- Email: [email protected]
Possible CP violating, PT conserving, scalar-pseudoscalar interaction
Volume 14, number 4
PHYSICS LETTERS
s u b s t a n t i a l l y with the n u c l e a r s u r f a c e r e g i o n w h e r e the d e n s i t y i s s m a l l e r . A f u r t h e r c a l c u lation w a s t h e r e f o r e c a r r i e d out f o r t h e ~ - d e c a y s of f o u r " s i m p l e " (i.e. doubly c l o s e d L S s h e l l p l u s o r m i n u s one nucleon) a s s u m i n g f o r p ( r ) the d e n s i t y d i s t r i b u t i o n obtained f o r n u c l e o n s in an o s c i l l a t o r p o t e n t i a l . The r e s u l t s a r e l i s t e d in t a b l e 1. Table 1 ~-decay of "simple" nuclei.
5(exeh) MA o15(~+)~15 F17(~)o17 Ca39(B~)K39 Se41(S4) Ca41
one-body
finite range
5.75% 4.20% 7.15% 5.40%
9.5% 6.0% 10.9% 7.0%
H e r e the m e s o n i c exchange c o r r e c t i o n i s defined as
15 Februar~1965
Of c o u r s e , the z e r o r a n g e a p p r o x i m a t i o n i s not too good f o r l i g h t e r nuclei, and so the c a l c u l a tions were repeated using fully anti-symmetrised L S - c o u p l e d wave functions f o r the n u c l e i and u s i n g the e x p r e s s i o n (1) f o r HAexch. The dominant t e r m s in t h i s Calculation (which i s s t a n d a r d ) only involve the combination (2) so that the s a m e p h e n o m e n o l o g i c a l f o r m F e - a r / a r can again be u s e d . The v a l u e s f o r the c o r r e s p o n d i n g m e s o n i c exchange c o r r e c t i o n 8(exch)finit e range a r e a l s o l i s t e d in t a b l e 1. It can be s e e n that a s one goes to h e a v i e r n u c l e i , the d i s c r e p a n c y b e t w e e n t h e two v a l u e s of 5 d e c r e a s e s a s i s to be expected. In s u m m a r y , m e s o n i c exchange e f f e c t s l e a d to an i n c r e a s e of the o r d e r 6-11% in a l l o w e d a x i a l v e c t o r m a t r i x e l e m e n t s , and a r e c e r t a i n l y not n e g l i g i b l e . A l t e r n a t i v e l y , in o r d e r to t a k e t h e s e e f f e c t s into account the a x i a l v e c t o r coupling c o n stant can be r e p l a c e d by an " e f f e c t i v e v' coupling c o n s t a n t i n c o r p o r a t i n g an e n h a n c e m e n t of t h i s o r d e r of magnitude.
( xeh A
5(exch) one-body
}one-body MA
(6)
w h e r e M A i s the single p a r t i c l e s h e l l m o d e l a x i a l v e c t o r m a t r i x e l e m e n t f o r a d e c a y and (MAeXCh)one bod~ i s the m a t r i x e l e m e n t of (HAeXCh)one_body a s s u m i n g an o s c i l l a t o r function f o r the odd p a r t i c l e . --
~¢
.
.
o
I) O S S I B L E CP VIOLATING, SCALAR-PSEUDOSCALAR
References 1. J.S.Bell and R.J.Blin--Stoyle, Nuclear Phys. 6 (195s) s7. 2. R.J.Btin-Stoyle and S.Papageorgiou, Nuclear Phys. 64 (1965) 1. 3. R . J . Blin-Stoyle, V. Gupta and H. Primakoff, Nuclear Phys. 11 (1959) 444.
PT CONSERVING, INTERACTION*
S. N. L O T S O F F The Enrico Fermi Institute f o r Nuclear Studies and the Department of Physics, The University of Chicago, Chicago, Illinois Received 20 January 1965
R e c e n t l y s e v e r a l [ 1 - 3 , 8 , 9] s u g g e s t i o n s have b e e n m a d e to account f o r C P v i o l a t i o n o b s e r v e d [4,5] in n e u t r a l K m e s o n decay. The p u r p o s e of t h i s note i s to p o i n t out t h e p o s s i b l e e x i s t e n c e of a _PT c o n s e r v ing s c a l a r iS) - p s e u d o s c a l a r (i)) CP v i o l a t i n g coupling a l s o c o n s i s t e n t with p r e s e n t l y a v a i l a b l e data. H i s t o r i c a l l y the m e a s u r e m e n t of the b r a n c h i n g r a t i o (B) between the muon and e l e c t r o n d e c a y of c h a r g e d p i o n s h a s s e r v e d to d e m o n s t r a t e the d o m i n a n c e of C P i n v a r i a n t V - A coupling in w e a k i n t e r a c t i o n s ; howe v e r , in view of the a p p a r e n t l y s m a l l d e g r e e of C P v i o l a t i o n a c l o s e r e x a m i n a t i o n of the effect of a s m a l l P T i n v a r i a n t s c a l a r - p s e u d o s c a l a r coupling on B i s j u s t i f i e d . If s o m e c o m b i n a t i o n of V - A and S - t ) coupling i s a s s t u n e d which a c c o u n t s f o r t h e o b s e r v e d d e g r e e of C P v i o l a t i o n the i n t e r a c t i o n H a m i l t o n i a n can be w r i t t e n p h e n o m e n o l o g i c a l l y f o r pion d e c a y into a lepton p a i r a s follows with the s u b s c r i p t L r e f e r r i n g t o e i t h e r m u o n o r e l e c t r o n f o r a r b i t r a r y a , b, a ' , b ' *This work is supported by the U~. Atomic Energy Cemmission, COO-264-Z33. 344
Volu~ie t4, number 4
PHYSICS LETTERS
15 February 1965
H~ : ~.Cfv(-~ ]) ~.~Q(~+ ~)~S)~v + m~c ,ls(-~]) ~ ( ~ ' + ~'~s) ~v + h,c. where Q r e p r e s e n t s the pion four m o m e n t a ; f0, f v the appropriate s c a l a r and v e c t o r f o r m f a c t o r s ; G ~- 10" 5 Mp-z (the usual V-A coupling constant with dimension of i n v e r s e proton m a s s squared); G the coupling constant a s s o c i a t e d with the S - P interaction having the s a m e dimensions as G. The expression f o r B obtained f r o m this Hamiltonian is rl- ( me~212 I f s ~ ' 5 - me ~ i2 + __me L
]
L
t~'~is'~,,~ivl 2
~-7~Svi
~m~j j
lisa'~-mu<~r 12 leafs'
~lv12
where 6 = G ' / G ; oz = ½(a+ b), ~ = ½(a- b); a ', B' s i m i l a r l y defined. In p a r t i c u l a r , some special c a s e s m a y be noted. F o r no S - P coupling (i.e. 8 = 0) the prediction f o r B is independent of the choice of a and b. Any t h e o r y which, at least phenomenologically would call for a phase difference between the v e c t o r and axial v e c t o r portion of the interaction in o r d e r to account f o r CP violation would not in any way change the calculated value of B. Let us, however, c o n s i d e r an S - P interaction violating C P in a m a x i m a l way, i.e. a = 1, b = 1, a' = 1, b' = i. The e x p r e s s i o n for B would then b e c o m e -
~'2Sv
+
~-
8/s~ l/v/
)
+½(i#~21 t~vvi '
This, however, m u s t a g r e e with the experimental value [6] of B, 1.247 ~0.028 x 10 . 4 which essentially does not differ f r o m the value p r e d i c t e d by a pure V - A t h e o r y (with radiative corrections). This can c o m e about f o r ½ 6 ( f s / f v ) ~ m e / r ~ = 3.66 x 10 -3. A s s u m i n g both s c a l a r and v e c t o r f o r m f a c t o r s to be of the s a m e o r d e r of magnitude, the CP breaking S - P interaction m u s t be weaker than the usual V,A int e r a c t i o n by a f a c t o r of 10 -3. Examination of the n e u t r a l K mes_on d e c a y a s s u m i n g a s c a l a r coupling of this strength leads to a n e s t i m a t e of the amplitude of the K~2 2~ d e c a y mode r e l a t i v e to the ~llg-L 2~ amplitude (}) which can be in a g r e e m e n t with the experimentally r e p o r t e d [4] value of about 2.3 x 10-3. Following S a c h s ' notation [1,7] we define
: [<~oIwl Ko>/
A(K° - ,
27r) - i g v ( m 2 - m 2 ) l m i +
A(K°-~ 2-)-~-igv(m2-m2)lm2
6g s
+ ~gs
The interaction Hamiltonian leading to the above was taken to be of the f o r m -
-
+ -
+
+
345
Volume 14, number4
PHYSICS LETTERS
I5February1965
so that
I
I(
gs/gv)
l - 2.3 ×
w h e r e gv(-mK2), g s ( - m ~ ) a r e the a p p r o p r i a t e v e c t o r and s c a l a r f o r m f a c t o r s ( a s s u m e d r e a l ) . We note that the r e q u i r e m e n t s on 6 f r o m both ] ~ ] and B a r e c o n s i s t e n t . C o n t r a r y to the S a c h s ' r e q u i r e m e n t of l a r g e v i o l a t i o n of t i m e r e v e r s a l i n v a r i a n c e (T) in leptonic d e c a y s of K m e s o n s , the S - P i n t e r a c t i o n s u g g e s t e d h e r e would p r o b a b l y lead to m e a s u r a b l e violation of T i n v a r i a n c e only in d e c a y s s p e c i f i c a l l y f o r b i d d e n by CP i n v a r i a n c e . A s s u m i n g r e a s o n a b l e f o r m f a c t o r b e h a v i o u r , h o w e v e r , a d i f f e r e n c e i n the widths (F) for K~ d e c a y r e l a t e d by the CP o p e r a t i o n would be expected, e . g . ] r ( K ~ - ~ g + p - P ) / r ( K ~ - ~ ~ - ~ + v ) - 1] ~ 10 -3. T h i s would r e s u l t f r o m i n t e r f e r e n c e of s c a l a r and v e c t o r couplings. The author would like t o thank P r o f e s s o r Y. Nambu f o r helpful d i s c u s s i o n s .
References 1. 2. 3. 4. 5. 6. 7. 8. 9.
R.G.Sachs, Phys.Rev. Letters 13 (1964) 286. J.S.Bell and J.K. Perring, Phys.Rev. Letters 13 (1964) 348. L, Wolfenstein, Phys.Rev.Letters 13 (1964) 562. J.H. Christenson, J.W. Cronin, V. L. Fitch and R. Turlay, Phys.Rev. Letters 13 (1964) 138. A.Abashian, R . J . A b r a m s , D.W. Carpenter, G.P. Fisher, B.M.K.Nefkens and J.H.Smith, Phys.Rev. Letters 13 (1964) 243. E.Di Capua, R.Garland, L.Pondrom and A. Strelzoff, Phys.Rev. 133 (1964) B1333. R.G.Saehs, Ann. Phys. (N.Y.) 22 (1964) 239. S.L.Glashow, Phys.Rev. Letters 14 (1965) 35. B. Laurent and M.Roos, Physics Letters 13 (1964) 269.
THE
BARYON-MESON SYMMETRY
COUPLING IN A THEORY IN A RELATIVISTICALLY
WHICH DESCRIBES INVARIANT WAY
SU(6)
W. RIJHL CERN, Geneva R e c e i v e d 20 J a n u a r y 1965
If we t r y to g e n e r a l i z e the SU(6) s y m m e t r y m o d e l in such a way that it b e c o m e s r e l a t i v i s t i c a l ly i n v a r i a n t , we a r e led in a s t r a i g h t f o r w a r d way to the d i s c u s s i o n of the group SL(6, C). T h i s group r e p r e s e n t s the h o m o g e n e o u s p a r t of a h i g h e r s y m m e t r y G which i n v o l v e s the c o m p l e t e inhomogeneous L o r e n t z group. A s i s known [1] the i n v a r i ant subgroup of t r a n s l a t i o n s has at l e a s t t h i r t y six independent g e n e r a t o r s . We study the m e s o n b a r y o n v e r t e x in a group t h e o r e t i c a l m o d e l G with the following p r o p e r t i e s . (1) The i n v a r i a n t subgroup of t r a n s l a t i o n s cons i s t s of two dual r e a l v e c t o r s p a c e s {xhb}, {Xhb} each one of d i m e n s i o n 36. (2) The m o m e n t u m m a t r i c e s P&b, ~ab" obey the relation 346
= -x 2
a
T h i s condition d e t e r m i n e s a c e r t a i n s u b s e t of 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 s of G with the p r o p_erty that t h e i r little g r o u p s a r e SU(6), (if x z > 0). F u r t h e r m o r e it g u a r a n t e e s that in the language of f i e l d equations only d i f f e r e n t i a l equations of second o r d e r a r e p e r m i t t e d . (3) We a s s u m e that the m o m e n t u m m a t r i c e s of p h y s i c a l p a r t i c l e s contain only the SU(3) s c a lax components
In t h i s m o d e l , which was e x a m i n e d with r e s p e c t to s o m e of i t s m a t h e m a t i c a l p r o p e r t i e s in a p r e -
PHYSICS LETTERS
s u b s t a n t i a l l y with the n u c l e a r s u r f a c e r e g i o n w h e r e the d e n s i t y i s s m a l l e r . A f u r t h e r c a l c u lation w a s t h e r e f o r e c a r r i e d out f o r t h e ~ - d e c a y s of f o u r " s i m p l e " (i.e. doubly c l o s e d L S s h e l l p l u s o r m i n u s one nucleon) a s s u m i n g f o r p ( r ) the d e n s i t y d i s t r i b u t i o n obtained f o r n u c l e o n s in an o s c i l l a t o r p o t e n t i a l . The r e s u l t s a r e l i s t e d in t a b l e 1. Table 1 ~-decay of "simple" nuclei.
5(exeh) MA o15(~+)~15 F17(~)o17 Ca39(B~)K39 Se41(S4) Ca41
one-body
finite range
5.75% 4.20% 7.15% 5.40%
9.5% 6.0% 10.9% 7.0%
H e r e the m e s o n i c exchange c o r r e c t i o n i s defined as
15 Februar~1965
Of c o u r s e , the z e r o r a n g e a p p r o x i m a t i o n i s not too good f o r l i g h t e r nuclei, and so the c a l c u l a tions were repeated using fully anti-symmetrised L S - c o u p l e d wave functions f o r the n u c l e i and u s i n g the e x p r e s s i o n (1) f o r HAexch. The dominant t e r m s in t h i s Calculation (which i s s t a n d a r d ) only involve the combination (2) so that the s a m e p h e n o m e n o l o g i c a l f o r m F e - a r / a r can again be u s e d . The v a l u e s f o r the c o r r e s p o n d i n g m e s o n i c exchange c o r r e c t i o n 8(exch)finit e range a r e a l s o l i s t e d in t a b l e 1. It can be s e e n that a s one goes to h e a v i e r n u c l e i , the d i s c r e p a n c y b e t w e e n t h e two v a l u e s of 5 d e c r e a s e s a s i s to be expected. In s u m m a r y , m e s o n i c exchange e f f e c t s l e a d to an i n c r e a s e of the o r d e r 6-11% in a l l o w e d a x i a l v e c t o r m a t r i x e l e m e n t s , and a r e c e r t a i n l y not n e g l i g i b l e . A l t e r n a t i v e l y , in o r d e r to t a k e t h e s e e f f e c t s into account the a x i a l v e c t o r coupling c o n stant can be r e p l a c e d by an " e f f e c t i v e v' coupling c o n s t a n t i n c o r p o r a t i n g an e n h a n c e m e n t of t h i s o r d e r of magnitude.
( xeh A
5(exch) one-body
}one-body MA
(6)
w h e r e M A i s the single p a r t i c l e s h e l l m o d e l a x i a l v e c t o r m a t r i x e l e m e n t f o r a d e c a y and (MAeXCh)one bod~ i s the m a t r i x e l e m e n t of (HAeXCh)one_body a s s u m i n g an o s c i l l a t o r function f o r the odd p a r t i c l e . --
~¢
.
.
o
I) O S S I B L E CP VIOLATING, SCALAR-PSEUDOSCALAR
References 1. J.S.Bell and R.J.Blin--Stoyle, Nuclear Phys. 6 (195s) s7. 2. R.J.Btin-Stoyle and S.Papageorgiou, Nuclear Phys. 64 (1965) 1. 3. R . J . Blin-Stoyle, V. Gupta and H. Primakoff, Nuclear Phys. 11 (1959) 444.
PT CONSERVING, INTERACTION*
S. N. L O T S O F F The Enrico Fermi Institute f o r Nuclear Studies and the Department of Physics, The University of Chicago, Chicago, Illinois Received 20 January 1965
R e c e n t l y s e v e r a l [ 1 - 3 , 8 , 9] s u g g e s t i o n s have b e e n m a d e to account f o r C P v i o l a t i o n o b s e r v e d [4,5] in n e u t r a l K m e s o n decay. The p u r p o s e of t h i s note i s to p o i n t out t h e p o s s i b l e e x i s t e n c e of a _PT c o n s e r v ing s c a l a r iS) - p s e u d o s c a l a r (i)) CP v i o l a t i n g coupling a l s o c o n s i s t e n t with p r e s e n t l y a v a i l a b l e data. H i s t o r i c a l l y the m e a s u r e m e n t of the b r a n c h i n g r a t i o (B) between the muon and e l e c t r o n d e c a y of c h a r g e d p i o n s h a s s e r v e d to d e m o n s t r a t e the d o m i n a n c e of C P i n v a r i a n t V - A coupling in w e a k i n t e r a c t i o n s ; howe v e r , in view of the a p p a r e n t l y s m a l l d e g r e e of C P v i o l a t i o n a c l o s e r e x a m i n a t i o n of the effect of a s m a l l P T i n v a r i a n t s c a l a r - p s e u d o s c a l a r coupling on B i s j u s t i f i e d . If s o m e c o m b i n a t i o n of V - A and S - t ) coupling i s a s s t u n e d which a c c o u n t s f o r t h e o b s e r v e d d e g r e e of C P v i o l a t i o n the i n t e r a c t i o n H a m i l t o n i a n can be w r i t t e n p h e n o m e n o l o g i c a l l y f o r pion d e c a y into a lepton p a i r a s follows with the s u b s c r i p t L r e f e r r i n g t o e i t h e r m u o n o r e l e c t r o n f o r a r b i t r a r y a , b, a ' , b ' *This work is supported by the U~. Atomic Energy Cemmission, COO-264-Z33. 344
Volu~ie t4, number 4
PHYSICS LETTERS
15 February 1965
H~ : ~.Cfv(-~ ]) ~.~Q(~+ ~)~S)~v + m~c ,ls(-~]) ~ ( ~ ' + ~'~s) ~v + h,c. where Q r e p r e s e n t s the pion four m o m e n t a ; f0, f v the appropriate s c a l a r and v e c t o r f o r m f a c t o r s ; G ~- 10" 5 Mp-z (the usual V-A coupling constant with dimension of i n v e r s e proton m a s s squared); G the coupling constant a s s o c i a t e d with the S - P interaction having the s a m e dimensions as G. The expression f o r B obtained f r o m this Hamiltonian is rl- ( me~212 I f s ~ ' 5 - me ~ i2 + __me L
]
L
t~'~is'~,,~ivl 2
~-7~Svi
~m~j j
lisa'~-mu<~r 12 leafs'
~lv12
where 6 = G ' / G ; oz = ½(a+ b), ~ = ½(a- b); a ', B' s i m i l a r l y defined. In p a r t i c u l a r , some special c a s e s m a y be noted. F o r no S - P coupling (i.e. 8 = 0) the prediction f o r B is independent of the choice of a and b. Any t h e o r y which, at least phenomenologically would call for a phase difference between the v e c t o r and axial v e c t o r portion of the interaction in o r d e r to account f o r CP violation would not in any way change the calculated value of B. Let us, however, c o n s i d e r an S - P interaction violating C P in a m a x i m a l way, i.e. a = 1, b = 1, a' = 1, b' = i. The e x p r e s s i o n for B would then b e c o m e -
~'2Sv
+
~-
8/s~ l/v/
)
+½(i#~21 t~vvi '
This, however, m u s t a g r e e with the experimental value [6] of B, 1.247 ~0.028 x 10 . 4 which essentially does not differ f r o m the value p r e d i c t e d by a pure V - A t h e o r y (with radiative corrections). This can c o m e about f o r ½ 6 ( f s / f v ) ~ m e / r ~ = 3.66 x 10 -3. A s s u m i n g both s c a l a r and v e c t o r f o r m f a c t o r s to be of the s a m e o r d e r of magnitude, the CP breaking S - P interaction m u s t be weaker than the usual V,A int e r a c t i o n by a f a c t o r of 10 -3. Examination of the n e u t r a l K mes_on d e c a y a s s u m i n g a s c a l a r coupling of this strength leads to a n e s t i m a t e of the amplitude of the K~2 2~ d e c a y mode r e l a t i v e to the ~llg-L 2~ amplitude (}) which can be in a g r e e m e n t with the experimentally r e p o r t e d [4] value of about 2.3 x 10-3. Following S a c h s ' notation [1,7] we define
: [<~oIwl Ko>/
A(K° - ,
27r) - i g v ( m 2 - m 2 ) l m i +
A(K°-~ 2-)-~-igv(m2-m2)lm2
6g s
+ ~gs
The interaction Hamiltonian leading to the above was taken to be of the f o r m -
-
+ -
+
+
345
Volume 14, number4
PHYSICS LETTERS
I5February1965
so that
I
I(
gs/gv)
l - 2.3 ×
w h e r e gv(-mK2), g s ( - m ~ ) a r e the a p p r o p r i a t e v e c t o r and s c a l a r f o r m f a c t o r s ( a s s u m e d r e a l ) . We note that the r e q u i r e m e n t s on 6 f r o m both ] ~ ] and B a r e c o n s i s t e n t . C o n t r a r y to the S a c h s ' r e q u i r e m e n t of l a r g e v i o l a t i o n of t i m e r e v e r s a l i n v a r i a n c e (T) in leptonic d e c a y s of K m e s o n s , the S - P i n t e r a c t i o n s u g g e s t e d h e r e would p r o b a b l y lead to m e a s u r a b l e violation of T i n v a r i a n c e only in d e c a y s s p e c i f i c a l l y f o r b i d d e n by CP i n v a r i a n c e . A s s u m i n g r e a s o n a b l e f o r m f a c t o r b e h a v i o u r , h o w e v e r , a d i f f e r e n c e i n the widths (F) for K~ d e c a y r e l a t e d by the CP o p e r a t i o n would be expected, e . g . ] r ( K ~ - ~ g + p - P ) / r ( K ~ - ~ ~ - ~ + v ) - 1] ~ 10 -3. T h i s would r e s u l t f r o m i n t e r f e r e n c e of s c a l a r and v e c t o r couplings. The author would like t o thank P r o f e s s o r Y. Nambu f o r helpful d i s c u s s i o n s .
References 1. 2. 3. 4. 5. 6. 7. 8. 9.
R.G.Sachs, Phys.Rev. Letters 13 (1964) 286. J.S.Bell and J.K. Perring, Phys.Rev. Letters 13 (1964) 348. L, Wolfenstein, Phys.Rev.Letters 13 (1964) 562. J.H. Christenson, J.W. Cronin, V. L. Fitch and R. Turlay, Phys.Rev. Letters 13 (1964) 138. A.Abashian, R . J . A b r a m s , D.W. Carpenter, G.P. Fisher, B.M.K.Nefkens and J.H.Smith, Phys.Rev. Letters 13 (1964) 243. E.Di Capua, R.Garland, L.Pondrom and A. Strelzoff, Phys.Rev. 133 (1964) B1333. R.G.Saehs, Ann. Phys. (N.Y.) 22 (1964) 239. S.L.Glashow, Phys.Rev. Letters 14 (1965) 35. B. Laurent and M.Roos, Physics Letters 13 (1964) 269.
THE
BARYON-MESON SYMMETRY
COUPLING IN A THEORY IN A RELATIVISTICALLY
WHICH DESCRIBES INVARIANT WAY
SU(6)
W. RIJHL CERN, Geneva R e c e i v e d 20 J a n u a r y 1965
If we t r y to g e n e r a l i z e the SU(6) s y m m e t r y m o d e l in such a way that it b e c o m e s r e l a t i v i s t i c a l ly i n v a r i a n t , we a r e led in a s t r a i g h t f o r w a r d way to the d i s c u s s i o n of the group SL(6, C). T h i s group r e p r e s e n t s the h o m o g e n e o u s p a r t of a h i g h e r s y m m e t r y G which i n v o l v e s the c o m p l e t e inhomogeneous L o r e n t z group. A s i s known [1] the i n v a r i ant subgroup of t r a n s l a t i o n s has at l e a s t t h i r t y six independent g e n e r a t o r s . We study the m e s o n b a r y o n v e r t e x in a group t h e o r e t i c a l m o d e l G with the following p r o p e r t i e s . (1) The i n v a r i a n t subgroup of t r a n s l a t i o n s cons i s t s of two dual r e a l v e c t o r s p a c e s {xhb}, {Xhb} each one of d i m e n s i o n 36. (2) The m o m e n t u m m a t r i c e s P&b, ~ab" obey the relation 346
= -x 2
a
T h i s condition d e t e r m i n e s a c e r t a i n s u b s e t of 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 s of G with the p r o p_erty that t h e i r little g r o u p s a r e SU(6), (if x z > 0). F u r t h e r m o r e it g u a r a n t e e s that in the language of f i e l d equations only d i f f e r e n t i a l equations of second o r d e r a r e p e r m i t t e d . (3) We a s s u m e that the m o m e n t u m m a t r i c e s of p h y s i c a l p a r t i c l e s contain only the SU(3) s c a lax components
In t h i s m o d e l , which was e x a m i n e d with r e s p e c t to s o m e of i t s m a t h e m a t i c a l p r o p e r t i e s in a p r e -