On incompatibility of the relativized SU(6) symmetry with unitary

Volume 16, number3

PHYSICS LETTERS

ON I N C O M P A T I B I L I T Y

1June1965

OF THE RELATIVIZED WITH UNITARITY

SU(6)

SYMMETRY

B. V. GESHKENBEIN, B. L. IOFFE, M.S. MARINOV, V. I. ROGINSKY Academy of Sciences of the USSR, Moscow R e c e i v e d 11 May 1965

The r e l a t i v i z a t i o n of the S U ( 6 ) - s y m m e t r y has been d i s c u s s e d in a n u m b e r of c o m m u n i c a t i o n s which r e c e n t l y appeared. They r e s u l t in a p r e s c r i p t i o n for c o n s t r u c t i n g L o r e n t z and SU(3) inv a r i a n t S - m a t r i x e l e m e n t s , their static l i m i t being SU(6) i n v a r i a n t as well (e.g. r e f s . 1, 2). With these t h e o r i e s one gets some l i n e a r r e l a t i o n s among the i n v a r i a n t a m p l i t u d e s at a r b i trary momenta. However, no one has up to now succeeded in c o n s t r u c t i n g a theory i n v a r i a n t u n d e r an exact group (not " s e l f b r e a k i n g " one as in the p r e s e n t s c h e m e s ) including the d i r e c t product L ® SU(3) a s a n o n t r i v i a l subgroup. The t r o u b l e is that it is difficult to r e c o n c i l e the existence of such a group with the free p a r ticle equations of motion. Yet the r e l a t i o n s which a r e not c o n s e q u e n c e s of i n v a r i a n c e under some group g e n e r a l l y c o n t r a d i c t u n i t a r i t y . The purpose of this l e t t e r is to c l e a r up the c h a r a c t e r and the extent of this c o n t r a d i c t i o n using some s i m p l e examples. We shall c o n s i d e r quark s i n g l e t - m e s o n and q u a r k - q u a r k e l a s t i c s c a t t e r i n g in the U(12) s c h e m e [I]. The difficulties a r i s i n g h e r e s e e m to be i n h e r e n t in any r e l a t i v i z a t i o n of

and 5~ are the s c a t t e r i n g phases for j = l * ½ c o r respondingiy. In the e n e r g y r e g i o n , where the e l a s t i c unit a r r y condition is valid, the phases 5~ a r e r e a l , which e n a b l e s one to solve each of the eqs. (2) for 5~ and 53+1. Thus we obtain two solutions

SU(6).

It can be e a s i l y shown that an a r b i t r a r y s u c c e s sion of the solution I and II leads to a sequence 5~ which does not tend to z e r o as l --- oo..~ The partial wave expansion c o n v e r g e s only if 5-~ -= 0 for all l. Even if it were a s s u m e d that the convergence were e n s u r e d owing to the b r e a k i n g of U(12) scheme at l a r g e a n g u l a r m o m e n t a the solutions of eq. (1) would n e v e r t h e l e s s be p h y s i c a l l y m e a n i n g l e s s b e c a u s e of their i n c o m p a t i b i l i t y with the threshold behaviour, of the phases: 5~ ~ p2/+l. It should be s t r e s s e d that the r e l a t i o n ~ (2) lead to paradoxical c o n c l u s i o n s even without u t i l i z i n g e l a s t i c u n i t a r i t y : f i r s t l y , (2) i m p l i e s that a r e sonance, if it e x i s t s , m u s t be p r e s e n t in e v e r y p a r t i a l wave. Secondly, u s i n g eq. (27 n e a r the n s p i n l e s s production t h r e s h o l d (in the case of equality of the p a r t i c l e i n t r i n s i c p a r i t y product in the i n i t i a l and final states) and taking into account the t h r e s h o l d behaviour of the jump Af~

C o n s i d e r the q u a r k - u n i t a r y singlet s p i n l e s s p a r t i c l e s c a t t e r i n g amplitude. This amplitude has a f o r m A + ½B(~1 + q2). According to the U(12) scheme B =0

(1)

Because of (1) the e l a s t i c u n i t a r i t y condition is a s y s t e m of two r e a l equations for one complex function A. We shall show that this s y s t e m has no soluUon. To this end we expand the s c a t t e r i n g a m p l i t u d e s into p a r t i a l wave s e r i e s (e.g. ref. 3), t r a n s f o r m these s e r i e s u n d e r the a s s u m p t i o n of t h e i r c o n v e r g e n c e and write down the condition (1) as a s y s t e m of equations for p a r t i a l waves: +

fl-1

-

+

- fl+l + K(fl-fl)

=0 ,

l = 1,2,...

,

where

/*,~ =

p-1 exp(i5~)

sin 5 ~ ,

~ : (~ + ~ ) / ( ~ - ~ )

(2)

(3)

I.

5/+ 1 = 5 / + 5 l + - 5 / _ + 1

sin{Sl+1 - 5+/_1)

= K

sin{S1 O~) (4)

H.

5;:5; -

+

5l+ 1 = 5/_ 1 Choosing for each l any of these two solutions we can e x p r e s s all the phases 5~ in t e r m s of two of them: 5o and 5~. F o r example, if one chooses the solution I for all values of l one obtains

57 -- l s i - (z- 1)50 , 5/ = ( l - 1 ) 5 +1 - ( z - 2 ) s o + a

(5)

tg c~ = (e./rn) tg(5~- 50) .

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Volume 16, number 3 4 -

and p o s i t i v e n e s s of Im Af~ one can show that

af~ ~ 0". The q u a r k - q u a r k e l a s t i c s c a t t e r i n g amplitude has in the U(12) s c h e m e a f o r m

~l~A(p~)~a

--B ,

(Pl)~

1 June 1965

PHYSICS LETTERS

(b2) ~B (P2) +

(6)

+ o 2 ~ d ( p 2 ) ~A(Pl) ~-B(P'1) ~B~b2) , where 01 and 0 2 a r e the i n v a r i a n t functions. Consider the e l a s t i c s c a t t e r i n g of two identical quarks. Using the notations of the paper [4] we e x p r e s s the F e r m i a m p l i t u d e s F 1 , F2, F3, F4,F5, in t e r m s of q51 and (P2: (7) F1 =½(~51+~2), F2=F4=~(d)l-q~2), F 3 = F 5 = 0 " E l a s t i c u n i t a r i t y i m p o s e s on two complex functions ~b1 and ~52 (i.e. four r e a l functions) five r e a l r e l a t i o n s . This s y s t e m of equations would have had a n o n t r i v i a l solution only if the equations were proved to be dependent. Using the p a r t i a l wave expansion one can show that the equations are independent and hence the s c a t t e r i n g a m p l i tude is identically zero. Here, as in the p r e v i o u s case, the existence of a r e s o n a n c e in one p a r t i a l wave i m p l i e s its p r e s e n c e in e v e r y p a r t i a l wave even without the e l a s t i c u n i t a r i t y condition. The e x a m p l e s c o n s i d e r e d above a r e s i m p l e , but they imply the existence of q u a r k s . However, one can hardly expect that r e c o n c i l i a t i o n of the 0(12) s c h e m e would be e a s i e r in the m o r e r e a l i s t i c cases. For example, in the U(12) s c h e m e each of

ON A M O D E L

OF

ELEMENTARY

the p r o c e s s e s 56 + 56 ~ 56 + 5_6 and ~6_+ 35 --~ 56 + 35 is d e s c r i b e d by four a m p l i t u d e s only instead of s e v e r a l h u n d r e d amplitudes in the conventional theory. It is unlikely that the u n i t a r y S - m a t r i x should b e a r such a violence. It would be i n t e r e s t i n g to cleat" up what m i n i real breaking of the s y m m e t r y s c h e m e is n e c e s s a r y if we a r e to provide u n i t a r i t y and whether the s y m m e t r y would not reduce to L C~SU(3) after such breaking. After this work has been finished the a u t h o r s r e c e i v e d a p r e p r i n t by B6g mad P a l s [5] where the p r o b l e m of u n i t a r i t y in the r e l a t i v i z e d SU(6) s y m m e t r y was c o n s i d e r e d and the difficulties a r i s i n g t h e r e b y were noted. The authors e x p r e s s t h e i r s i n c e r e gratitude to V. N. Gribov, I. Yu. Kobsarev, I. Ya. P o m e r a n c h u k and K. A. T e r - M a r t i r o s y a n for d i s c u s s i o n s and valuable r e m a r k s .

References 1. R. Delbourgo, A. Salam, J. Strathdee, Proc. Roy. Soc. A284 (1965) 146. 2. M.A.B. Beg, A. Pals, Phys. Rev. Letters 14 (1965) 267. 3. G. F. Chew, M. L. Goldberger, F.E. Low, Y. Nambu, Phys. Rev. 106 (1957) 1337. 4. M. L. Goldberger, M.T. Grisaru, S.W. Mac Dowell, D. Wong, Phys. Rev. 120 (1960) 2250. 5. M.A.B.Beg, A.Pais, preprint 1965. * The authors' attention was drawn to this fact by V. N. Gribov and I. Ya. Pomeranehuk.

PARTICLES

SYMMETRY

Yu. V. NOVOZHILOV

Leningrad State University Received 10 May 1965

Recently in a n u m b e r of p a p e r s [1-6] the inhomogeneous group P(SU6) was suggested to inc o r p o r a t e s p a c e - t i m e and i n t e r n a l s y m m e t r i e s . It includes 36 m o m e n t a Ppj, while the g e n e r a t o r s Ma (a = 1.. 35) of " s p a t i a l " r o t a t i o n s S U 6 - s u b group contain " o r b i t a l " and " s p i n " p a r t s . P h y s i cal models based on P(SU6) [1, 3-5] were different, however. In one case [3,4] s u p p l e m e n t a r y 32 mom e n t a Ppj, j ¢ O, were dropped as n o n p h y s i c a l , 348

while in the o t h e r s [1, 5] they were c o n s i d e r e d to be of g r e a t i m p o r t a n c e . In this l e t t e r we d i s c u s s the s y m m e t r y model built on the inhomogenous group P(SU6) [2, 3, 5] with " s p a t i a l " r e f l e c t i o n s in it. We suppose P(SU6) to d e s c r i b e all the phenomena with hadrons. The i m p o r t a n t point of hadron s y m m e t r y is its app r o x i m a t e c h a r a c t e r . That is why the true s y m m e t r y group m u s t contain a " b r e a k i n g " m e c h a n i s m