Factorizable dual model of pions

~

N u c l e a r P h y s i c s B31 (1971) 86-112.

North-Holland Publishing Company

FACTORIZABLE DUAL MODEL OF PIONS* A. N E V E U * *

a n d J . H. S C H W A R Z

Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540 R e c e i v e d 25 M a r c h 1971 A b s t r a c t : A new d u a l - r e s o n a n c e m o d e l of m e s o n s i s c o n s t r u c t e d t h r o u g h the use of c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s h a v i n g s i m p l e a n t i c o m m u t a t i o n p r o p e r t i e s a s well a s h a r m o n i c - o s c i l l a t o r type o p e r a t o r s of the c o n v e n t i o n a l m o d e l . T h i s m o d e l h a s the f o l l o w i n g v i r t u e s not s h a r e d by the c o n v e n t i o n a l one: (i) T h e l e a d i n g t r a j e c t o r y (p_fo) d o e s not m a k e a p a r t i c l e when it p a s s e s t h r o u g h z e r o . (ii) A ~ t r a j e c t o r y l i e s o n e - h a g unit b e l o w the p - t r a j e c t o r y . (iii) T r a j e c t o r i e s with a b n o r m a l - p a r i t y c o u p l i n g s to p i o n s , s u c h a s a~-A2 and 97', a l s o o c c u r . (iv) T h e ~, p, fo, ~ A 2 and 7 ' a r e a l l f o r c e d to h a v e the p r o u e r G - p a r i t y and i s o s p i n . (v) T h e a m p l i t u d e s f o r ~ - * ~ a n d 7r~ -~ ~ a r e p r e c i s e l y the o n e s t h a t h a v e b e e n s u g g e s t e d p r e v i o u s l y . (vi) T h e m o d e l c o n t a i n s n e i t h e r e m b a r r a s s i n g u n o b s e r v e d l o w - m a s s s t a t e s n o r a n e x c e s s i v e n u m b e r of h i g h - m a s s s t a t e s . (vii) ALL the p h y s i c a l s t a t e s t h a t we h a v e c h e c k e d h a v e p o s i t i v e n o r m s . To b e f a i r , we should a l s o List s o m e s h o r t c o m i n g s of o u r m o d e l : (a) T r a j e c t o r i e s with n o r m a l - p a r i t y c o u p l i n g s o c c u r o n e - h a l f unit too high. T h u s , f o r e x a m ple, ~D(0) = I and ~ 0 ) = ½, so t h a t the ~ ' i s a t a c h y o n and the p i s m a s s L e s s . On the oilier hand, the a b n o r m a l - p a r i t y t r a j e c t o r i e s a r e n i c e l y l o c a t e d . F o r e x a m p l e , ~ . ~ 0 ) = ½ and ~ , ( 0 ) = - 1 . (b) A s t r a i g h t f o r w a r d g e n e r a l i z a t i o n to SU(3) is u n s a t i s f a c t o r y b e c a u s e of the n o n - d e g e n e r a c y of the 0 and co. T h e s e f e a t u r e s of the m o d e l c l e a r l y i n d i c a t e t h a t i t is not a c c u r a t e l y d e s c r i b i n g the r e a ! world. P e r h a p s i t s m o s t i m p o r t a n t p r o p e r t y i s t h a t i t c o n t a i n s a gauge a l g e b r a l a r g e r t h a n the V i r a s o r o a l g e b r a of the c o n v e n t i o n a l m o d e l . We b e l i e v e t h a t a n u n d e r s t a n d i n g of t h i s a l g e b r a m a y p r o v e to b e v e r y i m p o r t a n t in the c o n s t r u c t i o n of m o r e r e a l lstic models.

I.

INTRODUCTION

I n t h e n e a r l y t h r e e y e a r s s i n c e V e n e z i a n o ' s c l a s s i c p a p e r [1] a g r e a t d e a l of e f f o r t h a s b e e n e x p e n d e d o n a t t e m p t s t o c o n s t r u c t d u a l - r e s o n a n c e models that either bear a resemblance to reality or else could serve as t h e b a s i s f o r a f u l l - f l e d g e d u n i t a r y ( a l b e i t u n r e a l i s t i c ) t h e o r y of h a d r o n s . C o n s i d e r a b l e p r o g r e s s t o w a r d t h e s e c o n d g o a l h a s g r o w n o u t of t h e s t u d y of

* R e s e a r c h s p o n s o r e d by US A t o m i c E n e r g y C o m m i s s i o n u n d e r c o n t r a c t AT (30-1)-4159. ** NATO F e l l o w , on l e a v e of a b s e n c e f r o m L a b o r a t o i r e de P h y s i q u e T h 6 o r i q u e , O r s a y , F r a n c e . W o r k p r e s e n t e d in p a r t i a l f u f f i I I m e n t of the r e q u i r e m e n t s f o r the d e g r e e of D o e t o r a t d e s S c i e n c e s P h y s i q u e s , U n i v e r s i t ~ P a r i s Sud.

A.Neveu, J.H. Schwarz, Factorizable dual model of pions

87

the N-point generalizations [2] of Veneziano's beta function. First, the one-loop graphs were constructed [3] and renormalized [4]. More recently, the general N-loop graphs have been constructed [5] and (in the planar case) renormalized [6]. Scherk has furthermore shown [7] that the renormalization is essentially unique when the intercept of the leading Regge trajectory s o = I, whereas in all other cases there is infinite ambiguity, i.e., the theory is non-renormalizable. The special role of s o = 1 was pointed out earlier by Virasoro [8] in connection with the problem of negative-norm states (ghosts). He showed that in this case there is a large gauge algebra that probably allows for the cancellation of all ghosts. We shall refer to the generalized Veneziano model with 0% = 1 as the "conventional model". Despite the manifold virtues of this model, the unitarization program encounters at least two serious problems. First, the model contains a tachyon, which causes severe difficulties, of course. Second, certain non-planar loop graphs give rise to a new singularity [9] that is not simply related to single-particle states. It has been suggested (probably correctly) that this singularity should be identified as the pomeron. Its precise character and intercept are determined by detailed characteristics of the spectrum of single-particle states. In the conventional model it turns out to be a branch point having intercept ¼(when the linear dependences are properly accounted for). It is our strong impression that such a singularity can be reconciled with unitarity only when it is a pole. One possible attitude, closely akin to the bootstrap philosophy [10], is that of all the many dual models that might be constructed, only the "right" one is free of g h o s t s and tachyons and gives a p o m e r o n pole. Veneziano has f u r t h e r sugg e s t e d [11] that even the coupling constant m a y be d e t e r m i n a b l e . However, this is an attitude we do not share. Another s h o r t c o m i n g of the conventional model is its lack of r e s e m blance to the r e a l world. At f i r s t , attempts to i n c o r p o r a t e m o r e r e a l i s t i c quantum n u m b e r s c e n t e r e d on attaching s u p e r s t r u c t u r e d e s c r i b i n g quark spin and internal quantum n u m b e r s to the conventional model [12]. This app r o a c h e n c o u n t e r e d i n s u p e r a b l e difficulties relating to p a r i t y doubling and ghosts. Recently, a n u m b e r of m o r e ambitious p r o p o s a l s have been made, but none of them has been v e r y s u c c e s s f u l so far. This p a p e r p r e s e n t s a new model containing r e a l i s t i c quantum n u m b e r s without p a r i t y doubling or ghosts (as f a r as we can tell). The b a s i c a l g e b r a of the model has been p r e v i o u s l y r e p o r t e d [13]. However, the v e r s i o n p r e sented in ref. [13] contains "pions" and " r h o s " of the wrong G - p a r i t y , and also has a n u m b e r of other inadequacies. Shortly a f t e r submitting that work we d i s c o v e r e d a slight but c r u c i a l modification that allows for substantial improvement. The new model has pions and rhos with the c o r r e c t quantum n u m b e r s . It also contains analogs of the V i r a s o r o gauges of the conventional model, and it a p p e a r s to be f r e e of ghosts. More r e m a r k a b l y , it p o s s e s s e s additional gauge o p e r a t o r s which have the a l m o s t incredible p r o p e r t y of decoupling the ground state and the leading Regge t r a j e c t o r y ! The a l g e b r a that is o p e r ative is not yet fully understood, but it is undoubtedly r a t h e r deep. The model contains a 7r, p, fo, 6o, A 2 and ~?', all with the p r o p e r quantum n u m -

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A. Neveu, J.H. Schwarz, Factorizable dual model of pions

b e r s . It does not c o n t a i n l o w - m a s s s t a t e s without an e x p e r i m e n t a l identification. In fact, it does not even h a v e a (r d e g e n e r a t e with the p o r an A 1. F o r h i g h e r m a s s e s the d e g e n e r a c y p r o b a b l y g r o w s e x p o n e n t i a l l y with m a s s as in the c o n v e n t i o n a l m o d e l , although the g a u g e a l g e b r a is so p o w e r f u l that it is not y e t c l e a r w h e t h e r it g r o w s that fast. T h e p - fo and co - A 2 t r a j e c t o r i e s a r e not "doubled", c o n t r a r y to p r e v i o u s s u g g e s t i o n s [14]. The a m p l i t u d e f o r ~Tr --. 7r~ is the one s u g g e s t e d by Shapiro and L o v e l a c e [15], while the one f o r ~rTr~ 7rw p r e c i s e l y c o i n c i d e s with V e n e z i a n o ' s o r i g i n a l s u g g e s t i o n [1]. An e m p i r i c a l r u l e o b s e r v e d in the m o d e l is that t h o s e t r a j e c t o r i e s with n o r m a l - p a r i t y c o u p l i n g s to pions (Tr, p, fo, etc.) o c c u r o n e - h a l f unit too high, w h e r e a s t h o s e with a b n o r m a l - p a r i t y couplings to pions (¢o, A2, 77', etc.) a r e n i c e l y located. This m e a n s that we have %r(0) = ½, otn(0) = Ot,o(0 ) = 1, otw(0 ) = ½, and o~?,(0) = -1. This p l a c e m e n t of t r a j e c t o r i e s ~ i m p l i e ~ that the 7r is a t a c h y o n and the p is m a s s l e s s . A n o t h e r difficulty is that the nond e g e n e r a c y of the p and co m a k e s a s t r a i g h t f o r w a r d g e n e r a l i z a t i o n to SU(3) m o s t u n s a t i s f y i n g . N a m e l y , it would r e s u l t in two s e t s of p - f and co - A 2 t r a j e c t o r i e s split by o n e - h a l f unit. One s e t would c a r r y the n o r m a l c o u plings and the o t h e r the a b n o r m a l ones. In o u r opinion, the m o d e l s h o u l d be r e g a r d e d as i n t r i n s i c a l l y l i m i t e d to SU(2). We s u s p e c t that a f u t u r e b e t t e r m o d e l m t g h t r e q u i r e including SU(3) as a b r o k e n s y m m e t r y . The p a p e r is o r g a n i z e d as f o l l o w s : sect. 2 is a r e v i e w of the c o n v e n t i o n a l m o d e l s t r e s s i n g t h o s e a p p r o a c h e s and notations that a r e r e q u i r e d in s u b s e quent s e c t i o n s . Sect. 3 p r e s e n t s the p r o p o s e d new m o d e l . In p a r t i c u l a r , it e x p l a i n s why the m a s s e s a r e c o n s t r a i n e d to be t h o s e m e n t i o n e d above. Sect. 4 d i s c u s s e s the b a s i c c o n j e c t u r e c o n c e r n i n g the e x i s t e n c e of e x t r a g a u g e o p e r a t o r s . The c o n j e c t u r e is not p r o v e d , but a s t r o n g c a s e is m a d e f o r it. In sect. 5 t h e r e is a r a t h e r d e t a i l e d i n v e s t i g a t i o n of the s p e c t r u m . All the s t a t e s at the f i r s t few m a s s l e v e l s and on the leading t r a j e c t o r i e s a r e d i s c u s s e d . In sect. 6 the f o r m u l a s f o r the s i x - p i o n a m p l i t u d e and the 7rp --. ~p a m p l i t u d e s a r e p r e s e n t e d . Sect. 7 is the c o n c l u s i o n .

2. R E V I E W O F C O N V E N T I O N A L

DUAL-RESONANCE

MODEL

T h e s p e c t r u m of the c o n v e n t i o n a l d u a l - r e s o n a n c e m o d e l is c o n v e n i e n t l y d e s c r i b e d in the F o c k s p a c e g e n e r a t e d by f o u r - v e c t o r c r e a t i o n o p e r a t o r s a~rn~, m = 1, 2 , . . . (ref. [16]). T h e s e o p e r a t o r s s a t i s f y the c o m m u t a t i o n r e lations

[a#rn'a:]=t

r a t ~ a'~U3 = , m' n, 0

[a~, a T v] = _g# u~ m

n

m,n '

(2.1a)

(2.lb)

w h e r e we u s e the t i m e l i k e m e t r i c gOO = _gii = 1. We find it helpful to r e -

place the a-operators and the momentum operator p by

A . Neveu, J.H. Schwarz, Factorizable dual model of pions

~IZn = -i4-na~n '

a ~ : i(-na ~ -n

n = 1,2,...

,

n

89

,

(2.2a)

n : 1,2 .....

(2.2b)

a/z = ~ f - ~ p g . o T h e a l g e b r a of eq. (2.1), r e - e x p r e s s e d

(2.2c)

for the or-operators,

takes the form

ot g , ot v] = _ingOt v 5 IvFt

/'t ~

(2.3) ~'t+n 9 0

F o l l o w i n g R a m o n d [17], we d e f i n e t h e o p e r a t o r

1 p u (r) = ~

~=~_~ol ~ e -in'r n-n '

(2.4)

whose • space average is just the momentum: 1 p / l = (p/z(.r)) : ~ - ~ f

27r P/~(T)dT.

(2.5)

o

T h e F o u r i e r m o m e n t s of P2(T) a r e of p a r t i c u l a r i m p o r t a n c e :

L a = - ( e i n r : p 2 ( r ) : ) =-7.1" ~ ot n m=-o~ -m'Oln+m: "

(2.6)

We u s e t h e s u p e r s c r i p t "a" a s a r e m i n d e r t h a t an o p e r a t o r i s d e f i n e d in t e r m s of t h e a o s c i l l a t o r s " s i n c e in t h e n e x t s e c t i o n we s h a l l i n t r o d u c e a "b" s p a c e a s w e l l . F o r n = 0, eq. (2.6) b e c o m e s

La 1 ~ C~-m'~m =R a _ p2 o = -2~o'~o - m=l

(2.7)

w h i l e f o r n > O, o~ n-1 La -~ o~ -'~ n = m=O - m ' ° t m + n - 2m=l ° t m ' ° l n - m "

(2.8)

a

The operators L n have the elegant algebra Lm a ' L an lj = ( m - n ) L m a 4-/$

(2.9)

'

e x c e p t when m+n = 0 a n u n i n t e r e s t i n g c o n s t a n t m a y n e e d to b e "tdded to t h e a a a c o m m u t a t o r . In p a r t i c u l a r , L _ l , Z o and L 1 f o r m an S U ( 1 , 1 ) a l g e b r a [18]. T h e c o n s t r u c t i o n of N - p o i n t f u n c t i o n s c a n b e a c h i e v e d in a n u m b e r of w a y s . T h e one t h a t i s m o s t c o n v e n i e n t f o r o u r p u r p o s e s i s to d e f i n e a v e r tex operator

V o ( k , z ) =e i k ' x e

n=l

n

e

n=l

n

z

-2k.p

,

(2.10)

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A. Neveu, d.H. Schwarz, Factorizable dual model of pions

to d e s c r i b e the e m i s s i o n of a p a r t i c l e of f o u r - m o m e n t u m k/~ satisfying a m a s s - s h e l l condition k2 = m2. The N-point function is then given by [19]

A N ( k l , k 2 , . . . ,kN) = f dgN(Z ) i~=l [zi×

<0[Vo(kl,

Zi+l l-m2 }

z 1) Vo(k 2, z 2 ) . . .

Vo(kN, ZN)IO) I ,

(2.11)

w h e r e ZN+ 1 - z 1 and

N - 1 0 ( z i _ Zi+l ) dZN_ 1 ~-1 (2.12) i=1 z i - Zi+l The i n t e g r a t i o n s in eq. (2.11) a r e along the r e a l axis with l i m i t s c o n t r o l l e d by the 0-functions of eq. (2.12); Zl, z 2 and z N a r e fixed n u m b e r s satisfying z I > z 2 > z N. The m o s t i m p o r t a n t f e a t u r e of A N is its i n v a r i a n c e under cyclic p e r m u t a t i o n of the m o m e n t a (duality). This f a c t depends c r i t i c a l l y on two p r o p e r t i e s of eq. (2.11). They a r e (1) the .invariance of the integrand under M6bius t r a n s f o r m a t i o n s and (2) the rule f o r c o m m u t i n g Vo(ki, zi) with dtZN(Z) : (z 2 - ZN)dZ 3 dz4 . . .

Vo(% zj).

Consider a Mbq~ius t r a n s f o r m a t i o n

az+b z--* z' = Tz - - cz+d' with a, b, c, d r e a l n u m b e r s satisfying ad - bc = 1. T h e s e t r a n s f o r m a t i o n s f o r m the group 0(2, 1), or, what is e s s e n t i a l l y the s a m e thing, SU(1, 1). When all the v a r i a b l e s Zl, z 2 . . . . . z N a p p e a r i n g in eq. (2.11) a r e s i m u l t a n e ously s u b j e c t e d to the s a m e t r a n s f o r m a t i o n , the r e s u l t i n g e x p r e s s i o n has the s a m e f o r m with z~, z~ . . . . , z ~ r e p l a c i n g Zl, z 2 , . . . , z N. To p r o v e this one f i r s t does an e a s y calculation to show that the m e a s u r e is f o r m i n v a riant, i.e., dbtN(Z ) = d~tN(Z' ). The i n v a r i a n c e of the r e s t of the integrand is d e m o n s t r a t e d by c o n s t r u c t i n g a u n i t a r y o p e r a t o r A(T) = e i~T" L a

(2.13)

w h e r e L a is made f r o m Lao, L a l and ~ T i S a t h r e e - v e c t o r a p p r o p r i a t e for T. The v e r t e x o p e r a t o r t r a n s f o r m s by A(T) Vo(k , z) A - I ( T ) = (a - cz') -2m2 Vo(k , z') .

(2.14)

The m o s t convenient way to p r o v e (2.14) is by f i r s t c o n s i d e r i n g the infini t e s i m a l t r a n s f o r m a t i o n s d e s c r i b e d by the f o r m u l a [19, 20] La n d [ n' Vo(k' z)] = z (Z-d-~ - n m 2) Vo(k, z ) ,

(2.15)

valid f o r all n. The proof of M6bius i n v a r i a n c e is c o m p l e t e d by noting that ei~ T ' L a l e a v e s the z e r o - m o m e n t u m ground state, [0}, invariant, and that N the f a c t o r s ~ (a - c z ' ) - 2 m 2 a r i s i n g f r o m the t r a n s f o r m a t i o n of the v e r t i c e s i=1

A.Neveu, J.H.Schwarz, Factorizable dual model of pions

91

a r e c o m p e n s a t e d f o r by e x t r a f a c t o r s a r i s i n g in the t r a n s f o r m a t i o n of N I~ ~ i - Zi+l[ - m 2 . The M6bius i n v a r i a n c e of the integrand i m p l i e s , in p a r i=1 t i c u l a r , that A N does not depend on the values chosen f o r Zl, z 2 and z N. The proof of cyclic s y m m e t r y is c o m p l e t e d by choosing the M~bius t r a n s . f o r m a t i o n T such that !

I

!

z 1 =z N,

z 2 =z 1 , zN=zN_ 1. Then, in o r d e r to obtain the f o r m u l a f o r Ahr(kN, k l , k 2 , . . . , k ~ 1), it is necf.,,e s s a r y to c o m m u t e Vo(kN, Zl) p a s t Vo(kl, z2) . . . Vo(kN_l, Z'N). The apv

-T

,

p r o p r i a t e f o r m u l a f o r this p u r p o s e is

Vo(k i, z i) Vo(k j, zj) = Yo(kj, zj) Yo(ki, z i) (-1) -2ki'kj .

(2.16)

T h e r e f o r e , using m o m e n t u m c o n s e r v a t i o n and the m a s s shell condition, the c o m p l e t e c o m m u t a t i o n of Vo(kN, Z'l) g i v e s r i s e to a p h a s e (-1)2 m2. F o r t u nately, this p h a s e is c a n c e l l e d by one o c c u r r i n g in the t r a n s f o r m a t i o n of N ~-[ I z i - Z/+ll - m 2 suitably redefined.

i=l Another i m p o r t a n t p r o p e r t y of A N is its f a c t o r i z a b i l i t y . This is d e m o n s t r a t e d by using

Vo(k , z) = z L a z m2 Vo(k)z - L a ,

(2.17)

(Vo(k) is a s h o r t e n e d notation f o r Vo(k , 1)), and c a r e f u l l y letting z 1 -. 0% z 1 -* 1, z N --, O. In this way, a f t e r a suitable change of v a r i a b l e s , one obtains

A g ( k 1, k 2 . . . . . k N) = <0;kll Vo (k2 )Do Vo (k3). . . Do Vo (kN_ l ) l O;kl~ , (2.18) w h e r e the p r o p a g a t o r D o is given by Do =

f1 dxxLao+m2_l

(1 - x) - m 2 - 1 , (2.19) o and ]0;k) r e p r e s e n t s a ground s t a t e with m o m e n t u m k. (In the f u t u r e we shall s i m p l y w r i t e [,0), leaving the m o m e n t u m implicit.) A c a s e of p a r t i c u l a r s y m m e t r y and beauty, d i s c o v e r e d by V i r a s o r o [8], o c c u r s when m 2 = -1. In this c a s e the leading Regge t r a j e c t o r y is ~(s) = 1 + s, and the ground s t a t e is a tachyon. Also, the p r o p a g a t o r t a k e s the simple form

Do = (Loal 1) -1 ,

(2.20)

so that the physical states satisfy the infinite-component wave equation (/.oa - 1)~, = 0 .

(2.21)

92

A. Neveu, J.H. Schwarz, Factorizable dual model of pions

Furthermore, conditions

t h e o n - s h e l l p h y s i c a l s t a t e s a r e s u b j e c t to the s u b s i d i a r y a LngO = 0 ,

n = 1,2 .....

(2.22)

~t a T o p r o v e t h i s it i s s u f f i c i e n t to s h o w t h a t L o - L n - 1 a n n i h i l a t e s a p h y s i c a l tree DoVo(k 2) DoVo(k 3) . . .

DoVo(kN_ 1) 107.

U s i n g eq. (2.9) a n d eq. (2.15) one h a s (Loa - L a - 1) 1 Vo (k) n L a- 1 o

La+n-1 o

(L:- 1. an + n 1

1) v o ( k )

Vo(k) ('La- La-n 1 ) .

(2.23)

La+n-1 o a

a

T h i s c a l c u l a t i o n , t o g e t h e r with the f a c t t h a t L o - L n - 1 a n n i h i l a t e s an o n s h e l l g r o u n d s t a t e , p r o v e s the v a l i d i t y of the V i r a s o r o s u b s i d i a r y c o n d i a a t i o n s . N o t i c e t h a t the g a u g e o p e r a t o r s L o - L n - 1 a n n i h i l a t e p h y s i c a l s t a t e s off m a s s s h e l l a s w e l l a s on m a s s s h e l l , w h e r e a s L an only a n n i h i l a t e s o n mass-shell physical states. One f i n a l c o n s t r u c t of t h e c o n v e n t i o n a l m o d e l that w e s h a l l n e e d i s the t w i s t o p e r a t o r [21]. By d e f i n i t i o n , the t w i s t o p e r a t o r , f~a, r e v e r s e s t h e o r d e r of the e x t e r n a l s t a t e s a l o n g a t r e e , i . e . , a t r e e and a t w i s t e d t r e e ( s e e fig. 1) a r e r e l a t e d b y

f~aVo(kl) Do Vo(k2) D o . . .

D o Vo(kg_ 1)]0)

= Vo(kN)DoVo(kN_l)Do...

DoVo(k2)lO).

(2.24)

A formula for ~a is a a 2 a a a = (-1~ ae -L-1 = (-1)/'o+p e -L-1 .

(2.25)

A g o o d w a y to p r o v e t h a t t h i s f o r m u l a s a t i s f i e s eq. (2.24) i s by l e t t i n g it a c t on the i n t e g r a n d of a f a c t o r i z e d f o r m of eq. (2.11). U s i n g t h e S U ( 1 , 1 ) k!

k2

kN_ {

.... (a)

f.,

k N kN_ ~

---)

t .... (b)

Fig. l a , b. Two N - p a r t i c l e states related by twisting.

k2 kt

93

A. Neveu, J.H. Schwarz, Factorizable dual model of pions

t r a n s f o r m a t i o n r u l e s , one shows in this way that ~ a V o ( k l , Zl) Yo(k2, z2) . . .

Vo(kN, ZN) [0)

: (-1) Loa+p2 Vo(kl, Z l - 1 ) Yo(k2, z2-1) . . .

= (-l)P2 Vo(kl, l-Zl) go(k2, l-z2) . . .

Vo(kN, ZN-1 ) [0)

Yo(kg,I-ZN) ]0)

= Vo(kN, 1 - z N) Vo(kN_ 1 , 1-ZN_ 1) . . . Vo(k 1 , 1 - z 1) 10).

(2.26)

The l a s t s t e p u s e d eq. (2.16). The v e r i f i c a t i o n of twisting can now be c o m pleted by making the change of i n t e g r a t i o n v a r i a b l e s z i = 1 - ZN_i+ 1. The m o s t i m p o r t a n t f a c t about the twist o p e r a t o r f o r our put, p o s e s h e r e l i e s in the o b s e r v a t i o n that p h y s i c a l s t a t e s a r e e i g e n s t a t e s of g2 , a

~t(p = (_l)Ra eL1 ~p = (_l)Ra~p .

(2.27)

F u r t h e r m o r e , when isospin is introduced by the C h a n - P a t o n p r o c e d u r e [22] (which s e e m s to be the only s e n s i b l e one) and s u m s on all the p e r m u t a t i o n s of e x t e r n a l lines of t r e e g r a p h s a r e included, the eigenvalue of ~2T m a y be identified with the c h a r g e conjugation quantum n u m b e r of the neutral m e m b e r of the m u l t i p l e t u n d e r c o n s i d e r a t i o n . T h e r e f o r e , if the G - p a r i t y of a p h y s i c a l s t a t e is known, its i s o s p i n m a y be r e a d i l y deduced f r o m its m a s s . The g e n e r a l i z a t i o n of this p r o c e d u r e to SU(3) is not difficult, but will not be r e q u i r e d in this paper.

3. NEW DUAL-RESONANCE MODEL The Fock s p a c e d e s c r i b e d in sect. 2 can be e n l a r g e d by the addition of a n t i c o m m u t i n g c r e a t i o n and annihilation o p e r a t o r s . Specifically let us cons i d e r the s p a c e g e n e r a t e d by the o s c i l l a t o r s of the p r e c e d i n g section to. ,,~" 1 3 g e t h e r wRh b~n, , m = ~, ~, . . . . We postulate, in addition to eq. (2.3) bm ~z' b ; } = _ggV ~

~/+n,

(3.1a)

O '

b~ v [ m ' an] = 0 ,

where, as a m a t t e r of convenience, we have defined b ~ introduce the o p e r a t o r - m H~Z('r) =

~ b ~ e -im7 m=-~o m '

(3.1b) = btrn~ We next "

(3.2)

with the s u m running o v e r all half i n t e g e r s , of c o u r s e . This o p e r a t o r has the a n t i c o m m u t a t i o n r e l a t i o n s { g ~ ( 7 ) , H V ( ~ ' ) } = - 2 ~ g u v 8(7- 7') .

(3.3)

A.Neveu, d.H.Schwarz, Factorizable dual model of pions

94

L e t t i n g / / ~ t (~.) : d / d T H ~z(T), define

Lb=-½i(einr:H(T).fI('r):)=-¼:

~

n

(n+2m)b_m.bn+m:.

(3.4)

~vl=_a o

F o r n = 0, eq. (3.4) b e c o m e s

L b =- ~ O

while f o r n = 1 , 2 , . . .

mb m.b m = R b ,

(3.5)

yn=l

we have

1

oo

Lb

1 S ½ (n+

n = -~

-¼ n - ~

2m)b-m'bn+m

(n- 2m)bm.bn_ m .

(3.6)

J u s t as in eq. (2.9), we have the a l g e b r a Lb

m'

L bl

Lb

n] = (m-n)

(3.7)

m+n'

a s i d e f r o m p o s s i b l e e x t r a c o n s t a n t s when m + n ~- 0. Eq. (3.1b) allows us to define

Ln =L a + Lb n

n

(3.8)

'

with the k n o w l e d g e that t h e s e o p e r a t o r s have the s a m e a l g e b r a . (This is a n a l o g o u s to addition of a n g u l a r m o m e n t u m . ) The L n o p e r a t o r s will enable us to d e m o n s t r a t e that o u r new m o d e l has m u c h the s a m e a l g e b r a i c s t r u c t u r e as the c o n v e n t i o n a l one. The next step is to f o r m u l a t e N - p o i n t f u n c t i o n s . F o r this p u r p o s e we r e p l a c e the T - v a r i a b l e by z = e iT and define

V(k, z) = k'H(Z)Vo(k, z) .

(3.9)

T h e N - p o i n t f u n c t i o n f o r N "pions" of m a s s m is then p o s t u l a t e d to be N 1 A N ( k l , k 2 . . . . , k N) = i=~IlIZ-~lzi-Zi+l[

fauN(Z)

× (OlV(kl, Zl)

½-rn2}

V(k2,z2)... V(kN,ZN)10),

(3.10)

w h e r e the i n t e g r a t i o n m e a s u r e is g i v e n in eq. (2.12). T h e p r o o f of c y c l i c s y m m e t r y p r o c e e d s as in the c o n v e n t i o n a l model. F i r s t one p r o v e s M6bius i n v a r i a n c e by c o n s i d e r i n g the t r a n s f o r m a t i o n of the v e r t e x o p e r a t o r . C o m bining d x [ L bn, Hl'*(z)] = zn(z~-~+-~n)HP'(z)

(3.11)

with eq. (2.15), one finds that f o r k 2 = m 2,

[Ln, V(k, z) ] = znlz ~---~+ n(½-m2) J Y(k, z) . T h u s , in a n a l o g y with eq. (2.14),

(3.12)

A. Neveu, J.H. Schwarz, Factorizable dual model of pions A(T) vvb,~Z_A_l tT) = ( a - c z ' ) 1 - 2 m 2 V(k, z ' )

95 (3.13)

N The e x t r a f a c t o r s 1-[ ( a - cz') 1 - 2 m 2 coming f r o m eq. (3.13) a r e p r e c i s e l y i--1 c a n c e l l e d by f a c t o r s that a r i s e in the t r a n s f o r m a t i o n of the product a p p e a r ing in eq. (3.10). The proof of cyclic s y m m e t r y is c o m p l e t e d by m e a n s of the a p p r o p r i a t e M~bius t r a n s f o r m a t i o n (see sect. 2) and a study of the c o m m u t a t i o n p r o p e r t i e s of the v e r t i c e s . B e s i d e s the p h a s e s of eq. (2.16) one obtains m i n u s signs f r o m the a n t i c o m m u t a t i o n of the k.H. (The 6-functions of eq. (3.3) do not contribute to the i n t e g r a l . ) The p h a s e (-1) N-1 that c o m e s about in this way is in fact r e q u i r e d in c o m p e n s a t i n g p h a s e s a r i s i n g f r o m the t r a n s f o r m a t i o n of { i=l ~ z ~ . ½ 1 ~ / - z/+l 1½-m2} . Thus duality of the m o d e l is proved. The amplitude A N n e c e s s a r i l y v a n i s h e s when N is odd, b e c a u s e the e v a luation of < 0 I k l . H ( Z l ) . . . k~/-/(ZN)J0> r e q u i r e s that b - o p e r a t o r s be paired. We m a y t h e r e f o r e define the G - p a r i t y o p e r a t o r by ~ b_m'b m G = (-1) m=½ ,

(3.14)

and a s s e r t that our pions have negative G - p a r i t y . By a c o n s t r u c t i o n analogous to that used to obtain eq. (2.18) we r e w r i t e the N - p o i n t function in the f o r m A l ~ k l , k 2 . . . . , kN) = <0[kl'b ½ V(k2)DV(k3)D. . . D V ( k N _ l )kN'b_½]O> , (3.15) with D = f

1

2 ~ 2 ~ r _3 d x ( 1 - x ) - m - ~ x m --~-o-2 ,

(3.16)

Lo = R a + Rb _ p 2 .

(3.17)

O

F o r the s p e c i a l value of the pion m a s s m 2 = -~, 1 one has O = (Lo-1)-I .

(3.18)

In this c a s e t h e r e a r e also V i r a s o r o - t y p e s u b s i d i a r y conditions, L n ~ = O, n = 1, 2 . . . . . In c o m p l e t e analogy with eq. (2.23), eq. (3.12) enables one to show that f o r m2 (Lo-L n-l)

_

V(k) - L o

+1n - 1 V(k) ( L o - L n -

1) "

(3.19)

Eq. (3.19), t o g e t h e r with the f a c t that L o - L n - 1 annihilates an o n - s h e l l pion ( d e s c r i b e d by k.b _~10;k> with k 2 = -½), p r o v e s the validity of the Ln s u b s i d i a r y c o n d i t i o n s . - f t is i m p o r t a n t to u n d e r s t a n d in this connection that

96

A. Neveu, J.H.Schwarz, Factorizable dual model of pions

we only c o n s i d e r s t a t e s of the F o c k s p a c e to be p h y s i c a l when they can be made from multipion systems. Now that the m o d e l h a s b e e n f o r m u l a t e d , let us b e g i n an e x p l o r a t i o n of its i m p l i c a t i o n s by c a l c u l a t i n g the f o u r - p o i n t function d e p i c t e d in fig. 2. A 4 : (0 [kl.b ½ V(k2) DV(k3)k4.b_½10 >

: / 1 dxxm2_s_l

( 1 - x) m2-t-1 l(kl.k2)(k3.k4)V-~x x

o + ( k 2 " k 3 ) ( h i ' k 4 ) ] / / l ~ x - ( k l " k 3 ) ( k 2 " k 4 ) l ¢ ~ T x - x)} = (kl-k 2)

(ka-k4) B(-a(s),

1 - a ( t ) ) + (k2.k3) ( k l . k 4 ) B ( 1 - a ( s ) , 1 - a ( t ) ) -

(kl.h 3) ( k 2 . h 4 ) B ( 1 - a ( s ) ,

1-a(t)) ,

(3.2O)

w h e r e a(s) = s - m 2 + ½ zs to be identified with the p - t r a j e c t o r y . If we r e q u i r e that t h e r e be no pole f o r a(s) = 0, then it is n e c e s s a r y that

k l ' k 2 : ½(s- 2m 2) = ½or(s) ,

(3.21)

and hence that m 2 = -~. R e m a r k a b l y enough, this is the s a m e m a s s condition r e q u i r e d f o r the V i r a s o r o g a u g e s . When it is s a t i s f i e d , eq. (3.20) m a y be s i m p l i f i e d by s o m e k i n e m a t i c s to A4 = _¼ F(1 - a(s)) F(1 - a(t))

(3.22)

T h i s is the f o r m u l a p r o p o s e d f o r w s c a t t e r i n g by L o v e l a c e and S h a p i r o [14] s o m e t i m e ago. T h e f a c t o r of ¼ h a s no d e e p s i g n i f i c a n c e s i n c e we a r e o m i t ting coupling c o n s t a n t s and o u r pion s t a t e is not n o r m a l i z e d : < ' l ' ) : <01k.b½k.b_½l0} : -k 2 - ½.

(3.23)

(Notice that it has positive norm only when it is a tachyon]) What we have succeeded in showing here is that for the special mass values m = -~ and m2= 0, eq. (3.22) arises from a fully factorizable scheme.

'l

kl r

\ k4

Fig. 2. Kinematicsof zr~"~ Ir?r.

4. BASIC CONJECTURE At first sight the model we have constructed appears to have an unreasonable spectrum of states. The state described by the ground state of all

A.Neveu, J.H.Schwarz, Factorizable dual model of pions

97

the o s c i l l a t o r s , which we shall call G, is evidently a s c a l a r m e s o n with M2 = -1 o c c u r r i n g for ap(s) = 0. F u r t h e r m o r e , with one b_~ and N - 1 a l l excitations, it should be possible to make a state of spin N ~nd M2 = N - ~. Such states would t h e r e f o r e lie on a t r a j e c t o r y one unit higher than the nt r a j e c t o r y - hence we call it the A - ( f o r a n c e s t o r ) t r a j e c t o r y . Our b a s i c c o n j e c t u r e is that G and the A - t r a j e c t o r y are spurious. If true, this would imply the existence of a r a t h e r profound gauge a l g e b r a in the model, going beyond the s t r a i g h t f o r w a r d g e n e r a l i z a t i o n of the V l r a s o r o gauges. We call it a c o n j e c t u r e b e c a u s e we have not yet fully proved it, but we have no doubt about its truth. In the previous section it was shown that f o r special m a s s values the G-pole does not o c c u r in 7rn s c a t t e r i n g , which implies that G does not couple to two pions. Another easy calculation is a d e m o n s t r a t i o n that the pion a n c e s t o r ~A (the f i r s t state on the A - t r a j e c t o r y ) does not couple to t h r e e pions. It is trivially decoupled f r o m an even n u m b e r of pions by G - p a r i t y , of course. The gA state is given by ~.b_410) with ~.k = 0 and k 2 = -½. T h e r e f o r e , the amplitude f o r nA --" 3~, found by a l m o s t identical a m t h m e t l c to that of eq. (3.20), is E

.

.

(0 it-b½ V(k2) DV(k3) k4.b_ ½ I0) = E .k2(k3.k4) B ( - a is), 1 - a (t))

+ • "k4(k2.k3) B ( 1 - a (s), - a (t)) - ¢ .k3(k 2 .k4) B ( 1 - a (s), 1 - a (t)) 1

= -~'(k2+k3+k4)

r(1-a(s))

r(1-a(t))

r(1-a(s)-a(t))

= 0

"

The spurious c h a r a c t e r of the ground state and A - t r a j e c t o r y a r e not independent phenomena. The following a r g u m e n t s show that it is sufficient to know that either the state G or the state n A is spurious to establish our conjecture. By definition, a spurious state is one that does not couple to any m u l t i p a r t i c l e s y s t e m whose constituents a r e all physical. T h e r e f o r e , a state which couples to a t w o - p a r t i c l e s y s t e m consisting of a physical state and a spurious state m u s t itself be spurious. We use this simple truth to establish the equivalence of 7rA and G being spurious by showing that t h e r e is a non-vanishing G - n - n A vertex. R e f e r r i n g to fig. 3a, we have (0 [V(k2)E .b_½ [0) -- -e.k 2 . But E.k 2 ¢ 0 is k i n e m a t i c a l l y possible, so 7zA We can f u r t h e r show that if G is spurious the ous. In the notation of fig. 3b, it is sufficient the states of the A - t r a j e c t o r y . In p a r t i c u l a r , state of spin N+ 1 f r o m the t e r m

(4.1)

is spurious if and only if G is. entire A - t r a j e c t o r y is s p u r i to show that V(k2)[0) eontains for k 2 = N - ½ we can obtain a

k2"b_½(~2 .a_X)N[0) , which a r i s e s in the expansion of V(k 2) [0). This t e r m includes a spin N + 1 p a r t b e c a u s e k 2 contains a non-vanishing component orthogonal to k. T h e r e f o r e , our c o n j e c t u r e would be established by proving G to be spurious. We have a l r e a d y s e e n that (G]2~r) = 0. In the appendix we show that (G14zr) = (G[67r) = 0. While sufficient to convince us that G is spurious, the c a l c u l a tions of the appendix do not constitute a complete proof of that fact.

98

A . Neveu , J. H. Schwarz , F a c t o r i z a b l e dual model of pions k2 k

G-

k2 -~ k1,~ "n" A

k

_ A-

(O)

kI

~G

(b)

Fig. 3. a. The G - T r - 7rA vertex, b. The A - ? r - G vertex. \

T h e n i c e s t w a y to p r o v e o u r c o n j e c t u r e w o u l d b e b y f i n d i n g a d d i t i o n a l g a u g e o p e r a t o r s ( b e s i d e s the /:n o p e r a t o r s ) . So f a r we h a v e b e e n u n s u c c e s s fu] in t h i s q u e s t . In o u r o p i n i o n , t h e s o l u t i o n of t h i s p r o b l e m i s r e q u i r e d to o b t a i n a c o m p l e t e u n d e r s t a n d i n g of w h a t i s g o i n g on in t h i s m o d e l . It i s q u i t e r e m a r k a b l e t h a t b y a d d i n g s u p e r s t r u c t u r e to t h e c o n v e n t i o n a l m o d e l , one c a n o b t a i n a r i c h e r g a u g e g r o u p . It i s n o t a s t r u c t u r e t h a t i s p r e s e n t and p r e v i o u s l y o v e r l o o k e d in t h e c o n v e n t i o n a l m o d e l . A l l t h e p h y s i c a l s t a t e s e v i d e n t l y i n v o l v e b - o p e r a t o r s in t h e i r d e s c r i p t i o n , s i n c e o t h e r w i s e one c o u l d i s o l a t e a s u b s p a c e i n v o l v i n g s t a t e s b u i l t f r o m s - s t a t e s only. It s e e m s l i k e l y t h a t one of the e x t r a g a u g e o p e r a t o r s , S, w i l l c o m m u t e w i t h £ o . T h e s p u r i o u s n e s s of the g r o u n d s t a t e w o u l d t h e n f o l l o w f r o m the f o r m u l a (01SI0) ¢ 0. Such a g a u g e o p e r a t o r is p o t e n t i a l l y v e r y p o w e r f u l in a s m u c h a s t h e r e i s no o b v i o u s l i m i t to t h e n u m b e r of s p u r i o u s s t a t e s t h a t S t c a n p r o d u c e . T h i s is in c o n t r a s t to t h e a c t i o n o f / : - n w h i c h c a n o n l y m a k e a s m a n y s p u r i o u s s t a t e s of a g i v e n m a s s s q u a r e d a s t h e t o t a l n u m b e r of s t a t e s n u n i t s l o w e r in m a s s s q u a r e d . On t h e o t h e r hand, s i n c e S t h a s t h e p o s s i b i l i t y of m a k i n g so m a n y s p u r i o u s s t a t e s it i s c o n c e i v a b l e t h a t it m a k e s a l l t h e s p u r i o u s s t a t e s , i n c l u d i n g t h o s e m a d e b y L_ n o p e r a t o r s . Such a g a u g e o p e r a t o r we w o u l d c h o o s e to c a l l a s u p e r g a u g e . In o u r u n s u c c e s s f u l a t t e m p t to f i n d t h e new g a u g e o p e r a t o r s , we w e r e l e d to c o n s t r u c t s o m e a d d i t i o n a l o p e r a t o r s . We s h a l l b r i e f l y m e n t i o n t h e m h e r e s i n c e t h e y m i g h t p o s s i b l y p r o v e u s e f u l to s o m e o n e s e e k i n g t h e new g a u g e s . T h e s e o p e r a t o r s (quite s i m i l a r to o n e s of R a m o n d ' s f e r m i o n m o d e l [23]) a r e G m : ~f2( eim-r p ('r).H(T)) =

~ n

and h a v e the c o m m u t a t o r s

~n.bm_ n ,

1 ± 3- ~ , . . . rn = ±-~,

, (4.2)

-~ - o o

and a n t i c o m m u t a t o r s [L m , Gn] = (½m - n) Gin+ n ,

(4.3)

{ V m , Gn} = 2 L m + n .

(4. 4)

T h e Grn a r e c e r t a i n l y n o t the new g a n g e s , but t h e y m i g h t p l a y a r o l e in their construction.

5. E X P L O R A T I O N O F T H E S P E C T R U M One of the f i r s t q u e s t i o n s t h a t a r i s e s in t h e c o n s i d e r a t i o n of q u a n t u m n u m b e r s i s how to d e t e r m i n e i s o s p i n s . F o r t h i s p u r p o s e t h e t w i s t o p e r a t o r

99

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is v e r y u s e f u l b e c a u s e , as was pointed out in sect. 2, its e i g e n v a l u e g i v e s the c h a r g e - c o n j u g a t i o n q u a n t u m n u m b e r f o r a p h y s i c a l state. T h u s , as the G - p a r i t i e s of the s t a t e s a r e e a s i l y d e t e r m i n e d by eq. (3.14), the i s o s p i n s a r e i m m e d i a t e l y deduced. It is i m p l i c i t in this r e a s o n i n g that the only p o s s i b i l i t i e s a r e I = 0 and I = 1 s i n c e we u s e the C h a n - P a t o n p r o c e d u r e [21]. A n a t u r a l g u e s s f o r the t w i s t o p e r a t o r is to r e p l a c e eq. (2.25) by (-1) Ra+Rb e - L - 1 . T h i s is a l m o s t , but not quite, c o r r e c t . F r o m eq. (3.11) it f o l l o w s that e

-Lb_l . / ~

e

Lb_I

-/-z- 1

'

(5.1)

(-1)RbH~t (z - 1) (-1) - R b = H~(1 - z) . Therefore, (-1) R b e

(5.2)

r e p e a t i n g eq. (2.26) f o r the b - p o r t i o n s of the v e r t i c e s ,

_L b -lk

H(Zl) k

T

H(z2)

H(zg)

I0>

, H(1 - Zl) H(I - z2) H(1 - ZN) = (-1)~Nk 1 - ~ k 2 " ~ i ~ z 2 . . . kN" ~ I0}

t ,,½N+½N(N-1) k . H ( 1 -- £ N ) H(1 -ZN_I) H(1 - Z l ) :'-*' g ~ kg-l'-~_ . . . kl" ~ _ ~ 1 [0}. ZN-1

(5.3)

In the l a s t s t e p we have u s e d eq. (3.3), d r o p p i n g the 5 - f u n c t i o n s s i n c e they do not c o n t r i b u t e to the i n t e g r a l s . We thus c o n c l u d e that the twist is g i v e n by

~2 = (-1)Ra+Rb-½N2 e-L-1 ,

(5.4)

f o r an N - p i o n s t a t e . Eq. (5.4) is not y e t in s u i t a b l e o p e r a t o r f o r m , h o w e v e r , s i n c e N has no o p e r a t o r r e p r e s e n t a t i o n . We t h e r e f o r e note that~ eq.

(5.4) is equivalent to (-I)Ra+Rb e-L-ifor even-G states and (-I) Ra+Rb-~ e-L-I for odd-G states. Thus we may re-express eq. (5.4) in the form ~2 = (-1~ Ra+Rb-¼(1-G) e - L - 1 .

(5.5)

We can now c o r r e l a t e i s o s p i n s with m a s s e s . States with M 2 = - ~~, ~3, ~7, . . . o r M 2 = 0 , 2 , 4 , . . m u s t be i s o v e c t o r , w h e r e a s s t a t e s with M 2 , 5 or •

= ~ , ~ , 2 , .

• •

M 2 =1,3,5 .... m u s t be isoscalar. In particular, the p and ~ are both isovector and m a y not be isospin doubled (i.e., degenerate with isoscalars). It

a l s o f o l l o w s that t h e r e c a n n o t be a a d e g e n e r a t e with the p o r a p ' d e g e n e r ate with fo. F u r t h e r m o r e , the a m p l i t u d e f o r ~r~ --* ~Tr cannot c o n t a i n odd d a u g h t e r t r a j e c t o r i e s . T h e r e a d e r m a y w o n d e r w h e t h e r we w e r e f o r c e d to i n t r o d u c e i s o s p i n o r w h e t h e r the m o d e l is m a t h e m a t i c a l l y tenable without it.

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A. Neveu, d.H. Schwarz, Factorizable dual model of pions

The answer is that when the o p e r a t o r of eq. (3.14) is identified as G - p a r i t y the isospins a r e determined. If we insist that all states a r e i s o s c a l a r , then this o p e r a t o r must c o r r e s p o n d to some o t h e r unphysical "modulo 2" rule. T h e r e is a slight ambiguity in the parity a s s i g n m e n t s of states. The r e a son f o r this is that P and GP are equally good o p e r a t o r s , which, in the absence of e l e c t r o m a g n e t i s m , f e r m i o n s and s t r a n g e m e s o n s , cannot be d i s tinguished. This fact leaves the f r e e d o m to switch the parity a s s i g n m e n t s of all the odd-G states. T h e r e f o r e , we may define the rr to be a p s e u d o s c a l a r . All other p a r i t i e s a r e then determined. We now e m b a r k on a detailed investigation of the s p e c t r u m . Not having the full gauge a l g e b r a in hand, another method for identifying spurious states is required. The method employed is to accept the validity of our c o n j e c t u r e that the ground state is spurious and then to deduce the e x i s t ence of other spumous states t h e r e f r o m . F o r example, we showed in sect. 4 that this implies that the A - t r a j e c t o r y is spurious. At the lowest m a s s squared, M 2 = -1, t h e r e is one spurious state, the ground state. At M 2 =-½ t h e r e is the physical pion given by k.b_¢ ]0>. This state has positive n o r m , but, as we mentioned before, this fact depends on its being a tachyon. The t h r e e spurious states, 7rA, a r e also at M 2 = -½. (We count a state of spin S as 2S + 1 states.)

At M2 = 0 there are ten candidates for states given by (~i [0> and One spurious state is given by

L_I [0) = V'2k. oz_1 [0>.

(5.6)

This spurious scalar happens to have zero norm (because k2 = 0). Additional spurious states can be found by studying the 7r-TrA system which can couple to spurious states only. Expanding out V(k2) El.b_ ½[0),

(5.7)

E.b_½k.b_½[0>,

(5.8)

e .vXale~ ( b_~b_-~lO> ,

(5.9)

with £1.kl = 0, one finds that

a r e a spurious v e c t o r and axial v e c t o r , respectively. They also have z e r o norm. The only r e m a i n i n g candidate for a physical state is

This state is orthogonal to the spurious vector for any choice of /3. To determine/3 one expands out the physical 27r state V(k 2) kl"b_ ½10). This yields the result/3 = -/2. The p is easily seen to have positive norm. At the next m a s s level, M 2 = ½, the states m u s t be odd-G and t s o s c a l a r . T h e r e are 24 candidates, which we now enumerate. F i r s t , we find the spurious states given by the s t a n d a r d g a u g e s : r~_l k.b_½[0> = ( ~ k . ~ -1 k.b_½+k.b_~)10>,

(5.11a)

A. Neveu, J.H. Schwarz, Factorizable dual model of pions L I£ .b_½{0) = (,/-2 k-Ot_lC.b_½+~-b ~){0 ) .

101 (5.11b)

Both of t h e s e s p u r i o u s s t a t e s have z e r o n o r m . T h e r e f o r e , as explained by Del Guidice and Di V e c c h i a [24], it follows that the conjugate z e r o - n o r m s t a t e s (obtained by k -~ -k) a r e also s p u r i o u s . Altogether this p r o d u c e s eight s p u r i o u s z e r o - n o r m states. We obtain f u r t h e r s p u r i o u s s t a t e s by cons i d e r i n g the t w o - p a r t i c l e s y s t e m consisting of a ~ and a ground state. E x panding out Vik2) ~), and doing s o m e k i n e m a t i c s , one finds the following additional s p u r i o u s s t a t e s : ~

{0) ,

(5.12a)

(spin 2)

C~_l.b ½]0) ,

(5.12b)

[~.b ~ - 42-(E.~ 1 k.b_½+k.~ 1 c.b ½)]{0>.

(5.12c)

The o c c u r r e n c e of the spin-2 s t a t e in (5.12a) was e x p e c t e d since it lies on the A - t r a j e c t o r y . We have now found a total of 17 s p u r i o u s s t a t e s , so 7 r e m a i n to be investigated. Six of the r e m a i n i n g s t a t e s with M 2 = ½ a r e

}co2>

--

v;~ c=I't b v ~b~ ½b~½10>. .

.

.

.

.

.

(5.13b)

T h e s e s t a t e s have the n o r m a l i z a t i o n (col {COl> = 1 and = - i and have quantum n u m b e r s a p p r o p r i a t e f o r the co. A s p u r i o u s combination is g e n e r ated by c o n s i d e r i n g the t w o - p a r t i c l e s t a t e consisting of a pion and a s p u r i ous a x i a i p , i.e.,

In this way one finds the s p u r i o u s s t a t e ]col) - "/'-2{co2), which has negative n o r m . A p h y s i c a l co is obtained f r o m a ~ plus a p h y s i c a l p, i.e., V(k2) (El.Or_ 1 +-/-2el.b_~kl.b_½)]0>. This calculation yields a s t a t e that is orthogonai to the s p u r i o u s co, n a m e l y

Ico> -- ¢ ]co1> - }co2> •

(5.14)

It is v e r y p l e a s i n g to d i s c o v e r that this s t a t e has p o s i t i v e n o r m . The final

k bff_½{0>. This s t a t e r e m a i n i n g s t a t e with M 2 = ~ is the s c a l a r ~p.vko kll Vb_½b_½ is s p u r i o u s , b e c a u s e it also couples to a pion and the s p u r i o u s axial p. Alt o g e t h e r , we have found that only 3 of the p o s s i b l e 24 s t a t e s with M2 = ½ a r e physical. Evidently a v e r y p o w e r f u l gauge group is at w o r k ! The a m p l i t u d e f o r the p r o c e s s co --. 37r can be c a l c u l a t e d f r o m (to {V(k3)DV(k2) k 1. b_½ I0). One e a s i l y finds

102

A.Neveu, J.H.Schwarz, Factorizable dual model of pions A(to--*37r)

' E~ klk2k v k 3(~B(1 - ~p(S), 1 - ~p(t)) . = ~Egvk~

(5.15)

Eq. (5.15) is r e c o g n i z e d to be the o r i g i n a l Veneziano f o r m u l a [1]. To extend the d i s c u s s i o n of the s p e c t r u m given above to higher m a s s e s would be a v e r y tedious p r o c e d u r e . F o r e x a m p l e , in the c a s e M 2 = 1 t h e r e a r e 55 s t a t e s to study. We t h e r e f o r e choose to switch our attention now to the leading Regge t r a j e c t o r i e s . We have a l r e a d y s e e n that the A - t r a j e c t o r y is s p u r i o u s . Another t r a j e c t o r y we can quickly d i s p o s e of is a p a r i t y double of the p _fo t r a j e c t o r y . Its o p e r a t o r d e s c r i p t i o n at spin N is

The c u r l y b r a c k e t s in (5.16) i m p l y the e x t r a c t i o n of m a x i m u m spin. T h i s m e a n s s y m m e t r i z i n g in ~ indices, and r e m o v i n g all t r a c e s and d i v e r g e n c e s . This t r a j e c t o r y is s p u r i o u s b e c a u s e it couples to the t w o - p a r t i c l e s t a t e consisting of a p h y s i c a l ~ plus a s p u r i o u s n A. The r e m a i n i n g t r a j e c t o r i e s with unit i n t e r c e p t a r e the p and fo. In p a r t i c u l a r one wishes to d e t e r m i n e t h e i r d e g e n e r a c y . T h e r e a r e t h r e e candid a t e s f o r spin N s t a t e s on these t r a j e c t o r i e s :

10)

(5.17c)

By c o n s i d e r i n g the p h y s i c a l two-pion s t a t e V(k2)kl.b_~lO), and doing s o m e s t r a i g h t f o r w a r d k i n e m a t i c s , one e s t a b l i s h e s that the f~ollowing s t a t e is physical:

For

N = 1, ( 5 . 1 8 ) r e p r e s e n t s the p h y s i c a l p, (ot~_ 1_ -4-~k.b_½b~)]O>.___ The

n o r m of each of the s t a t e s on the p h y s i c a l p - f o t r a j e c t o r y of (5.18) is p o s i tive. Consideration of the t w o - p a r t i c l e s t a t e consisting of a p h y s i c a l 7r and a s p u r i o u s ~A g i v e s r i s e to the following s p u r i o u s s t a t e

S i m i l a r l y , the t w o - p a r t i c l e s t a t e consisting of a p h y s i c a l 7r and the s p u r i o u s s t a t e e.Ot_lk.b_½10 > g i v e s the s p u r i o u s s t a t e

A. Neveu, J.H. Schwa~'z, Factorizable dual model of pions

103

The s p u r i o u s s t a t e s in (5.19) and (5.20) can be v e r i f i e d to be o r t h o g o n a l to the p h y s i c a l s t a t e in (5.18) as m u s t be the c a s e , of c o u r s e . We conclude that the p h y s i c a l p and fo t r a j e c t o r i e s a r e e x c h a n g e - d e g e n e r a t e with one a n o t h e r , but a r e o t h e r w i s e n o n - d e g e n e r a t e and c o n t a i n p o s i t i v e n o r m s t a t e s

only. Next l e t us c o n s i d e r the ~ - t r a j e c t o r y . R i s e x p e c t e d to be e x c h a n g e d e g e n e r a t e with an o d d - s i g n a t u r e i s o s c a l a r t r a j e c t o r y , although we found no J P = 1+ s t a t e a t M2 = ~. x T h e r e a r e f o u r c a n d i d a t e s of s p i n N,

_1

• • o _ flo

,

A s p u r i o u s c o m b i n a t i o n can be o b t a i n e d through u s e of the gauge o p e r a t o r L_I,

L-I~ -g

-i""

. . . .

" "

~

-~

-1"

" "

The states of (5.22) have zero norm, and therefore the conjugate states obtained by replacing b by -k are also spurious. A third spurious combination is contained in the two-particle state consisting of a ~ and a G, namely

•

I

-~

-i

" " "

-

"

The remaining candidate for the physical trajectory is the trajectory orthogonal to the three spurious ones,

(5.24) The s p i n - N s t a t e in (5.24) has p o s i t i v e n o r m . To s e e that it is in f a c t p h y s i c a l one can c o n s i d e r the s t a t e s coupling to a ~ plus a p h y s i c a l p. When t h i s

104

A . Neveu, J . H . Schwarz, Factorizable dual model of p~ons

is done, one finds (5.24) m u l t i p l i e d by N - 1 plus an i r r e l e v a n t a d m i x t u r e of a z e r o - n o r m s p u r i o u s s t a t e . One t h e r e f o r e d e d u c e s that (5.24) is p h y s i c a l e x c e p t f o r N = 1! T h i s e x c e p t i o n m e a n s that the R e g g e - r e s i d u e f u n c t i o n s have a z e r o at t h i s point. The a b s e n c e of this s t a t e is f o r t u n a t e f r o m a p h e n o m e n o l o g i c a l point of view s i n c e t h e r e is no good c a n d i d a t e f o r a J P = 1 +, IG = O- s t a t e with m a s s a r o u n d 1 GeV. On the o t h e r hand, t h e r e is a c a n d i date f o r the pion r e c u r r e n c e at m a s s 1640 MeV. Next we c o n s i d e r the co- A 2 t r a j e c t o r y , which happens to be d e g e n e r a t e with the n - t r a j e c t o r y in our m o d e l but is d i s t i n g u i s h e d by its p a r i t y . T h e r e a r e t h r e e c a n d i d a t e s f o r this t r a j e c t o r y :

EP-1

leVa k b~Y~ag2.

A s p u r i o u s c o m b i n a t i o n can be o b t a i n e d f r o m the s t a t e c o n s i s t i n g of a 7r and a s p u r i o u s p, and a d i f f e r e n t s p u r i o u s c o m b i n a t i o n can be found f r o m a 7r and a s p u r i o u s fo. Thus t h e r e is at m o s t one p h y s i c a l t r a j e c t o r y . P h y s i c a l s t a t e s a r e g e n e r a t e d by a n plus the p h y s i c a l p. One finds the r e s u l t --

_ ~Aff

0

i 0

1 0

10~

-~ -~ -~

+3q'2(N- I)

• .0~ -

"

I,,~

I

i

~

I

""c~

These states also have positive norm. The co and A2 trajectories are exchange degenerate and quite satisfactory in all respects. On the basis of the physical trajectories that have been found so far, we can formulate a remarkable empirical rule. Some of the trajectories of our model are approximately one-half unit higher than their experimental values, whereas others are very close to thier experimental locations. Furthermore these two classes of trajectories appear to be distinguished by their parities. The precise statement is that those states with normalparity couplings to pions, i.e., P = G(-1)S, where S is the spin ofthe state, occur 1 unit too high. Equivalently, the mass is given by M2 = M~exDt - ½. On the other hand, abnormal-parity states, those for which P = -G~-I)S, have M2 = M2xpt. One may conjecture that this rule will give the model predictive power for states that may not have been seen experimentally. The relationship between the model and the real world is quite intriguing. We conclude this section with a few comments about other states. Two states are conspicuous by their absence. They are the a-meson, which is

A . Neveu, J . H . Schwarz, Factorizable dual model of pions

105

g e n e r a l l y believed to be a b r o a d r e s o n a n c e d e g e n e r a t e with the p, and the A 1 meson. The e x p e r i m e n t a l evidence in each case is still g e n e r a l l y cons i d e r e d to be inconclusive. Whether our model constitutes an a r g u m e n t against them is debatable. T h e r e a r e two o t h e r physical states we shall c o m m e n t on. One is a s c a l a r m e s o n d e g e n e r a t e with the fo. Its existence is a l r e a d y evident f r o m eq. (3.22). We have not examined its d e g e n e r a c y o r norm. Another state of c o n s i d e r a b l e i n t e r e s t is a p s e u d o s c a l a r at the s a m e m a s s . Its quantum n u m b e r s a r e those of the ~?, but the e m p i r i c a l m a s s rule s t r o n g l y suggests that an identification be made with the 77'(960). T h e r e a r e two candidate~ f o r this state (5.27a)

IQ> : i

b',O ib b 10>. !

!

(5.2Vb) v

These states have the normalization (~1 1771> = 1 and 0/217/2> = -i. By constructing the coupling to a ~ and a member of the A2 trajectory it is possible to show that the physical combination is

which has positive norm. The orthogonal combination is spurious. 6. THE SIX-PION AMPLITUDE In o r d e r to a c q u i r e a m o r e c o n c r e t e understanding of the workings of the model, it is helpful to work out explicitly the six-pion amplitude. For the configuration shown in fig. 4 we have the f o r m u l a

A 6 = (0 Ikl" b½ V(k2) DV(k3) DV(k4) DV(k5) k6.b_½10> .

(6.1)

The evaluation of this e x p r e s s i o n gives r i s e to 15 t e r m s c o r r e s p o n d i n g to the different ways of pairing the six sets of b - o s c i l l a t o r s . After a s t r a i g h t f o r w a r d , but s o m e w h a t tedious, calculation one finds dUl 2 dUl 3 dUl4

--(~ (Sl 3)--I

A6 = f(1 -.12.13) (i -.13.14) "12P(s12)u13

~~ S 2 3 ~

u23

-Otp(S34) -Otp(S45) -~Tr(s24)-i -ot~(s35)-I -~p(Sl4) × u34 u45 u24 u35 u14 -°tp(S25) × u25 where the and

Y(k, u) ,

(6.2)

uij v a r i a b l e s c o r r e s p o n d to the notation of Goebel and Sakita [2]

106

A. Neveu, J.H. Schwarz, Factorizable dual model of pions

k~ Fig. 4. Kinematics of the six-pion amplitude.

1 +kl.k6k4.k5k2.k3 u23u45u25 1 1 Y(k,u) : kl'k 2k3"k 4k5"k 6 u12u14u34 u24

u35

+ kl'k2k3"k6k4"k 5 -u12u45 + k l'k 4 k2"k 3 k5"k 6 u23u14 +kl.k6k2.k5k3.k 4 u13

u34u25

1+ [kl'k4k2"k6k3"k6u13u24

+kl.k 3 k2.k 5 k4.k 6 u24u35 + kl.k 5 k2.k 4 k3.k 6 u13u35] - Ikl.k2k3.k5k4.k6 u24+kl.k 5 k2.k3k4.k 6 u35 1 u12

u23 .]

+kl.k5k2.k6k3.k4 u13 k .k k2.k6k4.k 5 u24 u34+ 1

u35

+ kl.k 3 k2"k 4 k5"k 6 ~

3

u45

u13 u25 2

+ kl.k 6 k3"k 5 k2.k 4 - -

- [kl.k4 k2.k5 k3.k6 u13u24u35 ] .

(6.3)

In this f o r m u l a we have grouped t e r m s together which t r a n s f o r m into one another under c y c l i c p e r m u t a t i o n s of the momenta. Thus the c y c l i c s y m m e t r y of A 6 is manifest. Fortunately there is a t r i c k that allows a significant simplification in the f o r m u l a f o r A 6. We know that the pion a n c e s t o r , hA, does not couple to five pions. (This is in fact proved since we show in the appendix that G does not couple to six pions.) On the o t h e r hand, the amplitude f o r ~A -~ 5n is obtained by replacing the m o m e n t u m k I in eq. (6.3) by a polarization v e c t o r E. The only way the v a r i o u s t e r m s can then add to z e r o is if the c o efficients of E.k2,¢.k3,... ,E.k6 a r e all equal so that the sum is p r o p o r t i o n al to E. (k 2 + k 3 + k 4 + k 5 + k6) = 0. Although it is r e a l l y not n e c e s s a r y to do so, we have checked that the five t e r m s in question are, in fact, equal. T h e r e f o r e , returning to the six-pion amplitude, we may r e p l a c e Y(k,u) by the coefficient of kl.k6, say. This amounts to replacing Y b y

A. Neveu, J.H. Schwarz, Factorizable dual model of pions Y'(k,u) = k4.k5k2.k 3

107

1 + k2.k5k3.k4 u13 + k3.k5k2.k4 u13 u23u45u25 u34u25 u25 (6.4)

The v e r i f i c a t i o n of cyclic s y m m e t r y when A 6 is e x p r e s s e d in t e r m s of Y is not manifest. One method is to d i s c o v e r judicious integrations by parts. The amplitude f o r p -~ 4n is obtained by taking the residue of A 6 at ap(S14 ) = 1. One finds (see fig. 5) A(p ~ 4n) = / d u l 2 d u l 3

a

~

I

a

--ap(S23)

u12 p(s12) u13

u23

1 -ap(S34) - a n ( s 2 4 ) - i [E .k4k2.k 3u23 × u34 u24

+E .k~ k~.k a u13 +E .k 3 k2.k 4 u131 .

(6.5) ,~ ,, --u34 The amplitude f o r np -~ np (see fig. 6a) then follows f r o m taking the r e s i d u e of eq. (6.5) at a p ( s l 2 ) = 1: .

A (np-~ rip) = -½E1.E 2 (1 - an(s ) - ap(t)) B(1 - a n(s), 1 - ap (t)) + El.k 3 ~2.k4B(- an(s), 1 - ap(t))

- El.k4~2.k3B(1 - an(s), 1 - Otp(t)) .

(6.6)

An a l t e r n a t i v e method of deriving eq. (6.6) is by d i r e c t l y evaluating (0 [(~ 1" a l ÷ ~Z~ vb½ kl.b½) V(k3) D r ( k 4 ) (E2.a_l + ~ZE 2.b_½ k2.b_½)[ 07 . (6.7) Eq. (6.6) c l e a r l y contains a n-pole in the s - c h a n n e l and a p-pole in the tchannel. Somewhat l e s s obvious is the nature of the pole at a n ( s ) = 1. According to the analysis of sect. 5 the only possible singularity at this m a s s is the w. The r e s i d u e of eq. (6.6) at an(S ) = 1 is

fl(t) = ½ a p ( t ) ~ l < 2 + ap(t)El.k3E2.k4 - ~l.k4E2.k3 .

(6.8)

On the other hand, by a s t r a i g h t f o r w a r d calculation one can show that for S-

1 2~

k2

k3

k4 k,~

kt Fig. 5. Kinematics for p -* 41r.

108

A.Neveu,

k3

kl'~l

J.H.Schwarz,

F a c t o r i z a b l e dual m o d e l of@ions

k4

t

k4

ki,Ei

kz'~2

ot

k3

(o)

~ k 2, E2

(b)

Fig. 6. a. Kinematics for lrp --* 7rp. b. Kinematics for 7rp --* py. ~Z, y. ~

(Y //'. U'. ~'

E~U~(rE1R1R3 £ ~ ' v ' k '

e2 R2 •4

= fl(t) .

(6.9)

Thus the r e s i d u e is c o m p l e t e l y a c c o u n t e d f o r by the w - r e s o n a n c e , without any a d m i x t u r e of 1 +, 0-, etc. The a m p l i t u d e shown in fig. 6b is o b t a i n e d by f a c t o r i z i n g eq. (6.5) at the pole ~p(S23 ) = 1: A(zrp ~ pTr) : -½El.E2(1 - a~(S) - aTr(u))B(1 - olaf(s), 1 - o~Tr(u)) + E l-k3 e 2 - k 4 B (- a~r(s), 1 - otlr(u)) + El'k4e2.k3B(1

- a~(s), - a~(u)) .

(6.10)

In c o n t r a s t to eq. (6.6), this e x p r e s s i o n c o n t a i n s o d d - G p o l e s dual to t h e m s e l v e s . It a l s o c o n t a i n s only an w - p o l e at o ~ ( s ) = 1, as is e a s i l y shown by the s a m e s o r t of a n a l y s i s as b e f o r e . M o r e c o m p l i c a t e d a m p l i t u d e s , such as p p -~ p p , could be obtained by m e t h o d s s i m i l a r to t h o s e in this s e c t i o n . H o w e v e r , a m o r e efficient r o u t e to such r e s u l t s would p r o b a b l y b e g i n by d e r i v i n g the g e n e r a l o p e r a t o r e x p r e s s i o n f o r the s c a t t e r i n g of N e x c i t e d s t a t e s . Such a f o r m u l a has b e e n found in the c o n v e n t i o n a l m o d e l [25]. In any c a s e , the m a i n p u r p o s e of this s e c t i o n was to v e r i f y s o m e of the a s s e r t i o n s of the p r e v i o u s s e c t i o n s f o r a non-trivial example.

7. C O N C L U S I O N

T h e dual m o d e l p r e s e n t e d h e r e b e a r s s o m e r e s e m b l a n c e to the r e a l w o r l d of m e s o n s , while still r e t a i n i n g the g o o d f e a t u r e s of the c o n v e n t i o n a l m o d e l . Its s p e c t r u m is a p p r o x i m a t e l y the s a m e as p r e d i c t e d by a naive m o d e l in which m e s o n s a r e m a d e f r o m q~l s t a t e s involving n o n - s t r a n g e q u a r k s only. Such a r u l e is r e q u i r e d f o r m u l t i p i o n s t a t e s in a dual q u a r k m o d e l , and it a c c o u n t s f o r finding w and fo t r a j e c t o r i e s and not q~ and f'. S t r e t c h ing the o b s e r v e d m i x i n g angle a bit, it could a l s o a c c o u n t f o r finding an ~?' and not an 77. In any c a s e , j u s t as with the e m p i r i c a l m a s s r u l e of sect. 5, t h e s e s t a t e m e n t s a r e only a d e s c r i p t i o n of what the m o d e l s e e m s to be doing and a r e not b a s e d on any p a r t i c u l a r l y d e e p u n d e r s t a n d i n g . A d e e p e r u n d e r -

109

A . N e v e u , J . H. S c h w a r z , F a c t o r i z a b l e dual m o d e l o f p i o n s

standing could be provided by a model f o r underlying q u a r k s , f o r example. Such a model might be useful f o r seeing how to make a sensible extension to SU(3). This is an i m p o r t a n t p r o b l e m , b e c a u s e a naive SU(3) extension of the p r e s e n t model gives a bad s p e c t r u m . The m o s t i m p o r t a n t f e a t u r e of our model is its gauge algebra. It is r e s ponsible f o r the a p p e a r a n c e of an e n o r m o u s n u m b e r of s p u r i o u s states. Specifically, it was shown in sect. 5 that only 3 of the 12 leading t r a j e c t o r i e s a r e physical. The new gauges f u r t h e r m o r e provide the key to obtaining a spurious state at the point w h e r e the p - fo t r a j e c t o r y c r o s s e s z e r o . An i m p o r t a n t p r o b l e m r e m a i n i n g to be solved is the explicit c o n s t r u c t i o n of the new gauge o p e r a t o r s . Such f o r m u l a s would both allow f o r a s i m p l i f i c a tion of the a l g e b r a in sect. 5 (and an elimination of the appendixX) and p r o vide a basis for a much b e t t e r understanding of some beautiful m a t h e m a t i c s that is at p r e s e n t hidden f r o m view. J u s t as the V i r a s o r o gauge a l g e b r a played an i m p o r t a n t role in m o t i v a t ing the c o n s t r u c t i o n of this model, so - we hope - will the new a l g e b r a i m plied by this model contribute to the c o n s t r u c t i o n of a still b e t t e r one.

A P P E N D I X

In this appendix we show that the ground state G decouples f r o m any state containing 4 or 6 pions. Using the notations of fig. 7a, we can write the amplitude f o r G going into four pions as: A (G--' 47r)

=

-Otp(12)-I -or ~r(13)-I -2k2.k 3 -2k3.k 4 4 J d x 1 dx 2 x 1 x2 (1 - Xl) (1 - x2) -2k2"k4 F (kl'k2) (k3"k4) (kl"k4) (k2"k3) × (1-XlX2) L xl ( 1 - x 2 ) + 1-x 1

-

(kl'k3) (k2"k4) 1 ~:~-2 J.

(A.1)

We then expand (1 -XlX2) - 2 k 2 " k 4 in a power s e r i e s and integrate over x 1 and x 2 to get

k4

k3

o.tt X2

k~

Xi

(a}

k6

k5

k4

o.ttfl X4

X3

X2

k3

k2

XI

t.k,

(b)

Fig. 7. a. The G ~ 41r amplitude, b. The G --~ 61r amplitude.

A. Neveu, d.H. Schwarz, Factorizable dual model of pions

110

A 4 : 4(k1.k2) (k3-k 4) ~ r(n+2k2"k4) n=0 n!F(2k2"k4) B[n- aTr(13) , - 2k3.k4]

F(n+2k2"k4) n=O n!r(2k2"k4)

× B [ n - Otp(12) - 1, 1 - 2k2-k3] + 4 ( k l . k 4 ) (k2.k3) ~

× B[n- O~T(13), 1 - 2k3.k4]B[n - otp(12), - 2k2.k3] - 2(kl.k3) x ~ r(n+2k2"k4+l) n=O n!F(2k2"k4) B[n- ot~(13), 1 - 2k3.k4] B i n - Otp(12), 1 - 2 k 2 - k 3 ] . (A.2) Next we e x p r e s s the B - f u n c t i o n s in t e r m s of F - f u n c t i o n s and u s e the f o l lowing k i n e m a t i c s : 2 k l - k 2 = [ ~ p ( 1 2 ) - n ] + [n] , 2 k l . k 4 = [- otn(13 ) - 2k3.k4+n ] - [n+ 2 k 2 . k 4 ] , 2 k l . k 3 = [ - Otp(12)- 2k2.k 3 + n + 1] + [otn(13)- n ] . It is then t r i v i a l to c h e c k that the t e r m s in s q u a r e b r a c k e t s in t h e s e t h r e e e q u a t i o n s g i v e r i s e to c o n t r i b u t i o n s which c a n c e l by p a i r s . In the a m p l i t u d e A 6 f o r G going into 6 p i o n s shown in fig. 7b, t h e r e a r e 15 t e r m s m u l t i p l y i n g the o r d i n a r y B a r d a k ~ i - R u e g g i n t e g r a n d [2]. T h e s e fifteen t e r m s a r e T1=8 T2 = 8

= 8 (kl"k6) (k4"k5) (k2"k3)xlx3

(kl"k 2) (k3"k 4) (ks"k6) ~-_~2~2)~ - x4 ) ,

78

(k 1" k 3) (k 2" k 5 ) (k 4. k6)XlX 3 (1 - XlX2X3) (1 - x3x4) '

%

T 3 : 8 (kl"k2) (k3"k6) (k4"k5)x3 (1 - x3) (1 - x2x3x4) '

T 4 8 (kl'k~l (k2.k 3) (k5"k6)x 1 :

-

x l ) (1 : ~ 4 )

(1 - x 1) (1 - x 3 ) ^ (kl'k2) (k3"k5) (~4-k6)x 3

rl°:-s

Tll

'

(k 1- k 4) (k 2. k 6) (k 3. ks)XlX2X 3 T5 = 8 _

T 6 = t~

(1 - x2x3) (1 - XlX2X3X4)

(kl"k5) (k2"k4) (k3"k6)XlX2X3 ~ - - XlX2) (1 - x2x3x4) ' (k I • k 6 ) (k 2 • k 5 ) (k 3" k4)XlX2X3

T7 = 8

'

(1 - XlX2X3) (1 - x2)

' '

( k l ' k 3) (k2"k 6) (k4"k5)xlx 3

'

(kl-k 4) (k3.k 6) (k2.k5)XlX2X 3 (1 - x2x3x4) (1 - XlX2X3)

T12 : - 8

(kl-ka) (k2-k4) (ks"k6)xl (1 - XlX2) (1 - x4) '

T13 = -8

(kX'k 5) (k2"k s ) (~3"~4)Xlx2xS (1 - XlX2X3X4) (1 - x2)

^ ( k l ' k 5 ) (k2"k3) (k4"k6)xlx3 T14 = -~ (1 - x l ) (1 - x3x 4) '

A. Neveu, J.H. Schwarz, Factorizable dual model of pions TI5 = -8

111

(kl.k6) (k3.h s) (~2"k4)xlx2x3 (a - x 2 x 3 ) ( i -

xlx2)

We then expand in power series all the factors of the Bardakci-Ruegg integrand which involve more than one x. This means expanding the following product: (1 - XlX2)-2k2"k4 (1 - x2x3)-2k3"k5 (1 - x3x4)-2k4" k6 (i - XlX2X3)-2k2"k5 × (1 - x2x3x4) -2k3"k6 (1 - XlX2X3X4),-2k2.k 6 L e t us call n24 , n35 , n46 , n25 , n36 , n26 the i n t e g e r s (running f r o m z e r o to infinity) i n v o l v e d in the e x p a n s i o n of e a c h f a c t o r . We then i n t e g r a t e e x p l i c i t l y o v e r Xl, x2, x3, x 4 e a c h t e r m of the s e r i e s , and u s e the f o l l o w i n g kinematics: in T1, T3, T9:

2 k l ' k 2 : [atp(12)- n 2 4 - n25 - n 2 6 ] + [n24 ] + [n25 ] + [n26 ] :

in T2, T 10, T12 : 2kl" k3 = [- Otp (12) - 2k 2. k 3 + n24 + n25 + n26.+ 1 ] + [n35] + [n36] + [ a ~ ( 1 3 ) - n 2 4 - n 2 5 in T4:

n 2 0 - " 3 5 - n36] ;

2 k l . k 4 = [-at rr(13) - 2k3.k 4 + n 2 4 + n35 + n25 + n36 +n26 ] + [n46] - [ n 2 4 + 2k2"k4] + [atO(14) - n46 - n35 - n25 - n36 - n26 ] ;

in T5, T l l :

2 k l . k 4 =[-atTr(13)-2k3.k4+n24+n35+n25+n36+n26+l ] + [n46] - [n24+2k2.k4] + [atp(14) - n46 - n35 - n25 - n36 - n26 - 1] ;

in T 6, T 13, T14: 2kl" k5 = [- Otp (14) + n35 + n46 + n25 + n36 + n26 + 1 - 2k 4. k 5 ] - [n25 + 2k~..ks] - ["35 + 9k3"k5] + [atTr(15) - n46 - n36 - n26 ] ; in TT, T8, T15:

2 k l . k 6 = [-at ~r(15) + n46 + n36 + n26 _ 2k5.k6] -

-

[n26 + 2k2-k6] - [n36 + 2k3-k6] [n46 + 2k4.k6] .

It is left a s an e x e r c i s e to the r e a d e r to s h o w that all the t e r m s in b r a c k e t s above, w h e n t a k e n with the a p p r o p r i a t e s e r i e s , c a n c e l one a n o t h e r by pairs.

112

A. Neveu, J.H.Schwarz, Factorizable dual model of pions

It s e e m s that s u c h a p e d e s t r i a n p r o o f of t h e d e c o u p l i n g of the g r o u n d s t a t e c a n n o t be g e n e r a l i z e d in p r a c t i c e to m o r e t h a n 6 p i o n s , o w i n g to t h e e n o r m o u s c o m p l e x i t y of the f o r m u l a s .

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