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U (3) × U (3) breaking and 2 γ - decays of pseudoscalar mesons
Volume 35B, number 7
U(3)
× U(3)
PHYSICS
BREAKING
AND
LETTERS
2 y-DECAYS
OF
19 July 1971
PSEUDOSCALAR
MESONS*
F . S T R O C C H I ** Scuola Normale Superiore, Pisa, Italy Istituto Nazionale di Fisica Nucleate, Sezione di Pisa, Pisa, Italy and R. V E R G A R A C A F F A R E L L I Istituto di Fisica dell~Universit~, Pisa, Italy Received 31 May 1971
The physical consequences of a U(3) ×U(3) breaking t e r m E 5 in the Hamiltonian H = Ho + E 5 + £iui are investigated. Mass formulae are obtained and the 77 - ~7' mixing is shown to be governed by this t e r m . The 2 y - d e c a y s of p s e u d o - s c a l a r m e s o n s are computed by using the above Hamiltonian. A significant test is the prediction of a very small rate E(1422) ~ 2y. T h e r e a r e s t r o n g i n d i c a t i o n s [1, 2] t h a t in t h e G e l l - M a n n - O a k e s - R e n n e r t h e f o l l o w i n g d e c o m p o s i t i o n of t h e s t r o n g H a m i l t o n i a n
(GOR) m o d e l [3, 4] b a s e d on
H=H o+EoU o+¢8U 8
(1)
t h e t e r m Ho, w h i c h i s i n v a r i a n t u n d e r SU(3) ® SU(3), c a n n o t b e U(3) ® U(3) i n v a r i a n t . O t h e r w i s e one would get the following r e l a t i o n s [5]*** zr =7}'
0 = -35 °
w h e r e 7r a n d 7' d e n o t e t h e s q u a r e s of t h e m a s s e s of t h e c o r r e s p o n d i n g p a r t i c l e s a n d 0 i s t h e 77 - 7/' m i x i n g a n g l e ~. T h e n a t u r a l c o n c l u s i o n i s t h a t U(3) ® U(3) i s b r o k e n at t h e l e v e l of H o. One m a y t h e r e f o r e s p l i t Ho in t h e f o l l o w i n g w a y [1,2] ?? H o=Hoo+e6 w h e r e H o o i s U(3) ® U(3) i n v a r i a n t a n d 6 b r e a k s U(3) ® U(3). T h e a i m of t h e p r e s e n t n o t e i s to t e s t eq. (2) a n d t o d e t e r m i n e t h e v a l u e s of ~ a n d <0] w h i c h a r e in f a c t e s s e n t i a l p a r a m e t e r s in t h e d e f i n i t i o n of t h e m o d e l ~ . T o this purpose s p e c i f y t h e 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 6 u n d e r F 0 5 . It i s r e a s o n a b l e to a s s u m e t h a t 5 a n 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 U(3) ® U(3), j u s t a s t h e t e r m s ~o Uo + c 8 u8 t r a n s f o r m i b l e r e p r e s e n t a t i o n of SU(3) ® SU(3):
(2) 6 ]07 --- (5} o, one h a s t o transforms as as an i r r e d u c -
* Supported by Istituto Nazionale di Fisica Nucleate, Sezione di P i s a , P i s a , Italy. ** P r e s e n t a d d r e s s : Department of P h y s i c s , University of Princeton. *** The f i r s t of the above relations has alo been d i s c u s s e d by Glashow [2]. It may be i n t e r e s t i n g to r e m a r k that the second relation r e m a i n s true also if a t e r m ¢ 3u3 is added in the Hamiltonian (1). A possible way out of the above difficulties is the introduction of unequal r e n o r m a l i z a t i o n constants Z i [2]. This s e e m s however an u n n e c e s s a r y complication and we will show later that one may get a perfectly consistent picture of the m e s o n spectrum by using Ward-like identities ~ la Glashow and Weinberg with all the Z i equal. We adopt the usual convention 1717 = cos 0 I v8~ + sin 0 Iron, I~7'~ = - sin0 [v8~ + cos0 I Vo~. ~f~" This splitting has been d i s c u s s e d mainly from the point of view of broken scale invariance. Our e m p h a s i s is on the breaking of U(3) ~ U ( 3 ) and t h e r e f o r e the t e r m E ~i appearing in eq. (2) is not a c - n u m b e r t e r m . The difference between the two points of view as well as the possible existence of c - n u m b e r t e r m s in Hoo will be d i s c u s s e d in a subsequent p a p e r [5]. ~'~'~" This is the case whenever the s y m m e t r y is spontaneously broken [4].
595
Volume 35B, n u m b e r 7
I F 0 5 , 6] = - i ~ - ~ d 6 - ;
PHYSICS
[F05, 6-] = i ~ d
LETTERS
19 July 1971
5
(3) [Fi, 5] = 0 = [Fz~ , 6];
[F i, 6 - ] = 0 = [ F 5, 6-1
(i : 1, 2 , . . . ,
8)
d i s a n u m e r i c a l c o n s t a n t a n d i t i s r e l a t e d to t h e " d i m e n s i o n s " of 6. S i n c e t h 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 U(3) ® U(3) c o n s t r u c t e d i n t e r m s of t h e (3, 3) ® (3, 3) r e p r e s e n t a t i o n a n d s a t i s f y i n g e q s . (3), t r a n s f o r m s [6] a s t h e trilinear t e r m s
+ i E~v+ol +~. +y . -c~..,-fi.,,-~ 1 3 =~eo~y (W~ W~ w u +w~ w g w v ) (4)
i3 = 2-eotflYi
~v
(W~"+or W~+Bw u'+y-w~" -c~ wp"'-fi~"-Yw v )
.(W~a= ½42 ~i(ki(ui +vi))~ a r e t h e c o m p o n e n t s of t h e (3, 3) + (5, 3) r e p r e s e n t a t i o n s o n e h a s d = 3. One m a y now c o n s t r u c t t h e f u n c t i o n a l W07), ~i -= {(~ luil 0), a n d w o r k out t h e W a r d - l i k e i d e n t i t i e s in t h e s t a n d a r d w a y [7]. Up to now w e h a v e d i s c u s s e d o n l y t h e 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 6 u n d e r U(3)X × U(3) a n d w e h a v e t a k e n f o r g r a n t e d t h a t , a s e v e r y t e r m in t h e L a g r a n g i a n ( o r H a m i l t o n i a n ) , it i s c o n s t r u c t e d i n t e r m s of t h e (3, 3) ® (3, 3). T h i s m a y g i v e r i s e to a n o n t r i v i a l p r o b l e m f r o m t h e p o i n t of v i e w of q u a n t u m f i e l d t h e o r y , b e c a u s e if ui(x) a n d vi(x) , i = 0 , . . . 8, a r e q u a n t u m f i e l d s , t h e r e i s a c e r t a i n a m b i g u i t y i n g i v i n g a m e a n i n g to t r i l i n e a r t e r m s l i k e I ~. T h i s a m b i g u i t y , h o w e v e r , a f f e c t s o n l y s u p e r f i c i a l l y t h e p h y s i c a l c o n c l u s i o n s . A s a m a t t e r of f a c t , ~what i s r e l e v a n t i s t h e s t r u c t u r e of t h e f u n c t i o n W(77) a n d o n e m a y s h o w [e.g. 8] t h a t to t h e t r i l i n e a r t e r m in t h e L a g r a n g i a n m a y b e g i v e n a m e a n i n g s o t h a t i t s e f f e c t s i n W(~?) i s t h e a p p e a r a n c e S : ~ of a t e r m g I~(r]). A c t u a l l y , in t h e t r e e a p p r o x i m a t i o n o n e h a s (,c 6) o = ~ I~ (77). ( T h i s a p p r o x i m a t i o n w i l l b e u s e d a t t h e e n d of t h e p a p e r ) . A s t r a i g h t f o r w a r d b u t l e n g t h y a l g e b r a g i v e s t h e f o l l o w i n g r e l a t i o n s S:~S FKFTr( 2 K - r / - 7 ? ' ) = 3 ~ ( F T r - 4 F g ) + (1 -FF~)0? ' -77) c o s 2 ( 0 + 0 )
(5)
0?'-~?) s i n 2 ( 0 + 0-) = 6 4 2 - ~ FTr
(6)
w h e r e 0-= a r c t g ½4"2 -~ 35o.. T h e y y i e l d a m i x i n g a n g l e a r o u n d 5 ° a n d * ~ = 4 . 1 + 0 . 1 mTr. Eq. (6) c l e a r l y s h o w s t h a t 0 = - 35 ° i H o i s U(3) ® U(3) i n v a r i a n t * * . It i s i m p o r t a n t to s t r e s s t h a t the above model of U(3) ® U(3) breaking may very well account for the77 -rl' mixing without recourse to
unequal renormalization constants. F u r t h e r i m p l i c a t i o n s of t h e a b o v e m o d e l m a y b e o b t a i n e d b y s t u d y i n g t h e 2 ~ d e c a y s of p s e u d o s c a l a r m e s o n s . T h e d i v e r g e n c e s of t h e a x i a l c u r r e n t s A ~ , A ~ a n d A ~ a r e e a s i l y o b t a i n e d b y c o m m u t i n g t h e c o r r e s p o n d i n g c h a r g e s a n d t h e H a m i l t o n i a n (1) a n d b y a d d i n g t h e A d l e r e l e c t r o m a g n e t i c c o r r e c t i o n [10]. ~ / a A ~ = ~ 43-c o ( ' / 2 + c ) q~Tr + ( e 2 / 1 6 ~ r a ) a(3) Fuu ~ u
(7)
S For the (3,3) + (3,3) r e p r e s e n t a t i o n F05 is defined in the usual way [F05, ui] = - i ~ v i [F05 vi] = i ~f~u i. S:~ A simple justification of the above s t a t e m e n t is that if the Lagrangian is a function of the field ui(x), vi{x}, the functional Wwill be only a function of t h e i r expectation values??: W = W(r]). Now as discussed above, the only invariant under SU(3)× SU(3) and parity, which b r e a k s U(3)× U(3) is I3+(77). Therefore, in the usual splitting W(??) = Wo(r/)+ E i ~7i one may' write quite generally Wo(~7)= Woo(T/)~ W~(77)13+(r/). The only a r b i t r a r i n e s s is in the function W'o(??), invariant under U(3)xU(3). This may be shown to reduce to a constant, if 6 is defined in the natural way [8]. $SS The f i r s t of the following relations is obtained by combining different Ward identities. It has the following advantages: it involves only quantities with d i r e c t physical meaning and it is approximately independent of 0 (a p a r a m e t e r which is r a t h e r unknown). In fact, the t e r m involving e is roughly 6 x 10 -2 s m a l l e r than the t e r m on the left hand side and it may be neglected as a f i r s t approximation. * After this work was completed we were informed that this value of ~ yields, in the free approximation, a mean baryon m a s s of 950 MeV. ** One may show [5] that in this limit 77¢ 77'. It may be i n t e r e s t i n g to note that an SU(2) @ SU(2) invariant theory (Frr =0) would imply 0 = - 3 5 ° .
596
Volume 35B, number 7 E° 8~A/~ = ~ [ ( ( ' / 2 2 c )
0~Ao~ = ~
'
PHYSICS
cos0+~/2 c sin0)oq+
LETTERS
19 July 1971
('/2 c cosO-(v~-c)sinO)q?,]
e2 a (8) +--167r 2
e2 ~o[(Sin 0 + c cos 0) ~0~+ (cos 0 - c s e n 0 ) o r / , ] - /-6 ~ 5- + - - a (0) 16~2
F~tu F g U
(8)
F~u ~ u
(9)
By t a k i n g the m a t r i x e l e m e n t s of the above e q u a t i o n s b e t w e e n the v a c u u m and the 2 y s t a t e a n d p r o c e e d i n g in the s t a n d a r d way [10,11] one has
(/-2+<') [/'~'rr_ : a(3)f 1ll.
(10)
[ ( / - 2 - c ) cos 0 + /-2 c s i n 01 .AIr ; r w~- + [/-2 c cos 0 - ( / - 2 - c ) s i n 0 ] ~ I."XITj w h e r e F~,. F~?, F•,,
M ~w A~~ . /~;$I;'
=a(8)fM"
(11)
a r e the 2• decay width of ~, ~7, ~'; Fi ~ m31(01(Dil 27}, a n d f is a c o m m o n f a c t o r
f < (- e2/167r2) (0 1F,pu - P ~ 1 2 7 ) ( 3 / c o i n c l u d i n g 2~ p h a s e s p a c e f a c t o r s . A f i r s t a n a l y s i s of eq. (10) g i v e s MTr a ( 3 ) f ~ 2 . 8 x 1 0 -5. Since a(3) and a(8) a r e of the s a m e o r d e r of m a g n i t u d e * * * , the t e r m a(8)fM~ is 1 0 - 2 s m a l l e r t h a n the f i r s t t e r m in eq. (11) and t h e r e f o r e it m a y be n e g l e c t e d . T h u s one o b t a i n s
{ ~ 3 [,/-2-c+]-2ctgO 2 F~?, = \M~ ) ~ (vr2- c)tgO-4-2c) r ~
(12)
T h i s q u e s t i o n p r e d i c t s $ FT?, ~ 10 keV. F o r c = - v ~ the above e q u a t i o n t a k e s the s i m p l e f o r m (Air/,)3 r v, = \ - ~ cotg 2 (0 +0) r~?
(12')
In o r d e r to e x p l o i t the c o n s e q u e n c e s of eq. (9) it is c o n v e n i e n t to n o r m a l i s e 5 i n s u c h a way that it has the d i m e n s i o n of a b o s o n f i e l d ~ . By w r i t i n g a W a r d i d e n t i t y ~ la Glashow a n d W e i n b e r g one gets E A 5 - 5 - (0) = L i m i~ p~o
f
= E (5} o + L i m i p-~0
fexp(-ipx)(T(civi(x ) 5-(0))}d4x (13)
One m a y show that the l a s t t e r m v a n i s h e s ~ J ' ~ w h e n c = -~-2 and t h e r e f o r e it is r e a s o n a b l e to n e g l e c t it as a f i r s t a p p r o x i m a t i o n . T h u s , one has A 515 . (5} o
o r e2
~
(eS}oA 5 - 5 -
(14)
If the two point f u n c t i o n A 5 - 5 - is d o m i n a t e d by one pole c o r r e s p o n d i n g to the m a s s M2 5- one has M 2 - <5}o = ~
(15)
and eq. (9) m a y be u s e d to p r e d i c t the r a t e F S - ~ 2y in t e r m of FT? a n d of (E 5}o. A c o n s i s t e n t p i c t u r e of the m e s o n s p e c t r u m is o b t a i n e d if 5 and 5- a r e a s s u m e d to be d o m i n a t e d b y the d ( 7 0 0 ) and the E(1422) m e s o n p o l e s [12]. In this c a s e one has r~-~2~
= (M5-)3 M~
(
~27c2/-~c
2
\(4-g-c)sinO-(~ccosO)
2
e2
E(~5) M~
F r / + O ( 1 0 .3
(16) F~7)
*** A quark model gives a(3) = ,/~ a(8). a(0) = ,/:~(3q2 +2 -4q), q being the quark charge. Eq. (12) does not depend on the model of U(3) x U(3) breaking. However. it could hardly be justified without realizing that the/7 -7/' mixing is governed by U(3) x U(3) breaking with no need of unequal renormalization constants. "[J" This simply amounts to redefine ~ as only the product ~ 5 has a definite meaning. For simplicity we will denote by £ and 5 the quantities so normalised. "~'~'~ This is easily obtained by writing the Ward identity for the charge /~0 + ~-,/2 F~. 597
Volume 35B, number 7
PHYSICS L E T T E R S
19 July 1971
/ I The 52 7 decay may be estimated by making the t r e e approximation for ~E5}0, (ES}o = ~I 3 (~7), One obtains a very small numberS: F 6 ~2~ ~ 3× 10-2 F T. This s e e m s a significant t e s t of the model d i s c u s s e d so far as there is no a p r i o r i reason why the E ~ 27 rate should be so small. As a m a t t e r of fact, the dominance of the 5 t e r m by the E(1422) meson pole implies that the E - 27 decay is strongly s u p p r e s s e d in the limit c = - ~f-2. The mechanism is the same that s u p p r e s s e s the matrix element <0} v o - v ~ v8127} when c = -~/-2, yielding eq. (12'). The basic fact is that in both c a s e s the fields 5 and u o - ~/2 u 8 dominate the divergences of the c u r r e n t s A ~ - ~ 2 A ~ a n d A s , respectively, and both the c h a r g e s A O - ~ 2 A 8 andA 8 commute with the e l e c t r o magnetic current. W e a r e v e r y m u c h g r a t e f u l to P r o f e s s o r cussions and for encouragement.
R. G a t t o a n d t o P r o f e s s o r
L. A. Radicati
for clarifying
dis-
S T h i s r e s u l t would not c h a n g e e v e n if one i n t r o d u c e s a m i x i n ~ b e t w e e n 11, 11 ' and 5.
References [i] M. Oell-Mann, University of Hawaii Summer School Lectures, 1969 (California Institute of Technology Report No CALT-68-244). [2] S. L. Glashow, Hadrons and their interactions, ed. A. Zichichi (AcademicPress, New York, 1968), [3] M. Gell-Mann, R.J. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195. [4] S. L.Glashow and S. Weinberg, Phys. Rev. Letters 20 (1968) 224. [5] F. Strocchi and R. Vergara Caffarelti, in preparation. [6] G. Cicogna, F. Stroechi and R. Vergara Caffare[li, Phys. Rev. Letters 22 (1969) 497. [7] B. Zumino, Brandeis Summer Institute Lectures, 1970. The main ideas go back to J. Schwinger, Proc. Natl. Acad. of Sciences 37 (1951) 452, 455; G. Jona-Lasinio, Nuovo Cimento 34 (1964) 1790; G. Parisi and M. Testa, Nuovo Cimento 67 (1970) 13. [8] H.D. Dahmen, G. Jona-Lasinio, Nuovo Cimento 52A (1967) 80'7. [9] G. Parisi and M. Testa, Rome University preprint. [10] S. Adler, Phys. Rev. 177 (1969) 2426. [11] S, L. Glashow, R. Jackiw and S. S. Shei, Phys. Rev. 187 (1969) 1916. [12] R.J. Oakes, presented at XVth Intern. Conf. on High energy physics, Kiev, 1970; G. Parisi and M. Testa, Rome University preprint.
598
U(3)
× U(3)
PHYSICS
BREAKING
AND
LETTERS
2 y-DECAYS
OF
19 July 1971
PSEUDOSCALAR
MESONS*
F . S T R O C C H I ** Scuola Normale Superiore, Pisa, Italy Istituto Nazionale di Fisica Nucleate, Sezione di Pisa, Pisa, Italy and R. V E R G A R A C A F F A R E L L I Istituto di Fisica dell~Universit~, Pisa, Italy Received 31 May 1971
The physical consequences of a U(3) ×U(3) breaking t e r m E 5 in the Hamiltonian H = Ho + E 5 + £iui are investigated. Mass formulae are obtained and the 77 - ~7' mixing is shown to be governed by this t e r m . The 2 y - d e c a y s of p s e u d o - s c a l a r m e s o n s are computed by using the above Hamiltonian. A significant test is the prediction of a very small rate E(1422) ~ 2y. T h e r e a r e s t r o n g i n d i c a t i o n s [1, 2] t h a t in t h e G e l l - M a n n - O a k e s - R e n n e r t h e f o l l o w i n g d e c o m p o s i t i o n of t h e s t r o n g H a m i l t o n i a n
(GOR) m o d e l [3, 4] b a s e d on
H=H o+EoU o+¢8U 8
(1)
t h e t e r m Ho, w h i c h i s i n v a r i a n t u n d e r SU(3) ® SU(3), c a n n o t b e U(3) ® U(3) i n v a r i a n t . O t h e r w i s e one would get the following r e l a t i o n s [5]*** zr =7}'
0 = -35 °
w h e r e 7r a n d 7' d e n o t e t h e s q u a r e s of t h e m a s s e s of t h e c o r r e s p o n d i n g p a r t i c l e s a n d 0 i s t h e 77 - 7/' m i x i n g a n g l e ~. T h e n a t u r a l c o n c l u s i o n i s t h a t U(3) ® U(3) i s b r o k e n at t h e l e v e l of H o. One m a y t h e r e f o r e s p l i t Ho in t h e f o l l o w i n g w a y [1,2] ?? H o=Hoo+e6 w h e r e H o o i s U(3) ® U(3) i n v a r i a n t a n d 6 b r e a k s U(3) ® U(3). T h e a i m of t h e p r e s e n t n o t e i s to t e s t eq. (2) a n d t o d e t e r m i n e t h e v a l u e s of ~ a n d <0] w h i c h a r e in f a c t e s s e n t i a l p a r a m e t e r s in t h e d e f i n i t i o n of t h e m o d e l ~ . T o this purpose s p e c i f y t h e 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 6 u n d e r F 0 5 . It i s r e a s o n a b l e to a s s u m e t h a t 5 a n 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 U(3) ® U(3), j u s t a s t h e t e r m s ~o Uo + c 8 u8 t r a n s f o r m i b l e r e p r e s e n t a t i o n of SU(3) ® SU(3):
(2) 6 ]07 --- (5} o, one h a s t o transforms as as an i r r e d u c -
* Supported by Istituto Nazionale di Fisica Nucleate, Sezione di P i s a , P i s a , Italy. ** P r e s e n t a d d r e s s : Department of P h y s i c s , University of Princeton. *** The f i r s t of the above relations has alo been d i s c u s s e d by Glashow [2]. It may be i n t e r e s t i n g to r e m a r k that the second relation r e m a i n s true also if a t e r m ¢ 3u3 is added in the Hamiltonian (1). A possible way out of the above difficulties is the introduction of unequal r e n o r m a l i z a t i o n constants Z i [2]. This s e e m s however an u n n e c e s s a r y complication and we will show later that one may get a perfectly consistent picture of the m e s o n spectrum by using Ward-like identities ~ la Glashow and Weinberg with all the Z i equal. We adopt the usual convention 1717 = cos 0 I v8~ + sin 0 Iron, I~7'~ = - sin0 [v8~ + cos0 I Vo~. ~f~" This splitting has been d i s c u s s e d mainly from the point of view of broken scale invariance. Our e m p h a s i s is on the breaking of U(3) ~ U ( 3 ) and t h e r e f o r e the t e r m E ~i appearing in eq. (2) is not a c - n u m b e r t e r m . The difference between the two points of view as well as the possible existence of c - n u m b e r t e r m s in Hoo will be d i s c u s s e d in a subsequent p a p e r [5]. ~'~'~" This is the case whenever the s y m m e t r y is spontaneously broken [4].
595
Volume 35B, n u m b e r 7
I F 0 5 , 6] = - i ~ - ~ d 6 - ;
PHYSICS
[F05, 6-] = i ~ d
LETTERS
19 July 1971
5
(3) [Fi, 5] = 0 = [Fz~ , 6];
[F i, 6 - ] = 0 = [ F 5, 6-1
(i : 1, 2 , . . . ,
8)
d i s a n u m e r i c a l c o n s t a n t a n d i t i s r e l a t e d to t h e " d i m e n s i o n s " of 6. S i n c e t h 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 U(3) ® U(3) c o n s t r u c t e d i n t e r m s of t h e (3, 3) ® (3, 3) r e p r e s e n t a t i o n a n d s a t i s f y i n g e q s . (3), t r a n s f o r m s [6] a s t h e trilinear t e r m s
+ i E~v+ol +~. +y . -c~..,-fi.,,-~ 1 3 =~eo~y (W~ W~ w u +w~ w g w v ) (4)
i3 = 2-eotflYi
~v
(W~"+or W~+Bw u'+y-w~" -c~ wp"'-fi~"-Yw v )
.(W~a= ½42 ~i(ki(ui +vi))~ a r e t h e c o m p o n e n t s of t h e (3, 3) + (5, 3) r e p r e s e n t a t i o n s o n e h a s d = 3. One m a y now c o n s t r u c t t h e f u n c t i o n a l W07), ~i -= {(~ luil 0), a n d w o r k out t h e W a r d - l i k e i d e n t i t i e s in t h e s t a n d a r d w a y [7]. Up to now w e h a v e d i s c u s s e d o n l y t h e 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 6 u n d e r U(3)X × U(3) a n d w e h a v e t a k e n f o r g r a n t e d t h a t , a s e v e r y t e r m in t h e L a g r a n g i a n ( o r H a m i l t o n i a n ) , it i s c o n s t r u c t e d i n t e r m s of t h e (3, 3) ® (3, 3). T h i s m a y g i v e r i s e to a n o n t r i v i a l p r o b l e m f r o m t h e p o i n t of v i e w of q u a n t u m f i e l d t h e o r y , b e c a u s e if ui(x) a n d vi(x) , i = 0 , . . . 8, a r e q u a n t u m f i e l d s , t h e r e i s a c e r t a i n a m b i g u i t y i n g i v i n g a m e a n i n g to t r i l i n e a r t e r m s l i k e I ~. T h i s a m b i g u i t y , h o w e v e r , a f f e c t s o n l y s u p e r f i c i a l l y t h e p h y s i c a l c o n c l u s i o n s . A s a m a t t e r of f a c t , ~what i s r e l e v a n t i s t h e s t r u c t u r e of t h e f u n c t i o n W(77) a n d o n e m a y s h o w [e.g. 8] t h a t to t h e t r i l i n e a r t e r m in t h e L a g r a n g i a n m a y b e g i v e n a m e a n i n g s o t h a t i t s e f f e c t s i n W(~?) i s t h e a p p e a r a n c e S : ~ of a t e r m g I~(r]). A c t u a l l y , in t h e t r e e a p p r o x i m a t i o n o n e h a s (,c 6) o = ~ I~ (77). ( T h i s a p p r o x i m a t i o n w i l l b e u s e d a t t h e e n d of t h e p a p e r ) . A s t r a i g h t f o r w a r d b u t l e n g t h y a l g e b r a g i v e s t h e f o l l o w i n g r e l a t i o n s S:~S FKFTr( 2 K - r / - 7 ? ' ) = 3 ~ ( F T r - 4 F g ) + (1 -FF~)0? ' -77) c o s 2 ( 0 + 0 )
(5)
0?'-~?) s i n 2 ( 0 + 0-) = 6 4 2 - ~ FTr
(6)
w h e r e 0-= a r c t g ½4"2 -~ 35o.. T h e y y i e l d a m i x i n g a n g l e a r o u n d 5 ° a n d * ~ = 4 . 1 + 0 . 1 mTr. Eq. (6) c l e a r l y s h o w s t h a t 0 = - 35 ° i H o i s U(3) ® U(3) i n v a r i a n t * * . It i s i m p o r t a n t to s t r e s s t h a t the above model of U(3) ® U(3) breaking may very well account for the77 -rl' mixing without recourse to
unequal renormalization constants. F u r t h e r i m p l i c a t i o n s of t h e a b o v e m o d e l m a y b e o b t a i n e d b y s t u d y i n g t h e 2 ~ d e c a y s of p s e u d o s c a l a r m e s o n s . T h e d i v e r g e n c e s of t h e a x i a l c u r r e n t s A ~ , A ~ a n d A ~ a r e e a s i l y o b t a i n e d b y c o m m u t i n g t h e c o r r e s p o n d i n g c h a r g e s a n d t h e H a m i l t o n i a n (1) a n d b y a d d i n g t h e A d l e r e l e c t r o m a g n e t i c c o r r e c t i o n [10]. ~ / a A ~ = ~ 43-c o ( ' / 2 + c ) q~Tr + ( e 2 / 1 6 ~ r a ) a(3) Fuu ~ u
(7)
S For the (3,3) + (3,3) r e p r e s e n t a t i o n F05 is defined in the usual way [F05, ui] = - i ~ v i [F05 vi] = i ~f~u i. S:~ A simple justification of the above s t a t e m e n t is that if the Lagrangian is a function of the field ui(x), vi{x}, the functional Wwill be only a function of t h e i r expectation values??: W = W(r]). Now as discussed above, the only invariant under SU(3)× SU(3) and parity, which b r e a k s U(3)× U(3) is I3+(77). Therefore, in the usual splitting W(??) = Wo(r/)+ E i ~7i one may' write quite generally Wo(~7)= Woo(T/)~ W~(77)13+(r/). The only a r b i t r a r i n e s s is in the function W'o(??), invariant under U(3)xU(3). This may be shown to reduce to a constant, if 6 is defined in the natural way [8]. $SS The f i r s t of the following relations is obtained by combining different Ward identities. It has the following advantages: it involves only quantities with d i r e c t physical meaning and it is approximately independent of 0 (a p a r a m e t e r which is r a t h e r unknown). In fact, the t e r m involving e is roughly 6 x 10 -2 s m a l l e r than the t e r m on the left hand side and it may be neglected as a f i r s t approximation. * After this work was completed we were informed that this value of ~ yields, in the free approximation, a mean baryon m a s s of 950 MeV. ** One may show [5] that in this limit 77¢ 77'. It may be i n t e r e s t i n g to note that an SU(2) @ SU(2) invariant theory (Frr =0) would imply 0 = - 3 5 ° .
596
Volume 35B, number 7 E° 8~A/~ = ~ [ ( ( ' / 2 2 c )
0~Ao~ = ~
'
PHYSICS
cos0+~/2 c sin0)oq+
LETTERS
19 July 1971
('/2 c cosO-(v~-c)sinO)q?,]
e2 a (8) +--167r 2
e2 ~o[(Sin 0 + c cos 0) ~0~+ (cos 0 - c s e n 0 ) o r / , ] - /-6 ~ 5- + - - a (0) 16~2
F~tu F g U
(8)
F~u ~ u
(9)
By t a k i n g the m a t r i x e l e m e n t s of the above e q u a t i o n s b e t w e e n the v a c u u m and the 2 y s t a t e a n d p r o c e e d i n g in the s t a n d a r d way [10,11] one has
(/-2+<') [/'~'rr_ : a(3)f 1ll.
(10)
[ ( / - 2 - c ) cos 0 + /-2 c s i n 01 .AIr ; r w~- + [/-2 c cos 0 - ( / - 2 - c ) s i n 0 ] ~ I."XITj w h e r e F~,. F~?, F•,,
M ~w A~~ . /~;$I;'
=a(8)fM"
(11)
a r e the 2• decay width of ~, ~7, ~'; Fi ~ m31(01(Dil 27}, a n d f is a c o m m o n f a c t o r
f < (- e2/167r2) (0 1F,pu - P ~ 1 2 7 ) ( 3 / c o i n c l u d i n g 2~ p h a s e s p a c e f a c t o r s . A f i r s t a n a l y s i s of eq. (10) g i v e s MTr a ( 3 ) f ~ 2 . 8 x 1 0 -5. Since a(3) and a(8) a r e of the s a m e o r d e r of m a g n i t u d e * * * , the t e r m a(8)fM~ is 1 0 - 2 s m a l l e r t h a n the f i r s t t e r m in eq. (11) and t h e r e f o r e it m a y be n e g l e c t e d . T h u s one o b t a i n s
{ ~ 3 [,/-2-c+]-2ctgO 2 F~?, = \M~ ) ~ (vr2- c)tgO-4-2c) r ~
(12)
T h i s q u e s t i o n p r e d i c t s $ FT?, ~ 10 keV. F o r c = - v ~ the above e q u a t i o n t a k e s the s i m p l e f o r m (Air/,)3 r v, = \ - ~ cotg 2 (0 +0) r~?
(12')
In o r d e r to e x p l o i t the c o n s e q u e n c e s of eq. (9) it is c o n v e n i e n t to n o r m a l i s e 5 i n s u c h a way that it has the d i m e n s i o n of a b o s o n f i e l d ~ . By w r i t i n g a W a r d i d e n t i t y ~ la Glashow a n d W e i n b e r g one gets E A 5 - 5 - (0) = L i m i~ p~o
f
= E (5} o + L i m i p-~0
fexp(-ipx)(T(civi(x ) 5-(0))}d4x (13)
One m a y show that the l a s t t e r m v a n i s h e s ~ J ' ~ w h e n c = -~-2 and t h e r e f o r e it is r e a s o n a b l e to n e g l e c t it as a f i r s t a p p r o x i m a t i o n . T h u s , one has A 515 . (5} o
o r e2
~
(eS}oA 5 - 5 -
(14)
If the two point f u n c t i o n A 5 - 5 - is d o m i n a t e d by one pole c o r r e s p o n d i n g to the m a s s M2 5- one has M 2 - <5}o = ~
(15)
and eq. (9) m a y be u s e d to p r e d i c t the r a t e F S - ~ 2y in t e r m of FT? a n d of (E 5}o. A c o n s i s t e n t p i c t u r e of the m e s o n s p e c t r u m is o b t a i n e d if 5 and 5- a r e a s s u m e d to be d o m i n a t e d b y the d ( 7 0 0 ) and the E(1422) m e s o n p o l e s [12]. In this c a s e one has r~-~2~
= (M5-)3 M~
(
~27c2/-~c
2
\(4-g-c)sinO-(~ccosO)
2
e2
E(~5) M~
F r / + O ( 1 0 .3
(16) F~7)
*** A quark model gives a(3) = ,/~ a(8). a(0) = ,/:~(3q2 +2 -4q), q being the quark charge. Eq. (12) does not depend on the model of U(3) x U(3) breaking. However. it could hardly be justified without realizing that the/7 -7/' mixing is governed by U(3) x U(3) breaking with no need of unequal renormalization constants. "[J" This simply amounts to redefine ~ as only the product ~ 5 has a definite meaning. For simplicity we will denote by £ and 5 the quantities so normalised. "~'~'~ This is easily obtained by writing the Ward identity for the charge /~0 + ~-,/2 F~. 597
Volume 35B, number 7
PHYSICS L E T T E R S
19 July 1971
/ I The 52 7 decay may be estimated by making the t r e e approximation for ~E5}0, (ES}o = ~I 3 (~7), One obtains a very small numberS: F 6 ~2~ ~ 3× 10-2 F T. This s e e m s a significant t e s t of the model d i s c u s s e d so far as there is no a p r i o r i reason why the E ~ 27 rate should be so small. As a m a t t e r of fact, the dominance of the 5 t e r m by the E(1422) meson pole implies that the E - 27 decay is strongly s u p p r e s s e d in the limit c = - ~f-2. The mechanism is the same that s u p p r e s s e s the matrix element <0} v o - v ~ v8127} when c = -~/-2, yielding eq. (12'). The basic fact is that in both c a s e s the fields 5 and u o - ~/2 u 8 dominate the divergences of the c u r r e n t s A ~ - ~ 2 A ~ a n d A s , respectively, and both the c h a r g e s A O - ~ 2 A 8 andA 8 commute with the e l e c t r o magnetic current. W e a r e v e r y m u c h g r a t e f u l to P r o f e s s o r cussions and for encouragement.
R. G a t t o a n d t o P r o f e s s o r
L. A. Radicati
for clarifying
dis-
S T h i s r e s u l t would not c h a n g e e v e n if one i n t r o d u c e s a m i x i n ~ b e t w e e n 11, 11 ' and 5.
References [i] M. Oell-Mann, University of Hawaii Summer School Lectures, 1969 (California Institute of Technology Report No CALT-68-244). [2] S. L. Glashow, Hadrons and their interactions, ed. A. Zichichi (AcademicPress, New York, 1968), [3] M. Gell-Mann, R.J. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195. [4] S. L.Glashow and S. Weinberg, Phys. Rev. Letters 20 (1968) 224. [5] F. Strocchi and R. Vergara Caffarelti, in preparation. [6] G. Cicogna, F. Stroechi and R. Vergara Caffare[li, Phys. Rev. Letters 22 (1969) 497. [7] B. Zumino, Brandeis Summer Institute Lectures, 1970. The main ideas go back to J. Schwinger, Proc. Natl. Acad. of Sciences 37 (1951) 452, 455; G. Jona-Lasinio, Nuovo Cimento 34 (1964) 1790; G. Parisi and M. Testa, Nuovo Cimento 67 (1970) 13. [8] H.D. Dahmen, G. Jona-Lasinio, Nuovo Cimento 52A (1967) 80'7. [9] G. Parisi and M. Testa, Rome University preprint. [10] S. Adler, Phys. Rev. 177 (1969) 2426. [11] S, L. Glashow, R. Jackiw and S. S. Shei, Phys. Rev. 187 (1969) 1916. [12] R.J. Oakes, presented at XVth Intern. Conf. on High energy physics, Kiev, 1970; G. Parisi and M. Testa, Rome University preprint.
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