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Neutrino oscillations schemes applied to neutrino-electron scattering Beaudry, Martin 1981

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NEUTRINO OSCILLATIONS SCHEMES APPLIED TO NEUTRINO-ELECTRON SCATTERING by MARTIN BEAUDRY B . S c , U n i v e r s i t e de M o n t r e a l , 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We ac c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December 1981 © M a r t i n Beaudry, 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t 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 reference and study. I f u r t h e r agree t h a t permission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood th a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of PHYSICS  The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 DE-6 (2/79) ABSTRACT In t h i s t h e s i s , a t h e o r y of n e u t r i n o masses and o s c i l l a -t i o n s i s b u i l t up i n the g e n e r a l c a s e . The work i s then r e s -t r i c t e d t o the t h r e e - s p e c i e s D i r a c c a s e , f o r which the r e q u i r e d p a r a m e t e r s , a mass s c a l e and a m i x i n g m a t r i x , a r e p r o v i d e d by phe n o m e n o l o g i c a l s o l u t i o n s and by a new H i e r a r c h i a l Mass Scheme c o u p l e d w i t h both Cabibbo and H i e r a r c h i a l m i x i n g s . The o s c i l l a t i o n s f o r m a l i s m and hypotheses a r e then a p p l i e d t o n e u t r i n o - e l e c t r o n s c a t t e r i n g e x p e r i m e n t s t o be performed a t Brookhaven, w i t h n e u t r i n o e n e r g i e s r a n g i n g from 20 t o 270 MeV. The v-e s c a t t e r i n g c r o s s - s e c t i o n s , the r e l e v a n t p h e n o m e n o l o g i c a l q u a n t i t i e s , and the r e s u l t i n g p r e d i c t i o n s a r e s u c c e s s i v e l y c a l -c u l a t e d and examined. i i i CONTENTS page A b s t r a c t . . . i i L i s t of T a b l e s v L i s t of F i g u r e s v i Ch a p t e r s I I n t r o d u c t i o n 1 I I N e u t r i n o Masses and M i x i n g s 3 1 Mass L a g r a n g i a n and E i g e n s t a t e s 4 2 N e u t r i n o O s c i l l a t i o n s 5 3 D i r a c N e u t r i n o s 7 I I I Mass a n d ' M i x i n g Schemes 10 1 N e u t r i n o Phenomenology ...10 2 The H i e r a r c h i a l Mass Scheme 11 3 Curves of P r o b a b i l i t y v e r s u s L/E 13 IV N e u t r i n o - E l e c t r o n S c a t t e r i n g 20 1 N e u t r a l - C u r r e n t C r o s s - S e c t i o n 21 2 Other C r o s s - S e c t i o n s 24 3 C r o s s - S e c t i o n s and the Weinberg Angle 26 V Phe n o m e n o l o g i c a l Q u a n t i t i e s 28 1 The Brookhaven Experiment 28 2 O s c i l l a t i o n s - r e l a t e d Q u a n t i t i e s 30 VI P r e d i c t i o n s f o r the Brookhaven Experiment 34 1 C o n s t r a i n t s on Accur a c y 34 2 C h o i c e of Curves and Schemes 35 3 Curves of & v e r s u s L 37 i v R e f e r e n c e s 46 Appendix 47 1 C o n v e n t i o n s and N o t a t i o n s 47 2 D i s c r e t e Symmetries 48 V LIST OF TABLES page T a b l e I : Summary of M i x i n g M a t r i c e s 16 v i LIST OF FIGURES page F i g u r e 1: vs L/E f o r S o l u t i o n A 17 F i g u r e 2: vs L/E f o r S o l u t i o n B 17 F i g u r e 3: P^, vs L/E f o r S o l u t i o n C 18 F i g u r e 4: P A e vs L/E f o r the H i e r a r c h i a l Mass Scheme 19 F i g u r e 5: An example of f a s t o s c i l l a t i o n s 19 F i g u r e 6: N e u t r a l - c u r r e n t e - vv e s c a t t e r i n g 21 F i g u r e 7: C h a r g e d - c u r r e n t e - i/e s c a t t e r i n g 21 F i g u r e 8: N e u t r a l - c u r r e n t v^e - {7 e s c a t t e r i n g 25 F i g u r e 9: C h a r g e d - c u r r e n t v e - v e s c a t t e r i n g 25 F i g u r e 10: F l u x d e n s i t y d i s t r i b u t i o n s 33 F i g u r e 11: Mass dependence of GL . 39 F i g u r e 12: (& vs L, a l l c h a n n e l s 40 F i g u r e 13: 61 vs L f o r both p r o p o s a l s 41 F i g u r e 14: Graph of 5^ v e r s u s L and E 42 F i g u r e 15: 61 vs L f o r S o l u t i o n s A and B 43 F i g u r e 16: & vs L f o r S o l u t i o n C 44 F i g u r e 17: 61 vs L f o r H i e r a r c h i a l masses 45 1 I . INTRODUCTION C u r r e n t l y , t h r e e s p e c i e s of n e u t r i n o s a r e b e l i e v e d to e x i s t , ve , , and vT , the l a t t e r s t i l l y et t o be obse r v e d . The o n l y d i s t i n c t i o n between them, i s t h e i r l e p t o n f a m i l y number L x , = 1 f o r oc~ and i / K , -1 f o r « + and v , and z e r o f o r e v e r y t h i n g e l s e ( oc = e, jj- , zr ) , which i s c o n s e r v e d by a l l known p r o c e s s e s , but might not be so by o t h e r s , such as n e u t r i n o o s c i l l a t i o n s . The i d e a t h a t n e u t r i n o s o s c i l l a t e from one s p e c i e s t o another was proposed by G r i b o v and Pon t e c o r v o [ 1 ] as an e x p l a n a -t i o n f o r the r e s u l t s of the I r v i n e S o l a r N e u t r i n o Experiment [ 2 ] , where the obser v e d f l u x was s i g n i f i c a n t l y lower than was ex p e c t e d from c u r r e n t models of the Sun. The i d e a has been deve-l o p e d and a p p l i e d t o v a r i o u s s i t u a t i o n s , such as the Savannah Rea c t o r a n t i n e u t r i n o e x p e r i m e n t s [ 3 ] and a c c e l e r a t o r - g e n e r a t e d f l u x e s . The whole s u b j e c t has been reviewed by B i l e n k y and Pon t e c o r v o i n r e f e r e n c e [ 4 ] . I f n e u t r i n o o s c i l l a t i o n s do not e x i s t , the whole known l e p -t o n i c w o r l d can be v e r y w e l l d e s c r i b e d by the Weinberg-Salam Model of E l e c t r o w e a k I n t e r a c t i o n s [ 5 , 6 ] , which s u c c e s s f u l l y en-compasses e l e c t r o m a g n e t i c and weak p r o c e s s e s w i t h i n one u n i f i e d framework. The model has been extended t o the quark w o r l d , and i s used as a b a s i s on which more g e n e r a l schemes a r e b u i l t t h a t attempt t o u n i f y l e p t o n and quark P h y s i c s i n t o some Grand-2 U n i f i e d Theory. T h i s l a s t e f f o r t w i l l be a f f e c t e d by the occurence of new f e a t u r e s such as n e u t r i n o o s c i l l a t i o n s , which c o u l d p r o v i d e new p r i n c i p l e s t o be c o n s i d e r e d and new e f f e c t s t o be examined. I t i s t h e r e f o r e i m p o r t a n t t o look f o r an answer t o the q u e s t i o n of whether n e u t r i n o s do o s c i l l a t e , and a v a s t e f f o r t i n v o l v i n g s e v e r a l e x p e r i m e n t s i s now under way i n t h a t d i r e c -t i o n . 3 I I . NEUTRINO MASSES AND MIXINGS As w i l l be shown l a t e r i n t h i s chapter, the existence of o s c i l l a t i o n s requires neutrinos to be massive. This requirement i s supported by other arguments, such as the fa c t that many Grand-Unified models c a l l f or massive neutrinos while no s a t i s -f a c t o r y p r i n c i p l e has been found that enforces a zero mass [ 7 ] . There i s a l s o some p r e l i m i n a r y evidence from experiment that favors the idea of massive neutrinos [8,9]. Neutrino masses could have the property of not conserving lepton number. This means that the p a r t i c l e created at a weak-i n t e r a c t i o n vertex ("weak ei g e n s t a t e " : i t i s the c u r r e n t l y known = ve ,v^,v^) would propagate not as a s.ingle plane wave, as i t would i f i t were a state of d e f i n i t e mass, but as a l i n e a r com-b i n a t i o n of mass eigenstates (to be l a b e l e d ). This w i l l make a pure weak eigenstate evolve i n t o a mixture of a l l p o s s i b l e v^s according to a p e r i o d i c f u n c t i o n of time: t h i s i s Neutrino O s c i l l a t i o n s . 4 1) Mass L a g r a n g i a n and E i g e n s t a t e s In a l a g r a n g i a n f i e l d t h e o r y of f e r m i o n s , the most g e n e r a l mass term p o s s i b l e i s f where, f o r N n e u t r i n o s p e c i e s , / ^  \ I 2/ v = I r 7 r e p r e s e n t s p o s i t i v e - e n e r g y n e u t r i n o s vc = j >#l i s f o r p o s i t i v e - e n e r g y a n t i n e u t r m o s and A,B,D,E are NxN complex m a t r i c e s w i t h d imensions of a mass. R e q u i r i n g CPT i n v a r i a n c e g i v e s (see Appendix) A = A B = B E = D w h i l e d e f i n i n g n = I u>% i; c j a l l o w s ( 2 . 1 ) t o condense i n t o a one-m a t r i x e q u a t i o n : £ = - n a M - n , + * . e . ( 2 . 2 ) where M = 1p g / i s a 2 N x 2 N mass m a t r i x . L e t the f i e l d s n t r a n s f o r m i n t o a new s e t $ t h r o u g h a u n i t a r y m a t r i x V: $ = : = Vtv. ( 2 . 3 ) t B e f o r e r e a d i n g t h i s s e c t i o n , the reader i s i n v i t e d t o c o n s u l t the Appendix f o r an o u t l i n e of the c o n v e n t i o n s used i n t h i s t h e -s i s . 5 E q u a t i o n (2.2) can be r e w r i t t e n as £ _ _ i v i M V + $ + L.c = - I H 1 + L.c. where M 0 = VMV*can be chosen d i a g o n a l , w i t h (M 0 )^ = m^+iy^ (m^ , y a r e r e a l ) , so t h a t A s k i n g M 0 t o be r e a l (which i s e q u i v a l e n t t o making £> CP i n v a r i a n t , see Appendix) g i v e s L = - H rvi CPCP (2.4) T h i s d e f i n e s each f as a mass e i g e n s t a t e , w i t h mass m . i « 2) N e u t r i n o O s c i l l a t i o n s L e t | Z f ( t = 0 ) ^ be a pure weak e i g e n s t a t e w i t h a d e f i n i t e momentum p: I Zj. (t = o) > = I n , (p)> l ^ U 4 ZrvJ At time t>0, t h i s e v o l v e s i n t o I V t ) > * e " i W t U ( p > > = e " ' H t I W + ) W ( p ) > 6 where H i s the f r e e - p a r t i c l e H a m i l t o n i a n , w i t h e i g e n s t a t e s l<f Cp)) and e i g e n v a l u e s E = -Jp2 +m2. The weak e i g e n s t a t e l ^ ^ p ) " ^ has thus become a s u p e r p o s i t i o n of s t a t e s l n p ( P ^ w i t h t r a n s i t i o n a m p l i t u d e s e * and c o r r e s p o n d i n g p r o b a b i l i t i e s ? u) = i < n . K ( t ) > r = i i / v „ e ' i E j L r ( 2 . 6 ) T y p i c a l l y , m^  w i l l be much s m a l l e r than p, so l e t E^  = p+m 2 / 2 p 2 and ^m2 = |m2-m,2|; ( 2 . 6 ) becomes = E l v i l | v f * 2 E V \ w % ' I ^ t / 2 f ( 2 . 7 ) T h i s p r o b a b i l i t y i s a p e r i o d i c f u n c t i o n of t i m e , which d e f i n e s n e u t r i n o o s c i l l a t i o n s . N o t i c e t h a t i f masses a r e z e r o or de-g e n e r a t e , &m?, = 0 and P ( t ) = P„„(0) = and no o s c i l l a t i o n s o c c u r . R e q u i r i n g T- (or CP-) i n v a r i a n c e a t t h i s l e v e l means P ( t ) = P ( - t ) . But - P (+ t ) a c c o r d i n g t o (2.7) To be t r u e f o r a l l t , t h i s i m p l i e s 7 for a l l i n d i c e s <x , p , J , k . I t can be shown [10] that a matrix s a t i s f y i n g t h i s c o n d i t i o n can be turned i n t o a r e a l matrix by r e d e f i n i n g the phases of neutrino f i e l d s : i) —» v e. " <f v e J (2.8) Such a transformation leaves Physics i n v a r i a n t . Any phase in V that cannot be suppressed by t h i s process causes CP v i o l a t i o n . 3) Dirac Neutrinos In the Lagrangian (2.1), the l a s t two terms represent the Dirac masses: X, - - v D v - vc D u c 4 - k . c (2.9) where p a r t i c l e s and a n t i p a r t i c l e s are represented by d i s t i n c t terms. The other two terms couple the neutrino f i e l d to i t s con-jugate. These are known as the Majorana mass terms: L = - y ' A u - u B u c + k.. c. In the case where only Dirac masses are considered, A and B are set to zero and the mass matrix becomes: / J> O M = ( O 5 8 L e t w h e r e X - L )' W = U t a r e N*N m a t r i c e s , so t h a t , and V = u 2 U M , where the £ = V u = and M = V M V t I t 1 » \ t i c u ; + u , 5 u H ' Let U 2 = U 3 = 0. T h i s g i v e s u,u.+ 0 and and X = U, v H 0 = U, 0 u.+ o M, o M.. i / M, and M 4 a r e N*N r e a l , d i a g o n a l m a t r i c e s . When f o c u s i n g on D i r a c P a r t i c l e s , one can r e s t r i c t the p i c -t u r e t o L = - V D v + k.c. (2.10) Let U,, M, be noted U, M so t h a t u> = L \j v and M = UDU +. A l l the f r e e parameters i n the t h e o r y a r e g i v e n i n m a t r i c e s M and U. They are counted as f o l l o w s . 9 The d i a g o n a l i z e d mass m a t r i x M g i v e s N-1 independent <Tm2 . The m i x i n g m a t r i x U has N 2 a m p l i t u d e s and N 2 phases, of the • l a t t e r 2N-1 are absorbed by The r e m a i n i n g ( N - 1 ) 2 phases and N 2 a m p l i t u d e s s u f f e r N 2 c o n d i t i o n s s e t by u n i t a r i t y , which s t a t e s where = T e "J . In the j * k c a s e , t h i s g i v e s (W/)' = I T T e { . 0 . E T T . « - , ' V * h I s j tj A redundancy has appeared, which a l l o w s a compression i n t o the j>k c a s e , l e a v i n g a N(N-1)/2 e q u a t i o n s system, and (N-1)(N-2)/2 phases remain f r e e . f There are N 2-N(N~1)/2 e q u a t i o n s l e f t which a c t as c o n s t r a i n t s on the N 2 a m p l i t u d e s , l e a v i n g N ( N - l ) / 2 of them independent. Thus, a model f o r N - s p e c i e s D i r a c N e u t r i n o O s c i l l a t i o n s w i l l be g i v e n by L a g r a n g i a n ( 2 . 1 0 ) , the p r o b a b i l i t i e s ( 2 . 7 ) , and r e q u i r e s N-1 squared mass d i f f e r e n c e s and a m i x i n g m a t r i x made of N(N-1)/2 a m p l i t u d e s and (N-1)(N-2)/2 C P - v i o l a t i n g phases. t N o t i c e t h a t f o r N = 2, no such phase can t h e r e f o r e e x i s t . 10 I I I . MASS AND MIXING SCHEMES 1) N e u t r i n o Phenomenology The phenomenology of n e u t r i n o o s c i l l a t i o n s can be t o t a l l y d e s c r i b e d g i v e n a s e t of mass s c a l e s and m i x i n g a n g l e s . T h i s t i o n , however, g i v e s but few i n d i c a t i o n s on the a c t u a l v a l u e of such p a r a m e t e r s , f o r the number of n e u t r i n o - i n d u c e d r e a c t i o n s observed i s s t i l l t oo s m a l l . With l i t t l e d e c i s i v e d a t a , d i f -f e r e n t s e t s of parameters can be proposed, e i t h e r out of p h e n o m e n o l o g i c a l c o n s i d e r a t i o n s , or as a by-product of t h e o r e t i -c a l frameworks or h y p o t h e s e s . Of the p h e n o m e n o l o g i c a l hypotheses p r e s e n t l y d e v e l o p e d , those of Barger [ 1 2 ] , de R u j u l a [ 1 3 ] , and c o l l a b o r a t o r s w i l l be c o n s i d e r e d . They c o n s i s t of t h r e e c l a s s e s of s o l u t i o n s , of which here i s an o u t l i n e . S o l u t i o n A accomodates both r e a c t o r and s o l a r e x p e r i m e n t a l d a t a , which d e f i n e bounds f o r 9, and 0 3 , w h i l e 02 i s s e t around 20-25° work f o c u s e s on the t h r e e - s p e c i e s c a s e , where one uses <£m2 <£m 2 3, and a Kobayashi-Maskawa m a t r i x [ 1 1 ] : 1 2 i 11 t o account f o r a s m a l l ve -v/T m i x i n g . The set of a n g l e s i s #i/0 2/03 = 45-50°/20-25°/30° w i t h £= 0°.t The mass s c a l e i s set a t 5 m j 3 = 1 eV 2 and ^m 2 2 = 0.05 eV 2, a l t h o u g h the l a t t e r i s but weakly c o n s t r a i n e d : a n y t h i n g over 10" 3eV 2 seems t o f i t . S o l u t i o n B i n t e r p r e t s r e a c t o r d a t a as measurements a t the lon g wavelength l i m i t . I t s e t s ^y/92/92 a t 55°/0°/45° and <£m 2 3 = 0.25 to 0.5 e V 2 , &m2}2 = 0.05 eV 2. Agreement of t h i s s o l u t i o n w i t h p r e s e n t data does not seem v e r y i m p r e s s i v e . S o l u t i o n C t a k e s the o p p o s i t e v i e w p o i n t and r e g a r d s r e a c t o r data as an average over s h o r t - w a v e l e n g t h o s c i l l a t i o n s . T h i s a l l o w s n e u t r i n o masses t o be l a r g e , though not t i g h t l y c o n s t r a i n e d : o~m 3^ = 10 t o 50 eV 2, and £ m 2 2 = 1 eV 2. A n g l e s are G,/02/O3 = 30 o/50°/55°. 2) The H i e r a r c h i a l Mass Scheme T h e o r e t i c a l p r o p o s a l s w i l l a l s o be c o n s i d e r e d i n t h i s work; they i n c l u d e a mass s c a l e and two m i x i n g schemes. Mass s c a l e s f o r elementary f e r m i o n s may be e s t a b l i s h e d w i t h i n the c o n t e x t of a u n i f y i n g scheme which p l a c e s q u arks and l e p t o n s i n a s i n g l e framework, such as those d i s c u s s e d i n [ 1 4 ] , or t r e a t s them i n p a r a l l e l , t h a t i s c o n s i d e r s a l e p t o n - o n l y mass t No CP v i o l a t i o n i s t o be c o n s i d e r e d i n t h i s work. 12 scheme b e f o r e a t t e n d i n g t o the q u a r k s . In the l a t t e r c a s e , the H i e r a r c h i a l Mass Scheme, where n e u t r i n o masses are s c a l e d w i t h the same r e l a t i v e r a t i o s as charged l e p t o n masses, t h a t i s m. ; m : m = m0 \ rn : m. ~ O, 511 ; i o 5. 6 •. 17 ?M i s among the f i r s t t o be c o n s i d e r e d . P r e s e n t l y , the o r i g i n of l e p t o n masses i s s t i l l u n c l e a r , and charged l e p t o n masses are about the o n l y a v a i l a b l e p o i n t t o s t a r t from; a scheme where n e u t r i n o masses d i f f e r from them by a c o n s t a n t f a c t o r , r e p r e s e n -t a t i v e of what g e n e r a t e s these masses, i s t o be regarded w i t h i n t e r e s t . f Such a scheme i n t r o d u c e s the n o t i o n of m a s s - r e l a t e d f a m i l i e s , \iej, 1 = 1,2,3, which c o i n c i d e w i t h the e l e c t r o w e a k f a m i l i e s [v] ' * = e'/*' T' a t t h e c h a r g e d - l e p t o n l e v e l , i . e . it= e, e t c . . , and not a t the n e u t r i n o l e v e l . A s i m i l a r f e a t u r e a l r e a d y e x i s t s w i t h the u-d quark p a i r , where the m a s s - i s o s p i n d o u b l e t ( j ) i s p a i r e d w i t h a weak c o u n t e r p a r t \ j / , w i t h u w = u . and d = d c o s $ - s s i n ( ? , where 0= 12.6° i s the Cabibbo A n g l e . The s i m i l a r i t y between t h i s mass scheme and the u-d quark p a i r may be f u r t h e r enhanced by a s k i n g l e p t o n s t o undergo the same Cabibbo m i x i n g as the s e quarks do. In t h i s c o n t e x t , Cabibbo m i x i n g i s d e f i n e d by c o n d i t i o n s t For a maximum ana l o g y w i t h charged l e p t o n s , n e u t r i n o masses w i l l be D i r a c - t y p e . 13 T h i s s t i l l a l l o w s many d i f f e r e n t m a t r i c e s t o be d e f i n e d , of which the f o l l o w i n g two are r e t a i n e d as an i l l u s t r a t i o n : M a t r i x D, d e f i n e d by ( £ ( * f , ( £ ) ' . t , * which g i v e s 9,/$2/93 = 17.55°/45°/45°, and M a t r i x E, d e f i n e d by ( — ) - = t ¥h) = V 0 l u : / l^rJ 3 & g i v i n g 9i/02/&3 = ^/0°/0°, which i s of co u r s e an extreme c a s e . One c o u l d a l s o c o n s i d e r the Cabibbo a n g l e not as an i n d e -pendent c o n s t a n t but as a m a s s - r e l a t e d q u a n t i t y , and ask f o r a m a t r i x of the same k i n d t o be b u i l t f o r l e p t o n s . An example of mass-dependent m i x i n g i s the H i e r a r c h i a l m i x i n g , d e f i n e d by where m < /m> i s the s m a l l e r / l a r g e r of m^  and m^. Such a scheme y i e l d s l i t t l e m i x i n g : 6^ = 4° a t b e s t . A d e f i n i t i o n s i m i l a r t o m a t r i x D's g i v e s M a t r i x F, w i t h @j92/62 = 4 .1 °/l 3 .7 °/i 3 . 7 0 . The m i x i n g m a t r i c e s used i n t h i s work a r e summarized i n Table I . 3) Curves of P r o b a b i l i t y v e r s u s L/E Curves of P^e vs L/E a r e shown on F i g u r e s 1 th r o u g h 4. With L/E r a n g i n g from 0.4 t o 6 m/MeV, phenomenological schemes A and B p r o v i d e l i t t l e o s c i l l a t i o n s w h i l e s o l u t i o n C y i e l d s s i g n i f i -1 4 can t e f f e c t s , as w i l l t he H i e r a r c h i a l Mass Scheme, p r o v i d e d the masses a r e p r o p e r l y chosen. W h i l e the c u r v e s have t h e i r L/E de-pendence s e t by the mass s c a l e , t h e i r a c t u a l h e i g h t depends on the s e t of m i x i n g a n g l e s , and here a g a i n e f f e c t s v a r y . I t i s w o r t h w h i l e t o remark t h a t Cabibbo m i x i n g s D and E show s i m i l a r e f f e c t s , so t h a t the freedom l e f t by d e f i n i t i o n s (3.2) has l i t t l e impact. The Cabibbo case can t h e r e f o r e be r e s t r i c t e d t o m a t r i x D. A remark has t o be made on the shape of the s e c u r v e s . As can be seen from the example of F i g u r e 5, the o s c i l l a t i o n p a t -t e r n can be i d e n t i f i e d as a ' c a r r i e r ' wave s u p p o r t i n g a s h o r t -wavelength ' s i g n a l ' , bound w i t h i n an 'envelope'. T h i s can be shown th r o u g h a lo o k a t e x p r e s s i o n (2.7) f o r P : PM,(«•/*; = L Lj U U U e " ' ( ^ ' ^ ) L / z E /*« j , k iJ* je ly. ke s L W S W J 1 + * E l{ U U U coiSvyJ- L / 2 E (3.3a) = + a z , c " i r * a » ^ f H | + \ ( 3 ' 3 b ) where « = 2 U I L U U +~ = 1 cSv*.2 and f£e - L 1^1* I ^ J 1 * s t f t e average p r o b a b i l i t y . The f i r s t two terms i n (3.3b) d e f i n e the c a r r i e r wave. W i t h f 3 1 = f 3 2 + f 2 1 / the l a s t two terms can be m o d i f i e d i n t o 1 5 S i n c e f 2 1 : f 3 2 = £ m 2 2 : <Sm22 = 42700:12462000, i t can be seen t h a t the f r e q u e n c y f 3 2 + f 2 1 / 2 = f 3 2 i s v e r y h i g h indeed compared t o f 2 i , and w i l l y i e l d merely unobservabl.e ' f a s t ' o s c i l l a t i o n s , bound w i t h i n an envelope by the f a c t o r s of f 2 i / 2 f r e q u e n c y , so t h a t one w i l l be c o ncerned w i t h the ' c a r r i e r ' p r o b a b i l i t y P°0 - P p -+- a cos f _ L / E m (3.4) p l u s the envelope (3.5) TABLE I: SUMMARY OF MIXING MATRICES Type Mixing Angles (degrees) p t*e. I Pheno. 50 25 30 0 .434 Pheno. 55 0 45 0 .221 Pheno. 30 50 55 0 . 1 57 Cabibbo 17.6 45 45 0 .085 Cabibbo 12.6 0 0 0 .091 H i e r . 4.1 13.7 13.7 0 .010 17 F i g u r e 1 : P^P vs L/E f o r S o l u t i o n A (Phenomenological) w i t h <fm22 = 0.05 eV 2 and om 2 2 = 1. eV 2 Arrow a t r i g h t shows p o s i t i o n of average p r o b a b i l i t y F i g u r e 2: P ^ evs L/E f o r S o l u t i o n B (phe n o m e n o l o g i c a l ) w i t h <5m2 2 = 0. 05 eV 2 and 6m 2 2 = 0. 25 eV 2 LO O o I I I I I I I I N 10-' 3 .5 7 10° 5 1 10 2 F i g u r e 3: P^e v s L/E f o r S o l u t i o n C (phen o m e n o l o g i c a l ) w i t h cfm 2 2 = 1. eV 2 a n d £ m 2 2 = 10. eV 2 Arrow a t r i g h t shows p o s i t i o n of average p r o b a b i l i t y in o F i g u r e 4: P r e vs L/E f o r the H i e r a r c h i a l Mass Scheme : w i t h M a t r i x D . : w i t h M a t r i x E : w i t h M a t r i x F A l l w i t h mt = 0.05 eV 1—I I I I I I ^ 7 10 Figure 5: An example of f a s t o s c i l l a t i o n s Curve of P^ e vs L/E: s i g n a l , c a r r i e r , and envelopes. H i e r a r c h i a l Mass Scheme, m, = 0.05 eV , Matrix B (phenomenological). Most of the region between the envelopes should be a l l black. 20 IV. NEUTRINO-ELECTRON SCATTERING The Weinberg-Salam Model has succeeded i n merging Quantum E l e c t r o d y n a m i c s and the Fermi f o u r - p o i n t Theory of Weak I n t e r a c t i o n s i n t o one framework; i t a d e q u a t e l y d e s c r i b e s the e x i s t e n c e of n e u t r a l weak c u r r e n t phenomena. I t has become the s t a n d a r d d e s c r i p t i o n f o r e l e c t r o w e a k p r o c e s s e s , and i t w i l l be taken as such i n t h i s work. i The new parameters i t i n t r o d u c e s a re m a i n l y the masses of weak gauge bosons, which w i l l be i r r e l e v a n t a t the e n e r g i e s con-s i d e r e d h e r e , and the Weinberg A n g l e , which g i v e s the m i x i n g between e l e c t r o m a g n e t i c and n e u t r a l c u r r e n t s . T h i s l a s t number w i l l be best measured i n a p u r e l y weak r e a c t i o n where no s t r o n g i n t e r a c t i o n s can mask i t s e f f e c t . The one chosen here i s n e u t r i n o - e l e c t r o n s c a t t e r i n g , whose c r o s s - s e c t i o n w i l l now be c a l c u l a t e d . 21 1) N e u t r a l - c u r r e n t C r o s s - s e c t i o n L e t c< be a s p e c i e s of weak e i g e n s t a t e s , i . e . c * = e r / M - o r - r . The v^e s c a t t e r i n g r e a c t i o n proceeds through e i t h e r n e u t r a l or charged c u r r e n t s , which a r e p i c t u r e d by F i g u r e s 6 and 7: F i g u r e 6 F i g u r e 7 N e u t r a l - c u r r e n t C h a r g e d - c u r r e n t vKe - vK e s c a t t e r i n g vK e - ve cx s c a t t e r i n g I t can be seen t h a t the c h a r g e d - c u r r e n t exchange p r o c e s s i n -v o l v e s t he c r e a t i o n of an ex l e p t o n ; u n l e s s oc = e, i t w i l l be s u b j e c t t o a k i n e m a t i c t h r e s h o l d . F u r t h e r m o r e , i f oc * e, r e a c -t i o n o u tput p r o d u c t s from n e u t r a l and charged c u r r e n t s a re d i s -t i n g u i s h a b l e , and i n t h i s case c r o s s - s e c t i o n s a r e always c a l c u -l a t e d s e p a r a t e l y . L e t the v e n e u t r a l - c u r r e n t c r o s s - s e c t i o n be c a l c u l a t e d f i r s t . The m a t r i x element a s s o c i a t e d w i t h F i g u r e 6 i s 71luc = | M* [ £ ( p ' ) ^ ( c w t c A y s ) e C p ) ] VfK) LviV) £ ( . -> / ) u ^ j ] (4.1) where G = 1.026. 10~ 5 M"2, i s the Fermi Constant proton M z^ 90 GeV i s the Z° Boson mass, k = p'-p = q-q' i s the 4-momentum t r a n s f e r , D ^ v(k) = -i - 3^"" U ^ /H z i s t h e zo p r 0 p a g a t o r , 22 c v = 2 s i n 2 # - 1 / 2 c„ = 1/2 (4.2) where s i n 2 # = 0.229, @, b e i n g the Weinberg a n g l e . In the e x p e r i m e n t s s t u d i e d i n t h i s work, the t o t a l i n i t i a l energy i s 270 MeV a t the v e r y most, so t h a t k « M z , and D^{k) can be reduced t o i g ^ / M 2 , w i t h v e r y good a p p r o x i m a t i o n , and Summing ]TTZ^  | 2 over o u t g o i n g and a v e r a g i n g over i n i t i a l e l e c t r o n s p i n s g i v e s t S = i | l T . l ^ y ^ c ^ c ^ ^ ^ ^ c ^ c ^ M J T . J ^ y ^ . - ^ ) ^ ^ ! - ^ ) } (4.4) E v a l u a t i n g t r a c e s y i e l d s °& = 16 j (^-S) lp.c| p'-V + ( c u + c j ' p..,' p'.c, ] (4.5) E q u a t i o n (4.5) i s the f i n a l e x p r e s s i o n f o r the s p i n - a v e r a g e d squared s c a t t e r i n g m a t r i x element; i t now can be e n t e r e d i n t o the c r o s s - s e c t i o n 2 5 (trr)5 (Z£') (£ff)» CZco') where p = (£,p), q = (co,q) , e t c . . . , and s = (p+ q ) 2 . E x p r e s s i n g g i v e s C / < T = _ g i _ £ i ' ^ ( p t v p ' - i ' l i K - ^ p i P ' - i ' * < s * o V i Y - i } ( 4 - 6 ) ZrTlS £' u>' A frame now has t o be chosen. S i n c e the experiment asks f o r i n -c i d e n t n e u t r i n o s t o s c a t t e r on e l e c t r o n s t r a p p e d i n m a t t e r , the 23 l a b frame can be taken as the one where p = (m e,0), and s_. ( p + ^ ) Z - rn/ + Z me w 2 ™ e (4.7) Wi t h ck> >20 MeV, the e l e c t r o n mass i s n e g l i g i b l e beyond the f i r s t o r d e r , and even more so f o r n e u t r i n o masses. Choosing p' such t h a t p'*q = |p'||q|-cos^ = t'wcos 0 and i n t e g r a t i n g over d 3 q ' u s i n g <£3(q-p'-q') g i v e s q'= q-p' and co = | q-p' I = "/co1 -* &'1 " 2£.'co COS& (4.8) T h i s y i e l d s ;+m| Le t d 3p'= dcpdcostf dp'p' 2 = d cf dcos 0 d t ' £' 2 v i a £ ' = Vp' 2-I n t e g r a t i o n over d g i v e s 2 , w h i l e u s i n g (4.8) and the r e l a t i o n S(F-{<.*)) = < f U - x J / | ^ c x j 3 (4.9) w i t h here | ( c o s ^ ) = c o ' , F = m+ w - e , m&«-^, y i e l d s an L e t y=£'/u> and E y = t-> . One can now w r i t e a d i f f e r e n t i a l c r o s s -s e c t i o n (4.10) . wc — — e o Err and the t o t a l c r o s s - s e c t i o n o Zi 24 2 ) Other c r o s s - s e c t i o n s Other c r o s s - s e c t i o n s a r e c a l c u l a t e d u s i n g a s i m i l a r method. In the (x * e c a s e , the f i n a l s t a t e of c h a r g e d - c u r r e n t exchange s c a t t e r i n g c o n t a i n s an o c l e p t o n w i t h a n o n - n e g l i g i b l e mass M. The c r o s s - s e c t i o n f o r t h i s p r o c e s s i s c a l c u l a t e d from m c c = - M ! f > < r " ^ 0 - y 5 > ^ t ) ] R T w U'i'J % («-y3) e ( P ) ] ( 4 . 1 2 ) w i t h D " V ) = -i U*»U"/m* ^ * T / V Ic'1 - M i s i n c e M w > 79 GeV » k'. W i t h s = (p+q) 2 = 2mw , t h i s y i e l d s a d i f f e r e n t i a l c r o s s - s e c t i o n J f c c = 9L { s - M2-) ( 4.13) j IT where a t h r e s h o l d appears a t s = M 2 ,• above which the t o t a l c r o s s - s e c t i o n i s SC. = CC ) ^ il = Gl ( j ^ ) 1 (4.14) In the oc = e c a s e , n e u t r a l and charged c u r r e n t s i n t e r f e r e ; t h i s i s r e p r e s e n t e d by a d d i n g up m a t r i x elements 7ft c, g i v e n by ( 4 . 3 ) , and TH , g i v e n above i n (4.12) and made c o m p a t i b l e w i t h the (4.3) format by s e t t i n g D w (k') = i g ^ * /M 2 W and by r e p o s i -t i o n i n g l e p t o n f i e l d s ( F i e r z T r a n s f o r m a t i o n ) t o make M>cc = i G [ 5 C p l ) y e c P ) ] [ uf,.j y ^ f i - y * ; u(<^)] (4.15) and 25 W a = ' % t + 7 ^ e = [ ec r-)^{lc y t o + ( c A - . ) ^ | efp)][u(<,"; y ^ f t - y ^ u f , ) ] (4.16) T h i s y i e l d s J r w e = 5! m e E j ( c v - c ^ O l -f ( c ^ c A ) M . - 3 ) 1 j (4.17) = ^ M E J ( C W - C a + 2 ) 1 + i ( c v + c j * } (4.18) Zr, T h i s study i s now completed by a loo k a t a n t i n e u t r i n o - e l e c -t r o n s c a t t e r i n g c r o s s - s e c t i o n s . C a l c u l a t i o n s s t a r t from diagrams i n F i g u r e s 8 and 9: F i g u r e 8 F i g u r e 9 N e u t r a l - C u r r e n t C harged-Current v« e - i/„ e S c a t t e r i n g Ue e - v„ e S c a t t e r i n g V e r t i c a l arrow shows d i r e c t i o n of time f o r both diagrams and the c o r r e s p o n d i n g m a t r i x e l e m e n t s , i n a ( 4 . 3 ) - t y p e format I K - = ^£ [e(p') y ^ K + ^ y ^ e CP>] I D c1'>y^( '+/5 ) ^  ci> 1 TJt^ = ^ [ 5Cp'j y (' + y s) e Cr>3 L 5 on y ^ h + y 5 ; i/^; 1 where the r i g h t - h a n d e d n e s s of a n t i n e u t r i n o s d e t e r m i n e d the use of (1 + y s ) / 2 i n s t e a d of the l e f t - h a n d e d p r o j e c t o r ()-/)/2. 26 Working out the c r o s s - s e c t i o n s g i v e s <r_ = 9l weE-u l C c „ , c A ) % i K - e j ' J (4.19) <r- a £ „ E, { ( c , + c j l * i ( s - w ) 1 1 (4.20) 3) C r o s s - S e c t i o n s and the Weinberg Angle U s i n g d e f i n i t i o n s ( 4 . 2 ) , one can r e w r i t e c r o s s - s e c t i o n s i n terms of the Weinberg a n g l e : _ G* m E . { - f i ^ e -Hs<nl9+-l\ (4.21a) II ^ e = ^1 <v,eEy { l i s i . " ^ w + M ^ 6 » w - 1 ] (4.21c) I t appears a t once t h a t the Weinberg a n g l e # w i s the o n l y parameter p r e s e n t t h a t comes from the Weinberg-Salam Model. T h i s makes the s e r e a c t i o n s an e x c e l l e n t t o o l f o r both measuring s i n 2 0W and a s c e r t a i n i n g the v a l i d i t y of the model, t h r o u g h a comparison of r e s u l t s i n v o l v i n g d i f f e r e n t t y p e s of n e u t r i n o s . F u r t h e r m o r e , w i t h a c u r r e n t l y a c c e p t e d v a l u e of s i n 2 . ^ = 0.229, 27 one has the o r d e r i n g ^Vc : ; ^be ' ^ e'-^c - °- 3 1 : °- 3 C : °32 : 2.2o : q.O (4.22) which w i l l be of importance when f l u x c o n t a m i n a t i o n s and o s c i l -l a t i o n s a re c o n s i d e r e d . 28 V. PHENOMENOLOGICAL QUANTITIES I t has been seen i n Chapter 4 t h a t v-e s c a t t e r i n g i s an e x c e l l e n t t o o l f o r measuring the Weinberg A n g l e . However, t h i s p r o s p e c t may be marred by the occurence of n e u t r i n o o s c i l l a -t i o n s . In t h i s c h a p t e r , n e u t r i n o m i x i n g and v-e c r o s s - s e c t i o n s are combined i n t o a s e t of phe n o m e n o l o g i c a l q u a n t i t i e s whose v a l u e s can be measured d i r e c t l y and p r o v i d e a c o r r e c t p i c t u r e of the n e u t r i n o o s c i l l a t i o n s d e t e c t e d , i f t h e r e a re any. 1) The Brookhaven Experiment The experiment s t u d i e d i n t h i s work c o n s i s t s of P r o p o s a l s E704 and E764, both t o be performed a t Brookhaven by a BNL-Brown-KEK-Pennsylvania-Stony Brook-Tokyo c o l l a b o r a t i o n . I t i s d e s c r i b e d i n r e f e r e n c e [ 1 5 ] , Here i s an o u t l i n e of i t . A f l u x of monoenergetic p r o t o n s s t r i k e s a p r i m a r y t a r g e t where p o s i t i v e p i o n s a r e e m i t t e d as a r e s u l t . They a r e c o l l i -mated and a l l o w e d t o decay i n t o ^ p a i r s . The beam i s then s t r i p p e d of any charged p a r t i c l e s i t c o n t a i n s , and a r r i v e s a t the secondary t a r g e t (the d e t e c t o r ) , 11Om downstream from the p r i m a r y , as a v r beam w i t h a 1.5% c o n t a m i n a t i o n of i / e and 17^  29 from p i o n ( T T +- e4" vt ) and muon e + v e 1^  ) decays. The t a r g e t i t s e l f i s a 23m-long composite d e t e c t o r . N e u t r i n o e n e r g i e s range r o u g h l y from 20 t o 270 MeV, w i t h d i s t r i b u t i o n c u r v e s shown on F i g u r e 10. A c c o r d i n g t o t h i s d e s c r i p t i o n , o b s e r v a b l e e v e n t s i n c l u d e v e - v e , V e ~ "e A*"" ' p l u s n e u t r i n o - n u c l e u s r e a c t i o n s . However, the t h r e s h o l d on - i/{oc g i v e n by (4.13) i s E u > (M 2-m ( 2)/2m e = 11 GeV f o r <* = /+., which i s f a r beyond the expe r i m e n t ' s energy range. T h i s l e a v e s v e - v e as the o n l y p u r e l y l e p t o n i c p r o c e s s a v a i l a b l e . The t o t a l number of v e - v e ev e n t s g i v e n by a ru n , a s s u -ming z e r o m i x i n g , i s where W = ^ Jfe $ /jE„ 3 ( E „ ) ( T C E J (5.1 ) c/4£ i s the t o t a l number of p r o t o n s h i t t i n g the p r i m a r y t a r g e t JQis the number of e l e c t r o n s i n the d e t e c t o r ( t a r g e t e l e c t r o n s ) <^p i s a d i m e n s i o n l e s s c o n s t a n t d e p i c t i n g the d e t e c t o r ' s ef f i c i e n c y ^ ( E w ) i s the neutrino-beam energy d i s t r i b u t i o n g(EU) i s the v^e - v^e c r o s s - s e c t i o n , which i s g i v e n by (4.21a) and can be w r i t t e n as where <r, = 9l m E IP (5.2) 30 Thus where I = jdE^ E u ^ ( e J and ^ = <^ 2 - M e gathers a l l the constants. S i m i l a r l y , a number of events per e l e c t r o n energy can be b u i l t out of d 6" /dy P ( E « ) = * > £ ^ JJE„ 7fE,) J£ ( E , ) 0 f E „ - E e ) (5.4) where £?(EU-E£) i s the Heaviside step f u n c t i o n . With 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 given by « l A ^ . + A ^ E e / E M - E^/e: ] (5.5) one gets D ( c e) = }( j A f t t l \ ) + A ^ ^ I ^ I + V ^ I(£e)j ( 5. 6) in an obvious n o t a t i o n . 2) O s c i l l a t i o n s - r e l a t e d Q u a n t i t i e s Upon f o l d i n g o s c i l l a t i o n s i n t o these expressions, i t has to be considered that what i s observed i s weak e i g e n s t a t e s , so that both the p r o b a b i l i t y of o s c i l l a t i n g i n t o some species i/* and the s c a t t e r i n g c r o s s - s e c t i o n s p e c i f i c to that species are taken 31 i n t o a c c o u n t . The f i r s t q u a n t i t y appears i n a s p e c i f i c beam d e n s i t y , d e f i n e d by where L i s the d i s t a n c e t o the ( p o i n t - l i k e ) s o u r c e , P ^ j E ^ L ) i s the v t o vK m i x i n g p r o b a b i l i t y , g i v e n by (3.3a) , ^(E^) = 2. R^{EU,L) i s the o r i g i n a l f l u x d e n s i t y . C r o s s - s e c t i o n s d i v i d e i n t o pure n e u t r a l - c u r r e n t f o r <x. * e and mixed NC-CC f o r <x = e, so t h a t i n f a c t a l l c r o s s - s e c t i o n s f o r <x * e are e q u a l . T h i s s i m p l i f i e s the p i c t u r e t o the f o l l o w i n g : L e t s- = ^ ! w E„ I? The t o t a l number of events ^ o S C = X ^ ^ - ^ L ^ where ' I„ ( O = (E,,l ) E u o< E u s p l i t s i n t o l N e S 6 = X ( O i I e + ^ ^ O and H s c , 7\ ( r p e i e + Z l i j lW ( e j + IW (5.7) s i n c e CP = F and I I , = I t I I = 1 S i m i l a r l y , 32 D o s c ( L , E c ) = 0 ( E . ) Thus, the experiment i s a c t u a l l y s t r e a m l i n e d i n t o merely comparing the observed numbers of ev e n t s fN/OJc and T2 O S c t o t h e i r p r e d i c t e d z e r o - m i x i n g v a l u e s fN and ~D , t h r o u g h t h e i r r a t i o s , e x p r e s s e d i n terms of the Weinberg a n g l e : 61(0 = H„ e = ± + ^ T * ( L ) IN ^ 1 = 1 + . *"«G" ^ l L ) (5.8) ! | W © w - H J I » * © w + I 1 ^ ( E e D = D « c _ d + ? s , ^ 0 w l ' " ( E . , L ) ( 5 ' 9 ) D A ( E e ) where A(e e) = L<Kn"6> + ( 2 s , ^ - i ) 1 ] 1'°' - ,^,^ (9 E I'"* W £ E l I f i' w i t h s i n 2 # = 0.229. Thanks t o the f a c t t h a t a l l a r e e q u a l f o r <*. * e, and t h a t A = A , A = A , c a l c u l a t i o n s towards p h e n o m e n o l o g i c a l p r e d i c t i o n s a r e s t r e a m l i n e d i n t o computing one set- of p r o b a b i l i -t i e s (P^.(E..,L)) and two t y p e s of mixing-dependent i n t e g r a l s , I (L) and i ' 0 1 ( E e , L ) . F i g u r e 10: F l u x d e n s i t y d i s t r i b u t i o n s S o l i d : E704 P r o p o s a l Dashed: E764 P r o p o s a l Curves r e p r e s e n t £J(E W) i n u n i t s of 10" 1 3cm~ 2proton"'MeV~ v e r s u s E y i n MeV. 34 V I . PREDICTIONS FOR THE BROOKHAVEN EXPERIMENT 1) C o n s t r a i n t s on Accur a c y The parameters p r o v i d e d by the hypotheses o u t l i n e d i n Chapter 3 a r e now combined w i t h the f l u x d e n s i t y c u r v e s f o r the Brookhaven e x p e r i m e n t s (see F i g u r e 10) t o compute q u a n t i t i e s 6{ih) and 5"(L,E^), d i s c u s s e d i n Chapter 5. The wide range of o s -c i l l a t i o n s parameters a v a i l a b l e and the l i m i t s t h a t e x i s t on the e x p e c t e d a c c u r a c y w i l l r e s t r i c t t h i s s t u d y i n t o something f a r from e x h a u s t i v e . The e x p e r i m e n t ' s a c c u r a c y i s c o n s t r a i n e d not o n l y by the l i m i t e d r e s o l u t i o n of the d e t e c t o r , but a l s o by f a c t o r s r e l a t e d t o the experiment i t s e l f . Here are the main c o n s t r a i n t s . The p o s i t i o n of the a c t u a l n e u t r i n o s o u r c e , which i s the p o i n t where p i o n s knocked out of the p r i m a r y t a r g e t decay i n t o a p a i r , depends on the p i o n s ' k i n e t i c energy, which i s d i s t r i b u t e d over a c e r t a i n range, and can make them t r a v e l q u i t e a d i s t a n c e b e f o r e they decay (~10m a t 160 MeV t o ~110m a t 1 GeV), or a t l e a s t they get s t r i p p e d o f f the beam. T h i s means a generous e r r o r has t o be added -to any v a l u e of the d i s t a n c e L. The f l u x d e n s i t y c u r v e s vs E„, F i g u r e 10), a r e merely c o m p u t a t i o n s worked out of pi o n - d e c a y P h y s i c s and the geometry of the e x p e r i m e n t a l s e t - u p . T h i s a f f e c t s the a c c u r a c y on E w . The number of events i t s e l f i s s u b j e c t t o an e r r o r , p a r t l y be-35 cause of the i n i t i a l contamination of the f l u x by ve and v 's. Since S" = 6^, , i t s e f f e c t i s relevant although i t s amount ("-'1.5%) i s not l a r g e . F i n a l l y , i t must be considered that the c u r r e n t l y a v a i l a b l e data on i/^e s c a t t e r i n g shows dis c r e p a n c i e s between d i f f e r e n t e x p e r i -ments, due in good part to the d i f f i c u l t y of o b t a i n i n g events, as a consequence of the very small c r o s s - s e c t i o n involved: 1.6-10" 3 7 cm2 MeV"1. Although the present generation of experiments i s expected to show a b e t t e r e f f i c i e n c y , a wide er r o r margin w i l l s t i l l e x i s t . A l l t h i s means that what i s to be looked f o r are patterns rather than curves and numbers of meaningless p r e c i s i o n ; our c a l c u l a t i o n s w i l l t herefore be r e s t r i c t e d to a few examples that w i l l h i g h l i g h t the main features of the proposed schemes. 2) Choice of Curves and Schemes We now narrow down to a small but r e p r e s e n t a t i v e set of parameters. The l a t t e r are already set i n phenomenological schemes, while i n the t h e o r e t i c a l ones a mass sc a l e s t i l l has to be f i x e d . In the H i e r a r c h i a l Mass Scheme, t h i s i s done by just g i v i n g one of the masses, say the l i g h t e s t , m,. Since the argu-ments of the cosine functions i n P have the form L/ , ' OK, C U u the magnitude of squared mass d i f f e r e n c e s a f f e c t s the dependence 36 of the c u r v e s on d i s t a n c e , as can be seen on F i g u r e 11 where p l o t s of the r a t i o ( d e f i n e d i n (5.8)) v e r s u s L a r e drawn f o r t h r e e d i f f e r e n t mass s c a l e s ; they are but the same c u r v e , s t r e t c h e d d i f f e r e n t l y . The a c t u a l h e i g h t of the c u r v e i s a mea-sure of the amount of m i x i n g , i . e . of the m i x i n g a n g l e s . T h i s a l l o w s the same p l o t s t o be s t u d i e d i n two independent ways i n o r d e r t o r e t r i e v e the o s c i l l a t i o n s p a rameters. The c u r v e f e a -t u r e s two o s c i l l a t i o n s w i t h a r a p i d damping towards the averaged ( l o n g - r a n g e ) m i x i n g v a l u e . A s i n g l e v a l u e f o r m, w i l l t h e r e f o r e be used t h a t w i l l e x h i b i t t h i s b e h a v i o u r w i t h i n a r e a s o n a b l e range of d i s t a n c e s , here 0 < L < 150m. The c u r v e used i n F i g u r e 11 i s a ' c a r r i e r ' c u r v e , t h a t i s , a p l o t i n v o l v i n g P^ e, d e f i n e d i n ( 3 . 4 ) , i n s t e a d of . I t l e a v e s out two o t h e r c h a n n e l s t h a t i n v o l v e the h e a v i e s t n e u t r i n o <f . These have been shown t o o s c i l l a t e v e r y r a p i d l y , so t h a t t h e i r c o n t r i b u t i o n s get averaged out as these o s c i l l a t i o n s a re i n t e g r a t e d over the energy spectrum. A graph of GL vs L - u s i n g M a t r i x A (phenomenological) shows how t h i s works ( F i g u r e 12). The envelope d e f i n e d i n Chapter 3 appears c l e a r l y t o have l o s t a l l i t s meaning. S i n c e no o s c i l l a t i o n s of comparable a m p l i t u d e are t o be encountered l a t e r on, the whole s t u d y can be r e s -t r i c t e d t o ' c a r r i e r ' c u r v e s . F i g u r e 13 shows p l o t s of (H vs L f o r b o t h E704 and E764 p r o p o s a l s . P a t t e r n s are a l i k e , and the d i f f e r e n c e between the c u r v e s i s a f a c t o r of f o u r i n the worst c a s e . G i v e n the s i m i l a r shapes and the expected a c c u r a c y of the e x p e r i m e n t , most of the 37 i n f o r m a t i o n can be o b t a i n e d w i t h one s i n g l e p r o p o s a l , here E764. Ap a r t from (R, , the o t h e r q u a n t i t y t h a t can be p l o t t e d i s (L,EC), d e f i n e d i n ( 5 . 9 ) . A study of can p r o v i d e some i n f o r m a t i o n about the a c t u a l shape of the n e u t r i n o f l u x - d i s t r i -b u t i o n , v i a _ Zf (L,E e) = 1 + where J and A a r e i n t e g r a l s over E w c o n t a i n i n g a £ ( E 0 - E e ) . U s i n g A - £ one g e t s where ^ and are i n t e g r a l s over the E e < EA < E e + £ i n t e r v a l . However, i n an experiment where the t o t a l number of e v e n t s i s but a few dozens, t h i s i s l i k e l y t o be of l i t t l e use. F i g u r e 14 g i v e s a sample of how Zf behaves. Due t o the shape of the ^ ( E u ) f l u x d e n s i t y , no s i m p l e p a t t e r n was t o be ex p e c t e d i n the energy or d i s t a n c e dependences. 3) Curves of Ow v e r s u s L The main q u a n t i t y chosen t o be p l o t t e d i s the r a t i o (R> of t o t a l number of e v e n t s , d e f i n e d i n e q u a t i o n ( 5 . 8 ) . G^> uses a l l the a v a i l a b l e e x p e r i m e n t a l d a t a and a l l o w s an e s t i m a t e of mass 38 and m i x i n g parameters t h r o u g h r e l a t i v e l y s t r a i g h t f o r w a r d r e l a -t i o n s h i p s , much s i m p l e r than those a study of V would i n v o l v e . F i g u r e s 15 through 17 p i c t u r e p l o t s of 51 vs L f o r the v a r i o u s s o l u t i o n s d i s c u s s e d i n t h i s work. F i g u r e s 15 and 16 show how p h e n o m e n o l o g i c a l s o l u t i o n s be-have. S o l u t i o n s A and B a r e dominated by a £ m 2 2 = 0.05 eV 2, which p l a c e s t h i s experiment i n the l o n g - w a v e l e n g t h l i m i t , and makes i t i n s e n s i t i v e t o such masses ( F i g u r e 15). F i g u r e 16 shows both the C s o l u t i o n and a c u r v e of Cabibbo m i x i n g w i t h h i e r a r -c h i a l masses, m, = 0.005 eV. With £ m 2 2 = 1 eV 2 i n both c a s e s , b e h a v i o u r s a r e s i m i l a r . The two t h e o r e t i c a l s o l u t i o n s a r e compared on F i g u r e 17. W h i l e Cabibbo m i x i n g l e a d s t o s i g n i f i c a n t --and d e t e c t a b l e - -r e s u l t s , the h i e r a r c h i a l case appears merely i n d i s t i n g u i s h a b l e from the no-mixing c a s e . Such a m i x i n g would r e q u i r e a v e r y c a r e f u l a n a l y s i s b e f o r e b e i n g r u l e d out by a n o - o s c i l l a t i o n v e r -d i c t . T h i s completes our study of the a p p l i c a t i o n of some t h r e e -s p e c i e s n e u t r i n o - o s c i l l a t i o n s schemes t o the Brookhaven e x p e r i -ment. I t appears t h a t t h i s experiment i s s e n s i t i v e t o mass s c a l e s S m 2 2 of the o r d e r of 1 t o 10 e V 2 ; s h o u l d n e u t r i n o masses f a l l w i t h i n t h i s range, i t would be a b l e t o p r o v i d e m e a n i n g f u l e v i d e n c e towards an answer t o the q u e s t i o n of the e x i s t e n c e of N e u t r i n o O s c i l l a t i o n s . 39 o CO in C\J o CM 0.0 40.0 80.0 120.0 160.0 200.0 F i g u r e 1 1 : Mass dependence of Curve of G{ v e r s u s L f o r the E764 P r o p o s a l , H i e r a r c h i a l masses, M a t r i x D. : in, = 0.005 eV : m, = 0.010 eV : m, = 0.015 eV Arrow a t r i g h t shows average ( l o n g - d i s t a n c e ) v a l u e f o r 40 F i g u r e 12: (2. vs L, a l l c h a n n e l s H i e r a r c h i a l masses, m, = 0.01 eV, M a t r i x A, E764 P r o p o s a l ; S o l i d : s i g n a l and c a r r i e r Dashed: the e n v e l o p e s 0.0 30.0 60.0 90.0 120.0 150 F i g u r e 13: ft vs L f o r both P r o p o s a l s S o l i d : E704 P r o p o s a l Dashed: E764 P r o p o s a l H i e r a r c h i a l masses, m, = 0.01 eV, M a t r i x D. 25o McV F i g u r e 14: Graph of H i e r a r c h i a l masses, m, = 0.01 Sample v a l u e s : 3"(L=0,E e) = 2^(70,250) = 2.04, tf v e r s u s L and E e eV, M a t r i x D, E764 P r o p o s a l . 1.0, V(L=l50,Ee=0) = 1.35, y( 150,250) = 1.01 43 CD CO in o CM in 0.0 40.0 80.0 - i r 120 .0 160 .0 200 .0 F i g u r e 15: ft vs L f o r S o l u t i o n s A and B E764 P r o p o s a l S o l i d : S o l u t i o n A, <Fm22 = 0.05 eV 2 Dashed: S o l u t i o n B, <Sm22 = 0. 05 eV 2 L a b e l e d arrows show averages of (R> f o r A and B 44 F i g u r e 16: Q> vs L f o r S o l u t i o n C S o l i d : S o l u t i o n C, <fm 2 2 = 1.0 eV 2 Dashed: H i e r a r c h i a l masses, m, = 0.005 eV, M a t r i x D. 45 LO o —1> 0.0 i i 1 1 r 30.0 60.0 90.0 120.0 150.0 F i g u r e 17: GL vs L f o r H i e r a r c h i a l masses S o l i d : M a t r i x F ( H i e r a r c h i a l ) Dashed: M a t r i x D (Cabibbo) m, = 0.01 eV 46 REFERENCES 1. V. G r i b o v and B. P o n t e c o r v o , Phys. L e t t . 28B (1969) 43 2. J.N. B a h c a l l and R. D a v i s , S c i e n c e 191 (1976) 264 3. F. R e i n e s et a l . , Phys. Rev. L e t t . 4_5 (1980) 1 307 4. S.M. B i l e n k y and B. P o n t e c o r v o , Phys. R e p o r t s 4J_ (1978) 225 5. E.S. Abers and B.W. Lee, Phys. R e p o r t s 9 (1973) 1 6. C. I t z y k s o n and J.-B. Zuber, Quantum F i e l d Theory, M c G r a w - H i l l , New-York (1980) 7. A. de R u j u l a , T a l k g i v e n a t the 9 t h ICOHEPANS C o n f e r e n c e , V e r s a i l l e s , J u l y 1 9 8 1 8. V.A. Lyubimov et a l . , Phys. L e t t . 94B (1980) 266 9. A.D. Dolgov and Ya.B. Z e l d o v i c h , Rev. Mod. Phys. 53 (1981) 1 10. N. Cabibbo, Phys. L e t t . 7_2B (1978) 333 11. M. Kobayashi and T. Maskawa, P r o g . Theor. Phys. 49 (1973) 652 12. V. Barger et a l . , Phys. Rev. D22 (1980) 1636 13. A. de R u j u l a et a l . , N u c l . Phys. B168 (1980) 54 14. L . W o l f e n s t e i n , T a l k a t N e u t r i n o Mass M i n i c o n f e r e n c e , C a b l e , W i s c o n s i n , Oct. 1980 15. A.K. Mann, T a l k a t LASL Workshop f o r N u c l e a r and P a r t i c l e P h y s i c s , Los Alamos, J a n . 1981 47 APPENDIX 1 ) C o n v e n t i o n s and N o t a t i o n s T h i s work uses the s t a n d a r d c o n v e n t i o n s f o r f e r m i o n f i e l d s and s p i n o r s , L a g r a n g i a n s , and Feynman r u l e s , as d e s c r i b e d i n I t z y k s o n and Zuber [6], w i t h the e x c e p t i o n of a d i f f e r e n t norma-l i z a t i o n f o r s p i n o r s and a n t i s p i n o r s , namely uu = 2m and vv = -2m i n s t e a d of uu = 1 and vv = -1. T h i s means H W( f ,s i i i (r,sl = (b + m) \r (p) V (f) - (*-•«.) With y s - + i y'v^y3' f e l i c i t y p r o j e c t i o n o p e r a t o r s and e i g e n s t a t e s are d e f i n e d as cl = IzX5 a. - "1 + I* 2 « " i -^ = ^ ^ ^ = % \ X s k)V = $ \ X = > M a t r i c e s a r e w r i t t e n u s i n g the c o n v e n t i o n M = M T f o r the t r a n s -pose of M, M f o r i t s complex c o n j u g a t e , and M T f o r i t s hermi-tean c o n j u g a t e . 48 2) D i s c r e t e Symmetries The d i s c r e t e symmetries i n v o l v e d i n t h i s t h e s i s a re charge c o n j u g a t i o n , C P - r e v e r s a l , and CPT. The f i r s t of them appears i m p l i c i t e l y t h r ough the use of the c h a r g e - c o n j u g a t e f i e l d of If- , , d e s c r i b e d as c r e a t i n g a p o s i t i v e - e n e r g y p a r t i c l e w i t h the quantum numbers of r e -v e r s e d . I t i s d e f i n e d by I L C = C ( Zf ) T = C y° Zf_* where C i s the charge c o n j u g a t i o n m a t r i x . H e l i c i t y p r o j e c t i o n s of V? are If-_C = (\f = % ^ (i.e. r i g h t - h a n d e d ) V = * ¥ The CP o p e r a t i o n i s used here on e x p r e s s i o n s l i k e cpMi^. and a c t s as f o l l o w s : = ( c ' V ) 3 m.l t - c ^ ] J'P ' P« Jit ty In p a r t i c u l a r , the e x p r e s s i o n x tf-cf + x Cp ^  becomes * <f 4- +• * 4 cp , so t h a t s e t t i n g i t i n v a r i a n t under CP r e q u i r e s x t o be r e a l . S i m i l a r l y , CPT o p e r a t e s as f o l l o w s : J « JL k « = ^ f y° M ^ j 1 * V Jk 7p = cfT C M C (c7) T - 4} M dpc 

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