On the commutation relations of weak currents

Volume 11, number 1

PHYSICS LETTERS

where

t

A(s) = (r~p-m) f de o

l~p(1- 8) + m

(2)

[ ~ ( 1 - ~ ) s - ~ m 2 - (1-~)~ 2] '

P = Pl + Pl', 07P) 2 = s , and (1) is evaluated b e tween f r e e field s p m o r s . The combined c a n c e l l a h o n c o n t r i b u t i o n s of the two fourth o r d e r diag r a m s of fig. 1 give a t e r m p r o p o r t i o n a l to

75[-17Pl+m-i7P2+m]75 t ' l ln2t ,

(3)

winch is zero between f r e e field s p m o r s . The a l t e r n a t i o n of u and t as the a s y m p t o t i c v a r i a b l e is c h a r a c t e r i s t i c of f e r m l o n t r a ] e c t o r l e s m p e r t u r b a t i o n theory. It i m p l i e s 1) that (t) m u s t be a s s o c i a t e d with two t e r m s , one of positive s i g n a t u r e and given by A(s), the other of negative s i g n a t u r e and given by -A(s). M o r e o v e r each of these t e r m s c o r r e s p o n d s to a pa~r of t r a j e c t o r i e s of opposite p a r i t y 3) winch may be s e p a r a t e d by u s e of the p r o j e c t i o n o p e r a t o r s (*#s - 17p)/2~/s 1). The p r o p e r t y to which we wish to draw attention is the a p p e a r a n c e of the l r p - m factor m (2). This i m p l i e s that the t r a j e c t o r y of opposite p a r i t y to the nucleon p a s s e s through l = -1 when s = m 2. It is the chrect analogue of the Reggelzation r e sult 1, 2) which s t a t e s m that case that the t r a j e c tory of the s a m e p a r i t y as the nucleon p a s s e s

1 July 1964

through l = 0 when s = m 2. The d i f f e r e n c e s m p a r i t y and a n g u l a r m o m e n t u m a r e due to the u s e of p s e u d o s c a l a r m e s o n s and vector m e s o n s r e spectively. It s e e m s that here we a r e dealing w~th a r a t h e r g e n e r a l p r o p e r t y of r e l a t w m t l c fermion tra]ectomes. It is not difficult to c o n s i d e r the higher o r d e r ladder t e r m s m the p s e u d o s c a l a r theory and show that they give the expected Ingher t e r m s m the expansion of the Regge c o n t r l b u t m n s , together with c a n c e l l a t i o n c o n t r i b u t i o n s . The v e r i f i c a t i o n of the r e m o v a l of the c a n c e l l a t i o n c o n t r i b u t i o n s when the c r o s s e d d i a g r a m s a r e c o n s i d e r e d p r e s e n t s the u s u a l t e c h n i c a l c o m p l i c a t i o n s . It i s hoped to d i s c u s s this e l s e w h e r e , together with an i n v e s t i g a t i o n of the behaviour of t r a j e c t o r i e s a s soclated with l = -n (n > 1) 1) M.Gell-Mann and M. L Goldberger, Phys Rev. Letters 9 (1962) 275, M Gell-Mann, M. L. Goldberger, F.E. Low and F. Zacharlasen, Physics Letters 4 (1963) 265, M Gell-Mann, M. L. Goldberger, F.E. Low, E. Marx and F. Zacharmsen, Phys.Rev. 133 (1964) B145 2) J.C. Polkinghorne, 'Asymptotm Behaviour of Feynman Integrals with Spm' (Cambridge D. A. M. T. P. preprmt), to be pubhshed . 3) V Gr~bov, Zh Eksperim.i Teor Fiz.43 (1962)1529 (English translation Soviet Phys , J. E. T. P. 16 (1963) 1080.

*****

ON

THE

COMMUTATION

RELATIONS

R. O E H M E

OF

WEAK

CURRENTS

* and G. SEGRI~. **

CERN, Geneva Received 3 June 1964

T h e r e a r e s e v e r a l ways m winch one can i n t r o d u c e the notion of u n i v e r s a l i t y for the weak i n t e r a c t i o n s . But t h e r e always a r i s e s the q u e s tion how to explam effects h k e the c o n s i s t e n t s u p p r e s s i o n of s t r a n g e n e s s changing leptomc decays of m e s o n s and b a r y o n s , wlnch s e e m to violate at l e a s t a nazve u n i v e r s a h t y p r i n c i p l e . If we a s s u m e that the c u r r e n t s winch e n t e r into the weak L a g r a n g i a n have well defined t r a n s f o r m a t i o n p r o p e r t i e s with r e s p e c t to the a p p r o x i m a t e s y m m e t r i e s of s t r o n g i n t e r a c t i o n s , then we may ask whether the b r e a k i n g of these s y m m e t r i e s could have s o m e t h i n g to do wtth d e w a h o n s f r o m u m v e r s a h t y . We want to e x a m i n e t i n s p o s s i b i l i t y

94

h e r e m connection with the c o m m u t a t i o n r e l a t i o n s of the c u r r e n t d e n s i t i e s and c e r t a i n a s s u m p t i o n s about the chvergence of the axial vector c u r r e n t . Let us a s s u m e that the c u r r e n t d e n s i t i e s V(~)(x) and A(Q(x) transform lake components of octets u n d e r SU3, and that they s a t i s f y the c o r r e s p o n d i n g c o m m u t a t i o n r e l a t t o n s 1). Then we c o n s i d e r the following models for the weak c u r r e n t mvo~vmg the s t r o n g l y i n t e r a c t i n g p a r h c l e s : A) j : cos 0(J(1)+IJ~2)) + s m 8(J(4)+tJ(5)) ,

(I)

* Guggenheim Fellow, o~ leave from the University of Chicago. ** National Science Foundation P~octoral Fellow

Volume 11, number 1

P HYS1CS LE TT ERS

w h e r e J = V, A, and where the angle 8 = OA =.0 V has notlnng to do w,th s t r o n g m t e r a c u- o n s 2 , 3) F o r s o m e u n s p e c i f i e d r e a s o n the weak coupling s e l e c t s t i n s component, winch r e m a m s e s s e n tially unchanged, if the s y m m e t r y b r e a k i n g p a r t of the s t r o n g i n t e r a c t i o n i s sw,tched on or off. In tins model 0 ,s s m a l l and d e s c m b e s the supp r e s s i o n of AS = 1 leptomc decays, except for r e n o r m a h z a t m n c o r r e c t m n s winch a r e p r e s u m e d to be negllg~ble m lowest a p p r o x i m a t i o n

B)

j = (j(1)+lj(2)) + (j(4) +~j(5)) ,

(2)

1 July 1964

we fred

5a(A(1) + 1A(2))a = _((/)I + y(_~)½)(p(1) + Lp(2)) 5a(A(4) + 1A(5))a = _((~)½ _ V(~)½)(p(4)+ Lp(5) .(5) Here S(0) Is the symmetric mass term, yS(8) the symmetry breaking interaction, and S(3)(x), P(Z)(x) are scalar and pseudoscalar denmhes respectively. Using eq. (5), we obtain two relevant sets of commutation relatmns The first relations are generated by I- spin and K- spin, they are

with

j (1) + 1j(2) = cos O (~(I) + 1~(2))

= 215(x - x' )((~)~ +~({)2) P(3)(x) , (6)

j (4) + ij(5) = sin 0 (~(4) + i~(5))

+

In tins case we assume that the angle 8 is due to the s y m m e t r y b r e a k i n g p a r t of the s t r o n g m t e r a c t m n s , and m such a way that 0 ~ 0 m the SU3 h m l t 4). The c a r e t denotes the c u r r e n t s m tins limit.

C)

j = (j(1) + ~j(2)) + (j(4) + 1j(5)) ,

but now the AS = 0 current does not change m the llmlt of SU3 symmetry, Le., j(1) + xj(2) = ~(1) + --''x3(2~, w h e r e a s the 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 has an a p p r o x l m a f e l y u m v e r s a ] dampxug factor ~ kV ~ kA such that

+

In a

=

+

(3) m J inK,

rough a p p r o x i m a t m n we have X ~ and hence h -~ 1 m the SU3 hmmt 1, 5) The par a m e t e r X is a " l a r g e " manffestatmon of the "small" symmetry breaking interaction, lust hke the p m n - k a o n m a s s difference. P h e n o m e n o l o g , cally, the m o d e l s A) and B) a r e e q m v a l e n t , w h e r e a s model C) d i f f e r s f r o m the o t h e r s only m so f a r as it does not affect the p r e s e n t sltuaflon m the 014 p r o b l e m We now want to c o n s i d e r these m o d e l s m conn e c t l o n with commutataon r e l a t i o n s and with the a s s u m p h o n that the ¢hvergence of the axxal v e c t o r c u r r e n t s is a p p r o x n n a t e l y p r o p o r t i o n a l to the app r o p m a t e o p e r a t o r s 2). T h e s e fxeld o p e r a t o r s p r o j e c t out single p m n o r k a o n . s t a t e s . W r i t i n g the s t r o n g H a m l l t o n l a n d e n s i t y m the f o r m H t ° t = Ho + S(O) + TS(8) = Ho + H ,

(4)

and u s i n g the r e l a t i o n

5a A~ 0)(x) = - x [ f d 3 y Ao0)(y), T-I(X)]Xo=Yo,

1

1

= 21~(,,-,:')((~)~-~(÷~)½{P(~)(~)+Z~ ~(8)(~)}, and s n n l l a r ones with V r e p l a c e d by A. We note that these f o r m u l a e do not revolve any s t r a n g e n e s s changing n e u t r a l c u r r e n t s o r d e n m t l e s . They a r e analogous to the f a m i l i a r c o m m u t a t o r s [I+, I ] = 2I 3 and [g+, K_] = I 3 + ~ Y 6 ) . In c o n t r a s t , our second set of c o m m u t a t i o n r e l a t m n s is r e l a t e d to the mLxed c o m m u t a t o r s [ I , K+] = L+ and [ K , I+] = L_, where L± = F 6 ± 1F7, and hence they revolve n e u t r a l AS = 1 t e r m s . We fred for these r e l a t i o n s [V~4)(x) - 1V~5)(x), 6/3(A(1)(x ' ) + n4(2)(xl)~]Xo=Xd = 16(x-,~')((~)~ + y Q ) ~ ) ( P ( ) ( x ) - IP(7)(x)) , C7) _1 ± 16~ = 16(~-x' )((2)2 - y(~)~)CP ~ JQ¢)+ IPCq)(x)) , and again s i m i l a r f o r m u l a e with V r e p l a c e d by A Let us now c o n m d e r matrLx e l e m e n t s of eqs (6) and (7) w, th r e s p e c t to m e s o n and vacuum s t a t e s Using the a s s u m e d p r o p e r t i e s of the d,v e r g e n c e of the axial c u r r e n t s , we obtain r e l a tions between the following matrLx e l e m e n t s :

;KO I

lv

)l w > --

FKK,

(8)

<01A(21A(a2)l ~- ) = 1~ a BTr , etc. Here the m v a r l a n t f o r m f a c t o r s a r e , of c o u r s e , functions of the a p p r o p m a t e mvarxants, but m the following we xgnore tins dependence We also 95

Volume 11, number 1

PHYSICS

h a v e a s s u m e d that the v e c t o r c u r r e n t s a r e F type only. F r o m eq (6), and the c o r r e s p o n d i n g o n e s l n V o l w n g the a x i a l v e c t o r c u r r e n t s , we obtain the a p p r o x i m a t e r e l a t i o n s 5) F~K(1

+

~ ) B K ~ m2 1 - ~ / 2 4 2

ryy By

~

1 +~/Z2 ' (9)

B2K

2

By

~

1-~/2~/2

2 1+~/~

mK

1-p/2~/2

i+o142 ,

w h e r e p ~ {S(8))o/(S(O)) o On the o t h e r hand, f r o m the mLxed c o m m u t a t o r s (7) we find a f o r mula hke 2 FyK(1-~)Sy mK 1 + y / ~ / 2 (10) FKKBK m~ 1 - ~ / 2 ~ / 2 " We do not c o n s i d e r the c o r r e s p o m h n g c o m m u t a t o r with the a x m l c u r r e n t , b e c a u s e if we u s e o u r app r o x i m a t i o n c o n s i s t e n t l y , it g i v e s r e l a t m n s m v o l v m g a l s o matrLx e l e m e n t s lnke (K- I A(4)_ 1A(5)l~+y- ) S i n c e we a s s u m e that << 1, and h e n c e a l s o p << 1, and s i n c e FKK(0) = F~y(0) = 1 a s a c o n s e q u e n c e of the c o n s e r v e d v e c t o r c u r r e n t , we s e e that e q s (9) and (10) a r e c o n s i s t e n t only in the h m l t of SU 3 s y m m e t r y In thlshmltwehave y = p = 0 , my = m K a n d ~ = 0 . Our a s s u m p t i o n s c o n c e r n i n g the d i v e r g e n c e 6nAn(Z), and t h e i r u s e in the c o m m u t a t m n r e l a t m n s of the c u r r e n t s , a r e , of c o u r s e , r a t h e r r e s t r ~ c t l v e . H e n c e ~t m a y not be s u r p r ~ s l n g that we a r e f o r c e d xnto the SU 3 l i m i t if we u s e all r e l e v a n t c o m m u t a t i o n r e l a t m n s On the o t h e r hand, ff we find s o m e r e a s o n that only the f i r s t s e t of c o m m u tatlon r e l a t m n s s h o u l d be u s e d in c o n n e c t i o n with the " a l m o s t c o n s e r v e d " axxal c u r r e n t , then the f o r m u l a (9) m d x c a t e an u m v e r s a l s u p p r e s s m n f a c -

96

LETTERS

1 July 1964

t o r ~ ~ h A ~ ~ V ~ m y / m K. We n o t e that t h e s e c o m m u t a t i o n r e l a t i o n s a r e r e l a t e d to t h o s e of the I - s p i n and K - s p i n o p e r a t o r s , w h i c h g e n e r a t e the c o r r e s p o n d i n g SU 2 s u b g r o u p s of SU3. L e t us now c o n s i d e r t h e s e r e s u l t s in v i e w of the m o d e l s A ) - C) T h e f i r s t m o d e l i s c e r t a i n l y c o n s i s t e n t with the SU 3 h m l t s of (9) and (10), but m add~tmn we m u s t a s s u m e that c h a n g e s due to rS(8) a r e s m a l l * The s e c o n d m o d e l r e q m r e s that the AS = 1 c u r r e n t v a m s h e s m the l i m i t y ~ 0 But ff the c u r r e n t s satxsfy c o m m u t a t i o n r e l a t i o n s l i k e (6) a n d / o r (7), we s h o u l d e x p e c t rather tg~ 1 than~ 0 However, tg0~ 1 in the SU 3 h m l t m a y be dxfflcult to r e c o n c i l e w i t h CVC. T h e t t u r d m o d e l m v e r y s u g g e s t i v e ff we c o n s i d e r only the r e l a t i o n s (9), but we m u s t add s o m e f u r t h e r a s s u m p t m n s in o r d e r to j u s t i f y the o m l s s m n of the mLxed c o m m u t a t i o n r e l a t m n s (7), at l e a s t w i t h i n the a p p r o x i m a t e s c h e m e we h a v e u s e d h e r e At p r e s e n t , we can only s p e c u l a t e about fins, but ~t m a y be of i n t e r e s t m c o n n e c t m n w~th the q u e s t m n of n e u t r a l c u r r e n t s m w e a k lnteractmns We w o u l d h k e to thank P r o f e s s o r f o r his kind h O s p l t a h t y at C E R N

L. Van Hove

References 1) M Gell-Mann, Phys Rev 125 (1962) 1067 2) M Gell-Mann and M L~vy, Nuovo Clmento 16 (1960) 705 3) N Cablbbo, Phys Rev Letters 10 (1963) 531. 4) N Cablbbo, Phys Rev Letters 12 (1964) 62 5) R Oehme, Phys Rev Letters 12 (1964) 550, 604 (E) 6) Compare J J De Swart, Rev Mod Phys 35 (1963) 916 7) M Gell-Mann, Proceedings of the Tenth Annual International Conference on High Energy Physms, 1960 (Interscmnce Pubhshers, New York, 1960). * There may also be other arguments whmh favour this model See, for example, ref 7