I L NUOVO CIME~-TO

Vor,. L H A, N. 4

21 Dicembre 1967

Is There a Lepton Octet? R. A. C~RmTENSEN

Department of Physics, University of California - Berkeley, Cal. (ricevuto il 5 0ttobre 1966; manoscritto revisionato ricevuto il 21 Febbraio 1967)

Summary. - - It is suggested that structure of an octet, and that the the hadrons participate may be of a current-parity selection rule in a

the leptons may weak interactions a current-current multidimensional

exhibit the strong in which they and form governed by eightfold way.

T h e f a c t s t h a t t h e w e a k i n t e r a c t i o n s conserve n e i t h e r t h e l e p t o n s do n o t a p p e a r to p a r t i c i p a t e in a n y o b s e r v e d m a k e s c o n s t r u c t i o n of S U~-type d e s c r i p t i o n s for t h e m a r a t h e r s p e c u l a t i v e business. Several 1/3 s c h e m e s of :Y a n d I a s s i g n m e n t for t h e leptons w i t h c o r r e s p o n d i n g selection rules for w e a k int e r a c t i o n s h a v e b e e n tried, w i t h v a r y i n g degrees of success a n d failure (1). One s u g g e s t e d (very t e n t a t i v e l y ) along these lines was t h e q u a r k l i k e -2/3 t r i p l e t s c h e m e of Gell-Mann (2). I t is s h o w n in 2~3 Fig. 1. T h e charge for p a r t i c l e s in this scheme is g i v e n b y t h e e q u a t i o n (1)

Q

=

13 ~ Y

2

:E- ~ z. T h i s t r i p l e t s c h e m e h a s a n u m b e r of serious difficulties, a m o n g which is t h e f a c t t h a t it

-1/3

Y nor I, and that strong i n t e r a c t i o n s e-~ \ ~

/ / lepton

v

/

A

/~antilepton~ :c/triplet ~ e* --~12 0 1/2 13

Fig. 1. - Lepton triplets.

(1) Cf., P. ROMAN: Theory o] Elementary Particles (New York, 1961), p. 517 and 548. (2) M. GELL-MANN: The Eight]old Way: A Theory of Strong Interaction Symmetry, CTSL-20 (1961), published in M. GELL-MANN and Y. NE'EMA~ (eds.), The Eightfold Way (New York, 1964).

IS TIIERE A L E P T O N

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OCTET~

contains only one t y p e of n e u t r i n o . B u i l d i n g u p o n some i m p l i c a t i o n s of t h e existence of two distinct n e u t r i n o s , one coupled to t h e m u o n a n d t h e o t h e r to t h e electron, G x ~ r o (3) s u g g e s t e d t h a t t h e l e p t o n s be classified as s h o w n in t h e w e i g h t d i a g r a m s of Fig. 2 for positive- a n d n e g a t i v e - h e l i c i t y leptons. Two d i s t i n c t 2 - c o m p o n e n t neutrinos Voa n d v~, b o t h l e f t - h a n d e d , .2 ~-) are t h e n o b t a i n e d in t e r m s of a %/ (+1 ~L (+) single 4 - c o m p o n e n t n e u t r i n o v b y 1 t h e definitions v. = 89 and ~ L~I/~ / L=I %= 89 ~ I n this scheme, the charge of a lepton is g i v e n b y t h e s a m e e q u a t i o n as t h a t for k/ hadrons, namely --2 V e~'+) (2)

XT Q = Z,+=, 2:

v(+) 1

e(§ +

L=-I

where Y(• is a q u a n t u m n u m b e r analogous to the h y p e r e h a r g e Xz -1 of hadrons. F u r t h e r m o r e , in this e~_) v(_) scheme t h e a d d i t i v e lepton n u m -2 P~,+) b e r c o n s e r v a t i o n law is a s s u m e d . I n order to e x p l a i n t h e non-1/2 0 ~/2 1/2 0 1/2 13 o b s e r v a t i o n of reactions such as some of those in the list (115)Fig. 2. - Definite-helicity lepton triplets. (136) below, t h e s e p a r a t e conserr a t i o n of electronic lepton n u m b e r L. a n d m u o n i c l e p t o n n u m b e r JS~ has also b e e n p r o p o s e d (4). H o w e v e r , t h e s m a l l n u m b e r of p a r t i c l e s i n v o l v e d in e a c h of these i n t e r a c t i o n s p e r m i t s us to m a k e a n c u t i r e l y different h y p o t h e s i s . I f we m a k e an isofermion-isoboson d i s t i n c t i o n b e t w e e n electronic a n d m u o n i c letptons, t h e rules of isospin v e c t o r a d d i t i o n can be used to exclude t h e s e reactions. The simplest integer charge e i g h t f o l d - w a y scheme, i.e. i r r e d u c i b l e r e p r e s e n t a t i o n of t h e f a c t o r g r o u p S U3/Z~, which i n c o r p o r a t e s t h e t w o n e u t r i n o s , m a k i n g one an isofermion a n d t h e o t h e r a n isoboson, is a l e p t o n octet. Since we k n o w t h a t v . r ~., a n d since n e u t r a l isobosons in our o c t e t m u s t be t h e i r o w n a n t i particles, therefore the electronic n e u t r i n o s m u s t be isofermions if t h e y are SU3 eigenstates. P u t t i n g t h e electron in t h e s a m e isotopic m u l t i p l e t as t h e

(3) R. GATTO: Nuovo Cimento, 28, 567 (1963). (4) L. B. O]s : Weak Interactions o/Eleme~tary -Particles (New York, 1965), p. 19.

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[~. A. CHI{IS'I'~]NSEN

electronic n e u t r i n o (in order to t a k e a d v a n t a g e of the isofermion-isoboson d i s t i n c t i o n to exclude u n o b s e r v e d reactions), this results in the basic lepton o c t e t s t r u c t u r e s h o w n in Fig. 3, w h e r e 7. is t h e n e u t r a l m e m b e r of the isotriplet (~+, X, ~-) a n d ~/ is an isosinglet. MARSHAK et. al. (5) s u g g e s t e d t h a t X, t h e n e u t r a l m e m b e r of t h e m u o n multiplet, be r e g a r d e d as t h e o b s e r v e d m u o n i c neutrino, v~, a n d be described b y a IKajorana field %(x)---- (1/v/2)[~(x)+v~(x)], where v(x) is a 4 - c o m p o n e n t D i r a c field a n d v*(x) is its c h a r g e c o n j u g a t e . T h e isosinglet ~(x) = (1/v/'2)[v(x)--v~ was a s s u m e d n o t coupled at all. Because of t h e low m a s s of t h e e+ 1 electron, m, ~ 0.511 MeV, t h e conj e c t u r e t h a t its m a s s is electromagnetic in origin h a s a m e a s u r e of reasonableness. H o w e v e r , no similar c o n j e c t u r e w i t h r e s p e c t to t h e m u o n ~- 0 pI~+ would be reasonable. I n p a r t i c u l a r , it seems u n r e a s o n a b l e to p u t ~z a n d v~ in t h e s a m e isotopic multiplet. Mass differences w i t h i n t h e k n o w n isotopic m u l t i p l e t s are all in the order -I ev of a few MeV. So t h e r e is v e r y -I -1/2 o I/2 1 little likelihood t h a t a n y t h i n g b u t strong i n t e r a c t i o n s can a c c o u n t for t h e b u l k of t h e m u o n m a s s m~ Fig. 3. - Lepton octet. 105.7 MeV. T h e f a c t t h a t t h e electronic n e u t r i n o is D i r a c (~, ~ ~) r a t h e r t h a n Major a n a (Vo--: v,) follows f r o m t h e f a c t t h a t t h e reaction ~.+n-+p+e-

does n o t occur, w h e r e a s if ~! were M a j o r a n a the p o s s i b i l i t y of this r e a c t i o n would be implied by the known reaction v.+n -~p+e-. S i m i l a r r e a s o n i n g should b e a p p l i c a b l e to t h e m u o n i c n e u t r i n o to deter-

(5) •. F~. MA.RSHAK, C. RYAN, T. K. RADItA and K. RAMAN: Phys. Rev. Left., 11, 396 (1963).

IS

TIIERE

zt- I . F . P T O N

mine whether or n o t it is Majorana.

1041

OCTET~

The reactions

~---~ ~ - + q ~ ~.§

--> ~ + + n ,

=+ -~ ~+ + % ~-~ v~ + n --> ~ - § have been observed at C E R N (+). Thus, if ~ ~ v~, we would e x p e c t to see the reactions 7v- --~ [z-+q~ L-~ q ~ + n - ~ [ z - + p , ~+ ~ ~+ § i ,~ v ~ + p - - + ~ + + n .

These reactions have not been o b s e r v e d (7). F o r these reasons it m a y be f r u i t f u l to investigate an a l t e r n a t i v e a s s i g n m e n t for Z and ~b. The possibility which we will consider is )~-----tL~ a n e u t r a l m u o n with a mass roughly the same as t h e charged m u o n mass, a n d (3)

1 ,p ---- - - ~ (v, + ~l,) , %/2

where v~ is the observed zero-mass m u o n i c neutrino.

The c o m b i n a t i o n

1

is t h e n regarded as the S Ua singlet a c c o m p a n y i n g this lepton octet, in the same sense ~s V/3 9 + v / ~ o j accompanies t h e v e c t o r m e s o n octet. B o t h [z~ a n d ~ are their own antiparticles, b u t only ,~ is ~ a j o r a n a since t~~ is massive. I n the scheme of M a r s h a k et. al. (8) t h e leptons are assigned a (c w e a k h y p e r c h a r g e >)analogous to b u t d i s t i n c t f r o m the s t r o n g h y p e r c h a r g e of h a d r o n s . L, B and Y~,,.k) are then assumed s e p a r a t e l y conserved b y each t e r m in s

(e) Sienna conference report o[ CERN neutrino experiment (1964). See H. MUIRHRAD: The Physics o] .Elementary Particles (New York, 1965), p. 337. (~) Sienna em~]erence report o] CERN neutrino experiment (1964). See H. ML'IRJIEAI): The Physics o] Elemen,tary Particles (New York, 1965), p. 377. (s) ]~. E. MARSHAK, C. RYAN, T. K. RADIIA and K. RAMA]~:Phys. l~ev. ~ett., l i , 396 (1963).

I~. A, CIIRIST~;NS~]N

1042

w h i c h is b u i l t u p out of c u r r e n t s consisting of a s u m of ehiral i n v a r i a n t lepton and baryon terms. I n t h i s i n v e s t i g a t i o n , we will n o t define a l e p t o n n u m b e r , we will m a k e no d i s t i n c t i o n b e t w e e n h y p e r c h a r g e for leptons a n d for hadrons, a n d we will s e e k selection rules focused u p o n t h e c u r r e n t s r a t h e r t h a n u p o n t h e particles. T h e w e a k i n t e r a c t i o n a p p e a r s to be r o u g h l y of the c u r r e n t - c u r r e n t f o r m (')

G j~j~ + h.c.

(5) where

(6) (7)

J'~

(8)

J~ = (o~',~ q- i~',~)(1 q- Ys) cos0, q- (~',~ q- i~-,,)(1 q- y6) s i n 0 , ,

=

~.~/~e-(i + y~) + ~.y~tz-(1 + Y~),

w h i c h is n o n - Y u k a w i a n . (The C a b i b b o angle (~0) is e x p e r i m e n t a l l y a b o u t 0o ~ 0.26 r a d ~ 15%) ( ~ o t e : (8) comes f r o m t h e S a k a t a m o d e l (x~) w h i c h considers a , b a s i c )) t r i p l e t of b a r y o n s , so t h a t 89 = ~ n a n d 89 =~A.) B a s e d on o u r h y p o t h e s i s of t h e existence of a l e p t o n octet, w h a t can we s a y a b o u t t h e f o r m of our w e a k c u r r e n t ? Suppose we set

(9)

J~ = Tr (/~ J~L),

where

(~0)

a n d s e e k a J~ such t h a t J~ is c o m p o s e d solely of ~.e- a n d + a - t e r m s (J~ will c a r r y t h e i r h.c.'s). T h e r e is such a possibility, a n d i t can b e s h o w n to be u n i q u e l y of the f o r m ('~)

(11)

J~ = y~(1 + ~) 89

i~)

(9) Cf. J. g. SAKtmAZ: Elementary _Particle Physics, Brandeis Summer Inst., 1961, .Lectures in Theor..Phys. (-New York, 1962), p. 324, for a brief discussion of possible corrections to the current-current form at very short distances, strong corrections and W-field (intermediate boson) corrections. (1o) ,N'. CABIBBO: Phys. t~ev..Left., 10, 531 (1963). (ix) S. SAKATA: .Progr. Theor. Phys., 16, 686 (1956). (1~) Including a 89(21 -t- i2~) current introduces the terms - - ~ / ~ e - -~ V~e-/a ~~- v.p-.

IS TIIERE A L E P T O N

OCTET?

(we h a v e m a d e it a V--A c u r r e n t b y including a f a c t o r 7=(1 +75)), which yields

(i2)

1043 (lS)

r

where t h e r a t i o of m u o n i c to electronic t e r m s is g i v e n b y t g 0 = v / ~- or 0 ~ 0.68 r a d ~ 39 ~ To get t h e angle b e t w e e n t w o c o m p o n e n t s such as V ~ a n d ~.e- it is n e c e s s a r y also to consider t h e c o n t r i b u t i o n of t h e singlet r w h i c h couples i n d e p e n d e n t l y . First, we n o t e t h a t this c u r r e n t does n o t include t h e h y p o t h e s i z e d ~o. This t h e n p r o v i d e s a possible grounds for c o m p a t i b i l i t y of t h e l e p t o n o c t e t s c h e m e w i t h n o n o b s e r v a t i o n of t h e ~0. H o w e v e r t h e 89162 c u r r e n t a p p e a r s to be seen in some reactions (t~). W e will r e t u r n to t h e p o s s i b i l i t y of a ~o later. Second, we n o t e t h a t 0 # 0 o. The t w o angles are q u i t e different, t h e Cab i b b o angle describing t h e r e l a t i v e p o r t i o n s of 89162 a n d 89 in t r i p l e t - t y p e h a d r o n i c w e a k currents, 0 d e s c r i b i n g t h e m i x t u r e w i t h i n a 89 4-i27) o c t e t - t y p e leptonic w e a k c u r r e n t . The c u r r e n t given b y eq. (11) is, of course, a s t r a n g e n e s s - c h a n g i n g c u r r e n t . B u t this does not conflict w i t h t h e l e p t o n o c t e t s c h e m e of s t r o n g c o n s e r v a t i o n rules, since t h e i n t e r a c t i o n (5) is c o m p o s e d of c o m p e n s a t i n g s t r a n g e n e s s - c h a n g i n g c u r r e n t s . (In a sense then, t h e f a c t t h a t t h e w e a k L a g r a n g i a n a p p e a r s to be of t h e c u r r e n t - c u r r e n t f o r m is a n o t h e r ~ch i n t )) t h a t t h e l e p t o n s m a y h a v e s t r o n g structure.) If, say, t h e p u r e l y leptonic w e a k i n t e r a c t i o n were n o t of t h e c u r r e n t - c u r r e n t form, b u t was of t h e f o r m of a c o n v e n t i o n a l s t r o n g i n t e r a c t i o n , t h e n e v e n w i t h the t r a n s f o r m a t i o n p r o p e r t i e s of (13)

it would p r o d u c e m a s s splittings in accord w i t h t h e Gell-:Vtann-Okubo m a s s f o r m u l a (~5). H o w e v e r , it does a p p e a r to be of t h e c u r r e n t - c u r r e n t f o r m , a n d the c o m p u t a t i o n of m a s s shifts due to such i n t e r a c t i o n s is m o r e c o m p l i c a t e d .

(13) The ratio t:g0~ v/~ for Qtx-:v.e- is fixed in this approximation regardless what J~ we choose, so long as we use the form (9) for the leptonie weak cuxrent with the assumption that the leptons form an octet. (If we use a lepton triplet

, we

get (7) with 7&c- coming from .~ (2~+ i t 2) and ~u~- from 89 + i2s)-) (t4) The occurrence of 89 iZz) currents will also be consistent with the odd-D occurrence of 89 currents which we will be examining shortly. (zs) S. OKUBO: Progr. Theor. Phys., 27, 949 (1962).

1044

R.

A. CHR, I S T E N S E N

T h e r e is a sort of n a t u r a l r e l a t i o n s h i p b e t w e e n such w e a k i n t e r a c t i o n s a n d t h e c o n v e n t i o n a l s t r o n g i n t e r a c t i o n s . S t r o n g i n t e r a c t i o n L a g r a n g i a n s are of a f o r m such as (14)

~'~ ~ ~

Tr

(HxOH~Hs)e t c . ,

(15)

= ~

Tr

(H~OH2H3H4)e t c . ,

w h e r e H~ is a h a d r o n (or a n t i h a d r o n ) s u p e r m u l t i p l e t . W e m i g h t s p e c u l a t e t h a t {{w e a k }) i n t e r a c t i o n L a g r a n g i a n s are of a f o r m such as (1~) (16)

~

= ~ Tr(LJZ)* Tr ~/~

(17}

l_ G .Ww - - %/-~ T r ( / , J L ) + T r

(18)

s

G~ T r V/--

(LJL)

(pure leptonic)

(BJB)

(leptonic) ,

(H1JHz)* Tr (H3JH4) ( n o n l e p t o n i c ) .

This process of s u b d i v i d i n g traces is o b v i o u s l y generalizable to orders of w e a k n e s s a n d e v e n to m i x t u r e s . The
-_ (]9)

B

Z-

~30

(20)

n

P

Z'+

~

A - - v/{ S O

~+

-B=

.

Z-

T h e n we can w r i t e s o m e of t h e h a d r o n i c w e a k c u r r e n t c o m p o n e n t s (omit-

(1~) Of course, if we use a baryon triplet b = superfluous for the baronic weak current.

then the trace operation becomes

~s Ttrr:ar: .t LErTOX OCTET?

1045

ring t k e ?~(l+y~) factors) as

(21)

Tr B 89()., + i22)B = -- ~ A ~ -

(22)

Tr 3 89

+ ~.,) B = ~ - ~ o +

-~- ~ A V~2r+

+ ~/~ ~52:~ + g Z - ,

+ ~/~ r o t + + V ~ r - A -

~ / ~ - 2:o.

(Of course, t h e m a t r i c e s 20, it3 a n d 20 give s e l f - i n t e r a c t i o n p a r t s , a n d t h e m a t r i c e s 89 and 89 j u s t give t h e a n t i p a r t i c l e s in t h e a b o v e c u r r e n t s . ) The difficulty is t h a t no m a t t e r w h a t we t a k e as t h e m a t r i x J we c a n n o t p i c k u p certain o b s e r v e d t e r m s , such a ~ n (it is necessarily l y i n g in t h e 23 p o s i t i o n off the diagonal), which are missed b y t h e t r a c e o p e r a t i o n . We can pick up the desired a d d i t i o n a l t e r m s , h o w e v e r , if we include r e p r e sentations in higher-dimensional m a t r i c e s . J u s t as we use t w o - d i m e n s i o n a l matrices

A = (a.)

(23)

w i t h inner a n d o u t e r p r o d u c t s defined b y

a~,,b,,,j

and

a.,~b~,,, ,

we can also use t h r e e - d i m e n s i o n a l m a t r i c e s

~ / = (a,k)

(2~)

with corresponding definitions of i n n e r a n d o u t e r p r o d u c t s

a~,~kbm~.,

and

ami,nbiml:

.

E m p l o y i n g 3-D matrices, we h a v e , u s i n g 3-D i n n e r p r o d u c t s

(25)

T,,(~J~) = ~.,,,j~.b~..

Due to the s y m m e t r y in t h e first a n d last indices, we are forced to p u t t h e physical w~tve functions ill positions su('h as those shown in Fig. 4. We h a v e set (26)

(2)~,~ = ( ~ ) , ~ ,

i, n = 1, 2, 3 .

Since t h e same symmetT T holds for the c u r r e n t c o m p o n e n t s ],~,, we c a n build t h e m out of obvious g e n e r a l i z a t i o n s of t h e 2~. W e shall r e f e r to t h e

1046

I~. A. CHR[ST~:~,~N

3-D ~'s as A's, a n d define t h e m so t h a t (27)

( A , ) . m . = (~,),~

m, n = 1 , 2 , 3 ;

i=0,1,...,8.

The r e m a i n i n g elements of the ~ a n d Ai are irrelevant.

'

3r /S

I

I

:

r-

I

1st i n d e x 4 [ I

2od

Tt~6/ A --1~/ 2 [o

11g----~ii I

z"

Fig. 4. - 3-D baryon matrix.

Corresponding to t h e 2 - D case, we can c o n s t r u c t the following 3-D c u r r e n t components:

(28) (29)

TrB89247 TrB89

x/~.~-A + x/~_ff-Zo+~oZ+,

Of all our c u r r e n t c o m p o n e n t s , the following occur in observed interactions :

24): 3-D:

-4-~-, iOA, zip, ~-A,

2 - Z o, Nox+, ~n,

The following do n o t ,

~-=o

2/):

~Z ~

3-D:

A~'-, 2X-,

3E+

Z~ +, Z--A, Z - Z ~

2+A, Z+ Z o. .$

IS

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OCTET ?

1047

H o w e v e r , t h e y are s t r o n g l y coupled to ~ or k, so t h e y a r e a n t i c i p a t e d u n d e r generalizations of the G e l l - M a n n - P u p p i t e t r a h e d r o n ( ' ) . The only listed c u r r e n t c o m p o n e n t n o t i n c l u d e d is g Z + (~8). T h e specific t y p e s of currents which we are e x a m i n i n g c a n n o t a c c o u n t for t h e e x i s t e n c e of this c o m p o n e n t . T h e r e is no k n o w n a priori r e a s o n w h y ~4J=i25 a n d A~4-izl5 c o m p o n e n t s should n o t occur ('~). (They include t h e u n o b s e r v e d n e u t r a l c u r r e n t s , which t h e c o n s e r v a t i o n laws we h a v e h y p o t h e s i z e d w o u l d exclude f r o m m a n y candidate i n t e r a c t i o n s a n y w a y . ) . W h e n we a d d t h e 3-D c u r r e n t to t h e 2 - D c u r r e n t , we m u s t n o r m a l i z e each b y its r e s p e c t i v e multiplicity. T h e 2-D c u r r e n t h a s m u l t i p l i c i t y 32, t h e 3-D cur= r e n t 33, a ratio of 89 L e t us e x a m i n e t h e h y p o t h e s i s t h a t (zo)

(3o)

J~ ~ J,,,= + 89

+ 89

+ ....

As a n e x a m p l e of w h a t this implies, let us c a l c u l a t e t h e C a b i b b o angle (2,)

(31)

tg 0o = ~ A s t r e n g t h V ~ - - ~ v/~ strengtl{ ~ ~a

(32)

~0.4,

(33)

0o ~ 22 ~ 9

A c t u a l l y this is fairly close to t h e e x p e r i m e n t a l value, considering t h e app r o x i m a t i o n a n d considering h o w s e n s i t i v e t h e r e s u l t is to t h e 3-D: 2-D relat i v e strength. An effective r e l a t i v e s t r e n g t h of 0.375 i n s t e a d of 0.333, s a y due to o t h e r t e r m s , would m a t c h t h e e x p e r i m e n t a l result 0 c ~ 15 ~ H o w e v e r , going to h i g h e r - D c u r r e n t s (22), we find t h a t we c a n n o t e x p l a i n the d i s c r e p a n c y on t h e s e grounds. F o r all e v e n - D c u r r e n t s look like (21) a n d (22) (and t h e i r conjugates), while all o d d - D c u r r e n t s l o o k like (28) a n d (29). So s u m m i n g to all orders s i m p l y m u l t i p l i e s (21) a n d (22) b y 1-~-~2_]_~4_~ . . . ~ ]

(17) M. GELL-MAxx-: Suppl. ~Vuovo Cimento, 4, 2, 848 (1956). (18) Reaction (157) was reported by GI.AZ~R at Kiev in 1959 as constituting about 0.3% of the X+ decays. I t seems to bc contradicted by the nonobservation of

v+n-+X++~ -. (19) A ~ ( R4 4-i75) current would introduce electron-muon terms. (20) The series obviously converges, since the absohlte value of each component of each J(~), is bounded by unity. (:1) The calculation of ]A/3[----~ enhancement ratios would proceed along similar lines. (22) The 4-D inner product, for example, is ai,~k,=b,,j,,z, etc. oo

1048

R.A.

CI[RISTENSEN

and (28) and (29) b y ~ + ~~ ~_ ... = ] (2a), leaving the odd-even relative s t r e n g t h at e x a c t l y 89 (Similarly, for the lepton c u r r e n t i t simply introduces an overall f a c t o r of 1-4- 89247 . . . . 23.) T h e r e are m a n y possible sources for the discrepancy, a few of which are the omission of higher orders in the ,~weakenization p r o g r a m ,, the assumption t h a t the physical currents are p u r e l y octet-type, and the a s s u m p t i o n t h a t the eightfold-way gives the physical c u r r e n t algebra exactly. T h e r e is a n o t h e r s o m e w h a t more c o n s e r v a t i v e and perhaps more sound i n t e r p r e t a t i o n which can be m a d e of the 3-D i n n e r product. This is t h a t i t is simply a t r i c k y w a y of t a k i n g a 2-D o u t e r product. W i t h this i n t e r p r e t a tion we would a b a n d o n the idea of an infinite series of higher-D currents, and with it a n y hope of calculating (from ~(u n i v e r s a l i t y )) under these s y m m e t r i e s alone) such p a r a m e t e r s as t h e Cabibbo angle. Recall the s t a n d a r d definitions for J ~ = V - - A :

34)

V -- G , [ ( 1 - - ~ ) F

+ a,/)]y=,

(35)

A :

+ ~D]~=7, 5 .

G~[(1--%)F

T a k i n g the baryons, for example, we have (24) (36) (37) (38) (39)

Fo~d = [ T r (/~89(A.-k i A , ) B ) - T r (B 89

i2,)B)] sin0c -----

• [ - - V ~ (,~Op _ ~ - 27o) _ x / ~ ( A p -- ~ - A )

-- , ~ - n -4- ~ o 2:+] sin

DSd : [ T r (B 89(A1 + iA2) B) + T r (1~ 89(21-- i~t.) B)] sin 0o -----

=[--V~(T-,~

s176

~ - A ) - k ~.-n-k ~~

(40)

F.R... ~ [ T r (B 89(A.-- iAT)B) -- T r (/~ 89()~ + i).7)n)] cos 0 c :

(41)

= [ - V 2 ( Z ~ X+ - - 2~- X o) - - ~ - S ~ + ~ p ] cos 0 o ,

(42) (43)

Oc,

D.B...: [ T r ( B ~ (A.--iA,)B) --k,T r (/~ 89

§ i2,)B)] cos 0c----

---- [x/~ (.dS+ + 2 - A ) + -~-X ~ + ~p] cos 0o,

where t h e isofermionic and isobosonic t e r m s t o g e t h e r give t h e t o t a l F and D

(23) It is difficult to believe that the 0.375 odd-D strength being just the effective relative strength needed to give the experimental Cabibbo angle is anything but coincidence. There appears to be no justification for suppressing all of the even-D ~A currents beyond the first. (24) Experimentally: G , ~ 1, a v ~ 0 , G~ ~ - - 1.18, a~'~ 0.67. The first two are explained by the weak vector current V being in the same octet as the conserved electromagnetic current, G~ by the Adler-Weisberger relation, and ~ by symmetry under the collinear SU3| SU 3 subgroup of SU 6.

is TH~]~ A L):I'TON OCTET?

1049

currents:

(44)

F = F~

+ F,,..,

(45)

D = Dodd A- D.,.,.

These are just t h e c o n v e n t i o n a l results o b t a i n e d in a slightly different language. Using the meson octet gives similar results. A p p l y i n g this interp r e t a t i o n to our h y p o t h e s i z e d l e p t o n octet we get:

(46)

/~aa = [-- X/~ (floe+-- ~-po) _ ~/~. (~2e+ _ $-v2 ) _ fi-~,o + +.p+] sin 0o,

(47)

Dora ---- [%/~ (fi~

(48)

F.L... ---- [-- V~ (flop+_ fi-flo) _ ~-v~ -~ roe+] COS0c,

(49)

.D,~,n= [V~a (~p+ + fry)) + $-v, + v,e +] cos 0o.

+ ~ - # o ) _ V~ (y%+_~ e-W) + P - ~.-}- ~oP+] sin 0c,

Since we h a v e t a k e n our w e a k v e c t o r l e p t o n c u r r e n t in t h e same o c t e t as t h e conserved electromagnetic c u r r e n t , we have G~-= 1, a Lv - - 0. Since t h e leptons do not p a r t i c i p a t e in the o r d i n a r y strong i n t e r a c t i o n s , t h e corresponding Adler-Weisberger relation (25) is trivial, giving G~ = - - 1 . Finally, in o r d e r to L 0. Thus the result of this a p p r o a c h is o b t a i n y , ( l + T s ) terms we take a~----

(50)

J'~ = r.( 1 + r.) { [ - V ~ ( j ~

~-po) _ V } ( ~ e + - ~-~) - fl-~. + ~.p+] 9

9sin 0 o + [-- V/2 (flop+_ fi-po) _ ~-v. + v.e +] cos 0c} . H o w e v e r this form contradicts e x p e r i m e n t a l evidence of the existence of v,p • c u r r e n t components. The t e r m we almost c e r t a i n l y w a n t is 7~(l+75)D.L.,. and, with the above O's, this requires a r'--- ~ 0. T u r n i n g n e x t to the nonleptonic weak interactions, we n o t e t h a t t h e y can be s u m m e d up in a L a g r a n g i a n of t h e f o r m

(51)

.W:'=%/2G_Tr~'I (.~,:J=iJ,)B)*Tr('I ,J,=J:i).~)B)-~ ~ 2 ""'

where we h a v e again o m i t t e d the y , ( l + y s ) factors for convenience. This is quite unlike the leptonic weak interactions, in which, as we h a v e m e n t i o n e d , c o m p e n s a t i n g pairs of strangness-changing c u r r e n t s seem always to appear, giving a Lagrangian of the f o r m

_G

(52)

t

- ]

4- i),~)B)

(25) ~. L. ADLER: Phys. Rev. Left., 14, 1051 (1965); W. I. WEISBERGER: Phys. 1~ev. Jbett., 14, 1047 (1965). 67

-

1l N u o v o

Cimento

A.

]050

:[(. A. C]IRIST].INS)'N

(By iml)lication we m e a n to include A - t y p e currents as well as ).-type.) Comparison of (51) and (52) suggests t h a t t h e r e is some conserved q u a n t i t y which we h a v e overlooked. As a possible hypothesis we m i g h t assign negative (( c u r r e n t - p a r i t y ~), to

.B 89

i22)B

and

L89

i2~).5

and

L89(,~d- i).2)L.

a n d positive c u r r e n t - p a r i t y to B 89(J~:~ i,~7)B

T h e n if we r e q u i r e t h a t c u r r e n t - p a r i t y be multiplicative, t h a t it change sign u n d e r t~ermitian conjugation, and t h a t .~f~ have positive c u r r e n t - p a r i t y , we r e p r o d u c e t h e results (51) a n d (52). H o w e v e r , this scheme contradicts the p u r e l y leptonic interactions, e.g. ~-deeays, which a p p e a r to have a Lagrangian of the f o r m

(53)

_--=Tr L

(2,4-i2~)L

)(1 +Tr

L-(2~:ki).~)L 2

e

)

+ ....

One possible out is to sacrifice :Y and I c o n s e r v a t i o n in p u r e l y leptonic w e a k i n t e r a c t i o n s t h o u g h r e t a i n i n g it for o t h e r leptonie weak interactions. This seems aesthetically a t t r a c t i v e . A second way is to construct a n o t h e r even more hybrid-looking q u a n t u m number. This l a t t e r approach m a y be more a p p r o p r i a t e , though, if a n y 89 currents must be included. I f we can v e r i f y in detail a c u r r e n t - p a r i t y t y p e selection rule, t h e n currentp a r i t y considerations will give us the p r o p e r n e u t r i n o assignments for reactions to go (if otherwise possible), and as a result will tell us which observed interactions violate :Y and I conservation and which do not (26). ~'ow let us t u r n to t h e reactions involving two or more mesons. Upon e x a m i n a t i o n , we find t h a t it is impossible to explain t h e m b y using the conv e n t i o n a l NIr B + B t e c h n i q u e of forming a 4-fermion interaction. An altern a t i v e is to t r e a t the mesons on the same footing as the leptons and baryons, using t h e m e s o n o c t e t

(54)

M ----

k-

+ Vi

(2e) Recall that reactions such as M ~ T,+L are usually put into such picture via virtual strong interactions of the type M ~5~B + B . Thus = ~ B ~ (26•

a

IS TIYERE A LEYfON

OCTET~

1051

We assign ~-I 89 n e g a t i v e c u r r e n t - p a r i t y , and ~'~ 89 positive c u r r e n t - p a r i t y . I n s u m m a r y , c u r r e n t p a r i t y assignments for particle supermultiplets seem to be given b y (55)

/ ' . = ( - - 1) ~+~+Ay

where B and J are supermultiplet q u a n t u m n u m b e r s and A 1r is t h e c u r r e n t h y p e r e h a n g e i n c r e m e n t . The p r o p e r rule for m u l t i p l y i n g c u r r e n t parities t h e n appears to be (56)

t

P.~ = 4-P,1P~ 2

with - - for E L and L M Lagrangians, -}- otherwise. The following p a t t e r n seems to be emerging: Conserves Ir and I : Tr (BOBM) , Tr(LJL) t

T r ( B J B ) (~7), etc.

T r (f~JL) t

Tr (LJL) ,

Violates :Y and I :

Tr (LJL) + Tr ( M J M ) , Tr ( M J M ) ~ T r ( J ~ J M ) , T r ( M J M ) ~ T r (/~,IB) , T r ( ~ J B ) ~ T r ( B J B ) , etc. In order to see the implications of c o m b i n i n g era'rent p a r i t y c o n s e r v a t i o n with the specification t h a t only 89 and 89177 c u r r e n t s exist, let us review the status of a n u m b e r of w e a k interactions. :1) Leptonic weak interactions allowed u n d e r b o t h t h e c u r r e n t p a r i t y selection rule and u n d e r the L~ and L e(mservation rule (neutrino a s s i g n m e n t s made below according to c u r r e n t p a r i t y selection rule) 1) Observed (57) (58)

~, + n --~ p -i-e-, a--~-P - ~ n + v ~ ,

(~7) Perhaps the ncutrhlo assignments in these reactions should be made so that they violate Y and I also. (Of course the sign rule for (56) would have to be altered accordingly.)

1052

R. A. CIIRISTENSEN

(59)

~=s

(6o)

[z~:-->e•

(ol)

=+ _~ ~ + + ~ ,

e-

, vtt~-vit

~

(62) (63)

-~T:~

(64)

7:- -> ~ - + v ~ ,

(65)

~e-+~.,

(oo)

__~=o + e - + ~ . ,

(67)

(68)

k + ~ ~.++~., ~=~ 4-e++Vo,

(69)

(70)

k - -~ ~.-+v.,

(71) (72) (73)

-+=~

k~ ~ + + e - + ~ . ,

(74)

-->~-+e++v.,

(75)

~-+t~++v~,

(76)

---~r

p,- + v~,

(77)

},~=-+e++v.,

(78)

~=++e-+~.,

(79)

~+4~-+v~,

(8o)

+~-+~++v,,,

(81)

n ~p+e-+~.,

~82)

A--~p A-e-+v~,

~83)

-~ p +~--4-v.,

(84)

E+ -+ A +e+-{-,J.,

IS TIIERE

A L:EI'TON OCTET T

(85)

Z - -+ n + e - + v~,

(86)

-+n+$-+v.,

(87)

-+ A + e - + ~ . ,

(88)

1053

E--+A+e-+v~,

(89)

-~ Z ~

(90)

E ~ -~ Z + + e - + v ~ .

We h a v e o m i t t e d observed reactions such as (91)

~=~-+ e+ @e+-~e- ~- v ~-v,

which simply add a p a r t i c l e - a n t i p a r t i c l e p a i r to a listed reaction. 2) Not y e t observed

(92)

k+-+ e++v,

(93)

k--~e-~-v,

(94)

~:o __>p @ e - n a y ,

(95)

E--~A+~-+v,

(96)

Eo _~ ?C+~ _ ~ - + ~ .

The nonoccurrence of these would be a m y s t e r y in t h e c u r r e n t p a r i t y s c h e m e as their nonoccurrence would also be in the l e p t o n n u m b e r c o n s e r v a t i o n scheme. B) Nonleptonic weak i n t e r a c t i o n s allowed u n d e r c u r r e n t p a r i t y listed below h a v e been observed): (97)

k+--> ~ + + ~ o ,

(98)

-+~++~++~-,

(99)

_>~+~_~o_~o,

(100)

k---~ ~-~-~o,

(101)

--> ~ - +~z+ + ~ - ,

(102)

-~ ~ - ~-~~~-:z~,

(i03) (104)

k,~ -+ ~ : + - ~ - , --> W~

~,

(all t h o s e

1054

1~. A. CHRISTENSE~

(105)

k ~ -+,:o §

+r:o,

(106)

.__~~+ + x - + x o ,

(107)

"-+7~+-1-'~-,

(108)

A -+ p + ~ - ,

(109)

--> n + v : ~ ,

(110)

Z + --->p +re ~ ,

(111)

~n+=+,

(112)

X--+n+~-,

(113)

E----> A + ~ - ,

(114:)

Eo .-,.A+~:o.

C) L e p t o n i c w e a k i n t e r a c t i o n s f o r b i d d e n u n d e r b o t h c u r r e n t p a r i t y a n d u n d e r iS, a n d J5~ c o n s e r v a t i o n (none of those listed below has b e e n observed): (115)

~ - + p --->p + e - ,

(116)

v ~ + p -+ n + e + ,

(117)

~ + n -+ p + e - ,

(118)

~, + n -+ p + e - ,

(119)

~ - -+ e - + y ,

(120)

-~ e - + e + + e - ,

(121)

rr ~ --+ y . + + e - ,

(122)

k + --~ ~ + + ~ + v ~ ,

(123)

-+ ~-+e++~+,

(124)

-~O+e++v~,,

(125)

--~ 7:~~-y.+-f-~ ~

(126)

-->~~ + e - + v ~ ,

(127)

-+ x~+~--}-%,

(128)

k ~ --->~++e-,

(129)

-+~-+e++~,

(130)

-->~- +~++~,,

(131)

--->~x++ f z - +~" ,

1055

(132)

~o --+ ~+ + e - + ' ~ ,

(133)

A ~n+~++e-,

(137)

E + -+ p + ~ - + e +,

(135)

-+p +tz++e-,

(136)

~ - --> Z +

+~-

+ e "- .

/9) ~onleptonic weak interactions forbidden under current p a r i t y (none of those listed below hase been observed): (137)

k o __+~o,

(138)

E - -'+ n + ~ - ,

(139)

~o _.).n

(140)

~-rc ~ ,

~p+T:-.

E) Leptonic interactions allowed under current p a r i t y but forbidden under L. and Z~ conservation: (141)

~ + _ + ,~o _,_ e + § v ~ .

Reaction (141) is allowed under cur r ent -pari t y via an .-~[+5,~-{-B mode. Whether or not it has been observed depends upon the neutrino assignment in the observed reaction =+-+~~ ,'--e++v. The following modification of the Brookhaven experiment (~) could determine which: (142a)

n~ -+ ~~ e+-+-v

(142b)

~

(142c)

- ~ v+n-+p+e-.

v+p-+n+e + ?

However, the branching ratio for this mode of ~+ decay is only abou~ 10-a! Another possibility is to look for v ~ + ~ + - ~ x ~ + or v ~ + e - - o ~ . - + x ~ using ~-+9• produced v~'s.

(2s) G. DANBY, J.-.~V[. GAILLARD, K. GOULIANOS, L. M. LED~RMA.N, N. ~ISTRY, M. SCI~WARTZ a~td J. S ~ I S ~ R G E R : Phys. Rev. Lett., 9, 36 (1962).

1056

R.A. CHRISTENSEN

F ) L e p t o n i e i n t e r a c t i o n s allowed u n d e r Zo and JS~ conservation b u t forbidden u n d e r c u r r e n t p a r i t y (forbidden regardless of neutrino assignment): (143)

k + --> ~+ + y ,

(144:)

"-->~+-~ ~ ~-v,

(145)

--> ~ + - ~ e + ~-e - ,

(146)

--> :~+ + ~+ + { z - ,

(147)

k - --> 7:--~y,

(148)

- - > T : - ~ ~-~,

(149)

- ~ rc--~e+-}-e- ,

(150)

-~-+~++~-,

(151)

k ~ --~ e+-~e - ,

(152)

~ ~.+ + ~ . - ,

(153)

A

- - > n ~ - e + + e- ,

(154)

-+ n + ~.+ + V.-,

(155)

E + --> p-~e+-i-e - ,

(156)

--~ p + a + ~ - a - ,

(157)

E + --+ n + y.+ +,~,

(158)

--~n+e++~,

(159)

~o - - > p - ~ e - ~ ,

(160)

8 - --> n + e - - ] - v .

R e a c t i o n s (143)-{160) obey all k n o w n conservation laws applicable to leptonic w e a k interactions. W h y t h e y h a v e n o t b e e n seen is a m y s t e r y . Thus we have a n o t h e r e x p e r i m e n t a l suggestion t h a t it m i g h t be fruitful to seek strong properties of leptons. Reactions (157) and (158) use t h e questionable ~ X + current. Although r e a c t i o n (158) was r e p o r t e d b y GLAZER at K i e v in 1959 as c o n s t i t u t i n g a b o u t 0.4 p e r c e n t of t h e E + decays, it is n o t r e p o r t e d in c u r r e n t l i t e r a t u r e among the E + d e c a y modes (~). I t also seems to be c o n t r a d i c t e d b y the f a c t t h a t t h e r e a c t i o n v - t - n - - ~ e - + E + has n o t b e e n seen. (~) Cf. L. B. 0KUN: Weak Interactions o/ Elementary Particles (New. York, 1965), p. 222.

I S TIII~:RE A L E P T O N

Allowed under L, and L~

Allowed under currentparity

Forbidden under currentparity

57 58 62 63 67 68 72 73 77 78 82 83 87 88 93 94

145 148 151 154 ~57 160

59 60 64 65 69 70 74 75 79 80 84 85 89 90 95 143 146 149 152 155 I58

Forbidden under /~. and /Su

6I 66 71 76 81 86 92

144 147 150 153 156 159

1057

OCTET ?

Nonleptonie

141

116 119 122 125 128 131 134

117 120 123 126 129 132 135

115 118 121 124 127 130 133 136

97

98

99

I00

I01

102 105 I0~ Ili 114

103 106 109 I12

104 107 I10 113

137 140

I38

139

Fig. 5. - A comparison of lepton n u m b e r a n d current-parity selection rules. N u m b e r s refer to reactions in the text. Underlined reactions have been observed. The questionable cases, 141, 157 a n d 158 are disenssed in the text. Results are s u m m a r i z e d in Fig. 5 above, where u n d e r l i n e d reactions h a v e b e e n o b s e r v e d . W e o m i t r e a c t i o n s s u c h as t h e f o l l o w i n g s i n c e we are, for s i m plicity, restricting our discussion to octets: (161)

~ - -~ n ~ - e - + ~ . ,

(162)

--> A + k - ,

(163)

-+ ~:~

+v~,

(164)

_+ ~ o + ~ : -

,

: F i n a l l y we c o m e to t h e u n o b s e r v e d r e a c t i o n

(165)

~.-+=+ - * ~ . + + e - + v ,

the observed reaction (166)

k •

--+=++=-§247

and the unobserved reaction

(167)

k + _, rc•177

]058

CHRIST}]NSEN

R.A.

D o w e j u s t dismiss t h e m , a n d a n y like t h e m , b y saying t h a t t h e y c a n n o t fit into a 4-particle i n t e r a c t i o n scheme of t h e t y p e (16) a n d (17)? Or do we analyze t h e m in t e r m s of some k i n d of h i g h e r - o r d e r weakness? A t this p o i n t we are n o t p r e p a r e d to a n s w e r such questions. I n s p e c t i o n of r e a c t i o n s in Fig. 5 reveals a v e r y n a t u r a l e x p l a n a t i o n w h y s o m e l e p t o n i c decays are AS = 0, 1AI3] = 1, some are AS/AQ = + 1 , IAI3] ~- 89 s o m e are AS/AQ = - - 1 , [AI3[----~, etc. I t is s i m p l y a m a t t e r of which reactions do n o t violate c u r r e n t - p a r i t y . (E.g., (157) is [AIsI----~.) N e x t we n o t e t h a t it is c o n s i s t e n t with the B r o o k h a v e n e x p e r i m e n t which showed that

(168a)

~• -+ ~ = + ~

/

(168b)

~+ ~+p

+. n + e +

(168c)

L+ v~-k-n -~ p + e - .

N o w l e t us r e t u r n briefly to t h e q u e s t i o n of t h e h y p o t h e s i z e d go. Such a ~o would be e x p e c t e d in e v e r y e n e r g e t i c a l l y possible r e a c t i o n where a v~ is p r e s e n t l y found. A s s u m i n g m~, ~ m , ~ to w i t h i n a few MeV, the following are t h e o n l y ones of t h e allowed r e a c t i o n s listed a b o v e which are energetically possible w h e n tz~ is s u b s t i t u t e d for v~ (~o): (169)

Ia-q-p --~n +~o,

(170)

~+ --+e++% q-~o,

(171)

~- --+e - + ~ , q-~o,

(172)

k + ---~~~ + e + -{-~o,

(173)

k - -+ ~~ + e - + ~ ~

(174)

k=~ -+ ~ - + e + +~. ~ ,

(175)

V:~ -+ ~ + + e - + ~ o ,

(176)

A

->p+e-+~z ~

(177)

Z - -+ n + e - + ~ ~ ,

(178)

.~- --~ A + e - +

~o,

(179)

- ~- Z~ + e - + ~ . ~ ,

(180)

E~ ~ Z++e-+~ ~

(181)

"~

--> E O + e - + ~ o .

(~) As we have seen, the ~o might not be observable via the g-capture reaction (169), due to the form of the weak current, leaving only (170)-(181) as candidates.

IS T,IEI~iF, A L E P T O N " O C T E T

1059

I f the ~z~ exists, we would e x p e c t to see decay modes such as the following (assuming mu, ~ m~,): (182)

[zo --~ e+-[-e-+ v~,

(183)

---~~. + v , + v ~ .

r

Considering only electromagnetic splitting of t h e l e p t o n octet, we h a v e mass formula

(184)

m~+ - - re,A_ = m,+ - - m ~ + m ~ , - - too_,

which is trivially satisfied e x a c t l y b y t h e e x p e r i m e n t a l masses. I f we assume %hat the strong splitting t~rm t r a n s f o r m s as an o c t e t m e m b e r , t h e n t h e f a c t t h a t it m u s t o b e y Xz and I c o n s e r v a t i o n means it m u s t t r a n s f o r m as 2s, t h u s giving us the mass formula (185)

3 m ~ + m , = 2mo.

This is not even close to the e x p e r i m e n t a l numbers. H e n c e t h e r e m u s t be a significant different c o n t r i b u t i o n to t h e strong splitting of the l e p t o n octet. This is not surprising since the leptons do not p a r t i c i p a t e in t h e o r d i n a r y s t r o n g interactions a n y w a y . Similarly, U-spin invariance of t h e electromagnetic i n t e r a c t i o n results in the following magnetic m o m e n t f o r m u l a e for the l e p t o n o c t e t :

086) (187)

,a._

~ ~a_,

(/88) 3

(189) (190)

i.

V~

pu.v = ;2 (flu~ Pv~) ,

where #~.~ is tile magnetic m o n l c n t of the t r a n s i t i o n ~o_+ v~ d e t e r m i n i n g the pl~)babitity of ~o _+ v~-t T decay. F u r t h e r , from the isotopic relation (191) we also get (192)

3

1060

R . A . CIIRISTENS]~N

Finally, the h a v e the P r e n t h i relation which yields (193)

/z~,. -t-/%~ : O.

W i t h t h e e x p e r i m e n t a l values g, : 2.002 3218 •

004 8,

g~ : 2.002 324-t-0.000 010,

g v < 1 0 -1~ ,

all of these relations are v i r t u a l l y e x a c t (slight inequalities are due to Laml) shift, etc.). The only a s s i g n m e n t of b a r y o n n u m b e r to the lepton octet consistent with b a r y o n c o n s e r v a t i o n a n d t h e observed interactions is B----0. This would m a k e t h e leptons the only k n o w n particles w i t h half-integer spin a n d even b a r y o n n u m b e r . (There are no k n o w n particles with integer spin a n d odd b a r y o n n u m b e r . ) I t is i n t e r e s t i n g t h a t this is a possible e x p l a n a t i o n w h y leptons d o n o t p a r t i c i p a t e in o r d i n a r y strong interactions. F o r one c a n n o t build a. l e p t o n w i t h even b a r y o n n u m b e r a n d half-integer spin out of c o m b i n a t i o n s of t h e h a d r o n s a n d a n t i h a d r o n s .

RIASSUNTO

(')

Si suggerisce che i leptoni possano presentare una rigorosa struttura di ottetto e che le interazioni deboli a cui essi e gli hadroni partecipano pos~ano essere della forma corrente-corrente governata da una regola di selezione della parit~ di corrente in una. via ottuplice multidimensionale. (') T r a d u ~ o n r a cura della Redaetone.

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