DUAL RESONANCE MODELS

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DUAL RESONANCE MODELS

Peter Goddard

To cite this version:

Peter Goddard. DUAL RESONANCE MODELS. Journal de Physique Colloques, 1973, 34 (C1),

pp.C1-160-C1-166. �10.1051/jphyscol:1973117�. �jpa-00215197�

P. GODDARD

DUAL RESONANCE MODELS Peter GODDARD

Department of Mathematics, University of Durham

1.- INTRODUCTION. - Giving a review of developments in dual models [I) presents problems, because the techniques that have been developed are not as wide- ly familiar as those of Lagrangian field theory, say.

This is particularly true this year when some of the most important developoents have had a highly tech- nical aspect. The field has advanced remarkably in the last few months and these developments have brought us to a very exciting point at this very moment. There now is the prospect in the near future of building consistent dual models for fermions, and perhaps, models which permit some direct comparison with nature.

The main areas in which our understanding has pro- gressed in the last year are the following :

i) the structure of physical states and the cons- truction of physical state projection operators.

(It is this technical development which initiated and facilitated much of the other recent progress)

ii) proof that the Pomeron singularity, generated by the model, is a pole (under suitable circunstan- ces) and proof that the consequent model for Pomeror?

scattering is ghost free.

iii) proof of the gauge propcrties of the dual fer- mion (Ramond) model, that it is ghost free if the

femion mass is zero, and that it couples consisten- tly to dual (Neveu-Schwarz) mesons.

iv) the detailed way in which the dual-model spec- trum and Born terms can be pictured in terms of a quantized model of interacting strings.

I shall explain something of the first three deve- lopments in this review. Unfortunately space (and time)limitations have not permitted me to deal with the fourth (see refs. 12-71).

Among the things these developments emphasize are i) the importance of critical dimensions for dual models : each model is really only satisfactory in a particular dimension of space-time, and ii) that the Pomeron singularity is an essential feature : it seems churlish to hope it will go away when the model goes to such efforts t~ make it consistent.

Before I can explain the solutions to problems lis- ted above, I shall first have to explain what the problems were and why they lead to significant ad- vances in building a dual theory of strong interac- tions.

2.- BACKGROUND : THE VENEZIANO MODEL. - ^{I start}

by discussing gauge conditions and the elimination of ghosts in the Veneziano N point function, the prototype dual model. In the operator formalism, which makes factorization evident, this N point function is defined as

where

(The integral is performed over a circle in the complex plane through the fixed points Za,Zb,Zc, the cyclic order of the Z's being maintained) and the function F is defined by the operator expression

evaluated in the Fock space defined by the annihila- tion and creation operators a? aV and position and

m' n momentum operators qt, pi :

[ a : . a

;

' 1 - ^{gV.' $1,". m,n > 0 ; [~~.P;I igWV} _{( 4 )}

aL10) = p p l ~ ) = 0 g = diag(-l,L,l, ...)

with V(Z,p) - :exp[ ip.Q(Z)j: (5)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973117

D V A L R E S O N K i C E MODELS C1-161

E q u a t i o n (1) t h e n d e f i n e s a Regge-behaved meromor- r a t o r s a i t . Thus r a t h e r t h a n t h r e e d i m e n s i o n s o f so- p h i c a m p l i t u d e which f a c t o r i z e s a t e a c h p o l e i n t o a

f i n i t e number o f terms, t h e maximum number needed b e i n g i n d e p e n d e n t o f t h e e x t e r n a l l e g s . a. i s t h e i n - t e r c e p t o f t h e l e a d i n g Regge t r a j e c t o r y . Independen- c e o f which t h r e e p o i n t s Za,Zb,Zc are k e p t f i x e d i s g u a r e n t e e d by t h e Moebius group i n v a r i a n c e p r o p e r t y o f F :

The p r e s e n c e o f t h e L o r e n t z m e t r i c t e n s o r g'V i n e q u a t i o n (4), ( n o t i c e we have one t i m e and a n unspe- c i f i e d , (D-1) s a y , number o f s p a c e dimensions), means t h a t we a r e n o t working i n a p o s i t i v e d e f i n i t e H i l b e r t space. Each t i m e component a: w i l l c r e a t e a negative-norm s t a t e :

We can o n l y a v o i d n e g a t i v e p r o b a b i l i t i e s i f i t i s n o t t h e whole Fock s p a c e which c o u p l e s a t t h e p o l e s b u t r a t h e r some s m a l l e r s p a c e o f " p h y s i c a l " s t a t e s which c o n t a i n s no n e g a t i v e - n o r m s t a t e s . The problem

i s one of r e c o n c i l i n g p o s i t i v i t y w i t h L o r e n t z i n v a - r i a n c e . I t a l s o o c c u r s i n a c o v a r i a n t t r e a t m e n t o f quantum e l e c t r o d y n a m i c s , where t h e F o u r i e r components o f t h e f r e e e l e c t r o m a g n e t i c p o t e n t i a l A a r e q u a n t i z e d a c c o r d i n g t o t h e c o n d i t i o n ,

[ a w ( k ) , a Y = ko gpv 6 ( k - k 1 ) . ⁽⁹⁾

Here t h e p o t e n t i a l n e g a t i v e norm o r g h o s t s t a t e s a r i s i n g from ao(kJt a r e removed by t h e L o r e n t z con- d i t i o n a . A = 0 which quantum-mechanically p r o d u c e s gauge c o n d i t i o n s

t o be s a t i s f i e d by t h e " p h y s i c a l " s t a t e s I*). Thus i n t h e d u a l model we need gauge c o n d i t i o n s t o remove t h e g h o s t s . Before we r e t u r n t o d u a l models l e t u s remember t h e s t r u c t u r e of t h e s o l u t i o n s t o (10).

k i s a l i g h t l i k e v e c t o r . We work i n t h e frame i n which k = (l/fi) (1,1,0,0,. . ^.) and u s e t h e n o t a t i o n

$ = (xO + X 1 ) ( 1 / n ) . C o n d i t i o n (10) becomes

+ t

and, a s [ a - ( k ) , a - ( k ) t ] - - ^{= O and} l a - $ , a (5) ^{1 1 0 ,}

i t f o l l o w s t h a t 14') must be b u i l t o f a i t (i ? 2 ) and a - t . The a - o p e r a t o r s do n o t m a t t e r . They commute w i t h t h e two ( i f we work w i t h D = 4 ) t r a n s v e r s e ope-

l u t i o n s t o (10) we h a v e e f f e c t i v e l y two. Any p a r t o f a s t a t e w i t h a' i n i t c a n be n e g l e c t e d . Such a p a r t i s null ^{: i t} i s o r t h o g o n a l t o a l l p h y s i c a l s t a t e s . T h i s f a c t , t h a t we o n l y g e t two p o l a r i z a t i o n s t a t e s f o r a p h o t o n (D-2 i n D d i m e n s i o n s ) i s f a m i l i a r . R a t h e r remarkably, t h e d u a l model a r r a n g e s a s i m i l a r t r a n s v e r s a l i t y f o r a s p e c t r u m i n c l u d i n g m a s s i v e p a r t i c l e s .

There i s always one gauge c o n d i t i o n i n d u a l models, and t h i s i s produced by t h e Moebius i n v a r i a n c e d e s - c r i b e d above. But one gauge c o n d i t i o n i s n o t s u f f i - c i e n t t o remove t h e g h o s t s produced by a n i n f i n i t y o f t i m e components. V i r a s o r o [8] p o i n t e d o u t t h a t i f a. = 1 t h e r e a r e a n i n f i n i t y o f gauge c o n d i t i o n s , which c o u l d p o t e n t i a l l y remove a l l t h e g h o s t s . T h i s means t h a t t h e s t a t e s t h a t c o u p l e a t a g i v e n mass

s h a l l s a t i s f y t h e c o n d i t i o n

and t h e V i r a s o r o c o n d i t i o n s

where

The L ' s c l o s e t o form t h e V i r a s o r o a l g e b r a

i n which t h e c-number term i s o f c r u c i a l i m p o r t a n c e . I t t u r n e d o u t t o be a h i g h l y non t r i v i a l problem t o d e s c r i b e w h e t h e r o r n o t e q u a t i o n s (12) and (13) have a n y s o l u t i o n s w i t h n e g a t i v e norm. The s t e p t h a t p r e - c i p i t a t e d t h e s o l u t i o n was t a k e n by Del G i u d i c e , Di Vecchia and F u b i n i r9) (DDF) who c o n s t r u c t e d o p e r a - t o r s A which behave l i k e a n n i h i l a t i o n and c r e a t i o n i

m

o p e r a t o r s 1101 :

i j * j t A j

IAm,Anl = m 6 i j 6m,-n

-n (18)

b u t g e n e r a t e o n l y p h y s i c a l s t a t e s , because t h e y com- mute w i t h t h e L n 3 s . The d e f i n i t i o n o f A : depends n o t o n l y on t h e momentum of t h e s t a t e b e i n g c o n s t r u c - t e d , h u t a l s o on a c h o i c e of a l i g h t l i k e v e c t o r k.

Then i r u n s o v e r t h e (D-2) v a l u e s t r a n s v e r s e t o t h e

momentum o f t h e s t a t e b e i n g c o n s t r u c t e d and k .

We c a n r e g a r d t h e DDF c o n s t r u c t i o n a s g i v i n g t h e com- p l e t e s o l u t i o n t o (12) and (13) t o g e t h e r w i t h t h e a d d i t i o n a l c o n d i t i o n s

We need a n o t h e r dimension o f o s c i l l a t o r s t o g e t t h e s o l u t i o n s t o (12) and (13) above. However when D t a k e s t h e c r i t i c a l v a l u e 26, t h e DDF s t a t e s a r e com- p l e t e i n t h e same s e n s e t h a t t r a n s v e r s e photon s t a t e s a r e complete - a l l t h e s o l u t i o n s o r t h o g o n a l t o them a r e n u l l p h y s i c a l s t a t e s and can be i g n o r e d . Thus we have t h e f o l l o w i n g no g h o s t theorem [11,12] : i f I$)

s a t i s f i e s t h e o n - s h e l l gauge c o n d i t i o n s (Lo-1) I+) ^{= 0 ,}

L,]$) = 0, n > 0 and D = 26,

where I f ) i s i n t h e s p a c e spanned by t h e DDF s t a t e s and Ins} i s a n u l l s p u r i o u s s t a t e ( o r t h o g o n a l t o a l l s o l u t i o n s of t h e c o n d i t i o n s ) .

For D < 2 6 we do n o t have t h i s t r a n s v e r s a l i t y b u t we can embed t h e problem i n t h e 26 dimensional one and deduce t h e a b s e n c e o f g h o s t s t h a t way. F o r D > 2 6 t h e r e a r e g h o s t s .

Before i t s o c c u r e n c e i n t h e g h o s t problem t h e magic dimension D = 2 6 and t h e i d e a t h a t two dimensions o f s t a t e s s h o u l d d e c o u p l e had a l r e a d y occured a s a r a - t h e r D e l p h i c p r e d i c t i o n o f L o v e l a c e [ 1 3 ] , a s a con- d i t i o n f o r t h e d u a l Pomeron t o he a p o l e . To s e e how t h i s p r o p e r t y came t r u e I move on t o c o n s i d e r d u a l l o o p s .

3. - LOOPS AND PRO.1ECTION OPERATORS. - The a m p l i t u - d e s I h a v e been d i s c u s s i n g up t o now a r e t o be r e - garded a s t h e Born terms i n a s t r o n g - i n t e r a c t i o n t h e o r y . These a m p l i t u d e s i n g e n e r a l a r e r e q u i r e d t o bc Regge behaved, a n a l y t i c w i t h s u i t a b l e c r o s s i n g symmetry, and L o r e n t z i n v a r i a n t . To o b t a i n u n i t a r i t y we add h i g h e r - o r d e r terms. F a c t o r i z a t i o n h a s been r e - q u i r e d s o t h a t t h i s can be done c o n s i s t e n t l y . Loops a r e formed by f a c t o r i z i n g N p o i n t f u n c t i o n s t o ob- t a i n a m p l i t u d e s f o r e x c i t e d s t a t e s , t a k i n g a t r a c e o v e r t h e s e e x c i t e d s t a t e s and p e r f o r m i n g a n i n t e g r a l o v e r t h e l o o p momentum i n a d d i t i o n t o t h e z - i n t e g r a - t i o n s ( s e e F i g . 1 ) .

Loops c o n s t r u c t e d i n t h i s Feynman d i a g r a m - l i k e way w i l l p o s s e s s t h e c o r r e c t n o r m a l - t h r e s h o l d - t y p e s i n -

g u l a r i t i e s . They may a l s o p o s s e s s o t h e r s i n g u l a r i t i e s n o t p u t t h e r e by d e s i g n , thrown up by o t h e r p a r t s o f t h e i n t e g r a t i o n r e g i o n .

Fokt Troce Planor Loop

I n a d d i t i o n t o t h e p l a n a r l o o p s i l l u s t r a t e d i n f i g u r e 2a t h e r e a r e a l s o non p l a n a r c o n t r i b u t i o n s c o n s t r u c t e d a s i n d i c a t e d i n f i g u r e 2b. I t i s t h i s non p l a n a r d i a g r a m t h a t p o t j s e s s e s a d d i t i o n a l s i n g u l a r i t i e s .

a ) Plonor Loop

X I - #

b) Non flanor loop

Fig. 2

I t h a s a Regge-Regge c u t i n t h e s - c h a n n e l , a f f e c t i n g t h e t - c h a n n e l h i g h e n e r g y b e h a v i o u r ( i n agreement w i t h Mandelstam's t h i r d double s p e c t r a l f u n c t i o n c r i -

t e r i o n ) . I n t h e t - c h a n n e l t h e r e a r e s i n g u l a r i t i e s w i t h vacuum quantum numbers l y i n g on t r a j e c t o r i e s w i t h h a l f t h e s l o p e o f t h e i n p u t Reggeons and a n i n -

t e r c e p t dependent on D. I n g e n e r a l , t h e s e s i n g u l a r i - t i e s a r e u n i t a r i t y - v i o l a t i n g c u t s . However, L o v e l a c e 1131 p o i n t e d o u t t h a t i f t h e dimension, D , o f s p a c e - t i m e were a l l o w e d t o v a r y , t h e s i n g u l a r i t y would p r o - b a b l y be a n i n t e r c e p t - 2 p o l e a t D = 2 6 i f t h e gauge c o n d i t i o n s removed two d i m e n s i o n s o f s t a t e s . T h i s , remarkably, i s e x a c t l y what happens when D = 26. As y e t one d o e s n o t u n d e r s t a n d

t h i s m i r a c u l o u s c o i n c i d e n c e o c c u r s , why t h e number D = 2 6 (and t h e t r a n s v e r s a l i t y of p h y s i c a l s t a t e s ) s h o u l d o c c u r both i n t h e c o n d i t i o n f o r t h e Pomeron t o be a p o l e and i n t h e proof o f t h e a b s e n c e of g h o s t s .

T h a t i t i s a p o l e d o e s n o t f o l l o w immediately from

t h e n o - g h o s t theorem. One h a s t o f i n d o u t how t o c a l -

c u l a t e l o o p s i n 26 dimensions e n s u r i n g . t h a t o n l y phy-

DUAL RESONANCE MODELS C1-163 s i c a l s t a t e s c o n t r i b u t e d t o t h e d i s c o n t i n u i t i e s . B r i n k

a n d O l i v e [14,15,16] showed how t o do t h i s e a r l y t h i s y e a r by c o n s t r u c t i n g a p r o j e c t i o n o p e r a t o r o n t o t h e

s p a c e o f p h y s i c a l (DDF) s t a t e s , which works o n s h e l l . The i d e a b e h i n d it i s s i m p l e and i t i s i m p o r t a n t be- c a u s e a number of o t h e r developments h a v e f o l l o w e d from it. C o n s i d e r t h e p r o j e c t i o n o p e r a t o r o n t o a g i -

1 2

v e n mass s h e l l p = 1 - N ; N i s a n i n t e g e r . T h e on- s h e l l s t a t e s s a t i s f y t h e c o n d i t i o n

S i n c e (L -1) h a s a n i n t e g r a l s p e c t r u m when p i s on some mass s h e l l we can c o n s t r u c t t h e p r o j e c t i o n ope- r a t o r o n t o mass s h e l l W , by u s i n g Cauchy's theorem.

(The i n t e g r a l i s o v e r a s m a l l wide a b o u t 2 x 0 ) . S i m i l a r l y we can c o n s t r u c t a p h y s i c a l - s t a t e p r o j e c - t i o n o p e r a t o r T i f we c a n f i n d a n o p e r a t o r E , w i t h a n i n t e g r a l s p e c t r u m such t h a t t h e DDF s t a r e s a r e

j u s t t h e s o l u t i o n s of

Then

I t i s e a s y t o s h o w - t h a t a s a t i s f a c t o r y E i s :

I f we w r i t e t h e N p o i n t ( t r e e ) a m p l i t u d e i n t h e v e r - t e x p r o p a g a t o r form(which f o l l o w s from (1)) :

(27) we s e e t h a t t h e n o - g h o s t theorem i s e q u i v a l e n t t o

where

and s i m i l a r l y f o r (m 1.

Brink and O l i v e [ 1 4 ] were a b l e t o show t h i s d i - r e c t l y ( t h u s c o n s t r u c t i n g a n o t h e r proof o f t h e a b s e n - c e of g h o s t s ) by showing

To do t h i s t h e y e x p r e s s e d E i n t e r m s o f gauge o p e r a - t o r s and t h e K n l s ( i m p l i c i t l y s e l e c t e d by t h e DDF s t a t e s ) :

The D ' s a r e f u n c t i o n s o f t h e K ' s a l o n e and form a c l o - s e d a l g e b r a w i t h t h e L ' s :

D 2

P,.Ln1 = (m-n)L,,,+n + n ( n - l ) 6 n , - m

Pm, ^{Dn] = -} (2m+n)Dmtn (33)

[Dm, D n l = 0

Then (30) f o l l o w s u s i n g t h e gauge c o n d i t i o n s s a t i s - f i e d by t h e ( o f f - s h e l l ) s t a t e s I+), 1 m) :

As T works o n l y on s h e l l , c a r e i s needed i n cons- t r u c t i n g l o o p s , w h e r e o f f s h e l l i n t e g r a t i o n s a r e p e r - formed. Brink and O l i v e [ I S ] circumvented t h i s d i f f i - c u l t y by u s i n g t h e Feynman t r e e theorem developed f o r d e a l i n g w i t h gauge c o n d i t i o n s i n Yang-Mills t h e o - ry. A c t u a l l y i t i s n o t e s s e n t i a l t o u s e t h i s theorem.

One may p r o v e d i r e c t l y t h a t t h e c o n j e c t u r e d l o o p i n - t e g r a n d h a s t h e r i g h t s i n g u l a r i t i e s . ( A l t e r n a t i v e l y , i t i s p o s s i b l e t o d e v e l o p a p r o j e c t i o n o p e r a t o r which works o f f s h e l l 1171, b u t so f a r i t h a s o n l y been

s u c c e s s f u l l y used f o r p l a n a r l o o p s ) . Using c l e v e r e x t e n s i o n s o f t h e t e c h n i q u e s developed f o r t r e e s , Brink and O l i v e 1161 w e r e a b l e t o p r o v e t h a t t h e Pomeron i s i n d e e d a p o l e i n 26 dimensions.

Having proved t h a t t h e Pomeron i s a p o l e , t h e n e x t s t e p i s t o d i s c u s s i t s f a c t o r i z a t i o n [ 1 8 , 1 9 ] , a n d , i n p a r t i c u l a r , w h e t h e r t h e r e a r e g h o s t s t a t e s i n t h e Pomeron s e c t o r . Having s o l v e d one g h o s t problem, one i s f a c e d w i t h a n o t h e r . At f i r s t i t a p p e a r e d t h a t t h e r e m i g h t be a g h o s t a t t h e second e x c i t e d l e v e l

[18], b u t a c a r e f u l a n a l y s i s by Ejeveu showed t h a t a l l was w e l l .

T h i s a n a l y s i s a l s o produced f u r t h e r o f e v i d e n c e o f t h e s i m i l a r i t y between t h e a m p l i t u d e s f o r t h e Pomeron and t h e v i r a s o r o - S h a p i r o (VS)mo&lr20,21]. T h i s model, which i s o b t a i n e d , r o u g h l y s p e a k i n g , by i n t e g r a t i n g

t h e Veneziano i n t e g r a n d o v e r a d i s c r a t h e r t h a n a c i r c l e , a l s o h a s l e a d i n g i n t e r c e p t 2. Both i t and t h e Pomeron s e c t o r o f t h e Veneziano model a r e f a c t o - ( a l . 4 t y E ~ 1") ⁼ o (30) r i z e d by two commuting s e t s o f a n n i h i l a t i o n and c r e a -

t i o n o p e r a t o r s l i k e t h e a ' s , a and a say. I n t h e VS from which (28) f o l l o w s by i n t e g r a t i n g t w i c e b e c a u s e

model, t h e gauge c o n d i t i o n s a r e v e r y s i m i l a r t o two

C1-164 P . GODDARD s e t s o f V i r a s o r o c o n d i t i o n s

(Lo -a> ^{lvs) = 0}

(where t h e Ln and En a r e made o u t o f a and a o p e r a - t o r s r e s p e c t i v e l y ) . The Pomeron gauge c o n d i t i o n s a r e r a t h e r d i f f e r e n t :

(L, - i n ) 1 ^{P ) =} o . ^{(3 6 )}

Another d i f f e r e n c e between t h e two c a s e s i s i n t h e p r o p a g a t o r s . The VS p r o p a g a t o r i s r a t h e r l i k e t h e c o n v e n t i o n a l one

w h i l e t h e Pomeron one h a s a n a d d i t i o n a l f a c t o r ,

O l i v e and Scherk [22] showed t h a t i n g e n e r a l t h e d i f - f e r e n c e s i n p r o p a g a t o r s and gauge c o n d i t i o n s e x a c t l y c a n c e l l e d . They proved t h a t

where t h e IPi) a r e Pomeron p h y s i c a l s t a t e s and T~~

i s t h e p r o p c t i o n o p e r a t o r f o r t h e VS model, cons- t r u c t e d i n t h e o b v i o u s way

T h i s means t h a t t h e Pomeron s e c t o r h a s t h e same spec- trum a s t h e VS model. I t i s s t i l l a n open q u e s t i o n w h e t h e r t h c s c a t t e r i n g a m p l i t u d e s a r e t h e same,

though t h e s h o r t a g e of a v a i l a b l e models makes i t v e r y l i k e l y .

4. - THE NEVEU - SCHIJARZ (PION) AND RAMOND (FERMION)

a. - Up t o now I h a v e been d e s c r i b i n g d e v e l o p - ments i n t e r m s o f t h e Veneziano model, p a r t l y f o r

s i m p l i c i t y . The c l a s s o f known models, which c a n a c h i e v e t h e d e g r e e of c o n s i s t e n c y I h a v e d e s c r i b e d f o r t h e Veneziano m o d e 1 , i s v e r y r e s t r i c t e d . The o t h e r model which h a s been most t h o r o u g h l y s t u d i e d i s t h e d u a l p i o n model o f Neveu and Schwarz [23,24]. T h i s model m a i n t a i n s t h e g h o s t k i l l i n g mechanisms o f t h e Veneziano model e n l a r g e d t o d e a l w i t h t h e l a r g e r

s p a c e o f s t a t e s . The model i s i n i t i a l l y f a c t o r i z e d i n a Fock s p a c e d e f i n e d by t h e a o p e r a t o r s o f t h e Veneziano model t o g e t h e r w i t h anticommuting b o p e r a - t o r s :

and t h e p h y s i c a l s t a t e s s a t i s f y gauge c o n d i t i o n s

and a mass s h e l l c o n d i t i o n

(Lo - 7 ) 1 19) = 0 . ^{( 4 3 )}

The L ' s a r e d e f i n e d s l i g h t l y d i f f e r e n t l y t o e q u a t i o n (14) and t h e L ' s and G ' s c l o s e t o form t h e a l g e b r a

T h i s model ( i n 4 dimensions a t l e a s t ) h a s a s p e c t r u m which i s more r e m i n i s c e n t o f t h e p h y s i c a l w o r l d . f u r t h e r m o r e , one can p r o v e e x a c t l y t h e same r e s u l t s a b o u t t h e a b s e n c e of g h o s t s and t h e t r a n s v e r s a l i t y o f p h y s i c a l s t a t e s [12,25,26], b u t t h i s t i m e t h e c r i t i c a l dimension i s 10. Again t h e r e i s a Pomeron s i n g u l a r i t y g e n e r a t e d w i t h t h e same i n t e r c e p t and s l o p e a s b e f o r e . Thus a g a i n we have t h e r e l a t i o n s

The r e l a t i o n a R(0) = 1 / 2 a (0) h a s o b v i o u s a t t r a c - P

t i o n s and p e o p l e have t r i e d t o a r g u e i t s model i n d e p e n - dence. T h i s model h a s t h e a t t r a c t i v e f e a t u r e t h a t t h e tachyon on t h e l e a d i n g Regge t r a j e c t o r y i s a b s e n t a n d i t i s p o s s i b l e t o f o r m u l a t e t h e v e r s i o n w i t h no t a c h y o n s a t a l l ( o u t s i d e t h e Pomeron s e c t o r ) . A l s o t h e model h a s a c o n s e r v e d G - p a r i t y - l i k e quantum num- b e r which e n a b l e s u s t o t h i n k o f one p a r t i c l e a s a p i o n .

One may c o n s t r u c t p r o j e c t i o n o p e r a t o r s f o r t h i s model a s w e l l . Here t h e Pomeron s e c t o r i s a l s o some- w h a t improved, w i t h t h e tachyon m i s s i n g on t h e l e a -

d i n g t r a j e c t o r y . U n f o r t u n a t e l y t h e r e i s a n e g a t i v e - G - p a r i t y Pomeron w i t h i n t e r c e p t 1. However, t h e r e a r e i n d i c a t i o n s t h a t theNeveu -Schwarz Pomeron may improve f u r t h e r when f e r m i o n s a r e t a k e n i n t o a c c o u n t . 1271

I t i s i n u n d e r s t a n d i n g f e r m i o n s t h a t t h e most

DUAL RESOti IANCE MODELS C1-165

r e c e n t p r o g r e s s h a s o c c u r r e d . S o o n a f t e r Neveu and Schwarz proposed t h e i r model f o r mesons t h e y and Thornr28,29] showed how t o p r o d u c e a model f o r f e r - mions w i t h t h e p h y s i c a l s t a t e s s a t i s f y i n g t h e gauge c o n d i t i o n s and g e n e r a l i z e d D i r a c e q u a t i o n proposed by Ramond [30]. The a m p l i t u d e f i r s t w r i t t e n down des- c r i b e d a d u a l f e r m i o n e m i t t i n g mesons (Fig.3). By

d u a l i s i n g t o t h e c o n f i g u r a t i o n of f i g u r e 4 t o f a c - t o r i z e a t a meson p o l e t h e y showed t h a t t h e meson s e c t o r o f t h e Ramond model was t h e Neveu-Schwarz model.

F i g u r e 4, o r , more g e n e r a l l y , f i g u r e 5, d e f i n e s a v e r t e x f o r f e r m i o n e m i s s i o n 131,321. Whereas t h e v e r - t e x f o r meson e m i s s i o n (used i n Fig.3) had s i m p l e p r o p e r t i e s w i t h r e s p e c t t o t h e f e r m i o n gauge condi- t i o n s , t h e f e r m i o n - e m i s s i o n v e r t e x o f f i g u r e 5 i s a v e r y much more c o m p l i c a t e d o b j e c t [ 3 3 , 3 4 ] . With some e f f o r t i t was shown t h a t f e r m i o r gauges c o n v e r t i n t o sums o f

Figs.3-4-5

meson gauges. But u n f o r t u n a t e l y meson gauges d i d n o t c o n v e r t i n t o sums o f f e r m i o n s ones. Thus t h e r e was a d a n g e r of b e i n g a b l e t o g e n e r a t e meson g h o s t s by f e r m i o n - a n t i f e r m i o n p a i r s . I n f a c t t h e f e r m i o n v e r - t e x changed a n " i n c i d e n t " meson gauge i n t o a sum o f

" t r a n s m i t t e d " f e r m i o n gauges and a sum o f " r e f l e c - t e d " meson gauges; T h i s r e f l e c t i o n and t r a n s m i s s i o n b e h a v i o u r had been o b s e r v e d between t h e Pomeron a n d Reggeon s e c t o r s o f t h e c o n v e n t i o n a l Veneziano model, where i t caused no problem 1341. Brink, O l i v e , Rebbi a n d Scherk [35] w e r e a b l e t o show t h a t i f t h e ground state f e r n i o n n a s s w e r e z e r o t h e t r a n s v e r s e meson s t a t e s would b e complete a t t h e p o i n t i n d i c a t e d i n

f i g u r e 5. T h i s m - 0 c o n d i t i o n had p r e v i o u s l y been deduced a s a n e c e s s a r y c o n d i t i o n f o r t h e d e c o u p l i n g o f meson s t a t e s , by Thorn1361 and a l s o by Schwarz

[ I ] , a s a c o n d i t i o n f o r t h e f e r m i o n n o - g h o s t t h e o - rem t o work. I n r e t r o s p e c t i t i s e a s y t o deduce i t a s a n e c e s s a r y c o n d i t i o n f o r t h e fermion s t a t e s t o be t r a n s v e r s e a t t h e f i r s t e x c i t e d l e v e l .

One r e a s o n t h a t i t was i m p o r t a n t t o g e t t h e f e r - mion e m i s s i o n v e r t e x c o r r e c t , p a r t i c u l a r l y i n r e l a - t i o n t o t h e meson s t a t e s which c o u p l e t o i t , i s t h a t i t i s n e c e s s a r y f o r t h e c o n s t r u c t i o n o f t h e a m p l i t u d e f o r fermion s c a t t e r i n g (Fig.6) .

Fig. 6

As Schwarz h a s p o i n t e d o u t [311, t h i s a m p l i t u d e i s i n t e r e s t i n g f o r t h e l i g h t i t t h r o w s o n t h e d u a l i - t y p r o p e r t i e s of o u r f e r m i o n s . Here t h e gauge p r o - p e r t i e s o f t h e f e r m i o n v e r t e x have a n o n - n e g l i g i b l e e f f e c t . F i g u r a t i v e l y s p e a k i n g , t h e y a r e r e f l e c t e d backwards and f o r w a r d s t o produce a c o r r e c t i o n f a c t o r . T h i s h a s been c a l c u l a t e d by O l i v e and S c t ~ e r k

[37]. They showed t h a t , f o r t h e a b s e n c e of meson g h o s t s , a f a c t o r ~ ( x ) h a s t o be i n t r o d u c e d i n t o t h e meson p r o p a g a t o r . Then

L -1

= ( 1 @ & ^A 1 ) . ^{( 4 6 )}

So a c a n d i d a t e f o r t h e a m p l i t u d e o f f i g u r e 6 i s

C o r r i g a n [38] h a s gone some way towards e v a l u a t i n g (47), and o b t a i n e d

{ ^{d x x} - a ( ~ ) - l ( ~ - x)k2k3 C(x) , (48) where

2 ) 5 D

C(x) = d e t ( 1 - A (X ) A (x)

k = O

(49) S a r e t e n s o r q u a n t i t i e s formed from t h e D i r a c ma-

k

t r i c e s ( n o r m a l l y y5, y', [y', y v ] , yPy5, 1 when

D = 4 ) , and

C1-166 P . GODDARD

~ ( x ) = d c t ( 1 - M(x) 2 ) (50) So, soon we s h o u l d known t h a t s i n g u l a r i t i e s a r e i n m + n + l t h e t - c h a n n e l o f f i g u r e 6, and c o n s e q u e n t l y what t h e

n + + ^{- 112 - 112}

Mm(x) ' (-XI 2 - m + n + l ( ) ( ) d u a l i t y p r o p e r t i e s o f Rarnond f e r m i o n s a r e . Schwarz (51) 1311 h a s s u g g e s t e d t h a t t h e y be i n t e r p r e t e d a s q u a r k s

a.>d g l u o n s and, i n a v e r y r e c e n t paper1411 shows how and (Am) and ( w T l m a r e t h e a n t i s y t m n e t r i c and sym-

t h e a d d i t i o n o f N c o l o u r d e g r e e s o f freedom t o t h e m e t r i c p a r t s o f M r e s p e c t i v e l y . Thus we have 3 un-

T 2 -1 Neveu Schwarz model p r o d u c e s a c r i t i c a l dimension known f u n c t i o n s , d e t ( 1 - ~ ~ ( x ) ) , ~ ( x ) and V (1-A ) V.

o f D s 1 0 - 2 N . Obviously N = 3 , D = 4 i s a n a t t r a c t i v e It i s p o i n t e d o u t t h a t t h e r e i s a t l e a s t one s i m p l e

r e l a t i o n between them [39] : p o s s i b i l i t y .

det - *) = 1 - I v T ( l . ~ ~ - l ~ I (52) 2 ACKNOWLEDGEMENT. - I am g r a t e f u l t o Edward CORRIGAN,

d e t ( 1 - A 2 ) David FAIRLIE, David OLIVE and John SCHWARZ f o r ex-

p l a i n i n g t h i n g s t o me and h e l p f u l c o n v e r s a t i o n s Nobody h a s a y e t shown how t o c a l c u l a t e t h e s e func-

t i o n s , b u t Schwarz and Wur40] found, u s i n g a computer t h a t

A(x) = ( 1 - ^{x ) 114} ,

a t l e a s t t o s i x decimal p l a c e s .

I n a comment a f t e r t h e t a l k C l a u d i o EBB1 drew a t t e n t i o n t o a p a p e r o f S. MANDELSTAM ("Manifestly d u a l f o r m l a t i o n o f t h e Ramond model" B e r k e l e y p r e p r i n t , August 1973) which c a l c u l a t e s f e r m i o n - f e r n i o n s c a t - t e r i n g a m p l i t u d e s u s i n g h i s s t r i n g formalism 171.

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