coo-1112-9 THE UNIVERSITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Physics Technical Report AN EXPERIMENTAL DETERMINATION OF THE K~ CHARGED TO S NEUTRAL BRANCHING RATIO AND JAII > 1/2 TRANSITIONS Jorge Guillermo Morfin ORA Project 04938 under contract with: U. S. ATOMIC ENERGY COMMISSION CHICAGO OPERATIONS OFFICE CONTRACT NO. AT(11-1)-1112 ARGONNE, ILLINOIS administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR May 1970

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1970.

TABLE OF CONTENTS Page ACKNOWLEDGMENTS vi LIST OF TABLES viii LIST OF FIGURES ix ABSTRACT xii CHAPTER I. PHYSICS OF THE Ks BRANCHING RATIO 1 S 1. Physical Interest in the Ks Branching Ratio 1 2. Isospin Analysis of K -' 2w Decays 8 3. Electromagnetic Corrections to the K~ S Branching Ratio 17 II. EXPERIMENTAL METHOD 19 1. Previous Experimental Values of R = r((s r(K ) 19 1.1 Eisler et al. 20 1.2 Crawford et al. 20 1.3 Baglin et al. 21 1.4 Brown et al. (1961) 22 1.5 Anderson et al. 24 1.6 Brown et al. (1963) 24 1.7 Chretien et al. 25 1.8 Summary of Older Generation Experiments 29 1.9 Gobbi et al. 33 2. General Method of the Present Experiment 34 III. THE EXPERIMENT AND REDUCTION OF THE DATA 38 1. The Beam and the Chamber 38 2. Scanning 42 ii

TABLE OF CONTENTS (Continued) Page CHAPTER 3. Editing 46 3.1 The K+ Sample 46 3.2 Editing of the n7 Sample 52 4. Measuring 53 4.1 Geometric Reconstruction 53 4.2 Kinematic Fitting 54 IV. CHECKS AND CORRECTIONS OF THE RAW SAMPLE 56 1. Monte Carlo Technique 56 2. The Charged Decays 58 2.1 Verification 58 2.2 Corrections to the Charged Sample 64 3. The Neutral Decays 70 4. The K~ Sample 82 V. RESULTS AND CONCLUSIONS 85 1. The K0 Charged to Neutral Branching Ratio 85 s 1.1 wr s-wave Phase Shifts 86 1.2 The Quantity Re(A2/A ) and AI > 1/2 Transitions 91 1.3 Ratio of AI = 5/2 to AI = 3/2 Transitions 92 2. Ratio of K0 to K0 Production 95 L s 3. Summary of Results: Conclusions 98 3.1 Branching Ratio and Associated Quantities 98 3.2 The Coherent Regeneration Phase, 0SL, in CF3Br 103 iii

TABLE OF CONTENTS (Continued) Page APPENDIX A. CHARGE EXCHANGE SCAN RULES 110 B. EDIT RULES 120 1. First Edit Rules 120 1.1 Type 1 Events 121 1.2 Type 2 Events 122 2. Second Edit Rules 123 2.1 With V 125 2.2 With y 127 2.3 With V and y 129 C. PHENOMENOLOGY OF K~ - 21r DECAY 131 1. The Decay K~ - 2r 131 S 1.1 The Decay Matrix 154 1.2 The Mass Matrix 136 1.3 The States lKS> and IKL>: Transition Matrix Elements 137 1.4 The Validity of the Abbud-Lee-Yang Approximation 143 2. Regeneration Phenomena 146 2.1 Coherent Regeneration 146 2.2 Incoherent Regeneration 149 35. K-K0 Interference 153 s L' D. wrr PHASE SHIFTS 157 1. Theory 158 1.1 Partial Wave Analysis of 7TT Elastic Interactions 158 1.2 Chew-Low Extrapolation: One-Pion Exchange 162 iv

TABLE OF CONTENTS (Concluded) Page APPENDIX 1.3 Durr-Pilkuhn and Benecke-Durr Parameterization 164 1.4 Model Dependent Predictions 165 1.4a Current Algebra Techniques 165 1.4b Veneziano Partial Wave Technique 166 2. Experimental Techniques 168 2.1 6~ and 62 - E. Malamud and O O P. Schlein 168.2 6~ - Hagopian et al. 171 2.3 6~ - Scharenguivel et al. 173 2.4 62 - Morse et al. 174 2.5 6' - Baton and Laurens 175 0 REFERENCES 178 V

ACKNOWLEDGMENTS I want to sincerely thank the following individuals for their assistance in this experiment and subsequent analysis. Prof. Daniel Sinclair for serving as chairman of my committee and making this thesis not only an educational but a truly enjoyable experience as well, Profs. John C. Vander Velde, Oliver Overseth, Robert Lewis, and Richard Teske for serving on my committee and willingly confronting the many questions that arose, Assoc. Prof. C. Thornton Murphy for giving me encouragement when I needed it most, Prof. Byron P. Roe and Dr. Jeremy Lys for their many helpful points and clarifications of the theory involved, Mr. Kenneth Bartley for his time and effort spent in the editing stages of the experiment, My fellow graduate students, both past and present, for their helpful discussions, in particular Charles L. Arnold, T. Michael Church, John Davidson, Anthony Fisher, Walter Gibbs, Robert Green, Hsien-Chih Hsiung, Richard Kiang, John Koschik, Lawrence Lovell, and Harvey Ring, The scanning and measuring personnel for processing the film through the earliest and most tedious stages of analysis. My special thanks to "the girls", Vivian French, Marianne Rauer, and Marietta Ross for not only helping in the scanning but also assisting in the necessary bookwork. Thanks also to the many part-time scanners and measurers, particularly vi

Mark Bendure, Jerry Malenfant, Italo Russo, and Fouad Saleh for both scanning and measuring, and Donna Buckley, Janet Gasior, and Karen Herholtz for their scanning and fantastic mini-skirts, Louise Peterson for programming assistance, John Allen and Nicholas Furkioti for keeping the scanning and measuring machines in working order, Carlee Murray for drafting most of the figures in this dissertation, Most significantly, my sincerest gratitude to my wife, Susie, for her unfailing patience and support throughout my thesis effort including the typing of this dissertation. vii

LIST OF TABLES Table Page I. Status of the AI = 1/2 Rule 2 II. AI = 1/2 Rule for K -+ 3w Decay 7 III. Detailed Event Breakdown After Second Edit 49 IV. Tabulation of Errors Found by Re-edit of 20% Sample of K+ Film 50 V. Disposition of Events in the K+ and 7r Film After Second Edit 52 VI. Corrections to the Ks -4 rr Sample 64 s VII. Corrections to the Ko - 70r~7r Sample 70 VIII. Detection of y Conversions 76 IX. Corrections to the K0 Sample 83 X. Reported Values of 60 and 62 88 0 0 XI. Summary of Total Ks and KL Production 96 S L XII. Re(A2/A ) and 5/2 for Three Values of R 96 viii

LIST OF FIGURES Figure Page 1. Graphical representation of the triangular relations for the Z decay. 4 2. Graphical summary of all reported values for the K~ charged to neutral branching s ratio. 32 3. The vertical and horizontal effects of each componet in the 28~ beam line when operating in Mode A. 39 4. Comparison of the cross sections for K+ + n - Ko + p — the primary interaction — and K+ + n - K~ + w~ + p, a possible source of false y events. 41 5. Distribution of the measured effective mass of the TT system. 59 6. Proper time distribution of measured sample K~ - + + T-. 61 7. Distribution of lab momentum for KO's produced via K+ charge-exchange reaction. 62 8. Distribution of separation between chargeexchange (C.E.) point and V-vertex as observed and as predicted via Monte Carlo methods. 63 9. Characteristics of "scattered-V" sample. 66 10. Separation between C.E. point and conversion point (nearest e e pair) as observed and as predicted by unweighted Monte Carlo. 72 11. Predicted y-ray dip and three hypotheses used to determine detection efficiency. 74 ix

LIST OF FIGURES (Continued) Figure Page 12. Predicted y-ray momentum and weighting 75 factor for three hypotheses. 13. Separation of C.E. point and nearest e e pair as observed, as predicted by unweighted Monte Carlo, and as predicted using Hypothesis B. 78 14. Separation of C.E. point and neutral decay point as observed and as predicted by Monte Carlo technique. 81 15. Dependence of Re(A2/A ) on value of 162 - 6 % using our value of R ~AR 2.117 ~0.064. 93 16. Dependence of ( —) on value of 62 - a3/2 I 2 ~1 using our value of R ~AR = 2.117 ~0.064. 94 17. The necessary accuracy of the quantity R to determine R - 2 to 1.0, 10.0%, and 25.0. 100 18. The behavior of the percentage error in Re(A2/A ) as a function of the percentage error of R - 2 and' cos (62 - 6 ). 102 19. A typical 0-prong Type 1 event with subsequent K0 - 2w7T0 decay. 112 s 20. A one prong Type 1 event with subsequent K0 -+ Fr r decay. 114 21. A Type 2 event composed of a Ks -4 T ~decay immediately after charge exchange. 115 22. Two Type 1 events within 20 cm (real space) of each other. 126

LIST OF FIGURES (Concluded) Figure Page 23. An illegal y event. 128 24. Standard one-pion exchange diagram for reaction 7T + N N + wr + r. 163 25. S-wave, isoscalar phase shift 6~ as 0 predicted by current algebra (Arnowitt) and Veneziano (Tryon) techniques. 169 26. Experimental determination of 6~ as a 0 function of M by Scharenguivel et al. and by Hagopian et al. 176 27. Experimental value of the I = 2 s-wave phase shift 62 as a function of M by Baton and Laurens and Morse et al. 177 xi

ABSTRACT AN EXPERIMENTAL DETERMINATION OF THE K~ CHARGED TO S NEUTRAL BRANCHING RATIO AND AII > 1/2 TRANSITIONS by Jorge Guillermo Morfin Chairman: Daniel Sinclair We have exposed the Michigan-ANL 40-inch heavy liquid bubble chamber filled with heavy Freon, CFIBr, to a separated K beam at the ZGS complex of Argonne National Laboratory. Approximately 150,000 pictures were taken at a transport momentum of 825 MeV/c. An auxiliary exposure of 20,000 pictures using a w beam of the same transport momentum was also obtained for determination of w+ contamination of the K beam. The resulting K+ sample was scanned for the charge exchange reaction K+ + n 4 K~ + p. Through subsequent edits those K~'s produced by the above reaction and decaying via the short-lived mode, K0, were categorized as to whether s their decay products were two charged pions (w + w-) or two neutral pions (V~ + ~o). The raw sample of 7500 K + 7 decays was then subjected to four corrections while the raw sample of 4280 Ks - W~ + 70 decays underwent six corrections. The corrected charged and neutral samples contained 7736 ~99 and 3654 ~100 events, respectively. With the resulting K0 charged to neutral branching ratio xii

of 2.117 ~0.064 we have investigated the difference in ww phase shifts 162 - 6~ where 62 and 60 are the s-wave isotensor 0 0. 0 0 and isoscalar phase shifts, respectively. Assuming no AI = 5/2 transitions we find 162 - 6 = (72 +We have -l o -11)' We have also investigated the relative magnitudes of I = 2 and I = 0 states in the composition of the 2w final state and find Re(A2/A ) = 0.022 ~0.014. Using the results of this last investigation we have examined the relative magnitudes of aI = 5/2 and AI = 3/2 transitions in the K~ - 2w decay. We find this ratio to be a/2/a3/2 = -0.391 ~0.510 or -2.18 ~2.85. As a check on our experimental method, a careful tabulation of all K~ production was kept. After five corrections were applied to the raw sample the total KL production was compared to the total Ks production and an estimate of the s phase of the amplitude for coherent regeneration in Freon, (-40 ~25)0, has been obtained. In the appendices we examine the phenomenology of K~ o 2w decays including regeneration and interference phenomena. We also give a resume of the status regarding 7wT phase shifts. xiii

CHAPTER I PHYSICS OF THE Ks BRANCHING RATIO S 1. PHYSICAL INTEREST IN THE Ks BRANCHING RATIO The function of the tAII = 1/2 selection rule in nonleptonic weak interactions has been of great interest to experimentalists within the field. since its formal introduction by Gell-Mann and Pais in 1954. On the other hand, the theorists' response has been justifiably non-commital since the validity or non-validity of this selection rule would not alter present weak interaction theory to any great extent as 2 long as any violation occurred in a prescribed manner. There are many predictions of the IAI = 1/2 rule that have been repeatedly checked through experimentation. Table I summarizes predictions of the (AI| = 1/2 rule along with the current experimental results; phase space corrections are included where significant. The decay of the Z hyperon has not been included since the usual test of the I AI i = 1/2 rule for the Z family is the closure of a triangle plotted on the s-wave and p-wave axis. By defining the amplitudes + _+ + E+ - - t = + t pTO A 0 2 n7r~ 0

2 TABLE I STATUS OF THE AI = 1/2 RULE AI = 1/2 * Ratio Prediction Experimental Value Kaons + + r(K - w+ro~) 0.0 (1.47 ~.03) x 10-3 r(K - 2r).-.-..-. -...- -2.0 See Chapter II F(KS Tr0 ) K - 3w See Table II See Table II Hyperons..I(_-AY -, P?7r - -).66.644 ~.008 r(A - pw-) + r(A - nw~) r(0 4 all).50.55 ~.o4 r(3" 4 all) r"_~ -+. Sr'. pQ8 events r( ~ 4 aT0~) ~2.0 8~3 events experimental values in ths table are take from Referental values in this table are taken from Reference 7.

3 the |I I = 1/2 prediction is A/1 21 +.0- - = 0 0 + + - The amplitudes 2+ and Z are fairly well known, and the orientation of + and Z- with respect to the s and p axis is also well known, being determined from the asymmetry parameters a, A, and y. However, the parameters of the decay Z+ pro have, at present, an inherent uncertainty reflected by the fact that a has been measured three times with results 0 = -0.80 ~0.163 = 0.986 ~0.074 = -0.88 ~0.055 Also, the sign of 7 is still uncertain necessitating two 0 solutions, both of which are shown in Figure 1. Note that closure of the triangle is a necessary but not sufficient condition for the assumption of AI = 1/2 dominance in Z 6 decay The JAI31 = 1/2 rule is equivalent to the AS = 1 rule since in the relation 2(Q - I3) = S + B (1.1) both baryon number (B) and electric charge (Q) are strictly

P-WAVE Figure 1. Graphical representation of the triangular relations for the Z decay. If AI 1/2 rule holds, the triangle should close.

5 conserved. The simplest way to insure IAI 1 = 1/2 in relation (1.1) is to assume I AII = 1/2. The observation that the IAI I = 1/2 rule is most likely only an approximate selection rule has been hypothesized, based on the presence of the decay mode K+, T+0o. As will be demonstrated shortly, this weak decay requires a tAII > 1/2 transition. Simplistic arguments lead to the result that if purely electromagnetic corrections ++ o allowed K+ V w+w then the ratio of K+ ww i to K~ o 27 s would be..... V- 2 ~) 1 05 R(+) x= r(+ - -+5) 10 F(K -. 2w) (137) 7 Experimentally we find7 (+) 1.7 x l0 1.4 x 10-3 R(). 1. 10 1.2 x 10 so that electromagnetic corrections fail by about two orders of magnitude to explain the abundance of K - w 7~. Thus the question before the experimentalist is not one of the validity of the I AI = 1/2 rule but rather by how much is the AI I = 1/2 rule violated and what is the composition of the JI I > 1/2 component of the weak decay Hamiltonian (i.e., is there also a AI = 5/2 as well as a AI = 3/2 component). There are several ways to experimentally answer these questions, and among the most direct methods of determination

6 are the K -+ 2v and K - 3v decays. By comparing the K - 3v decay rates, the existence of JAII = 3/2 and JAI = 5/2 amplitudes of the decay can be separately determined. However, due to the unreliability of necessary phase space calculations, there is no confidence in tests of jAI = 1/2 rule for K 4 3v decay to much better than 10%. The 3v decays of the K meson have been recently summarized by Aubert 9 and Overseth and Table II summarizes these decays. In Overseth's notation 1(+- =.o) y(+-o) = P(K, www) where p is the phase space factor due either to Trilling1 11 or Devlin as indicated. The results are divided into those ratios which indicate AI = 3/2 transitions or AI > 5/2 transitions. The table shows no clear evidence for any IAIJ > 5/2 amplitude but evidence consistent with a jAI I= 3/2 amplitude is indicated. The overall 10% uncertainty in any IAII = 1/2 test from K - 37w decay can be cut by more than half if one looks at the K - 2w decay. Unfortunately, the only thing that can be determined directly from K 4 2w decay is the existence of jAIJ) 2 3/2 amplitude and any measurement of the relative amount of JAIl = 3/2 and jAI| = 5/2 transitions must be based on knowledge of wT phase shift. However, it was our opinion that improved accuracy in the measurement of ww phase shifts

7 would soon be available, and for this reason we proposed to determine the branching ratio %(Ko +) - r(Ks~ - T r ) R =- -(- - (1.2) r(Ks -_ r~7r~o) in order to directly determine the AI I 3/2 amplitude, and ultimately the ratio of IAI = 3/2 to 1AII = 5/2 amplitudes of the weak interaction transition matrix which governs the K~ 4 2T decay. TABLE II AI = 1/2 RULE FOR K - 35r DECAY Experimental Value Ratio Prediction ~Trilling CDevlin y(~00o) 1.0 if no AI > 5/2 0.935 ~0.06 1.00 ~0.06 3/2 y(+-o) transitions y (++-) 1.0 if no AI > 5/2 1.03 ~0.03 1.00 ~0.03 4y (+ oo) transitions y(~+-o), 1.0 if no AI = 3/2 0.875 ~0.044 0.80 ~0.044 2y(+oo) transitions -y(.. 7( o- 1.0 if no AI = 3/2 0.79 ~0.04 0.79 ~0.04 7'(++-) - y(+oo) transitions

8 2. ISOSPIN ANALYSIS OF K -+ 27 DECAYS 12 The concept of isotopic spin was first introduced in low-energy nuclear physics to describe, phenomenologically, the similarity in nuclear force between any two nucleons. Thus the concept of charge was superfluous when discussing nuclear type forces, though electromagnetic corrections keep this statement from being totally valid. If we now go to isospin space (sometimes called SU(2) space) the w meson multiplet with its three charge states is represented by a single isovector (I = 1) with three projections on the I) axis. These states are (II, I)>), as follows: 11, +1> = ( (1.3a) Il, 0> =- ~ (1.3b)!1, -1> = n-. (1.3c) A similar analysis of the kaon family shows that the four members are divided into two doublets with isospin 1/2. The first doublet is |1/2, 1/2> = Kt (1.4a) 1/2, -1/2> = K~ (1.4b) while the second doublet is 11/2, 1/2> = v (1.5a) 11/2, -1/2> = K-. (l.5b)

9 In this experiment we are primarily interested in the 2w final states of the Ks decay, namely Tlv+ > and 1w~O~>. s Using Clebsch-Gordon coefficients7 we can construct these isospin amplitudes to be 1 - + IT, I3> = 10, 0> = 7 [iT+-> + IVT > I- 7~>] (1.6) l1, 0> =1 [1w - I - V >] (i.7) 12, 0> =:~ o> + I 2 ~~-> + 7-V+> + T >].(1.8) Note that I is always 0 since our final state has net charge 0, and that the 10,0> and 12,0> state vectors are symmetric while the 11,0> state vector is antisymmetric under the interchange of the isotopic spin coordinates of the two w mesons. Since the initial kaon state is that of a spinless boson, conservation of angular momentum restricts the final 2w spatial state to be of an s-wave nature and thus symmetric. Bose-Einstein statistics, which demand symmetry for the total 27w final state wave function, thus restricts the isospin wave function to be also symmetric. Thus the allowed final 2w isospin states are 10,0> and 12,0> while the initial isospin state is isospin 1/2. The AI = 1/2 selection rule says simply that the change in isospin between initial and final states in a weak decay is 1/2. If the weak interaction Hamiltonian governing this decay adheres strictly to this selection

10 rule then only the 10,0> final state is allowed and the branching ratio is (K' - v" r(Ks w ~7r) 1/3 Of course this result will be modified by a phase space factor due to the very slight mass difference between the charged and neutral modes of the final state pions. This turns out to be given by13 m 4m2 ^ -2 0.988 (1.10) /m - 4m! K~ Tr~ K0. 0 so that we may safely disregard this for now. A more dramatic effect will be observed if there is a part of the weak interaction Hamiltonian that allows IAIl = 3/2 or even iAII = 5/2 transitions. In this case the final state will be a mixture of 10,0> and 12,0> isospin amplitudes such that (27> = A j0,0> + A212,0>. (1.11) The relative amount of 10,0> and 12,0> amplitudes can be calculated with the aid of K+ 4 27r isospin analysis. In the decay K - 7r4 + the third component of isospin in the

11 final state will be 1, i.e., a final state of charge +1, so that the only allowed values of I are I = 2 and I = 1. Using Clebsch-Gordon Coefficients we can construct these states as 12,1> =' [IT+o> + 1rw O>] (1.12) and 1,1wwo> = ~>- TTro+> (1.15) where 12,1> is symmetric and 11,1> antisymmetric under the interchange of pairs of pions. Using arguments similar to those employed for the Ko -. 2r decay, we find that the boson nature of the final state again demands symmetry of the total wave function and thus symmetry of the isospin wave function. Thus the allowed final 2r state has isospin 2 while the initial K+ state has isospin 1/2, so that in no way can a AI = 1/2 amplitude contribute to this decay. Since this decay is governed solely by AI > 3/2, if charge independence of nuclear forces is valid and ignoring any possible second order weak interaction effects (AS = 2 transition in K~-~ complex) there is most likely a similar violntion of the IAI = 1/2 rule in K -4 2w decay. s We may now, simply with the use of Clebsch-Gordon coef-. ficients, derive the useful relation first specified by This relation is rigorously derived in Appendix C starting with the modified Weisskopf_Wigner treatment of the timedependent amplitudes K and K.

12 Abbud, Lee, and Yang A R - 2 = 6,r Re(A-) cos (62 - 6) + (Aem) (1.14) 0 0 which starts with the 27w state constructed as 12r, = A 10,0> + A2 r2,0>el(62-0) (1.15) o where AI = <(27T)I Hwk IK) 1(27r)I> is the stationary 2r eigenstate of Hst with total isospin I, while the 6I are the respective s-wave phase shifts. The branching ratio R becomes 2 A2 + 1 A2 + 4 Re A A cos (6 6 ) 22 122 4-A + A + Re A A cos (62 - ) 3 0 2 27T 2 2 0)' A+o +' A2 - Re.A 2 Dividing through by A we find 0 + A2- + 1 Re(A-) cos (62 - 60) 1 2 P - O 05 0 + - Re(7) cos (62 - 6) "'X-"''''::'' -' ~ -' -(6...6 ) We ignore the Im(A /A ) since we show (page 15, expression 1.19) that it is or t~e order of 10 to the minus four while the real part is two orders of magnitude larger. Thus the relative phase between A2 and A0 is, 0 or nwr.

13 Ignoring second order terms A2 4 A 2 +; Re(A-) cos (62 - 6) LR A0 =2 1 +; Re(A-) cos (62 6 - ] 4 A x + RRe() cos (62 6) 0 6 A2 A 2 = 2 1 + - Re(2) + 4(Re A-) 0 0 Again ignoring second order terms A R - 2 = 6J Re(~-) cos (62 - 6) 0 To determine the value of Re(A2/A ) we need only realize the relation between the standing wave amplitudes AI and the JAI 1 = 12 /2, /2and 5/2 transition amplitudes al/2, m3/2, and m5/2' respectively. Assuming a3/2 ~>> 5/2 we find A - / (1.16)

14 and A = a /2 1 * (1.17) for the K~ decay. For the K+ o 7 decay we define =<'(+ w k IK+> and in terms of a3/2 we find 2 1 A2 2 3/2;75 2 (1.18) From the decay rate of K+ 7 we can determine.0544 and from expressions above we see A+ - A22' * Neither the general Cabibbo theory nor any current-current form of the weak interaction based on octet x octet interactions can accommodate a AI = 5/2 term. The usual currentcurrent theory describes the K~ - 27 decays with products of currents of the form J~ S* where J is the AS - 0 current with AI = 1 and S is the AS = 1 current with AI = 1/2. Thus the product contains AI = 1/2 and AI = 5/2 but no AI = 5/2 component.

15 so that A2 = A (.0544) =.o44 0 Since we are interested in the Re(A2/A ) we can subtract |Im(A2/A) | by noting thatl5 jIm()I (I + (1.19) where Ir nq and h I are the CP violating parameters rl + - L (1.20) (oo) A(K O A ()(o)(o) Thus the imaginary part of A2 reflects the CP violation within the K0~- system and is consequently on the order of 10-3. Hence we may assume A2 Re(A'),.043 0 so that again, without prior knowledge of the wrwr s-wave phase shift, the limits on the branching ratio are given by expression (1.14) to be

16 1.64 < R < 2.36 We now consider the possibility of AI = 5/2 transitions taking place so that in expressions (1.17) and (1.18) the L5/2 terms are not zero. There are two possibilities. If we assume that c5/2 and a/2 are complex then we can merely set a5/2 an upper limit on the ratio -/. If, on the other hand, 0^/2 we assume that a5/2 and % 3/2 are both real then a quadratic equation exists for ( /2\ and there are thus two solutions. \a3/2/ In the first case, a5/2 and a3/2 complex, we find A2| A _ 1 0462 cos (62 - 6 ) /2 52 2 1I''-~R 2 1 a573/2 3 a5/2 I2 AA2 R -2 1a.3/2 + a5/2j A (1.21) Upon examining the relation between ly| and the relative phase between c5/2 and a3/2 we arrive at the inequality /2 a5/21 a5/2 2 V3/2 31 2. -----— /-2- 1 1 ^ |y| -- - - 2 (1.22) 1 + -25 a5 1 _/2 a3/2- a3/2 Current experimental values of (62 - 6 ) definitely indicate a first quadrant result s.o that we may possibly restrict the value of R to be 2.0 < R < 2.36. See Appendix D.

17 With this we can set an upper limit of 5/2L2 2 a5/2. I 0 (1.23) a3/2 M On the other hand, assuming a5/2 and a3/2 are real, we find - --' 353/2 2,1+ (=/2) The resulting solutions are 512 AF2 a 572 2 3/ 2 2'. /2* (1.25) c53/2 + a3/2 y 3. ELECTROMAGNETIC CORRECTIONS TO THE K~ BRANCHING RATIO S' In expression (1.14) the last term within parentheses (Aem) corresponds to the sum of corrections due to electro16 magnetic effects. Wu and Yang noted that these corrections lead to small changes in the ratio R. The explicit form of 17 the electromagnetic correction contains five terms7 aem =2 _ _n + al2n - - 0.015 + Va( 2 + v a) em m m 2V (1.26) These corrections include the previously mentioned phase space considerations.

18 The first term contains renormalization effects at the Kww vertex and pion wave function renormalization and is logarithmically divergent at A the cutoff momentum. The second term des18 t + - + cribes soft photon emission8 in K+ 4 r V (i.e., K+ -T 7+ y) with y energy less than w. The constant a is defined as ~ -20L 2 I+ V a = 2a V2n - ) 1] (1.27) 2V V + where V is the velocity of the w+ in the K rest system. The third term is due to phase space differences and was calculated earlier. The fourth term is a correction factor from the Coulomb interaction. The fifth and last term is a result of strong-interaction corrections to the electromagnetic correction. In addition some of the finite contributions of the first two terms are contained in C. There have been only four attempts at evaluating the term Aem. The first was only a partial estimation by Lee and Wu19 In 1967 Abbud, Lee, and Yangl attempted a complete evaluation of em and arrived at the estimate Aem -.4 ~.o4. However, they neglected to include the fourth term in equation (1.23), 17 and Belavin and Narodetsky7 recalculated Aem including this em i important Coulomb term. Their result was Aem =.006 with no 20 assigned error. Finally in early 1969 Nachtman and de Rafael recalculated Ae with a revised estimate of the last term. Their result, the result we will use in this paper, is Aem -.006. As can be seen the correction is small and is more than absorbed by the current experimental errors in both R and 62 - 6, so that we feel justified in ignoring it in our rough quantitative calculations.

CHAPTER II EXPERIMENTAL METHOD +0 + - 01 0 1. PREVIOUS EXPERIMENTAL VALUES OF R = r(Ks 7Tr )/r(Ks - r~r~) Although the existence of the charged mode of the Ks was experimentally confirmed2 as early as 1947, it was not until * ten years later that the state of the art, plus enhanced theoretical understanding of the phenomena, enabled the experimentalists to attempt an accurate measurement of R. Since 22 the experiment of Eisler2 in 1957 there have been essentially only nine experimental determinations of R including the present experiment under discussion. These experiments can logically be grouped into two temporal categories which we will call the "old" and the "recent" experiments. The first seven of the nine experiments make up the "old" group. All seven were performed between 1957 and 1963 and most quoted errors on R of 10% or greater. Between 1964 and 1968 no experiments were carried out, and the recent category did not begin until the experiment of Gobbi et al.23 in 1969. The experiment which we present here is the only other recent attempt at determining R. The last two "recent" experiments reflect the improved experimental techniques in assigning errors of 2-3% on R. There were several early experiments on V~ particles between 1947 and 1957. However, they observed only the charged mode, and this plus extremely limited statistics made their results only approximate at best. 19

20 1.1 Eisler et al. The experiment of Eisler et al. used a propane bubble chamber and was the first to observe the neutral decay Ks 4 s w~7w~ by observing the subsequent conversion (y - e+e-) of one or more of the resulting four y's (w0 - yy). The K~'s were produced in the associated reaction + p A~ + K~ (2.1) which was the predominant method of K~ production for the old category of experiments. The resulting statistics of 168 charged decays and only six observed neutral decays emphasized the very poor gamma detection efficiency in these early chambers. Based solely on the charged sample Eisler et al.. found R to be 5.25 ~3.37, while based on both the charged and neutral decays R was found to be 6.0 ~2.68. The results were self-consistent, but somewhat inconsistent with any AI = 1/2 prediction of 2.0 for R. 1.2 Crawford et al. 24 The experiment of Crawford et al.4 used reaction (2.1) in a hydrogen bubble chamber to compile a total of 1091 recorded events. In 227 events both the A and K~ decayed * A more detailed analysis of the general method used in these associative production experiments is given in the description of the Chretien et al. experiment (Section 1.7 of this chapter), the last of the "old" experiments.

21 within the fiducial volume via their charged modes, for 594 events only the A decayed via its charged mode, and for the remaining 270 events only the K~ decayed via its charged mode. After correcting for fiducial volume characteristics and scanning efficiency (300 double scan), the LRL group arrived at a value for R, based solely on the charged mode, of 2.11 ~.29 In addition, this experiment detected three events which were consistent with K o - w~r. Corrections for the extremely s poor conversion efficiency of hydrogen led to a value of R, based only on neutral decays, of 2.7 ~1.1. We quote this value of R for completeness only since we feel the assigned error is underestimated. 1.3 Baglin et al. The experiment of Baglin et al.5 used a mixture of propane and methyliodide with radiation length of about 10 cm in a small (3.4 x 2.4 x 2.0 radiation lengths) chamber to detect the K~ decays. Contrary to previous experiments, Baglin looked primarily for the neutral K~ decays. Again using reaction (2.1), only those events in which the associated A decayed via its charged mode and two of the four y's converted within the fiducial volume were used. To the raw sample of 16 events, four major corrections were made. The first In most experiments the Ks charged to neutral ratio is quoted as r(K~ 4 TW+W )/~(K, O all). For such cases we re-expressed the result in the form F(Ks 4 W+-w)/P(Ks * 7r~w~).

22 compensated for randomly scattered tracks simulating charged A decays. The second correction considered the possibility of alternate origins for the observed gammas. Thirdly, those charged A decays with a short line of flight or a slow secondary were accounted for. The last correction used Monte Carlo techniques to determine the y-ray detection efficiency. The resulting value of R was 2.85 ~.66, within 1.5 S.D. of being consistent with the AI = 1/2 rule. 1.4 Brown et al. (1961) 26 J. L. Brown et al. were the first experimentors to try and obtain a value of R using accurate counts of both the charged and neutral decays. The Michigan 21-liter bubble chamber was filled with a mixture of 99.5% Xenon and.5% ethylene with a resultant radiation length of 3.9 cm. Thus the cylindrical chamber was approximately 7.7 radiation lengths in diameter and 6.4 radiation lengths deep so that the average y detection efficiency was 70%. The source of K~'s again was a 7r beam and the associated production w-p - A~K~. As was customary in these associated production experiments, only those K decays accompanied by an easily scanned A~ - n'p were counted. The charged K decays were easily scanned, and the neutral decays were defined as the intersection of 2, 3, or 4 y-ray pairs in association with the charged decay mode of the A. The neutral decays of the K~ were subjected to additional checks in the form of 1. Comparison of decay length distributions. Average

23 decay length, L, of the two modes should be the same; however, it was found that L(Ks - + w ) = 1.7 ~.22 while L(Ks -, arr ) = 2.25 ~.34, a difference of two to three standard deviations. 2. Subtractions of stray-y events. A check on the number of events incorrectly identified as neutral events, because of the intersection of background y-ray pairs, is obtained by counting the number of times this happens in association with an event where both the A~ and K~ are observed to decay via their charged modes. In 120 T- + p. A~ + K~ I 1-. 7r- r+... T + p only one event had such a spurious y pair association. The resulting uncorrected statistics were 120 K~ 4 wV s decays which, when corrected for decays outside of the fiducial volume or too close to the origin, became a sample of 130.2 charged events. The Ks - r7rO raw sample contained 49 events, and after similar corrections the final neutral sample was 56.5. The resulting branching ratio was R = 2.533 ~0.27.

24 1.5 Anderson et al. During the 1962 CERN Conference on High Energy Physics 27 a value of R attributed to Anderson et al. was entered into the record with the only accompanying description being that it was carried out in the 72-inch hydrogen bubble chamber and only the charged mode was counted. Since there was never any follow-up publication, we give the Anderson et al. value of R = 2.85 ~.35 only for the sake of completeness. 1.6 Brown et al. (1963) 28 A Michigan-LRL collaboration, consisting of nearly the same physicists as the earlier Brown et al. experiment, redetermined the branching ratio in 1962. The xenon chamber was again used to get maximum gamma conversion, but for the first time the K~ production mechanism was the more efficient charge exchange reaction K+ + n - K~ + p rather than the associated production reaction. A separated K+ beam of.8 GeV/c was used and charge exchanged approximately 5% of the time. If there is negligible w+ contamination in the beam a reliable signature of K~ production is the charge exchange reaction itself which, in turn, is reliably identified by requiring all prongs from the interaction to stop in the chamber without decaying. Double scanning of the film resulted in ~ 12,000 charge exchanges of which 7,000 were within the fiducial volume. Hence around 3500 K~'s were in the initial s sample, which is about a twenty-fold increase in statistics over the earlier Michigan experiment.

25 The charged sample underwent four major corrections which accounted for scanning efficiency, kinematic failures (mislabeling V~'s), extremely short pion secondaries and negligible line of flight. The final sample contained 2477 events. The neutral sample was corrected for scanning efficiency, y conversion efficiency, and beam contamination and ultimately contained 1250 events. The resulting branching ratio stands alone in being the only result lower than the AI = 1/2 prediction of 2.0. It was found that R = 1.98 ~.12. 1.7 Chretien et al. The last of the experiments which fall into the "old" category was a four-institution collaboration henceforth 29 referred to as Chretien et al.2 This collaboration followed the methods previously established by branching ratio experiments prior to the Michigan-LRL experiment, described in the last section, in that they used a 1.144 GeV/c w beam to produce K~'s via the reaction w + p - A~ + K~. The chamber used was the 51.5-liter cylindrical chamber at the Cosmotron filled with a mixture of methyl iodide, propane, and ethane which resulted in a radiation length of 8.4 cm and an 82% probability of converting two of the four y rays. Since the purpose of this experiment, as well as the other associative reaction experiments, was not only to measure the branching ratio of the K~ but also the branching ratio RA of the A~

26 RA = r(A~ 4 p + w-) A r(A~ 4 n + w~) there were three types of events recorded. 1. The A~K~ sample, c c ~ + P - AO + K~ C c and AO p + K O + Tr C' c 2. The A~K0 sample c n v- + p * AOKO C n. with A~ as above and c Ko 7r~ o 70r~ 4y n with at least two of the y's converting. 3. The K A sample W + p - Kc + Al with K~ as above and c n s 7r~ + n A0 4 7~ + n n

27 with the two y's converting. The K~ branching ratio was determined using the A K and A K samples. The basis of the scanning was the search for c n all two prong V configurations associated with a r interaction and then a subsequent search for possible associated. Yrays. All charged V decays were then measured twice while neutral decays were measured as to projected line of flight of the converted gammas. A consistency check on the K~ sample was provided by a Bartlett S function determination of the mean K~ lifetime r = (0.87 ~0.05) x 101 compared to7 5s ~ 0 (.862 ~.006) x 10-1. Any possible bias toward the neutral (A Kn) or charged (AcKc) modes was determined by demanding similar properties of the charged A decay in both cases. The initial raw sample of charged events was 508 events while the initial neutral event count stood at 198. Five corrections were made to the raw charged sample: 1. A 16.4% correction was made to compensate for those Ko decays which occurred between 0 and.3 cm from c the w interaction. 2. A 3.8% correction for K~ decays.with secondaries of c.5 cm or less and/or K~ decays with opening angles less than 15~ or greater than 160~. 3. A 1.7% correction for K~'s having momentum less than c C 100 MeV/c. This correction is closely correlated to corrections 1 and 2 above and contains only the events not already corrected for in 1 and 2 above. 4. This correction compensates for those K~rs which C

28 interact in flight before decaying and therefore fail, kinematically, the requirement for Kc decays. The correction amounted to.7%. 5. This was an overall correction for background simulation of a Kc event (fake V's) and possible errors in the analysis procedure and was estimated as 0 ~1l. The final corrected number of charged decays was 628 ~35.9. The neutral sample underwent six corrections before arriving at the final quoted number. 1. Monte Carlo techniques were used to correct for the conversion efficiency of two or more y's from the Kn decay. A 22.1 correction resulted. 2. A 6.1 correction was made to compensate for associated y conversions accidentally pointing to alternate origins. 3. This is the complementary correction to 2 above and accounts for background Y's accidentally simulating Kn decays. This negative (subtractive) correction amounted to 2%. 4. This 1.6% correction was made to compensate for the difference in scanning efficiency for those events labeled as Kn decays and those labeled Kc decays. As might be expected, the charged decays were "easier" to find. 5. A.95 negative correction was made to account for charge exchange events (7- + p - n + 7~) in which the resulting neutron star simulated a V and both

29 y's converted. 6. As in the charged sample, this last overall correction was for analysis uncertainties but, in addition, this correction compensated for charged V's which satisfied both the A and K events and thus amounted c c to.8%. The final corrected neutral sample thus contained 255 ~21.2 events, which resulted in a final value of R = 2.47 ~.25. 1.8 Summary of Older Generation Experiments With the exception of the Michigan-LRL collaboration, all of the old generation experiments used w p production techniques to obtain their sample of Ks's and were thus characterized by low statistics. In addition, these early experiments were most likely hindered by a general lack of refinement in the measuring and subsequent analysis of both the charged and neutral modes. This is especially important when one realizes that to guarantee a reasonable gamma conversion efficiency high-Z elements have to be used. These very same high-Z elements increase multiple scattering effects, thus making tracks which would be easily measured smooth curves in hydrogen or deuterium, a nightmare collection of partially connected kinks. Thus even the charged decays of the K~, confirmed through measurement under these conditions, must be analyzed carefully, not to mention the much less massive (and thus more scattered) e-e+ conversion pairs. Even today with the much more sophisticated geometrical reconstruction

31 programs, relatively large uncertainties must be attached to tracks measured in heavy liquid chambers. The alternative to this measurement uncertainty is to use a hydrogen (or other low-Z) chamber and not count the neutral decays at all. This has a decided disadvantage in that a major quantity, the number of neutral decays, necessary for the branching ratio being determined is not seen but rather inferred by two other numbers and a constraint relation. Thus any error in the treatment of these two other charged samples is compounded when the number of neutral decays is inferred from these erroneous quantities. It is a well known observation of these branching ratio experiments that, due to less statistics and the necessity of greater corrections, the major portion of the final error attributed to the branching ratio comes from the neutral sample when it is directly measured. Thus the final error of those experiments which infer, rather than determine, the number of K~ - 2r~ decays might possibly be underestimated. As can be seen from Figure 2 all of the results, with the exception of the preliminary Anderson et al. value and the 6% determination of Brown et al. (1963), are consistent within errors. It can also be noted that, within errors, the value of the branching ratio as determined by these old generation experiments ranges from 1.83 to 3.5 with a world average7 of 2.165 ~0.08. Thus more refined experiments, in terms of beam, chamber, and analysis, were clearly needed. After a lapse of about five years, during which many of the

2.9 - ".- C +.8 c E 30" to, ^ ^ ^,> a 2.5 25 I U 2.4 a 0.b s 2.3- - -.-,r- 2.- 1 OLD GENERATION" EXPERIMENTS RECENT EXPE IME Figure 2. Graphical summary of all reported values for the K0 charged to neutral branching ratio. CDP CD e chargred to neutrall bra. nching~ ratio.

33 necessary refinements were realized, the experiment of Gobbi et al. began the second generation of experiments to determine R. 1.9 Gobbi et al. The first experiment of the new generation of determinations was unique in that it was a spark chamber experiment. They used a 2 BeV/c unseparated K+ beam coming down a beam line containing a counter telescope and differential Cerenkov counter set to reject +, p, and i. As in the Brown et al. experiment of 1963, the charge exchange reaction, K+ + n K~ + p, is used in this experiment. Detection equipment consisted. of a 24-gap, aluminum-foil spark chamber for the charged decays and a 4-radiation length, 49-gap, steel spark chamber for the detection of the neutral decays. In addition to the above, a third spark chamber was used to register the K point of entry into the target so that kinematical analysis of the charged decays could be performed. Due to the veto counters surrounding the target, front edge of the fiducial volumes for the charged and neutral modes were defined electonically and the effective front edge of the neutral fiducial volume is displaced upstream. The back edge of the fiducial volume is the last plate in the steel chamber while the sides are unbounded (detection efficiency quoted as constant and near 100 up to ~ 45~). All pictures were double scanned and the raw sample of charged and neutral decays was 6608 and 3016 events,

34 respectively, leading to an uncorrected branching ratio of 2.191 ~.048. Seven corrections were made on this raw branching ratio (quoted as a percentage). 1. +3.8 ~.1 to compensate for neutral decays in the downstream veto counter that did not convert and thus were accepted, while charged decays in this veto counter were rejected. 2. +1.0 ~0.5 to account for background to two Y events such as K n -+ p0~K~ and the charge exchange of beam contamination pions. 3. -0.5 for those events in which only one or no y's converted and thus were not counted. 4. +0.3 for neutral decays which escape the target. 5. -0.1 for y's with momentum back towards the target. 6. +0.1 for backgoing w interactions. 7. -0.3 for possible admixture of 3-body decays. The net result of these corrections was to raise the branching ratio to 2.285 ~.055. Thus the first of the recent experiments showed a violation of the AI = 1/2 rule of five standard deviations. 2. GENERAL METHOD OF THE PRESENT EXPERIMENT As in the experiment of Brown et al. (1963) and Gobbi et al. (1969), the KO~s in this experiment were produced via the charge exchange reaction K+ + n 4 K~ + p. Unlike the Gobbi et al. experiment, this experiment used a heavy liquid bubble chamber to observe the decays.

35 The basis of our method was to use the charge exchange of a K+ (K+ + n * K~ + p) as a signature for the creation of a K~. The vicinity of a charge exchange (Type I event) could then be examined for evidence of a charged decay of the K~ (a V whose arms were consistent with 7r+ ) or a neutral decay (one or more electron pairs). The topological definition of a charge exchange was a beam track interaction whose visible products, if any, were protons, i.e., heavily ionizing positive tracks which stopped in the chamber without evidence of a subsequent decay. There are two difficulties with the method as described so far. The first is that a w contamination in the beam produces w~'s by charge exchange and such events are not easily distinguished from K charge exchanges followed by a neutral decay of the K~. A second difficulty is that many of the K~'s decay so close to the charge exchange point that the separation between that point and the vertex of the V cannot be resolved by the scanner. In order to overcome these difficulties we asked the scanners to look for any beam interactions which produced a negative track (a Type II event). By subsequent examination of the Type II events we were able to classify them as follows: 1. The negative track was a Tr and was accompanied by a 7. Other tracks (if any) from the interaction vertex were protons. Since none of the events of this type were accompanied by V's we then deduced

36 that either they arose from a vr contamination in the beam or they were charged decays of K~'s very close to the charge exchange vertex. 2. The negative track was a 7 and all other tracks (if any) from the vertex were protons. Since none of the events of this type were accompanied by V's we then deduced that they were due to the interaction of a c contamination in the beam. 3. The negative track was a 7r and was accompanied by two r tracks and no other tracks. Such events were easily recognized as r decays in flight. 4. The negative track was an electron either from a Dalitz pair or from the conversion of a y-ray close to the interaction vertex. We used the "irt only" events described in (2) above as a measure of the + contamination in the K beam. The scanners scanned 9400 wT pictures taken at precisely the same momentum as the K+ pictures. The scan rules for the w+ film were the same as those for the K+ film. The sample was then subjected to four separate edits to further classify the events, a re-edit to check our editing accuracy, and a partial measurement of the charged decays to both confirm our classification of the Ks -* rr events and to give realistic input to our Monte Carlo program. The raw sample of charged decays was then corrected for such things as editing errors, and r+ contamination of the beam, while the raw sample of neutral decays was corrected for stray Y's,

37 editing errors, unobserved Y's, and w contamination of the beam. These corrections were based on both previous observations and Monte Carlo predictions. Our initial statistics and relatively small corrections enabled us to keep the error of the final branching ratio down to' 3%.

CHAPTER III THE EXPERIMENT AND REDUCTION OF THE DATA 1. THE BEAM AND THE CHAMBER The beam used for this experiment was the low momentum, two-stage separated 28~ beam at the ZGS installation of Argonne National Laboratory. The beam elements included seven quadrupoles, three bending magnets, and two electrostatic separators; in addition there were two preset mass slits and one set (horizontal and.vertical) of remotely controlled collimators. The beam was designed to operate in two modes by reversing the polarities of the first doublet (Q1 and Qp). In Mode A the ratio of separation-to-image size is maximized at the first mass slit while in Mode B this ratio is ~ 2.5 times less favorable but the solid angle of the beam is increased by, 50%. For this experiment beam contamination (primarily w, p, and L ) was crucial so that Mode A, as in Figure 3, was employed.. Essentially, the first two quadrupoles make the beam parallel in the vertical plane while focusing the beam horizontally at the first mass slit. The first bending magnet deflects the beam by 15~ while providing momentum dispersion. The first spectrometer provides mass separation by deflection in the vertical plane. Following this spectrometer a * For full details of the beam see M. Derrick and. G. Keyes, Low Momentum Separated. Beam, unpublished but obtainable from Argonne National Laboratory, Argonne, Illinois. 38

Ql CL2 VSI 04 82 05 MS CLI 02 Bl 03 MSI VS 2 06 n nnnnl in m m -- nn n Z 44.0 1.51.0o mU 1. / O> 5 2.0 Figure 3. The vertical and horizontal effects of each component in the 28~ beam line when operating in Mode A.

4o quadrupole is situated with a field gradient to give both horizontal and vertical focus at the first mass slit. Diverging particles at this point are made parallel in the vertical plane by the 40~ second bending magnet. The second stage introduces another separator, and a final doublet of quadrupoles to provide focus on the last mass slit. For monitoring the beam four cascade counters —though only three were used for this experiment —and one differential Cerenkov were available. The mass limit on the Cerenkov counter was set so that any particle of mass m < mK+ would not trigger the counter. Thus a coincidence circuit of Tl, T2, T3, and C would essentially count only K+ and p, while a circuit containing Tl, T2, T5 in coincidence with the Cerenkov in anti-coincidence (C) would essentially count only 7w and |I. Naturally, a circuit which excluded the Cerenkov entirely would count all charged particles. An inhibit circuit was employed using T1, T2, T3, and C such that when the number of 4- "-+ + -+ + was greater than two no picture was taken. The K beam was transported at 825 MeV/c. After traversing the monitoring equipment and the beam window, the K momentum was down to approximately 790 MeV/c upon entering the liquid, and was further reduced to 700 MeV/c in traveling - 23 cm to enter the fiducial volume. This momentum was chosen since it was high enough to overcome decay loss along the beam line, and low enough not to introduce w~ production from the interaction K+ + n - K~ + w~ + p as shown in Figure 4.

8.0 7.0 - 6.0 5.0- K+ N -K~+P SI ~4.0 - Ui 3.0 O * 2.0- K +K N-K-w'+P 1.0. 0. 2 03 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 P[K+] BeV/c Figure 4. Comparison of the cross sections for K+ + n * K~ + p — the primary interaction —and K+ + n -+ K~ + Tr + p, a possible source of false y events.

42 The 40-inch heavy liquid Michigan-Argonne bubble chamber, filled with heavy freon (CF3Br), served as both target and detector for the highly separated K+ beam described above. Heavy freon, radiation length h 11 cm, was chosen since we wished the y conversion efficiency to be as high as possible in order to detect the neutral decay of Ks - 2w0~ 47. It was determined that the conversion efficiency (y 4 e+e or y * Compton e ) of photons in the 40-inch HLBC filled with heavy freon is.862 ~.015. Using Monte Carlo techniques we also determined that 98.5% of the neutral decays have at least one gamma convert within a sphere of radius 20 cm centered on the charge exchange point. It was this observation which we incorporated into our edit rules. A total of 150,000 frames, each with four stereo views, were taken in this chamber using the K beam. In addition, 20,000 frames were taken of ft beam interactions for determination of wcontamination of the beam. 2. SCANNING An initial beam survey was carried out to restrict the definition of a "good" beam track. The position of entry, rate-of-change of angle and dip were measured from a sample of K beams and + beams. It was found to be impossible to confidently eliminate any T+ contamination using the results, but all off-momenta beams could be eliminated by constructing a two-piece template. The template consisted of a base whose upper portion was an arc of 58.4 cm radius of curvature, a

43 second piece whose bottom edge was an arc of 58.4 cm slid along the base and was used to define the entrance angle of a good beam within ~6~. Since the purpose of this experiment was to determine a branching ratio, scanning rules were of utmost importance. For this experiment, the possibility of missing an event, though unfortunate, was not disastrous as long as no bias towards the charged or neutral decays was involved. The scanning rules were designed with this in mind and because of their importance they have been reproduced, as used, in Appendix A of this paper. The most important requirement of our scan rules is the necessity of along-the-track scanning. If the scanner will consistently follow the beam track from the point it enters the fiducial volume to the point where it either interacts or leaves the fiducial volume there will be no bias toward one decay mode or the other. This method of scanning was demanded from the scanners and most effectively followed by them. In summary, the two main types of events scanned for were characterized as follows. (See Appendix A, Figures 19-21.) Type I 1. Any zero prong event if the beam track is not heavily ionized within 2 cm (low magnification) of its end. 2. Any 1, 2, 3..... n prong interaction whose outgoing tracks satisfy all of the following. a. All outgoing tracks must stop in the chamber

44 and the last 2 cm (high magnification) must be heavily ionized. b. No negatively charged prongs. c. The outgoing prongs must not interact or decay, i.e., no abrupt change in ionization. d. A prong may scatter through any angle as long as it satisfies (c) above and does not produce one or more prongs at the point of scatter. Type II 1. The interaction has an outgoing negatively charged prong longer than 4 cm (high magnification). 2. The beam track is minimum ionizing in the last 2 cm (low magnification) before interaction. All the film was double scanned for Type I and Type II events. The scanners were instructed to scan along the tracks from the point where they entered the fiducial volume to the point where they left it, typically a distance of ~ 55 cms. Only those tracks which entered the fiducial volume within ~6~ of the average beam direction were scanned. Since the huge fringing field of the chamber magnet bends the beam through, 80~ before entering the fiducial volume, this rule ensured that only "on momentum" tracks were scanned. All disagreements from the double scan were checked and resolved by a third party, either an experienced scanner or a physicist. At the conclusion of both scans and the disagreement check there were 29,444 recorded events. Of these 27,030 were Type I events and 2,414 were Type II events. This then

45 was the raw sample which had to be further classified. The question of scanning efficiency for this experiment was important only to the extent that we would want the detection efficiency of charged decays to equal the detection efficiency of the neutral decays. Of the three predominant models31 used in calculating scanning efficiencies, randommiss, correlated-miss, and two-scanner model, the random-miss model was employed for this experiment. This model is based on the hypothesis that all events are of equal difficulty for the scanner to detect and is the model most often used in bubble chamber work. Mathematically, the random-miss model treats all events missed by scanners as statistically independent events. When double scanning film, as we did in this experiment, the scanning efficiency for each roll is greatly improved. If we let NS be the number of events seen by a single scanner, NB the number seen by both scanners, then NS NS + NB is the number seen by either or both scanners. If the total number of events that could have been found, i.e., they passed the scan rules, is NT then we can relate the above quantities * to a single scan efficiency e as follows: NS - 2NTe(l - e) (3.la) This method of calculation is particularly attractive when many different scanners have processed the film. The efficiency e is indicative of the results of all scanners combined and does not necessarily reflect the result of any one of the scanners.

46 NB = NTe (3.lb) NSB = NT[ - (1 - e)2] =NT(2e e2).(3.1c) The double scan efficiency E is just NSB 2 NT = (2e - e ) = E. (2) NT (Se-e2). (3.2) Solving (3.1) for e and substituting in (3.2) we find 4NSBNB E — SB (3.5) (NsB + NB) Using this expression the double scan efficiency for K~ 4 7r w events was.994 while the double scan efficiency for Ko 7r~ro0 was.995. These correspond to single scan efficiencies of.923 and.929, respectively. Thus no scanning bias was detectable for the two scans. 3. EDITING 3.1 The K+ Sample A series of four edits was used to further classify the events. The rules governing these edits can be found in Appendix B. In the first edit all events found in the two scans were examined by experienced scanners. In the case of Type I events they checked to see that the charge exchange

47 vertex obeyed the scan rules and that there were no other Type I events within a distance of 10 cms on any of the four projected views. Ten centimeters on the scanning machine corresponds to 20 cms in real space. If the event passed both these tests a template consisting of two concentric circles of radii 5 cms and 10 cms, respectively, was centered on the charge exchange vertex and the area within the outer circle scanned on all four views for one or more electron pairs. Note was taken of any electron pair with an apex within the 10 cm circle on all four views. At this point no judgment was made concerning the origin of the Y-ray. The only other test applied to the electron pair was that its entire track length could not fit within a square 1 cm on a side. This imposed a low energy cutoff of about 10 MeV on the accepted electron pairs. Events which passed these tests were denoted as "y events" and were passed on to a second edit as such. In addition to searching for electron pairs, the first editor also searched for V's whose apex lay within the 5 cm circle on all four views. Such events were noted as V events. In order not to miss V's with wide opening angles the first editor was instructed to count as V's stray tracks intersecting the 5 cm circle provided they were not obviously beam tracks or connected to beam tracks. Events which passed both the Y-test and the V test were denoted as y-V events. In the case of Type II events the first editor examined them and classified them as described in Section 2 of this

48 chapter. Then the first editor searched for V's and electron pairs in the same way as for Type I events. Physicists carried out a second edit of the film. We examined all Y events, V events, and y-V events. We checked to see that the arms of each V were consistent with being Tw+ tracks and that the orientation of the V relative to the charge exchange vertex was consistent with conservation of momentum. All events whose arms were consistent with wT tracks were accepted, and the distance between the charge exchange point and the apex of the V was noted. Those events whose orientation was inconsistent with the conservation of momentum were denoted as scattered V events. In the case of y events and y-V events we examined each electron pair whose apex lay within the 10 cm radius on all four views to see if it had an origin other than the charge exchange vertex. The event remained a y-event if there was at least one electron pair within the 10 cm radius for which no origin, other than the charge exchange vertex, could be found and provided the electron pair(s) could have come from within the 5 cm circle on all four views. Again, the separation between charge exchange point and the nearest y was measured at the scan table and noted. It should be emphasized that the number of y rays per event in no way influenced our raw count of neutral decays. Table III contains a breakdown of all categories after the first two edits and the double scan. For the meaning of each classification see the edit rules in Appendix B.

49 TABLE III DETAILED EVENT BREAKDOWN AFTER SECOND EDIT First Edit Second Edit Scan - Classification Totals Classification Totals Reject 868 Reject 108 Neither V nor y 9187 V good. 5979 -P Associated V 6870 V fails 875 Si _____Undecided. 23 wX~~~~ ~~At least one H Associated y(s) 5616 good y 416 All y's fail 1394,.o _________Undecided 533 EPO y(s) and V fail 311.a c Associated V Good V, no y 1041 g ~and associated 1628 Goo y), no V 141 O. -;7r Both V and,; ~y)ls y(s) good 106 o Undecided 20 2 Type I within 20 cm 3195 Undecided 8 r decays 1230 r- with K+ 62 w- only 76 > V oIn good Type At least one 21 V on good. Type I event with 118 good y associated y(s) All y's fail 95 Hw~-associated y(sj_) _Undecided 1 V on good Type G 453 I event with 453 Good V e or e e pair 64 -on bad Type I 79 0 r Tr on illegitimate 22 Type I Undecided 7

50 To check the accuracy of these two major edits, we re-edited a 20% sample of the film and compared the results with the original edits. All conflicting categorizations were resolved by a third party during the fourth edit. In each event category we found errors (~ 3%) of both omission and comission approximately equal in numbers so that the corrections applied to each category for such errors are small. Table IV gives exact numbers for the re-edit-edit comparison. Classifications not listed in Table III showed no net change between edit and re-edit. TABLE IV TABULATION OF ERRORS FOUND BY RE-EDIT OF 20% SAMPLE OF K+ FILM Omission Comission V events 1st editor 15 4 2nd editor 25 24 y events 31 35 Y-V events 10 4 w- only events 0 1 Neither a V or a 56 67 y events The third edit was carried out by those scanners who had first edited the film and was twofold in purpose. Primarily we wanted to determine the distribution in spatial separation between the charge-exchange point and estimated copunctal

51 point for those events classified as associated Y events by the second editors. This distribution was to be checked later by Monte Carlo techniques described shortly. The secondary purpose of this edit was to insure that there indeed was no other Type I. event within 10 cm of those events classified as 24, 25, and 26 by the first editor. The fourth and last edit of the raw sample, carried on solely by physicists, was designed to serve several diverse purposes. All events in which the second editor had tossed out a V configuration were re-examined for possible regeneration effects (see Appendix C, Section 2) or Ks elastic scattering. A spot check was made of those events which, due to the first edit classification, were not passed on to the second edit and thus were not examined by physicists. In this spot check special attention was given to the distinguishing of any p contamination from those events categorized as having neither an associated V or y. As stated earlier all disagreements between editor and re-editor were resolved at this point. The final purpose of this fourth edit is related to a step in the analysis of the sample described in Section 4 of this chapter. This step was a further check of the V events obtained by measuring a randomly chosen sample of ~ 2000 events and testing to see if they constrained (tests applied are explained in Section 4) to the decay K~ - r. If they s did not constrain to this decay they were visually checked at this edit.

52 3.2 Editing of the 7 Film The r' sample went through the first two major edits only. All rules used in the K edit applied to the wT edit with the exception that even if there was another Type 1 event within 10 cm of the vertex in question it was still accepted. This was done since the important point was not which vertex the y was associated with but that the y was associated with either one. Table V contains a comparison of K+ and sample after the first two edits. In this table the sample is already grouped into its major divisions. TABLE V DISPOSITION OF EVENTS IN THE K' AND + FILM AFTER SECOND EDITING K+ Film w+ Film V events including Type II 7381 (535) 195 (195) events with w+r- at interaction vertex Y-V events 119 (22) 0 Total V events 7500 195 y events including Type II 4280 (26) 1025 (28) events whose negative track is an electron at the interaction vertex Type II events with wT but no 76 (76) 275 (275) r+ at the interaction vertex Events with neither a V nor 11747 (0) 1244 (0) a y Total of all events 23527 2739 The numbers in parentheses denote the number of Type II events in each category. We also demanded that the y point exactly to the C.E. point.

53 4. MEASURING To further guarantee that those events categorized as "associated V" events by the second editor were indeed Ks~ - w w decays, a random sample of 1000 (14%) such events were measured on digitized projection machines. These machines were equipped with encoding devices which transferred position measurements along with track and event identification to IBM cards for subsequent computer analysis. The resulting punched cards were processed through CAST, a measurement-checking program for Datex-machine measurements of bubble chamber events written at the University of Michigan. This program checks both for illegal and illogical control information on every card and compiles data for the event as a whole. Then CAST checks the event for consistency and unmeasured vertex points are extrapolated. Badly measured points are determined using a sliding circle fit and excluded. Finally CAST produces output in the necessary form to serve as input to SHAPE. 4.1 Geometric Reconstruction 32 SHAPE2 is a University of Michigan-created heavy liquid geometry program, designed to reconstruct the spatial coordinates of each measured track and to calculate, using rangeenergy relationships and the curvature implied by these spatial coordinates, the components of momenta and their associated errors plus all involved real space angles and errors. Briefly the structure of SHAPE is as follows.55

54 1. Reads all input constants which may vary from experiment to experiment and prints them. 2. Reads in the data (CAST output) for each event. 3. Tries to reconstruct all vertices as corresponding points. 4. Reconstructs tracks in three dimensions by calling, among other things, CURFIT —a large package of routines which obtain kinematical variables (range, angles, momenta) from the real space points of the track. 5. Reconstructs gamma tracks from e+e pairs. 6. Controls the calling of print and tape writing routines and further calculations. 7. When all events have been processed prints a list of events passed and. events failed. 4.2 Kinematic Fitting Fitting to the Ks -^ w hypothesis was accomplished with the program UKFIT written especially for this reaction and based on the deduction that the vector (I = ISI ) from charge exchange point to V-vertex should lie in the same direction as the vector (R = RIRI) found to be the kinematical vector sum of the 7TT + system. Thus S R = 1.0 (3.4) was one requirement for fitting the hypothesis. A second test

55 required that the effective mass of the wT system equal the mass of the K~ within 1 S.D. Of the 1000 events initially measured, 240 failed one or both of these tests and were subsequently visually examined. This rather high rate of failure (24%) is a consequence of the relatively poor reconstruction available for slow and extremely short tracks in heavy liquids plus the fact that we purposely set the requirements very tight so that any dubious events would be subject to reexamination.

CHAPTER IV CHECKS AND CORRECTIONS OF THE RAW SAMPLE 1. MONTE CARLO TECHNIQUE In addition to the check on the charged sample provided by the measurement described at the end of the last chapter, it is also possible to use Monte Carlo techniques to check both our samples. Monte Carlo may be defined as the use of the statistical properties of random numbers to solve problems. In this case we are trying to simulate the statistical nature of the K~ decay so that the application of Monte Carlo s methods is particularly well suited and consequently easy to apply. We used the kinematic properties determined from the random sample of measured charged decays as input to our Monte Carlo program. For each event that passed the fitting program we took the charge exchange point and momentum vector of the K~. The program then carried out the following steps. 1. Starting at the charge exchange space point the "particle" was stepped along its given direction.5 cm at a time. At each point a probability to decay is determined from the relation PD = DK 1 - e ( Ycr); a random number, r, is then generated and compared to PDK. If PDK < r then the "particle" has decayed and the program moves on to * An excellent reference for Monte Carlo techniques if F. James, Monte Carlo for Particle Physicists, CERN Reprint. 56

57 Step 2, while if PK > r Step 1 is repeated until it does decay. 2. We now have the decay point which is the apex of the V for the charged decay and the point of creation for the two w~'s of the neutral decay. Aside from geometrical projections and scaling factors to get from a 3-space point in the chamber to the screen of our measuring machines, we are now finished with the "particle" as a charged decay since we only need the projected distance from charge exchange point to decay point for comparisons. For the neutral decays, however, we are just beginning. Using K -4 w~7~ center of mass kinematics, we randomly decay (in terms of p and cos 8) each w~ to two gammas. Due to the extremely short w~ lifetime7 (r o =.89 x 1016 sec), we assume the K~ decay point and w~ decay points as identical. 3. Each of the four gammas is now stepped along its random, though kinematically constrained, direction 34 until a comparison of its conversion factor and a random number dictates a y 4 e + e conversion or a compton conversion. Naturally a check is made at each point to insure that the y is still within the boundaries of the chamber. 4. Having determined all items of interest —Ks decay point (V apex), the four gamma conversion points, and distributions of kinematical variables of the

58 particles involved —all 3-space points were projected to the screen of the scanning machine and relevant distances determined. Based on the quantities determined. with this Monte Carlo method we made the checks and some of the corrections to be discussed directly. 2. THE CHARGED DECAYS 2.1 Verification To determine the purity of our sample of observed charged decays we pursued two courses; the experimental determination of well known kinematic characteristics for the decay K~ s'w + T and comparison of observed distributions with Monte Carlo predicted distributions. First we measured and kinematically checked a random sample of charged decays as described * at the end of the last chapter. Of the 2200 events measured, 1940 events passed the kinematic checks or subsequent visual examination. In Figure 5 the effective mass of the wrT system is presented. There are 235 (12%) events in the central bin which rwus from.49 to.50 BeV while 892 events lie to the left of this bin and 813 events fall to the right. Thus the mean value of the distribution gives a mass of ~.495 BeV for the w +7 system with a full width at half-maximum of.050 BeV. In addition to the sample of 1000 events measured, fit, and visually inspected, a second sample of - 1200 events was measured and those that passed both fitting criteria were used to increase the statistics of our effective mass plot and proper time distribution. There was no further treatment of this second sample.

59 240 200 160 -~ 120_ Ll o 20 z 40- /.3 0.4 0.5 0. 6 0.7 0.8 M[7r+w BeV Figure 5. Distribution of the measured effective mass of the wTr+w system. Each bin is 10 MeV2 wide. The center of the peak is at, 0.495 BeV with F'WHM of, 0.050 BeV.

60 This effective mass of the 7T r7 system is to be compared with the mass7 of the K = (.49776 ~.00016) BeV. The proper time of each of these decays was also computed and the resulting distribution is presented in Figure 6. A straight line was visually fit to the points, and the intersecting dashed lines give the e1 point and the corresponding lifetime. This admittedly crude procedure yields a value for r(Ks) of.83 x 10 10 sec to be compared with7 (.862 ~.006) x -10 10 sec. Of the 1940 kinematically checked Ks -. 7TYr decays, s 580 were picked at random and used as input for the Monte * Carlo program described above. The momentum distribution of the 580 events chosen is shown in Figure 7. Since each event went through the procedure 20 times, the number of K~ decays generated was 11,600. The distribution of (charge-exchangepoint)-(+7r v- vertex) separation of the generated sample is compared to the distribution of distances noted during the second edit in Figure 8. The distributions are normalized to the first two bins, and it is found that the statistics match extremely well except for the last four bins. It is found that there is a 195 event surplus of actual events over predicted events; since 40% of these excess events occur in the last four bins this excess might be attributed to regeneration phenomena. Monte Carlo programs are notoriously inefficient and consequently outrageously expensive. This is the reason why we generated such a small sample.

61 w w a 600- \ 00 —z I 20I I 0.2 0.6 1.0 1.4 1.8 2.2 2.6 50 K r + r. The intersecting dashed lines give the point and the corresponding lifetime.

62 280 260 w 140 - o 140i 1202 1008060 4020 0.1 0.2 0.3 0.4 0.5 0.6 0.7 IP[MI BeV/c Figure 7. Distribution of lab momentum for K~'s produced via K~ charge-exchange reaction.

63 * DATA POINT'".MONTE CARLO 200oo POINT 1000aoo_ f0 _ 1 600 so 60 20I 2 3 4 5 6 7 4 9 0o DISTANCE Figure 8. Distribution of separation between charge-exchange (C.E.) point and V-vertex as observed and as predicted via Monte Carlo methods. Each unit of distance is 0.5 cm on the scan table which is, 1.0 cm in real space.

64 2.2 Corrections to the Charged Sample It was found necessary to make four minor corrections to the raw sample of 7500 charged decays. Table VI summarizes these corrections. TABLE VI CORRECTIONS TO K 7r 7t SAMPLE Total V events from 2nd edit 7500 ~87 Scattered V events 242 ~15 Due to wf contamination -53 ~ 8 Total edit corrections 90 ~45 Due to K~-K~ interference -43 ~ 7 L s Total K~ - w +7 events 7736 ~99 The largest correction was for scattered K~ particles which subsequently decayed as Ks -+ w. This class of events contains both K non-regenerative scatters and incohers ent regenerative effects. Our neutral sample includes such events while scattered K~ events decaying by the charged mode within the 5 cm scanning circle would most likely be eliminated at the second edit. To correct for this we examined all V events rejected by the second edit to find. out how many were charged decays which had failed the second edit because their configuration relative to the charge exchange point was inconsistent with the conservation of momentum. Of the 1126

65 events examined, 242 were found to be of this type and were consequently measured. The effective mass of the w + system is shown in Figure 9d. Of these 242 events, 80 were found to have a short proton track located close to the vertex of the V. The distribution of the projected distances of these scattered events from the charge exchange vertex is shown in Figure 9a, 9b, and 9c, respectively, for events with observed proton recoil, without observed recoil (containing those events scattering off neutrons and those events scattering off protons but imparting insufficient energy to the proton to render it visible), and total. These distributions are consistent with a flat distribution of KL regenerative effects plus the superposition of an expeonential fall-off from Ks scatters. An extremely crude visual estimate of the relative magnitudes of these two effects would attribute 150 to KL regeneration and L CD the balance (92) to K non-regenerative scattering. Because our neutral sample includes such scattered events we must therefore add 242 ~15 events to our raw sample. A second correction takes into account the results of both the sample measurement and the re-edit check. All of the 240 events which failed one or both of the fit requirements in the initial 1000 event sample were examined by physicists and in all but 22 of these cases the events were thought to be good K~ decays into w'w, having failed the kinematics tests either because of a measurably difficult * We attempt a more precise calculation of these two effects in the next chapter.

66 16 1J, 2 40 b Cwz,- 1! 32DISTANCE 10 > 30 40 c d > 24 3 4 X m: 16 < 1 7 N s _ 10 z z- 1234 6789 10 0. 0 0.45 5. 6 25 "STANCE MEfFY] BeV Figure 9. Characteristics of "scattered-V" sample. Separation between C.E. point and V-vertex for: a) events with observed point of scatter; b) events without observed point of scatter; c) total events observed. Figure 9d gives the measured effective mass for the scattered V sample. mass for the scattered V sample.

67 configuration such as a steeply dipping or scattered track or because of a slight K~ scatter before decay. The 22 failing events could in most cases be traced to errors by the second editor. The complement of this measurement check of the charged decays was executed when we examined all V's failing the second edit for the scattered K~ correction described above. It was found that 113 events went from rejected V's to good V's. In summary, the measurements showed that 22 events out of 1000 (13.5% of the 7500 initial V's) were incorrectly called charged decays while the re-examination of the rejected V events (the total sample of them) showed that 113 events were incorrectly called failing V's. We can now compare these results for consistency with the re-edit findings given in Table V. It is obvious that the 22 events of the 13.5% sample measurement is consistent with the 24 errors of comission by the second editor uncovered by the re-edit of the 20% sample. We also note that the 113 events which went from rejected V's to good V's is perfectly consistent with the 25 errors of omission uncovered by the sample re-edit. We have thus determined our V classification efficiency in two independent manners and found them to be consistent. Combining the results we apply a net correction of 90 ~45. The third correction applied to the charged sample accounts for the very minor w contamination of the beam. From Table IV we see that 76 "w- only" events were found in the K+ sample compared to 275 found in the sample of a+ film scanned, and since these events can be produced only by w+

68 interactions we deduce that the K film contained a contamination of 7+ mesons equal to 76/275 =.276 ~.032 of the f film. The w+ film contained 195 events simulating V events, hence the K film would be expected to contain 195 x.276 = 53 ~8 events due to f contamination which must be subtracted. Another correction to be considered is the possibility of interference between the KO - 2w and Ks~ 27 amplitudes. L 5 If it is determined that o = nL_ then this correction is unnecessary; however, since JT 1 is poorly known we are compelled to include the effect. From Appendix C, expression (C.41), we see that the integrated time dependence of the decay rate in the +w channel (assuming only cp 0) contains the major contribution -r T M ip|2 (1-0 -e ) M+- 2p12 and a correction term due to interference effects C_ - 14+_o [a cos cp_ - f sin cp ] where -rs/2 T r r a: e [Tsin (S T) - cos ( T) + 1.0] ([) 2 (+) (-) * - + - An event such as w n 4 T 7w p simulates a good Type II event and would thus enter our charged sample.

69 For our fiducial volume T = 6r so that s M+_ = (4.988 x lo 1)pl2 Then using7 1+T_1 = (1.92 ~.04) x 10-3 and35 p+ = 40~ ~6~ we can calculate the correction term to be c+ = (2.84 +0.07 x 10-) -0.10 x 10 so that the percentage of ultimately detected Tr 7r decays due to interference effects will be (.57 +.01)/pi The magnitude of the regeneration constant o, as defined, here, can be expressed in terms of the matrix elements Rij(x) as defined in Appendix C, Section 2, to be 1.0 + R12 1.0 - R1 Upon solving for o using heavy freon characteristics, we find p(CFjBr) = 1.04 which implies that the actual percentage is (.55 t -0:)8 Using the subtotal of 7804 events (raw sample plus all previous corrections), it is found that 43 ~7 must be subtracted as arising from interference effects. The final corrected number of K0~ - 7Tr W decays was s 7756 ~99 events.

70 3. THE NEUTRAL DECAYS The initial sample of events stood at 4280 ~65 and, as in the charged decays, certain corrections had to be made to this quantity. Since visual observation of the decay process Ks 7r. + 7r is contingent upon the detection of one of the s ultimately produced gammas, checks and corrections are very difficult and must reflect this uncertainty with larger assigned errors. A summary of all corrections for the neutral sample is given in Table VII. TABLE VII CORRECTIONS TO Ko - 77r0o SAMPLE Total y events from 2nd edit 4280 ~ 65 Due to imperfect detection 101 ~ 28 efficiency Edit corrections -20 ~ 40 Due to v+ contamination -284 ~ 36 Due to background y's -284 ~ 35 Outside fiducial volume -109 ~ 11 Due to K0-Ko interference -50 ~ 50 L s Total K -+ r~r0 events 3654 ~100 The techniques for reconstructing and fitting gamma conversions in heavy liquids need much improvement; therefore any checks that were made had to be made via Monte Carlo

71 predictions. We feel justified in pursuing this course since very few sources of extraneous y's are available. Even with the Monte Carlo predictions there is no off-hand check on the neutral sample because of the very intimate relationship between the corrections and these checks. To illustrate this Figure 10 presents the distance between charge exchange point and nearest Y as determined by both the second editor and the Monte Carlo methods. The two distributions are normalized to the total number of events so that there is surprisingly good agreement for the first two bins; however, beyond this, as the distance increases, predictions and reality tend to diverge. This does not indicate that there is a basic fault with our neutral sample. On the contrary, since the Monte Carlo points as plotted assume perfect (100%) gamma detection efficiency, we would be at a loss to explain agreement at this stage. Thus our first correction to the neutral sample, a positive correction for unobserved gamma events, should result in a more meaningful check between observed and predicted gamma. distributions. The probability of none of the four gammas from the neutral decay converting within the 10 cm (projected on the scan table) scanning circle is only about 1%. However, not all of the converted gammas will be observed by the editors. Some will be missed because they are dipping steeply in the chamber, others because their energy is too low. It is an important advantage of this experiment that this correction remains small even when low scanning efficiencies are assumed.

72 2000 I DATA POINT MONTE CARLO POINT o000 (100% efficiency) 80 600400 I (1) 200 z II —I w 100 c" 80 | 60 20 20 2 3 4 5 6 7 8 9 10 DISTANCE Figure 10. Separation between C.E. point and conversion point (nearest e+e pair) as observed and as predicted by unweighted Monte Carlo. One unit of distance is ~ 2 cm real space.

73 To determine this correction a small sample (12%) of neutral decays was examined by those scanners who were responsible for detecting the gammas initially. They were instructed to scan within a 10 cm radius of the charge exchange point and count the number of gammas. We then hypothesized three scanning models whose ultimately predicted gamma detection efficiency depended on a weighted product of a dip factor and an energy factor. Hypothesis A assumed that the detection efficiency for electron pairs varies between 0.0 and.95 as the gamma energy varies from 0 to 80 MeV but never gets above 0.95. Also the detection of dipped electron pairs is further inhibited by a factor which varies from 1.0 to 0.0 as the dip (measured up from the xy plane) varies from 30~ to 90~. Hypothesis B was designed similarly except that the energy factor reached. 0.95 at 40 MeV and the dip factor was 1.0 until 600. Hypothesis C assumed that the detection efficiency was both energy- and dip-independent and assumed a straight 95% detection efficiency. Figure 11 gives the distribution in dip of the gammas as predicted by the Monte Carlo program. Also contained in Figure 11 is a graphical description of the dip factor as used in the three hypotheses. Figure 12 gives the gamma ray energy distribution, again Monte Carlo predicted, as well as a graphical representation of the energy factor for the three hypotheses. Each of the hypotheses was subjected to Monte Carlo methods which resulted in a prediction for the number of unobserved gamma events, the ratio of 4-y to 3-y events and the ratio of (4-y + 35y) to (2-y + l-y) events as

74 1.0. 0. HYPOT4SIS C. 0.7 HYPOTHStS IB 0 OH.- HYWIHE~t$ A M Q0.4- / /S \ \ 13 0. 0.1 0. I a 84p J + > 72- w o90 70 50 30 -50 -70 -90 y-RAY DIP (degrees) Figure 11. Predicted y-ray dip and three hypotheses used to. 48determine detection efficiency. Weighting factor varies 2490 70 50 30 10 0 -10 -30 -50 -70 -90 linearly from 0.0 to O.. 100% detection efficiency waries assumed. to yield distribution.

75 1.0 YPOTHNES3S C HYPOTHESS S 0.6 kh k i E Q HYPOTHESIS A 5LL. 0.40.2 0.0 350 300250 - V) z 200L. L looz 50 0.1 0.2 03 0.4 0.5 06 0.7 y-RAY ENERGY (BeV) Figure 12. Predicted y-ray momentum and weighting factor for three hypotheses. 100 detection efficiency was Assumed to yield distribution.

76 well as the distribution of distances between charge exchange point and nearest y. The results for the three hypotheses as well as the ratios observed in the sample scan are summarized in Table VIII. TABLE VIII DETECTION OF y CONVERSIONS Hypo- Hypo- Hypothesis A thesis B thesis C Observed 4y/3y ratio.242.422.553.401 ~.050 (4y + 3)/(l + 2y).456 1.06 1.425 1.053 ~.085 Number of y events 286 96 50 missed It was then trivial to interpolate between the Monte Carlo predictions to arrive at our final correction for unobserved gamma events. From the 4y to 3y ratio, 10 0unobserved gamma events were predicted, while from the (4y + 3Y) to (2y + 1Y) ratio a consistent number 98 5 of unobserved +28 gamma events resulted. Thus, to the raw sample, 101 2 -20 gamma events were added. We can now also see how the unobserved gamma factor affects the distribution of nearest gammas as in Figure 10. Using the distribution of nearest y distances as predicted by Hypothesis B we find the distribution of Figure 13 which also contains the unweighted Monte Carlo points for ease in

77 comparison. As can be seen, agreement between the observed and predicted distributions has improved perceptibly. The next correction accounted for edit misclassifications as uncovered by the sample re-edit. From Table V this correction can be calculated to be minus 20 ~40 events. A correction for v contamination had to be made for the neutral sample as well as the charged sample. Using the same procedure as before and the results of Table IV we find that the number of y-events in reality due to w interactions was.276 x 1025 = 284 ~36 events. A further correction to the neutral sample is that which must be made to compensate for the background, unassociated y-rays which came from sources other than Ks - w7r~w decay. These background gammas have the following sources: (1) totally external y production; (2) w7 production along with the charge exchange; (3) a K interaction producing a A which decays by the neutral mode nv~; and (4) KL decay into three neutral pions. Of these effects, (4) is negligible within the * fiducial volume we are considering. Sources (1) and (2) can be compensated for by realizing that y + KL events, which make up the first two sources, should occur 1.47 (assuming charged to neutral ratio of about 2.12 and KL to K0 ratio of Lj s 1:1) times as often as the y + V events which we have counted and corrected. Hence we would expect sources (1) and (2) to A KL with a mean momentum of.325 BeV/c will decay into its 53~ mode within the first 10 cm after production only - 0.2% of the time. If we then fold in conversion probabilities we can safely ignore this correction.

78 2000I| DATA PONT |'.'!S> MONTE CARLO PONT Koo- 00% FFC IEWCY) oo00-; A MONTE CARLO.POINT 600: - (HYPOTHCSIS B) 600 ir. 400 t2O 100 z 60 40 20 10 I 2 3 4 5 6 7 8 9 10 DISTANCE Figure 15. Separation of C.E. point alnd nlearest e e pair a:s observed, as predicted by unweighted Monte Carlo, and as predicted using Hypothlesis B.

79 contributed (149 ~22) x 1.47 = 219 ~32. Source (3), A decays, was accounted for by scanning a sample of the film for the charged A decay (A * p- ). For the sample scanned 30 ~10 charged A decays were found which would indicate 112 ~22 in the whole sample. Taking the charged to neutral branching 7 ratio7 of the A~ as 1.88 ~.02 we estimate the number of A~ 4 nv~ decays as 65 ~12 events. Thus the total necessary correction for stray y's is 284 ~35 events. The complementary correction to the question of stray y's is the correction for an actual K~ -4 r~o decay whose only s * qualifying gammas have false alternate origins. To estimate this correction a sample of K+ decays in flight were found. w2 In the decay K +~ the decay vertex and thus the origin of the two y's is easily defined. Hence we know the true origin of the gammas and we could then find any alternate origins which also fit the gammas. A total of 1466 K 2 events were initially found and subjected to an edit which cut all but 461 of them. Next these 461 events were measured and geometrically reconstructed using CAST-SHAPE as described earlier; 309 passed the measurement stage. These 309 events were put through a K constraint program which.gave positive fits on 243 of them for a final sample of 486 gammas. It was found that 62 of these gammas had alternate origins which would seem to imply that we should increase our 1-y events by A full report of this phase of the experiment is contained in University of Michigan Bubble Chamber Group Research Note No. 14/69 by K. Bartley.

80 14.6%, our 2-Y events by 2.1% (14.6% squared), our 3-y events by 0.31%, and our 4-Y events by 0.045%. However, to perform this correction meaningfully, the distributions in e, p, and P should be identical for the K2 gammas and the Ks -, w~7r~ 7V2 S gammas. When the necessary cuts are imposed on the K sample to meet this condition it is found that the sum total (l-y + 2-y + 3-y + 4-y events) of gamma events missed in this manner is insignificant and thus negligible. The next correction to the neutral sample makes use of the fact that in the K~ -O w~70 events we do not observe the point of decay but rather the point of conversion for one or more of the resultant gammas. For this reason a separate check must be made to insure that we are using the same fiducial volume for both charged and neutral decays. To guarantee this all events accepted in the neutral sample were examined by experiencedscanners or physicists with the sole purpose of determining the copunctal point (point of origin) or distance of closest approach for l-y events and then measuring the distance between charge exchange point and copunctal point. Monte Carlo methods were used to predict what the distribution should resemble, and the results, both observed and predicted, are presented in Figure 14. From the observed distribution *. we see that 109 ~11 gamma events must be subtracted from the gamma total to insure equal fiducial volumes. Another interesting feature of Figure 14 is its consistency with Figure 8 *x. We have already subtracted the 284 stray y events before arriving at this figure.

81 10000 8000 6000- I DATA POINT 4000. MONTE CARLO POINT 2000800 l~ I- fiducial boundary 800 - zw 100 W 400 E l 0 2000 Z 80 - | 604020 I 2 3 4 5 DISTANCE Figure 14. Separation of C.E. point and neutral decay point as observed and as predicted by Monte Carlo technique. One unit of distance is ~ 5 cm real space.

82 (charge exchange-V vertex separation) in terms of the various pheonomena (both regenerative and non-regenerative scattering) we would most likely observe in the last half of our fiducial volume. To confirm that these various effects are not biased toward one or the other of the decay modes we can determine the branching ratio for the last half of the fiducial volume and compare it with the overall branching ratio. In the last 10 cm (real space) there are 378 ~19 V events and 288 ~17 y events. Correcting each sample as described throughout this chapter we obtain corrected samples of 455 ~21 V events and 227 ~19 y events. The resulting branching ratio is 2.00 ~0.19; certainly consistent with our overall branching ratio. As in the charged decays, a certain small percentage of those events we have detected are due to interference effects between the K. -. 2w~ and K~ - 2w~ amplitudes. In this I. s case we are interested in the parameters Ion and c 00 Since these quantities are very poorly known as of now, the error on this correction is quite large. Following the same procedure as used for the charged decays we find that 30 ~30 events must be subtracted from the neutral sample. Thus, with all correction considered, the final neutral sample stood at 3654 ~100 events. 4. THE K~ SAMPLE As a further check on our experimental method a count of K0 decays (charge exchange with neither V nor y associated)

83 was made. It follows'6 from the CPT theorem that the originally produced K~ in the charge exchange reaction K+ n 4 K~ + p has an equal chance of decaying as a Ks or as a KL. s L~ Thus, aside from uncalculable regeneration effects, the corrected number of K~ decays should equal the corrected number.... "''. s''' of K~ decays at the termination of our experiment. Table IX L summarizes the KL decays. L TABLE IX CORRECTIONS TO K0 SAMPLE L Total with neither V nor y 11747 ~108 from 2nd edit Due to t contamination -344 ~ 19 Due to background y's +284 ~ 35 Due to missed y conversions -101 ~ 28 Scattered V events -242 ~ 15 Re-edit corrections -55 ~ 55 Total K~ observed 11289 ~134 L The raw sample stood at 11747 ~108. The corrections to this sample are straightforward and in all cases except two are compensations for events shifted to or from the Kc~ decays in correcting the charged and neutral samples. The first exception is the correction for w+ contamination of the beam.

84 Following procedures previously established for the Ks decays s and using Table IV we must subtract 344 ~19 events from the raw sample. The other exception is the edit corrections. From Table V we see that the sample must be diminished by 55 ~55 events for this correction. The only other corrections are as follows: (1) 284 ~55 events which were determined to be KL + stray y and subtracted from the neutral sample are +28 added here; (2) 101 -20 events which were undetected as neutral decays because of imperfect gamma detection efficiency were added to the sample of neutral decays and must be subtracted from the KL decays; (3) 242 ~15 events whose associated V's failed to conserve momentum at the scan table were originally classified as KL decays. They have since been added to the charred sample and must therefore be subtracted from the KO charged sample and must therefore be subtracted from the Ks sample. The final K0 sample thus stands at 11264 ~154 events. iL

CHAPTER V RESULTS AND CONCLUSIONS 1. THE K~ CHARGED TO NEUTRAL BRANCHING RATIO From Table VI the corrected. total of K~ - + T s decays found in this experiment is 7736 ~99, while from Table VIII we note that the total K~ o w~ + w~ decays stands at 5s~~~~~~~s 3654 loo100. From these final figures we determine the Ks charged to neutral branching ratio to be F(K0 + + R (K -L X-+- ) - 7756 ~99 = 2.117 ~0.064 r(Ks 0.. v +'o)'3654 ~100 This is approximately two standard deviations away from 2.0 which we should obtain if the decay is pure AI = 1/2. It then appears that the odds are twenty to one in favor of some combination of AI = 3/2 and AI = 5/2 amplitude in the K~ decay. s Note also that although this result is consistent with the weighted average of the old generation experiments (2.16 ~.08) it is ~ 2.5 standard deviations away from the only other completed recent experiment of Gobbi et al. (2.285 ~.055). Therefore, an immediate and obvious result of this experiment is to point out the necessity for further high accuracy experiments to resolve the current substantial deviation in reported values. There are several attempts reported underway at this time. Of these the Hungarian group, Bozoki et al.37

86 has released the preliminary value of 2.12 +.17 in a recent preprint. Though it is questionable as to what value a weighted average of three such widely varying (2.285 +.055, 2.160 ~.080, and 2.117 ~.064) quantities is, the average obtained is 2.201..071. It is now possible, using expression (1.14), A2 R - 2 = 6/2. Re( —) cos (62 6 (5.1).0 as derived in Chapter I, to determine various quantities of interest depending on what we assume as known. 1.1 w7r s-wave Phase Shifts If we assume that there is no AI = 5/2 component of the decay amplitude, then it is possible to determine Re(A2/A ), as done in Chapter I, to be 0.044. Using this quantity, our value of R and expression (5.1) we can now solve for the difference in aw s-wave phase shifts 162 - 6 at Em = MK o Kcm o where the subscripts indicate the isospin of the Tw final state. It follows that R - 2.117 ~.064 2os(6 6) A A- --- =.3514 ~.172 2 3 6. A *.373 6. Re(A) which implies that the difference in s-wave phase shifts is

87 +10 162 6 = (72 1)~ The rrT s-wave phase shifts can be independently measured by studying the strong interaction r + N -* N + (n7r) Unfortunately the phase shifts derived, both experimentally and theoretically, are strongly model-dependent (as will be shown in Appendix D), so that agreement among various reported values is in even greater turmoil than the branching ratio we have determined. Table X summarizes the most recently reported experimental values, as well as those values predicted theoretically using current algebras, Veneziano and phenomenological S-matrix techniques. As can be seen, 162 - 6 | can 0 range all the way from (28 ~10)~ —using the Hagopian et al. value of 6~ and the Baton et al. value of 65 —to (73 ~10)0 0 0 using the Biswas et al. value of 60 and. the Malamud et al. O0 2 value of 6. An attempt at finding some sort of mean value 0 for these widely varying results ends with a value of 5G0 ~15)0 for the difference in phase shifts 162 - 6 1. ~~~. ~~~~.~~2 0 This value covers the entire range of theoretical values, from Arnowitt's current algebra prediction of (35 ~4)~ to Tryon's value, using a unitarized Veneziano partial wave analysis, of ~ (60 ~8)0~ A complete review of the current state-of-the-art in 7Tww phase shifts as well as a description of some of the theoretical models and experiments can be

88 TABLE X REPORTED VALUES OF 6~ AND 62 0 0 0 0 0 0 Reference (Degrees) (Degrees) (Degrees) Gutay8 and Carmony3' 19 ~6 (52 ~10) 40 Biswas et al. 4 56 ~10 -6 ~3 62 ~10 Malamud and Schleinl 41 ~5 -17 ~3 58 ~6 42 Hagopian et al.2 23 ~10 (75 ~10) Cline43 50 ~17 -10 ~3 60 ~17 44 Scharengiuel et al. 42 ~7 -7 ~3 49 ~8 Saxon et al.45 27 ~7 (50 ~8) Baton and Laurens4 -5 ~3 47 Katz et al.47 -8.7 ~0.9 48 Smithi and Manning -9.8 ~0.7 (-15.4 ~1.1) 4Q Current Algebra49 35 ~4 Veneziano50 54 ~5, -7 ~5 59 ~9 52 ~9 61 ~6

89 found in Appendix D of this paper. There are several implications of the quantity 162 - 6 j which we have determined and quoted as 162 - 6 [ = (72 ~10)~. First, and most important, is that in arriving at our value for 162 - 6 1 we assumed that the AI = 5/2 component of the decay amplitude was zero. If further experimentation shows that the 162 - 6 j lies in the vicinity of the mean value we have determined —(50 ~15)~ —or even lower, then only an admixture of AI = 5/2 amplitude will reconcile this lower value of 162 - 6 j and our experimental result for the K~ charged 2- 0 S to neutral branching ratio. This point will be examined in detail shortly. A second point of particularly current interest in which both our value of R and 162 - 6 I play a part is the question of time reversal invariance in the decay of KO~ - 2w. The breakdown of CP invariance for this decay has been experimentally known for some time. Thus, accepting the validity of the CPT theorem52, the violation of T invariance seems insured. Recently, however, R. C. Casella has shown 55 how, with the use of quantities determined from K~ decays, T invariance can be tested independently of CPT. Employing two related methods, a modified Wu-Yang triangle 56 and the Bell-Steinberger sum rules, Casella assumes T invariance and upon this assumption arrives at contradictions with the data. Since the method employing the modified Wu-Yang triangle can be shown to include the Bell-Steinberger sum rule analysis, we will briefly examine Casella's first method only. The

90 57,16 original Wu-Yang triangular relations were based on the assumption that the 27r decays were mainly responsible for the CP breaking effects (i.e., no large amount of Ks~ r), the validity of CPT, and the dominance of AI = 1/2 transitions in the K - 2rr decay. The resulting relations are _ -= e + e' (5.2) -= - 2e' (5.3) o0 where e and c' are related to the coefficients of K and KL (p and q in Appendix C) in the K~(Kr) state. The phase of e and. e' are given by [2(mL ms) 9= tan1 -- + Ae (5.4) S rL and, = - (6 -62) + 900 (+1800). (5.5) It is possible to modify these relations so that instead of being based on the validity of CPT, they are based on Tinvariance. The new relations 0 0

91 incorporate phases for e and e' which are rotated 90~ from the CPT phases e_ = - 90~ 0 = 0 - 90'0 These modified triangular relations are solved by putting in the measured value of 8 = -47~ ~10~ and value of'O r -- _0 rl-% which then results in a range of values for (6 - 6 ) where 0 2 the triangle does not achieve closure, hence implying Tnoninvariance. Thus, with our value of the branching ratio which reaffirms AI = 1/2 dominance —one of the bases for this whole analysis —and our value for j6 - 621 it can be shown 0 21 that T-noninvariance is ascertained for all values of r < 3.5 if (6 - 62) > 0 and for all values of r < 1.8 if (6 - 62) < 0. 1.2 The Quantity Re(A2/A ) and AI > 1/2 Transitions It is also possible using expression (5.1) to obtain the value of Re(A2/A ) by assuming that the phase shift 62 - 6 is known. For the sake of simplicity we will initially use the mean value of 16p - 60 =- (50 ~15)~ quoted in the last

92 section for computations and generalize the results to cover the full range of 162 - 6 later. We thus want to solve the 0 equation A2 R 2 Re(A-) = cos (62 ) (5.6) 0 2 0 using R = 2.117 +.064 and 162 - 6 i = (50 +15)~. We find 0 A2 Re() = 0.022 ~0.014 0 where the large error is basically due to the fact that it is the quantity R - 2.0 that determines the admixture of I = 0 and I = 2 states. Figure 15 generalizes this result in that it graphically shows how Re(A2/A ) behaves as a function of the difference in phase shift 162 - 6. Our error in R - 2 has already been folded in and is shown by the lower curve which gives the effective lower limit of Re(A2/A ) for any given value of 6 - 6 2 0 1.3 Ratio of AI = 5/2 to AI = 3/2 Amplitudes If we now assume that the Im(A2/A ) is less than 10-3 (recent measurements ofl j imply that Im(A2/A ) < 3 x 10 ) then we can find 1A2/Ao from the value of Re(A2/A ) we determined in the last section. Having this quantity and using the K+ -, Wrro decay rate as input we can determine the ratio

93 QOS 0.06_ o.o40r13 0.0 0.02 0. 10 20 30 40 50 60 70 80 90 I12 Q01 degrees Figure 15. Dependence of Re(A2/A ) on value of 162 - 6 using our value of R ~AR = 2.1I7 %0.064. Only median ~ value and lower limit (1 S.D.) are drawn.

94 -I.0 -1.0 -0.'7 -0... -0. 0 \ 0.1 10 20 3040 0 6 70 0 90 O.5 L5/26'igure 16. Dependence of 5 n value of 6 61 using our value of R ~ARt 2.117 ~0.064. Circles represent median values while upper and lower limit curves are at ~1 S.D. The equation yielding the smaller value for -5/ was used in determining the above curves. 3a/2

95 of AI = 5/2 to AI = 3/2 amplitudes as shown in Chapter I. We are again hampered by the large error in the quantity R - 2 (a 55%) and the large error in cos (62 - 6 ) ( 355f) and the fact that the resultant of these two errors, added in quadrature, appears both in the numerator and denominator of (5/2) \3/2 Assuming the amplitudes are real, the two solutions are X -01591 ~0.51 -0 -2.18 ~2.85 \15/2 2 35/2/ It is obvious that although these values are consistent with no AI = 5/2 transitions, the necessarily large errors preclude any positive statement. We have arbitrarily chosen the root smallest in magnitude for further calculations. Figure 16 5/2 generalizes our results and gives 5/- as a function of'3/2 162 - 61. 2. RATIO OF Ko TO Ks PRODUCTION L s From Tables VI and VIII, the total corrected number of Ks -* 2-T decays is 11390 ~141, while from Table IX we see S that the total corrected number of KL decays is 11289 ~134. Thus our final value for the ratio R is L,S R N(KS) _ 11190 ~141 = 0.503 ~o.oo8 LS N(Ks) + N(KL) 22679 ~195 while the absolute difference in the number of K~ and K~ s -JI

96 TABLE XI SUMMARY OF TOTAL K~ AND KL PRODUCTION Corrected Sample - K0s - rf + - 7736 ~ 99 S Corrected Sample - K0s - T0 + 1T~ 3654 ~100 Total Ks Decays 11390 ~141 S Corrected Sample - K0 -* all 11289 ~134 Corrected Total K~ Production 22679 ~195 T(K -, 27) ---— s - -0.503 ~0.008 r(Ks 4 27) + F(KL 1 all) TABLE XII Re(A2/A ) AND -5/2 FOR THREE VALUES OF R R Re(A/A) /25/2 R Re(A2/A) 5 ~3/2 2.117 ~0.064 0.02 ~o.o014 -0.391 ~0.510 2.201 ~0.071 0.037 ~0.018 -0.110 ~0.092 (Weighted Average) 2.285 ~0.055 0.052 ~0.020 -0.098 ~0.064 (Gobbi et al.) Assuming 162 - 6~1 = (50 ~15)~ 0 0

97 produced is N(Ks) - N(KL) = 101 ~195 The question then becomes what we expect the ratio R to be. L,S As was mentioned earlier, incoherent regeneration phenomena will tend to reduce the number of K" that we detect while supplementing the number of Ks decays by a similar amount. In the case of coherent regeneration, however, the observed number of K~ particles can either be enhanced or reduced depending on the regeneration phase OSL. Since the regeneration phase in heavy freon (CF3Br) is an unknown (we will shortly derive an estimation of eS L) we will, at this point, merely set limits on RLS In Appendix C, Section 2, we estimate that coherent effects can cause as much as a ~4.2% difference between the number of K~ observed in vacuum and the s number observed in our experiment. We have estimated the incoherent effects (Section 3.2 of this chapter) to be a 2.2% enhancement of the number of observed Ks at the expense of the KL sample. Thus, without prior knowledge of the coherent regeneration phase, the sum total of regeneration effects can cause as much as a 6.4% enhancement or 2% depletion of observed (in freon) K~ events over the corresponding number in vacuum. The limits on R, are then 0.9 <,S < 0 L,S

98 Our measured value of RLS = 0.503 ~0.008 does fall safely within this limit. 3. SUMMARY OF RESULTS: CONCLUSIONS In summary, the basic results of this experiment are as follows. ~1.- r(K~ 4 77r + T ) R = s_ = 2.117 ~0.064 r(K~ o TO + 7To) 2. Assuming no AI = 5/2 transitions, 162 61= (72 +10 0 ^ - = (72 -11) 3. Assuming a mean experimental value of the phase shift difference 162 - 6 j = (50 ~15)~ a. A Re(-) = 0.022 ~0.014 0 b. a/ b. 5/2 =-0.391 ~0.510 a3/2 4. In freon N(K~) RL = o.o0.3 ~0.00o8 L, S N(K ) + N(KO) 3.1 Branching Ratio and Associated Quantities There are several obvious conclusions to be drawn from these results. Strictly from the necessarily low accuracy of

99 our quoted values it may be surmised that more high precision determinations of the Ks charged to neutral branching ratio s are needed. They are necessary not only to resolve the final value of R, but also to reduce the error in R still further. Note that even though we have measured R to an accuracy of 3%, because of the value of R itself we have measured the quantity R - 2 to only R 55%. Since R - 2 is the quantity of interest we should aim for an experimental determination of this quantity to within 10% if not less. Figure 17 indicates the type of accuracy needed in determining R to yield 1%, 10%, or 25% accuracy in determining R - 2. As can be seen, if R is indeed, 2.1 then to measure R - 2 to 25%, 10%, or 1% we must measure R to 1.2%,.48%, or.048%, respectively. An analysis of our 3% error shows that the neutral sample contributes 2.75% in quadrature while the charged sample contributes 1.25%. If we in turn now break down each of these errors we find that we are essentially no longer statistics-limited in terms of minimum error attainable. We are rather at the stage where the necessary corrections dictate the major part of the error. Therefore, though it may be still possible to measure R to the accuracy necessary to determine R -2 to 25 with the present methods, it is almost inevitable that other techniques need be used if we wish to measure R - 2 to 10% or less. The attainment of 10% accuracy in the measurement of R - 2 will be a major step forward in eventually determining the quantity Re(A2/A ) to a desirable accuracy; however, it

100 3. 3.23.0 2.8 2.6 2.4 2.2 2.0 1.8 2 1.6'1C/.4/ 1.2 1~to.0 2.0 2.1 22 2.3 R Figure 17. The necessary accuracy of the quantity R to determine R - 2 to 1. 0, 10.0%, and 25.0%.

101 alone will not be sufficient to attain this goal. The present wide variance in the difference of wrr s-wave phase shifts, 162 - 6, must also be resolved. In using the mean value 162 - 6 0 = 50 ~15~ we are introducing a 35% error in cos ((62 - 6 )( which appears in the denominator of the expression for Re(A2/A ). This error plus our huge error in R - 2 combine to give us a 61% error in the value of Re(A2/A ). Figure 18 shows the dependence of the percentage error of Re(A2/A) on the percentage error of cos (62 - 6 ) for different values of the percentage error in (R - 2). We see from these curves that if R - 2 can be measured to 10% then the Re(A2/A ) can be determined to within 11% if cos (62 - 6 ) is known within 5%. A similar analysis can be carried out for the ratio of AI = 5/2 to AI = 3/2 amplitudes 5/, but it'3/2 only reconfirms the necessity for resolving the quantities R - 2 and 6 6 to a much higher accuracy than available now. 2. 0C:O Very little can be said about the quantities Re(A2/A ) 5/2 and 5/2 as determined by this experiment for obvious reasons. 03/2 If the phase shift difference 62 - 6 is ultimately found to be ~ 50~ then AI = 1/2 dominance is insured to the 97% level (from Re(A2/A ) as determined from expression (5.6)). However, it is still too early to make such claims. The - 130% error necessarily attributed to 5- precludes any discussion of the a3/2 result other than what was given at the time we calculated this quantity; thus although this ratio is consistent with 0 (i.e., no AI = 5/2 transition) it also brackets 0.90. To show how the branching ratio itself can influence the

102 35 30- I/ 0 $ a5201- 15.0 / o ej o I C( 0 2 4 6 8 10 12 14 16 18 20 Acos(.2 —,~) (%) ACOS (8- ) I 0 (%) cos (2"- 8o) Figure 18. The behavior of the percentage error in Re(A2/A ) as a function of the percentage error of R - 2 and cos ( 62 -.

103 quantities Re(A2/A ) and 5/2 we can calculate these quantities quanitie Re0A/3/2 using the weighted average for R determined as 2.201 ~0.071 and the Gobbi et al. value of 2.285 ~.055. We again use the mean value 162 - 6 f = (50 ~15)~. Table XII summarizes the 0 results for all three values of R. Under the assumption made about (62 - 6 ), it can be seen that both the weighted average 0 and the Gobbi et al. value are consistent with a small fraction of AI = 5/2 transition, while AI = 1/2 dominance is only assured at the 92% level. If indeed AI = 1/2 rule is shown to be so violated then the T-invariance arguments of Casella based on the modified Wu-Yang triangle and presented earlier, are better replaced by the arguments incorporating the BellSteinberger sum rules and not based on AI = 1/2 dominance. 3.2 The Coherent Regeneration Phase, 0SL in CF3Br We have found it possible to estimate the coherent regeneration phase, aSL, for CF3Br. We emphasize the fact that although the value we obtain is a probable value for SL due to the great sensitivity in eSL for the difference in N(K~) bJLi S and N(KL) a separate confirming experiment is most definitely needed.. From Appendix C we see that the intensity of Ks as a s function of., the distance from origin in mean decay lengths, is Is () = Constant x [e + 3.6 x 10- 2(e cos o - e'/cos (e - 6)]l. (5.7)

104 Thus, at any point., the correction to a pure exponential decay law is determined as C(,) = 3.6 x o102(e cos e - e cos (e - 68)) (5.8) whereas the pure exponential decay contributes the term E() = e. (5.9) Upon integrating over our fiducial volume we find 6 E _ f E(-3 d, = e~- dl; Fid. o Vol. and 6 6 C -- f C(i,) d( = 3.6 x 10'2 j [e cos 9 Vol. - e /2cos ( -6 C")] d Upon integration we find E = 0.9975 (5.10) and C = 0.0595 cos e - 0.0144 sin n (5.11)

105 so that the fraction of events due to the correction term is F = E = 0.0596 cos 9 - 0.0145 sin e. (5.12) We can determine F for our experiment by examining our totals of Ko and K~. From Table XI the corrected number of L s K~ particles (Ns) is 11390 ~141 while the similar total for K~ is N 11289 ~154. However, N contains a correction of L L s 242 ~15 events for "scattered V events" which contains scattered K~ events plus large angle incoherent regeneration effects. Thus, for the determination of F, these particles created as K~ but decaying as Ks must be subtracted from N L s s and added to NL. In Chapter III we estimated that 150 of these 242 events were due to incoherent scatters of 15~ or more. It is possible to make a rough numerical calculation of the relative amounts of K~ scatters and incoherent effects to s check our visual estimates. Let Y be the number of V's within two mean path lengths from the origin and X be the number of V's between two mean path lengths and the end of our fiducial volume. Sx and S are the number of Ks scatters in regions x y S X and Y, respectively, while Lx and L are the number of Ks y s incoherently regenerated from K0 particles. Then x = Sy + Lx and

106 Y =SyS+ Ly Since the scattering is relatively low energy, we assume it is isotropic so that those Ko scatters detected still roughly s follow an exponential distribution law, hence Sx e -2.0 -6.0 x e-20 - e - 0.14. - = - = 0.162 _-2.0. S 1.,0 - e20 0.86 The distribution of incoherently regenerated K~'s should remain flat over our fiducial volume so that L 2.0 y We thus have four equations and four unknowns and we can solve for L = (Lx + Ly) =1.635 x - 0.265 Y From Figure 9 we find X = 115 ~11, Y = 127 ~11 so that the total number of large angle incoherently regenerated Ko's which decay via the charged mode are calculated to be L = (158 ~18) - (54 ~3) = 154 ~18 which agrees quite well with our visual estimate. To this

107 total we must add the estimated number of small angle (< 15~) incoherent regenerations which were not caught at the edit stage. From our Gaussian approximation in Appendix C, Section 2, we find that the ratio of scatters of < 15~ to scatters of greater than 15e is w 1:10, hence the total number of regenerations is estimated to be 170 ~18 events. This would imply that the number of neutral decaying regenerations is.170 ~18 8o ~9 2.7117 ~64 80 ~and the total number of incoherent regenerations is 250 ~20 This quantity must be subtracted from N and added to NL. The corrected numbers for the two samples are Nc = 11140 ~142 and NL= 11539 ~132 with the difference in the two samples being NC NL = -399 ~196 N-C

108 We propose that the difference between D as calculated above and D as predicted by CPT (D = 0) is due entirely to coherent regeneration effects, all other effects being corrected for. Assuming our proposition, the quantity of CF3Br contained in our fiducial volume (a sphere of radius 10 cm) acted as an "anti-regenerator" subtracting 200 ~98 events from our Ks sample and adding them to our K0 sample. This being the case we determine F to be F ( 4 42)+ (200 98) = -o.0176 ~0.0086 (ll'1-40 ~1l42) + (200 ~96) From expression (5.12) we have -0.0176 ~0.0086 = 0.0396 cos 0 - 0.0145 sin 0 (5.13) so that the quadratic equation for cos 9 is 5.73 cos2 e + (4.5 ~2.21) cos 0 + (0.33 ~0.98). (5.14) Upon solving (5.13) for 0 and using the relation OSL'= 0e we find the only solution for 9SL compatible with expression (5.13) is eSL = (-40 ~25)~ (+90~)

109 We submit this then as a possible value of the phase for coherent regeneration in CF Br. This result can be compared with experimental determinations of the regeneration phase in other materials: ecu = (-36 ~12) (+900~). = (-43 ~8)0 (+900).0 = (-28 ~18)0 (+900~) These phases for copper (C. Alff-Steinberger et al., Phys. Letters 20, 207), carbon (M. Bott-Bodenhauser et al., Phys. Letters 20, 212), and hydrogen (M. Lusignoli et al. to be published in Nuovo Cimento) as well as our value in freon are only defined to within +90~ because of the behavior of the cosine of the phase.

APPENDIX A CHARGE EXCHANGE SCAN RULES These are the scan rules as used by the scanners in this experiment. 1. Beam Tracks. If a track satisfies all of the following, it is a beam track: a. Use the beam track template to determine whether a track enters the fiducial volume and has the correct entrance angle. b. The track must not be separated by more than 1 cm in views 1 and 4 at the point where the track enters the fiducial volume. Perform this test on low magnification with both views 1 and 4 on s imultaneously. c. The track must be minimum ionizing at the point of entrance of the fiducial volume. 2. Primary Interactions. Scan EVERY beam track along the path of flight (i.e., start at the point where it enters the fiducial volume and follow it as it moves deeper into the chamber) for a primary interaction. A primary interaction is any one of the following. a. Any scatter of the beam track of more than 10~. b. Any 0, 1, 2, 3,... prong interaction. Note that delta rays ("boil off" electron) are not considered prongs. c. Any decay of the beam track. 110

111 d. Any abrupt change in bubble density of the track. The importance of along-the-path scanning cannot be emphasized enough. Please strictly follow the above procedure. 3. Types of Recordable Events. If the primary interaction falls into any one of the following classes of interactions, record it on a mark sense card. a. Record any zero prong beam track interaction if the beam track is not heavily ionized within 2 cm of its end on low magnification. This is a Type 1 interaction. Figure 19 shows this type of event with a K0 -* ~~ + 7r~ decay. b. Record any 1, 2, 3, 4,... prong interaction if all of the outgoing tracks satisfy all of the following. i. All outgoing tracks must stop in the chamber, and the last 2 cm on high magnification of all prongs must be heavily ionized. ii. There are no negatively charged prongs (see (c) below). iii. The outgoing prongs must not interact or decay; thus, they may not have any abrupt change in ionization. iv. A prong may scatter through any angle as long as it satisfies the change in ionization criteria above (iii) and as long as it does not produce one or more additional

d. o " r~~~~~~~~~~~~o Ar~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r \/;.' ** -.' */* - - ^ i ~~~~~~~~.*~ -I *. i *, \~ ** /**. -^.. * *... *.... - * ~ *. ~/.I..-^,' t /.O *. * C.,^ *iur 19 A tyia 0 -pron"u. Type: I evn wit subsequen.K + ~' * *. * <C, * /^' *.' * ^'' K*.+ 2vo decay. / * -* *** L-.- - - r-^ —'i. * S.. * *..*' {.. * /'; - J - - "; **'' " *' * ~ *./. *"1 **~-*..'. / * ** * — ~ f' -i. /' >',*./*.~~ ~ ~ /: ^ * ^'.... * >: /.* ~6. *'..'<', * - /.'1 * /f. * ^ ^ v' ** ^~~. * /... * -r.. *'' * *' * ~~* * N'~~~~~~ ft i / *~' *. Fiue 9 Atpia 0-pr Tp I evn wit susqet K ~ -4 27r eay s

113 prongs at the point of scatter. If a primary interaction satisfies all of the above criteria (i, i iiiii, and iv) it is recorded as a Type 1 interaction. Figure 20 shows a typical one prong event with associated V. c. Record any primary interaction with a negatively charged prong that satisfies both of the following. i. The beam track must be minimum ionizing in the last 2 cm before interaction on low magnification. ii. The negatively charged prong must be longer than 4 cm on high magnification in any view. If an interaction satisfies both of the criteria above, it is recorded as a Type 2 interaction. Figure 21 shows a typical Type 2 event with V superimposed on charge exchange point. While performing disagreement scans for the charge exchange experiment, it has been noted that there are several criteria in the scan rules that are too subjective. Because of these points the scanners are "missing" events or recording extra events. These events which they are missing in no way reflect their true ability to find the desired topologies. Thus, the purpose of this paper is to try and make these offending criteria more objective. 1. Fiducial Volume - Left-hand Entrance Boundary. Because of the uncontrollable difference between

~~ ~. * *' *'..'.''.' **',.* *.:;::.. ** - **" " * *...' -.. *'...' "' *:':.''*- * **... r- <* - -. * *' * *. -. *': *. *** *: *...'* * * * *' *' **' ~* *- *..' *' * **. * * <'** * ** *'' y':.> * - **.. * ^ *.- *"..* *;;.:*'. * **......* * * " *. **.*' > ": * *. *': ** "- * *6 * * * * *.' ~.;..... * ~. * *..' ^' *. * *''' *\ *".*'**''.?. ^''" * ^.-'.' * *.; ^ *'. *..~ ~ "/ *- *' *. * *".':* ***: * * ** * *Ir. ".'* * *^' *.'. *. * * ^ r *' - * *, -^ * I **' *. *~~~, ** * *- - *' * **-' -** * t -/ * * * " <.**..*. *....* / **'*' ** *;. ~' *. *!. * *: * *: * **.: *. * ~. * /. *..' *E ~ * * *~*f *'' *. 1:~~~~ *~' *.. ~,. /..*. * *~' * ** *.'. * *.. *?~;.*' ">,' * ". -' *'.' *.''. ^' -'. *''.* * * ^.. -... ^ \ - *' /'.': * ^ ^ **.*~ "''' *'9.<. ***^*. **. \ z- - "" "'*~ * -*, *,~~ ~ ~ * *. ^ -*'" - **.**. *'*.*** *:' y-. - *~ "s * * / ^* **. ** *" f.'..; * / -^ **.;. ^ -- **. *, / -.., <.. /;*.* * ~..*^' /' - *~~~~~~~~~.. *. *.****~ ** *~ *- /.. - ^;.... ^ **** X^ - /':.?.' ** * ~;t * * -'* /. * "* / // * * ~~~( *~ *~ *'**\ *' -' - ^/ / - ^~ ~ **.- *\ *''. -.' *.. **'. * **'*..~ \ ~.. T ** -**** \ *** * *.; / ^> */*/ *.'\ /'., ~~ ** * ^ * *. ^' * *. ** * * >' * -^* ^'* *., -.:.'* ^ ^ __ *.^ * - /..~ ~~ ~ ~ *~..;~ -- ***.\. *r-*. *;.*/'' * ** *..' *......' / **-..: * j/. \ /,/~ *~~~~ r...* *. A * - / * *- * / -* ** *.,~ *** * * Q* * / * *.f Figur 20 AF on-rn yeIevn ihsbeuet'' K~ -~T'i~ d ca *

a....'......... * 5 *' *''"... - 9 % A * S 5.- &.**-~, 0 - 1~< 9^ >' -'.' / I.* / */. ~, *. -*.., * ~. 9'.,4~j 9 21. A T 2 e n composd / i. - * -: * A. *'... * **/ ~.l ^., * -. * *A -,..' - i'. * a I'C., < 1 ^~ ^ * r *-...-....'-'. "''~ 9..i \'' ~ ~:'''/(.'.$ *.' * H /-~~~~~~~~~^ N.:. * >'./' ~~ I d...', ~ ~. "''a' *'I'. v exchange~~~~~.~.r.-, ~ ((.* / Figr/ 1. K0 - w w decay immediately after charg e

116 machine settings and. templates, the templates presently provided do not give the same fiducial volume from machine to machine. To clear up this point, the left-hand boundary for entrace into the fiducial volume will be drawn from the frame itself rather than the template. A track will be considered within the fiducial volume at the left-hand limit if, at the point of entrance, it is still visible as a black track on a white background. 2. Pseudo One Prongs. In some instances a track may enter the fiducial volume and satisfy the beam criteria. It then appears to "come to rest" somewhere within the fiducial volume. Many scanners are listing these as Type 1, one prong interactions with questionable vertex. To remove these disagreements, I stress the fact that one must be able to detect an obvious vertex which correspons in at least two views. By obvious vertex we mean a change in direction of the track (angle change) or a definite, unquestionable change in density. In many cases, these pseudo one prongs are protons, so when these cases arise check the bubble density carefully and compare with other tracks on the same frame. 3. When and When Not are Two Vertices Connected. This question arises in two contexts. The first, and most important, is differentiating between a true 0

117 prong and the case where the last few bubbles of a track are missing. First of all, a general rule to follow is if two distinct vertices are obviously not connected in any one view then they are not connected. Conversely, two vertices must be connected in all four views to be deemed connected. To determine in any one view if two vertices are connected, use one of the three inscribed curves on the "angle definer" part of the template. If, when you match the curvature of the beam to the template, both vertices fall on the inscribed curve of the template, then the two vertices are connected in that particular view. Check all four views in this manner. The second. context in which this arises is in distinguishing Type l's from Type 2's. First of all, we have changed the scan program so that a scanner will no longer be charged with a miss if he or she incorrectly labels the event type. Let it suffice to say that the general rules —must be connected in all four views to be deemed connected —still hold.. If with this general rule you still cannot decide which type it is, write down your best guess and put a 1 in the picture column. 4. When Does Prong Leave Chamber. When you encounter a very steep track (moves around a lot from view to view) you might question whether it goes out the front or back of the chamber. To determine if it

118 goes out the front, measure the distance between the end of the prong and a convenient fiducial in view 4. Then in view 1 do the same with the corresponding fiducial. If the distance measured in view 1 is within 1 mm of the distance measured in view 4 then the prong goes out. 5. Ionization. This has proven to be the most subjective criteria of all. What is "heavily ionized." and an "abrupt change in ionization" eludes exact definition since it can vary from roll to roll, even frame to frame. To help clarify these points a scanner can use the following guides. The end of a prong is "heavily ionized" if the end of no other track on the same frame is obviously darker. If you have to stop and carefully compare the darkness of two tracks, then one is not obviously darker than the other. As to the question of an "abrupt change of density", let me begin by saying that an abrupt change occurs at a single point —from one bubble to the next — and cannot be spread out over even a 1/2 cm. Secondly, an abrupt change in density is repeated at other points along the track it is not an abrupt change. Note that decays usually have a heavy ionization to minimum ionization type change, while fast prongs that interact usually have a light ionization to medium ionization type change.

119 6. Finally note that the 4 cm rule for the negative track of Type 2's include all segments of the negative track if it should. happen to scatter. In conclusion, if after applying the above clarifications to any particular case you still have a question, record the event —do not throw it out. Also, we have noted that a majority of missed events have occurred. at the extremes of the fiducial volume, either just after entering or at the left-hand boundary. This indicates to me that the scanners are not following the procedure of along-the-track scanning. Be sure to follow every track that has any possibility of being a good beam from the point it enters the fiducial volume to the point where it leaves the fiducial volume if necessary.

APPENDIX B EDIT RULES 1. FIRST EDIT RULES 58 These edit rules, as written by Daniel Sinclair, were to be applied to the film after it had been scanned twice and all disagreements resolved. In. order to make a decision as to the first edit classification, apply the following tests. 1. Check to see that the event has been correctly identified by going over all of the scan rules except those concerned with beam position and entry angle. 2. In order to determine whether or not the event has a e associated with it, apply template to charge exchange point. Template has two circles radius 10 cm and 20 cm (real sphere equivalent). The event has a e if the apex of any "V" can be found within the 10 cm circle on all four views. A'V" is any V-shaped track configuration which is not directly connected to a beam track. One of the tracks should be negative and the other positive, but if they are so short that you cannot be sure of their sign, give them the benefit of the doubt. Be on the look-out for wide angle V's which might look like a single track with no apex. Study all such tracks carefully and if you think there might be an apex, include it. 3. In order to determine whether or not the event has 120

121 a y associated with it, place the template on the charge exchange point. Look on all four views to see if there is a y vertex either within the 10 cm circle or within the 20 cm circle and pointing to within the 10 cm circle. Accept the y ray whether or not it has an alternate origin. It should obey these rules on all four views. 4. If the event is Type 1 and has no 0 and no y's, check to see if another Type 1 is within 20 cm (real space projected) on at least one view. (See Second Edit Rules.) 1.1 Type 1 Events If the event is checked to be neither a Type 1 or Type 2 then no further work need be done and the first edit classification is 00. If the event is determined to be a Type 1, decide on one of the following first edit classifications. a. Event is checked to be a Type 1 but has no 8 or y pair associated with it and has no other Type 1 interaction with with 20 cm on any view.................................... 10 b. Event is checked to be Type 1 and has a 8 associated with it, but no y............. 1 c. Event is checked to be a Type 1 and has a y but no a associated with it.................. 14 d. Event is checked to be a Type 1 and has both a a and a y associated with it......... 15

122 e. Event satisfies (a) except that it has a Type 1 interaction with 20 cm on at least one view...................................... 16 f. It is impossible to make a decision on this event.......................................... 19 1.2 Type 2 Events If, on the other hand, the event is checked out to be a Type 2 event examine the interaction vertex to identify the secondaries and search for a 0 and y as described above. Then classify it as follows. a. Tr in flight................................... 20 b. w with K+.................................... 21 c. w secondary with no associated 0 or K... 22 d. w secondary with an associated.............. 23 e. 7 w on legitimate Type 1 with an associated 24 y.......................................... 24 f. w w on legitimate Type 1 and no associated y.. 25 g. e or e e pair on legitimate Type 1........... 26 h. e or e e pair on illegitimate Type 1........ 27 i. wr+w on illegitimate Type 1.................... 28 j. A decision cannot be made...................... 29 By "legitimate Type 1" is meant an event which would fit the rules for a Type 1 interaction but for the presence of the e or e e pair or r7r.

123 2. SECOND EDIT RULES The events to be examined in the second edit are those classified as 13-15, 23-25 in the first edit. If for some reason the second editor wishes to reject the event, the classification will be 000. To ultimately classify the event one or more of the following tests will have to be applied. 1. Second Edit s-Test. This test has three parts to it. a. The positive and negative track should be consistent with being pions. b. The straight line jointing the K charge exchange point to the apex of the V should be contained in the arms of the V. c. If one pion is obviously faster than the other then the faster pion should make the smaller angle with the straight line joining the charge exchange point to the apex. 2. Measurable Test for e's. Both the v+ and wv tracks must be measurable; that is, they must either stop without any inelastic interaction or they must have at least 15 cm of chord length (projected) on at least one view before interacting. (Note: 15 cm 7.5 cm low magnification.) 3. Second. Edit y. The test proceeds thus. First find that (those) y(s) which either lies within the 10 cm circle, or lies within the 20 cm circle and points to within the 10 cm circle on all four views. Place

124 the "pointing template" on each y in turn and check to see if any y has an alternate origin consistent with all four views. If at least one of these y's can be found without an alternate origin then the event has passed. Otherwise it has failed. An alternate origin may be the following. a. The interaction or decay of a beam track except when the interaction is another Type 1 meeting the requirements of b. The interaction or decay of a secondary except simple proton scatters. c. The stopping point of a w- (but not a w ). d. The stopping point of a K (but not of a proton). e. The tangent line to any electron or positron track from any source. f. The interaction point of w+ or wr tracks from any source. g. Any neutron star except single proton. For events which pass the test an estimate should be made of the total number of copunctal y's. To do this, scan the whole picture, not just the 20 cm circle. Use judgment as to the origin of a y-ray. A y-ray with an alternate origin may be included in this count if, in the editor's opinion, it is more likely to be a member of the copunctal set than it is to be associated with the alternate origin because of the longer conversion distance to the alternate origin.

125 4. Measurement of Distances. a. For e's measure the longest projected distance between the apex of the 0 and the charge exchange interaction in millimeters on low magnification. b. For y's measure the distance between the charge exchange point and the apex of the closest y. Make the measurement in centimeters on low magnification and on that view in which the distance is longest. Round off to the nearest centimeter. 5. Searching for Another Type 1 within 20 cm. Place the template on the event and search within the 20 cm circle on all views. Accept any event within this circle on at least one view whether or not it meets the fiducial volume and angle requirements of the scan rules. However, it must be a legal Type 1 in all other respects. See Figure 22 for this topology. 2.1 With V For events classified as 13 or 23 by the first edit, the following tests should be applied. 1. Second edit 0-test; does it pass? 2. Is the 0 measurable? (See later instructions.) 3. Is there another Type 1 vertex within 20 cm on any view? Code your result as follows: a. It is not a 9 and there is no other Type 1 within 20 cm on any view....................... Oij

~. ~. ~ ~. ~ tor." ~*' * ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~,, ***'.,%''.*^ ^ " /.-, e....~~~'% t *'' i?'.'''^' r, ~~~~~~~~~~~~~~~~~~~~~~~ V~ V4 0, l b 4,' ~~r * ****^ ^ *.*' If ~~~~~ ~ t l', /J, ~* ^ ~ ~ ~ ~ ~ &.,.' *.,~.... ~. / \, A. - / " - *. - * *"*' /~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~e..~..** -&..'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. ~*.ry.-l*l... ", > ff. ~~~/"'; P %.. I~~~ /~ %'ii, /..*' * * - 1 ~,. *-. *. *. *,. /. -* qjjr~~~~~~~~~~~~~~~~~~'q *~ ~ ~ ~ &,'.''.* *./. * *.' ~* * * r.~,~,r~~~~~~~~ ~ ~ ~. oI ~'~~/, ~~~.:':~~'.~~ ~~~~~~~. //* -. *.. /.* (.:,'*^ *.<..: OF~~~~~~~~~~~~~~~~~~.. ". *.'... ~~~~~~~~~~~~~~c~~~~~~~~~~~~~~ *.,~ \;....'.'* \ * * * -., ~ ~ O I r *. t....'." f"':.. do~~Y **.- -' * /;\A... /*': * i.-'.; /.^(. ^ <. ~* o f e a c h o t h e r. T h e l o w e r o n e i:' \ a K.. -7 d a * ~ ~~~~~~~~~~~~~~~~/'.'''..:. / ~. ~. / ^-~~~~~~ ~ ~ ~.r.,, *.,''' *.' **. \' / P o eac ote. Th~oe oei K~ o..r^T' ~eay ^./, the upper one a K~ V^ 07r~7~ decay. Neither were used. ~ z;' K 1'' i ~ 4~.'~' ~ e. ~ ~,~. I ~~~~~~~~~,'~ I/~~ ~ ~'. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~''~ "..q' ~~~~~~~~~~~~~~.I:.i r.~.. r. I ~"\ ~~ ~. ~. -.:. ~./ / i, r r II~I ~~~~~~~~~~.r. "'~~~~~~~.'~~,, -. +'... o~~~~~~~~~~~~~~~.' ~ac ~ 1, ~h oeroei \.; ~ $-T,,ry,' /'" the upper one a K" ~~~~~~~~r 07T. decay~. r~~~~~~~either ~~~e re us e,." "'I —~/ ~~ -._.:.. /.:..',%. ~-.,~..~~~~~~~~~~~~~~~~~~~~~~~~

127 b. It is a 9 and it is measurable and there is no other Type 1 within 20 cm on any view.... lij c. It is a e, it is measurable, and there is a Type 1 within 20 cm on some view............. 2ij d. It is a e, but it is not measurable and there is no Type 1 within 20 cm on any view.... 3ij e. It is a e, but it is not measurable and there is a Type 1 within 20 cm................. 4ij f. It is not a 0 and there is a Type 1 within 20 cm on some view............................ 5ij The last two. digits, i and j, are the distance from charge exchange point to the e vertex. If no decision can be made the classification is 999. 2.2 With y For those events classified as 14 or 24 in the first edit apply the following tests. 1. Second edit y-test. (See later instructions.) 2. Is there another Type 1 vertex within 20 cm on any view? Code the first digit of the classification as follows. Enter the estimated number of y's for events passing the test. If the event fails enter 0. Do not enter more than four unless, if the result of Test 2 is positive, add 5 to the entry. As the last two digits enter the distance of the closest y passing the y test. If none, enter 99. If no decision can be made, enter 999 as your final result. Fig. 23 shows illegal y event.

.u~~~~e, Ii~V A; -~,~. * \y *... * -'. -..'''". 7. U i i*..' - **. -. I' *. "..**. vd.:..-.. ^*. —-., S p ^^ >~... w *;, *' 9.... * * ~/ 1 * S..' *0* 0*"' * /5 9' %9' *9'~~~~~~~ ** * y -.***' > * / " \1 / -'.~.~~~~~~~~~~~~~~~~~~~~~~ /. **..-.'',;..-* Dp*'% \' V.'..., *...*.A * /' \ * / /'' / * **.9 *e.*. *,9. 9 / * /.0.t' *.,* * y~. C.I i ~ - /,...* * 0- / *.* *1 / /. -..V V I'~~ ~ * 7I f f n r C.' * 62. I. U..t * ~ ~~~~~~~~ *9'.. i I. S *r ~. I ~~'. S \ / r ** t/. ** \^y\' ^r - \ ^. - -. I ~ / ~ ~ ~ / p/. 5. / I~' * *.' * S. d * * /v i ^ * / a U. I* * * ~'H*./- C I ~c ^: ^^'1.~LT...*: - Z r~~~~~~~~~~~~~~~~~~~~~~~ \. i~~~~~~~~~~~~~ I,'V' K' ~ ~ ~ ~ ~ ~ ~ cag exhag wit posbe0s Hwvr y1s come from nearby K ^ decay.;b~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~U;.9~~~~~~~i 4~~t U I, 1' 0 S. S. —'. ~~~~~~~~~~I U'3 -, C Fiur 27. A lea > vet w-rn 8 chage xchngewit posibe ys. oweer y's com frmnab K2dcy

129 2.3 With V and y For those events classified as 15 in the first edit apply the following tests. 1. Second edit 0-test; does it pass? 2. Second edit y-test. 3. Is the e measurable? 4. Is there another Type 1 origin with 20 cm on any view? If the event passes the 0-test and no y's pass the y-test, code your result as described in Section 2.1. If the event fails the 6-test but passes the y-test, code your classification as a. No Type 1 origin within 20 cm................ 5ij b. There is a Type 1 origin with 20 cm............ 6iwhere the distance of the shortest y from the charge exchange point is digits ij. If the event passes both the 0-test and the y-test code your classification as a. There is no Type 1 origin within 20 cm......... 7ij b. There is a Type 1 origin within 20 cm.......... 8ij where the distance of the shortest y is again digits ij. If the event passes neither the 0-test nor the y-test, then enter 0 99 if no Type 1 within 20 cm; otherwise 088. If no decision can be made enter 999. Finally, if the first edit classification is 25, apply the "measurability test for 0's" to the w7 7T tracks leaving the origin. Code your result as follows. a. Fails the test and no Type 1 within 20 cm...... 200

130 b. Passes the test and no Type 1 within 20 cm..... 300 c. Fails the test and there is a Type 1 within 20 cm......................................... 400 d. Passes the test and there is a Type 1 within 20 cm.......................................... 500 e. No decision can be made....................... 999

APPENDIX C PHENOMENOLOGY OF K -4 2r DECAY S 1. THE DECAY Ks -, 2r s In this section we look at the K0~-K system from a phenomenological point of view with special emphasis on the 27T decay of K~. Only that analysis which bears direct effect to our experiment will be presented. Our ultimate goal in this section is to obtain the approximation formulated by Abbud-Lee-Yang as A. R - 2 = 6/2 Re(A) cos (62 - 6) + (Aem) (C.1) 0 where: r(Kxs tt ) r(Ks - Tr0rr) A2 and A are the I = 2 and I - 0 standing wave decay ampli-. 0 CD tudes, respectively, of K - w +?r; 62 and 6 are the r-I s-wave scattering phase shifts for the I = 2 and I = 0 states at E = mKo; Aem is the electromagnetic correction, mentioned at the end of the first chapter, which is composed of three factors: a) final state Coulomb interaction, b) radiative correction (due to transverse photons), and c) corrections from on-mass-shell decay of K~ - 27ry 4 2rr, with correction 131

1352 (a) being the most significant (see Chapter I, Section 4). To derive this expression we will startwith the initial time dependent amplitudes of K~ and K, and propagate them via a modified Weisskopf-Wigner treatment. 59 The original Weisskopf-Wigner59 description of unstable states held only for nondegenerate eigenstates and resulted in a differential equation for particle amplitude d-t (t (t) ) (C.2) where r is the total width of the state p which is the inverse of the lifetime. In 1934 Breit and Lowen modified the original Weisskopf-Wigner formulation to include the possibility of degeneracy and it is basically the Breit-Lowen method we must use since the scalar formula (C.2) will not describe the real and virtual transitions that cause K~-K mixing. A detailed application of the Breit-Lowen method (modified still further by Kallen61 in 1958) to the K0~62 system, can be found in Kabir. Essentially we must replace the decay constant r with a 2 x 2 decay matrix whose offdiagonal elements provide for the mixing phenomena, while the general neutral K state becomes a vector in a twodimensional Hilbert space with basis states IKO> _ ); |1 = (, ) (C.5)

133 so that |K(t)> = a(t) (0) + b(t) (1) = ( (C.4) Thus the general Schrodinger type equation which governs this system will be of the form - IK(t)> = DIK(t) (C.5) If we denote the various channels of decay as f and Af(E) eiEt is the amplitude of the decay product in channel f with energy E, then equation (C.5) breaks down to two coupled Schrodinger equations (in the notation of Wu and Yang) d - iEt i dt K(t) = HK(E) Af(E) e t (C.6) f,E d -Et it K(t) H (E) Af(E) eEt (C.7) with the amplitudes given by i-~A (Ft =.ie i.t Af(E) = -ieEt[HfK(E) K + H (E) K].(C.8) The various Hfi = <flHwk i) are the matrix elements which will later be expressed in terms of the decay and mass matrices so that further discussion of them will be left for that time.

134 Solving these via the modified Breit-Lowen treatment we obtain K -1/2 Xit (K.) =ee The amplitudes t and decay constants X are given by the eigenvalue equation (r + iM) i =?~. (C.9) 1.1'The Decay Matrix The Hermitian matrix r is called the decay matrix and its matrix elements are related to the weak Hamiltonian by the perturbation formulae rzz=~ X D= t.f!~HwkKo> I2 fr ~> 2 f r2 = r T Of I <f lHowk' K (C 10) 21 = 2 = 7r Pff J< Iw k f fwkl~ where f again is the final state and of is the density of final states per unit energy. It is possible to perform these sums and express the r matrix as the sum of the contributions

135 from the 7rr(I = 0), wrr(I = 2), leptonic, and 37r modes r= 1r + r + + r3 where 2 2 r = I ) A2 A2/ 0 0/ 2A22A2 2 2 * 2 A2 A A 2 2= y't 3vr -i aThe elements of r and were defined earlier. The elements 2 2 of r and Pjn represent the general expressions from page 154 with the sum over possible leptonic and 37w final states, respectively. respectively.

136 1.2 The Mass Matrix The Hermitian matrix M is called the mass matrix so that iM is an anti-Hermitian matrix which represents the effects of the mass shift. The elements of the mass matrix are much more complicated than those of the decay matrix since, in addition to the usual weak Hamiltonian perturbations, we now have both strong and electromagnetic couplings between various mass states. Thus we have (also in Wu-Yang notation) I,,nIHwk)KO>i2 11 = mK+ ( H KO> + P<IHwk + 2 n (mk - m) *M2 = mH +~ +FIHwk I H kin> Kn) wk (C)2 22 2 Kwk 0 0M21= M12 - IHwk KO + _ P I 21 M12kn (K mn) K n) where mK = <KOI Ht + H IK> = <KIHst + H 7i (assuming CPT allows the second equality). In the above elements P stands for the principle value while the sum over n extends over all unperturbed eigenstates of Hst + H with unperturbed energy mn.

137 The off-diagonal elements of M can be simplified by noting that the small mass difference between K' and K2 suggests that <(IHwkIK~) is essentially 0. This is also consistent with the hypothesis that the AS! = 2 componnt. of t.hb w.ATk lrrprnt JW is also essentially 0. ML 1.3 The States |Ks> and IK>: Transition Matrix Elements We can now construct the eigenvectors of the K0~- system which, in the notation of Lee, Yang, and Oehme, take the form i (= (Ipl2 + Iql2) 1/2 () (C.12) where P2 = i = A + + + A2 + i + x 12 12 2 3 + iyd3 + iM12 (C.13) and 2 +* * q 1`21. iM21 r12 + iM12 2 2 * A + A2 + XA - iyn + x37 - iy53 + iM12. (C.14) Thus, as our notation we shall take the Ks and K0 states to s be be:

138 Ko>= (P 12 + Iq2)-1/2 P) IK> - ( IP 12 q 12)-1/2 -q There are several other common representations for these states in use in today's literature. Of these the two most predominant are e - - q in which case iKS> = [2(1 + E 12)]-1/2 (~1 + and r = P/q so that the K~ state becomes s IKs> = (1 + rL2)-1/2 (+). (c.16) We are now in position to construct the 2rr decay amplitudes of Ks. We need first the isoscalar (I = 0) and isotensor (I = 2) standing wave amplitudes a and a2, respec0 a2 tively. Since 77r(I = 0)> = A (1) I7rTr(I = 2)>)- Hence

139 a2 = <(r(I = 2)I|K> = (A2A2)(jP2 + lql2)1/2 () a2 = ( p(2 + Iql2)-1/2 (A2P + A2q) (C.17) and a = ( |pI2 + lql2)-1/2 A (P + q) (C.18) We now construct the rr outgoing wave amplitude using 4+ - the Clebsch-Gordon coefficients. The 7 w state is constructed as follows (states are |I, I>): ll1,+1>; ~ - l1,l> Thus I+rV> = 1 l0,0> e ~ + 12,0> e (C.19) 3 a a2 O and i6 <7rwl IIKs> - ( lP|2 + Iql2)1/2 [ A e 0 -t T i6?wl r12 ib A+, e +2 Ar J+ /A2e ]p +L 4A e ~ + -A2e lq

14o [3( (P12 + Iq 2)]-1/2 ei6 x [^ A(P + q) + (A2P + A2q) e 2i 0 ] i6 Dropping the overall undetermined phase e 0, we have <7-7r- IIK~ > [3( IP2 + Iq 2) -1/2 [ A, (p + q) i(62-6 ) + (A2P + A2q) e ] (C.20) Similarly, <7070IIIKs> = [3( IPI+ q2)-12 [A (P + q) - 2 (A2P + A2q) ei(6260)]. (C.21) Or, in the more contemporary'r' notation < 7r | IIK> = [3( + Ir 2)] 1/2 [,A(1 + r) < IIK> = [3(1 + Ir12)i]1/2 A(1 + r) <^o7o iziK~> = [3(l + Irl2)]-~A2 [A0( + r)

141 - / (A2 + A2r) e 2. (C.23) By definition R= r(K +~ 7r ) _ rTIJIKS>O 2 r(KO - 7r~7r0) krw 7O1 II2K>1 Thus R A (1 + r) + (A2 + A2r) e 2 * —(-6 6-)-7. (C.24 ) A (1 + r) - (A2 + A2r) e 2 We now make the assumption, to be justified shortly, that the difference between r and 1 is negligible. Hence 2' A + 2 Re AF 2 R = 0 2 2 A - 22/2 Re A2F 0 where i(62-6 ) F = e Then 1 A2 2V2 [1 + ~ Re(s-) F] 2[1 - Re(-) F] 0

142 R =2 21 + a = 2 l + 2 Re a + a2 1 - 2a L - 4 Re a + 40cJ where A a =_ Re() 0 2 Neglecting terms of order a we find R =2[(1 + 2 Re a)(l - 4 Re a)1] Expanding the denominator R = 2[(1 + 2 Re c)(1 + 4 Re a)] 2[1 + 6 Re a + 8(Re a)2] From the definition of a and F above, cos (6 - 6) A Re a = --—; ---.Re (-) 0 Finally upon substitution, again neglecting terms of order a or smaller, R = 2 1 +; Re(A) cos (62 - 6 )] 0

143 A R - 2 = 6e Re(2) cos (62 - 6 ) 0 1.4 The Validity of the Abbud-Lee-Yang Approximation In the preceding derivation we have assumed limits on several quantities. To justify these we need, in addition to the Ko~ v r+\ (henceforth denoted as S ) and K*~ ~7r~ (Sn) S Co o 1 amplitudes, the corresponding K~ amplitudes. These are: C = <-w I IlKl~ > = [53(1 + 1r!2)-1/2 x [ Ao ( - r) - (A2 - Ar) F L- <7o o70IKL> - [3(1 + Ir12) 1/2 x [Ao(l - r) - (A2 - A2r) F] In addition we need the following nwnerical quantities7 1j+- - (1.92 ~.04) x 10 3 Lc/Sc oof - (3. ~1.2) x 10= - Ln/Sn R = 2.117 ~0.064 = S 2/ Sn 2 We have set the error in Vro I to reflect the significant deviations in reported values.

144 To determine the quantity r we use the four amplitudes L, Ln, Sc and S as follows. Let N = [(1. + -rI2)]-1/2 Then (J LC + Ln) = NAo(1 - r) and ( /Sc + Sn) = NA (1 + r) We define the quantity v, for later use, as 1 - r J Lc + Ln 1.+ r a Sc + Sn 1,+ rTL + A,] 0V = % + 0 (C.26) Since we are interested in the magnitude of r and v and we experimentally measure Irt+ and Iroo the best we can do is set an upper limit on v ^J2R + 1

145 (v -I'j.+ H. (c.27) This gives us a lower limit on r 1 -- v/ 1+ I Iv Using the values above we find Iv.< 6.61 x 10-3 and lr O>'0.987 The validity of the Abbud-Lee-yang approximation can be shown by solving for cos (62 - 6 ) exactly, that is dropping 0 no terms and assuming only that the CP violating component of the decay amplitude is small (r = 1). We find cos (62 - ) = R- 2 + (4R -- l)c12 (C.28) 0 A2 2/'. (R + 1) Re(A-) 0 which is quite similar to the Abbud-Lee-Yang form cos (62 - 6)= R -2- (c.28a) 6^2 Re (-) O

146 since l I2 = 9.68 x 10-4 2. REGENERATION PHENOMENA As early as 1955 regeneration of Ks from K0 was suggested 64 by Pais and Piccioni. Their basis for this hypothesis was the observation that if a K0 beam passes through matter its T component (S = -1) will be more strongly absorbed than the K~ amplitude (S = +1) since, due to strangeness arguments, K can make hyperons and hypernuclei while K~ can only elastic scatter and charge exchange. Hence, after traversing the matter, there will be relatively more K~ than I amplitude which. is equivalent to an admixture of K~. s 2.1 Coherent Regeneration Regeneration can occur with a scatter (incoherent regeneration) of the K~ beam or without a scatter (coherent regeneration). In coherent regeneration the amplitudes for Ks and K0 are obtained by solving a pair of coupled differential L CD 65 equations. Christenson et al.6 give the solution in matrix form s~(x Rl'(x) R12(x) s(o) ( = (-(C0.29) L(x)/ R21(X) R22(x)/ KL0) whe re

147 - -/2' 2wiNAsf2(O) (e /2 _ e. -Nax/~r2) s 2 - Nar(X/T ) 2 R(x) = e 0 e-i8' (C.30) The terms in R(x) are defined as follows: As = yCr - mean decay length of the Ks, 6 - K,-K0 mass difference described earlier, s r. aT - total cross section in material averaged between K~ and A, J = X/A - distance traversed by K in units of K mean decay length, K =PK /h wave number of K, s N number of nuclei per unit volume of material, f21(0) = 1/2 (fK(O) - f_(O)) _ forward regeneration K amplitude of the nucleus of the material. 66 D. Sinclair has applied these formulae specifically to heavy freon (CF3Br) by replacing Nf21(O) by Z Nj[f2(0)] where the sum is over the different elements in the material. Then putting in the initial condition Ks(O) = K,(O) -= 1 (C.~1) in equation (C.29)', we have

148 KO(x) = K [RE1(x) + R12(x)] (C.2) The resulting K~ intensity is given by s Is(x) = |K(x) 2 =- [ R11(x) + R2(x) 2 + 2Re Rll(x) R12 (x)]. (C.33) Since 1R12(x)j is of order 10-4 it may be neglected and is(x) = e -N'TX [' 4 NAs f2:(x)) Is (x.) = -!^ e e + ~-2 + NFIf2l(0)1I + NBrlf2l(o)lBr and 0 in (C.4) is defined as

149 where oSL is the phase of the regeneration amplitude. Upon substitution of numerical factors -NaTx.2 Is(X) =!_0 e [e +3.6 x 102 ^ x (e Cos e cos ( - )) (C.35) Upon integrating (C.35) over our fiducial volume we can determine the correction to our Ks sample for different values of s the regeneration phase eSL We estimate that this correction oSL may result in as much as a 4.2% enhancement or reduction over the corresponding number of K~ in vacuum and, although this s factor does not affect our charged to neutral ratio directly, it does affect our Ks/KL ratio. s L 2.2 Incoherent Regeneration Incoherent regeneration arises since the cross section for K~-nucleus and K-nucleus are different. The amplitude for scattering of the (per scattering center) is tsc(0,x) - CfK(0) K(x) + f_() K(x)]. (C.36) Solving (C.29) for Ks(x) and KL(x) and using the states % D. Sinclair extrapolates between measurements given in Christenson et al.05 and estimates errors at N 30%.

150 K(x) = E [Ks(x) + KL(X)] K(x) = [Ks(x) - KL(x)] equation (C.36) becomes ~sc( 0) = )f22('e)(R11K5(o0)> + R12IKL(O)>) + f21(e) R22 KL(0)>] SC(ex) = [f22(e)1Ks(x)> + f21(e)IKL()>)] where f21(8) is as defined before and 22(9) =2 - K(e) + f (e)] Thus the intensity of scattered- K is S s(Ox) = SC(o)2I N [ I f22() || KS(x)> I + f21(e)l2 IIKL(x)>12 + 2Re f22 () f21( x) ( K (x))>IK ( We can remove the x-dependence by integrating over our fiducial volume

151 ()= Fid. (e,x) dx = NC1f22(e)I2 N + N lscIf2l(e)2NL Vol. ILs + 2Ns Re f2(e) f22(K) (<K4(x) Ks(x)> dx (c.57) where Nsc is the total number of scattering centers, N the number of K0, and N the number of K0 The first term of s L 17 (C.37) comes from scattering of the Ks intensity. The third term reflects the difference in cross section of K-nucleon and K-nucleon and is insignificant when integrated over our fiducial volume. The second terms, however, come from regenerative scattering of the Kp intensity. The quantity I, can be calculated since every increment Ls dx of the regenerator contributes at its end (x = xf) the following number of events per steradian d i(dgra )) -N ovfr()12 fidcia vdxl Upon integrating over our fiducial volume ILs = NA (10 - e'6"0)NL J I f21(I)2 d Ls S. The quantity If21(e)(2 is, as of yet, undetermined in heavy freon; however, it can be approximated by a Gaussian

152 -2 da(e) ~ -e2/26 If21(9)1 = -Le 26 e / We can then roughly determine the parameters a and 6 by 0 demanding that at e = 0 we have f 21(0) as given from the earlier discussion on coherent regeneration (If21(0) jCFBr ~58 -24 2)*.58 x 10"4 cm ), while at e = r we let 1f21(r)12 be essentially 0( If21(7r) I2 1 barn). Under these conditions we find d-) a (.58 x 10-24) e-125922/ so that our expression for ILs becomes ILs 27NAs(.997) NL(.58 x lo024) J e-125 sin 0 de = 1.95 x 10 e125 /7 sin 1 de 1.95 x 10 e sin a d8 0 Thus this very crude approximation gives an estimate of 800 events due to incoherent K' regeneration. This number is only a factor of three from the visual estimate of 250 events determined at the end of Chapter III. This discrepancy no doubt comes from the assumptions we have made to determine the parameters a and 6 in the Gaussian approximation, compounded 0 * We emphasize the fact that this is an extremely crude estimate since it is most likely that If2a1(o)) 2 is highly peaked at 0~ and thus diverges from the Gaussian approximation.

155 by the approximation itself. 3. KO-K_ INTERFERENCE s L Since it is now known that KL can decay into 7 Tr and 7rw~0 channels, constructive interference between KL 4 2r and K.-* 2r amplitudes may become an important factor depending on the phase angle cp + = arg ( _ ). (oo) ( o) If we have a choherent superposition of KL and Ks states in the form IKO> -. [IK + pIK>] (C.38) + 7 IKe> + tKo > where p is a complex constant characteristic of the regenerative properties of the material at hand. If we take the eigenstates derived in the last section IKL(t)> = exp [-i(mL - i - ) tj IKL(O)> IKs(t)> = exp [-i(m - i -) t] IKs(O)> then the time dependence of K~ - Tw is given by P.K. Kabir57, pp. 66-78. Explicitly = r(l - e ( im ) where y' = Ys- Y is the wid.th of the eigenstates of Ks and KL in the material, m' = m -mL is the mass difference in the material, and r accounts for the difference in cross section between K~ and K".

154 I+(t)= -i(mL-iP/2)t K T- ITIK I+ (t) |e L <7 r+'- T KL > -i(m -ir /2)t < + p e s s <+ 2 -FTt 2 -Fst I+(t) IJ 12 ePLt + Ip 2 ers -(r +r )t/2 + 2 _po e ( L+ )t/2 cos (Amt - cp+ + CP ) (c.39) Since 2+ is on the order of 10 6, we can ignore this term so that the integrated intensity for the r T7 channel, assuming cp O, is T 2 - T 2+ |n+p R+_ = I+_(t) dt = -. - (1- e s ) + 2 - s -', + (An)2 x ~ cos cp [ e-' T(Am sin (AmT) - F' cos (AmT))+ rl] - F" T ~ - sin ip+- e T (r' sin (AmT) + Am cos (Amt)) - Amj (C.40)

155 whore r' = ((L + ) /2. Pxperimentally it has been shown that Arn rt 1 r /2 = r1 in which case s S,r i 12 -2rlT R+ - [ pI (1 - e s ) + r+-P -r' T x cos c L_[e s (sin (F'T) - cos (FsT))+ 1] A Zs ~ sin cp+Le s (sin (r'T) + cos (rFT)) - (c.41) For future use we need to know when the interference term will contribute the most as a function of c+. It can be shown that R+_ is maximum when cp = tansL (2p) (C.42) with the secondary condition tan p_ < (C.43) where a and p are defined in (C.41). For the case K~ 4- wr~r, one need only substitute the

156 amplitudes (7O~7r0T IK~ > for <+V-I TIKO > in equation (C.39). We find (L) (L) I (t) 2l eLt + p12 e s + pIe-(rL+r )t/2 + 2| o e s cos (Aomt - + cp )

APPENDIX D wn PHASE SHIFTS The pi-nucleon interaction is certainly one of the most studied aspects of experimental elementary particle physics. Unfortunately, due to several severe experimental problems, the determination of Tn phase shifts has not fully benfitted 68 from these years of study. As pointed out by Gutay at a recent conference on 7 w and KT interactions, the two major problems currently haunting the experimentalists are: 1. One must extract the wT phase shifts from inelastic TN scattering where the effects of the TT scattering are often clouded by other inelastic processes. 2. Since the reaction we must use w + N - N + (n7) necessitates the involvement of a virtual pion, the angular distribution of the pions depends both on the effective mass of the dipion system and the square of the four-momentum transfer. Thus to extract the th-wave contribution from background and absorption effects we must have knowledge of the wT distribution as a function of four-momenta transfer squared (A2) for each value of the dipion effective mass (IS). Much of the material in the first part of this resume comes from a paper delivered by L. J. Gutay at the recent Argonne Conference on w7wr and Kwr Interactions. 157

158 Gutay estimates that these restrictions demand more than 50,000 events in the NwwT final state at any single beam momenta; hence statistics are a problem. Because of these difficulties all experiments to date have used simplifying arguments to analyze the data. While the resulting phase shifts are consistent for I > 0 channels, the results for the s-wave zero isospin phase shift are as varied as the methods used to obtain them. 1. THEORY 1.1 Partial Wave Analysis of 7ww- Elastic Interactions The phase shifts themselves arise as a necessary parameter in the partial wave analysis of any given interaction since the scattering process changes not only the amplitude but the phase of the outgoing wave. Briefly, the principle used is to construct asymptotic solutions to the wave equation in regions far from the point of scatter so that the particle wave functions (1k) obey v2 + k2-= 0 (D.1) where k is the momentum of the scattered particle. Following the usual procedure69 the asymptotic form of the scattered wave is found to be tsc ~ 77 f(9) -- (D.2)

159 where V is the velocity of the particles and f(e) is the scattering amplitude given by f(e) = _ (2.+ 1) f, P(e). (D.3) In this last expression fp is usually referred to as the scattering function and is dependent on the phase shift in the form i 6. sin 6. e i -E — k (D.4) k' We can easily apply this method to obtain the wT phase shifts if we assume the following two things. First we assume that all pions are on the mass shell, while secondly we assume that all wT interactions are elastic. Although the Chew-Low extrapolation technique has shown these assumptions ultimately unnecessary (at least for reactions dominated by single pion exchange), assuming them at this early stage facilitates the theoretical description of wT scattering. Expressions (D.2)(D.4) give a total description of the scattered wave function in "momentum-space"; however, to accommodate the various charge combinations of two pions it becomes convenient to introduce "iso-space" wave functions that are eigenstates of the total isospin. Using methods described earlier in Chapter I and Appendix C, we can construct the various 27 states to be

160 Ir-> > = 2,2> (D.5a) Ir+7O> = 1 [12,1> + 1,1>] (D.5b) -7r~> = -/ [12,-> + 1l,-1>] (D.5c) 1+7r > = t 12,0> - |1,0> + 1 0,0 (D.5d) which are the most often observed dipion states in the reaction a p - N. (D.6) Combining the concept of isospin with the earlier constructed angular momentum states, equation (D.3) becomes c fz(e) = (2 1+ 1)[1 + (-I) ] f () (D.7) where now i5? s in 6I e tfJ-~~ k=~~~~~ -~((D.8) and k and 9 are specifically the pion momentum and the scattering angle in the dipion rest frame. Using (D.7) and (D.8)

161 we can now construct the scattering amplitude for any 27 state following the isostate mixing as given in (D.5a-d). For example _,.,~. 2 f() =.... (2,+ 1) e sin 6 [ + (-1) i6~ 1 + I e sin 6[l + (-1) P() (D.9) Since we are primarily interested in the region / = mKo we 70 may take advantage of the observation' that for energies such that JS' < m only the first two partial waves (1= 0,1) make P any significant contribution.. In this approximation the differential cross section obtained from (D.9) becomes d \1' 2 1 2 d — (0) =- - 1(9 sin 61) cos2 +n - q2 + 2[cos (62 -61) sin sn 1 0 0 1 cos cos (6~ _ 61) sin 6~ sin 6 ] cos e

162 + [sin2 6~ + cos (6 - 62) sin 6~ sin 62 sin2 62 + 4 ) * (D.10) There are, of course, other representations than that used here. It has been shown that the differential cross section may be expressed in terms of density matrix elements as done by Jacob and Wick71 or Csonka and. Gutay 72 1.2 The Chew-Low Extrapolation: One-Pion Exchange Thus life would certainly be simple if pion targets were available and 7rr elastic scattering dominated over a wider range of center-of-mass energies. Unfortunately such pion targets are not yet available. Because of these obvious limitations Goebel7 and later Chew and Low formulated a technique whereby. 7r- elastic scattering information could be extracted from higher energy inelastic wTN scattering data. Basically, if one assumes that the process rN - NTrr contains a one-pion exchange contribution then by extrapolating the 2 2 observed cross sections to the unphysical point A m we obtain the rrT scattering amplitude. In theory this can be done with confidence since the extrapolation need be carried. 2 2 out over a relatively short distance (m = 0.02 GeV2/c ). The relevant formulae can be derived from the one-pion exchange diagram (Figure 24). The contribution of this diagram to the

Fig-ure 24. Standard orne-pion exehange dliagrram.for realction'i~T ~+ fln N ~ + w ~~~TT ~ Tc hi~~~~~~ ~ Fiure 24. Standard one-pion exchang diagram for reaction Tr13-n -^N + r-t- r+ 7r

164 angular distribution of reaction (D.6) is found to be75 2 2,A2 22 2 d aS(S o ) S A P F2 2) = - p2S 2m22 a + _(S,2) FNw ) where S -= r rest mass, pi cm momentum of initial state, 0 2 P momentum of the exchanged pion and g = Nrp coupling conPe CD stant. We are then left with two unknowns: a +(S, \ - -T ~ o - - - + the cross section for wr T -.* w 7 with one of the incoming pions being virtual and FN (A ) - the "pionic" form factor of the nucleon. These must be "extrapolated". 1.3 Durr-Pilkuhn and Benecke-Durr Parameterization The result of Chew-Low extrapolation which we have briefly described in the last section is strongly dependent on the choice of extrapolation function. Of the various types of extrapolation functions the "linear" and the "quadratic" functions are those most often used by experimenters, and it 77 has been found77 that both functions fit the same data (for certain reactions) equally well yet yield widely variant extrapolated cross sections. Moreover Schlein et al. has described a case where neither of the functions gives the correct result. Because of these ambiguities Durr and Pilkuhn79 8o and recently Benecke and Durr have proposed parameterization methods which relate off-shell to on-shell cross sections. The results have been quite reliable when the Benecke-Durr

165 vertex correction factors are combined with the Chew-Low onepion exchange model extrapolation techniques. 1.4 Model Dependent Predictions In Appendix C, Section 1, it was shown how the phase 2 shifts 6~ and 6 enter the transition matrix elements for 0 0 Ks o wr + w~ and Ks -4 + w. Since these are the only s S phase shifts which enter this particular analysis we will describe the application of current algebra and Veneziano techniques with emphasis on the derivation of these quantities. 1.4a Current Algebra Techniques There are two types of analysis that have been applied to wT scattering. The first method involves the "soft-pion" 81 calculations of Weinberg and is based on four assumptions: 1. Conserved Vector Current (CVC) hypothesis. 2. The Partially Conserved Axial-Vector Current (PCAC) hypothesis. 3. Both vector and axial vector currents obey SU2SU2 equal time commutators. 4. A dynamical postulate restricting the change in the scattering amplitude if one restricts the pion momentum to zero, the "gentleness" hypothesis. In additin Weinberg made an additional assumption concerning the nature of a particular commutator which when combined with * As summarized by R. Arnowitt at the Conference on wT and KT Interactions, Argonne National Laboratory, 1969.

166 the first four assumptions led to a set of s- and p-wave scattering lengths. The second method was initiated in hopes of correcting errors introduced by the "gentleness" hypothesis of the soft-pion model. This new model, labeled the hard-pion 82 method, substitutes the concept of vector dominance for assumption number four. Weinberg's fifth hypothesis was modified only to the extent that it was made less stringent. The "hard-pion" results are simply the soft-pion calculations modified by a factor which represents the breakdown of Weinberg's "gentleness" hypothesis. 83 Arnowitt et al. have performed hard-meson calculations at energies comparable to the K~ mass. They have found that the scattering amplitude depends upon m and r which they a set at m 730 MeV and rF 200 MeV to obtain the best fit. With these parameters, the difference in phase shifts at the 2 K~ mass is found to be 56 - 6 (35 ~4)~. O - 2 1.4b Veneziano Partial Wave Technique84 85 Lovelace has proposed the following Veneziano representation for the w- s-wave elastic amplitudes 1 3 AI - A B.F(t,u) - [F(s,t) + F(s,u)], (D.lla) A - F(s,u) - F(s,t), (D.llb) A2 = -F(t,u), (D.llc)

167 where F(x,y) = Yr(l - C(x)) r(l - o(y)) r(l - o(x) - a(y)) +y r(l () (- ()) r(- (y)) D.12) r(2 - a(x) - a(y)) and a(x) = a + bx. (D.13) In general, the weakness of the Veneziano partial wave analysis is the failure of VI(v) —the partial wave projections — to satisfy unitarity. Tryon has recently proposed a modification of VI(v) such that the new VI(v) satisfy unitarity. Using the modified Veneziano partial wave projections and keeping only the first term of the Veneziano series (D.12) phase shifts have been computed by Tryon. The quantities a and b of expression (D.13) are the Regge parameters determined by Lovelace to be a = 0.483 and b = 0.017. Tryon has found two different solutions corresponding to the two indicated possible values for m. In the last section a current algebra analysis showed m = 730 MeV. Using this value the Veneziano model predicts a phase shift difference 6 - 6 = 0 0 (59 ~10)~. Another value of m equally probable in the analysis has m = 900 MeV which yields a difference in phase shifts of 62 - 6 = (61 +4)~. Thus, independent of the final

168 value for m the value of 6 - 60 from Veneziano techniques a o o lies in the range (59 ~10)~. Figure 25 compares 6~ as a 0 function of s as predicted by Arnowitt (current algebra) and Tryon (Veneziano). 2. EXPERIMENTAL TECHNIQUES There have been literally dozens of experiments carried out to measure, either directly or indirectly, the rww phase shifts. Even though each experiment may have some small difference to separate its method from other experiments it would be unnecessary to describe all experiments since there are only a few basic types of experiments. Also, with the exception of Malamud and Schlein all experimental values of 62 have fallen in the range of -5 to -10 degrees while (preferred) 42 values of 6~ have ranged all the way from2 (23 ~10)~ to 0o slightly over 50~. Thus, as representative of the whole class of experiments we will describe the experimental determination 2 of both 60 and 6 by Malamud and Schlein, the experimental 0 0 determination of 6~ by Hagopian et al. and Scharenguivel et al., 0 2 anf finally the values of 6 as determined by Morese et al. 0 and Baton and Laurens. Though in some cases the determination of the phase shifts was only a small part of the total experiment, we will concentrate only on that part of the analysis necessary for obtaining 60 and/or 62 0 0 2.1 6~ and 62 - E. Malamud and P. Schlein 0 0 This analysis used a mass compilation of 12 previous

60 TRYON 50 40 / UjL 0 30 ~C00 / ARNOWI'T H 20 10 0.30 Q34 0.38 0.42 0.46 0.50 0.54 0.58 i GeV Figure 25. S-wave, isoscalai phase shift 6~ as piedicted by current algebra (Arnowitt) at-d Veneziano ~ (Trycn) techniques. /S - 7T effective mass.

170 experiments containing a sum total of 68,356 events distributed as follows: p 4 v w n 20,555 events (D.14a) - p 4 W-~p 14,536 events (D.14b) w p 4 7 T p 14,638 events (D.14c) TTp,4 w w p 18,627 events (D.14d) Malamud and Schlein determined 6~ from reaction (D.14a), but 0 before they could attempt this they had to determine a value for 62 to use as input for the 6~ determination. There are 0 0 2 two ways to determine 6; either by examining s- and p-wave 0 interference in the w- wp reaction or by extrapolation tech_ - 4 +niques in v T A+ or wT n. Malamud and Schlein argued against using the w7-~p reaction for m less than m since there is increasing evidence for the dominance of wu exchange at these low energies which, of course, invalidates the onepion exchange hypothesis. This increasing importance of w exchange is particularly disastrous to interference analysis since it is suspected* that the Twr s-wave system results from. w exchange while the wT p-wave system is dominated by w exchange. Thus, without knowledge of the phases of the w- and ~T The evidience supporting this suspicion is, however, by no means conclusive (Frank Heney, private communication). Thus the Baton et al. value of 60 can not be disregarded simply on the basis of this argument.

171 r-exchange components s- and p-wave interference can yield no information. In light of this Malamud and Schlein used a Durr-Pilkuhn modified one-pion exchange model in the pole extrapolation to obtain the value of 62 over the mass region 0 considered. In the vicinity of the K-mass the pole extrapolation of 47rp w r-A++ yielded a value 62 -(17 ~3)~. Using this as 0 input for the master fit described earlier it was found 6(mK) = (41 ~5)~. Hence (6 62) = (58 ~6)~. -~mK):'0 0 2.2 6~ - V. Hagopian, E. Bogart, S. Marateck, W. Selove 0 This experiment, as in the Malamud and Schlein analysis, used a compilation of data. From 11 previous experiments they obtained 26,872 w p - r wr n and 17,440 o p 4 7r ~p events. From analysis of these two reactions Hagopian et al. hoped to determine 6~ and 61 O - Upon assuming that only s- and p-wave scattering contribute below 900 MeV they fit each angular distribution to a + al cos ~ + a2 cos 0 (D.15) where 9 is the angle between outgoing and incoming 7T in the dipion c.m. A new variable was defined, A, such that extrapolation of this function would yield r. The function determined was

172 (t2 + m2)2 A Q=~ ag --- -, (D.16) the a coming from (D.15). The extrapolated values found were labeled B_, and they determined that the product B u,where 2 2'1/2 1 u (= w- m2) (!.), (D.17) is directly proportional to the phase shifts as follows (see equation (D.10)). in2 o 1 2 2 B u = si sn sin 6 + 4 cos (5~ - 62) sin 6~ sin 62 N, (D.18) BU = 4 cos (6~ - 61) sin 6~ sin 6 LL ^ o I0 10 1 + 2 cos (62 - 61) sin 62 sin 6 N (D.19) o o 1 [9 $in2 61 B2u - 9 sin 61 N, (D.20) where N is a normalization constant. 61 was determined directly from B2u and 6~ was determined from the ratio of JLLd o O 0A I4, -4 j 4 BlU to B2u using as input the Baton and Laurens value of 62. The resulting equation for 62 is quadratic so there are two o

173 solutions for each m. At the K~ mass, using the Baton and Laurens value, 62 = -5 they obtain 6~ = (23 ~10)~ and (75 0 0 ~10)~ with the first quantity being preferred. 2.3 6~ - J. Scharenguivel, L. Gutay, D. Miller, F. Meiere, 0 S. Marateck This experiment differs from most others in that they investigate the forward-backward asymmetry in the dipion rest frame of the reaction 7 p - TT wr n where the asymmetry is defined as (S, ) Forward - Backward (D.21) Forward + Backward and then extrapolate (D.21) to obtain a(S,-m~). In order to extrapolate, a(S,A ) was plotted at given S values and a moderate A dependence was noted. It was found that a linear fit 2 2 m2 a(SA2) = (S) + y(S)(A2 + m+ ) (D.22) was sufficiently accurate. To obtain a(S,-m ), P(S) and y(S) had to be determined. This was done using threshold arguments to determine:(S) and current algebra constraints to fix y(S). Then using 62 as determined by Baton and Laurens and a previous o determination of 6 by essentially the same group the value of 6~ was determined for M -.26 to M =.6 GeV. It was ou a7" 7e KFiT found at the K mass that 6~ = (42 +?~)0. Figure 26 shows

174 experimental values of 6~. 0 2.4 6 - Morse et al.87 This Wisconsin-Toronto collaboration analyzed the three reactions 7r p T Wr -r A W p 7T 7r A T p 4 T-T A++ + MM using both the standard Benecke-Durr (BD) modified Chew-Low extrapolation and an s-wave modified Benecke-Durr calculation 88 as proposed by Olsson (OBD). Whereas the BD model gives u (q,...) o (Dq a (m, A ) ^ ------- - a (T m ) (D.25) 0 where q and q are the "off-shell" and "on-shell" momentum 0 and the u are modified Legendre functions of the second kind, the OBD model gives 2 + 2 2 m +A m q ar (mmA) = -V 2m2 q a (m ) (D.24) 2/ 7r7r Ae c iT'TT 7 W/ After cleaning the sample, a was calculated over the mass

175 2 2 2 range 0. -2.0 GeV, from which 6 62 and 64 were extracted for both the BD and OBD models. In the region of the K mass 2 we find their value of 6 to be (-15.4 ~1.1)~ using the BD 0 modification and (-9.8 ~.7) using the OBD model. 2 2.5 6 _ Baton and Laurens 0 This Saclay group used the reaction r p -, r~-~p to determine the I = 2 s-wave phase from m =.28 to m =.8 T7TT TTrF GeV. Note that this is the reaction that Malamud and Schlein 2 found inadequate for determining 6 (see Section 2.1). Fol0 lowing the standard extrapolation techniques they found that 2 in the region of the K-mass 6 = (-5 ~2)~. Figure 27 compares 0 experimental values of 6. 0

60oy 50R 0600 I I I I I 40120~' 0.3 0.35 0.40 0.45 0O5 0.55 0.6 M (GeV) Figure 26. Experimental determination of 6~ as a function of M by Scharenguivel et al. (-) and by Hagopian ~ et al. ( i ).

Mrr(MeV) 300 400 500 600 700 800 900 cu0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, -— 5 — 1 I —--— F -150 i 2Figure 27. Experimental value of the = 2 s-wave phase shift 62 as a function of M by Baton and Laurens ( —) and Morse et al. ) Curve fit to 3aton and Laurens is our own approximation.

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