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A measurement of the branching ratio, R=[mathematical description for a branching ratio] Berghofer, David 1979

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A MEASUREMENT OF T3E BRANCHING RATIO: R = ( I T -»• e V + IT + e v Y ) / ( it y v + ir + y v Y ) e e 1 y y by D a v i l Alan Berghofar B . S c , Massachusetts I n s t i t u t e Of Technology, 1971 H.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOB THE DEGREE OF DOCTOR OF PHIL3S3PHY IN THE FACULTY OF GRADUATE STUDIES (DEPARTMENT DF PHYSICS) BE ACCEPT THIS THESIS AS CONFORMING TO THE REQUIRED STANDARDS THE UNIVERSITY OF BRITISH COLUMBIA June, 1979 (tT)David Alan Barghofer, 1979 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . P h y s i c s Department o f • The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 O c t o b e r 1 0 , 1 9 7 9 i i A b s t r a c t The Branching Ratio E= ( TT+ -> e + V + 7T+ •> e + V Y ) / ( TT+ y + V + TT + U + V Y ) e e " . y y has been measured on the b a s i s of 100,000 iT - > e v events. Pions were provided by the TRIUMF c y c l o t r o n . The muons produced by pion decay themselves subsequently decay v i a + + — : i y -»• e v e v 1 J . . The mono-energetic p o s i t r o n s from pion decay and the " M i c h e l " d i s t r i b u t i o n of p o s i t r o n s from muon decay were si m u l t a n e o u s l y detected i n a l a r g e Nal c r y s t a l . . A r a y t r a c i n g experiment was a l s o performed t o a c c u r a t e l y determine the response of the N a l . . A f t e r r a d i a t i v e c o r r e c t i o n s , and c o r r e c t i o n s f o r bremsstrahlung, p o s i t r o n a n n i h i l a t i o n , and energy l o s s e s , a dis c r e p a n c y of about 4 % between the e m p i r i c a l T e v ^ l i n e s h a p e and the c a l c u l a t e d l i n e s h a p e remains.. The source of t h i s background has not yet been determined., Based on the number of p o s i t r o n s from pion decay and from muon decay detected d u r i n g s e v e r a l "time b i n s " , our r e s u l t i s R = (1.212±0. 017) x10-* The value p r e d i c t e d f o r V-A c o u p l i n g and muon-electron u n i v e r s a l i t y i s R (theory) = 1.233x10~* i i i TABLE OF CONTENTS ABSTRACT. ....,..,... v.... i i TABLE OF CONTENTS. . . . . . . . . . o . . . . . . . . . . . . . . o . . . . . . . . „ i i i LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V LIST OF T AELES. • ............ . • ... «..-.•»..»..«•••--..• .. x i ACKNOWLEDGEMENTS. ......,. . , . , ... - ,.,............ X i v CHAPTER I I n t r o d u c t i o n . ......................... , 1 Chapter I I Experimental: Method and Set-up. 2 3 A. I n t r o d u c t i o n . . j . . ..... ........... 23 B. Apparatus. ....................................... 3 2 1. The N a l ( T l ) d e t e c t o r . ................. .32 2., S c i n t i l l a t i o n c o u n t e r s . ................. 35 C. E l e c t r o n i c s L o g i c . ......................... 37 1. Gen er a l . ....... ...............:. . ... .,.... 37 2. Fast L o g i c . ........................... 45 3. Slow L o g i c . ............ . ............... ,49 D. , Timing C o n s i d e r a t i o n s * ............. .......... 50 1. Bin Widths and P o s i t i o n s . ............. .50 2. Blank i n g and Background S u b t r a c t i o n . .. 56 3. P i l e - D p R e j e c t i o n . .................... 62 E. Data A c q u i s i t i o n and Runtime Checks. ....... 67 1. General. ................ .... .......... .67 2. Runtime Tests of the Nal and S c i n t i l l a t i o n Counters. :....................... 74 3. General E l e c t r o n i c s T e s t s . ............ 76 4. Determination of tO. 77 5. T e s t s r e l a t i n g t o the Low Energy Peak and to backgrounds. .............. ........... 80 6.. Runtime Te s t s on the Data. ............. 86 Chapter I I I Data A n a l y s i s * ........................ 87 A., Lineshapes. ................................ 87 1. The I n t r i n s i c Nal Lineshape;. -.......>••. 87 2. C a l c u l a t e d s p e c t r a f o r ^-e^Yand T -e^. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 2 3. C a l c u l a t e d p o s i t r o n s p e c t r a frcm y + ->- E4- +v + v and y + -> e++ v + v + y decay. 9 3 B. C a l i b r a t i o n and F i t t i n g . ,...,*..........»,.,...,..,. 96 1. I n t r o d u c t i o n . .......................... ,96 2. Energy C a l i b r a t i o n of the Spectra* .... 102 3. R e s u l t s of F i t t i n g and F u r t h e r Background A n a l y s i s . .......................... 10 5 a. A p p l i c a t i o n of the S a y t r a c i n g i v l i n e s h a p e t o the Data. .,...,.10 5 b. F u r t h e r C o n s i d e r a t i o n o f Backgrounds. ..................................118 c. A p p l i c a t i o n o f the IT -e l i n e s h a p e t o the Data. ..................................129 C. Other C o r r e c t i o n s and U n c e r t a i n t i e s . ..L.... 135 1. Muon Leakage. .........................135 2. M u l t i p l e S c a t t e r i n g . ..................137 3. Bremsstrahlung;. .......................138 4. P o s i t r o n A n n i h i l a t i o n . .................139 Chayter IV R e s u l t s ; ....,.. .....................i„. 14 1 A. C a l c u l a t i o n of the Branching R a t i o . 14 1 B. Systematic C o r r e c t i o n s and U n c e r t a i n t i e s . ... 145 C. Semi-Final Results;. .........................147 D. Furth e r Comments. .......................... 147 Chapter V Comparison With Theory and I m p l i c a t i o n s . ..... . .....,,,......................153 Chapter VI C o n c l u s i o n s . ..........................158 Appendix A D e r i v a t i o n of the branching r a t i o formula.. .,. v....... . ,. .. . . ...., . , . . . .... .. ......161 Appendix B Runtime i n f o r m a t i o n recorded f o r each run. ..... .'. '.V. . . . . . . . ...... . . . . ....... ........... 167 Appendix C A n a l y s i s of the, data from the r a y t r a c i n g experiment. . ...................171 Appendix D E a d i a t i v e C o r r e c t i o n s t o fr-ev decay f o r the Nal d e t e c t o r . .,,......',.>.,........,...,. 189 Appendix E R a d i a t i v e C o r r e c t i o n s t o the y+evv Decay f o r the Nal Detector. ..................„,... 20 3 Appendix F E f f e c t of the Stopping D i s t r i b u t i o n on the v a r i a t i o n of the Nal R e s o l u t i o n with Energy; .22 9 Appendix G D e s c r i p t i o n and r e s u l t s of f i t s t o ^ Z£ data. ... . .'. . -. . . . •. . , . i ; . v . .. . ... 233 References .......................... ,,.... ........24 0 V LIST OF FIGURES FIGFIRE 1-1 Experiment of DiCapua et*. al± .. ..,.>..,.,..•......,..... 13 FIGURE II-1 •f-e experimental set-up. ........................... 26 FIGURE II-2 Timing l o g i c f o r the c o i n c i d e n c e b i n s . . ............... 30 FIGURE I i - 3 E l e c t r o n i c s Schematic. .............................. 38 FIGURE II-4 E l e c t r o n i c s r e l a t e d to pi-stops,* .................... 39 FIGURE I I - 5 E l e c t r o n i c s r e l a t e d to the time b i n s i . ............. 40 FIGURE II-6 E l e c t r o n i c s r e l a t e d to p o s i t r o n i d e n t i f i c a t i o n * ....... ,4 1 FIGURE II-7 E l e c t r o n i c s r e l a t e d to p i l e - u p r e j e c t i o n . . . . . . . . . . 4 2 FIGURE II-8 E l e c t r o n i c s r e l a t e d to the slow l o g i c (Nal s i g n a l ) . .,4 3 FIGURE II-9 Blanki n g d u r a t i o n vs c l e a n stops and a c c i d e n t a l s * ... 57 FIGURE 11-10 Re s u l t s of monte-carlo s i m u l a t i o n of a c c i d e n t a l c o i n c i d e n c e s . ..................... .61 v i FIGURE 11-11 E f f e c t of stop p i l e - u p r e j e c t i o n vs..Stop r a t e . ..... 6 5 FIGURE 11-12 Nal g a i n s h i f t s vs. Time of day*..............-,....,....- 69 FIGURE 11-13 Nal energy s p e c t r a f o r b i n one, run 760. ............ 70 FIGURE 11-14 Nal energy s p e c t r a f o r b i n s one to four* ............ 72 FIGURE 11-15 Determination of t o — t h e time i n t e r v a l between the p i - s t o p and b i n one. ................ 79 FIGURE 11-16 Low energy c a l i b r a t i o n s p e c t r a * . . . . . . . . . . . , . . . . . . . . . . . 82 , i FIGURE III-1 E f f e c t of s t r u c t u r e dependence and Landau s t r a g g l i n g cn the r a y t r a c i n g l i n e s h a p e . .. 90 FIGURE I I I - 2 Raytracing l i n e s h a p e a p p l i e d to the fr-e peak, background t o 70 MeV, FWHM=8.0%, f i t 1012.05. ...106 FIGURE I I I - 3 R a y t r a c i n g l i n e s h a p e a p p l i e d t o the IT -e peak, background t o 61 MeV, FWHM = 8.0%j f i t 1012.04. .*.107 FIGURE I I I - 4 Raytracing l i n e s h a p e a p p l i e d t o the it -e peak, background to 61 MeV, FWHM=7. 6%, f i t 1012.03. .,,108 v i i FIGURE III-5 F i t of the ff-e peak with two r a y t r a c i n g l i n e s h a p e s , F WHM=6. 8% ,£it 10 12. 02 . . . . . ... . . . . ....... 10 9 FIGURE II1-6 Raytracing l i n e s h a p e a p p l i e d to the u+ ->• e++v +v spectrum. ...................... 11 5 FIGURE I I I - 7 Prompt and delayed proton s p e c t r a from the TTp-e vy experiment. .................... 127 FIGURE I I I - 8 Prompt and delayed n e u t r a l s p e c t r a from the TT-eyy experiment. ............ 128 FIGURE I I I - 9 ir-e l i n e s h a p e a p p l i e d t o the TT -e peak ( f i t 1017.04)....,,...,,....,..., 131 FIGURE 111-10 TT-e l i n e s h a p e a p p l i e d t o the U+ + e++ v + v peak ( f i t 1112.01). 13 2 FIGURE IV-1 Comparison of energy s p e c t r a from the c u r r e n t experiment with t h a t of DiCapua. ........................... 151 FIGURE A.C-1 Experimental set-up f o r the r a y t r a c i n g r u n s . ........172 v i i i PI GO EE A. C-2 T y p i c a l multi-wire p r o p o r t i o n a l counter spectrum. ...174 FIGURE A.C-3 r a y t r a c i n g e x p e r i m e n t a l set-up f o r 20 degree i n c i d e n c e . ..........................................176 FIGURE A.C-4 Measured minus p r e d i c t e d wire numbers f o r MTJPC3. ..,.178 FIGURE A.C-5 Nal l i n e s h a p e determined by the 0 degree r a y t r a c i n g runs. . . . . . . . . . . . . , . . ... 184 FIGURE A.C-6 Raytracing l i n e s h a p e s f o r v a r i o u s e n e r g i e s . .........185 FIGURE A.C-7 Revised r e s u l t s f o r the Nal l ineshape of the r a y t r a c i n g runs at 20 degree i n c i d e n c e . ......................... 186 FIGURE A.D-1 Feynman diagrams f o r r a d i a t i v e c o r r e c t i o n s to i r - e 190 FIGURE A . D-2 C a l c u l a t e d vs. T h e o r e t i c a l TT -e V Y photon energy s p e c t r a . .......................... 194 FIGURE A.D-3 C a l c u l a t e d vs. T h e o r e t i c a l - r r - e v y p o s i t r o n - photon angular d i s t r i b u t i o n s . ........195 i x FIGURE A. D-4 T r - e v y p o s i t r o n energy s p e c t r a * ... >. ................ 197 FIGURE A. D-5 E f f e c t o f v a r y i n g the photon acceptance on the c a l c u l a t e d -rr-e lineshape;. ...,..* , ..201 FIGURE A.E-1 Feynman diagrams f o r r a d i a t i v e c o r r e c t i o n s to y + •+ e++ v + v . .......................204 FIGURE A. E-2 Schematic routes f o r c a l c u l a t i n g e f f e c t i v e y + -> e++ v + v r a d i a t i v e c o r r e c t i o n s . .........210 FIGURE A. E-3 Schematic of method B f o r c a l c u l a t i n g y + - ^ - e + + v + v r a d i a t i v e c o r r e c t i o n s . ......... 213 FIGURE A. E-4 Photon energy s p e c t r a f o r y + -> e + + v + v decay from a monte-carlo s i m u l a t i o n * .... .... ........ ... ........ 223 FIGURE A. E-5 Ehotc n - p o s i t r o n angular d i s t r i b u t i o n s f o r y + ->e++ v + v decay from a monte-carlo s i m u l a t i o n . ......................... 224 FIGURE A.E-6 R a d i a t i v e c o r r e c t i o n s t o the z e r o t h order y + ->e++ v + v shape, e x c l u d i n g detected photons* ................ ......................... 22 5 X FIG DEE A. E-7 R a d i a t i v e c o r r e c t i o n s t o the z e r o t h order u + ->• e++ v + v shape, i n c l u d i n g detected photons. .........,........,..,.,..,.,,.«.....,..,»-.,..-.....». 22 6 FIGURE A. E-8 u + -> e++ v + v r a d i a t i v e c o r r e c t i o n s method A vs. Method B2. ...227 FIGURE A.F-1 Se v e r a l schemes f o r v a r i a t i o n of the Nal r e s o l u t i o n with energy...................... 230 x i LIST OF TABLES TABLE 1-1 Experimental E r r o r s of DiCapua e t . a l . .. .............. 16 TABLE II-1 Counters used i n the - r r-e expe r i m e n t . . . . . . . . . . .. 27 TABLE II-2 Ein width t e s t s . ..,......,.,.,.,,..,,..,...,,..,....54 TABLE II-3 E a t i o s f o r a c c i d e n t a l s s u b t r a c t i o n . .,..,,,....,,.,..62 TABLE II-4 Values f o r p i l e - u p r e j e c t i o n from each b i n . .........64 TABLE I I - 5 Summary of TT-e Runs. ............................ 68 TABLE II-6 T y p i c a l r a t e s f o r counters and co i n c i d e n c e boxes f o r the TT-e experiment* ...... ................. 15 TABLE II-7 Area of the low energy peak i n the f o u r b i n s as a percentage o f y + -> e ++ v + \; p o s i t r o n s . ..84 1 ABLE I I I - 1 E f f e c t s of the stopping d i s t r i b u t i o n and Landau s t r a g g l i n g on the r a y t r a c i n g l i n e s h a p e . ..91 x i i TABLE I I I - 2 D i r e c t and i n d i r e c t T r-e backgrounds. ..,.....,..,.,121 TABLE IV-1 C a l c u l a t i o n of the branching r a t i o f o r v a r i o u s b e l i e f s ... . .......... .........144 TABLE IV-2 Systematic c o r r e c t i o n s t o the Tr-ev branching r a t i o * 146 TABLE IV-3 Present r e s u l t s f o r the n - e v branching r a t i o . ......148 TABLE A.C-1 Experimental r e s u l t s of r a y t r a c i n g runs a t 0 degree i n c i d e n c e . . ..................131 TABLE A.C-2 Bevised experimental r e s u l t s of r a y t r a c i n g runs at 2 0 degree incidence;*. ........................187 TABLE A. D-1 T T _ E v y / T T - e v branching r a t i o s . .....196 TABLE A.D-2 Percentage of the c a l c u l a t e d • TT-e li n e s h a p e below var i o u s energy c u t o f f s . ............................. 198 I ABLE A. D-3 E f f e c t of v -e r a d i a t i v e c o r r e c t i o n s on the Nal l i n e s h a p e . ............................202 T AELE A. E-1 Equations used to c a l c u l a t e r a d i a t i v e c o r r e c t i o n s to P«- + e++ v «• v decay (from Kallen) ......... 207 x i i i TABLE A. E-2 E e s u l t s of Monte C a r l o s i m u l a t i o n s o f the y+ -> e++v + v r a d i a t i v e c o r r e c t i o n s ; . . ........220 TABLE A.F-1 T a i l c o r r e c t i o n s of the u+ e + + v + v shape vs. . Energy................ 23 1 TABLE A.G-1 D e s c r i p t i o n of f i t s t o TT -e data. . ... .......... 23 7 TABLE A. G-2 H e s u l t s of f i t s to T r-e data. .......................238 4 x i v ACKNOWLEDGEMENTS The experiment t h a t became the t o p i c f o r t h i s t h e s i s r e q u i r e d l a r g e d o l l o p s of i n g e n u i t y , e f f o r t and time, which were c o n t r i b u t e d by many people. Doug Bryman, as the experimenter of r e c o r d , c o n s i s t e n t l y motivated the group t o reach toward the outer bounds of cur e x p e r i m e n t a l l i m i t a t i o n s . The other members of the U n i v e r s i t y of V i c t o r i a group — Madhu D i x i t , John McDonald, Art O l i n , Gren Mason, and Mike P i e r c e — c o n t r i b u t e d l a r g e q u a n t i t i e s o f i m a g i n a t i o n and s k i l l . , Many thanks to a l l of the above f o r a l l o w i n g my p a r t i c i p a t i o n i n such a dynamic and e x c i t i n g e f f o r t . I a l s o wich to thank David Measday, Mike H a s i n o f f , and Jean-Michel P o u t i s s o u f o r t h e i r many, many years of encouragement, a s s i s t a n c e and a d v i c e , without which the long trek of t h i s graduate student would not have been p o s s i b l e . . , 1 CHAPTER I INTRODUCTION The e l e c t r o n decay mode of the pion has long been a su b j e c t of i n t e r e s t , but the reasons f o r t h i s i n t e r e s t have evolved c o n s i d e r a b l y . . As e a r l y as 1949, Ruderman and F i n k e l s t e i n (RU49) poi n t e d out that a value f o r the branching r a t i o : TT->eV + TT-*eVY R = _ d-1) TT->UV + TT->UVY c f R=10 _* would i n d i c a t e that the pion i s pseudo-scalar, and the decay c o u p l i n g pseudo-vector, A pseudo-scalar c o u p l i n g would give a branching r a t i o of about R=5.5, whereas f o r s c a l a r , vector or ten s o r c o u p l i n g , the e l e c t r o n decay mode i s f o r b i d d e n . Ruderman and F i n k e l s t e i n noted t h a t these r e s u l t s do not depend on the strong i n t e r a c t i o n p a r t of the decay process, but only on the assumption t h a t the muon and e l e c t r o n couple i d e n t i c a l l y to the pi o n , an assumption we now c a l l muon e l e c t r o n u n i v e r s a l i t y . . The numerical p r e d i c t i o n s quoted above depend upon u l t r a - v i o l e t c u t o f f s f o r the r a d i a t i v e c o r r e c t i o n s , and t h e r e f o r e are s u s c e p t i b l e to adjustment (St49). I f the same c u t o f f s a re chosen f o r both the e l e c t r o n and the mupn, the values R = 10~* cr R = 5.5 are c o r r e c t (Tr56) . By 1951 , the pseudo-scalar nature o f the pion was e s t a b l i s h e d (Du51), and the nature of the weak i n t e r a c t i o n assumed the f o r e f r o n t . The experimental evidence gave a somewhat c o n f u s i n g p i c t u r e u n t i l , i n 1958, the : ! ' v e c t o r — a x i a l - v e c t o r (V-A) scheme was shown t o c o r r e c t l y e x p l a i n an enormous body of experimental data (Su58, Fe58, Sa58) . In f a c t , only t h r e e experimental r e s u l t s c o n t r a d i c t e d the V-A scheme., E l e c t r o n - n e u t r i n o angular c o r r e l a t i o n s i n 6He and 2 2 N a f o r Gamow-Teller t r a n s i t i o n s i n i t i a l l y seemed to i n d i c a t e a t e n s o r i n t e r a c t i o n (Eu55, Sh54)- These experiments were l a t e r shown to be i n e r r o r (Wu58, He58) . . A second problem was the absence of the l e p t o n i c decay modes of the hyperons. Although these decay modes were e v e n t u a l l y d i s c o v e r e d (see Go62 and r e f e r e n c e s t h e r e i n ) , t h e i r r a t e s are one to two orders of magnitude below the V-A p r e d i c t i o n . . Bather than d i s a l l o w i n g the u n i v e r s a l Fermi i n t e r a c t i o n , these r e s u l t s simply broadened t h i s concept by r e q u i r i n g the i n t r o d u c t i o n of the Cabibbo angle i n t o the weak i n t e r a c t i o n c o u p l i n g f o r strange p a r t i c l e s (Fe69). The l a s t discrepancy between V-A theory and experimental r e s u l t s i n v o l v e d the branching r a t i o given i n eguation 1-1 . The theory y i e l d e d an unambiguous p r e d i c t i o n o f : 3 B = f * 1 e 2 [m ) e KJ 3 « 10-* in - m 2 1 m. 2 _ where m e ' m u ' m T r = m a s s e s °f the e l e c t r o n , muon, pion g e , g u = weak i n t . Coupling constants f o r the e l e c t r o n , muon (Ok66 and Fe63) whereas e x p e r i m e n t a l l y the r e a c t i o n had never been seen and the pu b l i s h e d upper l i m i t was then about 1 0 _ s . T h i s d i s c r e p a n c y , as noted by Feynman and Gell-Mann (FE58) , was the most s e r i o u s f a i l u r e o f the V-A t h e o r y . E x p e r i m e n t a l l y , a measurement of the branching r a t i o B would be t r i v i a l because of the enormous d i f f e r e n c e i n Q value between the r e a c t i o n s Tr+«r>e++ v and TT + -> u + +v were i t not f o r the f a c t t h a t the muons thus produced a l s o decay v i a the r e a c t i o n u+-> e++ v +v . (The decay ch a i n Tr +->e++v w i l l h e r e a f t e r be r e f e r r e d to as the i r - e c h a i n . S i m i l a r l y , the TT+-> u++ v f o l l o w e d by y+->e++ v + v w i l l be r e f e r r e d t o as the T T - U -e chain.) However, the decay e l e c t r o n s from these two r e a c t i o n s can be d i s t i n g u i s h e d i n t h r e e ways: 1) t h r e e charged p a r t i c l e s are present i n the TT - y - e c h a i n , whereas on l y two are present i n the Tr-e decay; 2) e l e c t r o n s from TT decay are mono-energetic at 6 9.8 MeV, while e l e c t r o n s from y decay y i e l d a continuous " M i c h e l " s p e c t r u m — t h e a d d i t i o n a l degree of freedom a r i s i n g because the muon decays i n t o three p a r t i c l e s ; and 3) the T r-e decay r a t e decreases e x p o n e n t i a l l y a c c o r d i n g t o the pion l i f e t i m e of 26 nanoseconds ( n s ) , whereas the TT - y-e e l e c t r o n s e x h i b i t a "parent-daughter" time d i s t r i b u t i o n , showing a f a s t r i s e t i m e a c c o r d i n g to the pion l i f e t i m e , and a slow e x p o n e n t i a l f a l l - o f f a c c o r d i n g to the muon l i f e t i m e (2200 ns) . Ihe c r i t e r i a most f r e q u e n t l y used to d i s t i n g u i s h between the Tr-e and TT - y - e decay ch a i n s , p a r t i c u l a r l y i n the e a r l y experiments, was simply the d i f f e r e n c e i n e l e c t r o n e n e r g i e s . The experimental search f o r the r a r e decay TT ->e+v was begun i n 1951 by Friedman and Rainwater (Fr51) . By scanning photographic emulsions, they found zero or one 7 r->e— events, an i n c o n c l u s i v e r e s u l t . Lokanathan and St e i n b e r g e r (Lo55) continued the search i n 1955. They stopped a beam of TT + p a r t i c l e s i n a p o l y e t h e l e n e t a r g e t A one-half i n c h t h i c k counter stopped about 500 pions pe second. They measured the energy of the charged decay p a r t i c l e i n a range t e l e s c o p e c o n s i s t i n g o f i n t e r l e a v e d sheets of p l a s t i c s c i n t i l l a t o r and absorber. T h e i r acceptance was q u i t e small (0.3%). S i x days of running produced a branching r a t i o of R= (-0.3+0.9) 10-*, again no 5 p o s i t i v e evidence f o r t h i s decay. The l a s t e x p e r i m e n t a l search f o r T r-e*v before the strong case f o r V-A theory was presented, was done by Anderson and L a t t e s (An57) at the U n i v e r s i t y of Chicago. They used a double f o c u s s i n g spectrometer to detect the charged decay p a r t i c l e . T h e i r acceptance was about 2% and t h e i r pion stop r a t e about 2000 per second. Although e l e c t r o n s from both decay c h a i n s were measured i n the same d e t e c t o r , they c o u l d not be measured s i m u l t a n e o u s l y , s i n c e the spectrometer could only focus e l e c t r o n s of a s i n g l e momentum at a given time.. Six days running y i e l d e d a branching r a t i o R= (-0.4±9.0) 10~ 6, once again not f i n d i n g the r a r e decay TT -e + v . T h i s r e s u l t was the primary cause f o r concern r e g a r d i n g the V-A t h e o r y . T h i s c l a s s i c disagreement between experiment and a h i g h l y s u c c e s s f u l theory sparked i n t e r e s t both i n modifying the form of the V-A i n t e r a c t i o n (Hu58) and i n r e - c h e c k i n g the o r i g i n a l measurements.. By 1958, a group at CEEN (Fa58) had redone the experiment with a method s i m i l a r to Lokanathan and S t e i n t e r g e r (Lo55). Stopping pions i n a p l a s t i c counter, the CEEN group measured the energy of the charged decay p a r t i c l e with a range t e l e s c o p e i n f r o n t of a Nal c r y s t a l . F o r v a l i d decays i n t o t h e i r d e t e c t o r , they photographed the pulses with a f a s t t r a v e l l i n g wave o s c i l l o s c o p e . (Although the Nal c r y s t a l was p a r t of the experimental set-up, and the o s c i l l o s c o p e t r a c e s of the pulses seen i n the Nal were 6 a l s o photographed, no s y s t e m a t i c use of t h i s i n f o r m a t i o n was made.) T h i s method of energy a n a l y s i s , i f slow and t e d i o u s , does make use of a l l t h r e e c r i t e r i a p r e v i o u s l y mentioned to d i s t i n g u i s h the T r-e decays frcm the TT - y -e decays. Although e l e c t r o n s from both kinds of decays could be examined simultaneously i n the same d e t e c t o r , the T r-e's were p o s i t i v e l y i d e n t i f i e d by measuring a range curve, i . e . i n c r e a s i n g the t h i c k n e s s of absorber i n the tel e s c o p e u n t i l o n l y the e n e r g e t i c TT decay p o s i t r o n s pass completely through. With t h i s method, the CEEN group repo r t e d the f i r s t p o s i t i v e evidence f o r the ex i s t e n c e of pion e l e c t r o n decay: they found 40 T r-e events, of which 4 were estimated to be background. Thus, a lower l i m i t f o r the branching r a t i o , B > 4 » 1 0 - 5 , was e s t a b l i s h e d , i n c o n t r a d i c t i o n with the experiment of Anderson and L a t t e s . Almost immediately afterward, a'Columbia group confirmed the e x i s t e n c e of T r-e decay (Im58). By examining bubble chamber photographs r e c o r d i n g 65,000 pion decays, they were able t o f i n d s i x unambiguous TT -e decays, g i v i n g a branching r a t i o of E= (0. 93±0. 4) 1 0 _*, i n good agreement with the V-A p r e d i c t i o n . The CEEN group continued the experiment and t h e i r a n a l y s i s (As59), A f t e r 200 hours of running time, sto p p i n g about 600 pions per second, they scon reported f i n d i n g 73 TT -e events, of which an estimated 7 were background. T h e i r a n a l y s i s r e s u l t e d i n a branching r a t i o B= (1.22±0.30) •10-*. . 7 Any remaining doubt concerning the gross value o f the branching r a t i o was e l i m i n a t e d when the Chicago group, again l e d by Anderson, r e - d i d t h e i r e a r l i e r experiment (An60). They used t h e same d o u b l e - f o c u s s i n g spectrometer used i n the e a r l i e r experiment, and a very s i m i l a r method: the only major improvement was photographing the pulses seen i n t h e i r s t o p p i n g counter. T h e i r new measurement recorded 1346 i r - e events, which i n c l u d e d 148 background events. Thus, the branching r a t i o was determined to be: R= (1.21±0.07) «10-* The 6% u n c e r t a i n t y made t h i s value the most accurate measured a t t h a t time. The i n c o r r e c t r e s u l t s of the e a r l i e r experiment were a t t r i b u t e d to d i f f i c u l t i e s i n c o n s i s t e n t l y d i s t i n g u i s h i n g between the incoming pion and the decay e l e c t r o n , probably due to i n s t a b i l i t i e s i n the p h o t o - m u l t i p l i e r voltage or t o poor c l i p p i n g of the s i g n a l from the t a r g e t counter. Thus, the V-A form f o r the u n i v e r s a l Fermi I n t e r a c t i o n c o u l d account f o r e s s e n t i a l l y a l l experimental data r e l a t e d to non-strange weak i n t e r a c t i o n s . C o n t r i b u t i o n s from non V-A terms i n the i n t e r a c t i o n ( i . e . s c a l a r , t e n s o r or pseudo-scalar terms) were e x p e r i m e n t a l l y determined t o be small by p r e c i s e measurements of the energy spectrum f o r mucn decay (P160). With the f i r m e s tablishment of the V-A theory of weak 8 i n t e r a c t i o n s , the branching r a t i o R=(TT -ev )/(T T - U V ) was now being examined from a s l i g h t l y d i f f e r e n t p e r s p e c t i v e : were the e l e c t r o n and the muon s t r i c t l y i nterchangeable i n weak i n t e r a c t i o n s , or as l a t e r developed, could the p a i r s ( e , v ) and ( y , v ) be s u b s t i t u t e d f o r one another. T h i s e y hypothesis, known as muon-electron u n i v e r s a l i t y , i s important i n understanding s e v e r a l aspects of p a r t i c l e p h y s i c s — p a r t i c u l a r l y meson decays and the CVC h y p o t h e s i s . The most s e n s i t i v e t e s t of y - e u n i v e r s a l i t y p r e s e n t l y a v a i l a b l e i s a measurement of the TT -e branching r a t i o . The most p r e c i s e measurement of t h i s branching r a t i o , p r i o r to our present e f f o r t , was done by DiCapua e t . a l . . i n 1963 a t Columbia (Di63). Since the present experiment f o l l o w s the method devised by the Columbia group, I w i l l d e s c r i b e t h e i r experiment i n some d e t a i l . The use of a Nal c r y s t a l i n measuring the T r-e branching r a t i o has s e v e r a l a t t r a c t i v e f e a t u r e s . P o s i t r o n s from both the T r - e and the T T - U -e chains are detected i n the same counter. The dynamic range of the energy response o f the d e t e c t o r i s l a r g e enough t o analyze the e n t i r e r e g i o n of i n t e r e s t . F a c t o r s l i k e the s o l i d angle and the t o t a l number of pion stops w i l l c a n c e l i n the branching r a t i o c a l c u l a t i o n , and thus need not be known to high accuracy. The major problem with using a Nal c r y s t a l t o measure the T r-e branching r a t i o l i e s i n the accurate simultaneous determination of the number o f F-e decays and the number of T T - P -e decays from the 9 same c o l l e c t i o n of stopped pions ( a c t i v e l y d i s c o u n t i n g , f o r i n s t a n c e , the muon contamination of the beam). Background r e l a t e d t o the muon decay c h a i n , 10,000 times more l i k e l y than the pion decay c h a i n , would n e c e s s a r i l y swamp the T r-e energy r e g i o n i n any ungated Nal spectrum. However, the spectrum must be gated i n such a way t h a t the number of accepted y + e ++ v + v decays are g r e a t l y reduced and yet t h a t same t o t a l number o f y + -> e + + v +v decays are determined to allow c a l c u l a t i o n of the TT -e branching r a t i o . DiCapua e t . a l . achieved t h i s i n q u i t e an ingenious f a s h i o n , d e s c r i b e d below. The beam from the Nevis s y n c h r o - c y c l o t r o n had a muon contamination of 10%, and an equal e l e c t r o n contamination. The beam s p i l l c o n s i s t e d of uniformly populated 5.5 m i l l i - s e c o n d p u l s e s with a frequency of 60 Hz, g i v i n g a duty c y c l e of about 3. The incoming p a r t i c l e s were detected by two s c i n t i l l a t i o n counters, and approximately h a l f were stopped i n a t h i r d c o u nter, • measuring 6"x4"x1". T h i s s t o p p i n g counter was i n c l i n e d 45 degrees t o the beam. T h e i r stopping r a t e was 30,000/second. Decay p o s i t r o n s from the TT -e and TT - y - e chains were detected i n two p l a s t i c s c i n t i l l a t o r s , and t h e i r e n e r g i e s were measured i n a Nal c r y s t a l . The c r y s t a l had a r e s o l u t i o n of 13.4% at 72 MeV. In a d d i t i o n , the stopping d i s t r i b u t i o n had to be f o l d e d i n t o the Nal lineshape. (The "stopping d i s t r i b u t i o n " r e f e r s t o the v a r i a t i o n i n 10 the energy l o s t by decay p o s i t r o n s o r i g i n a t i n g from d i f f e r e n t s e c t i o n s o f the stop p i n g counter.) T h i s y i e l d e d a Nal l i n e s h a p e with a t o t a l f u l l - w i d t h at haIf-maximum (FWHM) of 15.3%. 5 high energy t a i l can be present i n the TT - y -e li n e s h a p e , due to p i l e - u p of a p o s i t r o n with another pulse. Since the branching r a t i o i s of the order of 1/10,000, even s m a l l backgrounds t h r e a t e n to obscure the 7 1 -e mono-energetic peak. However, the Tr-e 's can be enhanced r e l a t i v e to the TT - y - e p o s i t r o n s by a f a c t o r of 100 i f the i n s p e c t i o n p e r i o d f o r a c o i n c i d e n c e between the T stop and the r e s u l t i n g decay p o s i t r o n i s a s i n g l e p i c n l i f e t i m e — t h u s most of the TT - y -e events are not examined. T h i s procedure reduces the s i g n a l to noise r a t i o i n the T r - e energy r e g i o n , but then causes a n o r m a l i z a t i o n problem: p r e c i s e l y how many TT - y -e • s are being ignored? DiCapua e t . a l . . s olved t h i s by i n s p e c t i n g f o r two c o i n c i d e n c e b i n s ( c a l l e d "boxes" by DiCapua e t . a l . ) of equal width: the f i r s t a prcmpt " TT-e " bin c o n t a i n i n g p o s i t r o n s from both the TT -e and TT - y -e ch a i n s , and the second a delayed "dummy" b i n s e v e r a l p i c n l i f e t i m e s away c o n t a i n i n g very few p o s i t r o n s from the TT -e decay, but more from the TT - y -e c h a i n . The TT - y -e shapes i n both bins i n c l u d e an a c c i d e n t a l c o n t r i b u t i o n from " o l d " muons i n the st o p p i n g counter, i . e . those net r e l a t e d t o the c u r r e n t TT step, and from pions stopping elsewhere than the stop p i n g counter. 11 EiCapua e t . a l . showed t h a t , a f t e r t h i s a c c i d e n t a l c o n t r i b u t i o n i s s u b t r a c t e d , the branching r a t i o can be determined from the number of i r - y -e 's i n the " T r - e " and "dummy" b i n s , the number o f Tr-e 's i n " TT-e " b i n , and the time s e p a r a t i o n o f the two b i n s . E = N X , „ (X - A ) t , Tre y ( 1 - e V s) X t X - X e P S B 2 - B 1 * ^ where N = t h e number o f TT-e d e c a y s i n b i n one. TTe 3 B1 = t h e number o f TT- y - e d e c a y s i n b i n one. B2 = the number o f TT - y -e d e c a y s i n b i n two. t = the t i m e s e p a r a t i o n o f b i n s one and two. 1 / X = T = y l i f e t i m e = 2 1 9 7 . 1 ns . y y 1 / A = T = TT l i f e t i m e = 2 6 . 0 3 ns . TT TT (See f i g u r e 1-1 and the appendix A). The branching r a t i o i s then independent of the s o l i d angle o f the d e t e c t o r , the displacement of the b i n s from the time of the p i - s t o p , the width of the two bin s , and the mucn contamination o f the beam. In a d d i t i o n , the Columbia group i n s p e c t e d f o r muon decays d u r i n g two a d d i t i o n a l time b i n s . T h e i r " l o n g " box 12 was a lon g e r v e r s i o n of the "dummy" b i n - - i t i n c l u d e d e s s e n t i a l l y only TT - u -e decays, but with t e t t e r s t a t i s t i c s . An " a c c i d e n t a l s " b i n measures random muon decays not a s s o c i a t e d with the c u r r e n t pion stop. T h i s bin i s the same l e n g t h as the long b i n , again f o r improved s t a t i s t i c s . To i n s u r e the c o r r e c t n o r m a l i z a t i o n of T r-e • s to TT — u -e '.s i n the branching r a t i o c a l c u l a t i o n , b i n s one and two must have the same width to a high degree o f accuracy. By a t t a c h i n g an open ended delay l i n e to the inpu t of a d i s c r i m i n a t o r , DiCapua e t ^ a l , , were able t o generate two f o r t y nano-second b i n s , separated by 136 ns, tha t were i d e n t i c a l t o with i n 0.25%. Although the time between the pio n step and the beginning of the f i r s t b i n (tO) does not enter i n t o the branching r a t i o c a l c u l a t i o n , tO must be l a r g e enough t o i n s u r e t h a t prompt s c a t t e r e d p a r t i c l e s or r e a c t i o n p a r t i c l e s from the beam are not being accepted i n t o the II TT -e " b i n . Any background e n t e r i n g i n t c the " T r-e 11 b i n and not i n t o the "dummy" b i n , p a r t i c u l a r l y i n the low s t a t i s t i c s TT -e r e g i o n , can in t r o d u c e a d i s p r o p o r t i o n a t e l y l a r g e e r r o r i n t o the branching r a t i o . . The Columbia group s e t tO = 5 ns. To reduce the number of random muon decays i n t o the co i n c i d e n c e b i n s , the Columbia group blanked the e l e c t r o n c o i n c i d e n c e u n i t f o r 4200 ns a f t e r a p i - s t o p . Of c o u r s e , the blanking was delayed u n t i l a f t e r a c o i n c i d e n c e with 13 (a) TT 1 2 3 . 5 6 (b) - T i m e of t h e p i o n s t o p Tje d e c a y n u e d e c a y -300 ^200 -100 0 100 200 300 400 I I L_ * s ns A c c i d e n t a l s b o x ne b o x D u m m y box FIGDEE 1-1 Experiment Of DiCapua Et. a l . ICJ C O U N T S ( K t O » ) i 1 r 2 4 . 0 1 B . 0 H 1 2 . 0 • 6 . 0 H 0.0 — I 1 1 1 1 • " " >— IS.O 3 0 . 0 « . 0 B O . O 7 5 . 0 B O . O J 0 3 0 CHANNEL COUNTS 1 X 1 0 ' ) B O . O B O . O H «.oH 2 0 . 0 H 0.0 .o as.o I O O . O I O S . O uo.o C H R N N E L (A) EXPERIMENTAL SET-UP (B) TIMING CONSTRAINTS PLACED ON THE SPECTRA ( c ) ENERGY SPECTRA DETECTED BY THE NAI DURING THE PROMPT *-e COINCIDENCE BOX FIGURE 1-1 EXPERIMENT OF DICAPUA, ET. AL. (DI63) 14 any time bin cou l d occur. T h i s reduced t h e i r " a c c i d e n t a l s " r a t e by a f a c t o r of f i v e . T h e i r " a c c i d e n t a l s " r a t e , defined as the number of counts i n the " a c c i d e n t a l s " b i n d i v i d e d by the number of counts i n the "long " b i n , was about 15%. The Columbia group's "dummy" bin was i n c l u d e d i n t h e i r " l o n g " bin, and as such thu energy spectrum f o r t h i s b i n was not recorded separately,. Nal s i g n a l s r e l a t e d t o coi n c i d e n c e s f o r the " T r-e " b i n , the " l o n g " b i n and the " a c c i d e n t a l s " b i n were recorded i n separate 100 channel s e c t i o n s of an ana l y z e r . P o s i t r o n s with e n e r g i e s below approximately 7 MeV do not appear i n the Nal s p e c t r a . . As shown i n f i g u r e 1-1 (c) , the Tr-e *s i n the T r-e bin c o u l d not be e n t i r e l y separated frcm the TT - y -e • s,. An e m p i r i c a l l y measured l i n e s h a p e , with a gaussian f o r t i e stopping d i s t r i b u t i o n f o l d e d i n t o the i n t r i n s i c Nal li n e s h a p e , was i n t u r n f o l d e d i n t o the t h e o r e t i c a l Michel spectrum. T h i s procedure determined a " t a i l c o r r e c t i o n " f o r the nuuber of TT - y -e p o s i t r o n s below the 7 MeV c u t o f f . The TT — y - e shape from the "lon g " b i n was normalized t o the muon decay s p e c t r a i n the T r-e b i n ( a f t e r both had been c o r r e c t e d f o r a c c i d e n t a l s ) and s u b t r a c t e d channel ty channel, removing the high energy TT - y -e c o n t r i b u t i o n from under the TT -e peak. The TT -e r e g i o n was then summed, and a t a i l c o r r e c t i o n from the e m p i r i c a l l i n e s h a p e was a p p l i e d . . The TT - y -e region o f the TT -e bin 15 spectrum and the s c a l e r f o r the "dummy" b i n were c o r r e c t e d f o r a c c i d e n t a l s , and the proper t a i l c o r r e c t i o n s were a p p l i e d . T h i s i n f o r m a t i o n , together with the time s e p a r a t i o n of the " TT -e " and "dummy" b i n s , was then used t o c a l c u l a t e the branching ratio,. In 120 two hour runs, the Columbia group c o l l e c t e d approximately 10, 000 TT -e events. The s t a t i s t i c a l l y weighted average of the branching r a t i o was c a l c u l a t e d . T h i s number was lowered by 0.3% t o c o r r e c t f o r the muons which were l o s t from the stopping counter, and were thus u n a v a i l a b l e f o r TT - y -e decay. The branching r a t i o was then found t o be: E=( 1. 247±0. 028) • 10~* The e r r o r s c o n t r i b u t i n g t o the s t a t e d 2.2% u n c e r t a i n t y are i t e m i z e d i n Table 1-1. T h i s value agreed with the t h e o r e t i c a l p r e d i c t i o n s to w i t h i n cne-half standard d e v i a t i o n , and was thus c o n s i d e r e d a great success o f the V-A theory and a c o n f i r m a t i o n cf muon e l e c t r o n u n i v e r s a l i t y to bet t e r the 1 %. However, Bryman and P i c c i o t t o (Br75) pointed out th a t t h i s 1963 c a l c u l a t i o n used the then c u r r e n t l i f e t i m e of the pion of 25.5±0.3 ns. When the t h e o r e t i c a l branching r a t i o and t h e r e s u l t of DiCapua's experiment were r e c a l c u l a t e d using the present value f o r the pion l i f e t i m e , 26.030±0.023 ns, Bryman and P i c c i o t t o found: 16 i 1 T r — i | Quantity | Value | S t a t i s t i c a l | Systematic | I I I E r r o r | E r r o r | 1 E | 1 . 2 4 7 « 1 0 - * ( 1 . 3 % ~ | . . . . "| I Muon | 0.997 | | 0.5% | I l o s s I I I I | T T - e ~J" 1.091 ~t 0. 5% | 0.5% "J | Lineshape | | j I \ T r - y - e |~1u„037 ~ | | o . 3 % "| | Lineshape | | | | TABLE 1 - 1 Experimental E r r o r s Of DiCapua E t . a l . . . B (theory) =1. 233* 10~* ii R(exp.) = (1. 2 7 4 ± 0 . 0 2 4 ) « 1 0 - * The best p r e v i o u s measurement o f the branching r a t i o then d i f f e r s by one and one h a l f standard d e v i a t i o n s frcm the t h e o r e t i c a l p r e d i c t i o n , b r i n g i n g i n t o question t h i s r e - a f f i r m a t i o n of mucn-electron u n i v e r s a l i t y . The above mentioned t h e o r e t i c a l r e s u l t d e r i v e s from a s t r a i g h t f o r w a r d a p p l i c a t i o n of quantum f i e l d theory f o r weak i n t e r a c t i o n s with V-A c o u p l i n g (Mu65). The i n t e r a c t i o n i s assumed to be a p o i n t vertex, although i n t r o d u c t i o n of an i n t e r m e d i a t e v e c t o r boscn (IVB) , with 17 c u r r e n t l i m i t s on the mass of such a p a r t i c l e , makes no s i g n i f i c a n t d i f f e r e n c e . However, t h i s simple c a l c u l a t i o n must be c o r r e c t e d f o r r a d i a t i v e e l e c t r o - m a g n e t i c processes (Ki59). These c o r r e c t i o n s f o r weak i n t e r a c t i o n s have had long standing problems of r e n o r m a l i z a b i l i t y - - i . e . the succeeding terms i n the p e r t u r b a t i o n theory expansion c o n t a i n f a c t o r s which diverge f o r high energy v i r t u a l photons. The standard technique i s t o i n t r o d u c e u l t r a - v i o l e t c u t - o f f s , making the r e s u l t s f i n i t e but somewhat a r b i t r a r y . Eecent t h e o r e t i c a l c a l c u l a t i o n s u s i n g a r e n o r m a l i z a b l e gauge theory of weak i n t e r a c t i o n s have appa r e n t l y removed t h i s ambiguity (Ma76 and Go75). These r e s u l t s a l s o appear t o be l a r g e l y model-independent, i n d i c a t i n g t h a t an accurate measurement c f the branching r a t i o i s indeed a t e s t of muon-electron u n i v e r s a l i t y , r a t h e r than a v e r i f i c a t i o n of seme p a r t i c u l a r model f o r the i n t e r a c t i o n . In a more g e n e r a l way, r e c e n t advances i n weak i n t e r a c t i o n f i e o r y r e l a t i n g to gauge t h e o r i e s and to the d i s c o v e r y of heavy l e p t o n s has spurred i n t e r e s t i n improving the accuracy o f experimental measurements of weak decays. We have measured the branching r a t i o : using a technique s i m i l a r t o DiCapua e t . a l . The major improvements were: 18 1) Use of a much l a r g e r (18" diameter by 20") Nal c r y s t a l (TINA) with improved energy r e s o l u t i o n , thus b e t t e r s e p a r a t i n g p o s i t r o n s from the TT -e and TT - u -e decays. 2) Dse of the TEIOMF meson f a c t o r y with a juuch h i g h e r pion f l u x , which enabled the accumulation of 10 s TT -e events, an order of magnitude more than the p r e v i o u s measurement. The duty c y c l e was a l s o a f a c t o r o f t h r e e b e t t e r . In c o n c l u s i o n , I mention b r i e f l y o t h e r t e s t s of muon-electron u n i v e r s a l i t y . The weak k a c n i c decays are e n t i r e l y analogous to the pion decays s t u d i e d here, and so: 12 K -niV 9 e V - m e 2 ) 2 en — m * K V m2' P e = 2.57 x 1 0 " 5 ( t h e o r y ) T h i s branching r a t i o has r e c e n t l y been measured ty Heard e t . a l . (He75) and by Heintze et,. a l . (He76) by d e t e c t i n g both p o s i t r o n s and muons i n a magnetic spectrometer. Their combined r e s u l t s , based on 938 K* e+ events y i e l d s : R = (2. kS ± 0.11) x 1 0 ~ 5  K Z 2 = (0.95 + 0.04) x R R ( t h e o r y ) T h i s i m p l i e s a v e r i f i c a t i o n of muon-electron u n i v e r s a l i t y 1 9 to perhaps 10%. Another t e s t of u n i v e r s a l i t y i s the dete r m i n a t i o n of the r a t i o : K Z 3 K ^ T T 0 e + v For pure v e c t o r c o u p l i n g , the t r a n s i t i o n amplitudes are dependent upon the f o r f a c t o r s f + and f Q , which are u s u a l l y parameterized: f + ,0 <<*2) » f + ,o (°) n • A + , 0 q v i / j q = four-momentum t r a n s f e r Assuming muon-electron u n i v e r s a l i t y a llows the branching r a t i o t o be expressed as: B K Z 3 = 0.6«5[1-.3A + + 2.2 A Q . . j T h i s r a t i o was a l s o measured by Heintze, e t . a l . (He77) , simu l t a n e o u s l y with the measurements. The momentum of the charged decay l e p t o n was determined i n a magnetic spectrometer, and i t s energy was determined i n a bank of Nal d e t e c t o r s . They determine, a f t e r r a d i a t i v e c o r r e c t i o n s , 20 R = 0.670±0.014 Which, using the world average X + =0.029±0.003, y i e l d s : A Q(branching r a t i o ) = 0. 019±0.010 I n t e r e s t i n g l y , the q u a n t i t y X q can a l s o be determined d i r e c t l y from the K^3 D a l i t z p l o t , a method independent of muon-electron u n i v e r s a l i t y . , , For Heintze's data X Q ( D a l i t z p l o t ) = 0±0.009 Then | X Q (branching r a t i o ) - X Q ( D a l i t z plot) |<0.036 with a c o n f i d e n c e l e v e l of 90%. ; T h i s i m p l i e s an 8% check on muon-electron u n i v e r s a l i t y . & more p r e c i s e value of X Q ( D a l i t z p l o t ) from K°^ rr~ia+v data i s 21 X ^ ( D a l i t z plot) = 0. 020±O.G04 y i e l d i n g a 3% l i m i t on muon-electron u n i v e r s a l i t y . E e i n t z e notes t h a t t h i s i s based on the r a d i a t i v e c o r r e c t i o n s of Ginsberg (Gi66) I The r e l e v a n t r e s u l t s f o r u -p and e-p e l a s t i c s c a t t e r i n g seem somewhat confused (Ca69). Another promising approach i s the comparison of muon capture i n hydrogen with the analogous r e a c t i o n f o r e l e c t r o n s : y+p-;-->- n+v^ n->p+e +v A n a l y s i s of these decays i s complicated by the f a c t t h a t s t r o n g i n t e r a c t i o n e f f e c t s cause the r a t i o cf the c o u p l i n g c o n s t a n t s g f t /g to dev i a t e from u n i t y . For r e a c t i o n s i n v o l v i n g e l e c t r o n s , 1.258±0.015 (Kr75) g 3 / g 3 = y A ' y V 1.25910.017 (St78) An a n a l y s i s of the e x i s t i n g muon capture i n f o r m a t i o n by Mukhopadhyay (Mu77) y i e l d s : /g v = 1 - 2 3 ± ° - 0 7 22 which i s i n e x c e l l e n t agreement with the r a t i o f o r e l e c t i o n r e a c t i o n s , with an u n c e r t a i n t y of about 6 %. This i s yet another r e - a f f i r m a t i o n of muon-electron u n i v e r s a l i t y . A rec e n t p r o p o s a l by C.S. wu e t . a l . to re-examine muon capture i n hydrogen at the Nevis Laboratory could have s i g n i f i c a n t l y reduced t h i s u n c e r t a i n t y i n the r a t i o of c o u p l i n g c o n s t a n t s f o r muon capture.. I 23 Chapter I I EXPERIMENTAL METHOD AND SET-UP II-A. I n t r o d u c t i o n We wish to measure the branching r a t i o f o r -e versus normal pion decay to an accuracy of t e t t e r than 1 . 0 % . S p e c i f i c a l l y , we measure + + TT - • e V + TT +ev Y R = — - — ... ( I I - 1 ) TT - » - U V + TT -*yv y U P The muons are not themselves d e t e c t e d , but r a t h e r , t h e i r decay products from the r e a c t i o n P + " - e V e % , . . (11 -2 ) P o s i t r o n s from both these decay c h a i n s are detected i n a l a r g e Nal c r y s t a l . Since the p o s i t r o n s and photons from the decay + + TT -*-e v Y e u + - * e \ ' ~ \T Y . , . .(11-3) e u have a high degree o f angular c o r r e l a t i o n , the Nal u s u a l l y d e t e c t s the energy d e p o s i t e d by both the p o s i t r o n and the photon. 24 In philosophy and i n p r a c t i c e , t h i s experiment fo l l o w e d c l o s e l y t h at of DiCapua et. a l . _ (Di63), d e s c r i b e d i n the i n t r o d u c t i o n . Our aim was t o c o l l e c t 10 s T r-e events (using a Nal detector) with the best s i g n a l t o noise r a t i o we could achieve, while only using approximately one month of TRIOMF machine time.. We a l s o needed to c o l l e c t e q u a l l y c l e a n TT.-u.-e events, from the 53 MeV maximum energy down t o as low an energy as p o s s i b l e . We needed to minimize the background a r i s i n g from the decay of " e l d " muons i n the t a r g e t r e g i o n , and to completely e l i m i n a t e any background a r i s i n g from p i l e - u p of s e v e r a l s o r t s , or from the s c a t t e r of prompt p a r t i c l e s i n t o the Nal d e t e c t o r . We had to compensate f o r the e f f e c t s of muon contamination i n the beam. A l l t h i s had to be achieved i n such a manner t h a t the TT -e branching r a t i o c ould s t i l l be c a l c u l a t e d t o high accuracy. T h i s r e q u i r e d , among other t h i n g s , i d e n t i f i c a t i o n and timing i n f o r m a t i o n f o r the incoming "stopped" pions, i d e n t i f i c a t i o n and ti m i n g i n f o r m a t i o n f o r the p o s i t r o n s decaying i n t o a given acceptance, the dete r m i n a t i o n of the energy of the decay p o s i t r o n s , and some very p r e c i s e t i m i n g c o n s t r a i n t s i n the e l e c t r o n i c s l o g i c used t o analyze these s i g n a l s . Our source of pions i s the TRIUMF c y l o t r o n . The a c c e l e r a t o r has no macroscopic duty c y c l e , but the beam has a m i c r o - s t r u c t u r e c o n s i s t i n g of a beam b u r s t every 43 nano-seconds (ns) caused by the RF of 23.26 MHz,. Negative 25 hydrogen ions are a c c e l e r a t e d t o the r e q u i r e d energy (500 HeV maximum). The i o n s are then s t r i p p e d of t h e i r e l e c t r o n s , and the protons are swept out. cf the a c c e l e r a t o r tank and i n t o a beamline (BLI) which d e l i v e r s them to the meson production t a r g e t (T2)., The usual production t a r g e t s are made o f b e r y l l i u m or copper. The pions thus produced are guided down a secondary beamline (the stopped TT-U c h a n n e l - - M 9 ) T h e pion beam f o c u s s i n g was nominally achromatic. V a r i a b l e s l i t s , u s u a l l y set to 10 cm. or l e s s , c o l l i m a t e d the beam, y i e l d i n g a beam spot c f roughly 13 cm. FWHM. The pion energy was 30 MeV. A primary beam c u r r e n t of 3 micro-amps j • of protons produced a copious supply of p i c n s . . (The r a t e c f pions e x i t i n g from the M9 snout was about 400 K/s/ u A.. The pion stop r a t e i n the t a r g e t counter was about 4 0 K / S / u , A.) The experimental set-up i s shown i n f i g u r e I I - 1 . (The numbering of counters i s hen-consecutive because the l a t t e r stages of the j r - e experiment were run i n c o n j u n c t i o n with a u ->e +~Y experiment. . T h i s y -> e + y measurement r e q u i r e d a second Nal and a d d i t i o n a l c o u n t e r s which are not shown i n the f i g u r e . ) Dimensions of the counters are given i n Table I I - 1 . Incoming pions were detected i n counters S1 and S2, and degraded by one i n c h of CH i . The beam was then c o l l i m a t e d to a diameter of 3.8 cm,. The pions c o u l d then b e i d e t e c t e d i n c o u n t e r s S3, S4, and S5. The s i g n a t u r e f o r a stopped pion was 26 27 TABLE I I - l S C I N T I L L A T I O N COUNTERS USED IN THE n - e E X P E R I M E N T SECOND COUNTER DIMENSIONS E F F I C I E N C Y RF (CM) E F F I C I E N C Y 51 15xl5x.l6 0.996 0.986 52 10xl0x.32 0.984 0.956 53 lQxlQx.16 ? 0.870 54 7.5x7„5x.63 ? ° - 9 5 0 55 10xl0x.32 0.965 0.900 56 15xl5x.l6 0.996 57 8.7 DIAMETER 0.974 S l l 17.5 DIAMETEF* 0.999 x.32 S13 3Qx61x.32 0.988 S13' 30x61x.32 0.950 28 S1*S2»S3«»S4®S5. Under normal running c o n d i t i o n s , approximately 63% of the pions stopped i n counter S4. (These s h a l l be r e f e r r e d to as " l e g a l stops".) P o s i t r o n s produced from e i t h e r the I T-ev or i r * - * - y + + v f o l l o w e d by y + -> e++ v + v c h a i n s which decay i n t o the allowed s o l i d angle have the s i g n a t u r e S7«S 11® (S13 AS13'). Note t h a t counter S4 i s sandwiched as c l o s e l y as p o s s i b l e between co u n t e r s S3 and S5 to prevent "muon leakage" --that i s to prevent muons produced from TT decay (which have about 4 MeV k i n e t i c energy) from e s c a p i n g from the stopping counter, and thus a v o i d i n g d e t e c t i o n when they subsequently beta decay. The energy o f the p o s i t r o n s was measured i n a l a r g e N a l ( T l ) c r y s t a l known as "TINA" f o r Triumf Iodide of Na (see s e c t i o n I I - B ) . Timing i n f o r m a t i o n was taken e n t i r e l y from counter S11 r a t h e r than the Nal s i g n a l , u t i l i z i n g the counter's f a s t e r response time. P o s i t r o n s from the TT -e chain are mono-energetic, with a t o t a l energy c f 69.8 MeV. .. P o s i t r o n s from the T — y —e c h a i n can have any t o t a l energy from 0.511 to 52.8 MeV* with the h i g h e s t energy being the most l i k e l y . Because the TT r e decay i s four' orders of magnitude l e s s l i k e l y than the T r - y -e decay, even a very s m a l l high energy background from the TT - y -e shape measured i n the Nal, mostly r e l a t e d t o the p i l e - u p events, d r a s t i c a l l y worsens the s i g n a l t o noise r a t i o i n the TT -e r e g i o n . To a l l e v i a t e t h i s we do not r e c o r d the energy of a l l 29 p o s i t r o n s decaying i n t o the c r y s t a l , but only of those s a t i s f y i n g c e r t a i n time c o n s t r a i n t s . That i s , a c o i n c i d e n c e i s demanded between the p e s i t r c n decay and any of f o u r time b i n s , each generated by the pion stop.. Timing of the f o u r bins i s shown i n f i g u r e I I - 2 . The f i r s t time b i n i s about 25 ns long (a s i n g l e pion l i f e t i m e ) , and begins about 6 ns a f t e r the p i - s t o p . T h i s bin c o n t a i n s p o s i t r o n s frcm the - r r-e c h a i n and frcm the TT - y -e c h a i n . A l l other bins c o n t a i n e s s e n t i a l l y only p o s i t r o n s from the I T - y - e c h a i n . The seccnd time b i n begins seven pion l i f e t i m e s a f t e r the f i r s t , and has t i e same width t o w i t h i n 0.05%. The t h i r d time b i n begins well a f t e r the second. I t i s 170 ns l o n g , and y i e l d s a Michel spoctrum ( p o s i t r o n s from the T r ~ ; y - e chain) with good s t a t i s t i c s . The f o u r t h bin! i s formed well before the picn stop, has the same d u r a t i o n as b i n 3, and records events not r e l a t e d t c the c u r r e n t pion s t e p , i . e . a c c i d e n t a l s . .. (Timing c o n s i d e r a t i o n s are examined i n more d e t a i l i n s e c t i o n II-D). The muon l i f e t i m e of 2200 ns i s much longer than the time spanned by our f o u r bins. Subsequently, t h e r e were always " o l d " muons frcm e a r l i e r pion decays i n the t a r g e t area which c o u l d decay and be d e t e c t e d i n random co i n c i d e n c e with one of the time b i n s . Bin f o u r measures t h i s a c c i d e n t a l c o i n c i d e n c e r a t e , which i s used t o c o r r e c t the other t h r e e b i n s . In a d d i t i o n , t h i s randem r a t e was reduced by b l a n k i n g the " c l e a n " pion stop s i g n a l (from FIGURE II-2 TIMING LOGIC FOR THE COINCIDENCE BINS. -'100 -300 r200 TIME OF THE PI-STOP \ rlOO T S ( l - ' f ) = 3 0 0 N S B4 ' TwtJ=120NS ' NO PRIOR PI-STOP IN TARGET COUNTERS FOR 2.1 US - BLANKING DECAY *-U-e DECAY .100 200 Ts(1-3)-301NS ,300 400 ; Ts=170NS Bl B2 B3 •Tw=25NS T w 3=120 NS • EXPONENTIAL DECAY.OF B4 STOP PILE-UP REJECTION FOR 430 NS UNREPLENISHED" ACCIDENTALS MEASURED IN B4 4- + |B1 ! P2 I B3 : RF STRUCTURE OF BEAM: |= TIME OF ARRIVAL OF T T , j = TIME OF W, jjc = TIME OF « (NS) OJ o 3 Tl which th«j time b i n s were generated) f o r a f i x e d p e r i o d a f t e r each "raw" pion stop.. T h i s p e r i o d was determined by o p t i m i z i n g the clean/raw pion stop r a t i o versus the random/real pion decay r a t e , and was set to 2.1 micro-seconds (see s e c t i o n I I - D ) . . The average " c l e a n " pion stop r a t e was 80 thousand per second (K/s). Two types of p i l e - u p r e j e c t i o n were performed. Counters 13 and 13' t o g e t h e r cover the e n t i r e f r o n t f a c e of TINA, and were c a l l e d TINA Veto (TV) counters. TV p i l e - u p was done f o r an i n s p e c t i o n time of 430 ns. T h i s e f f e c t i v e l y prevented charged-charged p i l e - u p f o r p a r t i c l e s coming from the st o p p i n g area ( l i m i t e d by the response time f o r r e - t r i g g e r i n g the c o u n t e r s ) , and thus improved the r e s o l u t i o n and the TT -e and TT - y -e s e p a r a t i o n . Raw stop p i l e - u p was a l s o done f o r an i n s p e c t i o n time of 430 ns,, T h i s was necessary to i n s u r e t h a t a l l f c u r time b i n s — p a r t i c u l a r l y b i n s 1 and 2—were t r e a t e d i d e n t i c a l l y . A problem a r i s e s i f , a f t e r a " l e g a l " pion stop generated the f o u r time b i n s , a second pion a r r i v e s before tho second time b i n has commenced i n (real) time. A p o s i t r o n from t h i s second p i o n can then be c o i n c i d e n t with bin two, but not with b i n one. That i s , the p o s i t r o n s i n b i n s one and two would not be products of the same set o f parent pions. , The branching r a t i o i s u n f o r t u n a t e l y s e n s i t i v e t o t h i s type of f a l s e event, (See s e c t i o n n - D . ) 32 Events from each of the f o u r b i n s which s a t i s f y t i e above d e s c r i b e d c r i t e r i a are analyzed and recorded i n one of the f o u r s e c t i o n s of an ND2400 analyzer,. The s p e c t r a were w r i t t e n onto magnetic tape u s u a l l y every hour,. F i n a l a n a l y s i s was done on the TRIUMF E c l i p s e mini-computer or the U.B.C. Compute!:, i n i t i a l l y an IBM 370/168, l a t e r upgraded t o an Amdahl 470 V/6 computer. II-B. Apparatus 1. The N a l ( T l ) Detector A l a r g e Nal(Tl) c r y s t a l - - T I N A — w a s used to measure the energy of the p o s i t r o n s from the TT - e cr TT - y - e decay char.ns which s a t i s f i e d c e r t a i n c o n d i t i o n s . The c r y s t a l measured 45.7 cm. i n diameter by 50,i8 cm. i n le n g t h . The c r y s t a l was manufactured by the Harshaw Chemical Company, Ohio, U.S.A., and i s used f o r the d e t e c t i o n o f high energy photons and e l e c t r o n s . The c r y s t a l was viewed by seven BCA 4522 phototubes. The phototubes have a diameter of 12.7 cm,, and a r i s e t i m e of about 20 ns. However, the r i s e t i m e of the summed pulse i s about 50 ns, due to the v a r i a t i o n i n path l e n g t h that 3 3 the l i g h t must t r a v e l b efore i t reaches the phototubes. The anode pulses of the i n d i v i d u a l tubes were summed (using an LSS f a n - i n ) . The tubes were powered by separate power s u p p l i e s . The summed pu l s e was c l i p p e d t o approximately 300 ns, which reduced p i l e - u p e f f e c t s while i n t r o d u c i n g e s s e n t i a l l y no d e t e r i o r a t i o n i n the energy r e s o l u t i o n . The s i g n a l was then passed through a l i n e a r gate and i n t o an a m p l i f i e r , which i n t e g r a t e d the s i g n a l f o r a time constant of 2 micro-seconds. The l i n e a r gate v a s t l y reduced the number of s i g n a l s being a m p l i f i e d . WhAle the use of s e v e r a l photo-tubes to view the l i g h t avoids the problem of non-uniformity of the p h o t o - s e n s i t i v e area encountered by very large phototubes, the c o l l e c t i o n of tubes must be balanced to e f f e c t i v e l y make use of t h i s advantage. B a l a n c i n g the phototubes was made r e l a t i v e l y simple f o r t h i s experiment by the prominence of the p o s i t r o n s from muon decay. although these p o s i t r o n s are not mono-energetic, they are peaked at the maximum energy, and the midpoint of the high energy edge i s r / e l l d e f i n e d . By connecting the photo-tubes i n d i v i d u a l l y at the f a n - i n and varying the high voltage of each, the photo-tubes could q u i c k l y be balanced or re-checked. 34 The a c t u a l d e t e c t o r response at a given energy depends on a v a r i e t y of f a c t o r s - - i n c l u d i n g : 1) the area of the c r y s t a l ' s f r o n t f a c e which p a r t i c l e s are p e r m i t t e d to enter (the f r o n t face i l l u m i n a t i o n ) ; 2) the count r a t e i n the c r y s t a l ; and 3) the accuracy with which the tubes have teen balanced. Although the response a t 70 MeV was worse than expected (FWHM of 6% i n s t e a d of 4%), i t was s t i l l a vast improvement over the e a r l i e r experiment (Di63 and Fo63). A more d e t a i l e d examination of the Nal response f u n c t i o n i f g i ven i n s e c t i o n III-A* During the f i n a l run of the experiment, frequent g a i n s h i f t s of the s p e c t r a occured. T h i s was e v e n t u a l l y discovered to be a d i u r n a l e f f e c t (see f i g u r e 11-12). A temperature s e n s i t i v i t y of the zener doides i n the f i n a l stages of the p h o t o - m u l t i p l i e r bases was the cause.. T h i s made data a n a l y s i s more d i f f i c u l t , but s i n c e our s p e c t r a are s e l f - c a l i b r a t i n g , we were a b l e to c o r r e c t f o r t h i s e f f e c t . 35 II-B-2. S c i n t i l l a t i o n Counters P l a s t i c S c i n t i l l a t i o n counters (NE110 made by Nuclear E l e c t r o n i c s ) were used to d e f i n e the incoming pions and the decay p o s i t r o n s . The counter arrangement was shown i n i f i g u r e I I - 1 . The counters were viewed by BC& 8575 photo-tubes. Dimensions and e f f i c i e n c i e s o f the counters are given i n t a b l e I I - 1 . Since the p l a s t i c counters s u p p l i e d a l l t i m i n g i n f o r m a t i o n , which was c r i t i c a l t o t h i s experiment, g r e a t care was taken to in s u r e a l l counters were f u n c t i o n i n g p r o p e r l y . . Counters were c a r e f u l l y plateaued, u s u a l l y using a small e f f i c i e n c y counter to y i e l d the a b s o l u t e e f f i c i e n c y . That i s , t o measure the e f f i c i e n c y of counter 52, a small e f f i c i e n c y counter, SX, was p l a c e d behind S2. The absolute e f f i c i e n c y of counter S2 was then the r a t i o of the number of c o i n c i d e n c e s (S1»S2»SX)/(S1«SX). . The e f f i c i e n c y was then maximized by a d j u s t i n g the counters high v o l t a g e and d i s c r i m i n a t o r s e t t i n g . . Counters S1, S2, 53, S4, S5 and S6 were plateaued using the incoming pion beam-. Counters S7 and S11 were plateaued using p o s i t r o n s i n the incoming beam (by i n s e r t i n g enough absorber upstream to stop pions and muons) , while S13 and S13' used p o s i t r o n s from muon decay. T h i s i n s u r e d that the p c s i t r c n i t e l e s c o p e c o u l d a c t u a l l y d e t e c t the p o s i t r o n s , which d e p o s i t c o n s i d e r a b l y l e s s energy than pions i n the p l a s t i c 36 counters. The experiment was r a t h e r s e n s i t i v e t o p i l e - u p e f f e c t s , p a r t i c u l a r l y t o p i l e - u p of stopped p a r t i c l e s . Beam p a r t i c l e s could a r r i v e i n time "buckets" separated by 43 ns, due t o the m i c r o - s t r u c t u r e of the c y c l o t r o n beam. To i n s u r e t h a t the counters i n the pion t e l e s c o p e could d e t e c t incoming p a r t i c l e s i n the f i r s t bucket f o l l o w i n g a p i - s t o p , the s i g n a l s frcm counters S i , S2, S3, S4, and S5 were c l i p p e d from a width of about 50 ns at the base of the s i g n a l t o a width of about 20 ns.. The e f f i c i e n c y of the counters t o see p a r t i c l e s i n t h i s second bucket (second EF e f f i c i e n c y ) was measured by performing a co i n c i d e n c e of a counter with i t s e l f delayed by 43 ns, and comparing t h a t to a co i n c i d e n c e of a counter with i t s e l f delayed b y 86 ns* T h i s second EF e f f i c i e n c y i s a l s o given i n t a b l e I I - 1 . (Counter S5 had an odd shape t h a t was not amenable to c l i p p i n g . ) ; Counting r a t e s of the . s c i n t i l a t o r s , and of v a r i o u s c o i n c i d e n c e s , were p e r i o d i c a l l y checked through out the run to p r o t e c t a g a i n s t counter f a i l u r e s , l i g h t l e a k s , d i s c r i m i n a t o r f a i l u r e s or d r i f t s i n the high voltage or d i s c r i m i n a t o r t h r e s h o l d s f o r the var i o u s counters. These checks a l s o p r o t e c t e d against the e r r a n t f i n g e r , which has been known to appear from time t o time. 5 7 II-C. E l e c t r o n i c s Logic 1. General E l e c t r o n i c s l o g i c f o r p a r t i c l e p h y s i c s experiments tends t o grow i n an o r g a n i c , though not n e c e s s a r i l y s t r a i g h t f o r w a r d , f a s h i o n . Not onl y does the l o g i c evolve but, due t o emergencies, equipment f a i l u r e s , brainstorms and r h e t o r i c a l successes and d e f e a t s , i t can be s a i d t o devolve, mutate, become aberrant, and c o n t a i n a number cf dead ends and u n f u l f i l l e d i d e a s . The schematic diagrams of the e l e c t r o n i c s presented shows the a c t u a l l o g i c used, r a t h e r than an i d e a l i z a t i o n . .. Only the l o g i c r e l a t e d to the s c a l e r s (except f o r the p i l e - u p s c a l e r s ) has been emitted. F i g u r e II-3 shows the conceptual l o g i c . A pion stop i s i d e n t i f i e d , and produces a s e t of f o u r time b i n s . (Only two are shown here.) Care i s taken' t o i n s u r e bins one and two have the same d u r a t i o n and c o n s t a n t s e p a r a t i o n * A charged p a r t i c l e (assumed t o be a p o s i t r o n ) e n t e r i n g the Nal c r y s t a l i s i d e n t i f i e d . , T e s t s f o r co i n c i d e n c e between the charged p a r t i c l e and each of the f o u r time b i n s are performed. . I f a c o i n c i d e n c e i s found, the anode s i g n a l from the Nal c r y s t a l i s a m p l i f i e d , 38 FIGURE 11-3 SCHEMATIC DIAGRAM OF THE ELECTRONICS TINA LINEAR GATE ROUTER A M P L I F I E R ND2400 ANALYZER • [ > D o a 0 DELAY DISCRIMINATOR AND OR FAN IN ( F / l ) FAN OUT (F/O) P . U . G . G . G . SCALER P I L E UP GATE GATE GATE GENERATOR & DELAY LEGEND 3 9 FIGURE II-** ELECTRONICS RELATED TO PI-STOPS S 6 H 1—I si-i S 2 H S3-1 ST S 5 1 CLEAN VETO F/ l ' T -L^  pi e-NEGATIVE CLEAN D STOP DELAY F / 0 • > D O O 0 I P.U.G. 6 . G . DELAY DISCRIMINATOR AND OR FAN IN FAN OUT SCALER PILE UP GATE GATE (F/ l ) ( F / 0 ) GATE • GENERATOR & DELAY LEGEND FIGURE 11-5 ELECTRONICS RELATED TO THE TIME BINS ko TIMING B I N OR (BOR) -VvV- • [ > D o a DELAY DISCRIMINATOR AND OR FAN IN (F/I) FAN OUT (F/O) 0 SCALER P.U.G, PILE UP GATE G.G. GATE GENERATOR G & D GATE & DELAY LEGEND 4 1 FIGURE 11-6 ELECTRONICS RELATED TO POSITRON IDENTIFICATI S 7 H sl3H ELECTRON F/O sl3-4 DELAYED ELECTRON FOR B4 • [ > D D ( ] 0 0 ra DELAY DISCRIMINATOR AND OR -WV-FAN IN FAN OUT (F/I) (F/O) PILE UP 6ATE GATE GATE GENERATOR & DELAY LEGEND 42 FIGURE 11-7 ELECTRONICS RELATED TO PILE-UP REJECTION BlNl B l P > - — _ G.Gi DELAYED 1 B l STOP 2 L 0 — • t>D D a o DELAY DISCRIMINATOR AND OR IP .U .G. G . G . FAN IN FAN OUT (F/I) (F/O) SCALER PILE UP GATE GATE GATE 6ENERAT0R & DELAY LEGEND 4 3 FIGURE 11-8 ELECTRONICS RELATES TO THE SLOW LOGIC (NAI SIGNAL) Sl l Iff UNDANT G.G. T.V.P.U. F/O STOP P.U. r, & -D TINA GATES BlNl LINEAR BIN2 BIN3 LINEAR GATE BlNl ROUTER AMP-LIFIER f— V ND2400 ANALYZER ANTI-COINCIDENCE • I>DD a 0 a 0 DELAY DISCRIMINATOR AND OR FAN IN FAN OUT SCALER FILE UP 6ATE GATE (F/ l ) (F/O) 6ATE GENERATOR S DELAY LEGEND d i g i t i z e d , and histogrammed i n one of the f o u r s e c t i o n s of the a n a l y z e r . S u r e l y t h i s should be a s t r a i g h t f o r w a r d procedure. The e l e c t r o n i c s l o g i c i s given i n f i v e schematics: 1) l o g i c f o r the p i - s t o p ; 2) l o g i c f o r the time b i n s ; 3) l o g i c f o r the p o s i t r o n s ; 4) l o g i c f o r pile-^up r e j e c t i o n ; and 5) slow l o g i c f o r the Nal s i g n a l . Those boxes which i n t e r f a c e the f i v e schematics are r e p l i c a t e d i n each., The experiment was switched on and o f f with the ND2400 analy z e r by u t i l i z i n g the ADC busy output. Both CAMAC and v i s u a l s c a l e r s were used dur i n g the experiment* A f t e r each run, CAMAC s c a l e r s were read and p r i n t e d out. Other i n f o r m a t i o n was manually r e c o r d e d (see s e c t i o n I I - E ) . The energy histograms were then w r i t t e n from the a n a l y z e r onto magnetic tape f o r l a t e r a n a l y s i s . 45 I I - 0 2 . Fast L o g i c F i g u r e s II-3 and II-4 give schematics cf the e l e c t r o n i c s used to i d e n t i f y the p i - s t o p and generate the f o u r time t i n s . A v a l i d stop i s given by S1«S2«S3*S4»S5 However, stop b l a n k i n g and stop p i l e - u p were both performed on S1«S2* (S3 AS4 AS5)*S6. . During the b l a n k i n g p e r i o d , a second pion stop i s not acknowledged a " c l e a n " stop, and does not generate any of the f o u r b i n s . T h i s b l a n k i n g p e r i o d was set to 2.1 micro-seconds (see s e c t i o n II-D) . The timing f o r the p i - s t o p i s determined by counter S3. T h i s counter had a low count, and was l e s s l i k e l y to allow an incoming pion to be confused with an outgoing charged decay product. The n e g a t i v e tO delay a f f e c t s a l l f o u r b i n s i n an i d e n t i c a l f a s h i o n . An open ended delay l i n e on the input t o the f i r s t 'double box' d i s c r i m i n a t o r generated b i n s one and two* . The f o l l o w i n g two d i s c r i m i n a t o r s i n s u r e t h a t the a t t e n u a t i o n cf the r e f l e c t e d s i g n a l t h a t generated b i n two does not a f f e c t the t i m i n g i n an a r b i t r a r y f a s h i o n * . The "Timing" box searches f o r a c o i n c i d e n c e between e i t h e r bin one or bin two, generated by the p i - s t o p , and a decay p o s i t r o n e n t e r i n g the Nal c r y s t a l . T h i s i n s u r e s t h a t the t h r e s h o l d f o r a c o i n c i d e n c e , which could vary from box to box, does not change the e f f e c t i v e width of bins one and two i n an 46 asymmetric fashion._ Which time b i n caused the " t i m i n g " box to f i r e i s determined i n a subsequent c o i n c i d e n c e with "ta g " pulses f o r bins one and two which are a l s o generated by the p i - s t o p . The widths of time bins t h r e e and f o u r are f a r l e s s c r i t i c a l . These widths are s e t using the output width adjustment of the " t a g " d i s c r i m i n a t o r s . Only a s i n g l e p i - s t o p — p o s i t r o n c o i n c i d e n c e i s then r e q u i r e d f o r these bins. Bins one, two and three are fanned i n and delayed, such t h a t t h e p o s i t r o n s i g n a l routed through each of these b i n s a r r i v e s a t the Bin OB (BOB) i n time with the p o s i t r o n s i g n a l coming through bin four (see f o l l o w i n g paragraph). The BOB i s then used to gate the Nal energy s i g n a l i n t o the a n a l y z e r . . Figure I I - 5 shows the e l e c t r o n i c s used t c i d e n t i f y a charged p a r t i c l e e n t e r i n g the Nal c r y s t a l * A p o s i t r o n i s d e f i n e d as S7«S 11» (S 13^S13») . The Nal s i g n a l i t s e l f i s not used i n e i t h e r the p o s i t r o n or the p i l e - u p c i r c u i t r y , s i n c e any d i s c r i m i n a t o r t h r e s h o l d s e t high enough to give u s e f u l t i m i n g i n f o r m a t i o n would have the e f f e c t of l o s i n g v a l u a b l e i n f o r m a t i o n contained i n the very low energy p a r t of the s p e c t r a . A l s o , s i n c e the Nal s i g n a l a r r i v e s much l a t e r than t i e s i g n a l from the s c i n t i l l a t o r s , the e l e c t r o n i c s would have been n e e d l e s s l y ccimplicated. I n s t e a d , counter S11, which d e f i n e d the s o l i d angle f o r p o s i t r o n acceptance, was used to determine the t i m i n g of 47 the p o s i t r o n s i g n a l . I t was s h i e l d e d and thus had a low count r a t e . Counters S13 and S13' (also r e f e r r e d t o as TV c o u n t e r s , f o r TINR Veto) together covered the e n t i r e f r c n t face of the Nal, e n a b l i n g i n s p e c t i o n f o r charged p a r t i c l e p i l e - u p to be performed., The p o s i t i v e tO delay box, which se t s the time i n t e r v a l between the p i - s t o p and the beginning of the f i r s t time b i n , a f f e c t s a l l four bins i n the same manner. In c r d e r to measure the r a t e of random decays not a s s o c i a t e d with the c u r r e n t p i - s t o p , the p o s i t r o n s i g n a l i s delayed and regenerated f o r the b i n f o u r c o i n c i d e n c e . F i g u r e II-7 shows the e l e c t r o n i c s f o r p i l e - u p r e j e c t i o n . Two very d i f f e r e n t types of p i l e - u p r e j e c t i o n s are performed. . The p i l e - u p of charged p a r t i c l e s i n t o the Nal c r y s t a l w i l l cause any a s s o c i a t e d s i g n a l to be r e j e c t e d , s i n c e a c c e p t i n g these s i g n a l s would e f f e c t i v e l y worsen the energy r e s o l u t i o n and i n c r e a s e the high energy t a i l i n the l i n e s h a p e s . Since the TV p i l e - u p has the t i m i n g of the p o s i t r o n , the r e j e c t i o n can be performed i n the f a s t l o g i c , immediately downstream o f the Bin OB. The i n s p e c t i o n time i s 430 ns, which i s much longer than the 300 ns gate a p p l i e d to the Nal l i n e a r s i g n a l . , The p i l e - u p of s t o p p i n g p a r t i c l e s i s quite a s e r i o u s problem f o r d i f f e r e n t reasons. T h i s method f o r measuring the Tr-e branching r a t i o i s c r i t i c a l l y dependent upon the time s t r u c t u r e of the s y s t e m — p a r t i c u l a r l y the t i m i n g 48 between the p i - s t o p and the time bins during which data can be accepted. The number of- TT - p -e events i n bins one and two can be used as a n o r m a l i z a t i o n cnly because they have, t o high accuracy, the same width, and the time d i f f e r e n c e between them i s known. T h i s presupposes that the b i n s are a c c e p t i n g decay p o s i t r o n s from the p i - s t o p t h a t generated the b i n s , except f o r the a c c i d e n t a l s ' c o n t r i b u t i o n , which i s monitored. However, a s p u r i o u s c o n t r i b u t i o n t h a t i s not random i n time can a r i s e from p i - s t o p s t h a t p i l e - u p over the p e r i o d of i n s p e c t i o n times f o r a p o s i t r o n coincidence;. That i s , a second pion coming t o r e s t i n t h e s t o p p i n g counter a f t e r the f i r s t time b i n i s c l o s e d and b e f o r e the second time b i n opened can decay and c o n s i s t e n t l y be d e t e c t e d i n b i n two but not b i n one. T h i s i s an i n s i d i o u s e f f e c t t h a t can t h r e a t e n the i n t e g r i t y o f the measurement. To guard a g a i n s t t h i s , s i g n a l s i n any of the four time bins f o r which a second pion stopped ever a period c f 430 ns were r e j e c t e d . . I f a stop p i l e - u p occurs, then any c o i n c i d e n c e of the f o u r time b i n s with a p o s i t r o n was r e j e c t e d . Because t h i s p i l e - u p has the t i m i n g of the p i - s t o p r a t h e r than the p o s i t r o n , t h i s r e j e c t i o n i s performed i n the s l o w - l o g i c — a s an a n t i - c o i n c i d e n c e with the ND2400 analyzer gate. • Timing c o n s i d e r a t i o n s and runtime checks of the p i l e - u p c i r c u i t r y are c o n s i d e r e d ' i n s e c t i o n s II-D and I I - E . 49 II-C-3. Slow l o g i c F i g u r e II-8 i s a schematic of the e l e c t r o n i c s f o r p r o c e s s i n g the anode s i g n a l of the photo-tubes viewing the Mai c r y s t a l , which c o n t a i n s a l l the a v a i l a b l e i n f o r m a t i o n concerning the energy of the p o s i t r o n s . , S i g n a l s from each c f the seven photo-tubes were brought independently from the experimental area v i a low l o s s c a b l e s . The s i g n a l s were combined using an a c t i v e f a n - i n , and the summed s i g n a l was then fanned-out to allow checks t o be made without i n t e r r u p t i n g data a c q u i s i t i o n . , The s i g n a l was then d ^ l i v n r e d to a l i n e a r gate which was used to prevent r i l e - u p problems i n the a m p l i f i e r s . A TINA-gate was generated f o r a l l events which generated a Bin OB (that i s , a p o s i t r o n c o i n c i d e n c e with any of the f o u r time bins) and!^ which did net correspond to a TV p i l e - u p . A redundant c o i n c i d e n c e with counter S11 was performed (S11 was a l s o r e q u i r e d f o r the p o s i t r o n i d e n t i f i c a t i o n ) . Although the s i g n a l s frcm the p o s i t r o n t e l e s c o p e c o u n t e r s c o n t a i n e d a c c u r a t e enough t i m i n g i n f o r m a t i o n f o r g a t i n g the Nal s i g n a l , t h i s timing i n f o r m a t i o n was degraded somewhat when the s i g n a l s were routed through d i f f e r e n t modules, depending on which time bin c o i n c i d e d with vhe p o s i t r o n s i g n a l s . The redundant S11 c o i n c i d e n c e i n s u r e d t h a t the g a i n would be the same i n a l l the f o u r t:\me bins. , 50 The gated Nal s i g n a l was then a m p l i f i e d and d e l i v e r e d to an ND2400 analyzer.. B i n gates, i d e n t i f y i n g which time bin c o i n c i d e d with the p o s i t r o n s i g n a l , a r r i v e d at the BOOTEE before the Nal s i g n a l reached the a n a l y z e r . I f nc stop p i l e - u p o c c u r r e d d u r i n g the p r o c e s s i n g o f t h i s event, the energy i n f o r m a t i o n was then histogrammed i n one of the fou r s e c t i o n s of the an a l y z e r . At the end of each run, data was t r a n s f e r r e d from the a n a l y z e r to magnetic tape f c r l a t e r a n a l y s i s . II-D. Timing C o n s i d e r a t i o n s 1. Bin Widths And P o s i t i o n s In order to a c c u r a t e l y determine the number of TT -e decays which r e s u l t from our stopped pion p o p u l a t i o n , we i n s p e c t f o r charged p a r t i c l e decays f o r a time i n t e r v a l spanning a s i n g l e pion l i f e t i m e . The i n t e r v a l commences soon a f t e r the pion a r r i v e s * ; T h i s method all o w s us to accept the g r e a t e s t number of TT -e decays that are simultaneously "contaminated" with the fewest number of TT — p —e 's, which c o n t r i b u t e a background under the TT -e 's. With our b i n width of 25 ns, the r a t i o of 51 T T-e / T r - y - e events i n bin one was approximately 1:100. As d e s c r i b e d i n the appendix, the i n f o r m a t i o n needed to determine the branching r a t i o i s c o n t a i n e d i n the energy s p e c t r a f o r two time b i n s : the f i r s t c o n t a i n i n g Tf-e and TT - y -e decays, the second c o n t a i n i n g only TT - y -e uecays. These two time b i n s must have, the same du r a t i o n to a very high degree of a c c u r a c y — a discrepancy of 0.05% i n the b i n widths r e s u l t s i n a systematic e r r o r of 0.1% i n the branching r a t i o . , A t h i r d b i n of l o n g e r d u r a t i o n commences a f t e r b in two, and y i e l d s a good s t a t i s t i c s measurement of the TT - y -e p o s i t r o n spectrunu A f o u r t h b i n of the same d u r a t i o n as b i n t h r e e but beginning before the c u r r e n t p i - s t o p , measures the a c c i d e n t a l s background r e s u l t i n g from the decay of " o l d " muons i n the t a r g e t area. The method f o r generating two time b i n s of the same d u r a t i o n was d e s c r i b e d i n s e c t i o n II-C-2. . To determine the degree of accuracy t o which the d u r a t i o n s were the same, we generate f a l s e time b i n s and p o s i t r o n s s i g n a l s . We count the number c f c o i n c i d e n c e s f o r each b i n — t h e s e s c a l e r s should be i d e n t i c a l , to w i t h i n s t a t i s t i c s , f o r i d e n t i c a l b i n s . . O b t a i n i n g adequate s t a t i s t i c a l p r e c i s i o n , however, i s a problem with t h i s methods . We need to acquire 10 8 counts per bin to reduce the s t a t i s t i c a l e r r o r to about 0.014%. And yet, we must avoid pushing the count r a t e s too high which might cause p i l e - u p problems or to d e t e r i o r a t e the 52 response of the e l e c t r o n i c s modules* Our g e n e r a l procedure was to pulse the " c l e a n stop" c o i n c i d e n c e box a t a constant r a t e * The " p o s i t r o n s " were generated by p l a c i n g a strong r a d i o a c t i v e cesium source adjacent to a p l a s t i c s c i n t i l l a t i o n counter. T h i s random s i g n a l i s passed through a d i s c r i m i n a t o r to allow f o r s e t t i n g both the t r i g g e r i n g l e v e l and the output s i g n a l width. In a d d i t i o n , the proximity of the r a d i o a c t i v e source to the com t e r and the high v o l t a g e a p p l i e d t o the counter's photo-ttibe could a l s o be adjusted t c optimize the "random" rate while a v o i d i n g p i l e - u p and m u l t i p l e f i r i n g s . C o n d i t i o n s f o r a good b i n width t e s t were determined e m p i r i c a l l y . T y p i c a l l y , the c l e a n stops were pulsed at a rate of about 0.3 megahertz*. The p o s i t r o n box f i r e d at an average "random" r a t e of about 0*03 megahertz, with a d i s c r i m i n a t o r width deadtime of about 400 ns. T h i s r e s u l t e d i n a c o i n c i d e n c e r a t e f o r b i n cne cf 320 per second. Thus, to a c q u i r e reasonable s t a t i s t i c s , a b i n width t e s t r e q u i r e d a f u l l 24 hour day. , T h i s meant, f o r the most p a r t , t h a t b i n t e s t s were run bef o r e the beam time f o r t h i s experiment began, a f t e r i t f i n i s h e d , c r during scheduled shutdowns. Before s t a r t i n g a bin width t e s t , we checked t h a t the random " s i n g l e s " and pulsed " s i n g l e s " i n d i v i d u a l l y counted i d e n t i c a l l y i n a l l f o u r s c a l e r s * T h i s r e q u i r e d , f o r example, t h a t the output pulses of the c o i n c i d e n c e boxes 53 f o r b i n s one to f o u r have the same width. T h i s i s t y p i c a l of the kinds of c o m p l i c a t i o n s i n t r o d u c e d by the need to complete a b i n width t e s t i n a reasonable l e n g t h o f time--these output widths have nothing t o do with the i d e n t i t y of the two b i n s , nor probably with the o v e r a l l v a l i d i t y of the TT- e experiment. Quite p r c i a b l y the b i n widths were the same to a degree of accuracy even h i g h e r than we s h a l l quote, hut e s t a b l i s h i n g t h i s i s d i f f i c u l t . Table I I - 2 g i v e s the r e s u l t s of a number of b i n width t e s t s f o r the f i n a l e l e c t r o n i c s c o n f i g u r a t i o n s of July-August of 1977.. The t e s t of J u l y 30 shows a bin width d i f f e r e n c e of about 0.09%, but t h i s very l i k e l y r e s u l t s from some marginal e f f e c t . o f the very high random rate* The other r e s u l t s e s t a b l i s h t h a t the discrepancy i n the widths of b i n s one and two was l e s s than 0.05%. The discrepancy c o n s i s t e n t l y corresponds to the width of b i n one g r e a t e r than the width of b i n two> which would r e s u l t i n an experimental value f o r the branching r a t i o s m a l l e r than i t should be by 0.1%. The r e l a t i v e p o s i t ions o f : t h e b i n s i n time were measured f i r s t on a cathode ray o s c i l l o s c o p e (CEO) and then, more a c c u r a t e l y , on an Ortec 437A time to amplitude c o n v e r t e r (TAC), again using the ND2400 a n a l y z e r . The TAC was c a l i b r a t e d using an Ortec 462 p r e c i s i o n time c a l i b r a t o r , nominally accurate t o 15 pico-seconds. The s e p a r a t i o n time between bins one and two (which i s the q u a n t i t y t s t h a t e n t e r s the branching r a t i o TABLE 11-2 B I N W I D T H T E S T S " P O S I T R O N " R A N D O M R A T E (MHz) " C L E A N S T O P " P U L S E D R A T E (MHz) Bl C O I N . R A T E (Hz) W I D T H ( B I ) N - W T D T H (B2) W I D T H ( B I ) (%) W I D T H (B4) W I D T H ^ B l ^ D A T E 0.036 0.388 320. 0.014+0.02 4.72 6/18/77 0.034 0.307 290. 0.049+0.013 4.84 6/20/77 0.031 0.309 250. 0.044+0.013 4.85 6/29/77 0.123 0.309 1000. 0.088+0.006 4.84 6/30/77 0.040 0.320 340. 0.030+0.050 4.93 7/12/77 0.041 0.300 340. 0.033+0.024 4.82 7/13/77 55 formula) as measured at the output of the B1 and B2 co i n c i d e n c e boxes, was ts=170.2±0.1 ns T h i s time s e p a r a t i o n was determined by the l e n g t h of the open ended d e l a y cable attached t o the input of the f i r s t 'double box' d i s c r i m i n a t o r * T h i s was set to be ne a r l y an i n t e g r a l number of EF p u l s e s t o i n s u r e t h a t anj primary-beam r e l a t e d background would be the same i n both b i n one and b i n two- The same delay c a b l e was used i n a l l the ir-e p r o d u c t i o n runs. The time s e p a r a t i o n between b i n one and three as measured at the output of " t i m i n g " and the B3 c o i n c i d e n c e , i s t(s1-3) = 301.2±0.2 ns This time d i f f e r e n c e was a l s o s e t, using a number of delay cables and boxes, t o be.approximately an i n t e g r a l number of EF cycles;. , The time s e p a r a t i o n between bins one and f o u r as measured at the output of " t i m i n g " and the E4 c o i n c i d e n c e , i s t(s1-4) = 300.2±0.2 ns 56 T h i s time d i f f e r e n c e was s e t i n a f a s h i o n s i m i l a r t c t ( s 1 - 3 ) , except t h a t f o r B4, the delay was cn the p o s i t r o n s i g n a l i n s t e a d of the p i - s t o p t a g . The above time q u a n t i t i e s were a l s o checked at the £in-OE (BOB) and found t o he i n agreement* Tc i n s u r e t h a t the above measurements a c t u a l l y mean what they imply, we a l s o v e r i f i e d t h a t the p o s i t r o n s i g n a l a r r i v e s at the " t i m i n g " and B3 c o i n c i d e n c e at the same time, and th a t the p i - s t o p s i g n a l a r r i v e s at the " t i m i n g " and B4 c o i n c i d e n c e s i m u l t a n e o u s l y . II-C-2. Blanking And Background S u b t r a c t i o n A complete s e t o f fou r time bins was generated by a p i - s t o p i n the t a r g e t counter S4* . However, to reduce the a c c i d e n t a l r a t e of the y + -> e+ + v + v decays i n t o the Nal, we "blanked" or vetoed the c l e a n stops ( i . e . those generating a set of time bins) f o r a time i n t e r v a l a f t e r each raw stop. The t r a d e - o f f here i s between the a c c i d e n t a l / r e a l r a t i o (as measured by B i n 4 / B i n 3) and the c l e a n stop/raw stop r a t i o . . These q u a n t i t i e s , versus the d u r a t i o n of b l a n k i n g , were measured during the 1976 run, and are p l o t t e d i n f i g u r e XI-9. , Our u n c e r t a i n t i e s could be minimized by a c q u i r i n g good s t a t i s t i c s on the TT -e decays, at the co s t of i n c r e a s i n g our a c c i d e n t a l s 57 FIGURE I I-9 BLANKING DURATION VS. CLEAN STOPS AND ACCIDENTALS to o < cu a. o < U J o 1 . 0 i .8 + ^ .6 A .2 (CLEAN ST0PS)/(RAW STOPS) 0 0 . 4 0 3 0 . 3 0 t 2 0 . 2 0 t * o.io 4 H h H h -I 1 1 1 1 1 H ACCIDENTALS (BlNVBlN3) -H 1 H 8 LENGTH OF BLANKING IN MICRO-SECONDS 58 rate somewhat.. A c c o r d i n g l y , we s e t the d u r a t i o n of H a n k i n g to be 2.1 micro-seconds. During normal running c o n d i t i o n s , the raw stop r a t e was about 130 K/s, and the clean stop r a t e about 84 K/s. Our s o l i d angle was approximately 3 SS. At the time of the experiment, u n f o r t u n a t e l y , we d i d not f u l l y a p p r e c i a t e the e f f e c t of t h i s b l a c k i n g upon our " a c c i d e n t a l s " s u b t r a c t i o n . Language, of course, i s used not only t o communicate but t o dece i v e — w e c o n s i s t e n t l y r e f e r r e d to a c c i d e n t a l s as "randoms", a c c e p t i n g the i m p l i c a t i o n t h at t h i s background was randcm i n time. Thus, i n order to s u b t r a c t t h i s background, we need only s c a l e down the number of counts i n B4 by a f a c t o r of the r a t i o (width of B1)/(width of B4) . Even then, s i n c e the branching r a t i o depends upon A t A t e y S B2' - B1' E e ( B 2 ( r a w ) - a c c ) - ( B 1 ( r a w ) - a c c ) (where a c c = a c c i d e n t a l s — s e e the appendix A f o r other d e f i n i t i o n s ) the "randcm s u b t r a c t i o n e n t e r s i n c n l y at the 10% l e v e l . A t ( 1 _ e V s ) a c c = 0 . 0 8 ( a c c ) U n f o r t u n a t e l y , t h i s f a c i l e reasoning i s not a c c u r a t e . The b l a n k i n g s i g n a l was generated by S1» S2« (S3 A S4^S5) »S6. Thus, a pion s t o p p i n g i n any of the th r e e t a r g e t counters 5 9 generates a bl a n k i n g s i g n a l . . C o n v e r s e l y , when a set of f o u r b i n s i s generated, then no p r i o r p i - s t o p has occured i n any of the t a r g e t counters f o r at l e a s t 2.1 micro-seconds. I f we assume, momentarily, t h a t a l l the a c c i d e n t a l s r e s u l t from o l d muons coming from p i - s t o p s i n the t a r g e t counters, then t h i s background i s not random i n time. Indeed, t h i s f i x e d p o p u l a t i o n of o l d muons i s not being r e p l e n i s h e d f o r the d u r a t i o n of the f o u r time b i n s . Thus, t h i s a c c i d e n t a l s r a t e i s f a l l i n g o f f acc o r d i n g to the muon l i f e t i m e of 2*2 micro-seconds. (This p o r t i o n of the a c c i d e n t a l background r a t e can then be r e f e r r e d to as the unreplenished background.) T h i s i n i t s e l f i s not a major p r o b l e m — t h e i n f o r m a t i o n i n B4 would s t i l l be s u f f i c i e n t — w e need simply e x t r a p o l a t e the a c c i d e n t a l s r a t e using the muon l i f e t i m e * .. However our i n i t i a l a s s u m p t i o n — t h a t a l l a c c i d e n t a l r e s u l t from p i - s t o p i n the t a r g e t c o u n t e r s — i s not v a l i d . A c c i d e n t a l decays from muons i n the s h i e l d i n g , l i g h t p i p e s , e t c . i s then a r e p l e n i s h e d background t h a t has no time dependence. The r a t i o of a c c i d e n t a l s from r e p l e n i s h e d / u n r e p l e n i s h e d sources c o u l d have been determined e x p e r i m e n t a l l y , had we r e a l i z e d the problem while we were running the experiment. Now, we have t o r e l y on c a l c u l a t i o n s to determine the re p l e n i s h e d / u n r e p l e n i s h e d r a t i o f o r a c c i d e n t a l s . A s e r i e s of Monte C a r l o c a l c u l a t i o n s were done to determine the number of unreplenished a c c i d e n t a l s t h a t would occur from the known t a r g e t counter stop r a t e s . 6 0 T h i s was done f o r both the 1976 data, when the d u r a t i o n cf b l a n k i n g was v a r i e d , and f o r the 1977 data, when i t was not- However, i n 1976, b l a n k i n g was only dene on stops i n counter S 4 — o n l y on 63% of the t o t a l stops i n a l l three t a r g e t c ounters.. In 1977, blanking was done on 100% of t a r g e t counter s t o p s . R e s u l t s of the Monte C a r l o c a l c u l a t i o n s and of experiment are given i n f i g u r e 11-10. These r e s u l t s i n d i c a t e and a c c i d e n t a l r a t e of 3.8±1.0 % (of the t o t a l number of T r - y y - e p o s i t r o n s ) from sources other than the t a r g e t counters f o r the 1976 date. (The t o t a l a c c i d e n t a l r a t e f o r t h i s data was approximately 29 i %) For the 1977 data, we f i n d t h i s " r e p l e n i s h e d a c c i d e n t a l s " rate to be 4.6±0.4% (out of a t o t a l r a t e of 19,.2 % ) . The e r r o r r e s u l t s from our u n c e r t a i n t y i n the e f f e c t i v e pion stop rate--we f e e l t h a t i s l i e s between 175 and 225 thousand p i - s t o p s / s e c o n d . . The r a t i o s f o r a c c i d e n t a l s s u b t r a c t i o n and the corresponding change i n the value of the branching r a t i o i s given i n t a b l e I I - 3 . The u n c e r t a i n t y i n r e p l e n i s h e d / u n r e p l e h i s h e d a c c i d e n t a l s adds a s y s t e m a t i c u n c e r t a i n t y of 0.04 % t c our branching r a t i o c a l c u l a t i o n . X 1976 DATA (BLANKING ON 63 % OF THE STOPS) MONTE CARLO SIMULATION (ON 63 % OF THE STOPS) LENGTH OF BLANKING IN MICRO-SECONDS FIGURE 11-10 RESULTS OF MONTE CARLO SIMULATION OF ACCIDENTAL COINCIDENCES (FROM UNREPLENISHED SOURCES) 62 COMPLETELY R E P L E N I S H E D —RANDOM A C C I D E N T A L S R A T I O FOR B l ° - 2 0 5 0.1834 0.1886 +0.0007 A C C I D E N T A L S R A T I O FOR B2 ° ' 2 0 5 0.1698 0.1782 ± 0 . 0 0 1 0 CHANGE I N BRANCHING R A T I O + 2 ' 3 % "0.7% 0+0.04% TABLE I I - 3 Rat i o s For A c c i d e n t a l s S u b t r a c t i o n . COMPLETELY U N R E P L E N I S H E D M I X E D — E X P O N E N T I A L I I - D - 3 . Pile-Up R e j e c t i o n fts d e s c r i b e d i n s e c t i o n I I - C , two types of p i l e - u p r e j e c t i o n were performed: raw stop p i l e - u p and TINA-Vetc (TV) p i l e - u p . The raw stop p i l e - u p i n p u t had the signature S1 »S2« (S3 AS4 AS5) »S6. The TV p i l e - u p i n p u t was the fanned i n s i g n a l of the two TV counters which covered the e n t i r e f r o n t face c f the Nal c r y s t a l (with a s m a l l overlap) . Stop p i l e - u p was necessary t o prevent a second stopped pion from decaying i n t o b i n two. These counts are are not " a c c i d e n t a l s " i n the way t h a t b i n fou r measures a c c i d e n t a l s . N e i t h e r are they v a l i d decays, because they cannot be c o i n c i d e n t with b i n one, on l y with b i n two. 63 There i s a 3.3 % p r o b a b i l i t y f o r a second p i - s t o p (at an average incoming rate of 200 K/s) which could decay i n t c E2, r e s u l t i n g i n a change i n the branching r a t i o of 6.0%. T h i s o r i g i n a l l y motivated our implementation of the p i l e - u p c i r c u i t r y f o r the 1976 run. For the 1977 runs, we c l i p p e d the p h o t o - m u l t i p l i e r s i g n a l s f o r counters S1, S2, S3, and S5 to improve d e t e c t i o n of p i - s t o p s a r r i v i n g i n the f i r s t subseuent EF burst* The s i g n a l s were c l i p p e d t o a width of about 20 ns at the base, with seme p o s i t i v e overshoot. The second EF e f f i c i e n c y was measured to be about 90 % (see t a b l e I I - 1 ) ; E i t h e r type of p i l e - u p caused a r e j e c t i o n of the event i n t o any of the f o u r time b i n s . ., The number of r e j e c t i o n s were counted f o r each run, and t h i s i n f o r m a t i o n i s summarized i n t a b l e I I - 4 . , Thus, i n 1977, 8.0 % of the decays i n t o b i n one were r e j e c t e d due to stop p i l e - u p . The excess of E2 stop p i l e - u p r e j e c t i o n s over B1 r e j e c t i o n s i s 3,. 1 %. T h i s i s the number of second pions which would have decayed i n t o B2 had they not been r e j e c t e d . . T h i s number depends upon the time s t r u c t u r e of the time b i n s and upen the pion stop r a t e . These excesses f o r both B2 and B3 over B1 as a f u n c t i o n of stop r a t e were c a l c u l a t e d with a Monte-Carlo program. We d i d not have a good o n - l i n e record of the average stop r a t e , but the absolute value of B1 stop p i l e - u p events was used as a measure of the stop r a t e , and the s c a l e was determined by known high- and low-stop r a t e 64 TABLE I M P I L E - U P R E J E C T I O N N O R M A L I Z E D T O T H E N U M B E R O F C O U N T S I N E A C H B I N D A T A T . V . P I L E - U P ( % ) B l B2 B3 B4 S T O P P I L E - U P (%) B l B2 B3 B4 1976 6 . 0 5 . 9 6 . 1 6 . 6 6 . 5 7 .1 9 . 3 6 . 6 1977 2 . 3 2A 2 . 6 2 . 8 8 . 0 11.1 13.9 8 . 3 TABLE II-4 Values For P i l e - u p R e j e c t i o n From Each B i n . runs. T h i s c r i t e r i o n agreed f a i r l y w e ll with t a k i n g the number of p i - s t o p s and d i v i d i n g by the d u r a t i o n of the run as given by channel zero of the a n a l y z e r . Comparison of the c a l c u l a t e d and experimental r e s u l t s are given i n f i g u r e 11-11. The d i f f e r e n c e i n r a t e s f o r the 1976 and 1977 data r e f l e c t s only the change of the input t o the stop p i l e - u p b o x — f r o m S1•S2»S3«S4«S5 i n 1976 to S1«S2« (S3 AS4 AS5) •"S6~ i n 1977. The time i s the le n g t h of time a f t e r which the p i l e - u p c i r c u i t r y can d e t e c t a second p i - s t o p . ( i . e . , =70 means th a t p i - s t o p s i n the f i r s t subsequent EF burs t w i l l not be r e j e c t e d . ) He see tha t f o r the 1976 data, the stop p i l e - u p c i r c u i t r y was not s e n s i t i v e to p i - s t o p s i n the f i r s t or second subsequent IF b u r s t s . Thus, the 1976 data must be c o r r e c t e d f o r these spurious counts i n b i n two. In 1977, we were much more s u c c e s s f u l i n r e j e c t i n g a l l stop p i l e - u p events. (Two FIGURE 11-11 EFFECT OF STOP PILE-UP REJECTION VS. STOP RATE. NUMBER OF SPURIOUS EVENTS IN BIN TWO RELATED TO A SECOND PI-STOP. THE TIME T, IS THE LENGTH OF TIME AFTER WHICH THE PILE-UP CIRCUITRY CAN DETECT A SECOND PI-STOP. ALSO GIVEN IS THE CORRESPONDING CORRECTION THAT MUST BE APPLIED TO BIN TWO. 1 1 SPURIOUS COUNTS IN BIN TWO (B2-BD/62 MONTE CARLO CALCULATIONS ' T„> 0 NS 100 150 200 250 PION STOP RATE ( K / s ) 66 p i ' s a r r i v i n g i n the same EE b u r s t and generating the time .bins do not give s p u r i o u s decays s i n c e they can decay i n t o e i t h e r B1 or B2 with the expected p r o b a b i l i t y . ) No c o r r e c t i o n need be a p p l i e d to the 1977 data f o r stop p i l e - u p . Values f o r second EF e f f i c i e n c y (see Table II-1) i n d i c a t e t h a t the system could d e t e c t 90 % cf the stops from the second BF bu r s t . The stop p i l e - u p y i e l d s an assigned s y s t e m a t i c u n c e r t a i n t y c f ±0.1% to the branching r a t i o . The purpose of TV p i l e - u p was to improve the Nal r e s o l u t i o n somewhat--the TV counters could not r e t r i g g e r g u i c k l y enough to r e j e c t prompt s c a t t e r e d events from p i l i n g up with B1 events. From t a b l e I I - 4 , we see t h a t the TV p i l e - u p r e j e c t e d the same percentages from a l l f o u r b i n s ; There i s no good reason why the TV r e j e c t i o n f o r the 1976 data was 2.5 times l a r g e r than f c r the 1977 data. Improved c o l l i m a t i o n of the beam or d i f f e r e n t l e v e l s on the s c i n t i l l a t i o n counter power s u p p l i e s or d i s c r i m i n a t o r s could c o n t r i b u t e to t h i s e f f e c t . 67 I I - E . Data A c q u i s i t i o n And Runtime Checks 1. , General The c u r r e n t ^-e experiment was run e n t i r e l y on the M9 stopped pion/muon channel i n the meson area of the TEIUMF c y c l o t r o n . & summary of the runs i s g i v e n i n Table I I - 5 . The runs i n February, May and August of 1976 were devoted t o red u c i n g the backgrounds under the -> e++ v * v p o s i t r o n energy s p e c t r a , improving the e l e c t r o n i c s l o g i c , o p t i m i z i n g the d e t e c t o r arrangement, e t c . In October of 1976, the f i r s t p r o d uction run occurred, accumulating 18,000 TT -e decay events. T h i s run i l l u m i n a t e d a number of p r o b l e m s - — p r i m a r i l y with energy c a l i b r a t i o n , d e t e r m i n a t i o n of background l e v e l s , and p i l e - u p s u b t r a c t i o n s . I t a l s o i n c l u d e d a somewhat mysterious low energy peak superimposed cn the y + - > e + + v + v shape* The f i n a l p r o d u c t i o n runs, i n June, J u l y and August of 1977, were done i n c o n j u n c t i o n with the y e + y experiment. The t a r g e t of 100 thousand TT —e events was achieved. As p r e v i o u s l y mentioned, troublesome g a i n s h i f t s o ccurred throughout the 1977 pr o d u c t i o n run. As shown i n 68 TABLE 11-5 S U M M A R Y O F n - e R U N S D A T E R U N N I N G N U M B E R N U M B E R O F R U N T I M E . O F n - e C L E A N S T O P S N U M B E R S ( S H I F T S ) E V E N T S 2/76 6 0 O . O l x l O 1 0 1-200 5/76 6 TOO 0 . 4 x l 0 1 0 1-100 8/76 10 6000 0 . 6 x l 0 1 0 1-250 10/76 . 1 0 18000 1 . 5 x l 0 1 0 350-444 6/77 28 35000 2 . 1 x l 0 1 0 1000-1219 7-8/77 26 63400 3 . 9 x l 0 1 0 1220-1651 T O T A L 86 126,400 9 . 5 x l 0 1 0 F I G U R E 11-12 NAI G A I N S H I F T S V S . T I M E OF DAY 270 U l m §280 < X 290 0 4 8 12 16 20 T I M E OF DAY ( H O U R S ) 71 f i g u r e 11-12, the g a i n s h i f t s were c o r r e l a t e d with the time of day, presumably r e p r e s e n t i n g a dependence of the p h o t o - m u l t i p l i e r response to temperature. S p e c t r a were w r i t t e n onto magnetic tape at l e a s t once every hour, although the s p e c t r a were erased from the analyzer only every two or t h r e e hours*„ Subsequently, a l a r g e segment of the data was s h i f t e d and summed to produce a good s t a t i s t i c s run (run 760). F i g u r e 11-13 shows b i n one from run 760, i l l u s t r a t i n g the low energy peak c o n t a i n i n g 60,000 counts, the y+ -> e++ v +v shape c o n t a i n i n g 4,700,000 counts, and the TT-€ peak, c o n t a i n i n g 38,000 counts* F i g u r e 11-14 shows b i n s one to f o u r f o r the g a i n s h i f t e d run 760, and f o r a s i n g l e run, 1538. Also given i s the TT -e region f o r b i n s one and two, both the raw data and the r e s u l t s a f t e r a c c i d e n t a l s have been s u b t r a c t e d . A v a r i e t y of runtime t e s t s were performed before, during and a f t e r the production runs. These i n c l u d e d both the usual t e s t s on performance of the e l e c t r o n i c s modules and on s i g n a l t i m i n g , plus more s p e c i f i c t e s t s r e l a t i n g t o the low energy peak, the presence of backgrounds, and the e f f i c i e n c y of p i l e - u p r e j e c t i o n . Some of the l e s s mundane runtime t e s t s w i l l be d e s c r i b e d i n the f o l l o w i n g s e c t i o n s . 72 FIGURE 11-14 ( A ) E N E R G Y S P E C T R A FOR RUN 760 T H E G A I N S H I F T E D RUN BIN ONE BIN TWO BIN THREE BIN F O U R REGION «-e REGION A F T E R A C C I D E N T A L S S U B T R A C T I O N FIGURE 11-14 (B) ENERGY S P E C T R A FOR RUN 1538 C O U N T S I K 1 C * ) O . O -I ^ i i . . . O.O BO.O 120.0 ICO .0 Z40.0 »O0 .0 3 6 J 0 420 .0 B I N O N E B I N TWO -T-B I N T H R E E C O U N T S (X101 ) O.O ftu.O 120.0 100.D 2*u.O ?O0 .0 3G0.0 420.0 H^ON N t U B I N FOUR C O U N T " O B I N 1 * + B I N 2 .0 n i .o 110.0 K S . D J«.O "g^jjg/ 0 S B S 0 O B I N 1 + B I N 2 • • • cao.o zas.o »io.o i».i 3«o.o "g^^Jg-0 "s'° n-e R E G I O N w-e R E G I O N A F T E R A C C I D E N T A L S S U B T R A C T I O N 74 II-E-2. Runtime Te s t s Of The NaT And  S c i n t i l l a t i o n Counters Since t i m i n g c o n s i d e r a t i o n s were of the g r e a t e s t importance i n t h i s experiment, proper f u n c t i o n i n g of the counters was c o n s i s t e n t l y necessary* The p l a s t i c s c i n t i l l a t i o n c o u n t e r s were c a r e f u l l y t e s t e d . . A problem t h a t s u r f a c e d d u r i n g the p r e l i m i n a r y runs was undetected f a i l u r e of counter s , or d i s c r i m i n a t o r or c c i n c i d e n c e box f a i l u r e * F u r t h e r problems occ u r r e d when the e l e c t r o n i c s l o g i c was changed f o r some t e s t and not r e s e t p r o p e r l y to the i n i t i a l s t a t e . A good safeguard against these problems (which d i d not i n t e r f e r e with data a c q u i s i t i o n ) was a r e g u l a r check of the r a t e s f o r the counter and c o i n c i d e n c e boxes., T y p i c a l r a t e s are given i n t a b l e I I - 6 . The p h o t o - m u l t i p l i e r s viewing the Nal c r y s t a l were balanced during the run by examining the y + e++ v + v shape with a s i n g l e photo-tube and a d j u s t i n g the high v o l t a g e l e v e l . , The r e s o l u t i o n of the Nal d i d not vary d r a m a t i c a l l y over the course of the run* 75 TABLE 11-6 T Y P I C A L R A T E S F O R C O U N T E R S A N D C O I N C I D E N C E B O X E S F O R T H E i r - e E X P E R I M E N T C O U N T E R R A T E / S E C O N D C O I N C I D E N C E / P I L E - U P B O X R A T E / S E C O N D S I 1 . 4 x l 0 6 S1*S2 1 . 2 x l 0 6 S2 1 . 2 x l 0 6 S 1 » S 2 S3 S4 0.19x10 3 S3 0.32x10 3 S 1 » S 2 S3 S4 S5 67.X10 3 S4 0.33x10 3 S>S2 S3 S4 S5 129.X10 3 S5 0.18x10 3 C L E A N S T O P 84x103 S6 0.13x10 3 S7.S11 T V 5.8x10 3 S7 3 0 . x l 0 3 B0R 240. S l l l l . x l O 3 S3 A S4 A S5 4 9 0 - x l O 3 S13 1 2 . x l 0 3 S > S 2 « (S3 A S4 A S5) S6 191.X103 S13' 2 0 . x l 0 3 S T O P P . U . 14.X10 3 C E R E N K O V 75.X10 3 S T O P P . U . 30. X ( B 1 , 2 , 3 , 4 ) T . V . P . U . 700. 76 II-E-3. General E l e c t r o n i c s T e s t s The usual e l e c t r o n i c s t e s t s checking s i g n a l t i m i n g and the f u n c t i o n s of the NIM modules were performed. Seme t e s t s r e l a t i n g to the t i m i n g c o n s i d e r a t i o n s were d e s c r i b e d i n s e c t i o n II-D. L i n e a r i t y of the &DC was checked by a t t a c h i n g an a t t e n u a t o r to the i n p u t s i g n a l before a m p l i f i c a t i o n . No s i g n i f i c a n t departure from l i n e a r i t y was found. Timing the p i l e - u p r e j e c t i o n s i g n a l s was somewhat cf a problem. T h i s timing was t e s t e d by t a k i n g the busy output of the p i l e - u p gates i n place of the p i l e - u p output, thus r e j e c t i n g e s s e n t i a l l y e v e r y t h i n g . For the TV p i l e - u p busy t e s t s , the leakage ( i . e . the r a t i o of counts g e t t i n g i n t o the s p e c t r a to the number of gates) i n s p i t e c f the p i l e - u p busy r e j e c t i o n was c o n s i s t e n t l y about 1 0 - s . The stop p i l e - u p busy t e s t s y i e l d e d approximately 1% leakage. T h i s i s due to an i n e f f i c i e n c y i n the counter S5. ( E e c a l l t h a t a set of time b i n s was generated f o r each S1«S2«S3«S4»S5, whereas the stop p i l e - u p was done on S1«S2»(S3 AS4 S5) •S6.) 77 II-E-4. Determination Of TO The time displacement between the a c t u a l p i - s t o p and the commencement of b i n one i s r e f e r r e d to as tO- tO mcst be l a r g e enough t o i n s u r e t h a t no prompt beam r e l a t e d events get i n t o b i n one. Too l a r g e a value cf tO reduces the number of T r - e • s accepted i n t o b i n one, and a l s o causes a danger of a c c e p t i n g prompts from a second EF bu r s t . The p r e c i s e time of the p i - s t o p , and thus the absolute value of tO, i s d i f f i c u l t t o determine* I n s t e a d , the r e l a t i v e time as s e t on a delay box, tOD, i s used as a measure* to was s e t i n the e a r l i e r experiment cf DiCapua (Di63) by removing the degrader from the incoming beam and simply s c a l i n g t h e number c f counts i n b i n cne while varying tOD. For the present experiment, t h i s procedure y i e l d e d only a b a r e l y s i g n i f i c a n t i n c r e a s e i n b i n cne c o i n c i d e n c e s , due to the low energy o f the incoming pions. Our a l t e r n a t i v e procedure wasd t o move counter S11 frcm i t s p o s i t i o n i n f r o n t of the Nal c r y s t a l (see f i g u r e II-1) t o a p o s i t i o n i n the incoming beam downstream o f the t a r g e t counter. We co u l d then measure tOD e f f e c t i v e l y , although the mix of p a r t i c l e s i n the beam proved another problem. Our p r a c t i c e was to measure the in-beam tOD delay curve with no absorber between S1 and S2 (thus seei n g e v e r y t h i n g i n the beam); with the usual amount o f 78 CH a absorber, thus viewing the e n e r g e t i c p a r t i c l e s i n the beam; and with an a d d i t i o n a l one i n c h of aluminum degrader, thus d e t e c t i n g only the very e n e r g e t i c e l e c t r o n s i n the beam. R e s u l t s are shown i n f i g u r e 11-15. We see t h a t the u n c e r t a i n t y i n the tOD timing i n t r o d u c e d by the above e f f e c t s i s only the order cf one ns» Our p r a c t i c e was to set tOD approximately 6ns away from the prompt edge, at the p o i n t i n d i c a t e d by the arrow cn f i g u r e 11-15. At the same time, we noted t h a t when tOD was v a r i e d i n t o the prompt r e g i o n (tOD>25ns), a r e l a t i v e l y f l a t h i g h energy background appeared i n the 70-90 MeV r e g i o n cf b i n one. The change i n t h i s background as a f u n c t i o n of tOD i s a l s o shown i n f i g u r e 11-15. , T h i s high energy (HE) background curve approximates the other curves used f o r tO deter m i n a t i o n . I f any t h i n g , the prompt "edge" i s b e t t e r d e f i n e d by the HE background curve.. Our procedure during the 1977 p r o d u c t i o n runs was t o determine tOD u t i l i z i n g only the HE background variation,. (For f u r t h e r d i s c u s s i o n of the HE background, see s e c t i o n III-B-3-b.) FIGURE 11-15 D E T E R M I N A T I O N OF T O ~ T H E T I M E I N T E R V A L B E T W E E N T H E P I - S T O P A N D B I N O N E X N 0 A B S O R B E R O 5/8 I N C H CH2 D E 6 R A D E R • 1 . 0 I N C H A L . D E G R A D E R ijt H I G H E N E R G Y B A C K G R O U N D X o ° o o o o o - | o NORMAL S E T T I N G O F T Q D •X o *  x x * J x ° X j O - L — j , j 1 1 £ \ 1 1 1 1--10 0 10 20 30 40 T 0 D ( N S ( 80 II-E-5. T e s t s R e l a t i n g To The Low Energy Peak And To Backgrounds A number of t e s t s were done to determine the source of the low enegy peak and to a s c e r t a i n the p o s s i b i l i t y of a low energy background* Many were i n c o n c l u s i v e or not well thought out. The t e s t s which proved r e l e v a n t w i l l now be d e s c r i b e d . . A v a r i e t y of t e s t s were done to b e t t e r d e f i n e the energy c a l i b r a t i o n . Our thoughts on the source of the low energy peak centered on two p o s s i b i l i t i e s : 1) low energy p o s i t r o n s producing a v a l i d s i g n a t u r e , but having i n s u f f i c i e n t energy t o reach the N a l , thus f i x i n g the low energy peak at zero MeV (these can be c a l l e d f a l s e g a t e s ) ; and 2) s i m i l a r l y low energy p o s i t r o n s which a n n i h i l a t e i n the dead l a y e r between the TV counters and the a c t u a l Nal c r y s t a l , producing two 0.511 MeV gamma ra y s , one of which gets detected by the Nal. To a c c u r a t e l y f i x the energy of the lew energy peak, we r e q u i r e d s e v e r a l good c a l i b r a t i o n p o i n t s below 5 MeV that could be seen s i m u l t a n e o u s l y with the low energy peak, and t h a t entered the s p e c t r a with the same g a t i n g on the Nal l i n e a r s i g n a l . T h i s was achieved using a 2*Na source, which decays with a p o s i t r o n i n prompt c o i n c i d e n c e 8(1 with s e v e r a l cascade gamma r a y s . A. s u f f i c i e n t l y a c t i v e 2*Na source was produced by i n s e r t i n g an A l s t r i p i n t o the proton beam* As seen i n f i g u r e 11-16, s e v e r a l low energy l i n e s c o u l d e a s i l y be i d e n t i f i e d i n the Nal spectrum when no pion beam was present. When p i ' s were s t o p p i n g i n the t a r g e t , the 2*Na l i n e s , while being routed i n t o a s i n g l e s u b-section of the a n a l y z e r , c o u l d s t i l l be i d e n t i f i e d and used t o determine the energy o f the low energy peak. The spectra i n f i g u r e 11-16 were a c q u i r e d with a much higher gain than normal.. The low energy peak then appears t o be the sum of two peaks, with energies of 60±100 keV and 480±100keV. Thus both p o s s i b l e sources appear t o c o n t r i b u t e t o the low energy peak i n about equal amounts. The weighted energy of the low energy peak i s then E (low energy peak) = 0.265±0.100 MeV The low energy peak, then, y i e l d s very v a l u a b l e i n f o r m a t i o n . . Under normal gain, our zero channel was determined f o r each run to b e t t e r than 1/5 of a channel. However, the problem has not been e n t i r e l y s o l v e d . E s t i m a t i n g the number of f a l s e gates and 0.511 MeV gammas we expect from the u * -> e + + v + v s p e c t r a , i n c l u d i n g the e f f e c t s of Landau s t r a g g l i n g and of the Nal l i n e s h a p e , y i e l d s a value f o r (# of counts i n the low energy peak)/(# of y + ->• e++ v + v ) of about 0.6 ± 0.2 %. T h i s does not in c l u d e the e f f e c t s of b a c k s c a t t e r i n g p a r t i c l e s i n the 8 2 FIGURE M-16 LOW E N E R G Y C A L I B R A T I O N S P E C T R A COUNTS -1 16.0 12.0 —i 8.0 —\ O .0 2.72±.05 MEV 3.W*.1 MEV + D A T A F I T O B A C K G R O U N D 4 , 1 6 ± . l MEV CHRNNEL ( A ) ^ 'N A S P E C T R U M AT H I G H G A I N C O U N T S CX10 1 ) 40 .0 30.0 -H 20 .0 H IO.O H o.o + D A T A — F I T O B A C K G R O U N D 88 ;„ «:o 9 S . o »4.o l - i - " 1 0 ! H 0 R N ^ 2 L - ° , , 0 - ° ( B ) LOW E N E R G Y P E A K I N T H E M I C H E L S P E C T R U M * A T H I G H G A I N 83 Nal, which w i l l i n c r e a s e the c a l c u l a t e d value somewhat. The measured numbers, when a f l a t background i s e x t r a p o l a t e d beneath the low energy peak are given i n t a b l e I I - 7 . Thus, the number of counts i n the low energy peak can probably be e x p l a i n e d . However the s m a l l but' d i s t i n c t time dependence remains a mystery. S e v e r a l t e s t s were done t o examine the p o s s i b i l i t y cf a low energy background. The very f i r s t of the p r e l i m i n a r y 1 1 -e runs c o n s i s t e d of y + e++ v + v" p o s i t r o n spectra with an o b v i o u s l y erroneous upturn at lew energies* By decreasing the t h i c k n e s s of the s h i e l d i n g at the f r o n t f a c e of the Nal and moving the c r y s t a l c l o s e r to the t a r g e t counter, by r e d u c i n g the p o s i t r c n acceptance, and by d e f i n i n g the p o s i t r o n t r a j e c t o r i e s more s t r i n g e n t l y , the low energy r e g i o n of the " M i c h e l " s p e c t r a was reduced s u b s t a n t i a l l y * The q u e s t i o n then arose as to whether the background had been e l i m i n a t e d e n t i r e l y or p a r t i a l l y . We f e l t the most l i k e l y source of t h i s low energy background was those p o s i t r o n s which t r i g g e r e d counter S7 (see f i g u r e II-1) but h i t the remaining 12.5 cm, of s t e e l s h i e l d i n g the Nal. These c o l l i s i o n s c o u l d produce a "shower" of charged p a r t i c l e s , t r i g g e r i n g counters S11 and S13-S13' while d e p o s i t i n g l i t t l e energy i n the Nal. These are c l e a r l y s p u r i o u s events. During the f i n a l p r o d u c t i o n run, a f t e r the set-up and 84 TABLE II-7 A R E A O F T H E L O W E N E R G Y P E A K I N T H E F O U R B I N S A S A P E R C E N T A G E O F ir-y-e P O S I T R O N S . B l B2 B3 B A C K G R O U N D S U B T R A C T I O N A R E A O F L E P E A K / # O F y-e 1.66% 1.39% 1.32% 2.36% T R I - A N G U L A R B A C K G R O U N D A R E A O F L E P E A K / # O F y-e 1.33% 1.08% 1.01% 1.97% F L A T B A C K G R O U N D W I T H R A N D O M S S U B T R A C T E D 1.13% 0.92% 0.79% 1.97% // w A R E A O F L E P E A K W I T H R A N D O M S S U B T R A C T E D 39.3K 59. IK 220. K 131K F L A T B A C K G R O U N D P E A K C H . 18.5 19.0 19.0 18.5 // // FWHM(CH) 8 . 2 8 . 5 8 . 5 9 . 0 II II e l e c t r o n i c s had been f i n a l i z e d , we r e - i n v e s t i g a t e d t h i s probable source of the low energy background* By a l l o w i n g l e s s w e l l d e f i n e d p o s i t r o n t r a j e c t o r i e s to c o n t r i b u t e t o the s p e c t r a , we f e l t we should be a b l e t o i n c r e a s e the low energy background. T h i s was accomplished by simply not r e g u i r i n g a c o i n c i d e n c e with counter S7 (see f i g u r e II-1) f o r a v a l i d p o s i t r o n s i g n a l . When S7 was ( e l e c t r o n i c a l l y ) removed, the number of counts i n the M i c h e l s p e c t r a i n the energy r e g i o n 3-6 MeV became 1.3% of a l l the y + ^ e + + v + v ' s, vs. only 0.67% when a c o i n c i d e n c e with £7 was r e q u i r e d . Having e s t a b l i s h e d the p o s s i b l e presence of a low energy background, we then wished to a s c e r t a i n whether, under our normal running c o n d i t i o n s , we had completely e l i m i n a t e d t h i s background from our s p e c t r a . Since the background c o u l d be induced by d e f i n i n g the p o s i t r o n t r a j e c t o r i e s l e s s s t r i n g e n t l y , we f e l t a l i k e l y t e s t was to d e f i n e the t r a j e c t o r i e s more s t r i n g e n t l y and examine any changes i n the " M i c h e l " spectrum. Under normal running c o n d i t i o n s , counter S11-—a c i r c u l a r counter 17.5 cm. i n diameter by 0.16 cm. t h i c k — w a s the d e f i n i n g counter f o r p o s i t r o n s * We i n s e r t e d an a d d i t i o n a l s c i n t i l l a t i o n counter-—5 cm. i n diameter by 0.32 cm. t h i c k — b e t w e e n counters S11 and S13-S13*. By demanding a c o i n c i d e n c e between t h i s counter and the other p o s i t r o n c o u n t e r s , we reduced our acceptance to about 8% of normal. A two hour run under these c o n d i t i o n s 86 ( y i e l d i n g about 110,000 counts i n the y + -»- e + + v + v shape of hin t h r e e , and about 140 counts i n the TT -e r e g i o n of b i n one) produced a M i c h e l s p e c t r a e s s e n t i a l l y i d e n t i c a l t o those a c q u i r e d under normal running c o n d i t i o n s . (That i s , the M i c h e l shapes i n the b i n s were i d e n t i c a l to w i t h i n s t a t i s t i c s , about 0.3 %) This r e s u l t i n d i c a t e d t h a t the s p e c t r a a c q u i r e d under our normal running c o n d i t i o n s were the c l e a n e s t t h a t we could reasonably expect, and t h a t perhaps we had e l i m i n a t e d the low energy background e n t i r e l y . II-E-6 , . Runtime. Tests On The Data To safeguard a g a i n s t f a i l u r e o f the e l e c t r o n i c s modules, d e t e c t o r s or p h o t o - m u l t i p l i e r s , or any u n i n t e n t i o n a l change i n the experimental set-up, the TT -e group monitored a number of parameters throughout the f i n a l production run, which extended over s e v e r a l months. T h i s procedure i s b r i e f l y d e s c r i b e d i n appendix B. Ey c o n s i s t e n t l y r e c o r d i n g and comparing these q u a n t i t i e s , we became aware of and c o r r e c t e d a number of spontaneous problems, and were able t o keep the q u a l i t y of our measured date u n i f o r m l y good throughout the f o u r months of the f i n a l production run. 87 Chapter I I I DATA ANALYSIS A. Lineshapes 1. The I n t r i n s i c Nal Lineshape To a c c u r a t e l y determine the response f u n c t i o n of the Nal, and j u s t f o r the fun of i t ^ we performed a r a y t r a c i n g experiment i n the s p r i n g of 1977.. A d e s c r i p t i o n o f t h i s experiment, and of the a n a l y s i s of the d a t a , i s given i n appendix C. The s t r o n g i n f e r e n c e given by the a n a l y s i s of the r a y t r a c i n g data i s t h a t the Nal l i n e s h a p e changes very l i t t l e with i n c i d e n t p a r t i c l e mcmenta between 30 and 80 MeV/c, and f o r a n g l e s of i n c i d e n c e o f l e s s than 20 degrees. T h i s i s c o r r o b o r a t e d by the agreement between the r a y t r a c i n g r e s u l t s and the 130 MeV n-gamma l i n e s h a p e . The e x p o n e n t i a l f a c t o r n (see appendix C) i s very poorly d e f i n e d by the r a y t r a c i n g data to be n=0.5±0.5 88 T h i s l a c k of p r e c i s i o n r e s u l t s frcm s e r i o u s g a i n s h i f t s i n the Nal energy s p e c t r a , and from problems with the m u l t i p l e s c a t t e r i n g c o r r e c t i o n s and the v e r t i c a l d i s p e r s i o n of the e l e c t r o n beam. The l i n e s h a p e d e r i v e d frcm the r a y t r a c i n g data i s net e x a c t l y the " i n t r i n s i c " Nal l i n e s h a p e f o r the T r - e experiment, due t o the e f f e c t s of the stopping d i s t r i b u t i o n and the e f f e c t s o f a d d i t i o n a l Landau s t r a g g l i n g . The stopping d i s t r i b u t i o n e f f e c t s r e s u l t frcm i the f i n i t e volume of the t a r g e t counter.. That i s , decay p o s i t r o n s t h a t o r i g i n a t e on the " f a r s i d e " cf the t a r g e t , away from the N a l , must t r a v e r s e a g r e a t e r d i s t a n c e of CH 2 and thus l o s e more energy than p o s i t r o n s o r i g i n a t i n g on the near s i d e . Given the known pion s t o p p i n g d i s t r i b u t i o n and the energy l o s s f o r p o s i t r o n s i n CH22 , t h i s can be e a s i l y c a l c u l a t e d . (For f u r t h e r i n f o r m a t i o n on the pion s topping d i s t r i b u t i o n , see the s e c t i o n oh muon leakage, s e c t i o n I I I - C - 1 ) . . "Landau s t r a g g l i n g " i s a d e s c r i p t i o n of the d i s p e r s i o n i n energy l o s t by charged p a r t i c l e s t r a v e r s i n g matter (see Wi29, La44, Go52-, and Se64)» The primary r e s u l t of t h i s c a l c u l a t i o n i s a u n i v e r s a l s t r a g g l i n g f u n c t i o n - w h i c h can be used f o r any charged p a r t i c l e . - The Landau theory agrees with experiment very w e l l r e g a r d i n g the shape of the energy l o s s f u n c t i o n , although there are some d i s c r e p a n c i e s i n the mean energy l o s t (Go52). The s t r a g g l i n g c o r r e c t i o n t o the Nal l i n e s h a p e i s p o t e n t i a l l y 89 important s i n c e , because there i s some p r o b a b i l i t y f o r t i e p o s i t r o n s to l o s e a great d e a l of energy, the t a i l cf the lineshape could be s i g n i f i c a n t l y i n c r e a s e d . Note t h a t some of the r e l e v a n t Landau s t r a g g l i n g i s already i n c l u d e d i n the r a y t r a c i n g r e s u l t s . The " f r o n t f a c e " m a t e r i a l of the TIN A e n c l o s u r e , the TV counters and the seven i n c h counter were a l l i n p l a c e during the r a y t r a c i n g run.. Thus Landau s t r a g g l i n g caused by t h i s m a t e r i a l i s a l r e a d y represented i n the r a y t r a c i n g l i n e s h a p e . A Monte C a r l o s i m u l a t i o n program was w r i t t e n t o f o l d the e f f e c t s of the stopping d i s t r i b u t i o n and r e s i d u a l Landau s t r a g g l i n g i n t o the r a y t r a c i n g l i n e s h a p e . (For f u r t h e r i n f o r m a t i o n on Monte C a r l o c a l c u l a t i o n s , see Op77 and Fi71.) The r a y t r a c i n g l i n e s h a p e f o r p o s i t r o n s of 61.3 MeV and zero degrees entrance angle was used. E e s u l t s cf the s i m u l a t i o n are shown i n f i g u r e I I I - 1 . Given are the r a y t r a c i n g l i n e s h a p e , the l i n e s h a p e i n c l u d i n g the e f f e c t of the stopping d i s t r i b u t i o n , and the l i n e s h a p e with the e f f e c t s of the st o p p i n g d i s t r i b u t i o n and Landau s t r a g g l i n g . The e f f e c t of the l a t t e r i s very s l i g h t . . These d i s t o r t i o n s of the l i n e s h a p e are summarized i n t a b l e I I I - 1 . 9 0 FIGURE I I I - l EFFECT OF THE STOPPING DISTRIBUTION AND LANDAU STRAGGLING ON THE RAYTRACING LINESHAPE, DETERMINED BY A MONTE CARLO CALCULATION. 3xl0 : co t-z L U > o 2xl0 3 cu < o T. 1x10" t o o u • RAYTRACING LINESHAPE RAYTRACING LINESHAPE WITH X STOPPING DISTRIBUTION CORRECTION RAYTRACING LINESHAPE WITH STOPPING DISTRIBUTION AND ° LANDAU STRAGGLING CORRECTIONS X o • . X x • x o x • X . o X o • 220 260 T 300 340 CHANNEL S1 R A Y T R A C I N G W I T H S T O P P I N G W I T H S . D . L I N E S H A P E D I S T R I B U T I O N A N D L A N D A U ( S . D . ) S T R A G G L I N G P E A K E N E R G Y R E S O L U T I O N 61.3+.2 MEV 59.6+.2 MEV 59.5+.2 MEV 6.8+.2Z 7.6+.2% 7.91.2% TABLE III-1 E f f e c t s Of T i e Stopping D i s t r i b u t i o n And Landau S t r a g g l i n g On The B a y t r a c i n g Lineshape. Thus, the " i n t r i n s i c " Nal l i n e s h a p e used i n the a n a l y s i s of the v-e data i s the 61.3 MeV/c, 0 degree i n c i d e n c e r a y t r a c i n g l i n e s h a p e with the c o r r e c t i o n s f o r the pion s t o p p i n g d i s t r i b u t i o n and the r e s i d u a l Landau s t r a g g l i n g a p p l i e d . 92 III-A-2. C a l c u l a t e d Spectra For TT -ev And TT-g vy P o s i t r o n s from the decay of ir-ev are mcno-energetic at a t o t a l energy of 69.786 MeV. However^ the corresponding r a d i a t i v e decay T r - e v y can produce p o s i t r o n s with energy l e s s than the above. The combined p o s i t r o n s p e c t r a produced by the r a d i a t i v e and n o n - r a d i a t i v e e l e c t r o n decay of the pion has been examined many times (see Br64, B160, Be58, So72, Gc75, and Ma77), U n f o r t u n a t e l y , these r e s u l t s cannot be d i r e c t l y a p p l i e d to the a n a l y s i s of the present data, s i n c e the Nal c r y s t a l s i m u l t a n e o u s l y detected seme of the hard photons, i n a d d i t i o n to d e t e c t i n g the decay p o s i t r o n s . E x i s t i n g c a l c u l a t i o n s y i e l d e x p r e s s i o n s f o r the energy s p e c t r a f o r p o s i t r o n s only, or f o r photons cnly* New c a l c u l a t i o n s were required to produce the s p e c t r a f o r the mix of p o s i t r o n s and photons seen by the Nal c r y s t a l . D e t a i l s of the c a l c u l a t i o n o f r a d i a t i v e c o r r e c t i o n s to Tr-e are given i n appendix D. The o v e r a l l e f f e c t s are very s l i g h t , i n c r e a s i n g the " t a i l " of the lin e s h a p e below 55 MeV by l e s s than 0.5 %. .. 9 3 I I I - A - 3 . C a l c u l a t e d P o s i t r c n Spectra Frcm u 1 e++ v + v _And y * e * + v + v y Decay P o s i t r o n s from the decay y + -»• e++ v • v and from the a s s o c i a t e d r a d i a t i v e decay y • -»- e++ v + v y vary i n t o t a l energy from 0.511 t o 52,831 MeV. The p c s i t r c n s p e c t r a , a l s o r e f e r r e d to as the Michel s p e c t r a , i s maximized a t the h i g h e s t energy, and f a l l s smoothly t o zero at the minimum energy. In the approximation of zero e l e c t r o n mass, the shape of the p o s i t r o n energy s p e c t r a f o r the decay of u n p c l a r i z e d muons c a l c u l a t e d t o zer o t h order i s given by = 2 x 2 { | ( 1 - x ) + p ( * x - 1 ) } / T dE 2 - 3 V ... (III-1) where T = l i f e t i m e o f t h e muon P x = E / E e max p = M i c h e l p a r a m e t e r , a f u n c t i o n o f t h e c o u p l i n g c o n s t a n t s = 3/4 f o r V-A t h e o r y 94 The shape of the spectrum* i n c l u d i n g r a d i a t i v e c o r r e c t i o n s , has been c a l c u l a t e d many times (see Mu65, Be56, Be58, Ok58, Ec59, Fr59, Ki59a, Ki59b, Du63 and Ka68j. Once a g a i n , the r e s u l t s o f these c a l c u l a t i o n s cannot be d i r e c t l y a p p l i e d to our data, s i n c e the Nal can det e c t hard photons i n c o i n c i d e n c e with the decay p o s i t r o n s . Due t o d i s c r e p a n c i e s between the experimental and t h e o r e t i c a l l i n e s h a p e s f o r p o s i t r o n s from muon decay (described i n s e c t i o n I I I - B - 2 ) , we f e l t a c a r e f u l c a l c u l a t i o n of the r a d i a t i v e c o r r e c t i o n s t o muon decay was r e q u i r e d . T h i s c a l c u l a t i o n , and s e v e r a l e n t e r t a i n i n g problems encountered, are de s c r i b e d i n appendix E.. The r e s u l t i n g change i n the t h e o r e t i c a l s p e c t r a from y + +e ++ v + v decay seen by the Nal i s very s l i g h t . The above r e s u l t i s the expected t h e o r e t i c a l spectrum seen i n the Nal. . To make use o f t h i s , we must take i n t o account two a d d i t i o n a l f a c t o r s : 1) The i n t r i n s i c Nal l i n e s h a p e (described i n s e c t i o n I I I - A - 1 ) , and 2) The r e l a t i v e v a r i a t i o n i n the e f f e c t of the stopping d i s t r i b u t i o n . Again, the stop p i n g d i s t r i b u t i o n i s the v a r i a t i o n i n the energy l o s t i n the t a r g e t counter due to i t s f i n i t e volume* Although the Nal l i n e s h a p e at 65 MeV i n c l u d e d t h i s e f f e c t , which makes the e f f e c t i v e Nal r e s o l u t i o n worse than the i n t r i n s i c r e s o l u t i o n , a f u r t h e r adjustment must be made because the e f f e c t of the stopping d i s t r i b u t i o n on the e f f e c t i v e Nal r e s o l u t i o n i s energy 95 dependent. For p r a c t i c a l purposes, the stepping d i s t r i b u t i o n f o l d s a gaussian i n t o the Nal l i n e s h a p e . , T i e width of t h i s gaussian i s determined by the average energy deposited i n the t a r g e t counter by the p o s i t r o n s , a g u a n t i t y which v a r i e s s l o w l y with the energy of the p o s i t r o n . , However, the e f f e c t i v e Nal r e s o l u t i o n , d e f i n e d by (FWHM i n MeV) /(Peak Energy i n MeV) i s i n v e r s e l y p r o p o r t i o n a l t o the p o s i t r o n energy. Consequently, the n e a r l y constant amount added to the FWHK-by the stopping d i s t r i b u t i o n does induce an a d d i t i o n a l energy dependence i n t o the e f f e c t i v e r e s c l u t i c n f u n c t i o n of the Nal* C a l c u l a t i o n of the e f f e c t s of the stepping d i s t r i b u t i o n and of the Nal l i n e s h a p e i s presented i n appendix F. Even very extreme v a r i a t i o n s i n the parameters used t o estimate these e f f e c t s change the •u+ -> e + + v + v l i n e s h a p e by l e s s than ± 0.5 % belcw 20 MeV. The u n c e r t a i n t i e s i n the muon decay l i n e s h a p e seen ty the Nal are then e n t i r e l y dominated by u n c e r t a i n t i e s i n the mono-energetic NaT l i n e s h a p e and i t s energy dependence, t o be d i s c u s s e d below* 96 I I I - B . C a l i b r a t i o n And F i t t i n g 1. . In t r o duction In order to u t i l i z e the c a l c u l a t e d l i n e s h a p e s d e s c r i b e d i n the previous two s e c t i o n s , we must be able to compare them t o the e m p i r i c a l data i n seme standardized f a s h i o n . He do t h i s i n the customary manner — by f i t t i n g the data with our l i n e s h a p e s . The computer r o u t i n e used here f i t by minimizing c h i - s g u a r e d i n the usual f a s h i o n . The absolute value of the minimum chi - s q u a r e d i s o f t e n not a completely r e l i a b l e measure of the goodness of f i t f o r good s t a t i s t i c s Nal s p e c t r a , because f r e q u e n t l y the Nal l i n e s h a p e s are not known to high accuracy* An a l t e r n a t i v e f o r comparing c a l c u l a t e d l i n e s h a p e s and e m p i r i c a l s p e c t r a i s to i n t r o d u c e a p o s s i b l e background i n t o the f i t . . I f the comparison between l i n e s h a p e and s p e c t r a i s good, the background w i l l f i t to zero. The amount of background with some reasonable shape r e q u i r e d f o r "gocd" f i t s i s a u s e f u l measure of the discrepancy between the l i n e s h a p e s and the s p e c t r a . The f i t t i n g r o u t i n e used f o r t h i s study, known as REPTILE, was implemented i n FORTRAN on the IBM 370 and Amdahl computers a t U n i v e r s i t y of B r i t i s h Columbia., The 97 heart o f the f i t t i n g r o u t i n e was taken more or l e s s d i r e c t l y from Bevington's book "Data Seduction and E r r o r a n a l y s i s f o r the P h y s i c a l S c i e n c e s " (Be69).. The method c f searchi n g f o r the minimum chi - s g u a r e d i s known as the gradient expansion a l g o r i t h m , which combines the best f e a t u r e s of: 1) the g r a d i e n t search, which f e l l o w s the d i r e c t i o n of ste e p e s t descent o f the ch i - s g u a r e d s u r f a c e , and thus i s very f a s t when f a r from the minimum, but converges very s l o w l y when near the minimum; and 2 ) the method of l i n e a r i z i n g the f i t t i n g f u n c t i o n , which approaches t h e minimum ch i - s g u a r e d r a p i d l y from nearby, but i s u n r e l i a b l e when f a r away. The f i t t i n g procedure i s standard. A d e r i v a t i v e matrix i s c a l c u l a t e d f o r the x2 surface and the parameters to be v a r i e d . T h i s matrix i s i n v e r t e d t o give an e r r o r matrix which i n d i c a t e s the d i r e c t i o n and amount the parameters are t o be v a r i e d , and als o the u n c e r t a i n t i e s a s s o c i a t e d with the c o l l e c t i v e v a r i a t i o n of the parameters., T h i s procedure i s repeated u n t i l X 2 ceases to decrease, when the f i t i s e i t h e r terminated, i f the rec e n t change i n x 2 i s w i t h i n some l i m i t ( u s u a l l y 0. 1%) or a new d e r i v a t i v e matrix i s c a l c u l a t e d , and the procedure begun a g a i n . Parameters v a r i e d were u s u a l l y the amplitude (or area) o f the l i n e s h a p e , the p o s i t i o n (or energy) , and parameters r e l a t e d t o a reasonable background shape. The r e s o l u t i o n (or FWHM) of the l i n e s h a p e was never v a r i e d a u t o m a t i c a l l y , because o f the complexity and cost S 8 r e q u i r e d . Instead, f i t s were . performed f o r a s e r i e s of r e s o l u t i o n s , and the best r e s o l u t i o n i d e n t i f i e d "by hand." Note t h a t the e r r o r s c a l c u l a t e d from the e r r o r matrix i n c l u d e the e f f e c t s of the u n c e r t a i n t i e s i n the other parameters. They cannot account f o r any u n c e r t a i n t y i n the s u p p l i e d l i n e s h a p e . The parameters — a m p l i t u d e s , e n e r g i e s , background v a r i a b l e s — c o u l d be f i x e d or allowed to vary to minimize X 2 . Another o p t i o n permitted one set of parameters t c he v a r i e d while minimizing x 2 o v e r a c e r t a i n energy r e g i o n , f o l l o w e d by a second f i t over a p o s s i b l y d i f f e r e n t energy r e g i o n v a r y i n g another s e t of parameters, while those parameters v a r i e d f o r the f i r s t f i t c c u l d e i t h e r be held f i x e d or allowed once again t o vary. In a d d i t i o n t o the f u n c t i o n s d e s c r i b e d above, the REPTILE f i t t i n g r o u t i n e performed a v a r i e t y of other t a s k s , i n c l u d i n g c a l c u l a t i n g Nai l i n e s h a p e with a d e s i r e d c a l i b r a t i o n and r e s o l u t i o n , background s u b t r a c t i o n s , energy c a l i b r a t i o n s , sums and t a i l c o r r e c t i o n s over v a r i o u s r e g i o n s , e t c . The REPTILE f i t t i n g program was used to f i t both the TT —e v and y + e*+ v + v r e g i o n s of the Nal s p e c t r a , although not s i m u l t a n e o u s l y . . The procedure f o r f i t t i n g the TT -e r e g i o n was as f e l l o w s . The a c c i d e n t a l s s p e c t r a i n bin f o u r was s u b t r a c t e d channel by channel i n t h e . c o r r e c t r a t i o s from b i n s one and two (see s e c t i o n II-D-2). Then the TT - y - e S9 shape from B2 was normalized t o t h a t i n B1 and s u b t r a c t e d channel by channel from B1. . T h i s was done to remove the high energy t a i l t h a t i s r e l a t e d to the TT . - y - e events from beneath the TT -e r e g i o n . T h i s t a i l i n c l u d e s both charged-charged and charged-uncharged p i l e - u p events, and any e f f e c t of a high energy t a i l on the mono-energetic Nal l i n e s h a p e * The v a l i d i t y of t h i s " y + e ++ v + v shape s u b t r a c t i o n " i s based on the assumption that the y * ->- e + + v + v shapes are the same i n b i n s one and two. Care was taken to assure t h i s by making B1 and B2 " i d e n t i c a l " with r e s p e c t to muon decay, and a v a r i e t y of t e s t s , d e s c r i b e d i n the next s e c t i o n , support t h i s assumption. Presumably, what then remains i n B1, w i t h i n the s t a t i s t i c a l u n c e r t a i n t i e s of the background s u b t r a c t i o n s , should be p u r e l y the Tr-e shape. , (Note t h a t a s l i g h t c o r r e c t i o n to account f o r the s u b t r a c t i o n of a s m a l l number of TT -e events i n B2 must be a p p l i e d to the branching r a t i o . ) However, because the y + -> e + + v + v 's are two orders of magnitude l a r g e r than the T r - e »s, t h i s s t a t i s t i c a l u n c e r t a i n t y becomes very l a r g e i n the y *• -> e ++ v . + v r e g i o n -- 0 t c 50 MeV. Thus, u s e f u l TT -e i n f o r m a t i o n i s contained e n t i r e l y i n the r e g i o n above the y + -> e*+ v + v edge. T h i s spectrum, a f t e r a c c i d e n t a l s and y + e ++ v + v shape have been s u b t r a c t e d , i s f i t with the c a l c u l a t e d l i n e s h a p e p r e v i o u s l y d e s c r i b e d , and a smocthly varying background. A p a r a m e t r i z a t i o n f o r an unknown background 100 shape t h a t has been found extremely u s e f u l i s a q u a d r a t i c background which i s f o r c e d to zero at some energy. T h i s background shape i s d e f i n e d : B(E) = Bl ( E - c u t o f f ) 2 + B2 ( E - c u t c f f ) T h i s form has the advantages t h a t i t can be f o r c e d smoothly to zero i n a r e g i o n where the background i s known t o be ze r o , and i t reduces the p r o b a b i l i t y cf p h y s i c a l l y u n l i k e l y "bumps" or n e g a t i v e - p o s i t i v e backgrounds t h a t can occur when f i t t i n g with a purely polynomial form f o r a background. T h i s form was suggested by Dr. D., Measday (Ee74). The energy c a l i b r a t i o n f o r the TT-e f i t s was taken from the automated c a l i b r a t i o n of the y + ->e++v + v s p e c t r a , d e s c r i b e d i n the f o l l o w i n g s e c t i o n . A f t e r the f i t has been performed, " t a i l c o r r e c t e d sums" are c a l c u l a t e d f o r a s e l e c t i o n of c u t o f f e n e r g i e s . These are sums of the f i t t e d s p e c t r a over a c e r t a i n energy region*. These sums are then c o r r e c t e d f o r a " t a i l " , t h a t part of the c a l c u l a t e d l i n e s h a p e t h a t l i e s outside- the summed r e g i o n . T h i s c o r r e c t i o n then depends only on the f i t t e d p o s i t i o n (or energy) of the l i n e , net cn the f i t t e d amplitude. E f f e c t s of the f i t t e d background can e i t h e r te i n c l u d e d or ignored. ft very s i m i l a r procedure was f o l l o w e d when f i t t i n g the y+ -> e ++ v + v re g i o n i n any of the f o u r b i n s . Of 101 course no p + •+ e + + v + v s u b t r a c t i o n was performed, a y + -». e + + v + v l i n e s h a p e was used f o r the f i t , and a d i f f e r e n t energy r e g i o n was examined. The energy c a l i b r a t i o n was determined a u t o m a t i c a l l y as d e s c r i b e d i n the f o l l o w i n g section;. . Parameters f o r c a l c u l a t i n g the u + -»- e ++ v +v spectra from the r a d i a t i v e c o r r e c t e d M i c h e l s p e c t r a and the Nal l i n e s h a p e with a p p r o p r i a t e energy dependence were de s c r i b e d i n s e c t i o n III-A-3.. T a i l c o r r r e c t e d sums were al s o .performed f o r y+ -»- e ++ v + v f i t s * The TT - e peaks i n our s p e c t r a r e s u l t i n g from one t c two hour runs co n t a i n e d s t a t i s t i c s much tec poor to enable c a r e f u l i n s p e c t i o n of the p r o p e r t i e s o f the TT -e mono-energetic peak. In order t o perform these i n v e s t i g a t i o n s more q u i c k l y and with l e s s expense, a l a r g e segment of the data (about 40%) was g a i n - s h i f t e d and summed to produce a s i n g l e good s t a t i s t i c s s e t of s p e c t r a . T h i s set of s p e c t r a was i d e n t i f i e d as run 760 (as p r e v i o u s l y mentioned i n s e c t i o n I I - E - 1 ) . The g a i n - s h i f t i n g procedure was c a r e f u l l y t e s t e d to i n s u r e no energy dependent or o v e r a l l d i s t o r t i o n of the spectra occured. T h i s procedure d i d not compensate f o r changes i n the Nal r e s o l u t i o n * In a d d i t i o n , comparative f i t s were done f o r run 760, a subset of u n s h i f t e d runs i n c l u d e d i n run 760, and another e x c l u s i v e subset of runs. A l l these r e s u l t s are i n good agreement, r e - a f f i r m i n g the b e l i e f t h a t the g a i n s h i f t e d run 760 i s a good r e p r e s e n t a t i o n of 102 the o v e r a l l 7 1 -e data. A subset of the data independently g a i n - s h i f t e d by Dr.. W.S. D i x i t i s a l s o i n e x c e l l e n t agreement with run 760. The process of f i t t i n g the data has continued ever a twelve month p e r i o d . . As p r e v i o u s l y d e s c r i b e d , many parameters a f f e c t the f i t s , and a great many f i t s have been performed to a s c e r t a i n t h e i r importance. The most important r e s u l t s of these f i t s w i l l be d e s c r i b e d i n the f o l l o w i n g s e c t i o n . A s t a n d a r d i z e d d e s c r i p t i o n of some c f the comparison f i t s i s presented i n appendix G. II I - E - 2 . Energy C a l i b r a t i o n Of: The Spe c t r a Accurate energy c a l i b r a t i o n of our s p e c t r a was necessary t o enable reasonably p r e c i s e a n a l y s i s of the Mi c h e l s p e c t r a * P o s i t r o n s from muon decay range i n energy from 0.511 to 52.83 MeV. The presence of a lew energy peak t h a t permitted an accurate determination of the channel of zero energy was then very f o r t u i t i o u s . (See s e c t i o n II-E-5.) As mentioned above, the low energy peak was f i t to determine an energy c a l i b r a t i o n p o i n t . The shape of t h i s peak cannot be c a l c u l a t e d to any great c e r t a i n t y , nor can the e x t r a p o l a t e d background from the y + e++ v + v spectrum. In p r a c t i c e , a gaussian was used t o f i t the 1C3 peak, with a l i n e a r or h o r i z o n t a l background,. These unknowns c o n t r i b u t e an u n c e r t a i n t y o f 0.25 channels t o the peak p o s i t i o n of the low energy peak. The c a l i b r a t i o n r c i n t f o r the f o u r bins i s very s l i g h t l y d i f f e r e n t : Low E n e r g y peak i n b i n : B3 > B2 > B1 > B4 d i f f e r e n c e (ch) 0.12 0.23 0 . 30 --d i f f e r e n c e (kev) 20 39 5 1 --T h i s u n c e r t a i n t y i n zero p o i n t c a l i b r a t i o n c o n t r i b u t e s only 0.05 5! to the u n c e r t a i n t y i n the branching r a t i o . Besides the low energy peak, two other c a l i b r a t i o n p o i n t s are a v a i l a b l e — the P + •+ e++ v + v high energy edge, and the mono-energetic T r-e peak. In p r a c t i c e , t i e p o s i t i o n of the TT -e peak could not be a c c u r a t e l y determined i n the one to two hour runs, because of poor s t a t i s t i c s , although i t co u l d be determined i n the gain s h i f t e d run. Determination of the u * •+ e*+ y + v "edge" r e q u i r e s t h a t the r e s o l u t i o n of the Nal c r y s t a l at t h a t energy be known. T h i s r e s o l u t i o n was determined by minimizing x 2 i n the upper energy r e g i o n of the M i c h e l s p e c t r a (40-53 MeV). Now, to a s s i g n an "energy" t o t h i s p o i n t (that i s , the energy seen by the N a l ) , the amount of energy l o s t i n 104 the t a r g e t and other counters and i n the Nal e n c l o s u r e must be s u b t r a c t e d from the known endpoint energy, and an e l e c t r o n mass must be added, due to the a n n i h i l a t i o n of the M i c h e l p o s i t r o n i n the c r y s t a l . C a l c u l a t i o n of the energy l o s s , using the known t h i c k n e s s e s of the c o u n t e r s , estimates o f t h i c k n e s s e s of the counter wrappings, and i n f o r m a t i o n on the e n c l o s u r e cf the Nal from Harshaw Chemical Company, y i e l d e d E - l o s t ( 5 0 MeV) = 5.1 ± 0.1 MeV Use of t h i s v alue f i x e s the c a l i b r a t i o n p o i n t s f o r both the TT -e and the TT- U -e l i n e s h a p e s . When a n a l y z i n g the data, the f i t t i n g r o u t i n e determined the c a l i b r a t i o n using the low energy peak and the u + -> e++ v + v shape, and then used t h i s f o r both the TT - u -e and TT -e f i t s . Using t h i s procedure, the energy of the T r-e peak was found to be about 400 keV higher than expected. An a l t e r n a t i v e method i s to take the i n f o r m a t i o n on the TT-e and T T - U -e shapes and the low energy peak and to c a l c u l a t e the energy l o s s . T h i s y i e l d s E - l o s t ( 5 0 MeV) = 5.6 ± 0.2 MeV T h i s u n c e r t a i n t y i n the energy l o s s r e s u l t s i n an u n c e r t a i n t y of 0.2 % i n the branching r a t i o * Note t h a t 105 a l l e n e r g i e s quoted i n the f o l l o w i n g d i s c u s s i o n , u n l e s s otherwise noted, are energies seen i n the Nal c r y s t a l , a f t e r energy l o s s and p o s i t r o n a n n i h i l a t i o n have been taken i n t o account. III-B-3,. R e s u l t s Of F i t t i n g And F u r t h e r  Background A n a l y s i s III-B-3-a. A p p l i c a t i o n Of The R a y t r a c i n g Lineshape To  The, Data The r a y t r a c i n g l i n e s h a p e , c o r r e c t e d f o r Landau s t r a g g l i n g , the stopping d i s t r i b u t i o n , and the e f f e c t s of !-eW ( r e f e r r e d t o as TG3) has been f i t t c the ir-e mcno-energetic peak as d e s c r i b e d i n the p r e v i o u s s e c t i o n . The r e s o l u t i o n , as determined by minimizing the.counters i n the peak r e g i o n , i s FWHM = 7.7 ± 0.2 X. (See f i t s 1012.03 and 1012»04, and appendix G. ) However, the l i n e s h a p e does not agree w e l l with the data i n e i t h e r the i peak region nor the t a i l * F i g u r e I I I - 2 shows a f i t t o run 760 with the r a y t r a c i n g l i n e s h a p e ( f i t 1012.05). The r e s o l u t i o n used was 8.0 % — somewhat l a r g e r than measured 1 0 6 FIGURE 111-2 RAYTRACING LINESHAPE APPLIED TO THE n~e PEAK, BACKGROUND TO 70 MEV, FWHM=8.0%, FIT 1012.05 144.0 , COUNTS + DIFFERENCE: D R T f l - L I N E 9 6 . J • • F I T T E D BACKGROUND 48.0 J 0.0 FIGURE 111-3 107 RAYTRACING LINESHAPE APPLIED TO THE t l - e PEAK, BACKGROUND TO 61 MEV, FWHM = 8.0%, FIT 1012.04 120.0 n 96.0 72.0 48.0 24.0 COUNTS CX101 ) + DflTfl — FIT O BRCKGROUND F I T ; RESOLUTION 8.0* BRCKGROUND: SECOND FIT 61 MEV CUTOFF 0.0 55.0 60.0 65.0 ENERGY 70.0 60.0 40.0 0.0 -40.0 A -80.0 COUNTS BRCKGROUND + DJFFERENCE=DRTR-L1NE 55.0 60.0 65.0 ENERGY • * 4 *70.ff 1 0 8 FIGURE I I 1 - 4 RAYTRACING LINESHAPE APPLIED TO THE H-e PEAK, BACKGROUND TO 61 MEV, FWHM - 7.6%, F I T 1012.03 150.0 100.0 50.0 0.0 -50.0 -100.0 + DIFFERENCE=DBTR-LJNE COUNTS BACKGROUND V 1 , • • . _ * , , r — 1 — — — ,55.0 60.0 1 i 11 1 65.0 TP^ fl ENERGY i • • 109 FIGURE II1-5 F I T OF THE I l - e PEAK WITH TWO RAYTRACING LINESHAPES, FWHM = 6.8%, FIT 1012.02 120.0 96.0 72.0 48.0 24.0 0.0 1 COUNTS (X101 ) + DRTR — FIT 0 BRCKGROUND FIT: TWO LINES. RES. 6.8 % BRCKGROUND: SECOND FIT ENERGY 150.0 100.0 50.0 0.0 -50.0 -100.0 C0UNT5 + DIFFERENCE=DRTR-LINE BRCKGROUND 55.0 6 0 • t V + ENERGY 110 to account f o r the excess counts i n the t a i l region;. T i e f i r s t f i t , r e g i o n 61 - 71 MeV, v a r i e d only the p o s i t i o n and amplitude. The second f i t , r e g i o n 52*5 to 72 MeV, i n t r o d u c e d a q u a d r a t i c background, c u t o f f a t 70 MeV, and allowed only the background t o vary* . P l o t t e d i s the dat a , the f i t , and the f i t t e d background, and a l s o the d i f f e r e n c e between the data and the f i t t e d l i n e s h a p e . We see t h a t the peak region of the TT -e data has a very u n l i k e l y shape. The f i t i n t h i s r e gion takes the middle values, l e a v i n g p o s i t i v e and ne g a t i v e bumps i n the d i f f e r e n c e curve that sum to almost zero.. In the t a i l r e g i o n , the d i f f e r e n c e i s c o n s i s t e n t l y p o s i t i v e , n e c e s s i t a t i n g a p o s i t i v e background.. The discrepancy i n the t a i l r e g i o n i s the more d i s t u r b i n g problem.. The v a r i a t i o n i n the t a i l c o r r e c t e d sum f o r c u t o f f s between 52.5 and 58 MeV was 2.4%. (See appendix 6.) A d e c i s i o n must be made whether these counts are genuine ^-e events, i n which case they should be i n c l u d e d i n the branching r a t i o c a l c u l a t i o n , cr are they background counts of some s o r t , which should be excluded? In other words, do we b e l i e v e the sura down to 52.5 MeV, or must we c u t o f f a t 58 MeV or higher? F i g u r e I I I - 3 shows a s i m i l a r f i t with the background c u t o f f = 61 MeV.. ( F i t 1012.04)*. T h i s p a r a m e t e r i z a t i o n cf the background g i v e s a somewhat b e t t e r f i t , by a l l o w i n g the f i t t e d background t o r a p i d l y i n c r e a s e beneath the t a i l r e g i o n , without r e q u i r i n g a s i m i l a r i n c r e a s e under the 111 peak* The X 2 i s somewhat improved, but the f i t and the t a i l c o r r e c t e d sums are unchanged., F i g u r e III-4 g i v e s a s i m i l a r f i t , but with the Nal r e s o l u t i o n = 7.6 % ( f i t 1012*03). T h i s f i t has a s i g n i f i c a n t l y lower X 2 , at the expense of i n c r e a s i n g , the amount of background r e q u i r e d , and a l s o the, v a r i a t i o n i n the t a i l c o r r e c t e d sums. The sum t o 52.5 MeV i s 0.3 % lower than the p r e v i o u s case.. To 58 MeV, i t i s 0.7 % lower. The oddly shaped peak of the TT -e data could have a v a r i e t y of e x p l a n a t i o n s . . Remember t h a t t h i s spectrum r e p r e s e n t s n e a r l y 40 5? of a l l the IT -e data, a c q u i r e d over two weeks of running. We hoped to minimize the g a i n s h i f t s by w r i t i n g s p e c t r a onto magnetic tape every hour but of course some e f f e c t s of the g a i n s h i f t s could s t i l l be present. Another p o s s i b i l i t y i s t h a t a skewed s t o p p i n g d i s t r i b u t i o n r e l a t e d to the f o c u s s i n g of the pion beam might y i e l d a double peaked energy l o s s d i s t r i b u t i o n , r e s u l t i n g i n a double Nal peak. A l e s s l i k e l y but s t i l l p o s s i b l e e x p l a n a t i o n i s t h a t some obscure malfu n c t i o n i n the e l e c t r o n i c s e l e c t i o n and a m p l i f i c a t i o n c f the pulses induced t h i s odd mono-energetic li n e s h a p e * (The f a c t t i e EiCapiia a lso measured an odd TT -e peak .shape i s an odd coincidence.) Any of these e x p l a n a t i o n s suggests f i t t i n g the data with two or more mono-energetic peaks.. A f i t with two Nal peaks of r e s o l u t i o n 6.8 % i s i l l u s t r a t e d i n f i g u r e I I I - 5 112 ( f i t 1012.02). Note t h a t , although the n e g a t i v e and p o s i t i v e peaks i n the d i f f e r e n c e p l o t are g r e a t l y reduced, the f i t t e d background i s e s s e n t i a l l y unchanged. The t a i l c o r r e c t e d sums are reduced (not increased) from the s i n g l e peak f i t of r e s o l u t i o n 7.6 % by 0.4 55 at 52.5 MeV and 0.9 % at 58 MeV. Although the second peak accounts f o r seme events at a lower energy, the r e s o l u t i o n c f the peaks mcst be reduced t o give improved f i t s , thus reducing the t a i l c o r r e c t i o n s from the s i n g l e peak ( r e s o l u t i o n 7.6 %) values. These two peaks f i t with e n e r g i e s and areas : l i n e 1 l i n e 2 E n e r g y A r e a 66.5 MeV 62.6 MeV 32,550 3,450 Other attempts at e x p l a i n i n g t h i s d i s c r e p a n c y , such as e x p l i c i t e l y a p p l y i n g a number of "reasonable" g a i n s h i f t f u n c t i o n s , were not f r u i t f u l . The question a r i s e s whether some flaw i n the g a i n s h i f t i n g process might have d i s t o r t e d the s p e c t r a . Although the s t a t i s t i c s i n the mono-energetic Tr-e peak f o r any i n d i v i d u a l two hour run are too poor to permit any comparison, a s e r i e s of i n d i v i d u a l u n s h i f t e d runs can be f i t , and these r e s u l t s compared with the g a i n s h i f t e d runs,. The main c r i t e r i a here i s the d i f f e r e n c e between the 52.5 113 MeV and the 58 MeV t a i l c o r r e c t e d sums. Two s e t s of u n s h i f t e d runs were examined. The f i r s t s e t , runs 1563 -1604, was a subset of the g a i n s h i f t e d r u n s . The second s e t , runs 1183 - 1219, had not been g a i n s h i f t e d . R e s u l t s of the i n d i v i d u a l runs are as f e l l o w s ( f o r d e f i n i t i o n of the parameters, see appendix G) : j TT—e Run(s) t y p e £(52.5) 1(58) A E 760 g a i n s h i f t e d 38041 37196 2.4% 1183-1219 n o t s h i f t e d 15807 15290 3.2% 1563-1604 n o t s h i f t e d 8451 8206 2.9% The d i s c r e p a n c i e s i n the A E are not s t a t i s t i c a l l y s i g n i f i c a n t . We see the same tendency i n a l l t h r e e s e t s , s t r o n g l y i m p l y i n g t h a t t h e discrepancy between the r a y t r a c i n g l i n e s h a p e and the TT -e mono-energetic peak i n the t a i l r e g i o n i s not an a r t i f a c t of the g a i n s h i f t i n g procedure. Nothing concerning the unusual peak shape can r e a d i l y be determined. 114 For the previous f i t s , X 2 was c a l c u l a t e d by weighting the data p o i n t s i n v e r s e l y with the number of counts a t each p o i n t . T h i s "normal" s t a t i s t i c a l weighting does not take i n t o account the increased u n c e r t a i n t y caused by the background s u b t r a c t i o n s performed on the s p e c t r a . When the t r u e s t a t i s t i c a l weighting i s used, the values of X 2 are s i g n i f i c a n t l y reduced, but the r e s u l t s of the f i t are only s l i g h t l y a f f e c t e d . (Compare f i t s 1012.03 and 1112,03.) The r a y t r a c i n g l i n e s h a p e , c o r r e c t e d f o r Landau S t r a g g l i n g and the stopping d i s t r i b u t i o n (lineshape TLE) i s somewhat more s u c c e s s f u l when a p p l i e d to the y + e + + v + v spectrum.. I n i t i a l t e s t s were performed to check the i d e n t i t y c f the f o u r b i n s . S e v e r a l Nal r e s o l u t i o n s were used t c f i t the high energy r e g i o n of the y • + e++ v + v shape — 35 t o 50 MeV. The r e s o l u t i o n of b i n s one, two, and three was determined to be 8.6 ± 0 . 2 %, and of b i n f o u r was 9.0 ± 0.2 %m The somewhat worse r e s o l u t i o n of E4 can be a t t r i b u t e d t o the extended source of the a c c i d e n t a l decays. Other comparisons between bins one, two and t h r e e , i n c l u d i n g f i t t e d backgrounds, t a i l c o r r e c t i o n s , and peak p o s i t i o n s are a l l i n e x c e l l e n t agreement. F i g u r e I I I - 6 shows the r e s u l t s of a f i t to the y + -> e+ + v + v s p e c t r a , u s i n g the r a y t r a c i n g l i n e s h a p e f o l d e d i n t o the c a l c u l a t e d Michel s p e c t r a , as d e s c r i b e d i n s e c t i o n I I I - f t - 3 . ( F i t 311.04) a q u a d r a t i c background FIGURE II1-6 1 1 5 RAYTRACING LINESHAPE APPLIED TO THE MICHEL SPECTRUM C O U N T S C X 1 D 2 ) 2 0 0 . H 1 50 . — 1 OD . — 5 0 . O . T + DATA — FIT 0 BACKGROUND T —1MB llttOOtf^W 0.00 7.50 T 15.00 22.50 30.00 37.50 45.00 52.50 E N E R G Y C O U N T S t X l O 1 ) 5 0 2 5 . H o . — 1 - 2 5 . H T T T + DIFF=DATA-LINE _ FITTED BACKGROUND -50 . O.OO 7.50 ISIOO 22150 30.00 3 7 E g ° p R £5 • 0 0 52.50 116 f o r c e d t o zero at 50 MeV has been i n t r o d u c e d . To the eye, the f i t i s reasonably good, although t h e r e i s again some disagreement i n t h e high energy "peak" r e g i o n . . T h i s , I b e l i e v e , i s to be expected' one of the advantages of t h i s experimental method of determining the branching r a t i o i s t h a t p o s i t r o n s frcm both T -e and IT - y -e ch a i n s are detected s i m u l t a n e o u s l y i n the same Nal de t e c t o r . However, t h i s i m p l i e s t h a t the same i n t r i n s i c Nal l i n e s h a p e , with s u i t a b l e m o d i f i c a t i o n f o r the change i n p o s i t r o n energy, should be a p p l i e d to ,both the T r -e and y + -> e++ v + v peak. . The r a y t r a c i n g r e s u l t s , and most pr e v i o u s experience, i n d i c a t e s t h a t the energy dependence of the Nal l i n e s h a p e i s adequately accounted f o r by a d j u s t i n g the FWHM. : s i n c e the same >NaI l i n e s h a p e should be a p p l i e d t o both r e g i o n s , the d i s c r e p a n c y cf the TT -e e m p i r i c a l l i n e s h a p e and c a l c u l a t e d l i n e s h a p e suggests t h a t a s i m i l a r discrepancy i n the y + -> e++ v • v shape should be present*. The l a t t e r i s much l e s s pronounced than the fcrmer because the T r - y y - e shape i s smeared over a wide energy r e g i o n . Any reasonable source of i n f l u e n c e on the TT -e peak -- gain s h i f t s , odd sto p p i n g d i s t r i b u t i o n , uneven energy l o s s e s --should have an e n t i r e l y analogous e f f e c t on the y + -* e + + v + v shape. Once again, the disc r e p a n c y should matter l i t t l e i f the s p e c t r a can be shown to co n t a i n l i t t l e or no background, i n which case only a small t a i l c o r r e c t i o n would be necessary. . As we see frcm 117 f i g u r e III-6 and t a h l e A.G-2, a background cf 0.4 ± 0.5 % of the .y +• e++ v + v p o s i t r o n s i s r e q u i r e d t o get a reasonable f i t . (The l a r g e u n c e r t a i n t y here r e s u l t s frcm the u n c e r t a i n t i e s i n the e x t r a p o l a t i o n of the Nal lineshape.) Another i n d i c a t i o n t h a t the c a l c u l a t e d l i n e s h a p e dees not reproduce the e m p i r i c a l y + •+ e++ v + v l i n e s h a p e with complete s a t i s f a c t i o n i s the v a r i a t i o n i n the t a i l c o r r e c t e d sum f o r d i f f e r e n t e n e r g i e s . V a r y i n g the low energy c u t o f f between 3 and 20 MeV causes the t a i l c o r r e c t e d sum to change by 0.3 SL For c u t o f f s between 20 and 35 MeV, the sum changes by about 0.8 . Varying the r e s o l u t i o n parameters used f o r f o l d i n g the Nal li n e s h a p e i n t o the Michel s p e c t r a does not d r a m a t i c a l l y a l l e v i a t e these problems. Again the o r i g i n of these p o s s i b l e background counts i s both important and u n c e r t a i n . Runtime t e s t s d e s c r i b e d i n s e c t i o n II-E-5 e s t a b l i s h e d the p o s s i b i l i t y o f a low energy background s t r e t c h i n g perhaps up t c 50 MeV. But these same t e s t s seemed t o i n d i c a t e t h a t t h i s background had been e l i m i n a t e d by the standard geometry used during the run. As p r e v i o u s l y mentioned, the shape of the M i c h e l s p e c t r a i s very s i m i l a r i n B1, B2 and B3.. Once a g a i n , f i t t i n g u n s h i f t e d runs s t r o n g l y i m p l i e s t h a t the background i s not an a r t i f a c t of the g a i n s h i f t i n g procedure. Other t e s t s on the data do not y i e l d any 118 i n f o r m a t i o n on the source of these "background" counts. For example, a run a c q u i r e d d u r i n g a very low p i o n stop r a t e (15 K/s vs. 75 K/s) produced a M i c h e l s p e c t r a t h a t r e q u i r e d an i d e n t i c a l amount of "background" f o r a good f i t . Nor i s t h i s improved by changing from "mixed" a c c i d e n t a l s s u b t r a c t i o n s to e n t i r e l y f l a t c r e x p o n e n t i a l s u b t r a c t i o n s . although the r a y t r a c i n g l i n e s h a p e does net f i t the U + ->. e ++ v + \J spectrum i n an e n t i r e l y s a t i s f a c t o r y manner, the d i s c r e p a n c i e s seem c l e a r l y r e l a t e d to the u n c e r t a i n t i e s present i n the v a r i a t i o n of the Nal l i n e s h a p e with energy. The presence of a background beneath the y +• -* e + + v + v shape i s not s t r o n g l y implied* I I I - B - 3 - b . F u r t h e r C o n s i d e r a t i o n Of Backgrounds The r e s u l t s given i n the p r e v i o u s s e c t i o n i n d i c a t e the major problems with the present stage of the a n a l y s i s of the TT -e data. although the l i n e s h a p e s f o r the mono-energetic T r-e peak was c a l c u l a t e d with g r e a t care, i t d i s a g r e e s with the e m p i r i c a l l i n e s h a p e by s e v e r a l percent* The c r u c i a l question i s whether the excess counts i n the T r-e peak that cannot be accounted f o r with the r a y t r a c i n g l i n e s h a p e are genuine TT -e events t h a t 119 must be i n c l u d e d i n the branching r a t i o c a l c u l a t i o n or, i n s t e a d , are some kind of background events t h a t should be dis c a r d e d . an analogous que s t i o n can be posed f c r the y+ -> e + + v + v r e g i o n and y e t , even i f both d i s c r e p a n c i e s are background r e l a t e d , they cannot r e s u l t frcm the same source. R e c a l l t h a t the y + -»- e++v + v> shape i s very s i m i l a r i n b i n s one, two and th r e e (to the order of 0.1 % ) . T h i s i m p l i e s t h a t the "background" must have a time d i s t r i b u t i o n s i m i l a r t o t h a t f o r muon decay, a p o s s i b l e source f o r t h i s background, as d i s c u s s e d i n s e c t i o n I I - E - 5 , i s y + e++ v + v decays t h a t produce a shower frcm the Nal f r o n t s h i e l d , and d e p o s i t o n l y a s m a l l amount o f energy i n the c r y s t a l . Runtime experiments e s t a b l i s h e d the p o s s i b i l i t y o f t h i s type of background., although f u r t h e r t e s t s i n d i c a t e d t h a t i t had been e l i m i n a t e d to the order of 0. 1 %» , On the other hand, i f a "background" i s present beneath the TT -e peak, i t must have a time d i s t r i b u t i o n s i m i l a r to pion decay — i t must be prompt* . Were t h i s not t r u e , these spurious counts would have been removed by the Michel shape s u b t r a c t i o n of B2, which removes any high energy extension of the y + e++ v + v shape r e s u l t i n g from p i l e - u p or from the i n t r i n s i c Nal l i n e s h a p e . . Thus, any p o s s i b l e c o n t r i b u t i o n of prompt background processes must be c l o s e l y examined. We know that the presence of a high-energy background i s a p o s s i b i l i t y . In f a c t , the high energy background was 120 r o u t i n e l y measured to determine the time of the p i ^ s t o p — tO (see s e c t i o n I I - E - 4 ) . T h i s background seemed g u i t e f l a t i n the r e g i o n 50-90 MeV, and perhaps beyond. I n the TT-e s p e c t r a , a f t e r a c c i d e n t a l s and p i l e - u p s u b t r a c t i o n s , the energy r e g i o n between 74 - 84 MeV i s e s s e n t i a l l y empty ( i . e . 9 counts vs. 38,000 TT.-e »s, or 0.02 %) . Although t h i s i n d i c a t e s t h a t the tO suppression -- that i s the approximately 6 ns i n t e r v a l a f t e r the p i - s t c p u n t i l the f i r s t time b i n was opened — was very e f f e c t i v e i n removing the "prompts", the p o s s i b i l i t y of seme time dependence to the background energy s p e c t r a must be f u r t h e r i n v e s t i g a t e d . A prompt background beneath the TT -e peak c o u l d be i n t r o d u c e d i n two ways: 1) d i r e c t l y , where the background would a r i s e from charged p a r t i c l e s with t i m i n g and energy s i m i l a r t o TT -e events; or 2) i n d i r e c t l y , where the background event p i l e s up with a genuine y + e ++ v + v event. Since any prompt background event i s almost c e r t a i n l y , r e l a t e d to the incoming pion beam, the genuine y + -> e + + v + v event must be a c c i d e n t a l . These types c f background are compared i n t a b l e I I I - 2 . . Note t h a t care must be taken when c a l c u l a t i n g c r o s s s e c t i o n s f o r an i n d i r e c t background.. Although they enter the spectrum i n a p i l e - u p f a s h i o n , these processes are c o r r e l a t e d i n time with a l l the events c o n t r i b u t i n g t o E1 -- a l l promptly f o l l o w a p i - s t o p . . A simple method f o r c a l c u l a t i n g r a t e s of time c o r r e l a t e d p i l e - u p events i s to 121 TYPE OF BACKGROUND TYPE OF TIME ENERGY CROSS PARTICLE DISTRIBUTION DISTRIBUTION SECTION (MB) DIRECT INDIRECT CHARGED 6 NS AFTER Tn (50MEV<) (LOOM U E<74MEV CHARGED/ ANY PROMPT n / r s n r M \i on NEUTRAL (INCLUDING TQ) 0<E<25nEV 2 0 . TABLE I I I - 2 D i r e c t And I n d i r e c t 7 1 -e Backgrounds. c a l c u l a t e the p r o b a b i l i t y f o r a given process t o c o n t r i b u t e t o the Nal s p e c t r a f o r each p i - s t o p . T h i s p r o b a b i l i t y can be a p p l i e d to any c o l l e c t i o n of p i - s t o p s , and i n p a r t i c u l a r , t o the stops which generate bins f o r which an a c c i d e n t a l c c i n c i d e n c e f o r B1 c c c u r s . Thus, knowing the rate of a c c i d e n t a l s i n B1 allows immediate c a l c u l a t i o n o f the i n d i r e c t background r a t e . The i n d i r e c t c r o s s s e c t i o n s presented were c a l c u l a t e d t o account f o r about 3 % of the TT -e events, which i s the number of counts i n question. T y p i c a l values f o r a s i n g l e run are 122 u + -> e ++ v + v events i n B1 , = 50,OOO/hr a c c i d e n t a l y * •+ e++ v events i n B1 = 13,20 0/hr TT -e events i n E1 = 500/hr "background" = 15/hr cl e a n p i - s t o p s = 300,000,000/hr raw p i - s t o p s = 375,000,000/hr We see from t a b l e I I I - 2 , t h a t f o r a d i r e c t process, cnl y a very small c r o s s s e c t i o n of 0,.4 micrc-barns i s re g u i r e d t o account f o r the suspected background. However, the charge, energy and time requirements place severe c o n s t r a i n t s on these processes. A l t e r n a t i v e l y , the i n d i r e c t processes are much l e s s c o n s t r a i n e d , but the c r o s s s e c t i o n r e g u i r e d i s 5,000 times l a r g e r . C a l c u l a t i o n s f o r a v a r i e t y of p o s s i b l e background sources were performed. The r e s u l t s are now presented.,, 123 E l a s t i c s c a t t e r i n g TT + , TT + The incoming pions have o n l y 30 MeV k i n e t i c energy, and are suppressed by the f a c t t h a t b i n cne does not open f o r f i v e ns f o l l o w i n g the p i - s t o p . , Thus, t h i s type of background e n t e r s i n d i r e c t l y . . The c r o s s s e c t i o n f o r TT + s c a t t e r i n g near 90 degree i s o = 5 mb/sr rr I f a l l pions reached the N a l , the background r a t e would be : Rate = 10-3 x (13,200 a c c i d e n t a l u + ->• e+* v + v events/hr) Rate = 13/hr However, most of the s c a t t e r e d pions w i l l have l o s t too much energy t o reach the N a l . . Taking t h i s i n t o account, the most generous estimate f o r s c a t t e r e d pions e n t e r i n g the c r y s t a l i s then about 3/hr, a f a c t o r of 5 too small. A more r e a l i s t i c e stimate i s l e s s than 1/hr, Charge Exchange TT+,TT0 f o l l o w e d by TT0,2Y 124 Neutral p a r t i c l e s , thus an i n d i r e c t hack ground,. a =0.02 mb/sr Prob = (0.02X10 - Z 7 cmVsr) (2*0.167*4 sr) (1.5 cm) (4.6X1022 atoms/cm*) Prob = 0.6x10 -6 Background r a t e = 0*1/hr The c r o s s s e c t i o n i s much too s m a l l . E l a s t i c S c a t t e r i n g of p o s i t r o n s i n the beam A s t r a i g h t f o r w a r d c a l c u l a t i o n y i e l d s a (90 degree ) = 2 x 1 0 - 3 2 c m z = 0.02 micro barns The c r o s s s e c t i o n i s f a r too s m a l l to be seen d i r e c t l y cr i n d i r e c t l y . Nuclear Reaction Tr+p,px E x t r a p o l a t i n g from recent experimental d a t a , we estimate = 0.1 + 0.1 mb/sr f o r production of proton o f energy 70 < E (proton) < 80 MeV which, a f t e r enegy l o s s , r e s u l t s i n protons of 57 < E (protons) < 66 MeV. For a d i r e c t r e a c t i o n , i . e . when t o was v a r i e d t o i n c l u d e prompt p a r t i c l e s , the high energy background detected c o u l d a r i s e v i a a process with a c r o s s 125 s e c t i o n of about 40 micro barns/sr, Howe and time dependence c f the proton s p e c t r a well enough to draw any r e a l c o n c l u s i o n s d i s c u s s i o n below). N e u t r a l Backgrounds: TTr + ,nX and other beam r e l a t e d neutrons Very l i t t l e was known r e g a r d i n g e f f e c t i v e c r o s s s e c t i o n f o r these r e a c t i o n s * . ver, the energy were not known (see f u r t h e r To f u r t h e r examine gu e s t i o n s r e l a t i n g to p o s s i b l e backgrounds, s e v e r a l t e s t s were done during a run of the % - e vy expe riment i n the summer of 1978. . The experimental set-up f o r t h i s was n e a r l y i d e n t i c a l to that of the Tr-e experiment-. Data was recorded event by event using a PDP 11/40 computer. "Subsequent a n a l y s i s was done by Dr. M. S. . D i x i t and Dr* J . M. , P o u t i s s o u . Protons were i d e n t i f i e d by examining the amount of energy d e p o s i t e d i n a p l a s t i c s c i n t i l l a t i o n counter, The prompt proton energy s p e c t r a (within a window of 16 ns about the stop) and the delayed proton energy s p e c t r a are given i n f i g u r e I I I - 7 . The delayed spectrum appears to be the leakage of M i c h e l - e l e c t r o n s . , The prompt s p e c t r a i s r e l a t i v e l y f l a t from the d i s c r i m i n a t o r c u t o f f near 30 MeV to the ADC c u t o f f near 85 MeV:* A ten MeV b i n above 55 MeV 126 y i e l d s c =0.11 mb/sr T h i s s p e c t r a i s strong evidence t h a t the prompt to high energy background i n the TT -e experiment was indeed protons. I t a l s o i n d i c a t e s t h a t a d i r e c t background of protons i s not a l i k e l y e x p l a n a t i o n f o r our TT -e background, s i n c e the r e g i o n above 74 MeV i n the TT -e s p e c t r a i s very near zero. Note, however, that the s p e c t r a g i v e s no i n f o r m a t i o n on the c r o s s s e c t i o n below 30 MeV. We cannot, then, e n t i r e l y r u l e out the p o s s i b i l i t y t h a t the TT +, px c r o s s section' r a p i d l y i n c r e a s e s below 30 t MeV and i n t e g r a t e d between 0 and 25 MeV i s about 20 m i l l i barns. I f t h i s i s the case, these protons could account f o r our TT -e background i n d i r e c t l y , by p i l i n g up with a c c i d e n t a l P + e++ v + v decays* Tests were a l s o done to i n s p e c t the n e u t r a l background. Prompt and delayed spectra are given i n £igure I I I - 8 . We see a r a p i d l y f a l l i n g s p e c t r a above the d i s c r i m i n a t o r s c u t o f f near 10 MeV. . The delayed time window i s about 21 ns, the prompt i s about 7 ns,. For purposes of TT -e , a l l these events are p-rcmpt, and the t o t a l c r o s s s e c t i o n above 10 MeV i s o* = 1 mb Although t h i s energy d i s t r i b u t i o n could account f o r the background i n the TT -e s p e c t r a by p i l e - u p with a c c i d e n t a l FIGURE 111-7 P R O M P T A N D D E L A Y E D " P R O T O N " S P E C T R A 1 2 7 COUNTS 32.0 H 2 * 4 . 0 1 6 . 0 H 8 . 0 O .0 C X 1 0 * ) I 4- • ^ • • ^ 4 -1 1 I 1 1 1 1 — 0.00 10.00 2D.00 30.00 40.00 50.00 50.00 70.00 E N E R G Y ( A ) P R O M P T P R O T O N S P E C T R U M COUNTS ( X 1 0 1 ) 1 6 . 0 H 1 2 . 0 H B.O 4 . 0 H 0 . 0 O . 00 n—1 r 10. OC 20.00 30.00 40.00 50.00 60.00 70.00 E N E R G Y ( B ) D E L A Y E D " P R O T O N " S P E C T R U M FIGURE 111-8 PROMPT AND DELAYED NEUTRAL SPECTRA COUNTS ( X 1 0 2 ) 1 28 9.6 H 7 . 2 H 4 . 8 H 2 . 4 H Q.O — i i , , 1 1 r O.DO 1D.OO 2D.DO 30.00 40 . 00 50.00 BO.00 70.OO E N E R G Y (A) PROMPT NEUTRAL SPECTRUM COUNTS 3 2 . 0 H 2 4 . 0 1 6 . 0 8 . 0 H O . 0 O . 0 0 i 1 r 10.00 20.00 30.00 40.OD 50.00 60.00 E N E R G Y 70 .00 (B) DELAYED NEUTRAL SPECTRUM 129 ^ + -»- e+ + v + v ' s, the c r o s s s e c t i o n i s a f a c t o r of 20 too small* Note, however ^ t h a t the r a p i d l y i n c r e a s i n g c r o s s s e c t i o n helow 10 MeV might be s u f f i c i e n t t o account f o r the prompt background. III-B-3-c. . A p p l i c a t i o n Of The TT -e Lineshape To The Data As d i s c u s s e d i n the previous s e c t i o n s , the r a y t r a c i n g l i n e s h a p e does not f u l l y account f o r the lew energy t a i l seen i n the TT -e mono-energetic peak, . However, c a l c u l a t i o n s and measurements r e l a t e d to p o s s i b l e background sources have not yet s u c c e s f u l l y e x p l a i n e d these d i s c r e p a n c i e s . An a l t e r n a t i v e approach to a n a l y s i s of the data i s to accept the TT -e l i n e s h a p e (rather than the r a y t r a c i n g lineshape) as the genuine moho-energetic Nal l i n e s h a p e — a f t e r r a d i a t i v e e f f e c t s have been taken i n t o account* C a l c u l a t i o n s of the e f f e c t s of the more obvious i f a c t o r s which might account f o r the d i s c r e p a n c y — g a i n s h i f t s and/or unusual stopping d i s t r i b u t i o n s f o r the incoming p i o n s — i n d i c a t e t h a t these f a c t o r s could not s u f f i c i e n t l y i n c r e a s e the low energy t a i l . . One can i n v e n t a number of other e x p l a n a t i o n s which, although l e s s 130 l i k e l y , are not completely i m p l a u s i b l e . , For example, cne of the photo-tubes viewing the Nal may have f a i l e d i n t e r m i t t e n t l y — p e r h a p s f o r o n l y one second per minute. .. This might be caused by seme marginal c a b l e connection* s o l d e r i n g j o i n t , e t c . A l o s s of 1/7 of the energy s i g n a l — w e l l smeared out by v a r i a t i o n s i n the entrance angle of the p o s i t r o n and thus of the r e s u l t i n g shower—-would s h i f t genuine TT -e e v e n t s . i n t o the t a i l r e g i o n i n j u s t the r i g h t manner. / Suggestions of t h i s type i n v a r i a b l y seem to be easy to hypothesize but i m p o s s i b l e i ' t c v e r i f y . F i g u r e I I I - 9 shows the Tr-e monoenergetic peak f i t with i t s e l f a f t e r smoothing ( f i t 1017.04). The f i t i s q u i t e good, and r e q u i r e s l i t t l e background, as expected.„ F i g u r e 111-10 shows a f i t t h a t a p p l i e s the TT -e mono-energetic l i n e s h a p e , a f t e r TT -e r a d i a t i v e e f e c t s have been taken i n t o account, t o the U + ->-e++v + v energy spectrum. A l a s , we do not f i n d here the c o n f i r m a t i o n that the TT -e l i n e s h a p e t r u l y r e p r e s e n t s the Nal mono-energetic lineshape* , Whereas the r a y t r a c i n g lineshape had perhaps too l i t t l e low energy t a i l , the 7 7 -e l i n e s h a p e apparently has too much t a i l and r e q u i r e s a n e g a t i v e background of n e a r l y 8 %. The c a l c u l a t e d U + + e + + v + v shape here has a v a r i a t i o n i n the t a i l , c o r r e c t e d sum of 2.7 % between 3 and 35 MeV. Thus the T - e l i n e s h a p e i s much l e s s s u c c e s s f u l i n f i t t i n g the y+ +e ++ v + v shape than the r a y t r a c i n g l i n e s h a p e . FIGURE 111-9 1 3 1 II - e LINESHAPE APPLIED TO THE IJ-e PEAK, FIT 1017.04 96.0 72.0 48.0 24.0 -0.0 , COUNTS CX101 ) + ORTR FIT 0 BACKGROUND LINESHAPE = PI-E BoeBoogoammaiagBPirBiffliDapoBooo qooaeeoQoaooooopoooaoo^ l 55.0 60.0 65.0 70.0 ENERGY COUNTS 80.0 40.0 0.0 -40.0 J -80.0 • + • • • + •% * \ + + —K I T i - f t i * * «.+55.0 60.0* T f * r - * f T T , , . 65.0 7Q-+0 + • * • ENERGY • •* + + DIFFERENCE = DATA - LINE BRCKGROUND FIGURE I11-10 II-e LINESHAPE APPLIED TO THE MICHEL SPECTRUM 400.0 300.0-J 200.0-4 100.0-J COUNTS (X102 ) + DRTR FIT 0 BRCKGROUND LS=PI-E-NU. RES=8.BZ 0.0 ENERGY B O . O COUNTS (X101 J 0.0 -60. OH -120.©J -JBO.W -240.0] 1 5 . 0 30.0 45 ENERGY + DIFF=DRTR-LINE FITTED BRCKGROUND B O . O 133 Of course, f o l d i n g the mono-energetic Nal l i n e s h a p e i i n t o the c a l c u l a t e d M i c h e l spectrum encompasses s e v e r a l parameters t h a t a re not e s p e c i a l l y w e l l d e f i n e d . Although the c a l c u l a t i o n o f the r a d i a t i v e c o r r e c t i o n s f o r the y+ -»• e++ v + v spectrum seems q u i t e f i r m , the. piocess used to f o l d i n the mono-energetic l i n e s h a p e i s somewhat open to qu e s t i o n . The r a y t r a c i n g data i n d i c a t e d the t a i l cf the mono-energetic l i n e s h a p e does not change d r a m a t i c a l l y as a f u n c t i o n of p o s i t r o n energy* The v a r i a t i o n i n FWHM with energy i s not w e l l known, though l i m i t s are i n d i c a t e d . The e f f e c t of the stopping d i s t r i b u t i o n on the e f f e c t i v e Nal r e s o l u t i o n can be r e a d i l y c a l c u l a t e d . (See s e c t i o n I I - A - 3 ) . Allowing these r e s p e c t i v e parameters to vary by more than reasonable amounts (from n=0.5, RSD=1* 1%, n=1. 5 t o n=0.3, RSD = 0.5?5, m=0.5 — see appendix F) does decrease the v a r i a t i o n i n the t a i l c o r r e c t e d sum to about 1.2 % (from 3.6 %) f o r c u t o f f s between 3 and 35 MeV at the expense of worsening the l o c a l x 2 i n the region o f the high energy s l o p e frcm 0.5 t o 3.3. Thus, generous adjustments i n hew the mono-energetic l i n e s h a p e i s f o l d e d i n t o the M i c h e l spectrum do not d r a m a t i c a l l y improve the agreement between the e m p i r i c a l p + + e++ v + v l i n e s h a p e and the c a l c u l a t e d r e s u l t u s i n g the TT -<E l i n e s h a p e . A negative background beneath the Mi c h e l spectrum i m p l i e s a s i g n i f i c a n t energy dependent e f f i c i e n c y of some sort* I n t e r e s t i n g l y , the previous experiment of Di Capua 134 (Di63) notes the l o s s of some events near t h e i r low energy c u t o f f , about 10 MeV. DiCapua a t t r i b u t e s t h i s to an energy dependent e f f i c i e n c y of t h e i r • e l e c t r o n t e l e s c o p e . T h i s , however, seems most u n l i k e l y , s i n c e the dE/dx of p o s i t r o n s i n the r e l a t i v i s t i c energy r e g i o n between 20 and 50 MeV changes by very l i t t l e * , The e f f e c t s of m u l t i p l e s c a t t e r i n g and p o s i t r o n a n n i h i l a t i o n are much too s m a l l to account f o r such and energy dependent e f f i c i e n c y (see f o l l o w i n g s e c t i o n ) . Other a l t e r n a t i v e s f o r an energy dependent e f f i c i e n c y i n our experimental s e t up seem very sparse. Thus, at t h i s present stage, our a n a l y s i s of the l i n eshape f o r the TT -e data i s l a r g e l y u n s u c c e s s f u l . The most r e a d i l y a v a i l a b l e c o n s i s t e n c y check — s i m u ltaneously f i t t i n g the T r-e and the u •+ ->!e++ v + v shapes with the same mono-energetic l i n e s h a p e — p o i n t s out a g l a r i n g i n c o n s i s t e n c y i n our c u r r e n t understanding cf the TT -e data. . 13 I I I - C . Other C o r r e c t i o n s And U n c e r t a i n t i e s I I I - C - 1 . Muon Leakage The q u a n t i t y of i n t e r e s t i n t h i s experiment i s the r a t i o o f the number of pions decaying v i a Tr-e vs. The number decaying by the TT-U -e' chain* As such, a c o r r e c t i o n must be a p p l i e d f o r the number cf muons which escape from the t a r g e t area (and from our acceptance) before decaying. The 4 MeV muons produced by pion decay can t r a v e r s e approximately 1.5 mm of CH2 , or s e v e r a l c e n timeters i n a i r * We must r e l y on c a l c u l a t i o n s using known parameters d e s c r i b i n g the t a r g e t c ounters and the incoming beam to determine what percentage of muon have "leaked" away before decaying,.. As p r e v i o u s l y d e s c r i b e d , the incoming pions are i d e n t i f i e d by two counters, degraded by CH , c c l l i m a t e d , and stopped i n a t a r g e t counter. The t a r g e t counter i s sandwiched betweeen a downstream veto counter and an upstream counter with the primary purpose of minimizing leakage. The s i g n a t u r e f o r a p i - s t o p — S1«S2»S3«S4«S5 -r e g u i r i n g t h a t the pion come to a h a l t i n the t a r g e t counter (S4) or i t s downstream wrapping, or the upstream wrapping of counter S5. In order f o r the decay muon to 136 escape (that i s , to t r a v e r s e l e s s than 1.5 mm of C H 2 ) , the parent pion must stop very near the surface of the counters* Thus, most of the escapes come from a p i - s t o p i n the t a r g e t wrapping. The t h i c k n e s s of the t a r g e t counter and the wrapping, the depth d i s t r i b u t i o n of the stopping pions, and the g e o m e t r i c a l arrangement of the sandwich c o u n t e r s , a r e the most important f u n c t i o n s a f f e c t i n g the muon leakage. U n f o r t u n a t e l y , because of the very t i g h t packing of the Nal c r y s t a l s and s c i n t i l l a t i o n counters r e g u i r e d f o r the y -> e + y experiment, the counters were mounted i n a r a t h e r ad-hoc f a s h i o n . Although the counter p o s i t i o n s were measured s e v e r a l times, they were not s t u r d i l y h e l d i n p l a c e . Thus we must allow a generous u n c e r t a i n t y i n the p o s i t i o n s of the s c i n t i l l a t i o n counters., T h i s , i n t u r n , induces s i g n i f i c a n t u n c e r t a i n t i e s i n the number of muon t h a t escaped d e t e c t i o n * The c a l c u l a t e d c o r r e c t i o n f o r muon leakage to the branching r a t i o i s (-0*15 ± 0.15) %. 137 I I I - C - 2 . M u l t i p l e S c a t t e r i n g Decay p o s i t r o n s from the t a r g e t counters must t r a v e r s e the p o s i t r o n t e l e s c o p e before e n t e r i n g the Nal c r y s t a l . Thus, p o s i t r o n s can m u l t i p l e - s c a t t e r both out c f and i n t o our acceptance* Since the e l e c t r o n s c a t t e r i n g angle i s a f u n c t i o n c f p o s i t r o n energy, the number of p o s i t r o n s s i g n i f i c a n t l y s c a t t e r e d i s energy dependent. M u l t i p l e s c a t t e r i n g t h a t occurs i n the t a r g e t c o u n t e r s i s i s o t r o p i c and has no e f f e c t * . Likewise, s c a t t e r i n g i n the TV counters cannot d e f l e c t p o s i t r o n s s u f f i c i e n t l y f c r them to miss the Nal. Thus, the o n l y m u l t i p l e s c a t t e r i n g t h a t can have a s i g n i f i c a n t e f f e c t occurs i n the s c i n t i l l a t i o n counters S7 and S11. . M o l i e r e ' s theory of m u l t i p l e s c a t t e r i n g g i v e s r e s u l t s i n c l o s e s t agreement with experiment (Mo48). The s c a t t e r i n g d i s t r i b u t i o n resembles a normal curve, but with a long non-gaussian t a i l . The d i s t r i b u t i o n used i n t h i s a n a l y s i s was taken from G a l b r a i t h (Ga65). A monte-carlo s i m u l a t i o n was performed t o c a l c u l a t e the e f f e c t s of m u l t i p l e s c a t t e r i n g at v a r i o u s e n e r g i e s . D i f f i c u l t i e s concerning t h i s c a l c u l a t i o n are almost e n t i r e l y r e l a t e d to a c c u r a c y . : For the vast m a j o r i t y of p o s i t r o n t r a j e c t o r i e s , the e f f e c t s of m u l t i p l e s c a t t e r i n g are not l a r g e eneough to d e f l e c t the p a r t i c l e i n t o or out of the Nal acceptance. At the same time, the number of 138 t r a j e c t o r i e s t h a t are d e f l e c t e d s u f f i c i e n t l y are almost i n v a r i a b l y those d e f l e c t e d by aricmously l a r g e a n g l e s . Thus, r e s u l t s are c r i t i c a l l y dependent on good s t a t i s t i c a l knowledge concerning the long non-gaussian t a i l . The r e s u l t i s t h a t the e f f e c t Of m u l t i p l e s c a t t e r i n g weighted by the P + •+ e ++ v + v d i s t r i b u t i o n , i n t e g r a t e d from 0 t o 50 MeV, i s +0.19 ± 0.05 %. The e f f e c t on the TT -e events i s +0.15 ± 0.04 %m Thus, the e f f e c t on the branching r a t i o i s -0.04 ± 0.09 5L III-C-3. Bremsstrahlung The e f f e c t o f the r a d i a t i o n energy l c s s . c n the Nal. lin e s h a p e was d i s c u s s e d i n s e c t i o n s I I I - A - 1 , I I I - B and I I I - C . The decay p o s i t r o n s can r a d i a t e away some p o r t i o n of t h e i r energy while t r a v e r s i n g the p o s i t r o n t e l e s c o p e . T h i s can c o n c e i v a b l y induce an energy dependent d i s t o r t i o n i n t he p o s i t r o n energy s p e c t r a i n the N a l . However, T h i s e f f e c t i s minimized, because the Nal c r y s t a l d e t e c t s these hard photons i f they e n t e r the c r y s t a l . Thus, we need only be concerned with determining how many of the bremsstrahlung photons miss the Nal c r y s t a l . A rough c a l c u l a t i o n was performed, using a bremsstrahlung angular d i s t r i b u t i o n formula from Jackson (Ja62). . The percentage of photons s c a t t e r i n g at angles 139 g r e a t e r than s e v e r a l values was c a l c u l a t e d f o r s e v e r a l d i f f e r e n t p o s i t r o n e n e r g i e s . The h a l f angle f o r the cone of our acceptance was about 36 degrees* For a rough estimate, we then examined photons with a s c a t t e r i n g angle > 18 degrees* Although at 2 MeV, about 7 % are s c a t t e r e d by more than 18 degrees, only 0.06 S& of 5 MeV p o s i t r o n s have a l a r g e r s c a t t e r i n g angle. . Ey 30 MeV, t h i s drops t o 3 x 1 0 - 9 , by 50 MeV, to 8 X 1 0 - H . Thus we conclude t h a t the e f f e c t of bremsstrahlung energy l o s s cn the TT -e branching r a t i o i s e n t i r e l y n e g l i g i b l e . I I I - I : - 4 . P o s i t r o n A n n i h i l a t i o n Decay p o s i t r o n s can a n n i h i l a t e i n f l i g h t before they reach the Nal c r y s t a l . Again, t h i s can r e s u l t i n an energy dependent e f f i c i e n c y f o r our d e t e c t i o n system, but once again, the e f f e c t i s minimized because the Nal can detec t the r e s u l t i n g photons. However, i f a n n i h i l a t i o n occurs before the p o s i t r o n s i g n a t u r e i s generated, the energy w i l l not be recorded i n the measured spectrum. The c r o s s s e c t i o n f o r p o s i t r o n a n n i h i l a t i o n i n f l i g h t i s g iven by H e i t l e r (He5U)* The number cf p o s i t r o n s a n n i h i l a t e d a t the r e l e v a n t e n e r g i e s , f o l d e d i n t o the i energy d i s t r i b u t i o n f o r u + e ++ v + v and ir-ev were r e a d i l y c a l c u l a t e d , We f i n d t h a t p o s i t r o n 140 a n n i h i l a t i o n reduces the number of V + + e++ v + v events by 0.65 %, and the number of T -e v events by 0.36 X. . Thus, the e f f e c t on the branching r a t i o cf p o s i t r o n a n n i h i l a t i o n i s -0.29 ± 0.05 JL . am Chapter IV RESULTS A. C a l c u l a t i o n Of The, branching, r a t i o The 7 7 -e branching r a t i o i s c a l c u l a t e d using an ingenious formula given by DiCapua e t . a l . . ( D i 6 3 ) and r e d e r i v e d i n appendix A. The v a l i d i t y of t h i s formula i s based on the f o l l o w i n g : 1) The f a c t t h a t time bins one and two are of e g u a l d u r a t i o n . 2) The f a c t t h a t the energy s p e c t r a of b i n s one and two ( a f t e r c o r r e c t i o n f o r a c c i d e n t a l coincidences) r e s u l t from decay p o s i t r o n s of the same s e t of stopping p a r t i c l e s . .. To then c a l c u l a t e the branching r a t i o , we need to know: 1) The l i f e t i m e s of the pion and mucn 2) The time s e p a r a t i o n o f b i n s one and two 3) The number of TT -ev events i n bin one 4 ) the number of U + e ++ v + v events i n bin one and bin two As d e s c r i b e d i n the previous s e c t i o n s , the major d i f f i c u l t y i n the a n a l y s i s of t h i s data i s a c c u r a t e l y determining the a b s o l u t e numbers f o r the TT -e and 142 T T-y -e decay c h a i n s . In f a c t , at p r e s e n t , we cannot c l a i m to have accomplished t h i s with complete s a t i s f a c t i o n . Severe d i s c r e p a n c i e s e x i s t between c a l c u l a t e d and e m p i r i c a l 7 7 -e l i n e s h a p e s . I f we accept the r a y t r a c i n g l i n e s h a p e as a c c u r a t e l y d e s c r i b i n g the Nal response, we must a l s o accept a background of between 3 and 4 % cf the t o t a l number of TT -e events i h the r e g i o n between the TT -e and TT - y-e shapes. . iWe have not yet i d e n t i f i e d a source of t h i s background* A background of about 0.4% beneath the TT - y -e shape i s i m p l i e d but net r e g u i r e d . Even with t h i s u n c e r t a i n t y , however, I b e l i e v e the weight of the a n a l y s i s performed to date c l e a r l y i n d i c a t e s t h a t t h i s i n t e r p r e t a t i o n i s c o r r e c t . F u r t h e r i n v e s t i g a t i o n s cf the source of the background are now being done.. As shown i n t a b l e IV-1, the a n a l y s i s u t i l i z i n g the r a y t r a c i n g l i n e s h a p e y i e l d s a branching r a t i o of 1.212x10 -*. A l t e r n a t i v e l y , i f we accept the Tr-e l i n e s h a p e at face value, we f i n d t h a t t h i s cannot be s u c c e s s f u l l y used to f i t the TT - y -e shape unl e s s we allow a n e gative background of about 8 %. T h i s a n a l y s i s would then y i e l d a branching r a t i o of 1. 150x10-*,. A t h i r d o p t i o n i s t o accept the r e s u l t s . c f the runtime t e s t s f o r the i r - e v and TT-evy experiments, and of the subsequent background c a l c u l a t i o n s . These a l l i n d i c a t e t h a t no l a r g e backgrounds are present anywhere. We can them sum the TT - y -e shape down to 3 MeV and apply 143 a s m a l l t a i l c o r r e c t i o n , and a l s o sum the TT -e peak do«n to 52.5 MeV and a p p l y a s i m i l a r l y s m a l l c o r r e c t i o n . T h i s a n a l y s i s would y i e l d a branching r a t i o of 1.246x10-*. Table IV-1 g i v e s the r e s u l t s of the branching r a t i o c a l c u l a t i o n f o r the r a y t r a c i n g a n a l y s i s . The s y s t e m a t i c c o r r e c t i o n s are d i s c u s s e d i n the f o l l o w i n g s e c t i o n . . The s t a t i s t i c a l u n c e r t a i n t i e s i n c l u d e the e f f e c t s of a c c i d e n t a l and background s u b t r a c t i o n s . These s t a t i s t i c s are based only on the 40 % of the TT - e data t h a t has been g a i n s h i f t e d and analyzed to date. The r a y t r a c i n g l i n e s h a p e r e s u l t s are d e r i v e d from f i t 315.03. The u n c e r t a i n t y i n the TT-e lineshape r e s u l t s from an u n c e r t a i n t y of 0.2% i n the FWHM and of an u n c e r t a i n t y of 1.0 channel i n the p o s i t i o n * The r a y t r a c i n g l i n e s h a p e does not f i t the T r-e shape w e l l , e i t h e r i n the peak or the t a i l r e g i o n s . The u n c e r t a i n t y i n the TT - y -e l i n e s h a p e i s d e r i v e d from an u n c e r t a i n t y i n the Nal i n t r i n s i c r e s o l u t i o n near 50 MeV, and an u n c e r t a i n t y i n the v a r i a t i o n of the e f f e c t i v e Nal r e s o l u t i o n with energy. We then see t h a t the a n a l y s i s of the TT -e data u t i l i z i n g the TT -e l i n e s h a p e y i e l d s a branching r a t i o t h a t i s 5. 1 % l e s s than the r e s u l t when the r a y t r a c i n g l i n e s h a p e i s used. T h i s r e s u l t , i n t u r n , i s 2.8 % below the branching r a t i o a r r i v e d at i f no backgrounds are assumed to be present. TABLE IV-1 CALCULATION OF THE Tr-ev BRANCHING RATIO FOR VARIOUS BELIEFS CONDITIONS N y e N y e N f f e B.R. B.R. STATISTICAL LINESHAPE B l B2 B l W I T H UNCERTAIN s UNCERTAIN'S CORR. Y E a ) ^ ^ o ) ^ a ) ^ a ) RAYTRACING LINESHAPE 3 . ^ 1 8 x l 0 6 6 . 4 9 M 0 6 36623 1.2182 1.2117 0.16 0.7 0 . 4 0 . 3 xlCT 4 x l ( T 4 145 IV-B. Systematic C o r r e c t i o n s and U n c e r t a i n t i e s Systematic c o r r e c t i o n s and u n c e r t a i n t i e s have been discus s e d i n some d e t a i l throughout t h i s document. . A summary of these c o r r e c t i o n s and u n c e r t a i n t i e s i s g i v e n i n t a b l e IV-2. The s e c t i o n t h a t d i s c u s s e d each source of c o r r e c t i o n and u n c e r t a i n t y i s given. . The sources are f o r the most part s e l f - e x p l a n a t o r y . Ihe b i n widths and b i n p o s i t i o n u n c e r t a i n t i e s were a s c e r t a i n e d from run-time t e s t s * The r e s t r e s u l t from s t r a i g h t f o r w a r d c a l c u l a t i o n s ^ or monte-carlo s i m u l a t i o n s . " A c c i d e n t a l s s u b t r a c t i o n " r e f e r s t o the u n c e r t a i n t y i n the r a t i o of f l a t to e x p o n e n t i a l l y decaying a c c i d e n t a l backgrounds. The s t a t i s t i c a l e f f e c t of these s u b t r a c t i o n s i s i n c l u d e d i n the s t a t i s t i c a l e r r o r s . The " s u b t r a c t i o n of B2 f"-e events" r e f e r s to the f a c t t h a t , while doing a p i l e - u p s u b t r a c t i o n of the B2 u + -* e ++ v • v shape frcm E1, a s m a l l number of t r u e T r-e events present i n B2 i s a l s o s u b t r a c t e d from the Tr-e peak. The u n c e r t a i n t i e s have been added l i n e a r l y , p r i m a r i l y to compare with the r e s u l t s of DiCapua, whc handled h i s s y s t e m a t i c s i n t h i s f a s h i o n . . The sum i n quadrature i s also given. 1 4 6 T A B L E I V - 2 SYSTEMATIC CORRECTIONS TO THE ^ r - e v BRANCHING RATIO REFERENCE SECTION CORRECTION UNCERTAINTY BIN WIDTHS I I - D - 1 0 + 0 . 1 BIN POSITIONS I I - D - 1 0 + 0 . 0 2 ACCIDENTALS SUBTRACTION I I - D - 2 0 ± 0 . 0 3 STOP P.U. REJECTION I I - D - 3 + 0 . 1 ± 0 . 1 SCINTILLATION COUNTER EFF. I I - B - 2 0 0 MUON LEAKAGE I I I - C - 1 - 0 . 1 5 ± 0 . 1 5 MULTIPLE SCATT I I I - C - 2 - 0 . 0 4 ± 0 . 0 9 e + ANNIHIL. I I I - C - 4 - 0 . 2 9 ± 0 . 0 5 BREMSSTRAHLUNG I I I - C - 3 I I I - B - 2 - c - 0 . 2 ± 0 . 0 2 ENERGY CALIBRATION I I - E - 5 0 ± 0 . 2 * -LIFETIME 0 ± 0 . 1 V -LIFETIME 0 0 BIN 2 ff-e SUBTRACTION I I - D - 3 + 0 . 0 5 0 TOTAL - 0 . 5 3 ± 0 . 8 6 (IN QUAD.) ± 0 . 3 2 147 IV-C. Semi-Final R e s u l t s The r e s u l t s of the 1 1 -e data a n a l y s i s i n i t s present stage are given i n t a b l e IV-3, together with the most widely accepted t h e o r e t i c a l p r e d i c t i o n . The s t a t i s t i c a l u n c e r t a i n t i e s were combined l i n e a r l y , as were the lineshape u n c e r t a i n t i e s . These were then combined with the s y s t e m a t i c u n c e r t a i n t i e s i n quadrature* R e s u l t s f o r the case of adding the systematic u n c e r t a i n t i e s i n quadrature are a l s o given, The experimental r e s u l t i s then s l i g h t l y more than one standard d e v i a t i o n l e s s than the t h e o r e t i c a l p r e d i c t i o n . IV-D. F u r t h e r Comments In summary, arguments supp o r t i n g each of the suggested modes of data a n a l y s i s are: 1) B a y t r a c i n g Lineshape - - The a n a l y s i s of the r a y t r a c i n g experiment y i e l d s Nal l i n e s h a p e s t h a t do not change s i g n i f i c a n t l y with energy or entrance angle, and a l s o agree well with the 130 MeV TT p,n¥ l i n e s h a p e . . T h i s l i n e s h a p e can be used t o reproduce the TT - u. -e l i n e s h a p e with TABLE IV-3 PRESENT RESULTS FOR THE *-ev BRANCHING RATIO BRANCHING RATIO B.R./THEORY - 1 (xlO~ 4 ) (%) THEORY RAYTRACING LINESHAPE 1.233 1.212 + .017 -1.70 + 1.4 ( ± .014) ( + 1 . 2 ) UNCERTAINTIES IN PARENTHESES HAVE BEEN COMBINED IN QUADRATURE. 149 r e s o n a b l e success* The presence of a s m a l l prompt background i s a reasonable p o s s i b i l i t y . 2) The -rr -e L i n e s h a p e . — The mono-energetic TT —e peak r e p r e s e n t s , i n the absence cf backgrounds, the a c t u a l response of the Nal c r y s t a l during the 7 7-e experiment* I f , i n f a c t , no background i s present, the r a y t r a c i n g l i n eshape i s perhaps not a p p l i c a b l e , because of the balance of the Nal tubes or some marginal response i n the e l e c t r o n i c s system, or some other reason unknown. The 7 7 -e monc-energetic l i n e s h a p e should then be used t o analyze the TT -e data* 3) No Backgrounds — T h e o r e t i c a l c a l c u l a t i o n s and runtime t e s t s d u r i n g both the TT -ev and TT -e vy experiments have not yet y i e l d e d an obvious source of the background. I f no background i s present, the s p e c t r a should be summed over the u s e f u l r e g i o n of data a c q u i s i t i o n , and c o r r e c t e d f o r the s m a l l t a i l s of the l i n e s h a p e s t h a t are out s i d e the summinq re g i o n . . Note t h a t the three types of a n a l y s i s presented are 150 not n e c e s s a r i l y l i m i t i n g cases. I f , f o r very obscure reasons, a p o s i t i v e background i s present beneath the T r - p . - e shape, and no background i s present beneath the Tr-e peak, the branching r a t i o could i n c r e a s e by s e v e r a l percent. another, perhaps more l i k e l y p o s s i b i l i t y , i s that, i f no backgrounds are present anywhere, the sum f o r the TT - y - e should be extended down t o 0 MeV to i n c l u d e the low energy peak. T h i s would reduce the no background r e s u l t by 0.5 - 1V.0%, b r i n g i n g i t i n t o b e t t e r agreement with theory. although the r a y t r a c i n g a n a l y s i s seems by f a r the most probable, we f e e l t h a t the source o f t h i s prompt background must be i d e n t i f i e d , and i t s shape c a l c u l a t e d c r measured before we can c a l l these r e s u l t s c o n c l u s i v e . In t h i s case, the l i n e s h a p e u n c e r t a i n t i e s c c u l d be somewhat reduced, as would the s t a t i s t i c a l u n c e r t a i n t i e s when the e n t i r e body of data i s analyzed. Perhaps some f u r t h e r r e d u c t i o n i n the sy s t e m a t i c u n c e r t a i n t i e s i s p o s s i b l e , The o v e r a l l e r r o r s c o u l d then be reduced to l e s s than 1.0 Our o v e r a l l e r r o r s , then, w i l l only be about a f a c t o r of two s m a l l e r than the pre v i o u s best r e s u l t s of DiCapua, even though we have ten times the s t a t i s t i c s of t h a t e a r l i e r experiment and the q u a l i t y of our data i s much improved, as can bee seen i n f i g u r e IV-1. I b e l i e v e the small decrease i n the s t a t e d e r r o r i s the r e s u l t of more s t r i n g e n t and comprehensive a n a l y s i s of the energy 151 FIGURE IV-1 COMPARISON OF ENERGY SPECTRA FROM THE CURRENT EXPERIMENT WITH THAT OF DICAPUA, ET. AL. (Dl63) COUNTS CX10 3 ) T i i i i i i ' — O.O 75.0 150.0 225.0 300.0 375.0 450.0 525.0 CHANNEL (A) COMPARISON OF PROMPT ENERGY SPECTRA COUNTS CXlQg ) H T i 1 1 1 r i ~i i i 1 1 1 1 — 300.0 32:5.0 350.0 375.0 400.0 425.0 450.0 475.0 CHRNNEL (B) * - e REGION OF THE PROMPT ENERGY SPECTRA 152 s p e c t r a . Many o f the g l a r i n g problems i n the a n a l y s i s c f the present data would be f a r more d i f f i c u l t t o i d e n t i f y with a 10 MeV energy c u t o f f and an energy r e s o l u t i o n of 153 Chapter V COMPARISON WITH THEORY AND IMPLICATIONS A t h e o r e t i c a l c a l c u l a t i o n o f the branching r a t i o (TT -»-e +V ) + (TT ->-e +V +Y) R = e e , + + . - + + (TT -*u + C 7 "*"u + V y + Y ) i s s t r a i g h t f o r w a r d , u t i l i z i n g the techniques of quantum f i e l d t h eory. Okun (Ok66) presents an easy to f o l l c w e x p o s i t i o n of t h i s c a l c u l a t i o n - The V-A theory of weak i n t e r a c t i o n s y i e l d s a p r e c i s e p r e d i c t i o n f o r the branching r a t i o i f 1) The strong i n t e r a c t i o n matrix elements are the same f o r bcth the 7 1 - e and TT - y - e decays. 2) Muon-electron u n i v e r s a l i t y holds t r u e . Ordinary phase space arguments y i e l d a value of R=3.5 However, h e l i c i t y s e l e c t i o n r u l e s (or maximal p a r i t y v i o l a t i o n ) reduces t h i s by a f a c t o r o f (mass of electron/mass of muon) 2, which then y i e l d s 154 R = CM e m - m - nr 1. 278X10-* T h i s number must now be c o r r e c t e d f o r r a d i a t i v e processes, and h e r e i n a r i s e s s e v e r a l long standing problems. When e l e c t r o - m a g n e t i c c o u p l i n g i s added to t i e i n t e r a c t i o n Lagrangian and p e r t u b a t i o n t h e c r y a p p l i e d , s e v e r a l i n t e g r a l s are found to be d i v e r g e n t . As with the r a d i a t i v e c o r r e c t i o n s to muon decay p r e v i o u s l y d i s c u s s e d , i n f r a r e d d ivergences a r i s e , r e l a t e d t o zero-energy photons i n the v i r t u a l c o r r e c t i o n s , which are c a n c e l l e d by s i m i l a r terms i n the inner-bremsstrahlung c o r r e c t i o n s . However, f o r pion decay, as with n u c l e a r beta decay, the v i r t u a l r a d i a t i v e c o r r e c t i o n s a l s o g i v e r i s e t o i n t e g r a l s which d i v e r g e i n the u l t r a - v i o l e t (DV) l i m i t . In the past, t h i s problem was viewed as a r i s i n g from the treatment of the composite s t r o n g l y i n t e r a c t i n g p a r t i c l e s as p o i n t p a r t i c l e s . The hope was t h a t when the s t r u c t u r e of the s t r o n g i n t e r a c t i o n p a r t i c l e s was b e t t e r kncwn, a more exact treatment would remove the DV divergences. In the i n t e r i m , a g e n e r a l l y accepted method of handling these d i v e r g e n t terms was t o int r o d u c e some u l t r a - v i o l e t c u t o f f , u s u a l l y p r o p o r t i o n a l t o the square of the mass of the decay p a r t i c l e . When c a l c u l a t i n g r a d i a t i v e c o r r e c t i o n s to the Tr-e branching r a t i o using 155 t h i s scheme, terms p r o p o r t i o n a l t o t h i s DV c u t o f f (A ) were found to c a n c e l i f the c u t o f f was taken to he the same f o r both fr -e and TT - y decay c h a i n s . . (See Ka68, Ki59a, Ki59b, B160, Be58, and So72.). This y i e l d s E(theo.) = 1.233 X 10-* (A / A = 1) 1 e ' y ' Note, however, t h a t i f the 0V c u t o f f s f o r TT -e and TT- y decay are i n s t e a d chosen to be i n the r a t i o of the r e s p e c t i v e p a r t i c l e masses, the branching r a t i o becomes (Er75) E(theo.) = 1.2 58 X 10-* ^e/Ay= m / mp ) Eecent advances i n the a p p l i c a t i o n of gauge t h e o r i e s to weak i n t e r a c t i o n s appear to have a l l e v i a t e d the need f o r t h i s a r b i t r a r y c u t o f f . Gauge t h e o r i e s c f weak i n t e r a c t i o n s e x p l o i t a n a l o g i e s to the f o r m u l a t i o n o f ele c t r o - m a g n e t i c i n t e r a c t i o n s , but with s e v e r a l c o m p l i c a t i o n s : 156 1) Weak c u r r e n t s are net conserved, as opposed to charged currents;. 2) Weak i n t e r a c t i o n s are c o n f i n e d to s m a l l d i s t a n c e s . 3) The i n t e r m e d i a t e v e c t o r boson (IVE) has charged s t a t e s . Over the l a s t 20 years, problems with weak i n t e r a c t i o n gauge t h e o r i e s have been s o l v e d . Spontaneous symmetry breaking allowed the i n t r o d u c t i o n of massive bosons i n t o the t h e o r y , while the Higgs mechanism removed the n e c e s s i t y f o r massless bosons.. The proof by Van Hoof't of the r e n o r m a l i z i b i l i t y of gauge t h e o r i e s i n 1971, and the l a t e r evidence f o r n e u t r a l c u r r e n t s , made the gauge t h e o r i e s even more a t t r a c t i v e . , Weinberg had a l r e a d y used gauge t h e o r i e s t c p o s t u l a t e n e u t r a l IVB's. . With f u r t h e r suggestions of p o s s i b l e v i o l a t i o n s of l e p t o n number c o n s e r v a t i o n , gauge t h e o r i e s of weak i n t e r a c t i o n s have a t t r a c t e d a great d e a l of a t t e n t i o n by both t h e o r i s t s and e x p e r i m e n t a l i s t s . Returning to our present endeavor, gauge t h e o r i e s seem t o have produced a more f i r m p r e d i c t i o n f o r the Tr-e branching r a t i o . Goldman and Wilson (Go75) showed t h a t , i n a gauge t h e o r y , the r e n o r m a l i z a t i o n c o n s t a n t s are not independent and, i n t h e i r model, y i e l d once again a branching r a t i o R (theo.) = 1.233 X 10-* 157 The model of Goldman and Wilson, however, s t i l l a l l o w s the branching r a t i o to be modified by s t r o n g i n t e r a c t i o n c o r r e c t i o n s -- i . e . the s t r u c t u r e of the pion. More r e c e n t l y , Marciano and S i r l i n (Ma77) c l a i m to have demonstrated t h a t the l e p t o n mass s i n g u l a r i t y f o r pion decay does not depend on s t r o n g i n t e r a c t i o n e f f e c t s , and thus can be r i g o r o u s l y computed.. T h e i r r e s u l t , a g a i n , agrees with the e a r l i e r p r e d i c t i o n B (theo.) = 1. 233 X 10-* T h i s , then, i m p l i e s t h a t a p r e c i s e measurement o f the T T - e / T r - u branching r a t i o i s indeed a measure of mucn-electron u n i v e r s a l i t y , and not a probe of the pion s t r u c t u r e , which makes an accurate d e t e r m i n a t i o n even more d e s i r a b l e . 158 Chapter VI CONCLOSIONS Our measurement of the branching r a t i o TT+ev + TT+evy  R = TT+yv + Tf+yvy was very s u c c e s s f u l i n many ways. The t a r g e t of 1 0 s Tr-e events was achieved. The q u a l i t y o f the data was uniformly good over the e n t i r e s e v e r a l months of the run. The s p e c t r a are r e l a t i v e l y c l e a n , and the Mic h e l shape seen i n the Nal c r y s t a l has been recorded e s s e n t i a l l y down to zero energy. B a d i a t i v e c o r r e c t i o n s t o both TT -e and the Vi-e decay have been c a l c u l a t e d and found t o be s m a l l . Other c o r r e c t i o n s to the hi anchii g r a t i o have been c a l c u l a t e d and found to be s m a l l . The weight of the a n a l y s i s i n d i c a t e s the presence cf background counts i n the region between the T r-e and the TT - y - e shapes. The area of t h i s background i s 3 cr 4 % of the t o t a l area of the Tr-e peak. The elements of the a n a l y s i s i n f a v o r of t h i s c o n c l u s i o n a r e : 1) The c h a r a c t e r of the Nal li n e s h a p e dees not change d r a m a t i c a l l y (by more than t e n t h s of a percent) over the r e l e v a n t p a r t i c l e e n e r g i e s and entrance an g l e s . Thus, adjustments to the FWHM are alone s u f f i c i e n t t o produce r e p r e s e n t a t i v e Nal l i n e s h a p e s . T h i s i s supported by comparison o f the l i n e s h a p e s from the v a r i o u s r a y t r a c i n g runs and from the 15 (p,n) experiment. The T r-e l i n e s h a p e d i s a g r e e s with these others by 3-4 % of the t o t a l , i n the energy r e g i o n below the peak. 2) The Nal l i n e s h a p e can be a p p l i e d to f i t t i n g the TT - u -e l i n e s h a p e with l i t t l e or no background. The T r-e l i n e s h a p e r e q u i r e s a background of -8 %. 3) An e n t i r e l y p l a u s i b l e source c f the background events beneath the prompt T r-e peak i s some prompt beam r e l a t e d process. 4) A somewhat weaker point -- a kncwn d i f f e r e n c e between the s p e c t r a f o r bins one and two (besides the obvious TT -e peak) e x i s t s --the d i s c r e p a n c y between the areas and e n e r g i e s of the low energy peak. Elements of the a n a l y s i s t h a t do not f a v o r t h i s c o n c l u s i o n a r e : 1) The source of the background events has not been determined. 2) No background e x i s t s i n the energy r e g i o n higher than the TT -e peak, to the order of 0.01%. Based on the present a n a l y s i s , u s i n g the r a y t r a c i n g l i n e s h a p e and a l l o w i n g a 3 or 4% background beneath the Tr-e peak, our r e s u l t f o r the ir-ev branching r a t i o i s : B = (1.217 ± 0.017)x10-* I h i s number i s 1.3 ± 1.4 % below a f i r m t h e o r e t i c a l p r e d i c t i o n . T h i s u n c e r t a i n t y i s not d r a m a t i c a l l y b e t t e r than DiCapua e t . a l . ' s s t a t e d e r r o r of 2*2 % (Di63) . Our 160 r e s u l t i s three standard d e v i a t i o n s below t h a t of DiCapua. A n a l y s i s of our TT -e data c o n t i n u e s . C u r r e n t l y , emphasis i s being placed on more comprehensive a n a l y s i s o f : 1) the prompt n e u t r a l background measured i n the rr -e vy experiment, 2) babbha s c a t t e r i n g , and 3) p o s s i b l e TH-yvY p i l e - u p c o r r e c t i o n s . The f u t u r e i s wrapped i n mystery. 161 Appendix A PERIvaTION OF THE BRANCHING EaTIO F O R M O L a . r I g n o r i n g r a d i a t i v e c o r r e c t i o n s , which are e s s e n t i a l l y m o d i f i c a t i o n s to the l i n e s h a p e s , we wish to determine R = Tf->-ev T T + e v y TT->y v Tr+y v y where the muons decay f u r t h e r v i a y + -»• e ++v + v . T h i s r a t i o i s of the or d e r 10-*, To prevent the mono-energetic 7 7-e p o s i t r o n peak from being l o s t i n a background f r o n T T - y - e decays (range 0.511 - 52.8 MeV) caused by p i l e - u p and the r e s o l u t i o n f u n c t i o n o f the d e t e c t o r , we i n s p e c t f o r decays during a time i n t e r v a l of roughly cne pion l i f e t i m e . To prevent prompt s c a t t e r e d p a r t i c l e s from e n t e r i n g the s p e c t r a as background, the time i n t e r v a l must begin s e v e r a l nano-seconds a f t e r the p i - s t c p . The number of Tr-e p o s i t r o n s measured i h t h i s i n t e r v a l i s then 162 N = R N Q. e " X l T T 0 ( 1 - e X * T ) ( A D TTe TT where E = the branching r a t i o ir-ev / T T-yv = the number of p i - s t o p s = the s o l i d angle X^ = the i n v e r s e pion l i f e t i m e tO = the time delay a f t e r the p i - s t o p before the f i r s t time b i n commences T = d u r a t i o n of the time b i n Many of these q u a n t i t i e s are d i f f i c u l t t c measure a c c u r a t e l y — we should e l i m i n a t e as many as p o s s i b l e fxcm the branching r a t i o r a t i o formula. We can immediately e l i m i n a t e N °- by u t i l i z i n g the number of TT - y-e p o s i t r o n s measured i n t h i s f i r s t time b i n . 163 B 1 = N (B 1 ) TT-y-e X X = N n ^ ^ { ( i + f ) - ^ -TT X - X A . 1 1 V - X _ T „ Y - X t y s - X T 1-e V ) - X T ( 1 - e 1 7 ) } ( A 2 W h e r e X^ = the i n v e r s e muon l i f e t i m e f = the muon contamination of the ream Equation A-2 i s d e r i v e d by c o n s i d e r i n g the number of pions which decay t c muons i n the time i n t e r v a l d t ' about t ' , and f u r t h e r , the number of these muons t h a t decay to p o s i t r o n s between t and t+dt, and then i n t e g r a t i n g over t i e r e l e v a n t time i n t e r v a l s . Using equations &1 and A2, we can c a l c u l a t e E independent of N ^ , but we have now i n t r o d u c e d a dependence on muon contamination, and E i s s t i l l dependent on tO and T i n a f a i r l y s e n s i t i v e way. These time q u a n t i t i e s are very d i f f i c u l t t o measure a c c u r a t e l y : tO depends upon when the pion i s s a i d t o have stepped, as determined from the s i g n a l from the s c i n t i l l a t i o n counter. Ihe width T ± S i n f a c t determined by a c o i n c i d e n c e between a b i n generated by the p i - s t o p and a s i g n a l from a 164 p o s i t i o n e n t e r i n g a d e t e c t o r , and thus i s very dependent cn the p a r t i c u l a r response of a c o i n c i d e n c e box. DiCapua (Di63) proposed an i n g e n i o u s method f o r av o i d i n g these messy time dependences. CiCapua measured the number of p o s i t r o n s from TT- U -e decay i n a second time b i n commencing s e v e r a l pion l i f e t i m e s a f t e r the p i - s t o p , but with the same d u r a t i o n . T h i s second b i n c o n t a i n s very few TT -e decays, but the number of TT - y - e events, i n analogy to equation A2, i s given by B 2 E N , s T T - y - e ( B 2 ) A A -A ( t + t ) , 0 TT y y s O - A T V V A Y - A ( t +t ) , . (A3) TT s 0 , , - A T ) , e ( 1 - e TT ) y A TT where t s = the time s e p a r a t i o n of b i n one and b i n two. A t M u l t i p l y i n g equation A3 by e ^ s and then s u b t r a c t i n g equation A2 y i e l d s : , A t - A t - A T : V 5 B 2 - B 1 = [ N i r f l e * ° ( 1-e " )] x ) (A..-X_) t_ (A4) JL VA -A V U TT ( 1 - e V 7 1 S ) 165 We see the q u a n t i t y i n square b r a c k e t s i s i d e n t i c a l to a group of terms i n equation At. We can s o l v e f o r t h i s g u a n t i t y , s u b s t i t u t e i n equation A1, and then s o l v e f o r fi, y i e l d i n g N A t V - X , e » 8B2 -B1 * y - (X - X ) t * (1 - e V V 8 ) 16 Q u a n t i t a t i v e l y : Tv = V A p = 2 1 9 7 . 1 ± . 0 8 n s T TT = 1 / A T T = 2 6 . 0 3 ± . 0 2 n s tS = 170.2 ± 0.1 ns Then N R = 1 . 1 9 5 x 1 0 ~ 2 — ( A 5 ) [ 1 . 0 8 0 5 ( B 2 ) - B 1 ] 1 ' Eguation A5 i s q u i t e an a t t r a c t i v e l i t t l e s e t of numbers when compared with i t s parentage. The only time dependence i s a f a i r l y i n s e n s i t i v e one on the s e p a r a t i o n cf the two b i n s . A c t u a l l y , the q u a n t i t y tS i s more s u s c e p t i b l e t c measurement than e i t h e r tO cr x . A l s o , the f a c t o r c o n t a i n i n g the muon contamination of the beam has been s u b t r a c t e d away l i k e an i n f i n i t y . Equation A5, then, t o g e t h e r with the e x p e r i m e n t a l l y determined q u a n t i t i e s the number o f 7 r -e 's, B1, B2, and t S , and with e x i s t i n g data on pion and muon l i f e t i m e s y i e l d s our numerical r e s u l t s f o r the ir-e branching r a t i o . 167 Appendix B RUNTIME INFORMATION RECORDED FOR EACH RUN A form was generated s p e c i f y i n g i n f o r m a t i o n t c be recorded a t the end of each run. The i n f o r m a t i o n i n c l u d e d : 1) Eun number (s) 2) Experimenter 3) Date 4} Time 5) Times the run was w r i t t e n onto magnetic tape. ( T y p i c a l l y , a run l a s t e d three hours b e f o r e being erased, but was copied t o tape every hour.) 6) The nominal beam c u r r e n t on the production t a r g e t (T2) 7) The s e p a r a t i o n of the s l i t s on the M9 stopped pion-muon channel. 8) The r a t e of the Cerenkov counter (a measure of the a c t u a l proton c u r r e n t on the p r o d u c t i o n t a r g e t ) 9) The " c l e a n s t o p " r a t e 10) The magnetic tape number 11) The value of tOD In a d d i t i o n , sums over v a r i o u s energy r e g i o n s f o r each of the f o u r b i n s were recorded, along with the "window" s i z e , i n terms c f channels of the a n a l y z e r . These r e g i o n s i n c l u d e d : 168 1) The low energy peak (-2 to +2 MeV) 2) The TT - y - e shape (2 t o 54 MeV) 3) The v a l l e y (54 to 56 MeV) 4) The T r - e peak (56 to 75 MeV) 5) The high energy r e g i o n (75 to 90 MeV) 6) The TT - y - e re g i o n o f fewest counts (2 to 4 MeV) 7) the TT - y - e r e g i o n of g r e a t e s t counts (42 to 44 MeV) 8) The TT - y - e c a l i b r a t i o n p o i n t f c r b i n three (approximately 1/3 from the base on the high energy slope of the TT - y -e ) i 9) The T r-e peak channel i n b i n one. The form s p e c i f i e d a number of r e s u l t s t c be c a l c u l a t e d from t h i s i n f o r m a t i o n . S p e c i f i c a l l y : 1) (# of counts i n the low energy peak)/ (# of counts i n TT - y -e ) f o r B3 2) (# i n v a l l e y ) / ( # i n TT - e ) f o r E1 3) (# i n the TT - y -e r e g i o n of g r e a t e s t counts)/(# i n the TT - y - e r e g i o n of fewest counts) 4) .(# i n the TT -e r e g i o n of B l ) / ( # i n the T r-e r e g i o n of B2) 5) (# i n the EE region) / (# i n the TT. - y - e region) f o r B1 and B2 Other i n f o r m a t i o n was recorded only when changes were made. This i n f o r m a t i o n i n c l u d e d : 169 1) Current s e t t i n g s on the magnets cn beamline M9 ( e s s e n t i a l l y the focus of the pion beam) 2) Voltages on the Nal p h o t o - m u l t i p l i e r s 3) The production t a r g e t used. A BASIC computer program, w r i t t e n by Br. M.S. D i x i t , read and p r i n t e d out a number of CAMAC s c a l e r s . The s c a l e r s were gated u s i n g the ADC Busy s i g n a l from the a n a l y z e r , and so r e l a t e d to each measured s p e c t r a . The s c a l e r s measured: 1) The # o f times the Cerenkov counter t r i g g e r e d 2) The # of c o i n c i d e n c e s between counters SI and S2 3) The # of S1»S2»S3»sa 4) The # of raw stops 5) The # of c l e a n stops ( a f t e r blanking) 6) The # of c o i n c i d e n c e s f o r each of the f o u r bins 7) The # of Bin OS's 8) The # of TIN A Gates 9) The # o f TV (TINA Veto) p i l e - u p s r e j e c t e d from each of the f o u r bins 10) The # of stop p i l e - u p events r e j e c t e d from each of the f o u r b i n s 170 A l s o , the computer program requested seme of the above recorded i n f o r m a t i o n from which i t c a l c u l a t e d : 1) The # o f TT - y - e events i n B1 and E2 a f t e r ( i n c o r r e c t ) a c c i d e n t a l s s u b t r a c t i o n s (B1* and E2*) 2) The e f f e c t i v e # of T T-e ,'s a f t e r TT — y —e shape s u b t r a c t i o n 3) The branching r a t i o and s t a t i s t i c a l u n c e r t a i n t y 4) The e f f e c t i v e # of T r-e /B1 * 5) E1«/B2« 6) t of ir - y - e "s i n B3/# of TT - y -e ' s i n B2 7) # o f TT - y - e • s i n B4/# of TT - y -e «s i n B3 ( a c c i d e n t a l s rate) 171 Appendix C ANALYSIS OF THE DATA FROM THE RAYTRACING EXPERIMENT To a s c e r t a i n the response of the Nal to p o s i t r o n s c f va r i o u s energies, we performed, i n March of 1977, a r a y t r a c i n g experiment using p o s i t r o n s from the M20 muon channel at TRIDMF, and using the "Chalk E i v e r " magnet (which has a gap of 5.5 i n c h e s , a pole f a c e 16 inches i n diameter, and a maximum f i e l d o f 5.7 k i l o g a u s s ) to improve upon the momentum d i s p e r s i o n of the beam. The experimental set-up f o r the r a y - t r a c i n g runs i s shown i n f i g u r e A.C-1. The M20 heamline at TRIUMF, although designed as a stopped muon ch a n n e l , has s u r p r i s i n g l y good o p t i c s f o r p o s i t r o n s . (Re are g r e a t l y indebted t o firs. D. Fleming , J . Brewer, and K. . Nagamine and the the whole msr group f o r making the M20 beamline a v a i l a b l e t c us at a very busy time i n t h e i r schedule.) P a r t i c l e s e x i t i n g from the beamline were detected by a p l a s t i c s c i n t i l l a t i o n counter, and were then t r a c k e d along t h e i r t r a j e c t o r i e s using f o u r Multi-Wire P r o p o r t i o n a l Counters (MWFC's). They were then detected by another p l a s t i c s c i n t i l l a t i o n counter. (The same counter used as S11 d u r i n g the T r-e run, i n the same p o s i t i o n r e l a t i v e t o the Nal._ The s t e e l s h i e l d i n g i n f r o n t of the Nal used d u r i n g the TT -e run sas a l s o i n place during the r a y t r a c i n g runs.) The p a r t i c l e s then -172 FIGURE A.C-1 E X P E R I M E N T A L S E T - U P F O R T H E R A Y T R A C I N G R U N S R E G I O N O F F I E L D S U R V E Y R E G I O N F O R W H I C H F I E L D ; WAS R A D I A L L Y E X T E N D E D ; S2 — MWPC1 • MWPC2 MWPC3 ^ C H A L K R I V E R M A G N E T MWPM . / X S l l 173 d e p o s i t e d t h e i r e n e r g i e s i n the Nal c r y s t a l . The MWPC's have been d e s c r i b e d i n d e t a i l elsewhere (Er77). P r i o r t o the experiment, the magnet, wire chambers, and the Nal were c a r e f u l l y a l i g n e d by the TBIDMF alignment group. The p a r t i c l e beam i n t o the Nal under "wide open" c o n d i t i o n s c o n s i s t e d of T r / y / e ' s i n the r a t i c 0.04/1/66 as determined by the time of f l i g h t (TOF).. "Accepted e v e n t s " r e q u i r e d a c o i n c i d e n c e between counters S1 and S2 and a l l four of the MWPC's. The MWPC timing i n f o r m a t i o n was passed t o an LBS 222 8 CAMAC Time to D i g i t a l Converter (TDC), and s i m i l a r l y the Nal energy i i n f o r m a t i o n to a CAM AC Analog t o D i g i t a l Converter (ADC), and then s t o r e d on tape f o r each event. , To convert the MWPC t i m i n g i n f o r m a t i o n t o i n f o r m a t i o n cn the p a r t i c l e ' s p o s i t i o n r e q u i r e d knowledge o f the gain and c a l i b r a t i o n of the TDC s p e c t r a . The gain was s e l f determined, as the wire s e p a r a t i o n was e x c e l l e n t . , (See f i g u r e A.C-2). T y p i c a l l y , a r a y t r a c i n g run c o n s i s t e d of a f u l l tape c o n t a i n i n g roughly 900,000 events. On-line a n a l y s i s was l i m i t e d t o histogramming the MWEC time s p e c t r a and the Nal energy s p e c t r a , and the energy s p e c t r a t h a t r e s u l t e d when var i o u s c u t s were placed on the MWPC i n f o r m a t i o n f o r the event. B a y t r a c i n g runs were done f o r seven p o s i t r o n momenta between 20 and 80 MeV/c. In order t o a s c e r t a i n the e f f e c t FIGURE A.C-2 TYPICAL MULTI-WIRE PROPORTIONAL COUNTER SPECTRUM C O U N T S C X 1 0 3 ) 1 i 1 i T 1 1 I I ' l l I I I ' 1 1 1 1 1 11 i i n i i i 11 i i i T i I I i i i i i i i i i i 11 i i i 1 1 1 11 11 j 11 »i I I i 1 1 j i 11 i - j 16.0 -d • . 1 2 . 0 -g 8 . o -q 4 . 0 -d O . O I l l M n i i | i i i i n i i i | i i n n n i p i i i n i T T - ] r r i i T T I I I ] I t i i i T i 11 | I I l i I r i i r'| i 1 1 i 0 . 0 3 0 . 0 6 0 . 0 9 0 . 0 1 2 0 . 0 1 5 0 . D 1 8 0 . 0 2 1 0 . 0 CHRNNEL 175 of the f r o n t face i l l u m i n a t i o n of the Nal and o f the detected p a r t i c l e ' s angle c f e n t r y i n t o the N a l , the f o l l o w i n g two t e s t s were made. For one momentum (72.5 MeV/c) , the 17 . 5 cm. diameter counter i n f r c n t of the Nal c r y s t a l was r e p l a c e d by a 5 cm. diameter counter. For ttio momenta (49.7 and 72.5 MeV/c) , the Nal was r o t a t e d t o 20 degrees t o normal from the c e n t e r l i n e of the beam e n t r y (see f i g u r e A.C-3)., ( S i m i l a r data was recor d e d f o r the sm a l l e r Nal c r y s t a l cn s i t e , MINA.). T y p i c a l l y , cne hour •was r e q u i r e d to f i l l a tape when the beamline was set f cr 70 MeV/c p o s i t r o n s . T h i s time i n c r e a s e d t o 2.5 hours f o r 20 MeV/c p o s i t r o n s . The magnetic f i e l d of the chalk r i v e r magnet was v a r i e d between 1.1 and 4.2 k i l o g a u s s . Only t h r e e p o i n t s along the t r a j e c t o r y are r e q u i r e d to determine the momentum of the charged p a r t i c l e t r a v e r s i n g a magnetic f i e l d . The f o u r t h c c i n t (determined by the MWPC's) can be used to estimate the e f f e c t s c f m u l t i p l e s c a t t e r i n g , v e r t i c a l beam d i s p e r s i o n , and the f i n i t e r e s o l u t i o n of the MWPC's. A f t e r the experiment, a c a r e f u l survey o f the magnetic f i e l d produced by the chalk r i v e r magnet f c r s e v e r a l c u r r e n t s was performed by D. . Evans o f TRIUMF. . (The survey of the magnetic f i e l d d i d not e n t i r e l y cover the needed area. We l a t e r e x t r a p o l a t e d the survey of the magnetic f i e l d r a d i a l l y . ) A " r a y t r a c i n g " program ( o r i g i n a l l y ceded by Arthur Haynes of TRIOMF) was used to t r a c e t he p a r t i c l e s FIGURE A.C-3 EXPERIMENTAL SET-UP FOR THE RAYTRACING RUNS AT 20 DEGREE INCIDENCE 1 7 7 t r a j e c t o r i e s through a magnetic f i e l d using a Bunga-Kutta i n t e g r a t i o n method. T h i s program was adapted t o u t i l i z e the magnet survey and the p o s i t i o n i n f o r m a t i o n given by the MWPC's t o produce a "look-up" t a b l e . Thus the p o s i t r o n i n f o r m a t i o n from the MWPC's could be t r a n s l a t e d i n t o i n f o r m a t i o n on the momentum of the p a r t i c l e . Lookup t a b l e s were generated f o r momentum steps of 1.0% and 0.5%. E x i t t r a j e c t o r i e s were c a l c u l a t e d t c one tenth of a wire accuracy. These c a l c u l a t i o n s i n d i c a t e d that a change i n momentum of 1 % corresponded t o a displacement of 3.6 wires on MWPC3 , although t h i s v a r i e d with incoming d i r e c t i o n from 2.6 to 4.5 wires. S i m i l a r l y , the average f o r MWPC4 f o r 1% momentum b i t e s was 7.5 wire s , varying between 5.6 and 9.4 wires. A "lookup" program was then w r i t t e n to determine the p a r t i c l e ' s momentum frcm the p o s i t i o n i n f o r m a t i o n . Osing gain and c a l i b r a t i o n i n f o r m a t i o n s u p p l i e d , the program would determine which wire had been t r i g g e r e d on each of the MWPC's. I t would then a s c e r t a i n whether a t a b l e entry e x i s t e d f o r th a t "incoming d i r e c t i o n " - - a s determined by MWPC1 and MWPC2 ,. . I f an ent r y e x i s t e d , the program c o u l d then use MWPC3 or MWPC4 t o determine the momentum. The. discrepancy between the p r e d i c t e d and measured wire number f o r the a d d i t i o n a l MWPC was a l s o examined. F i g u r e A.C-4 shows a histogram f o r the (measured wire of MWPC3 ) - (p r e d i c t e d wire of MWPC3 ) wfien MWPC4 was used to determine the momentum. The average momentum of FIGURE A.C-4 MEASURED MINUS PREDICTED WIRE NUMBER FOR MWPC3 WIRE.NUM oo 179 the incoming p a r t i c l e s was 80 MeV/c. We see t h a t the c e n t r o i d of t h i s shape agrees very w e l l with the p r e d i c t e d value. The spread of the d i s t r i b u t i o n i s about 9.6 wires TWHM, or about 2.6% spread i n the momentum, . T h i s i s a r e s u l t of m u l t i p l e s c a t t e r i n g i n the wire chambers and the a i r , the v e r t i a l d i s p e r s i o n o f the beam, and also of the qu a n t i z a t i o n of the p o s i t r o n i n f o r m a t i o n i n t o an i n t e g r a l wire number. The corresponding curve r e s u l t i n g when the lookup t a b l e f o r a 0.5% mcmentum b i t e was used y i e l d e d a FWHM of about 1*8%. . The lookup program generated histograms of the energy seen i n the Nal f o r ten p a r t i c l e momentum re g i o n s d e f i n e d by t h e eleven momenta i n the lockup t a b l e . The c e n t r o i d of the Nal l i n e s h a p e i n c r e a s e d p r o p o r t i o n a l l y with the p a r t i c l e momenta, as expected. These histograms were then gain s h i f t e d and summed, y i e l d i n g a l i n e s h a p e with improved s t a t i s t i c s . During data a c q u i s i t i o n , severe gain s h i i t s occured i n the Nal energy s p e c t r a , again r e l a t e d t c problems i n the p h o t o - m u l t i p l i e r bases* To minimize the e f f e c t of these g a i n - s h i f t s , the data was analyzed i n s i x t e e n segments, and each segment was g a i n - s h i f t e d i n d i v i d u a l l y . , S i x t e e n was about the g r e a t e s t number t h a t c o u l d be used while the peak p o s i t i o n was s t i l l d e f i n e d with s u f f i c i e n t ac Tur acy. We a l s o decided to e l i m i n a t e p a r t i c l e s whose t r a j e c t o r i e s i n d i c a t e d severe m u l t i p l e s c a t t e r i n g . To do 180 t h i s , we accepted only those p a r t i c l e s whose absolute value of (measured wire - p r e d i c t e d wire) was l e s s than 3. (For MWPC3 , t h i s corresponded to +0.8% change i n momentum.) Table A.C-1 g i v e s the r e s u l t s of the a n a l y s i s cf the r a y t r a c i n g data f o r zero degrees i n c i d e n t angle on the Nal, for s i x average momenta. Given are 1) The t o t a l number of events on tape 2) The percentage of the t o t a l events e l i m i n a t e d due to a) MWPC1 out of range b) MWPC2 out of range c) MWPC4 out of range d) e x c e s s i v e m u l t i p l e s c a t t e r i n g 3) The percentage of t o t a l events histogrammed 4) The percentage o f "good events" a) f o r each of the momenta ranges b) f o r momenta out of range ( i . e . above or below the c a l c u l a t e d value f o r MWPC4 . These are not "good e v e n t s " ) . 5) The uncorrected r e s u l t s from a) FWHM of the Nal li n e s h a p e b) FWTM of the Nal l i n e s h a p e c) e x p o n e n t i a l f a c t o r n, de s c r i b e d below 6) The r e s u l t s c o r r e c t e d f o r m u l t i p l e s c a t t e r i n g f o r a) the FWHM b) the e x p o n e n t i a l f a c t o r n The e x p o n e n t i a l f a c t o r i s used t o parametrize the v a r i a t i o n i n r e s o l u t i o n (meaning FWHM) c f the Nal linehshape with energy as given i n the equation TABLE A . C - l 1 8 1 EXPERIMENTAL RESULTS OF RAYTRACING RUNS AT ZERO DEGREE INCIDENCE. (SEE TEXT FOR EXPLANATION.) MOMENTUM (MEV/C"' 21.5 32.4 49.7 61.3 72.5* 82.1 l/l MWPCl LO 1.5 1.7 1.5 1.7 1.8 1.6 Ui t -t -z " " HI 9.0 7.0 7.4 5.5 5.0 4.9 z UJ > Ul > MWPC2 LO 27.4 14.4 12.5 5.4 3.9 3.8 Ui _] " " HI 5.6 • 5.4 3.3 3.0 5.9 3.5 o Ul »-1 < MWPC4 OUT OF RANGE 19.9 18.8 16.8 13.0 19.2 13.3 (J UJ —} u-o EXCESSIVE 23.6 30.5 31.6 34.2 28.3 32.8 UJ CC u x MULTIPLE SCATT. GOOD EVENTS ( X OF ALL EVENTS) TOTAL NUMBER OF EVENTS ( x l 0 3 , 13.0 601 22.2 861 26.9 485 37.2 715 35.6 801 40.1 799 BINNED MOMENTUM D I S P E R S I O N : p<-5 50.7 42.4 37.6 35.7 43.2 28.5 -5<p<-4 3.7 4.0 2.6 3.0 4.8 4.6 -4<p<-3 5.3 5.4 4.1 4.9 7.0 7.2 -3<p<-2 6.2 7.1 6.6 7.6 11.1 9.8 -2<P<-1 7.6 9.9 9.9 11.0 16.7 14.2 -l<p< 0 10.6 12.0 13.4 15.1 19.7 18.3 0<p< 1 12.4 14.2 15.6 17.7 19.7 18.3 1<P< 2 13.7 16.0 16.7 16.8 13.4 14.3 2<P< 3 14.9 13.7 11.5 12.8 5.9 8.3 3<P< 4 13.8 10.3 10.8 7.6 1.5 3.5 4<P< 5 11.7 7.5 5.8 3.6 0.1 1.5 "5<P " 89.6 44.5 34.1 13.0 2.0 6.6 UNCORRECTED RESULTS: FWHM (Z) 14.9+.6 10.6+.3 8.7±.3 6.5+.2 6.9+.1 5.8+.1 FWTM (%) 37+4 25+2 21+2 15+1.5 18+.5 13.6+.5 F A C T O R ^ ° ' 7 ± ' 1 ° - 6 I ' 1 0 , 8 ± ' 2 0 A ± ' 1 L 4 ± ' 3 — RESULTS CORRECTED FOR MULT. SCATT. FWHM (2) 12.6+1.7 8.7+1.4 8.1+.5 6.0+.4 6.5+.4 5.4+.4 F A C T O R N T N A L ° ' 6 I ' 2 ° ' 5 I ' 3 0 ' 8 ± ' 7 0 A ± ' ' I 1 , 5 ± ' 5 ™ * THIS RUN USED A TWO INCH DIAMETER DEFINING COUNTER, INSTEAD OF THE NORMAL SEVEN INCH DIAMETER DEFINING COUNTER. THESE ARE NOT "GOOD EVENTS" 182 R(E) = E • R(E') These r e s u l t s are not e n t i r e l y s a t i s f y i n g . The uncorrected e x p o n e n t i a l f a c t o r n i s very poorly determined. C o r r e c t i n g f o r m u l t i p l e s c a t t e r i n g i n c r e a s e d the u n c e r t a i n t i e s , but doesn't improve the r e s u l t . We f e e l t h a t the l i k e l y cause of these d i f f i c u l t i e s l i e s i n the severe Nal g a i n s h i f t s which c o u l d not be completely compensated. However, t h i s data i s net e n t i r e l y devoid of i n t e r e s t . Although the g a i n - s h i f t s s e v e r e l y degrade the i n f o r m a t i o n concerning the Nal r e s o l u t i o n , the low energy t a i l r e g i o n of the l i n e s h a p e i s a f f e c t e d t o a much l e s s e r degree. Accurate knowledge cn the shape of t h i s t a i l i s r e q u i r e d to generate the shape f o r p o s i t r o n s from ]\ * -»• e + + v • v decay seen i n the N a l , and to c o r r e c t f o r the number of TT -e p o s i t r o n s l o s t beneath the TT -y -e shape. The Nal l i n e s h a p e f o r 61.3 MeV, zero degree i n c i d e n c e angle i s shown i n f i g u r e A.C-5. For comparison, a line s h a p e p r e v i o u s l y measured by the TINA group f o r gamma rays from the r e a c t i o n Tr~(p,n)Y i s g i v e n . These gamma rays are mono-energetic at 130 MeV,. T h i s l i n e s h a p e has been s h i f t e d , normalized, and the r e s o l u t i o n s t r e t c h e d to match the r a y t r a c i n g l i n e s h a p e . The agreement i s e x c e l l e n t , which lends some credence to be b e l i e f t h a t the r a y t r a c i n g data may be somewhat u s e f u l . R a y t r a c i n g l i n e s h a p e s f o r three p a r t i c l e momenta a t 183 zero degree i n c i d e n c e are given i n f i g u r e A.C-6. A f t e r allowance i s made f o r u n c e r t a i n t i e s caused by g a i n - s h i f t s and m u l t i p l e s c a t t e r i n g , we can say t h a t the v a r i a t i o n i n the Nal l i n e s h a p e with p o s i t r o n s energy i s net dramatic. I n i t i a l a n a l y s i s of the twenty degree r a y t r a c i n g data y i e l d e d l i n e s h a p e s with a long f l a t t a i l down t o zero energy. Examining the experimental set-up f o r the r a y t r a c i n g runs when the c r y s t r a l was r o t a t e d t o 20 degrees (see f i g u r e A.C-3), we see t h a t n e a r l y h a l f of MWPC4 should not be t r i g g e r e d f o r "good events" corresponding to s t r a i g h t l i n e t r a j e c t o r i e s , because counter S2 would not be f i r e d . . . D i v i d i n g MWPC4 i n t o g u a r t e r s and l a b e l i n g them as i n f i g u r e A.C-3, we see t h a t a l l of qu a r t e r 4 and most of quarter 3 o f MWPC4 should be ou t s i d e our acceptance. M u l t i p l e s c a t t e r i n g can all o w seme genuine "good e v e n t s " to i n c l u d e t h i s r e g i o n , but a possibe source o f background events i s these p a r t i c l e s impinging on the Nal s h i e l d and producing an e x t e r n a l shower which could t r i g g e r the s c i n t i l l a t i o n counter and dep o s i t some small b i t o f energy i n the c r y s t a l . These spu r i o u s events produced a t a i l i n the 20 degree r a y t r a c i n g data. T h i s t a i l was e l i m i n a t e d by r e j e c t i n g any. p a r t i c l e t r a j e c t o r i e s t h a t i n c l u d e d the t h i r d or fo u r t h q u a r t e r s o f MWPC4 f o r t h i s data.„ E e s u l t s f o r the 72.5 MeV, 20 degree i n c i d e n c e angle r a y t r a c i n g run are shown i n f i g u r e A*C-7. . The 20 degree data i s i n good agreement with the zero degree data f o r a FIGURE A . C - 5 NAI LINESHAPE DETERMINED BY THE ZERO DEGREE RAYTRACING RUNS COUNTS CX10 1 ) — RRYTRRCING DRTR 6 1 . 3 MEV. 0 DEGREES + N-GAMMR C O I N . 130 MEV 0 . 0 2 0 . 0 4 0 . 0 ENERGY 6 0 . 0 FIGURE A.C-6 C O U N T S C X 1 0 2 } 1 1 1 r I I I I 1 1 1—^nfHiTimj, 3 0 0 . 0 3 2 0 . 0 3 4 0 . 0 3 6 0 . 0 3 8 0 . 0 4 0 0 . 0 4 2 0 O 4 4 0 0 CHANNEL RAYTRACING LINESHAPES AT ZERO DEGREE INCIDENCE, FOR VARIOUS ENERGIES. THE GAIN, FWHM, AND INTEGRAL OF THE LINESHAPES HAVE BEEN NORMALIZED CD Ul FIGURE A.C-7 o . o 9.6 7 . 2 —\ 4 . 8 —\ 2 . 4 —\ COUNTS f — C X 1 0 2 ) T T RAYTRACING LINESHAPE AT TWENTY DEGREE INCIDENCE,, 72.5 MEV RAYTRACING LINESHAPE AT 0 TWENTY DEGREE INCIDENCE, 49.7 MEV RAYTRACING LINESHAPE AT + ZERO DEGREE INCIDENCE, 61.3 MEV 3 0 0 . 0 3 2 0 . 0 3 4 0 . 0 3 6 0 . 0 3 8 0 . 0 4 0 0 . 0 4 2 0 . CHRNNEL 4 4 0 . 0 RAYTRACING LINESHAPES AT TWENTY DEGREES INCIDENCE, FOR TWO ENERGIES. THE GAIN, FWHM, AND INTEGRAL OF THE LINESHAPES HAVE BEEN NORMALIZED. A 61.3 MEV LINESHAPE AT ZERO DEGREE INCIDENCE HAS BEEN GIVEN FOR COMPARISON. TO TABLE A.C-2 REVISED EXPERIMENTAL RESULTS OF RAYTRACING RUNS AT TWENTY DEGREES I N C I D E N C E . ( S E E TEXT FOR E X P L A N A T I O N . ) MOMENTUM (MEV/C) 49.7 72.5 MWPCl LO 1.6 1.4 9.0 6.9 MWPC2 LO 15.8 6.1 " " HI 3.6 25.5 MWPC4 OUT OF RANGE 11.1 11.0 E X C E S S I V E MULTIPLE SCATT. 37.8 11.5 in Z Q -I Ul < o u . u i o ~3 Ul •« OC GOOD EVENTS « OF A L L EVENTS) 21.1 34.6 TOTAL NUMBER OF EVENTS ( X l 0 3 ) 8 8 5 8 8 5 1 OF GOOD EVENTS I N EACH QUARTER: QUARTER 1 32.8 36.2 QUARTER 2 55.7 56.0 QUARTER 3 7.0 1.6 QUARTER 4 4.4 3.1 u i > a o o (9 o BINNED MOMENTUM D I S P E R S I O N : p<-5 • 41.5 29.4 -5<P<-4 4.8 4.6 -4<P<-3 7.1 7.7 -3<p<-2 10.4 11.8 -2<P<-1 13.7 16.8 -1<P< 0 17.0 20.4 0<p< 1 17.5 19.2 l<p< 2 14.7 12.7 2<p< 3 9.0 5.0 3<p< '-i 4.3 1.4 4<p< 5 1.6 0.4 5<p • 14.1 3.0 UNCORRECTED R E S U L T S : FWHM (Z) 8.1±.3 6.4+.1 FWTM (%) 21+2 16+.5 EXPONENTIAL 0 7+2 f) R+ S FACTOR N U l / i ' ^ RESULTS CORRECTED FOR MULT. SCATT. FWHM (Z) 7.5+.5 6.0+.4 S N T i A L 0-63±.7 0.85±.8 * THESE ARE NOT "GOOD E V E N T S . " 188 nearby energy* R e s u l t s of the a n a l y s i s o f the r a y t r a c i n g data f o r an i n c i d e n c e angle of twenty degrees are given i n t a b l e &.C-2. In a d d i t i o n to the i n f o r m a t i o n given f o r the z e r o degree data, the percentage of good events f o r each of the g u a r t e r s of MWPC4 i s given. 189 Appendix D RADIATIVE CORRECTIONS TO I~E DECAY FOR THE NAI DETECTOR The r a d i a t i v e c o r r e c t i o n s are a mixture cf i n n n e r bremsstrahlung terms (I. B. — see f i g u r e A.B-1a), s t r u c t u r e dependent terms (S.D.--see f i g u r e A i D - T b ) # and terms a r i s i n g v i a i n t e r f e r e n c e between the;se two : proceses. The ex p r e s s i o n f o r the r a d i a t i v e c o r r e c t i o n s t c the T -e which most r e a d i l y l e nds i t s e l f to c a l c u l a t i o n i s given i n a paper by Brown and Bludman (Br64). The expre s s i o n i s FIGURE A.D-1 FEYNMAN D I A G R A M S F O R R A D I A T I V E C O R R E C T I O N S T O (A) I N N E R - B R E M S S T R A H L U N G P R O C E S S E S ( B ) S T R U C T U R E D E P E N D E N T P R O C E S S E S 191 d 2W = 1-y+(m/y) 2 dxdy IB x z ( x + y - 1 - m 2 / y 2 J r v 2 . , , . 1 v W 1 „ 2 / 1 | 2 , , 2 x(m/y) 2 ( l - m 2 / y 2 ) [x + 2 C 1 - X ) ( 1 - m /y )+ x + y _ 1 _ m ^ / p 2 1 + A [ ( 1 + Y ) 2 { . . . } + ( 1 - Y ) M. . .}] b ... (A. D - 1) + A i N T [ ( 1 + Y ) { * • * } ] where x E k /E Y max y E E /E e max y = mass o f t h e p i o n m mass o f t h e l e p t o n A = i n n e r b r e m s s t r a h l u n c o u p l i n g c o n s t a n t IB AS D E s t r u c t u r e d e p e n d e n t c o u p l i n g c o n s t a n t A I N T ~ i n t e r f e r e n c e t e r m c o u p l i n g c o n s t a n t 192 The parameter y encompasses our r e g a r d i n g the s t r u c t u r e of the pion. measurements set Y = 0.44±0.12 -2. 1+. 11 (Ig73) Although the s t r u c t u r e dependent and i n t e r f e r e n c e terms are themselves of i n t e r e s t , f o r the present work, the IB terms alone c o n t r i b u t e s i g n i f i c a n t l y i These terms r e s u l t from a s t r a i g h t f o r w a r d quantum electrodynamic c a l c u l a t i o n , and are w e l l understood., To c a l c u l a t e the " t h e o r e t i c a l " spectrum seen by the Nal c r y s t a l f o r p o s i t r o n s from i r - e v decay plus those photons from TT -evy decay which enter the c r y s t a l , a Monte C a r l o computer s i m u l a t i o n of the experiment was dene. T h i s work was o r i g i n a l l y performed by Dr. M.S. D i x i t . The program simulated pions stopping i n the t a r g e t counter which produced decay p o s i t r o n s t h a t enter the c r y s t a l . With a p r o b a b i l i t y given by equation A. D-1, the pi-decay produces a hard photon. The s i m u l a t i o n program determines whether t h i s photon e n t e r s the Nal c r y s t a l . In Dr. D i x i t * s c a l c u l a t i o n , the p o s i t r o n energy and angle between the p o s i t r o n and the photon were chesen randomly, and the photon energy then determined;. . In a l a t e r u n c e r t a m t x e s E x i s t i n g 193 c a l c u l a t i o n by the author, the p o s i t r o n and photcn e n e r g i e s where chosen, and the angle was then determined. The two methods i n v o l v e d d i f f e r e n t weightings f o r the Monte C a r l o c a l c u l a t i o n , but the r e s u l t s were i n e x c e l l e n t agreement.. Unless otherwise noted, the f o l l o w i n g r e s u l t s are from the author's c a l c u l a t i o n . . As a t e s t of t h e Monte C a r l o s i m u l a t i o n , r e s u l t s are compared with t h e o r e t i c a l r e s u l t s of V S o e r g e l (So72),„ These t h e o r e t i c a l e x p r e s s i o n s ignored terms the order of (me/mfr)2, and so f r e q u e n t l y diverged f o r very s m a l l angles or e n e r g i e s . Thus, the minimum angle or energy was an a d j u s t a b l e parameter i n the t h e o r e t i c a l e x p r e s s i o n s which could be used to normalize the curves* In t h i s sense, the theory i s approximate. The shapes given below a r e the r e s u l t s of a Monte C a r l o s i m u l a t i o n with a minimum photon energy of 0.1 MeV. C a l c u l a t i o n s with a minimum of 0..001 MeV show no s t a t i s t i c a l l y s i g n i f i c a n t change. Fig u r e ft. D-2 shows the TT - evy photcn energy s p e c t r a as determined by the approximate theory and the Monte Carlo s i m u l a t i o n . The agreement i s very gccd, F i g u r e A.D-3 shows the angular d i s t r i b u t i o n s f c r ""-e VY as a f u n c t i o n of e — the angle between the ey p o s i t r o n and the photon. Again, the agreement between the approximate theory and the s i m u l a t i o n i s good, although the Monte C a r l o r e s u l t s are perhaps s l i g h t l y lower a t l a r g e r a n g l e s . ( T h i s does not seem to be s t a t i s t i c a l l y F I G U R E A.D-2 CALCULATED VS. THEORETICAL * - e v Y PHOTON ENERGY SPECTRA PHOTON ENERGY (MEV) 1 9 5 FIGURE A.D-3 CALCULATED VS. THEORETICAL « - " T POSITRON - PHOTON ANGULAR DISTRIBUTIONS, 10. < i -> c r < o o 1. Ul CC 0.1 APPROXIMATE THEORY (S072) ~] X MONTE CARLO RESULTS ( K M I N = 0.1 MEV) J I I i i 1. 0.8 0.6 0.1 0.2 0. -0.2 -0.4 -0.6 -0.8 -1. cos(ee Y) 196 TABLE A.D-1 ir-eVY / tf-ev BRANCHING RATIOS AS A FUNCTION OF PHOTON CUTOFF MOMENTUM K „ I N (MEV) APPROXIMATE THEORY (%) MONTE CARLO RESULTS (%) MONTE CARLO RESULTS (DR. DIXIT'S J CALCULATION) (%) 0 . 1 9 . 0 14.2 ± 6 . 5 9.02 0 . 5 7 . 3 1 3 . 3 ± 6 . 5 1 . 0 6 . 4 11.6 ± 6 . 5 2 . 0 5 . 4 6 . 8 + 1 . 5 5 . 1 4 . 0 3 . 8 ± 0.4 10.2 2 . 8 2 . 9 + 0 . 3 20.4 1 .68 1 .60 ± 0 . 1 3 0 . 7 1 .05 0 . 9 3 ± 0.05 4 0 . 9 0.65 0.60 + 0.04 51.1 0 . 3 8 0.30 ± 0.01 1 9 7 FIGURE A.D-1 CALCULATED VS. THEORETICAL " - e v Y POSITRON ENERGY SPECTRA POSITRON ENERGY (MEV) 198 ENERGY CUTOFF (MEV) MONTE CARLO RESULTS FOR T A I L CORRECTION (%) MONTE CARLO RESULTS FOR T A I L CORRECTION (DR. D I X I T S CALCULATION) (%) 40 0.22 + .01 0.22 45 0.27 ± .02 0.28 50 0.33 ± .03 0.35 55 0.11 ± .04 0.45 60 0.54 ± .05 0.58 TABLE A.D-2 Percentage Of The C a l c u l a t e d T T - e Lineshape Eelow V a r i o u s Energy C u t o f f s . s i g n i f i c a n t . ) Table A.D-1 g i v e s the branching r a t i o f o r TT -e decay as a percentage of a l l T r-e decays. These are given f o r the s i m u l a t i o n and t h e o r y as a f u n c t i o n of minimum photon e n e r g i e s . Agaiu the agreement i s reasonably good. Note t h a t the l a r g e u n c e r t a i n t i e s f o r s m a l l photon energy c u t o f f s seem to r e s u l t frcm the use of a Monte C a r l o method to perform an i n t e g r a l near a divergence. F i g u r e A.D-4 shows the p o s i t r o n energy spectrum f o r TT—e vy decays. Again, the approximate theory and Monte . C a r l o r e s u l t s agree reasonably w e l l . Also given i s the energy spectrum seen i n the Nal when both the p o s i t r o n s and the photon are d e t e c t e d . T h i s simultaneous d e t e c t i o n 199 r e s u l t s i n counts from the t a i l r e g i o n (of l e s s energy than the n o n - r a d i a t i v e decay) being s h i f t e d t o an energy higher than the mono- e n e r g e t i c decay. T h i s l a s t spectrum i s t h a t seen i n the Nal c r y s t a l when the pion decays to an e l e c t r o n . Table Am E-2 g i v e s the percentage of the lineshape i n the t a i l below the given energy ( p o s i t r o n and detected photon) c u t o f f . Thus, i f we can i n t e g r a t e the Tr-e peak dcwn to 55 MeV, the t a i l c o r r e c t i o n s f o r t h i s t h e o r e t i c a l l i n e s h a p e becomes 0.4±0.04%. In the above c a l c u l a t i o n , a photon was " d e t e c t e d " i f i t entered any p a r t of the 45 cm., diameter f r o n t f a c e of the Nal c r y s t a l . T h i s i s o v e r - o p t i m i s t i c , s i n c e photons which graze the edge of the c r y s t a l do not d e p o s i t a l l t h e i r energy.. To account f o r t h i s e f f e c t , we reduced the " f r o n t f a c e acceptance of the N a l " from the p h y s i c a l value of 45 cm. i n diameter t o 38 cm. i n diameter. T h i s r e g u i r e d t h a t 10 MeV photons t r a v e r s e at l e a s t one i n t e r a c t i o n s l e n g t h of N a l . Reducing t h i s acceptance tends to i n c r e a s e the t a i l of the t h e o r e t i c a l l i n e s h a p e . A c a l c u l a t i o n was a l s o done f o r an acceptance of 17.5 cm.--an extreme ex a g g e r a t i o n . , These t h e o r e t i c a l s p e c t r a are then f o l d e d i n t o the i n t r i n s i c Nal l i n e s h a p e t o produce a l i n e s h a p e t o be used f o r f i t t i n g the experimental TT - e peak. R e s u l t s of t h i s f o l d i n g f o r the v a r i o u s acceptances are given i n f i g u r e a.D-5. We see. that the r a d i a t i v e c o r r e c t i o n s to TT -e decay i n c r e a s e the 200 t a i l of the l i n e s h a p e seen i n the Nal s i g n i f i c a n t l y , hut not d r a m a t i c a l l y . Table A.D-3 shows how the t a i l c o r r e c t i o n (the percentage of the lineshape below the given energy) v a r i e s f o r the d i f f e r e n t l i n e s h a p e s . The l i n e s h a p e we a n t i c i p a t e to be c o r r e c t has an acceptance of 38 cm. The corresponding u n c e r t a i n t y i n the l i n e s h a p e should be about FIGURE A.D-5 EFFECT OF VARYING THE PHOTON ACCEPTANCE ON THE « - « V T LINESHAPE 370 CHANNEL NUMBER TABLE A.D-3 THE EFFECT OF RADIATIVE CORRECTIONS TO w-«v DECAY ON THE INTRINSIC NAI LINESHAPE FOR VARIOUS PHOTON ACCEPTANCES. (HERE, THE ACCEPTANCE IS THE DIAMETER OF A CIRCLE DEFINING THE ALLOWED FRONT FACE ILLUMINATION OF THE N A I . ) PERCENTAGE OF THE LINESHAPE BELOW THE ENERGY CUTOFF ENERGY RAYTRACING LINESHAPES WITH RADIATIVE CORRECTIONS (MEV) LINESHAPE 18 INCH 15 INCH 7 INCH (NO RADIATIVE ACCEPTANCE ACCEPTANCE ACCEPTANCE CORRECTIONS) (%) (%) (%) 20 0 0.1 0.2 0.3 30 0.02 0.3 0.3 0.6 40 0.2 0.6 0.7 1.1 50 1.3 1.9 2.1 2.6 55 3.3 4.0 4.3 5.2 60 10.6 11.5 11.7 12.7 Appendix E RADIATIVE: CORRECTIONS TO THE U+ev V . g y DECAY FOR THE NAI DETECTOR The f i r s t p ublished attempt to c a l c u l a t e the r a d i a t i v e c o r r e c t i o n s t o muon decay was dene by Behrends, F i n k e l s t e i n and S i r l i n i n 1956 (Be56), and was f u r t h e r analyzed by K i n o s h i t a and S i r l i n i n 1959 (Ki59a). . Both v i r t u a l (V — see f i g u r e A.E-1a) and i n n e r brehmstrahlung (IB — see f i g u r e A.E-1b) c o r r e c t i o n s were c a l c u l a t e d . I n d i v i d u a l l y , these terms diverge f o r very lew energy photons. However, when combined, t h i s i n f r a r e d divergence e x a c t l y c a n c e l s . (A s i m i l a r e f f e c t occurs when c a l c u l a t i n g e l e c t r o n - e l e c t r o n s c a t t e r i n g — the d i f f e r e n t i a l c r o s s s e c t i o n diverges near zero degrees unless the emission of hard photons i s i n c l u d e d i n the c a l c u l a t i o n . See Jackson - Ja56.) The c o r r e c t i o n terms a l s o diverge f o r p o s i t r o n s of maximum energy. T h i s i s c o n v e n t i o n a l l y swept under the t h e o r e t i c a l carpet with the argument that f o r e l e c t r o n s of t h i s energy, nc r e a l photons can be emitted and so the IB term does not apply. For the purposes of c a l c u l a t i o n , a s m a l l d e t e c t o r - r e s o l u t i o n i s i n t r o d u c e d which averages the spectrum near the upper end p o i n t . (See K a l l e n - Ka68, page 90). The r e s u l t s of these c a l c u l a t i o n s were to decrease the muon l i f e t i m e by 3.5 % and to decrease the e f f e c t i v e value of the rho parameter f o r a V-A i n t e r a c t i o n 2 0 4 FIGURE A.E-1 FEYNMAN DIAGRAMS FOR RADIATIVE CORRECTIONS TO y+e+v e +v y +Y (A) VIRTUAL CORRECTION PROCESSES (B) INNER BREMSSTRAHLUNG PROCESSES 205 from 0,75 to 0.69, However, these e a r l y c a l c u l a t i o n s made a s u b t l e but important mistake, as Berman noted i n 1958 (Be58) . Indeed, combining the V and IB c o r r e c t i o n s dees e l i m i n a t e the i n f r a r e d divergence, but t h i s e n t a i l s "postponing" the divergence i n the f o l l o w i n g manner—a " s m a l l " photon mass was i n t r o d u c e d i n t o the v i r t u a l c o r r e c t i o n terms, which allowed i n t e g r a t i n g over v i r t u a l photon e n e r g i e s . The same photon mass term was then in t r o d u c e d i n t o the IB c o r r e c t i o n s . Then the photon mass terms, which would diverge as the (probably) f i c t i t i o u s g u a n t i t y was decreased t o zero, c a n c e l l e d each other when the c o r r e c t i o n s were combined. Berman emphasized t h a t "the e l i m i n a t i o n cf the (IB) divergence should be done i n a manner c o n s i s t e n t with the v i r t u a l p r o c e s s . " Thus, we must in t r o d u c e a " s m a l l " photon mass i n t o the IB c o r r e c t i o n s . . Massive photons, however, would have t h r e e d i r e c t i o n s of p c l a r i z a t i o n - - t w o l o n g i t u d i n a l d i r e c t i o n s p l u s the t r a n s v e r s e p o l a r i z a t i o n t h a t i s forbidden f o r massless photons. The c a l c u l a t i o n s done p r i o r to Berman*s d i d not allow f o r t h i s a d d i t i o n a l degree of freedom. E f f e c t i v e l y , they e l i m i n a t e d the v i r t u a l divergence by way of a photon mass, and the IB divergence by way of a minimum photon momentum, i n s t e a d of c o n s i s t e n t l y using one method or the o t h e r . Those i n v o l v e d i n the e a r l i e r c a l c u l a t i o n s v e r i f i e d t h a t t h i s mistake had been made, and r e - c a l c u l a t i e n s 206 confirmed Eerman's r e s u l t s (Ki59b). The e f f e c t of the new r a d i a t i v e c a l c u l a t i o n s were t o i n c r e a s e the muon l i f e t i m e by +0.42 % (vs. . -3.5 ?J given e a r l i e r ) and to decrease the e f f e c t i v e value of rho t o only 0.71 (vs. 0.69). T h i s "pre-gauge theory" understanding of r a d i a t i v e c o r r e c t i o n s i s well summarized i n a paper by K a l l e n (Ka68). In a d d i t i o n , K a l l e n presents equations f o r the r a d i a t i v e c o r r e c t i o n s i n a form r e a d i l y useable i n numerical c a l c u l a t i o n s . Table A.E-1 g i v e s K a l l e n ' s eguations which were used i n our c a l c u l a t i o n of the muon decay r a d i a t i v e c o r r e c t i o n s . (Note t h a t the e r r o r i n K a l l e n ' s equation 5.1a has been c o r r e c t e d — +34X2 i n s t e a d of -34X 2. See Ki59a and Du63.) There are a number of ways to use the t h e o r e t i c a l e x p r e s s i o n s i n the s i m u l a t i o n program to produce the r e q u i r e d Nal s p e c t r a . In p a r t i c u l a r , two r o u t e s , shown s c h e m a t i c a l l y i n f i g u r e A.E-2, seemed to us most a t t r a c t i v e . As shown, the p o s i t r o n s p e c t r a f c r y+ e + + v + v decay with r a d i a t i v e c o r r e c t i o n s i s c a l c u l a t e d from the t h e o r e t i c a l expressions f o r the zeroth order s p e c t r a , the v i r t u a l c o r r e c t i o n s and the IB c o r r e c t i o n s . . T h i s s p e c t r a assumes "no phctcns d e t e c t e d " — t h e IB terms have been i n t e g r a t e d over a l l a l l o w a b l e photon e n e r g i e s and a n g l e s . The r e g u i r e d Nal s p e c t r a with "some photons d e t e c t e d " could be c a l c u l a t e d i n a very s i m i l a r f a s h i o n . The only d i f f e r e n c e i s t h a t , when i n t e g r a t i n g over a l l photon e n e r g i e s and angles, we TABLE A.E-1 EQUATIONS USED TO CALCULATE RADIATIVE CORRECTIONS TO MUON DECAY (FROM KA68). ZERO'TH ORDER MICHEL SPECTRUM EQ. § 2.27 EFFECT OF VIRTUAL CORRECTIONS EQ. # 3 .23 d 2 a d f t d x = ( 1 + 2 F i ( x ) ) I 0 ( x ) - 2 x3 ( F 2 ( x ) + F 3 ( x ) ) — 4 T T T EQ. § 3.21 F i ( x ) EQ. n 3.19A EQ. # 3.19B - ^{( logVy - f ) ( l o g % +1) -•<-x>+ y z my 1 1 . , 1 - x , m 2 , + l o g ( x ) ( - - ; - - + l o g ~ ~ x ' l o g u > TT2" F 2 (X) 1 I 2 ) 2 ' 4 a -1 2TT( 1 - x ) 2 { x l o g ( x ) + 1 - x } F 3 ( X ) = ^ T T ^ { ( 2 x " 1 , l o g ( x , + 1 " x } TABLE A.E-1 (CONT.) PHOTON SPECTRUM E Q < # FROM I , B . CORRECTIONS POSITRON SPECTRUM WITH V. AND I . B . CORRECTIONS d 3a 3 2 = 2 £ { A * - - V - ( 2 + y ) + dftdxdy TT 4 + A (6-5y-2y 2-2x(4+3y)) -xy(3-y-y 2) - (3-y-4y 2)+2x 3(1+2y) 1 - f „ A „ W J _ o i „ w S ^ l 1 _ 1 _ 1 + j (y 2(3-2y)+6xy(1-y)+2x 3(3-4y)-4x 3) x A 2 ( x + y ) ( 3 - 2 x - 2 y ) ( ^ ) 2 } - -, n = 2 x 1 3-2x+—— f (x) }— — dMdx 2TT T 4TT V f(x) - 2(3-2x)R(x)-31og(x)+ m u +-^f 2 ( (5 + 17x-34x 2) log(x )-22x+34x 2 } J x m _ e m R(x) - {log(x-H-1 } { 2 1 o g ( — +-|} + m x 2 e . +log(1-x){log(x)+1—} - l o g ( x ) - 2 * ( - x ) T T 2 1 O 03 TABLE A.E-1 (CONT.) WHERE: x • .E +/E e^ max y T . E /E Y max • 192 m g U g = c o u p l i n g constant *(-x) = an adaption o f the spence f u n c t i o n (see Ka68 fi Mi49) me = p o s i t r o n mass m = muon mass u EQ. § 5.11B A = - * S^L . , . O O S , + 2 T5T7J ^e'^y = 4-momenta of the p o s i t r o n , photon ®ey ~ angle between the p o s i t r o n and photon note a l s o the r e l a t i o n s h i p among x, y, and A 0 4 y 1-1xA £ 1 EQ. 5.11B WITH MASS CORRECTION: 2 m e 2 2 m v 2 A « 1 - cos6 + - 2 (—) + (—1) (y^m =photon) ^ mass NJ O 2 1 0 FIGURE A.E-2 SCHEMATIC ROUTES FOR CALCULATING EFFECTIVE " * e v « v i i Y RADIATIVE CORRECTIONS FOR A NAI DETECTOR. THEORY PRESENT CALCULATIONS 211 must sum the photon and p o s i t r o n e n e r g i e s f o r t h a t region of the phase space when the photon e n t e r s the Nal- . T h i s procedure--method A — h a s the advantage of accomodating a wide v a r i e t y of t e s t s on the n u m e r i c a l c a l c u l a t i o n . S p e c i f i c a l l y the "no photon detected" s p e c t r a can s i m u l t a n e o u s l y be c a l c u l a t e d , as can the photon energy s p e c t r a , the p o s i t r o n - photon angular d i s t r i b u t i o n , the branching r a t i o f o r photons above some c u t - o f f energy, and the t o t a l c o r r e c t i o n t c the muon l i f e t i m e . These can a l l be compared with well e s t a b l i s h e d t h e o r e t i c a l r e s u l t s . T h i s method has the disadvantage of d e a l i n g d i r e c t l y with the e x p r e s s i o n s f o r both V and IB c o r r e c t i o n s , and of not u t i l i z i n g the r e s u l t s c f e x i s t i n g c a l c u l a t i o n s . A second route f o r c a l c u l a t i n g the Nal s p e c t r a --method B — was o r i g i n a l l y suggested to us ty Dr. Harold F e a r i n g of TEIDMF, and was l a t e r independently suggested and advocated by Dr. C h a r l e s P i c c i o t t o of the U n i v e r s i t y of V i c t o r i a . T h i s procedure begins with the "no photon d e t e c t e d " s p e c t r a with r a d i a t i v e c o r r e c t i o n s , and simply m o d i f i e s t h i s s p e c t r a f o r those photons which w i l l be. detected by the Nal but were not accounted f o r i n the e x i s t i n g c a l c u l a t i o n s . The a t t r a c t i v e f e a t u r e of t h i s method i s that i t produces the r e g u i r e d r e s u l t while doing the l e a s t v i o l e n c e t o an e s t a b l i s h e d c a l c u l a t i o n s . The drawback i s t h a t ony one numerical check can be made--that the i n t e g r a l of the d i s t o r t i o n caused by the detected photons i s zero. 212 I n i t i a l l y , we opted f o r method A. However, when seme problems arose i n i n t e r p r e t i n g the t h e o r e t i c a l e x p r e s s i o n s (discussed below) , we decided t o check our r e s u l t v i a method £. Method A i s a s t r a i g h t - f o r w a r d combination of the zero t h order s p e c t r a and the V and IB c o r r e c t i o n s , as shown i n K a l l e n . Method B r e q u i r e s somewhat more e x p l a n a t i o n . ( T h i s d i s c u s s i o n i s s c h e m a t i c a l l y i l l u s t r a t e d i n f i g u r e A.E-3.) We wish t o a d j u s t the P+ e ++ v + v p o s i t r o n spectrum to account f o r those photons detected i n the Nal. The s u g g e s t i o n of Drs. Fea r i n g and P i c c i o t t c was t o c a l c u l a t e a c o r r e c t i o n term f o r detected photons C (E N a l ) = f ( E -k,k,9) - f ( E ,k,6) e e ... (A.E-1) where F ( E ,k , 6) = I.B. c o r r e c t i o n t e r m = t h e p r o b a b i l i t y f o r p r o d u c i n g a e o f e n e r g y E g and a p h o t o n o f momentum k, w i t h an o p e n i n g a n g l e 0 E N a l = e n e r g y d e p o s i t e d i n N a l by e and Y 2 1 3 FIGURE A . E -3 SCHEMATIC OF METHOD B FOR CALCULATING RADIATIVE CORRECTIONS TO MUON DECAY METHOD B l SPECTRUM INCREASED~ BY AMOUNT F(E e-k.*.e) FOR COUNTS SHIFTED FROM A LOWER ENERGY METHOD E2 SPECTRUM REDUCED BY AMOUNT F ( E e , k . 6 ) FOR COUNTS SHIFTED TO A HIGHER ENERGY SPECTRUM REDUCED AT E e BY AMOUNT F(E e»k , 6 ) FOR COUNTS SHIFTED TO A HIGHER ENERGY SPECTRUM INCREASED AT E e+k BY AMOUNT F(E e.*,e) FOR COUNTS SHIFTED FROM A LOWER ENERGY 214 The f i r s t term cn the r i g h t hand s i d e (EHS) accounts f o r events which have been s h i f t e d up i n the s p e c t r a because the photon was d e t e c t e d . The p o s i t r o n energy E (e) < E(N a l ) , but the detected photon accounts f o r the d i f f e r e n c e E(NaI)=E (e) +k The second term accounts f o r events f o r which E(NaI)=E(e) , but have now been pushed up i n energy, s i n c e the photon was detected. We see t h a t , although the f u n c t i o n F di v e r g e s f o r smal l photon e n e r g i e s , the f u n c t i o n C i s w e l l behaved, s i n c e the EHS goes t o zero as k does. T h i s form f o r the c o r r e c t i o n term, method B1, was suggested by the Drs. P i c c i o t t o and Fearing* In the midst of n u m e r i c a l l y c a l c u l a t i n g the c o r r e c t i o n f u n c t i o n C, I r e a l i z e d t h a t we were c a l c u l a t i n g two terms when one would s u f f i c e . . For a given p o s i t r o n energy, photon energy, and opening angle, we can reach the same end by c a l c u l a t i n g the f o l l o w i n g c o r r e c t i o n f u n c t i o n f o r each simulated event: C(E ) = C(E )-F(E ,k ,0 ) e e e C(E +k)=C(E +k)+F(E ,k , 0 ) : e e e 215 (These equations f o l l o w FORTRAN conventions) Thus, we c a l c u l a t e a s i n g l e IB c o r r e c t i o n s term and " p h y s i c a l l y " s h i f t the events from a lower energy t o a higher energy. I n one sense, t h i s i s more s t r a i g h t f o r w a r d , but the f a c t t h a t C i s w e l l behaved near k=0 i s s l i g h t l y l e s s obvious. Dr. P i c c i c t t c has agreed that the two methods are the same. The l a t t e r procedure method B2 — was implemented s i n c e i t had s e v e r a l advantages f o r numerical c a l c u l a t i o n . ( P r i m a r i l y , method E2 could be implemented while re d u c i n g l e s s of s i g n i f i c a n c e . ) To implement these methods, a Monte C a r l o s i m u l a t i o n program was w r i t t e n , s i m i l a r t o that d e s c r i b e d i n the previous s e c t i o n . , A muon from the t a r g e t counter decays, y i e l d i n g a p o s i t r o n t h a t e n t e r s the Nal c r y s t a l . An IB photon energy and angle are randomly chosen, and the p r o b a b i l i t y f o r such a decay i s c a l c u l a t e d . The program determines whether the photon e n t e r s the Nal c r y s t a l . In e f f e c t , we perform the i n t e g r a l over photon energy and angle n u m e r i c a l l y , v i a a Monte C a r l o method. To avoid computer o v erflows which c o u l d occur f o r the dive r g e n t V and IE c o r r e c t i o n terms, we implemented a v a r i a b l e "photon mass" (m^) i n an e f f o r t t c d u p l i c a t e the a n a l y t i c method f o r han d l i n g these divergences. Thus, f c r any simulated event f o r which the photon energy was l e s s 216 than the photon mass, n e i t h e r V nor IB r a d i a t i v e c o r r e c t i o n s were c a l c u l a t e d . We f i r s t used method A, which allowed a v a r i e t y of numerical checks. Although many of the g u a n t i t i e s c a l c u l a t e d agreed very well with the a n a l y t i c a l r e s u l t s , t h e r e were very important d i s c r e p a n c i e s . . A f t e r a f a i r l y i n t e n s i v e e f f o r t , the f o l l o w i n g t e s t s on the r e s u l t s of the numerical c a l c u l a t i o n s gave p o s i t i v e r e s u l t s : 1) Comparison of the branching r a t i o _ f o r r a d i a t i v e decay _( y + -»- e ++ v + v + y ) / ( y + -> e++ v + v ) with the theory o f Ki59b 2) Comparison of the i n t e g r a l of the v i r t u a l c o r r e c t i o n s terms as a f u n c t i o n of the photon mass (m ) with the numerical r e s u l t s of EPS (Be56) . . 3) Comparison of the Monte C a r l o r e s u l t s f o r the i n t e g r a l . o f the IB terms with r e s u l t s of standard numerical i n t e g r a t i o n r o u t i n e s cn the IBM 370. 4) comparison of the p o s i t r o n energy s p e c t r a with Ki59b 5) Comparison of the photon energy s p e c t r a with Ki59b 6) Comparison of the p o s i t r o n - p h o t o n angular d i s t r i b u t i o n with Ec59 217 Other t e s t s g i v i n g p o s i t i v e agreement were: 1) Comparison of the form cf the IB c o r r e c t i o n terms from K a l l e n with those given by Fr59 and Be58. (As K a l l e n notes, h i s ex p r e s s i o n c o n t a i n s an a d d i t i o n a l term p r o p o r t i o n a l t o T h i s term i s dropped i n many t h e o r e t i c a l e x p r e s s i o n s , as are other terms the order o f (me) 2. Note, however, that t h i s term becomes s i g n i f i c a n t f o r smal l angles and positron, e n e r g i e s . . Removal of t h i s term d i d not e l i m i n a t e the problems d e s c r i b e d below.), 2) Hand c a l c u l a t i o n s demonstrating t h a t the di v e r g e n t terms i n the V and IB c o r r e c t i o n s e x a c t l y c a n c e l l e d . (Note t h a t to do t h i s , we must i n c l u d e K a l l e n * s " a d d i t i o n a l " term.) However, the f o l l o w i n g d i s c r e p a n c i e s came to l i g h t : 1) The change i n the t o t a l muon l i f e t i m e c a l c u l a t e d by the Monte C a r l o s i m u l a t i o n program was -3.5 % vs. The expected value of +0.42 % as given by the a n a l y t i c i n t e g r a t i o n of the same ex p r e s s i o n s . 2) The shape of the r a d i a t i v e c o r r e c t i o n s to the "no photon detected" s p e c t r a was wrong, becoming ne g a t i v e at X=E (e)/E (max) = 0.95 vs X=0.69 as c a l c u l a t e d by Ka68, Be58, Ki59b. A f t e r checking and re- c h e c k i n g the implemented t h e o r e t i c a l e x p r e s s i o n s , and the Monte C a r l e s i m u l a t i o n program, we e v e n t u a l l y r e a l i z e d t h a t our r e s u l t s were i n very good agreement with the i n c o r r e c t t h e o r e t i c a l c a l c u l a t i o n of BFS. In some way, we were making the same mistake they d i d , although we were using p r e s e n t l y accepted e x p r e s s i o n s f o r the r a d i a t i v e c o r r e c t i o n s . 218 The problem, then, was t h a t we were t r e a t i n g the photon energy c u t o f f as a minimum photon momentum i n the IB c o r r e c t i o n terms, i n s t e a d of as a photcn mass as i n the V c o r r e c t i o n s terms.. With t h i s p e r s p e c t i v e , we were able to l o c a l i z e the source of our d i f f i c u l t y . Equation 5.11b i n K a l l e n (given i n t a b l e A.E-1) d e f i n e s the q u a n t i t y A which i s e s s e n t i a l l y the dot product of the 4 momenta f o r the p o s i t r o n and the photon, normalized by t h e i r r e s p e c t i v e e n e r g i e s . The angular dependence o f the IB c o r r e c t i o n s can be c o n v e n i e n t l y expressed i n terms of A . Note that the approximate expansion of A keeps the u s u a l l y n e g l i g i b l e term (2/X2) *(me/my) 2 T h i s term becomes important f o r very s m a l l angles and a l s o prevents A going to zero which would cause problems when i n t e g r a t i n g the IB terms i n v e r s e l y p r o p o r t i o n a l t o A „ But we now see t h a t the p o s i t r o n and photon are being handled d i f f e r e n t l y . . T h i s , of course, i s reasonable s i n c e the IB e x p r e s s i o n r e l a t e s to the f l u x of "hard" photons. However, f o r our purposes, we must t r e a t the photon as a massive p a r t i c l e . Thus, we must add a c o r r e c t i o n t c equation 5.11b, making i t symmetrically dependent on p o s i t r o n s and photons. T h i s "photon mass c o r r e c t i o n term" i s simply 219 (2/Y2) (a /ml-a The complete c o r r e c t e d e x p r e s s i o n f o r A i s given i n t a b l e A.E-1.. Although t h i s c o r r e c t i o n i s only s i g n i f i c a n t f o r very s m a l l angles and photon e n e r g i e s , i t has a very d i s c e r n a b l e e f f e c t , s i n c e these c o n d i t i o n s are by f a r the most l i k e l y . I n c l u s i o n of the "photon mass c o r r e c t i o n term" does so l v e the above mentioned problems. Table A.E-2 summarized the r e s u l t s of the Monte C a r l o s i m u l a t i o n program with r e s p e c t to the e f f e c t o f the r a d i a t i v e c o r r e c t i o n s on the mucn l i f e t i m e and the ( u *• -> e+* v + "v + y ) / ( y + e + + v * v ) branching r a t i o as a f u n c t i o n of minimum photon energy.. Here, the F r o n s d a l l S D b e r a l l form drops a l l terms the erder cf (m /m ) 2 i n c l u d e d i n K a l l e n * s equations* The K a l l e n , e u 3 form with no photon mass c o r r e c t i o n f o l l o w s e x a c t l y K a l l e n * s f o r m u l a t i o n * . " K a l l e n with photon mass c o r r e c t i o n " i n c l u d e s the above d e s c r i b e d m c d i f i c a t i o n . As shown i n t a b l e A.E-2, the e f f e c t of the r a d i a t i v e c o r r e c t i o n s as c a l c u l a t e d by t h e Monte C a r l o s i m u l a t i o n program changes frcm a decrease of about 4.8±1.8% to an i n c r e a s e of 0.41±0.1%, i n agreement with the accepted t h e o r e t i c a l p r e d i c t i o n . The shape of the "nc photons detected" p o s i t r o n spectrum a l s o comes i n t o good agreement with theory, as i n d i c a t e d by a f i t t e d value of TABLE A.E-2 RESULTS OF MONTE CARLO SIMULATIONS OF THE RADIATIVE CORRECTIONS v. THEORY 1 FRONSDALL S UBERALL KALLEN (mY=0) KALLEN (WITH « V CORRECTION) (KI59B (FR59) (KA68) (%) (SEE TEXT) (%) (%) S I N (MEV) SB K M I N (MEV) = "Y (MEV) -0.1 0.01 0.001 0.1 0.01 0.001 0.1 0.01 0.001 CHANGE IN 0.4205 -2.8 -4.3 -5.5 -2.4 -4.8 -1.9 0.41 0.63 1.4 MUON LIFETIME +0.3 ±1.5 ±2.5 ±0.2 ±1.8 ±1.3 ±0.1 ±0.22 ±1.2 0.1 9.0 8.6±.2 8.9±.3 8.7±.2 8.2±.2 8.2±.2 8.5±.2 5.5±.l 8.0±.2 8.8±.3 0.5 6.1 6.1±.l 6.0±.l 6.1+.1 5.7±.l 5.7±.l 5.6+.1 4.7±.l 5.4±.l 5.6+.1 1.0 4.9 4.8±.l 4.7±.l 4.9±.l 4.5±.l 4.5+.1 4.4±.l 4.0±.l 4.3±.l 4.3±.l 5.1 2.3 2.4+.06 2.4±.06 2.4i.06 2.2+.07 2 .2± . l 2.2+.07 2.2±.04 2.2±.05 2.2±.04 10.2 1.3 1.5±.04 1.4±.04 1.5±.04 1.3±.04 1.3+.1 1.3±.04 1.3+.02 1.3±.03 1.3±.03 20.4 0.5 .55±.02 .53±.02 .56±.02 .50±.02 .50 .49±.02 .48±.01 .48±.01 .48±.01 30.7 0.19 .19+.01 .19±.01 .20±.01 .18±.01 .18 .17+.01 .18±.01 .18+.01 .18+.01 40.4 .04 .04 .04 .04 .04 .04 .04 .04 .04 .04 221 rho=0.72±0.01 (using a c h i - s q u a r e d m i n i m i z a t i o n f i t t i n g r o u t i n e — s e e s e c t i o n I l l - B ) ; , U n f o r t u n a t e l y , t h i s i s net the only e f f e c t o f the "photon mass c o r r e c t i o n term." Frcm t a b l e A.E-2, we see the branching r a t i o f o r ( y + + e + + v + v + Y ) / ( y + + e + + v • ^  ) decay no lenger agrees with t h e o r y . The discrepancy i s as l a r g e as 40% f o r photons down to 0.1 MeV. This e f f e c t i s s m a l l at l a r g e r photcn e n e r g i e s , as can a l s o he seen i n f i g u r e A* E-4 which shows the photon energy s p e c t r a f o r e+ + v +v decay i n t e g r a t e d ever a l l angles. Here, a l l t h r e e curves have been normalized t o an i n t e g r a l o f one. Viewed i n t h i s way, the discrepancy i s s l i g h t . . F igure A.E-5 gives the angular d i s t r i b u t i o n f o r photons with r e s p e c t to the p o s i t r o n d i r e c t i o n f o r y + e++ v + v +y decay. , Again, the curves have been normalized to a u n i t i n t e g r a l . Here, the discrepancy seems more pronounced, with the "photon mass c o r r e c t i o n term" r e s u l t s about 1535 l a r g e r at backward angles. Note t h a t these d i s c r e p a n c i e s i n the f l u x and angular d i s t r i b u t i o n compensate each other at backward a n g l e s . Only i n t h i s region has the theory f o r IB c o r r e c t i o n s been e x p e r i m e n t a l l y v e r i f i e d , and even then, not t o high accuracy (see Be64). The t h e o r e t i c a l p r e d i c t i o n s f o r the branching r a t i o f o r r a d i a t i v e decay are extremely dependent on the f l u x at low photon e n e r g i e s and small angles, regions t h a t are f o r the most p a r t e x p e r i m e n t a l l y 222 u n a v a i l a b l e . S t i l l , the p r e d i c t i o n s of a theory as f r e q u e n t l y v e r i f i e d as quantum electrodynamics are not e a s i l y bent. Thus we were not completely s a t i s f i e d with these r e s u l t s * Bt the same time, as i n d i c a t e d p r e v i o u s l y , the c a l c u l a t e d p o s i t r o n spectrum was i n e x c e l l e n t agreement with they t h e o r y . , F i g u r e A..E-6 g i v e s the c a l c u l a t e d vs. T h e o r e t i c a l c o r r e c t i o n s to the z e r o t h order p o s i t r o n s p e c t r a f o r t h r e e d i f f e r e n t values of "phctcn mass". Given are the v i r t u a l c o r r e c t i o n s , the IB c o r r e c t i o n s , and the sum, compared with theory. We see the agreement i s e x c e l l e n t , although the s t a t i s t i c a l f l u c t u a t i o n s i n the Monte C a r l o r e s u l t s grow i n c r e a s i n g l y worse f o r s m a l l e r values of m^ . T h i s decrease i n the accuracy of the c a l c u l a t i o n i s not caused by any t h e o r e t i c a l problem with the i n t r o d u c t i o n of photon mass, but r a t h e r with numerical problems of attempting to i n t e g r a t e a f u n c t i o n near a divergence v i a a Monte C a r l e method.. These r e s u l t s i n d i c a t e t h a t we should use the l a r g e r value of iry = 0.1 MeV t o c a l c u l a t e our c o r r e c t i o n s with good accuracy while not d i s t o r t i n g the photon s p e c t r a too v i o l e n t l y . F i g u r e A.E-7 g i v e s the c o r r e c t i o n i n the z e r o t h order s p e c t r a which r e s u l t i n the s p e c t r a seen by the Nal. That i s , the IB terms now i n c l u d e the photon energy i f the photon was detected by the Nal c r y s t a l . . Here we see t h a t the r e s u l t i n g c o r r e c t i o n s are very n e a r l y zero over the e n t i r e energy r e g i o n . Thus, the spectra seen i n the Nal FIGURE A.E-4 PHOTON ENERGY SPECTRA FOR u - e u v y DECAY FROM A MONTE CARLO SIMULATION U I QJ H E V ) • THEORY (Kl59B) MONTE CARLO RESULTS O FRONSDALL & UBERALL FORM (NO PHOTON hASS C O R R . ) X MONTE CARLO RESULTS KALLEN FORM (WITH PHOTON MASS C O R R . ) 9 X 9 X o X o o h A o X ? x.° X o x X 10 20 30 PHOTON ENERGY (MEV) 40 50 224 FIGURE A.E-5 P O S I T R O N - P H O T O N A N G U L A R D I S T R I B U T I O N S F O R M-ev v Y D E C A Y F R O M A M O N T E - C A R L O S I M U L A T I O N < M = 0.1 MEV) X 8 X 9 . X - * - *x 0.1 « 3 B I M A T E T H E O R Y M O N T E C A R L O R E S U L T S O F R O N S D A L L & U B E R A L L F O R M ( N O P H O T O N M A S S C O R R . ) -M O N T E C A R L O R E S U L T S A K A L L E N F O R M ( W I T H P H O T O N M A S S C O R R . ) X d A X X X e X^ _ i | -0.8 -0.4 0. cos(eey) 0.4 0.8 FIGURE A.t-G RADIATIVE CORRECTIONS TO THE ZERO'TH ORDER MICHEL SHAPE FOR VARIOUS PHOTON M A S S CUTOFFS, EXCLUDING DETECTED PHOTONS. THEORETICAL RADIATIVE CORRECTIONS TO THE MICHEL SHAPE X MONTE CARLO VIRTUAL CORRECTIONS (V) ° TlONTE CARLO INNER-BREMSSTRAHLUNG CORRECTIONS (IB) • SUM OF MONTE CARLO V AND IB CORRECTIONS 1.0 0.0 -1.0 1 "PHOTON — i 1 1 1 — ~ :»ss" = 0.100 MEV f p „ r 0 O 0 o o « 0 o o 0 0 * 8 0 0 ° 0 0 . ° ' ' • ' 1 • — 1 1 1 p~ 226 FIGURE A.E-7 RADIATIVE CORRECTIONS TO THE ZERO'TH ORDER MICHEL SHAPE, INCLUDING DETECTED PHOTONS. "PHOTON MASS" = 0.100 MEV 1.0 H T T T T H E O R E T I C A L R A D I A T I V E CORRECTIONS TO T H E M I C H E L S H A P E (NO PHOTONS D E T E C T E D ) X MONTE CARLO V I R T U A L C O R R E C T I O N S (v) o MONTE CARLO I N N E R - B R E M S S T R A H L U N G C O R R E C T I O N S (IB') I N C L U D I N G D E T E C T E D PHOTONS • SUM OF MONTE CARLO V AND IB' C O R R E C T I O N S 0 . o ° e n ° ° o ° o ° o o o ° o ° ° ° o *XX * * x XX - 1 . 0 0 10 n 1 20 30 ENERGY (MEV) T 40 50 FIGURE A.E-8 w _ e V e V y RADIATIVE CORRECTIONS — METHOD A VS. METHOD B2 TOTAL RADIATIVE CORRECTIONS TO THE ZERO'TH ORDER MICHEL SPECTRUM AS A FUNCTION OF THE SUMMED ENERGY OF THE DETECTED POSITRONS AND PHOTONS. T o o • • o o 2 • o 5 L -• METHOD A o METHOD B2 { 10 20 30 40 50 ENERGY OF POSITRONS AND DETECTED PHOTONS (MEV) 228 c l o s e l y resembles the z e r o t h order s p e c t r a . To f u r t h e r reassure o u r s e l v e s about these r e s u l t s , we developed a Monte C a r l o s i m u l a t i o n program u t i l i z i n g method B2, p r e v i o u s l y d e s c r i b e d . F i g u r e A.F.-8 shows the r e s u l t s of t h i s s i m u l a t i o n vs. . the r e s u l t s of method A. We see that the agreement i s q u i t e good, although the s t a t i s t i c a l f l u c t u a t i o n s are somewhat worse. F u r t h e r , the r e s u l t s o f both methods agree very w e l l with a l a t e r "rough hand c a l c u l a t i o n " done by Dr. , C. , P i c c i o t t o . Thus we f e e l f a i r l y c o n f i d e n t c f the v a l i d i t y cf our c a l c u l a t i o n s of the e f f e c t of r a d i a t i v e c o r r e c t i o n s on the p o s i t r o n s p e c t r a from muon decay seen i n the Nal c r y s t a l . 229 Appendix F EFFECT OF THE STOPPING DISTfllBPTION ON THE VARIATION OF THE NAI BESOIOTTON WITH ENERGY As d e s c r i b e d i n s e c t i o n III-A-1, tbe energy dependence o f the Nal r e s o l u t i o n f u n c t i o n ( i . e . the FWHM) i s c o n v e n i e n t l y parametrized as R (E) = ( I ' / E ) n R' (E») An obvious p o s s i b i l i t y f o r i n c l u d i n g the e f f e c t of the stopping d i s t r i b u t i o n i s to augment t h i s r e l a t i o n s h i p as f o l l o w s R(e) = ( E' / E ) n R ' (E') + (E '/E) m RSD (E ') where the parameters m and RSD can be determined e m p i r i c a l l y In p r a c t i c e , the r e s o l u t i o n ( for the TT -e data) a t the high energy p o i n t of the Mic h e l s p e c t r a can be determined by minimizing chi-sguared (see s e c t i o n I I I - B ) . The FWHM i s then e x t r a p o l a t e d down from t h i s f i x e d p o i n t . z o or LU a . x < < < I Q 60 J 50 E ~ N - 0 . 5 , STOPPING DISTRIBUTION FOLDED IN EXPLICITELY — F - N - 0 . 5 , M=1.5, RSD=1.1% 0 FIGURE A . F - 1 G - N - 0 . 5 , M - 1 , 3 , R . n - l . l X 40 30 J 20 J 10 A & B — N=0.5 C - - N=0.3 10 40 20 30 ENERGY (MEV) SEVERAL SCHEMES FOR VARIATION OF THE NAI RESOLUTION WITH ENERGY 50 to OJ o TABLE A.F-1 TAIL CORRECTIONS TO THE MICHEL SPECTRUM VS. ENERGY FOR VARIOUS METHODS OF VARYING THE NAI LINESHAPE RESOLUTION WITH ENERGY. METHOD OF TAIL CORRECTION ( i N %) BELOW ENERGY CUTOFF E(MEV) = VARYING FWHM (SEE TEXT) 3 10 15 20 25 30 35 A 1.3 4.6 9.5 16.8 27.1 40.0 54.6 B . 1.2 4.5 9.5 16.8 27.1 40.0 54.5 C 1.2 4.4 9.3 16.5 26.8 39.7 54.4 D 1.3 4.7 9.8 17.1 27.4 40.2 54.7 E 1.1 4.2 9.0 16.2 26.4 39.2 53.9 F 1.3 4.7 9.7 17.0 27.3 40.1 54.6 G 1.2 4.7 9.7 17.0 !' 27.2 40.0 54.6 H 1.3 4.8 9.8 17.1 27.3 40.1 54.6 232 F i g u r e A.F-1 shows the v a r i a t i o n of FWHM f o r the Nal lineshape with energy f o r a v a r i e t y of cases, a l l of which have a r e s o l u t i o n of 9.5% at 50 MeV. The cases a r e : A) The zeroth order Michel spectrum with n=0.5 (no energy dependence on stepping d i s t r i b u t i o n B) The c a l c u l a t e d M i c h e l spectrum seen by the Nal with n=0.5 C) Nal Michel spectrum, n=0.3 D) Nal M i c h e l spectrum, n=0.7 E) Nal Mic h e l spectrum, e x p l i c i t e l y f o l d s in gaussian t o account f o r stopping d i s t r i b u t i o n . F) Nal M i c h e l spectrum, n=0.5, parametrized e f f e c t of stopping d i s t r i b u t i o n , ESD=1.1%, m=1.5 G) Nal Michel spectrum, n=0.5, ESD=1.1X, m=1.3 B) Nal Mi c h e l spectrum, n=0.5, RSD=1.1?i, m=1-7 Case E e x p l i c i t e l y f o l d s a gaussian i n t o the " i n t r i n s i c " Nal l i n e s h a p e , and f o l d s the r e s u l t i n t c the M i c h e l s p e c t r a (a computer-time consuming p r o c e s s ) . T h i s , then, i s the f u n c t i o n we wish t o d u p l i c a t e with our p a r a m e t r i z a t i o n . As we see, the c l o s e s t approximation i s case F. Table A i F - 1 g i v e s the t a i l c o r r e c t i o n (ii,e. The percentage o f the l i n e s h a p e i n c l u d e d i n the t a i l ) f o r s e v e r a l c u t o f f e n e r g i e s . 233 Appendix G DESCRIPTION AND RESULTS OF PITS-TO n -E .. DATA The parameters d e s c r i b e d are as f o l l o w s : D e s c r i p t i o n of F i t s Nal l i n e s h a p e : T L R — r a y t r a c i n g l i n e s h a p e with Landau s t r a g g l i n g and sto p p i n g d i s t r i b u t i o n f o l d e d i n TG3—TLR with the TT - e l i n e s h a p e f o l d e d i n TPI-- TT -e e m p i r i c a l l i n e s h a p e from run 760, with s l i g h t l y reduced t a i l T PD—TPI with t h e o r e t i c a l -j\-e unfolded TNG—130 MeV TT~(p,n)Y l i n e s h a p e Nal R e s o l u t i o n : FWHM Michel Spectra l i n e s h a p e : R S 2 — M i c h e l s p e c t r a with r a d i a t i v e c o r r e c t i o n s , photons d e t e c t e d , smoothed by s p l i n e f i t . R G — M i c h e l s p e c t r a with r a d i a t i v e c o r r e c t i o n s , no photons de t e c t e d R O — z e r o t h order M i c h e l s p e c t r a M i c h e l Spectra Nal R e s o l u t i o n Parameters: Parameters n,m RSD d e s c r i b e d i n s e c t i o n III-A-3 S u b t r a c t i o n s : A ( * ) - - A c c i d e n t a l s (B4) where * = M—mixed s u b t r a c t i o n r a t i o s * = E— e x p o n e n t i a l s u b t r a c t i o n r a t i o s * = E—randpm s u b t r a c t i o n r a t i o s P ( * ) - - P i l e up s u b t r a c t i o n , where * = Bx--bin number used F i t t e d Background: Q C ( * ) — q u a d r a t i c c u t o f f , where * = energy of cut (MeV) Type of F i t : S ( E - E 1 ) - - s i n g l e f i t over energy region E-E' T1 ( E - E ' ) - - f i r s t f i t over s p e c i f i e d energy r e g i o n T 2(E-E?)—second f i t Parameters v a r i e d : A — a m p l i t u d e or area of l i n e P — p o s i t i o n o r energy of l i n e B1,B2—background parameters Weighting: S — s t a t i s t i c a l ( i n c l u d i n g the e f f e c t o f s u b t r a c t i o n s ) L--low s t a t i s t i c s weighting (equal weighting of a l l data) N — n o r mal weighting ( s t a t i s t i c a l , but e x c l u d i n g the e f f e c t of 2 3 5 subtr actions) A s t a n d a r d i z e d method f o r i n t e r p r e t i n g the r e s u l t s of the f i t s has a l s o been developed. R e s u l t s of F i t s X 2 : reduced x 2 reduced ^ 2 In the r e g i o n of the peak F i t t e d Area: Area of the l i n e d e r i v e d from the minimum X 2 r e s u l t s . (These are a u t o m a t i c a l l y c o r r e c t e d f o r any p o r t i o n o f the l i n e s h a p e outside o f the f i t t e d region.) Energy L o s t : ELOST=Theofetical energy - f i t t e d energy ( c o r r e c t e d f o r mass c f a n n i h i l a t e d e l e c t r o n ) Peak Channel: PEAK = f i t t e d peak channel T a i l C o r r e c t e d Sums: E (*) = sum of counts i n the s p e c t r a above * MeV, c o r r e c t e d f o r t h a t p a r t o f the t a i l o f the f i t t e d l i n e s h a p e o u t s i d e o f the f i t t e d r e g i o n . . (This depends on the f i t t e d peak channel o n l y , not on the amplitude.) V a r i a t i o n i n the T a i l C o r r e c t e d Sums: AE (*-*») = v a r i a t i o n i n the t a i l c o r r e c t e d sums f o r low energy c u t o f f s between * and *' MeV 236 T a i l C o r r e c t e d Sums with Background Subtracted: E — B(*) = sum of counts i n f i t t e d s p e c t r a above * MeV reduced by the f i t t e d background i n t h i s r e g i o n , c o r r e c t e d f o r the t a i l . Neither the menu f o r d e s c r i p t i o n s nor the menu f o r r e s u l t s i s e n t i r e l y comprehensive f o r a l l the f i t s performed. When necessary, more complete d e t a i l s of i n d i v i d u a l f i t s are d e s c r i b e d i n the t e x t . The d e s c r i p t i o n and r e s u l t s of some of the f i t s performed are given i n t a b l e s A.G-1 and A.G-2. 237 TABLE A.6-1 DESCRIPTIONS OF SOME FITS TO THE DATA. FIT § TYPE BINS NAI L.S. NAI RES. z MUON N RES. RSD (Z) SUBTRACTIONS FITTED BACK. TYPE PARAMETERS OF FIT VARIED WT 1012.03 i r e Bl TG3 7.6 A(M)+P(B2) QC(61) Tl(61 -71) T 2 ( 5 3 B l B2 -72) B l ' B 2 N 1012.04 a 0 m 8.0 - - - • m * • * mm i r 1012.05 n a m m - - - m m QC(70) mm m tt w 1017.04 w m TPI m - - - m m oc(61) s(53-72) APB1B2 it 1112.03 * m TG3 7.7 - - - m m m mm m II s 311.04 l i e Bl TLR 8.6 0.5 1.1 1.3 A(M) ec(50) s(3-51) " • tt 315.03 «r Bl,2 n * * 1.5 tt m * * It tl tt 318.01 T T e B l TG3 7.7 - - - A(M)+P(B2) oc(65) s(53-72) " " m 318.02 n « It tl - - - m H oc(71) U It II It M 318.06 l i e H TPU 8.6 0.5 1.1 1.5 A(M) oc(50) s(3-51) " " It 318.08 II tl H 0.3 0.5 0.5 m m n it mm tt 320.01 it m It 7.5 m * * K m mm it tt U T A B L E A.G-2 RESULTS OF SOME FITS TO THE »-«v DATA. (A) RESULTS OF FITS TO THE *-e REGION FIT # 2 X 0 AREA E-LOST (MEV) PEAK B/*e CH. tt) K 5 2 . 5 ) s ( 5 8 ) AE (%) £ ( 5 2 ) N (X) *<58) N tt) 1 0 1 2 . 0 3 3.4 4.3 3 6 0 1 1 4.6 397.7 3.8 3 7 9 3 5 3 6 9 3 9 2.7 0.3 0.7 1 0 1 2 . 0 4 4.5 5.2 3 6 4 0 6 4.8 3 9 6 . 8 3.0 3 8 0 4 1 3 7 1 9 5 2.4 - -1 0 1 2 . 0 5 4.8 6.1 3 6 4 1 6 4.8 3 9 6 . 8 4.9 3 8 0 4 1 3 7 1 9 4 2.4 - -1 0 1 7 . 0 4 1.4 0.2 3 7 9 5 6 4.4 3 9 8 . 8 0.3 3 8 2 1 6 3 8 1 2 1 0.2 0.5 2.5 1 1 1 2 . 0 3 1.4 2.7 3 6 4 3 1 4.7 3 9 7 . 5 3.3 3 7 9 5 6 3 6 9 7 9 2.8 0.2 0.6 3 1 8 . 0 1 1.1 2.4 3 5 6 0 2 5.5 3 9 7 . 7 4.9 3 7 8 0 8 3 6 6 1 4 3.2 -.2 -.6 3 1 8 . 0 2 1.4 2.8 3 5 7 1 2 5.6 3 9 7 . 5 4.9 3 7 8 1 1 3 6 6 3 1 3.2 -.2 -.6 TABLE A.6-2 RESULTS OF SOME FITS TO THE DATA. (B) RESULTS OF FITS TO THE U-e REGION FIT # 2 X 0 AREA (xlO6) E-LOST (MEV) PEAK CH. 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