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Mass of kappa meson from spectral function sum rules
_8.A.SJ Nuclear P h y s i c s B1O (1969) 208-212. North-Holland Publ. C o m p . , A m s t e r d a m
MASS OF KAPPA MESON FROM SPECTRAL FUNCTION SUM RULES R. ACHARYA Institut f l i t theoretische Physik der Universit~t Bern, Bern, Switzerland
Received 21 October 1968 Abstract: A modified form of the second Weinberg sum rule is proposed for s t r a n g e ness-changing vector and a x i a l - v e c t o r c u r r e n t s . Together with the f i r s t sum rule, it is used to p r e d i c t the kappa mass. The estimate is compared with the value obtained from chiral s y m m e t r y considerations of Glashow and Weinberg.
In t h e p r e s e n t n o t e , we o b t a i n a n e s t i m a t e of t h e k a p p a m e s o n m a s s (strange, scalar meson) from spectral function sum rules. We find that m K ~ 1130 M e V in e s s e n t i a l a g r e e m e n t w i t h t h e p r e d i c t i o n of b r o k e n c h i r a l s y m m e t r y [1]. W e s t a r t w i t h a m o d i f i e d f o r m of t h e s e c o n d W e i n b e r g s u m r u l e b u t r e t a i n t h e s t a n d a r d f o r m of t h e f i r s t s u m r u l e f o r t h a t p a r t i c u l a r SU(2) ® SU(2) s u b g r o u p of SU(3) ® SU(3) w h i c h p e r t a i n s to t h e s t r a n g e n e s s - c h a n g i n g v e c tor and axial vector currents: cO
of
Id~2
PK*(p2)-PKA(~2)~ 2 ~2 ~ = F K
F2 ~,
(1)
2
(2)
o Eq. (1) i s , of c o u r s e ,
the usual first sum rule with
(3a) PKA(/~2 ) ~ g 2 A 5(~ 2 - m 2 A ) ,
(3b)
where inK* = 890 M e V and m K A = 1250 MeV. Eq. (2), however, differs from the standard form of the second s u m rule [2] by the presence of extra terms on the right-hand side of this equation. $ On leave of absence
from
the American
University
of Beirut,
Beirut,
Lebanon.
KAPPA MESON MASS
209
T h e s e t e r m s m a y be r e g a r d e d as ' c o r r e c t i o n ' t e r m s a r i s i n g due to the finite m a s s e s of the kaon and the k a p p a m e s o n . It has b e e n pointed out r e cently by J a c k i w [3] that finite m a s s c o r r e c t i o n s m u s t be e x p e c t e d in the SU(3) ® SU(3) g e n e r a l i z a t i o n of the second s u m rule. Such c o r r e c t i o n s a r e not e x p e c t e d to be s e r i o u s in the c a s e of the SU(2) ® SU(2) s u b g r o u p which d e a l s with strangeness-conserving V and A c u r r e n t s , in view of the s m a l l n e s of the pion m a s s [4], in c o n t r a s t to the p r e s e n t c a s e of slrangenesschanging V and A c u r r e n t s , n o n - v a n i s h i n g d i v e r g e n c e s of which a r e p r o p r o t i o n a l to the k a p p a and the kaon m a s s e s r e s p e c t i v e l y . Although t h e r e e x i s t no known m o d e l s of field t h e o r y f r o m which eq. (2) m a y b e r i g o r o u s l y d e r i v e d , it is, n o n e t h e l e s s , p o s s i b l e to advance p l a u s i bility a r g u m e n t s in f a v o u r of eq. (2) f r o m a s y m p t o t i c s y m m e t r y c o n s i d e r a tions a d v o v a t e d by Okubo and c o a u t h o r s $. Following r e f . [5], one w r i t e s K*u(q) _ AKA(q) -= F(q2)6~ u + G(q2)q~qu , Ap.
(4)
where F(q 2) =
fd~ 2 PK*(t12)-
PKA(P 2) , q2 + p2
G(q 2) = fdl~ 2 PK*(lX2)-PKA(~2) + F2 ~2(q2 + ~2) q2+m2
(5)
F2
(6)
q2+m2"
The a s s u m p t i o n that G(q 2) ~ 1/q 4+~ g i v e s us the modified second s u m r u l e . One s e e s this i m m e d i a t e l y by o b s e r v i n g that
G(q2._>~) ~ _
{f[pK,(,2)_OKA(~Z2)]+m2~
_mKFK } 22 + O 1 .
(7)
The leading t e r m of O ( l / q 2) is a b s e n t in eq. (7) in view of the f i r s t W e i n b e r g s u m r u l e eq. (1). It is i m p o r t a n t to notice that the s i m u l t a n e o u s a s y m p t o t i c r e q u i r e m e n t F(q 2) oc 1/q2+E and G(q 2) oc 1/q4+E on the i n v a r i a n t functions ( c o r r e s p o n d i n g to the s u p e r v a l i d i W of a s y m p t o t i c s y m m e t r y ) will be mutually c o n s i s t e n t only if F2m 2 = F2m 2. T h i s s p e c i a l c a s e will be c o n s i d e r e d l a t e r . F o r the p r e s e n t , we shall c o n c e r n o u r s e l v e s with eqs. (I) and (2). Pole dominance
of eqs. (I) and (2) yields g2,
g~(A
2
F2
m2. and
2
2
2 2 _ m2F2
g K * - g K A = mK'FK
K g "
See ref. [ 5]. In view of eq. (I) we are ignoring the Schwinger term in eq. (4).
(9)
210
R. AC HARYA At this point, we take the f i r s t s u m r u l e [6] g2
g2,_
m2
m2,
F2
(10)
K'
and the K S F R r e l a t i o n [7], 2
2
go = 2F21rr a p ,
(11)
into c o n s i d e r a t i o n . F r o m eqs. (8) - (11), we obtain m2: F2 FI~ , 2 mg2 = inK*2 + 2 ~F ~~ ( m 2K A - i n 2K , ) - ~ t m K A -
2 . inK)
(12)
In v i e w of the r e l a t i o n [8] 2 = F2 FK ~ ,
(13)
which holds to all o r d e r s of S U ( 3 ) - b r e a k i n g i n t e r a c t i o n u n d e r pole d o m i nance ofJ=0andJ= 1 m e s o n s , eq. (12) r e d u c e s to 2 F2 2 2 - -{m=KA- 2 i n K , + mI¢}" m2 = mK*+F
(14)
We o b s e r v e that the f i n i t e kaon m a s s c o r r e c t i o n to the kappa m a s s a r i s e s e n t i r e l y f r o m the s e c o n d t e r m ( r e c a l l that m K A ~ ( 2 m K * ) , the k a p pa and the K*(890) b e c o m i n g d e g e n e r a t e in the l i m i t of vanishing kaon m a s s . To e s t i m a t e the k a o n m a s s c o r r e c t i o n , one m u s t in t u r n e s t i m a t e F "2/w2 We do this by a p p e a l i n g to the G l a s h o w - W e i n b e r g [9] r e f i n e m e n t of I T I ± K" the A d e m o l l o - G a t t o t h e o r e m : f + ( 0 ) = (F 2 + F 2 - F 2 K ) ( 2 F K F ) - 1
(15)
w h e r e f+(0) d e n o t e s the r e n o r m a l i z e d Ke3 f o r m f a c t o r at q2 = 0, and by u s i n g the e x p e r i m e n t a l r e s u l t -
FK FTr-~(0)
= 1.28 .
(16)
Eqs. (13), (15) and (16) give F2
Eq. (14) then y i e l d s m K
F2 ~2.2.
(17)
1130 MeV. T h i s value is c o m p a r a b l e with the
$ See r e f . [ 1 0 ] . Actually ~ = 1.28±0.06, but we are working with 1.28 to obtain an estimate of m K.
KAPPA MESON MASS
211
p r e d i c t i o n m K ~ 1080 M e V f r o m b r o k e n c h i r a l s y m m e t r y [1]. A l t h o u g h t h e a g r e e m e n t i s not p e r f e c t I , it i s e n c o u r a g i n g t h a t t h e two p r e d i c t i o n s a r e f a i r l y c l o s e to e a c h o t h e r . F i n a l l y , l e t u s c o n s i d e r t h e p o s s i b i l i t y of r e c o v e r i n g t h e u s u a l f o r m of t h e s e c o n d s u m r u l e f r o m eq. (2): t h i s h a p p e n s when
T h e n c o n s t r a i n t eq. (18) i m p l i e s t h e v a l i d i t y of t h e s t a n d a r d s u m r u l e g 2 . = = g f ( A f o r a s p e c i f i c c h o i c e of t h e k a p p a m a s s . T h i s c a s e h a s b e e n t r e a t e d b y K a n g [12]. H o w e v e r , o u r v i e w p o i n t d i f f e r s in one i m p o r t a n t r e s p e c t : W e 2 -- F 2 t a k e t h e a t t i t u d e t h a t one m u s t n e c e s s a r i l y e m p l o y t h e r e l a t i o n F K w i t h i n t h e f r a m e w o r k of p o l e d o m i n a n c e of s p e c t r a l f u n c t i o n s u m r u l e s , a s s o o n a s one a c c e p t s t h e v a l i d i t y of t h e K S F R r e l a t i o n eq. (11). T h i s p o i n t i s d i s c u s s e d in d e t a i l in r e f . [8]. Now, in t h e e a r l i e r t r e a t m e n t s of t h e r a t i o F2K//F2~ b a s e d on both t h e W e i n b e r g s u m r u l e s f o r t h e (~, K ~ ) s y s t e m , one d o e s e m p l o y K S F R b u t t h i s h a s t h e c o n s e q u e n c e t h a t F K / F ~ i s not u n i t y . It i s , t h e r e f o r e , c l e a r t h a t t h e u s e of t h e e q u a l i t y F 2 = F 2 w o u l d not b e c o n s i s t e n t w i t h t h e a s s u m e d v a l i d i t y of b o t h t h e W e i n b e r g s u m r u l e s f o r t h e (p, KA) s y s t e m . H o w e v e r , we a r e d e a l i n g w i t h b o t h t h e W e i n b e r g s u m r u l e s f o r t h e chiral p a r t n e r s K* a n d KA, t o g e t h e r w i t h t h e f i r s t s u m r u l e f o r p a n d K*, a n d a l s o K F S R . In t h i s c a s e , if t h e e q u a l i t y F 2K = F 2 w e r e not i m p o s e d , we w o u l d o b t a i n -
A
(19/
T h e a b o v e e q u a t i o n by itself d o e s not d e t e r m i n e t h e r a t i o F 2 / F 2 in t e r m s of t h e m a s s e s alone, in v i e w of t h e p r e s e n c e of theo r a t i o F2'/F 2. H e n c e , t h e r e i s no o b v i o u s i n c o n s i s t e n c y in u s i n g F 2 = F ~ in t h e p r e s e n t c o n t e x t . If w e do t h i s , we o b t a i n
F r o m e q s . (18) a n d (19'):
4
2
~nK
2
~K*
2 2K ~ 50 (with i n K , = 890 M e V a n d m K A = 1250 But, t h e n eq. (20) g i v e s F~(/F $ F o r experimental attempts, see ref. [11].
212
R. ACHARYA
MeV), c o n t r a d i c t i n g eq. (17). One m a y , i n s t e a d , fix the k a p p a m a s s $ f r o m eqs. (17) a n d (18) g i v i n g m g ~ 730 MeV. T h e n , eq. (20) g i v e s m K A 1110 MeV w h i c h is l o w e r t h a n the c u r r e n t l y a c c e p t e d v a l u e of 1250 MeV. H e n c e , t h i s p o s s i b i l i t y is not f a v o u r e d by e x p e r i m e n t . In c o n c l u s i o n , we w i s h to p o i n t out t h a t a s t r a i g h t f o r w a r d e x t e n s i o n of the m o d i f i e d s e c o n d s u m r u l e eq. (2) to the full SU(3) ® SU(3) g r o u p l e a d s to i n c o r r e c t p r e d i c t i o n s : f o r e x a m p l e , w h e n a p p l i e d to t h e a x i a l - v e c t o r This may be an indication c u r r e n t s of ' ~ ' a n d ' K ' t y p e , it g i v e s m~(A < t h a t the m e c h a n i s m of s y m m e t r y - b r e a k i n g is f a r m o r e s o p h i s t i c a t e d in the SU(3) ® SU(3) g e n e r a l i z a t i o n of t h e s e c o n d s u m r u l e t h a n i s i m p l i e d by a n a i v e e x t e n s i o n of eq. (2). In any c a s e , a n e x p e r i m e n t a l t e s t of eq. (14) m u s t b e a w a i t e d b e f o r e one p a s s e s j u d g e m e n t ~ on the v a l i d i t y of eq. (2). I t a k e t h i s o p p o r t u n i t y to thallk P r o f e s s o r s A. M e r c i e r , H. L e u t w y l e r a n d Dr. H. B e b i ~ f o r the w a r m h o s p i t a l i t y e x t e n d e d to m e at the U n i v e r s i t y of B e r n e .
RE FERENCES [i] L. Chang and Y. Leung, Phys. Rev. Letters 21 (1968); R. Acharya and H. Bebig, Mass of kappa meson and relative sign of F K and FTT from broken chiral symmetry, University of Berne, preprint, Aug~tst 1968. [2] S. Weinberg, Phys. Rev. Letters 18 (1967) 507. [;~] R. Jackiw, Phys. Letters 27B (1968) 96; CERN Report, TH 896, May 1968. [4] S. P. De Alwis, Weinberg sum rules and the pion electromagnetic mass differences, University of Cambridge, preprint, September 1967; C. Cook, L. Evans, M. Han, N. Lipshutz and N. Straumann, Nucl. Phys. B5 (1968) 140. [5] T. Das, V. Mathur and S. Okubo, Phys. Rev. Letters 18 (1967) 761. [6] S. Glashow, H. Schnitzer and S. Weinberg, Phys. Rev. Letters 19 (1967) 139. [7] R. Acharya, H. H. Aly, H. A. Mavromatis and K. Schileher, Nuovo Cimento 54A (1967) 179 and references therein. [8] R. Acharya and H. H. Aly, Phys. Letters 27B (1968) 166. [9] S. Glashow and S. Weinberg, Phys. Rev. Letters 20 (1968) 225. [i0] N. Brene, M. Roos and A. Sirlin, Nucl. Phys. B6 (1968) 255. [ii] R. Bland, G. Goldhaber, B. Hall and G. Trilling, Phys. Rev. Letters 21 (1968) 173; E. Fowler, L. Montanet and R. Bizzarri, Phys. Rev. Letters 21 (1968) 833. [12] K. Kang, Nuovo Cimento 56 (1968) 1147. [13] T. G. Trippe et al., Phys. Letters 28B (1968) 203.
This violates the Glashow-Weinberg bounds m K --< 525 MeV or :> 945 MeV obtained with F2 =F 2 anu~ F2/F 2 ~ z K ~ 2.2; see ref. [9]. :~ A recent experiment by Trippe et al. (ref. [13]) has indicated the existence of the kappa meson around 1100-1200 MeV. if this is confirmed, it would provide good support for the modified sum rule eq. (2).
MASS OF KAPPA MESON FROM SPECTRAL FUNCTION SUM RULES R. ACHARYA Institut f l i t theoretische Physik der Universit~t Bern, Bern, Switzerland
Received 21 October 1968 Abstract: A modified form of the second Weinberg sum rule is proposed for s t r a n g e ness-changing vector and a x i a l - v e c t o r c u r r e n t s . Together with the f i r s t sum rule, it is used to p r e d i c t the kappa mass. The estimate is compared with the value obtained from chiral s y m m e t r y considerations of Glashow and Weinberg.
In t h e p r e s e n t n o t e , we o b t a i n a n e s t i m a t e of t h e k a p p a m e s o n m a s s (strange, scalar meson) from spectral function sum rules. We find that m K ~ 1130 M e V in e s s e n t i a l a g r e e m e n t w i t h t h e p r e d i c t i o n of b r o k e n c h i r a l s y m m e t r y [1]. W e s t a r t w i t h a m o d i f i e d f o r m of t h e s e c o n d W e i n b e r g s u m r u l e b u t r e t a i n t h e s t a n d a r d f o r m of t h e f i r s t s u m r u l e f o r t h a t p a r t i c u l a r SU(2) ® SU(2) s u b g r o u p of SU(3) ® SU(3) w h i c h p e r t a i n s to t h e s t r a n g e n e s s - c h a n g i n g v e c tor and axial vector currents: cO
of
Id~2
PK*(p2)-PKA(~2)~ 2 ~2 ~ = F K
F2 ~,
(1)
2
(2)
o Eq. (1) i s , of c o u r s e ,
the usual first sum rule with
(3a) PKA(/~2 ) ~ g 2 A 5(~ 2 - m 2 A ) ,
(3b)
where inK* = 890 M e V and m K A = 1250 MeV. Eq. (2), however, differs from the standard form of the second s u m rule [2] by the presence of extra terms on the right-hand side of this equation. $ On leave of absence
from
the American
University
of Beirut,
Beirut,
Lebanon.
KAPPA MESON MASS
209
T h e s e t e r m s m a y be r e g a r d e d as ' c o r r e c t i o n ' t e r m s a r i s i n g due to the finite m a s s e s of the kaon and the k a p p a m e s o n . It has b e e n pointed out r e cently by J a c k i w [3] that finite m a s s c o r r e c t i o n s m u s t be e x p e c t e d in the SU(3) ® SU(3) g e n e r a l i z a t i o n of the second s u m rule. Such c o r r e c t i o n s a r e not e x p e c t e d to be s e r i o u s in the c a s e of the SU(2) ® SU(2) s u b g r o u p which d e a l s with strangeness-conserving V and A c u r r e n t s , in view of the s m a l l n e s of the pion m a s s [4], in c o n t r a s t to the p r e s e n t c a s e of slrangenesschanging V and A c u r r e n t s , n o n - v a n i s h i n g d i v e r g e n c e s of which a r e p r o p r o t i o n a l to the k a p p a and the kaon m a s s e s r e s p e c t i v e l y . Although t h e r e e x i s t no known m o d e l s of field t h e o r y f r o m which eq. (2) m a y b e r i g o r o u s l y d e r i v e d , it is, n o n e t h e l e s s , p o s s i b l e to advance p l a u s i bility a r g u m e n t s in f a v o u r of eq. (2) f r o m a s y m p t o t i c s y m m e t r y c o n s i d e r a tions a d v o v a t e d by Okubo and c o a u t h o r s $. Following r e f . [5], one w r i t e s K*u(q) _ AKA(q) -= F(q2)6~ u + G(q2)q~qu , Ap.
(4)
where F(q 2) =
fd~ 2 PK*(t12)-
PKA(P 2) , q2 + p2
G(q 2) = fdl~ 2 PK*(lX2)-PKA(~2) + F2 ~2(q2 + ~2) q2+m2
(5)
F2
(6)
q2+m2"
The a s s u m p t i o n that G(q 2) ~ 1/q 4+~ g i v e s us the modified second s u m r u l e . One s e e s this i m m e d i a t e l y by o b s e r v i n g that
G(q2._>~) ~ _
{f[pK,(,2)_OKA(~Z2)]+m2~
_mKFK } 22 + O 1 .
(7)
The leading t e r m of O ( l / q 2) is a b s e n t in eq. (7) in view of the f i r s t W e i n b e r g s u m r u l e eq. (1). It is i m p o r t a n t to notice that the s i m u l t a n e o u s a s y m p t o t i c r e q u i r e m e n t F(q 2) oc 1/q2+E and G(q 2) oc 1/q4+E on the i n v a r i a n t functions ( c o r r e s p o n d i n g to the s u p e r v a l i d i W of a s y m p t o t i c s y m m e t r y ) will be mutually c o n s i s t e n t only if F2m 2 = F2m 2. T h i s s p e c i a l c a s e will be c o n s i d e r e d l a t e r . F o r the p r e s e n t , we shall c o n c e r n o u r s e l v e s with eqs. (I) and (2). Pole dominance
of eqs. (I) and (2) yields g2,
g~(A
2
F2
m2. and
2
2
2 2 _ m2F2
g K * - g K A = mK'FK
K g "
See ref. [ 5]. In view of eq. (I) we are ignoring the Schwinger term in eq. (4).
(9)
210
R. AC HARYA At this point, we take the f i r s t s u m r u l e [6] g2
g2,_
m2
m2,
F2
(10)
K'
and the K S F R r e l a t i o n [7], 2
2
go = 2F21rr a p ,
(11)
into c o n s i d e r a t i o n . F r o m eqs. (8) - (11), we obtain m2: F2 FI~ , 2 mg2 = inK*2 + 2 ~F ~~ ( m 2K A - i n 2K , ) - ~ t m K A -
2 . inK)
(12)
In v i e w of the r e l a t i o n [8] 2 = F2 FK ~ ,
(13)
which holds to all o r d e r s of S U ( 3 ) - b r e a k i n g i n t e r a c t i o n u n d e r pole d o m i nance ofJ=0andJ= 1 m e s o n s , eq. (12) r e d u c e s to 2 F2 2 2 - -{m=KA- 2 i n K , + mI¢}" m2 = mK*+F
(14)
We o b s e r v e that the f i n i t e kaon m a s s c o r r e c t i o n to the kappa m a s s a r i s e s e n t i r e l y f r o m the s e c o n d t e r m ( r e c a l l that m K A ~ ( 2 m K * ) , the k a p pa and the K*(890) b e c o m i n g d e g e n e r a t e in the l i m i t of vanishing kaon m a s s . To e s t i m a t e the k a o n m a s s c o r r e c t i o n , one m u s t in t u r n e s t i m a t e F "2/w2 We do this by a p p e a l i n g to the G l a s h o w - W e i n b e r g [9] r e f i n e m e n t of I T I ± K" the A d e m o l l o - G a t t o t h e o r e m : f + ( 0 ) = (F 2 + F 2 - F 2 K ) ( 2 F K F ) - 1
(15)
w h e r e f+(0) d e n o t e s the r e n o r m a l i z e d Ke3 f o r m f a c t o r at q2 = 0, and by u s i n g the e x p e r i m e n t a l r e s u l t -
FK FTr-~(0)
= 1.28 .
(16)
Eqs. (13), (15) and (16) give F2
Eq. (14) then y i e l d s m K
F2 ~2.2.
(17)
1130 MeV. T h i s value is c o m p a r a b l e with the
$ See r e f . [ 1 0 ] . Actually ~ = 1.28±0.06, but we are working with 1.28 to obtain an estimate of m K.
KAPPA MESON MASS
211
p r e d i c t i o n m K ~ 1080 M e V f r o m b r o k e n c h i r a l s y m m e t r y [1]. A l t h o u g h t h e a g r e e m e n t i s not p e r f e c t I , it i s e n c o u r a g i n g t h a t t h e two p r e d i c t i o n s a r e f a i r l y c l o s e to e a c h o t h e r . F i n a l l y , l e t u s c o n s i d e r t h e p o s s i b i l i t y of r e c o v e r i n g t h e u s u a l f o r m of t h e s e c o n d s u m r u l e f r o m eq. (2): t h i s h a p p e n s when
T h e n c o n s t r a i n t eq. (18) i m p l i e s t h e v a l i d i t y of t h e s t a n d a r d s u m r u l e g 2 . = = g f ( A f o r a s p e c i f i c c h o i c e of t h e k a p p a m a s s . T h i s c a s e h a s b e e n t r e a t e d b y K a n g [12]. H o w e v e r , o u r v i e w p o i n t d i f f e r s in one i m p o r t a n t r e s p e c t : W e 2 -- F 2 t a k e t h e a t t i t u d e t h a t one m u s t n e c e s s a r i l y e m p l o y t h e r e l a t i o n F K w i t h i n t h e f r a m e w o r k of p o l e d o m i n a n c e of s p e c t r a l f u n c t i o n s u m r u l e s , a s s o o n a s one a c c e p t s t h e v a l i d i t y of t h e K S F R r e l a t i o n eq. (11). T h i s p o i n t i s d i s c u s s e d in d e t a i l in r e f . [8]. Now, in t h e e a r l i e r t r e a t m e n t s of t h e r a t i o F2K//F2~ b a s e d on both t h e W e i n b e r g s u m r u l e s f o r t h e (~, K ~ ) s y s t e m , one d o e s e m p l o y K S F R b u t t h i s h a s t h e c o n s e q u e n c e t h a t F K / F ~ i s not u n i t y . It i s , t h e r e f o r e , c l e a r t h a t t h e u s e of t h e e q u a l i t y F 2 = F 2 w o u l d not b e c o n s i s t e n t w i t h t h e a s s u m e d v a l i d i t y of b o t h t h e W e i n b e r g s u m r u l e s f o r t h e (p, KA) s y s t e m . H o w e v e r , we a r e d e a l i n g w i t h b o t h t h e W e i n b e r g s u m r u l e s f o r t h e chiral p a r t n e r s K* a n d KA, t o g e t h e r w i t h t h e f i r s t s u m r u l e f o r p a n d K*, a n d a l s o K F S R . In t h i s c a s e , if t h e e q u a l i t y F 2K = F 2 w e r e not i m p o s e d , we w o u l d o b t a i n -
A
(19/
T h e a b o v e e q u a t i o n by itself d o e s not d e t e r m i n e t h e r a t i o F 2 / F 2 in t e r m s of t h e m a s s e s alone, in v i e w of t h e p r e s e n c e of theo r a t i o F2'/F 2. H e n c e , t h e r e i s no o b v i o u s i n c o n s i s t e n c y in u s i n g F 2 = F ~ in t h e p r e s e n t c o n t e x t . If w e do t h i s , we o b t a i n
F r o m e q s . (18) a n d (19'):
4
2
~nK
2
~K*
2 2K ~ 50 (with i n K , = 890 M e V a n d m K A = 1250 But, t h e n eq. (20) g i v e s F~(/F $ F o r experimental attempts, see ref. [11].
212
R. ACHARYA
MeV), c o n t r a d i c t i n g eq. (17). One m a y , i n s t e a d , fix the k a p p a m a s s $ f r o m eqs. (17) a n d (18) g i v i n g m g ~ 730 MeV. T h e n , eq. (20) g i v e s m K A 1110 MeV w h i c h is l o w e r t h a n the c u r r e n t l y a c c e p t e d v a l u e of 1250 MeV. H e n c e , t h i s p o s s i b i l i t y is not f a v o u r e d by e x p e r i m e n t . In c o n c l u s i o n , we w i s h to p o i n t out t h a t a s t r a i g h t f o r w a r d e x t e n s i o n of the m o d i f i e d s e c o n d s u m r u l e eq. (2) to the full SU(3) ® SU(3) g r o u p l e a d s to i n c o r r e c t p r e d i c t i o n s : f o r e x a m p l e , w h e n a p p l i e d to t h e a x i a l - v e c t o r This may be an indication c u r r e n t s of ' ~ ' a n d ' K ' t y p e , it g i v e s m~(A < t h a t the m e c h a n i s m of s y m m e t r y - b r e a k i n g is f a r m o r e s o p h i s t i c a t e d in the SU(3) ® SU(3) g e n e r a l i z a t i o n of t h e s e c o n d s u m r u l e t h a n i s i m p l i e d by a n a i v e e x t e n s i o n of eq. (2). In any c a s e , a n e x p e r i m e n t a l t e s t of eq. (14) m u s t b e a w a i t e d b e f o r e one p a s s e s j u d g e m e n t ~ on the v a l i d i t y of eq. (2). I t a k e t h i s o p p o r t u n i t y to thallk P r o f e s s o r s A. M e r c i e r , H. L e u t w y l e r a n d Dr. H. B e b i ~ f o r the w a r m h o s p i t a l i t y e x t e n d e d to m e at the U n i v e r s i t y of B e r n e .
RE FERENCES [i] L. Chang and Y. Leung, Phys. Rev. Letters 21 (1968); R. Acharya and H. Bebig, Mass of kappa meson and relative sign of F K and FTT from broken chiral symmetry, University of Berne, preprint, Aug~tst 1968. [2] S. Weinberg, Phys. Rev. Letters 18 (1967) 507. [;~] R. Jackiw, Phys. Letters 27B (1968) 96; CERN Report, TH 896, May 1968. [4] S. P. De Alwis, Weinberg sum rules and the pion electromagnetic mass differences, University of Cambridge, preprint, September 1967; C. Cook, L. Evans, M. Han, N. Lipshutz and N. Straumann, Nucl. Phys. B5 (1968) 140. [5] T. Das, V. Mathur and S. Okubo, Phys. Rev. Letters 18 (1967) 761. [6] S. Glashow, H. Schnitzer and S. Weinberg, Phys. Rev. Letters 19 (1967) 139. [7] R. Acharya, H. H. Aly, H. A. Mavromatis and K. Schileher, Nuovo Cimento 54A (1967) 179 and references therein. [8] R. Acharya and H. H. Aly, Phys. Letters 27B (1968) 166. [9] S. Glashow and S. Weinberg, Phys. Rev. Letters 20 (1968) 225. [i0] N. Brene, M. Roos and A. Sirlin, Nucl. Phys. B6 (1968) 255. [ii] R. Bland, G. Goldhaber, B. Hall and G. Trilling, Phys. Rev. Letters 21 (1968) 173; E. Fowler, L. Montanet and R. Bizzarri, Phys. Rev. Letters 21 (1968) 833. [12] K. Kang, Nuovo Cimento 56 (1968) 1147. [13] T. G. Trippe et al., Phys. Letters 28B (1968) 203.
This violates the Glashow-Weinberg bounds m K --< 525 MeV or :> 945 MeV obtained with F2 =F 2 anu~ F2/F 2 ~ z K ~ 2.2; see ref. [9]. :~ A recent experiment by Trippe et al. (ref. [13]) has indicated the existence of the kappa meson around 1100-1200 MeV. if this is confirmed, it would provide good support for the modified sum rule eq. (2).