 reaction for a P—wave K—Tr interaction using aflig. extension. 59 Table 14. Branching ratio at E = léfihrfor the K—fl' final state inter- action using a‘/F} extension. Reaction Statistical Model F.S. S—waves ,t _ /< 77 2° 1.2 13.4 o O O k 7’ 5 1.0 6.6 1‘ o ' k 7T Z 1.2 11.3 4 _ k°77 z 1.5 19.2 a - + K 77 Z 1.5 1.0 F.S. = Final state F .‘S . P—wave s 5.8 4.2 3.2 5-5 60 smaller resonances caused by interference terms in the T-matrix appear. The branching ratio shows significant differences between the 4 _ S and P—wave cases, the main ones involving the K 7720 and KOTroZ'O - ”#7“ ‘ . reaction products. Unfortunately, the ratio Z If”): which is the easiest to observe experimentally is around 1.7 for both the S and P- wave case. p F. On the energy shell scattering. ‘ Taking fit]: ,'we have i 2 Z i; W =F- Uga- a). + ma t where 7-. :: SW25 " Q (106) I}; : (MSW/5Q and Q [“4” *F" d : .—-—- 1 z MK m 53+qu x (107) "I 2 W247 _ Z Mk3 ((03% ”712) (7742+ 292,} +2 77W) 642+ m; +2 "7.”; The momentum Spectrum for the hyperon at E = 167’)” and a S-wave resonance X is shown in Figure 19-for the reaction 773/0 -'> k°+ rr" + 2:- . The final state interaction now seems to cause a suppression in the spectrum. This comes about for values of Mli>0 . The momentum spectrum for the reaction 77"4- ’D -—-—> K; + 17" + 20 has values of Wk<0. This spectrum has the usual sharp peak (see Figure 20). Covsr A (fir/did, l5- 0_ 86' 61 I l l I I 8.7 8.9 9d 93 7.5 z0r o + - Figure 19. Momentum spectrum at E = 167’}, for the K 77 Z reaction for a S-wave K—TT interaction neglecting off the energy shell contributions. 62 GOA/$73 x (If/dad), A i 01 l o I I I I I 56' 8-7 8»? 9. I 9:3 $5" ”V I - a Figure 20. Momentum spectrum at E = 167m for the K ” 2 reaction for a S-wave K—fl' interaction neglecting off the energr shell contributions . 63 In general, one can write the T matrix square as - J I z _ _2 /‘ _ W 2 1M 72 )Tl—JF/ — 7— 4/ “FAQ/N73] +[_Tr‘/l(”p~'-7r%2// (108) where JR and )j; refer to the real and imaginary parts of the off the energy shell contributions and i and Z‘ represent the real and imaginary parts of the on the energy shell contributions. By neglecting JR and k' , we have just equation 105. Of the two remaining terms, 7;: is the dominant one. By observing the first term in equation 105, we can easily see why the sign of W1 /2 is so important in determinating the shape of the momentum spectrum. Of the two terms neglected, )1; is the dominant one. Furthermore, 14);; . Thus the second term in equation 108 is the dominant one. Since the second term does not depend on the sign of wl/Z’ it will always give a positive contribution. The branching ratio is listed in Table 15. The main effect seems to be the suppression of KOTT-Z+ while enhancing the (”#020 reaction. There appears to be no noticeable difference between the S and P-wave cases. 0. Isobar Model. As a final consideration, we shall calculate the branching ratio — if using the isobar model. The reaction will now be 7"? —7 K + Z where K* is an isospin doublet and Z an isospin triplet. Table 15. Branching ratio at E = 16PM for the K-fl' final state interaction neglecting off the energy shell contributions. Reaction S.M. F.S.-S F.S.—P k‘n'z‘ 1.2 1.6 1.7 /<°n°z° 1.0 1.7 1.8 "‘ ° ’ l 2 1'6 1 5 [(11 Z 0 o o Hut. 1.5 1.6 1.5 KOTT'Zf 1.5 1.0 1.0 S.M. Statistical model F.S.-S F.S.-P Final state S—wave Final state P—wave 65 Table 16. K* and Z quantum numbers {If *0 + O '- Particle K K Z Z Z IZ +l/2 -1/2 +1 0 -1 The pertinent charge wave functions in terms of isospin wave functions are listed in Table 17. The only non—zero matrix elements are the following (77> MW?) e I/3 a, + 2/3 RV, (109) (m was» : «a e. — I27. 0. where R3/2 and Rl/2 are the I=3/2 and 1/2 matrix elements. The isobar model only tells us about the relative weights of the various states I I in our system. It tells us nothing about the matrix elements formed (R3/2 and Bil/2). To evaluate these, we shall go to Fermi's statistical theory whereby we neglect the cross terms and equate [53%] to /El/z/. Using the above approach, the ratio Kit/3°50 becomes 5/4. Let us now allow the K* s to decay. + o /<*'+ /, k' + H' -¥o - /< ——>/’<*+Tr \K°+TI’° The wave functions for K* in terms of K and 77' are K*+ : f; WWK+ +'J%-WJKO Kyo : vflé; ”riff -V[gr'zrfif° The branching ratios for the K* decays are thus , K¥+ W0K+/TT¢—Ko = [/2 (110) for 2 [(+00 W-k4/n-0K0 Table 17. K*.. 2 "’P *+ + K2: ¥+- K 2: k"? k3" K+IT+I’ + - I .. give a K07! Z/Kéroz ratio of approximately two and essentially no Kfihii+ reactions. This is in good agreement with our final state 4. - _ calculations for both 8 and P-wave resonances which give a FIFE/((502 ratio of 1.7 and a small amount of KOIT‘Z" reactions (see Table 14) .8 71 The isobar model from Table 18 also is in good agreement with the experi- ment, predicting a K°1T+Z-/k+7T°z_ ratio of 2.0 and no leaf-2+ reactions. To see if the K* meson corresponds to the K' particle conjectured by Gell-Mann,15 we must determine the spin. From Table XIII, within the validity of our approximations, we see that knowledge of the branching ratios involving the K TT-Zo or Kdfi'oZ-o reactions can help us determine whether the resonance is in the S or P state (in terms of spin, spin = O or 1). However, these reactions are the most difficult ones to determine experimentally, and no results have been obtained on them as yet. Since the square of the T_matrix (equation 97) is of the same form for both S and P—wave resonances and the arbitrary parameters were fitted to the same experimentally observed values of full width at half maximum and equivalent mass of the K* particle, it is perhaps difficult to see at first why our final results should be so different for the S and P—wave resonances. However, the phase shift dependence on momentum is quite different for the two cases (equations 103 and 104). This can lead to entirely different IR and Ii functions for S and P—waves and hence different T—matrices and branching ratios. we can conclude that with the information known so far, the reaction #9? -’> K-f 1T+ Y at a total center of mass energy of 16 "Hr can be satisfactorily explained in terms of a strong I =‘1/2 K—TT resonance with an equivalent mass of approximately 880 mev. VII. 1. 2. 3. 4. 10. ll. 12. 13. l4. l5. l6. 17. 18. 19. 20. 72 Bibliography 0 c Franzinetti and G. 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