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On the existence of axial vector mesons
Volume 21, number4
PHYSICS LETTERS
ON THE
EXISTENCE
OF
AXIAL
1June 1966
VECTOR
MESONS
*
B. R E N N E R
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England Received 23 April 1966
Certain current algebra commutators can only be saturated if the existence of axial vector mesons is assumed. As possible candidates the A1(1072 ) and Kc(1215 ) resonances are examined.
The g r e a t s u c c e s s e s of c u r r e n t a l g e b r a s [1] in r e l a t i n g o b s e r v e d quantities** e n c o u r a g e us to u s e t h e m f o r p r e d i c t i o n s . The p a t t e r n i s e a s i l y d e s c r i b e d . We look for a c o m m u t a t o r which cannot be s a t u r a t e d by the known s t a b l e and r e s o n a n t p r a t i c l e s t a t e s . Taken between a p a r t i c l e and the vacuum, s p i n s and p a r i t i e s of the contributing i n t e r m e d i a t e s t a t e s a r e l i m i t e d and f u r t h e r s e l e c t e d by the a p p l i c a t i o n of d i s p e r s i o n t h e o r e t i c techniques [2]. We c o n s i d e r c o m m u t a t o r s between axial v e c t o r c h a r g e s and c u r r e n t s p r o d u c i n g a v e c t o r current. A
A
[ Qi ; Jj (x)l.tI = ifijk iV(x)
(1)
between a v e c t o r m e s o n and the vacuum ( V I . . . 10). The r i g h t hand s i d e i s p r o p o r t i o n a l to the p o l a r i zation v e c t o r ~V, the left hand side i s r e p r e s e n t e d by an i n t e g r a l lim q~0
F/~(pV, q) = (2)
fd4xO(-Xo)(Vl[n~/(x),jA(0)] [0)
lira exp iqx. q--*0 with-the d i v e r g e n c e D~/(x) of the a x i a l c u r r e n t . A c c o r d i n g to the g e n e r a l method of Fubini et al. [2j, we i n t r o d u c e the v a r i a b l e v = (pV. q)/m V, fix q,2 = 0 and w r i t e Fix a c c o r d i n g to k i n e m a t i c invariants. * The research reported in this document has been sponsored in part by the Air Force Office of Scientific research, OAR under Grant AF EOAR 65-36 with the European Office of Aerospace Research, United States Air Force. ** We apologize for not enumerating here interesting papers and preprints in this field, now already over one hundred.
F~ (pV,q)
= A(v) eV + B(v)(eV.
q)q~ + C(P)(eV. q)pV . (3)
F o r the i n v a r i a n t m a t r i x e l e m e n t s we will p r o p o s e u n s u b t r a c t e d d i s p e r s i o n r e l a t i o n s in v and the m o s t n a t u r a l choice i s to have t h e m f o r A(v), B(v) and C(v). The a b s o r p t i v e p a r t s a(v'), ~(v') and ~(v') a r e the c o e f f i c i e n t s of eV, (eV. q,)q~ and (EV. q,)pV to be p i c k e d f r o m [2] ½i (Z~)4 n~ (V] ¢ ( 0 ) I n)(n [jA(0)• [0 )64(pV+ q,_ with (q,)2 = 0 . We take the c o m m u t a t o r
pn) (4)
A 'gr oA [Q[3 ÷] = jV+ between
(p+] and i0). Only s t a t e s o f J P = (0-) and (1 +) can c o n t r i b u t e to (4). The e s s e n t i a l point i s now that 0-(Tr) s t a t e s c o n t r i b u t e with unique m a t r i x e l e ments (vIDAI~)_ V.q, ;(~ijAl0)_p; = p ~V+ q ~, (5) only to B(u) and C(v). Only (1+) states can contribute to A(v I. Introducing an invariant m a s s distribution p~(M) w e find ¢
GA+p +7ro(M)
(o) where w e have expressedC¢ The singularity of (M 2 -M~) -1 is not taken seriously: we believe that it merely indicates a limit to the single particle approximation for p+ and should be avoided by replacing M p by M p + iI'p . ~'¢ The other notation is:
(.+lo~+)~ J0>= :.1.~; v " :~o =:~/./2; ~o+ bx+),10>=
453
Volume 21, number 4
PHYSICS LETTERS
1June 1966
the A 1 r e s o n a n c e w o u l d be the p o s t u l a t e d 1 + m e s o n - c) the A 1 r e s o n a n c e i s a k i n e m a t i c e f f e c t and the 1 + m e s o n i s s t i l l u n d i s c o v e r e d and h a s a l a r g e width. In any c a s e t h e s e a r g u m e n t s s h o u l d be w e i g h t e d t o g e t h e r w i t h e s t i m a t e s of F u r l a n et al. [5], who s u g g e s t (1 +) m e s o n s a r o u n d 1100 M e V f r o m c u r r e n t a l g e b r a p r e d i c t i o n s f o r a x i a l v e c t o r c h a r g e r a d i i of b a r y o n s . A s i m i l a r r e a s o n i n g c a n be a p p l i e d to the c o m A ~ 1.V mutator : ~ j K + b e t w e e n (K*+I and 10 ) .
[rn 2 - (q,)21K~0(q')2 and a p p l i e d P C A C in s e t t i n g K p A ( 0 ) ~ 1. So if t h e r e a r e no a x i a l v e c t o r m e s o n s with I= 1, Y= 0 we n e e d s u b t r a c t i o n c o n s t a n t s in the dispersion relations or current algebras are disproved * Eq. (6) d o e s not s p e c i f y w h e t h e r the (1 +) m e son c o n t r i b u t i o n s c o m e f r o m r e s o n a n c e s o r b a c k g r o u n d . But b e c a u s e we want to e x a m i n e the A1(1072) s t a t e [4] a s a p o s s i b l e c a n d i d a t e , l e t us a s s u m e that it d o m i n a t e s t h e s u m r u l e . We know MA1 and GAt]~tp~no f r o m the width, but not f A 1 C u r r e n t a l g e b r a s c a n d e t e r m i n e it; c o n s i d e r
[QI3,]K+]
We find ' :f,<.
=
dM
riO)
(M2 - M2,)
I
the c o m m u t a t o r
A ,jV] [Q[3
:j[A
c a l l i n g f o r a (1 +) m e s o n with 1 = ½, Y = i-½. We e x a m i n e the Kc(1215 ) r e s o n a n c e a s a b o v e . A p -
b e t w e e n the A I
plying
s t a t e and the v a c u u m . I n t e r m e d i a t e s t a t e s a r e only the (1-) m e s o n s due to c u r r e n t c o n s e r v a t i o n . We get in a n a l o g y to (6)
1 rf SvA (M) fA1 = - ~
3 (M 2 _ M 2 1)
fv (M) pV(M) dM
find f~¢
T h e a u t h o r w i s h e s to t h a n k D r . J. C. P o l k i n g h o r n e f o r i n t e r e s t i n g d i s c u s s i o n s and u s e f u l c o m m e n t s on t h i s w o r k . He i s f u r t h e r g r a t e f u l to the f o u n d a t i o n C u s a n u s w o r k and G o n v i l l e and C a i u s C o l l e g e , C a m b r i d g e , f o r g e n e r o u s s u p p o r t of h i s studies.
(9)
F r o m eq. (9) we e s t i m a t e t h e e f f e c t i v e c o u p l i n g c o n s t a n t GAip+~O = 6.1 GeV. F r o m the width of t h e A~ r e s o n a n c e F A = 125 M e V we find GA~p+~o = 2.56 G e v . l w e s h o u l d e x p e c t to get a p r e d i c t i o n l a r g e r t h a n o b s e r v e d , b e c a u s e the w e i g h t i n g f a c t o r (M 2 - M2) - 1 e m p h a s i z e s the low e n e r g y t a i l and the b a c k g r o u n d and l e a d s to M a y < MA1 , but i t s e f f e c t s h o u l d not be s o l a r g e . T h r e e p o s s i b i l i t i e s r e m a i n open: a) the d i s c r e p a n c y is due to the s i n g l e p a r t i c l e a p p r o x i m a t i o n ; b) t h e r e a r e s t r o n g b a c k g r o u n d c o n t r i b u t i o n s , p a r t i c u l a r l y of l o w e r e n e r g i e s ; - in t h e s e c a s e s
References 1. M. Gell-Mann, Phys.Rev. 125 (1962) 1067. 2. S. Fubini, G.Furlan and C.Rossetti, Nuovo Cimento 40 (1965) 1171. 3. Riazuddin and Fayyazudin: University of Pennsylvania preprint. 4. A.H. Rosenfeld et al., University of California p r e print, UCRL-8030 (1965); Xuong, Conf. on Nuclear structure and elementary particles, Oxford 1966. 5. G.Furlan, R.Jengo, E.Remiddi, University of Trieste preprint; Physics Letters 21 (1966) 679.
• In this connection we r e m a r k that Riazuddin and Fayyazudin [3] deduced very interesting results from this commutator, by not setting in equation (2), but regardin~ the ~ntegral as dominated by the ~'-meson pole at q~ = rnTr .
~' We do not use SU(3) symmetry or conservation of the strangeness changing current; the scalar states (K 725) do not contribute to A(V). ~+ Not considering the competing decay Kc ~ p + K for phase space reasons.
q2= 0
454
= -fT¢* + and in a n a l o g y to (9) we get
3.75 GeV, to be c o m p a r e d w i t h the v a l u e 1.7 G e V f r o m t h e e x p e r i m e n t a l d e c a y width ¢+. The s a m e c o m m e n t s a p p l y h e r e a s w i t h the A 1 r e s o n a n c e .
(8)
•
(Kcl and 1 0 ) w e
( M 2 v - M2K,)~/2 = f~" GK,~ K+ ~o , giving GK *+K c+ 7r° =
and in the p - d o m i n a n c e a p p r o x i m a t i o n f p = f A l " T h i s r e s u l t i s r e m a r k a b l e , b e c a u s e we did not m a k e any a s s i g n m e n t s to m u l t i p l e t s of a p p r o x i m a t e s y m m e t r y g r o u p s . We r e w r i t e (6) ~/2 : - Gp~A{yof~ " (M2av - M2"-lp)
[QIA33,jV+]=½jA+ b e t w e e n
*
*
*
*
PHYSICS LETTERS
ON THE
EXISTENCE
OF
AXIAL
1June 1966
VECTOR
MESONS
*
B. R E N N E R
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England Received 23 April 1966
Certain current algebra commutators can only be saturated if the existence of axial vector mesons is assumed. As possible candidates the A1(1072 ) and Kc(1215 ) resonances are examined.
The g r e a t s u c c e s s e s of c u r r e n t a l g e b r a s [1] in r e l a t i n g o b s e r v e d quantities** e n c o u r a g e us to u s e t h e m f o r p r e d i c t i o n s . The p a t t e r n i s e a s i l y d e s c r i b e d . We look for a c o m m u t a t o r which cannot be s a t u r a t e d by the known s t a b l e and r e s o n a n t p r a t i c l e s t a t e s . Taken between a p a r t i c l e and the vacuum, s p i n s and p a r i t i e s of the contributing i n t e r m e d i a t e s t a t e s a r e l i m i t e d and f u r t h e r s e l e c t e d by the a p p l i c a t i o n of d i s p e r s i o n t h e o r e t i c techniques [2]. We c o n s i d e r c o m m u t a t o r s between axial v e c t o r c h a r g e s and c u r r e n t s p r o d u c i n g a v e c t o r current. A
A
[ Qi ; Jj (x)l.tI = ifijk iV(x)
(1)
between a v e c t o r m e s o n and the vacuum ( V I . . . 10). The r i g h t hand s i d e i s p r o p o r t i o n a l to the p o l a r i zation v e c t o r ~V, the left hand side i s r e p r e s e n t e d by an i n t e g r a l lim q~0
F/~(pV, q) = (2)
fd4xO(-Xo)(Vl[n~/(x),jA(0)] [0)
lira exp iqx. q--*0 with-the d i v e r g e n c e D~/(x) of the a x i a l c u r r e n t . A c c o r d i n g to the g e n e r a l method of Fubini et al. [2j, we i n t r o d u c e the v a r i a b l e v = (pV. q)/m V, fix q,2 = 0 and w r i t e Fix a c c o r d i n g to k i n e m a t i c invariants. * The research reported in this document has been sponsored in part by the Air Force Office of Scientific research, OAR under Grant AF EOAR 65-36 with the European Office of Aerospace Research, United States Air Force. ** We apologize for not enumerating here interesting papers and preprints in this field, now already over one hundred.
F~ (pV,q)
= A(v) eV + B(v)(eV.
q)q~ + C(P)(eV. q)pV . (3)
F o r the i n v a r i a n t m a t r i x e l e m e n t s we will p r o p o s e u n s u b t r a c t e d d i s p e r s i o n r e l a t i o n s in v and the m o s t n a t u r a l choice i s to have t h e m f o r A(v), B(v) and C(v). The a b s o r p t i v e p a r t s a(v'), ~(v') and ~(v') a r e the c o e f f i c i e n t s of eV, (eV. q,)q~ and (EV. q,)pV to be p i c k e d f r o m [2] ½i (Z~)4 n~ (V] ¢ ( 0 ) I n)(n [jA(0)• [0 )64(pV+ q,_ with (q,)2 = 0 . We take the c o m m u t a t o r
pn) (4)
A 'gr oA [Q[3 ÷] = jV+ between
(p+] and i0). Only s t a t e s o f J P = (0-) and (1 +) can c o n t r i b u t e to (4). The e s s e n t i a l point i s now that 0-(Tr) s t a t e s c o n t r i b u t e with unique m a t r i x e l e ments (vIDAI~)_ V.q, ;(~ijAl0)_p; = p ~V+ q ~, (5) only to B(u) and C(v). Only (1+) states can contribute to A(v I. Introducing an invariant m a s s distribution p~(M) w e find ¢
GA+p +7ro(M)
(o) where w e have expressedC¢ The singularity of (M 2 -M~) -1 is not taken seriously: we believe that it merely indicates a limit to the single particle approximation for p+ and should be avoided by replacing M p by M p + iI'p . ~'¢ The other notation is:
(.+lo~+)~ J0>= :.1.~; v " :~o =:~/./2; ~o+ bx+),10>=
453
Volume 21, number 4
PHYSICS LETTERS
1June 1966
the A 1 r e s o n a n c e w o u l d be the p o s t u l a t e d 1 + m e s o n - c) the A 1 r e s o n a n c e i s a k i n e m a t i c e f f e c t and the 1 + m e s o n i s s t i l l u n d i s c o v e r e d and h a s a l a r g e width. In any c a s e t h e s e a r g u m e n t s s h o u l d be w e i g h t e d t o g e t h e r w i t h e s t i m a t e s of F u r l a n et al. [5], who s u g g e s t (1 +) m e s o n s a r o u n d 1100 M e V f r o m c u r r e n t a l g e b r a p r e d i c t i o n s f o r a x i a l v e c t o r c h a r g e r a d i i of b a r y o n s . A s i m i l a r r e a s o n i n g c a n be a p p l i e d to the c o m A ~ 1.V mutator : ~ j K + b e t w e e n (K*+I and 10 ) .
[rn 2 - (q,)21K~0(q')2 and a p p l i e d P C A C in s e t t i n g K p A ( 0 ) ~ 1. So if t h e r e a r e no a x i a l v e c t o r m e s o n s with I= 1, Y= 0 we n e e d s u b t r a c t i o n c o n s t a n t s in the dispersion relations or current algebras are disproved * Eq. (6) d o e s not s p e c i f y w h e t h e r the (1 +) m e son c o n t r i b u t i o n s c o m e f r o m r e s o n a n c e s o r b a c k g r o u n d . But b e c a u s e we want to e x a m i n e the A1(1072) s t a t e [4] a s a p o s s i b l e c a n d i d a t e , l e t us a s s u m e that it d o m i n a t e s t h e s u m r u l e . We know MA1 and GAt]~tp~no f r o m the width, but not f A 1 C u r r e n t a l g e b r a s c a n d e t e r m i n e it; c o n s i d e r
[QI3,]K+]
We find ' :f,<.
=
dM
riO)
(M2 - M2,)
I
the c o m m u t a t o r
A ,jV] [Q[3
:j[A
c a l l i n g f o r a (1 +) m e s o n with 1 = ½, Y = i-½. We e x a m i n e the Kc(1215 ) r e s o n a n c e a s a b o v e . A p -
b e t w e e n the A I
plying
s t a t e and the v a c u u m . I n t e r m e d i a t e s t a t e s a r e only the (1-) m e s o n s due to c u r r e n t c o n s e r v a t i o n . We get in a n a l o g y to (6)
1 rf SvA (M) fA1 = - ~
3 (M 2 _ M 2 1)
fv (M) pV(M) dM
find f~¢
T h e a u t h o r w i s h e s to t h a n k D r . J. C. P o l k i n g h o r n e f o r i n t e r e s t i n g d i s c u s s i o n s and u s e f u l c o m m e n t s on t h i s w o r k . He i s f u r t h e r g r a t e f u l to the f o u n d a t i o n C u s a n u s w o r k and G o n v i l l e and C a i u s C o l l e g e , C a m b r i d g e , f o r g e n e r o u s s u p p o r t of h i s studies.
(9)
F r o m eq. (9) we e s t i m a t e t h e e f f e c t i v e c o u p l i n g c o n s t a n t GAip+~O = 6.1 GeV. F r o m the width of t h e A~ r e s o n a n c e F A = 125 M e V we find GA~p+~o = 2.56 G e v . l w e s h o u l d e x p e c t to get a p r e d i c t i o n l a r g e r t h a n o b s e r v e d , b e c a u s e the w e i g h t i n g f a c t o r (M 2 - M2) - 1 e m p h a s i z e s the low e n e r g y t a i l and the b a c k g r o u n d and l e a d s to M a y < MA1 , but i t s e f f e c t s h o u l d not be s o l a r g e . T h r e e p o s s i b i l i t i e s r e m a i n open: a) the d i s c r e p a n c y is due to the s i n g l e p a r t i c l e a p p r o x i m a t i o n ; b) t h e r e a r e s t r o n g b a c k g r o u n d c o n t r i b u t i o n s , p a r t i c u l a r l y of l o w e r e n e r g i e s ; - in t h e s e c a s e s
References 1. M. Gell-Mann, Phys.Rev. 125 (1962) 1067. 2. S. Fubini, G.Furlan and C.Rossetti, Nuovo Cimento 40 (1965) 1171. 3. Riazuddin and Fayyazudin: University of Pennsylvania preprint. 4. A.H. Rosenfeld et al., University of California p r e print, UCRL-8030 (1965); Xuong, Conf. on Nuclear structure and elementary particles, Oxford 1966. 5. G.Furlan, R.Jengo, E.Remiddi, University of Trieste preprint; Physics Letters 21 (1966) 679.
• In this connection we r e m a r k that Riazuddin and Fayyazudin [3] deduced very interesting results from this commutator, by not setting in equation (2), but regardin~ the ~ntegral as dominated by the ~'-meson pole at q~ = rnTr .
~' We do not use SU(3) symmetry or conservation of the strangeness changing current; the scalar states (K 725) do not contribute to A(V). ~+ Not considering the competing decay Kc ~ p + K for phase space reasons.
q2= 0
454
= -fT¢* + and in a n a l o g y to (9) we get
3.75 GeV, to be c o m p a r e d w i t h the v a l u e 1.7 G e V f r o m t h e e x p e r i m e n t a l d e c a y width ¢+. The s a m e c o m m e n t s a p p l y h e r e a s w i t h the A 1 r e s o n a n c e .
(8)
•
(Kcl and 1 0 ) w e
( M 2 v - M2K,)~/2 = f~" GK,~ K+ ~o , giving GK *+K c+ 7r° =
and in the p - d o m i n a n c e a p p r o x i m a t i o n f p = f A l " T h i s r e s u l t i s r e m a r k a b l e , b e c a u s e we did not m a k e any a s s i g n m e n t s to m u l t i p l e t s of a p p r o x i m a t e s y m m e t r y g r o u p s . We r e w r i t e (6) ~/2 : - Gp~A{yof~ " (M2av - M2"-lp)
[QIA33,jV+]=½jA+ b e t w e e n
*
*
*
*