Volume 26B. number 8 P H Y S I C S L E T T E R S 18 March 1968

Q U A R K M O D E L C A L C U L A T I O N S O F M E S O N B A R Y O N S C A T T E R I N G *

R a m e s h CHAND Department of Physics, Wayne State University, Detroit, Michigan 48202, USA

Received 16 February 1968

In an SU(3) symmetr ic quark model, the high energy scattering of pseudo-scalar mesons with the baryons is investigated in the crossed t-channel. Under the assumption that the physical baryon octet is given by: 8_ (physical) = 8 ' cos 0 +8 sin{?, where 8' and 8 baryons a r i se from 3 ® ~ and 3 ~, 6 respect ively, we find good agreement between the experimental data and the quark model predictions for 0 20 °.

R e c e n t l y , B a r g e r et a l . h a v e a n a l y z e d the h igh e n e r g y da ta on m e s o n nuc leon s c a t t e r i n g and found d i s a g r e e m e n t be tween the q u a r k m o d e l p r e - d i c t ions and the e x p e r i m e n t a l r e s u l t s [1]. H o w - e v e r , m o s t of the p r e v i o u s qua rk m o d e l c a l c u l a - t ions a r e p e r f o r m e d wi th in the f r a m e w o r k of SU(6) s y m m e t r y [2]. In SU(6) s y m m e t r y s c h e m e , b a r y o n s a r e a s s i g n e d to the s y m m e t r i c 5.66 r e p r e - + sen ta t ion which con t a in s only one ½ b a r y o n oc te t . In th is no te we wi sh to i n v e s t i g a t e the m e s o n b a r y o n p r o c e s s e s in an SU(3) s y m m e t r i c q u a r k mode l .

In our SU(3) s y m m e t r i c q u a r k mode l , the p s e u d o - s c a l a r m e s o n s even though c o m p o s i t e s of q u a r k - a n t i q u a r k p a i r s (Q ~)), a r e t r e a t e d a s b a s i c en t i t i e s . We deno te the p s e u d o - s c a l a r m e s o n oc t e t by the m a t r i x (P~), w h e r e in our no - ta t ion , the l o w e r i ndex r e f e r g to the qua rk and the uppe r index to the a n t i q u a r k ; the i n d i c e s 1, 2, and 3 r e f e r to the Po, no, and ~o q u a r k s r e s p e c - t i ve ly . The ba ryon s t a t e s a r e c o m p o s i t e s of t h r e e q u a r k s :

3 ® 3 ® 3 = 1 0 ® 8 0 8 ' O 1 , (1)

w h e r e 8 and 8' a r i s e f r o m 3 ® 3 and 3 ® 6 r e s - p e c t i v e l y . S ince in the SU(3) s y m m e t r i c q u a r k mode l , t h e r e a r e two b a r y o n o c t e t s , the p h y s i c a l b a r y o n oc t e t can be t aken as a l i n e a r comb ina t i on , of the f o r m :

8 (phys ica l ) ~ _8' c o s 0 + 8 s i n 0 , (2)

w h e r e fo r s i m p l i c i t y , we c h o o s e the mix ing ang l e 0 to be r e a l . The b a s i c d i f f e r e n c e be tween o t h e r qua rk m o d e l s [2,3] and o u r s l i e s in the e x t r a d e g r e e of f r e e d o m which we have in t e r m s of the c h o i c e of 0.

* Work part ial ly supported by the U.S .Atomie Energy Commission through Syracuse Universi ty.

In t e r m s of the q u a r k s t a t e Qi (k) fo r i n t e r n a l m o m e n t u m k, we de f ine the t h r e e qua rk s t a t e s by the r e l a t i o n :

Bijk ~ Qi ( k l ) Qj (k2) Qk (k3) • (3)

In th is p a p e r , we sha l l c o m p l e t e l y s u p p r e s s the s p a c e - t i m e d e p e n d e n c e of Bij k. Now, n o r m a l i z - ing each Bijk to uni ty, the n o r m a l i z e d ba ryon s t a t e s con ta ined in (1) can be wr i t t en as:

S = e i j k B i j k / 4 - 6 , (4.1)

N'ji = iron (Bran j + Bmjn ) / 4-~ , (4.2)

N i " 1 i ] = (~zmnBjm n - - ~ S j e k m n B k m n ) / V ~ , (4.3)

DOk = (Bij k +Bikj +Bjki +Bjik +Bk 0 + Bkji)/g-6- ,

(4.4)

w h e r e N'], N i , and Dij k a r e g iven in t e r m s of the ba ryon st~ttes by the SU(3) s y m m e t r y [4,5]. The non e x i s t e n c e of the ba ryon s i ng l e t S i m p l i e s a c o n s t r a i n t on the BOle'S , the r i gh t hand s ide of (4.1) equal to z e r o . E x p r e s s i o n s (4) on i n v e r s i o n g ive B 0 k ' S in t e r m s of the ba ryon s t a t e s .

The b a s i c a s s u m p t i o n of our qua rk mode l i s " the add i t i v i t y " . Under th i s a s s u m p t i o n , the m e s o n ba ryon i n t e r a c t i o n can be r e p r e s e n t e d by the f i g u r e shown; Qc = Qd fo r e l a s t i c s c a t t e r i n g . S i n c e at high e n e r g i e s , the c r o s s e d l - c h a n n e l e f f ec t s d o m i n a t e , the b a s i c i n t e r a c t i o n is g o v - e r n e d by the exchange of the SU(3) s ing le t and the oc te t o b j e c t s only. T h e r e f o r e , the / - c h a n n e l e f f e c t i v e L a g r a n g i a n can be w r i t t e n as :

L e f t = A 1BabCBabcPs d + (BabCBab d + (5.1)

~CBabe B , d 1 d P s f ) + A 8 a • d - :~°d a b e l × [A8s(Psc - ~ 5 c / a c ] '

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Volume 26B. number 8 P H Y S I C S L E T T E R S 18 March 1968

Meson - ~ ~ ~ Meson

Fig. I. Typical diagram for the elastic and inelastic scattering of pseudo-scalar mesons with baryons in a

quark model.

where

Ps ~ ½(PP + PP) , Pa =- ½(PP- PP) . (5.2)

The ove ra l l ampl i tudes A1, A8s and A8a which a r e in genera l complex, a r e the in tegra l s over the s p a c e - t i m e v a r i a b l e s and hence contain al l the spin and k inemat ic dependences . In our no- tation, with B abc -:/~abc, the b a r r e d s ta tes r e f e r to the incoming pa r t i c l e s and the unbar red to the outgoing pa r t i c l e s .

Combining the ma t r ix e l emen t s given by (5.1) with the definit ion of the physical baryon octet , we at once get the ma t r ix e l emen t s for the s ca t - t e r ing of p s e u d o - s c a l a r mesons with the phys ica l baryons . The ma t r ix e l ements , not p re sen ted h e r e [7], sa t is fy al l the SU(3) re la t ions [8], i r - r e s p e c t i v e of the va lue of the mixing angle 0. This of course is not su rp r i s i ng s ince SU(3) s y m m e t r y is a built in fea ture of our quark model. However , SU(3) s y m m e t r y breaking ef- fects can be eas i ly in t roduced, by t r ea t ing the exchange' of S = ± 1 objects d i f ferent ly f r o m the S = 0 objects . This is equivalent to t r ea t ing the X o quark di f ferent ly f r o m the Po and n o quarks.

In this paer , our main ob jec t ive i s t o d i scuss those f ea tu res of our quark model for which the re exis ts cons ide rab le d i sc repancy between the expe r imen ta l data and the pred ic t ions of o ther quark models [1,3]. Fo r this purpose , we define

A(AB) = ~t(AB) - ~t(AB) . (6)

Combining the ma t r ix e l emen t s obtained f rom (5.1) with the definit ion of the physical baryon octet g ives

A(Kp) = 2F(4+a) ,

A(Trp) = 4F(I +a) ,

A(Kn) = 2 F ( 2 - a ) ,

(7.1)

(7.2)

(7.3)

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Table 1 A comparison between the calculated and the experi- mental values of the ratios of A(Kn), £~(?rp) and ½A(Kp),

given by (7).

Experiment [1] Quark model SU(6) [3] 0 = 20 ° ~) = 15 °

A (Kn) 1.8 1.8 0 1 1 A(~p)

A(Kp) 1.4 1.40 1 1 2A(/rp)

A(Kp) 0.8 0.78 1 1 2A(Kr{)

where

a - - - c o s 2 0 - f 3 sin 20 , F = (41r/k) .AImA8a. (7.4)

In these express ions , for s impl ic i ty , the word physical has been dropped f r o m the proton p and the neutron n s ta tes . We a s s u m e that at the e n e r - g ies of i n t e r e s t to us, the c .m. momentum k is the same for al l the above t h r ee p r o c e s s e s . F r o m (7), independent of the value of 0, we at once get the weak J o h n s o n - T r e i m a n re la t ion [3]:

A (Kp) = ~ (Trp) + A (Kn) , (8)

which is reasonably well obeyed exper imenta l ly [1]. The ca lcula ted and the expe r imen ta l va lues of the ra t ios of the quant i t ies of (7) a r e p resen ted in the table. One immed ia t e ly no t ices that for 0 = 20 ° , our quark model predic t ions ag ree a l - most pe r f l ec t ly with the exper imenta l data where as 6 = 15 ° r ep roduces the SU(6) r e su l t s of John- s o n - T r e i m a n [3]. In addition to these , we a lso find that for 0 = 20 o, our quark model p r e d i c - t ions a r e cons is ten t with o ther exper imenta l r e - sui ts d i scussed in ref . 1. In o rde r to de t e rmine if t he re is deep s igni f icance in the value 0 = 20 °, at the p re sen t t ime, inves t iga t ions a r e on the way on other e l emen ta ry pa r t i c l e reac t ions in our SU(3) s y m m e t r i c quark model.

It is a p l ea su re to thank P r o f e s s o r E. C. G. Sudarshan for his hospi ta l i ty at Syracuse Uni- v e r s i t y during the s u m m e r of 1967 when this work was ini t iated. Our specia l thanks to Drs . N. Mukunda and A. M. Gleeson for in t e re s t ing conversa t ions .

Rcfe~'cnccs

1. V.Barger and L.Durand, Phys. Rev. 156 (1967) 1525.

2. H.Lipkin, Phys. Rev. Letters 16 (1966) 1015; G. Joshi, V. Bhasin and A. Mitra, Phys. Rev. 156 (1967) 1572.

Volume 26B, number 8 P H Y S I C S L E T T E R S 18 March 1968

3. K. Johnson and S. B. T re iman , Phys . Rev. Le t t e r s 14 (1965) 189.

4. S. Okubo, Lec tu r e s on uni tary s y m m e t r y , Univers i ty of Roches te r Rept. (unpublished).

5. G. Zweig, Sy m m et r i e s in e l emen ta ry par t ic le phys- ics, 1965, Internat ional School of Phys i c s "Ettore Majorana" (Academic P r e s s . 1965) p. 192.

6. R. Chand, Non-leptonic hyperon decays in a quark model, to be published.

7. R. Chand, A detailed account of me son -ba ryon s c a t - t e r ing in a quark model will be published.

8. M.Gourdin , Uni tary s y m m e t r i e s (North-Holland, A m s t e r d a m , 1967). Other re la ted r e f e r e nc e s can be found in this book.

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