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Tachyon-free dual model with a positive-intercept trajectory
Volume34B, number 6
TACHYON-FREE
PHYSICS
DUAL
MODEL
WITH
LETTERS
29 March 1971
A POSITIVE-INTERCEPT
TRAJECTORY*
A. N E V E U ** and J. H. SCHWARZ
Joseph Henry Laboratories, Princeton University, Princeton, New Jersey
08540, USA
Received 15 February 1971
A dual-reasonance model with two kinds of trajectories split by one-half unit is constructed, The lowest mass state on the leading trajectory has spin one, although the G-parity assignments are unsuitable for an identification with ~ 's and p~s.
One of the i m p o r t a n t u n s o l v e d p r o b l e m s in the c o n s t r u c t i o n of a r e a l i s t i c d u a l - r e s o n a n c e m o d e l f o r m e s o n s i s the i n t r o d u c t i o n of t a c h y o n - f r e e p o s i t i v e - i n t e r c e p t t r a j e c t o r i e s . In t h i s l e t t e r we r e p o r t s o m e p r o g r e s s in t h i s d i r e c t i o n by e x h i b i t i n g a m o d e l with two k i n d s of t r a j e c t o r i e s , a l ( S ) and a2(s), s u c h that a l ( S ) = a2(s) + 1/2 = s + 1 - m 2
(1)
and the f i r s t s t a t e on a l ( S ) i s a v e c t o r m e s o n at s = m 2. T h e s e two f e a t u r e s a r e j u s t what one w o u l d l i k e f o r a d u a l m o d e l of the n - p s y s t e m . U n f o r t u n a t e l y , the m o d e l a s s i g n s n e g a t i v e G p a r i t y to a 1 and p o s i t i v e G - p a r i t y to a 2 so that an i d e n t i f i c a t i o n with ~ ' s and p ' s c a n n o t be m a d e . C o n c e i v a b l y a s i m i l a r s c h e m e m i g h t h a v e the p h y s i c a l G - p a r i t y , although the l a c k of an u n d e r l y i n g q u a r k d e s c r i p t i o n of the s t a t e s m a k e s it doubtful that the u l t i m a t e m o d e l w i l l be c o n s t r u c t e d a l o n g the s a m e l i n e s . In a d d i t i o n to the h a r m o n i c o s c i l l a t o r a n n i h i l a t i o n o p e r a t o r s a ~ of the s t a n d a r d V e n e z i a n o m o d e l [1], we i n t r o d u c e a n t i c o m m u t i n g o p e r a t o r s bm~, w h e r e the i n d e x m r u n s o v e r h a l f i n t e g e r s f r o m 1/2 to +~o. T h e f o l l o w i n g a l g e b r a i s p o s t u l a t e d (our m e t r i c i s s p a c e l i k e ) :
[b~, any] : [b~, any] = {bm~, b~} : 0
(2a)
[a~, a~ ~f] = { b~n, b~ ~f} : gl~U6mn.
(2b)
~ /q~ = -
b ~ exp ( i m r )
77
:
f
dr exp(inr)H(r).
(3)
/~(r) :
(4)
-TT
w h e r e t h e s u m in (3) r u n s o v e r half i n t e g e r s , b~m =- bt~Im, a n d / ~ t ~ ( r ) = d H t ~ r / d r . T h e f o l l o w i n g a l g e b r a i s Readily v e r i f i e d :
b b = (n-m) Lb+n [ Lm,Ln] {Hg(r), HV('r') } = 27rg~xv5(T-z')
(5) (6)
[Lb,H"('r)] = exp (inr)(THg('r)- i/tlx('r)).(7) F r o m t h e s e f o r m u l a s one o b s e r v e s that L b and L b l a r e the i n f i n i t e s i m a l g e n e r a t o r s [2] of f r a c t i o n a l - l i n e a r t r a n s f o r m a t i o n s in the v a r i a b l e z = exp ( i t ) , as r e q u i r e d f o r the c o n s t r u c t i o n of Mt~bius-invariant N-point functions. In o r d e r to c o n s t r u c t N - p o i n t f u n c t i o n s in a way that m a k e s f a c t o r i z a t i o n and c y c l i c s y m m e t r y m a n i f e s t s i m u l t a n e o u s l y , we i n t r o d u c e a v e r t e x o p e r a t o r in the K o b a - N i e l s e n v a r i a b l e s [3]. F o r the e m i s s i o n of a v e c t o r m e s o n of m o m e n t u m k g and p o l a r i z a t i o n v e c t o r ct~ s a t i s f y i n g k2 = -m 2 andk.~ = 0weuse
(8)
V(k,z) : ~.H(z) VO(k,z)
w h e r e V0 i s the v e r t e x o p e r a t o r of the s t a n d a r d Veneziano model
Vo(k,z) = z -2D'p exp
Next we construct the operators H~Z('r) =
i
L b = -~
exp
(ik.x)
(9)
{~f2n~__l=k.atnz--~nl exp 1~
n=l ~ k'an
Z---n~nl
oo
* Research sponsored by the Air Force Office of Scientific Research under Contract AF 49 (638)-1545. ** NATO Fellow, on leave of absence from Laboratoire de Physique Theorique, Orsay, France.
p and x a r e m o m e n t u m and s o m e t i m e s r e p r e s e n t e d by delicate limiting procedure function for vector mesons
position operators " z e r o m o d e s " and a [2]. T h e N - p o i n t i s g i v e n by
517
Volume 34B, number 6
PHYSICS
N
N (i0)
G : (-1) r e : l / 2
where z~, I.,.~ r+ ~ z I and d~N(Z) is the usual M6biusinvariant volume element
d~ZN(Z) :
(11)
N-1 (O(Zi_Zi+l)~ . (z2-ZN)dZ3dz4... dZN_ 1 i~=2- ~i-~i+l-i w i t h Zl, z2, z N a r b i t r a r i l y c h o s e n as fixed p o i n t s . C y c l i c s y m m e t r y f o l l o w s f r o m the M S b i u s - i n v a r i a n c e of the i n t e g r a n d and the c o m m u t a t i o n p r o p e r t i e s of V w i t h i t s e l f * . F a c t o r i z a t i o n i s b e s t d e m o n s t r a t e d by r e c a s t i n g eq. (10) into the f o r m
A N = (0 I~1.51,/2 V(k2, 1)DV(k3, 1) . . . DV(kN_I, 1 ) ~ N ' b 1~/21 0)
(12)
w h e r e the p r o p a g a t o r D i s g i v e n by 1
D = f dx(1-x) -m2-1/2 x rn2+LO-3/2 0
(13)
and
LO =p2+ n=l ~
nan.a n + ~
rn =1/2
rnb~.brn.
(14)
It i s e a s y to s e e that A N v a n i s h e s u n l e s s N i s e v e n b e c a u s e the c a l c u l a t i o n i n v o l v e s p a i r i n g the b o p e r a t o r s . In p a r t i c u l a r , one finds f o r the four-point function
A4 =
f
1
29 March 1971
One n o t i c e s that s t a t e s with an e v e n n u m b e r of b e x c i t a t i o n s lie on a2(s) o r one of i t s d a u g h t e r s w h e r e a s s t a t e s with an odd n u m b e r of b e x c i t a t i ons lie on al(S) or one of i t s d a u g h t e r s . H e n c e one i n t r o d u c e s the G - p a r i t y o p e r a t o r
AN = fd~N(Z)~=~ 1 l z ~ l / 2 ' z i - z i + l 'l/2-rn2}
× (0 i i~I=l V(ki , z i ) [ 0 ) ,
LETTERS
(15)
(16)
which a n t i c o m m u t e s with the v e r t e x V. T h u s the a l s t a t e s h a v e odd G - p a r i t y and the a 2 s t a t e s h a v e e v e n G - p a r i t y . T h i s i s o p p o s i t e f r o m what i s r e q u i r e d for the p - v s y s t e m . The d e g e n e r a c y of the s p e c t r u m , which i s q u a l i t a t i v e l y r a t h e r s i m i l a r to the u s u a l c a s e , can a l s o be d e d u c e d f r o m the H a m i l t o n i a n . One finds, in p a r t i c u l a r , that the l e a d i n g R e g g e t r a j e c t o r y J = a l ( S ) is n o n d e g e n e r a t e . The n e x t one, J = a2(s) , h a s a s i n g l e s c a l a r p a r t i c l e at a2(s) = 0, but a s m a n y a s f o u r s t a t e s along the r e s t of the t r a j e c t o r y , t h r e e of t h e m h a v i n g the n a t u r a l p a r i t y . In the c a s e rn 2 = - 1 / 2 the o p e r a t o r s L b can be added to the Ln'S of V i r a s o r o [4] and u s e d to g e n e r a t e an i n f i n i t e set of W a r d i d e n t i t i e s . T h i s m a s s v a l u e c o r r e s p o n d s to a l ( 0 ) = 3/2 and a2(O) = 1, which i s not s u r p r i s i n g o n c e one not i c e s that the a m p l i t u d e f o r s c a t t e r i n g N of the l o w e s t - m a s s e v e n G - p a r i t y s c a l a r s i s j u s t the u s u a l V e n e z i a n o f o r m u l a . (The odd- G s t a t e s do modify the c o r r e s p o n d i n g loop f o r m u l a s , h o w ever. ) In conclusion, our model has all the difficulties of the simpler one. It does, however, succeed in incorporating additional "odd G" trajectories whose leading state is a vector meson without introducing too many new peoblems or drastically increasing the degeneracy of the spectrum. As our understanding of the algebraic requirements of duality improves one may hope that more realistic models for meson scattering can eventually be formulated.
r
dx x rn2-s-1 (l-x) rn2-t-1 ~ ( e l . e 2 ) ( e 3 . e 4 ) l ~ x
+
0 ( e 2 " e 3 ) ( e 1 " ¢ 4 ) ~/l~-x- (e l " e 3 ) ( e 2 " e 4 ) ~ Z ~
J.
T h e f i r s t p o l e of t h i s a m p l i t u d e i s a s c a l a r at s = r n 2 - 1 / 2 c o r r e s p o n d i n g , in the n o t a t i o n of eq. (1), to a2(s) = 0. T h e g e n e r a l s t r u c t u r e of the s p e c t r u m can be d e d u c e d f r o m the e i g e n s t a t e s and e i g e n v a l u e s of the H a m i l t o n i a n rn 2 - 1 / 2 + I43" * In the anticommutation of H's the 6 functions that arise are evaluated at the edges of the integration region and make no contribution.
518
References [1] s. Fubini, D.Gordon and G. Veneziano, Phys. Letters 29B (1969) 679. [2] S. Fubini and G. Veneziano, Nuovo Cimento 67A (1970) 29; L. Clavel[i and P. Ramond, Phys. Rev. D2 (1970) 973; K. Bardakci and M. B. Halpern, "New Dual Quark Models" Berkeley preprint (1970). [3] Z.Koba andH.B.Nielsen, Nucl. Phys. BI0 (1969) 633. [4] M.A.Virasoro, Phys. Rev. D1 (1970) 2933.
TACHYON-FREE
PHYSICS
DUAL
MODEL
WITH
LETTERS
29 March 1971
A POSITIVE-INTERCEPT
TRAJECTORY*
A. N E V E U ** and J. H. SCHWARZ
Joseph Henry Laboratories, Princeton University, Princeton, New Jersey
08540, USA
Received 15 February 1971
A dual-reasonance model with two kinds of trajectories split by one-half unit is constructed, The lowest mass state on the leading trajectory has spin one, although the G-parity assignments are unsuitable for an identification with ~ 's and p~s.
One of the i m p o r t a n t u n s o l v e d p r o b l e m s in the c o n s t r u c t i o n of a r e a l i s t i c d u a l - r e s o n a n c e m o d e l f o r m e s o n s i s the i n t r o d u c t i o n of t a c h y o n - f r e e p o s i t i v e - i n t e r c e p t t r a j e c t o r i e s . In t h i s l e t t e r we r e p o r t s o m e p r o g r e s s in t h i s d i r e c t i o n by e x h i b i t i n g a m o d e l with two k i n d s of t r a j e c t o r i e s , a l ( S ) and a2(s), s u c h that a l ( S ) = a2(s) + 1/2 = s + 1 - m 2
(1)
and the f i r s t s t a t e on a l ( S ) i s a v e c t o r m e s o n at s = m 2. T h e s e two f e a t u r e s a r e j u s t what one w o u l d l i k e f o r a d u a l m o d e l of the n - p s y s t e m . U n f o r t u n a t e l y , the m o d e l a s s i g n s n e g a t i v e G p a r i t y to a 1 and p o s i t i v e G - p a r i t y to a 2 so that an i d e n t i f i c a t i o n with ~ ' s and p ' s c a n n o t be m a d e . C o n c e i v a b l y a s i m i l a r s c h e m e m i g h t h a v e the p h y s i c a l G - p a r i t y , although the l a c k of an u n d e r l y i n g q u a r k d e s c r i p t i o n of the s t a t e s m a k e s it doubtful that the u l t i m a t e m o d e l w i l l be c o n s t r u c t e d a l o n g the s a m e l i n e s . In a d d i t i o n to the h a r m o n i c o s c i l l a t o r a n n i h i l a t i o n o p e r a t o r s a ~ of the s t a n d a r d V e n e z i a n o m o d e l [1], we i n t r o d u c e a n t i c o m m u t i n g o p e r a t o r s bm~, w h e r e the i n d e x m r u n s o v e r h a l f i n t e g e r s f r o m 1/2 to +~o. T h e f o l l o w i n g a l g e b r a i s p o s t u l a t e d (our m e t r i c i s s p a c e l i k e ) :
[b~, any] : [b~, any] = {bm~, b~} : 0
(2a)
[a~, a~ ~f] = { b~n, b~ ~f} : gl~U6mn.
(2b)
~ /q~ = -
b ~ exp ( i m r )
77
:
f
dr exp(inr)H(r).
(3)
/~(r) :
(4)
-TT
w h e r e t h e s u m in (3) r u n s o v e r half i n t e g e r s , b~m =- bt~Im, a n d / ~ t ~ ( r ) = d H t ~ r / d r . T h e f o l l o w i n g a l g e b r a i s Readily v e r i f i e d :
b b = (n-m) Lb+n [ Lm,Ln] {Hg(r), HV('r') } = 27rg~xv5(T-z')
(5) (6)
[Lb,H"('r)] = exp (inr)(THg('r)- i/tlx('r)).(7) F r o m t h e s e f o r m u l a s one o b s e r v e s that L b and L b l a r e the i n f i n i t e s i m a l g e n e r a t o r s [2] of f r a c t i o n a l - l i n e a r t r a n s f o r m a t i o n s in the v a r i a b l e z = exp ( i t ) , as r e q u i r e d f o r the c o n s t r u c t i o n of Mt~bius-invariant N-point functions. In o r d e r to c o n s t r u c t N - p o i n t f u n c t i o n s in a way that m a k e s f a c t o r i z a t i o n and c y c l i c s y m m e t r y m a n i f e s t s i m u l t a n e o u s l y , we i n t r o d u c e a v e r t e x o p e r a t o r in the K o b a - N i e l s e n v a r i a b l e s [3]. F o r the e m i s s i o n of a v e c t o r m e s o n of m o m e n t u m k g and p o l a r i z a t i o n v e c t o r ct~ s a t i s f y i n g k2 = -m 2 andk.~ = 0weuse
(8)
V(k,z) : ~.H(z) VO(k,z)
w h e r e V0 i s the v e r t e x o p e r a t o r of the s t a n d a r d Veneziano model
Vo(k,z) = z -2D'p exp
Next we construct the operators H~Z('r) =
i
L b = -~
exp
(ik.x)
(9)
{~f2n~__l=k.atnz--~nl exp 1~
n=l ~ k'an
Z---n~nl
oo
* Research sponsored by the Air Force Office of Scientific Research under Contract AF 49 (638)-1545. ** NATO Fellow, on leave of absence from Laboratoire de Physique Theorique, Orsay, France.
p and x a r e m o m e n t u m and s o m e t i m e s r e p r e s e n t e d by delicate limiting procedure function for vector mesons
position operators " z e r o m o d e s " and a [2]. T h e N - p o i n t i s g i v e n by
517
Volume 34B, number 6
PHYSICS
N
N (i0)
G : (-1) r e : l / 2
where z~, I.,.~ r+ ~ z I and d~N(Z) is the usual M6biusinvariant volume element
d~ZN(Z) :
(11)
N-1 (O(Zi_Zi+l)~ . (z2-ZN)dZ3dz4... dZN_ 1 i~=2- ~i-~i+l-i w i t h Zl, z2, z N a r b i t r a r i l y c h o s e n as fixed p o i n t s . C y c l i c s y m m e t r y f o l l o w s f r o m the M S b i u s - i n v a r i a n c e of the i n t e g r a n d and the c o m m u t a t i o n p r o p e r t i e s of V w i t h i t s e l f * . F a c t o r i z a t i o n i s b e s t d e m o n s t r a t e d by r e c a s t i n g eq. (10) into the f o r m
A N = (0 I~1.51,/2 V(k2, 1)DV(k3, 1) . . . DV(kN_I, 1 ) ~ N ' b 1~/21 0)
(12)
w h e r e the p r o p a g a t o r D i s g i v e n by 1
D = f dx(1-x) -m2-1/2 x rn2+LO-3/2 0
(13)
and
LO =p2+ n=l ~
nan.a n + ~
rn =1/2
rnb~.brn.
(14)
It i s e a s y to s e e that A N v a n i s h e s u n l e s s N i s e v e n b e c a u s e the c a l c u l a t i o n i n v o l v e s p a i r i n g the b o p e r a t o r s . In p a r t i c u l a r , one finds f o r the four-point function
A4 =
f
1
29 March 1971
One n o t i c e s that s t a t e s with an e v e n n u m b e r of b e x c i t a t i o n s lie on a2(s) o r one of i t s d a u g h t e r s w h e r e a s s t a t e s with an odd n u m b e r of b e x c i t a t i ons lie on al(S) or one of i t s d a u g h t e r s . H e n c e one i n t r o d u c e s the G - p a r i t y o p e r a t o r
AN = fd~N(Z)~=~ 1 l z ~ l / 2 ' z i - z i + l 'l/2-rn2}
× (0 i i~I=l V(ki , z i ) [ 0 ) ,
LETTERS
(15)
(16)
which a n t i c o m m u t e s with the v e r t e x V. T h u s the a l s t a t e s h a v e odd G - p a r i t y and the a 2 s t a t e s h a v e e v e n G - p a r i t y . T h i s i s o p p o s i t e f r o m what i s r e q u i r e d for the p - v s y s t e m . The d e g e n e r a c y of the s p e c t r u m , which i s q u a l i t a t i v e l y r a t h e r s i m i l a r to the u s u a l c a s e , can a l s o be d e d u c e d f r o m the H a m i l t o n i a n . One finds, in p a r t i c u l a r , that the l e a d i n g R e g g e t r a j e c t o r y J = a l ( S ) is n o n d e g e n e r a t e . The n e x t one, J = a2(s) , h a s a s i n g l e s c a l a r p a r t i c l e at a2(s) = 0, but a s m a n y a s f o u r s t a t e s along the r e s t of the t r a j e c t o r y , t h r e e of t h e m h a v i n g the n a t u r a l p a r i t y . In the c a s e rn 2 = - 1 / 2 the o p e r a t o r s L b can be added to the Ln'S of V i r a s o r o [4] and u s e d to g e n e r a t e an i n f i n i t e set of W a r d i d e n t i t i e s . T h i s m a s s v a l u e c o r r e s p o n d s to a l ( 0 ) = 3/2 and a2(O) = 1, which i s not s u r p r i s i n g o n c e one not i c e s that the a m p l i t u d e f o r s c a t t e r i n g N of the l o w e s t - m a s s e v e n G - p a r i t y s c a l a r s i s j u s t the u s u a l V e n e z i a n o f o r m u l a . (The odd- G s t a t e s do modify the c o r r e s p o n d i n g loop f o r m u l a s , h o w ever. ) In conclusion, our model has all the difficulties of the simpler one. It does, however, succeed in incorporating additional "odd G" trajectories whose leading state is a vector meson without introducing too many new peoblems or drastically increasing the degeneracy of the spectrum. As our understanding of the algebraic requirements of duality improves one may hope that more realistic models for meson scattering can eventually be formulated.
r
dx x rn2-s-1 (l-x) rn2-t-1 ~ ( e l . e 2 ) ( e 3 . e 4 ) l ~ x
+
0 ( e 2 " e 3 ) ( e 1 " ¢ 4 ) ~/l~-x- (e l " e 3 ) ( e 2 " e 4 ) ~ Z ~
J.
T h e f i r s t p o l e of t h i s a m p l i t u d e i s a s c a l a r at s = r n 2 - 1 / 2 c o r r e s p o n d i n g , in the n o t a t i o n of eq. (1), to a2(s) = 0. T h e g e n e r a l s t r u c t u r e of the s p e c t r u m can be d e d u c e d f r o m the e i g e n s t a t e s and e i g e n v a l u e s of the H a m i l t o n i a n rn 2 - 1 / 2 + I43" * In the anticommutation of H's the 6 functions that arise are evaluated at the edges of the integration region and make no contribution.
518
References [1] s. Fubini, D.Gordon and G. Veneziano, Phys. Letters 29B (1969) 679. [2] S. Fubini and G. Veneziano, Nuovo Cimento 67A (1970) 29; L. Clavel[i and P. Ramond, Phys. Rev. D2 (1970) 973; K. Bardakci and M. B. Halpern, "New Dual Quark Models" Berkeley preprint (1970). [3] Z.Koba andH.B.Nielsen, Nucl. Phys. BI0 (1969) 633. [4] M.A.Virasoro, Phys. Rev. D1 (1970) 2933.