Meson-meson scattering

MESON-MESON SCATTER]ING

j. L.PETERSEN CERN, Genevc~, Switze,~'IJ~~d

NORTH-HOLLA~.~D PUB LBSH ING COMPANY" - AMSTE R OAiY~

PHYSICS REPORTS (Section C of Physics Letters) 2, no. 3 (1971) 155-252. NORTH-HOLLAND

MESON-MESON

SCATTERING

J. L. P E T E R S E N *

CERN, Geneva, Switzerland Rect~ived 1 June 197~

Ca..tents: 1. ~nttocluction 2. ]~asic f o r m a l i s m .: 1. A m p l i t u d e n o r m a l i z a t i o r s . C r o s s - s e c t i o n s ".2. U n i t a r i t y . P a r t i a l wavo e x p a n s i o n s ~.3. I s o s p i n conv~nticnz 2.4. I s o s p i n c r o s s i n g n m t r t c e s 3. T h e o r i e s of m e s c n - m e s o n s c a t t e r i n g 3.1. S o f t - m e s o n t h e o r y of low e n e r g y p a r a m , , t e r s 3.2. R e s o n a n c e couplings apd K S F I relation.~ 3.3. V e n e z i a n o t h e o r y 3.4. G e n e r a l p r i n c i p l e t h e o r i e s on -cw e n e r g y ~'n i n t e r a c t i o n s 3.5. The ~7 p r o b l e m 3.6. 2 l o s i n g r e m a r k s 4, I n f o r m a t i o n f r o m p e r i p h e r a l d i - m e ~ o n p r o d u c t i o n 4.1. P h e n o m e n o l o g i c a l f r a m e w o . ' . 4.2. O f f - s h e l l e f f e c t s 4.3. A b s o r p t i v e effects 4.4. C r o s s - s e c t i o n e x t r a p o l a t i o n 4.5. A m p l i t u d e p a r a m e t r i z a t i ) n 4.6. E x p e r i m e n t a l r e s u l t s ,m n~, scaLr ~ring 4.7. E x p e r i m e n t a l informaLion t : o m Kl~ p r o d u c t i o n 4.8. l~:xperimental r e s u l t s on K7 s c a t t e r i n g 5. I n f o r m a t i o n f r o m o t h e r s o u r c e s 5.1. Low e n e r g y i n e l a s t i c p i o n - n u e l e o n i n t e r a c t i o n 5.2, Low e n e r ~ r e l a s t i c p i o n - n u c l e o n i n t e r a c t i o n 5.3, E x p e r i m e n t a l i n f o r m a t i o n f r o m Ke4 d e c a y 5.4 M i s c e l l a n e o u s additional r e a c t i o n s 5.5. D i s p e r s i o n r e l a t i o n p h e n o m e n o l o ~ ' 6. Co1~clusion

157 159 160 160 162 163 167

Refe f e n c e s

24~

A T.mx.'-lc-x~r ; m p r . ~ a ~ n t ~ e l

.,,~ v,Y-,-,m~nt t h , ~ r a ~ t l r - m l

ran,!

nhonm~t~ntdno';oml

172 175 1:}1 193 193 193 199 2O3 207 20J 211 220 222 228 22~ 230 232 235 239 244

l:nt'~L~'!p~i:rt.

of p , ~ e u d o s c a l a r m e s o n - m e s o n i n t e r a e t U m s .

Single orders for this issue PHYSICS R E P O R T S {Section C of PHYS]. S L E T T E R S ) 2, no. 3 (19711 155-251.

Copies of this i s s u e m a y be o b t a i n e d at the p r i c e given below. All o r d e r s should be s e n t d i r e c t l y to tl'e P u b l i s h e r . O r d e r s m u s t be a( c o m p a n i e d by c h e c k . Single i s s u e p r i c e Hfl. 3 0 . - ,

$ 8.50, post ige i n c t u d e d .

* O~ leave of a b s e n c e from the Nie|~ Bohr i n s t i t u t e , Copenhagen, D o n m a r k .

l~!ES()N-MESON SCATTERI NG

;i57

1. INTRODUCTION The fundamenta.~ i m p o r t a n c e of m e s o n - m e s o n s c a t t e r i n g for p a r t i c l e p h y s i c s w?,s evidenced t o w a r d s the end ~f the fifties wben it w a s d i s c o v e r e d fi~at the beh"~riour of n u c l e a r f o r m f a , : t o r s r e f l e c t e d the p r e s e n c e of e s s e n t i a l m e s o n c!~.~l ir~eractions u n e x p l a i n a b l e ~,n t e r m s ol the n u c l e o n p r o p e r . At the , a m e t i m e , theoretieians s t a r t e d to r e a l i z e theft ~ s c a t t e r i n g s h o u l d be the s i m p l e s t e x a m p l e where truly r e l a t i v i s t i c e f f e c t s a r e c f d o m i n a t i n g i m p o r t a n c e . T h i s is so b e c a u s e the participating p a r t i c l e s all h a v e the s a m e ( s m a l l r e l a t i v e to the er~ergies inv01ved) m a s s so t h a t c r o s s i n g F r o p e r t i e s b e t w e e n p h y s i c a l l y d : s c o n n e c t e d t h a n uels give c r u c i a l c o n s t r a i n t s . Thi~ m a y be o p p o s e d f o r e x a m p l e to p~on ~ c l e o n scattering w h e r e n o n - . r e l a t i v i s t i c c o n s i d e r a t i o n s can a l r e a d y lead to an i ~ t e r e s t ing first ~ r d e r m o d e l M a n d e l s t a m , with his introd~acing of doubte dispersion r~/at/o~.s, s u g g e s t e d how to d e a l with t h i s s i t u a t i o n in a v e t ] g e n e r a l f r a m e w o r k based on a n a l y t i c i t y p r o F e r t i e s , u n i t a r i t y and c r o s s i n g . '['he b e g i n n i n g o[ the s i x ties then saw a m a s s i v e effort to u n d e r s t a n d low and m e d i u m e n e r g y m e s o n meson i n t e r a c t i o n in t h e s e t e r m s . It w a s soon c l e a r , h o w e v e r , that the prod~:cc,~ ~, models w e r e u n a b l e to a c c o m m o d a t e i~ a s a i i s f a c t o r y way ,,¢ii newly discovere,,i crucia! iea.tures oi ~n i n t e r a c t i o 2 : ~ an¢l f m e s o n s p r o d u c e d in high e n e r g y c o t i i sions and d e c a y i n g into two p i o n s , and a s t r o n g low e n e r g y s w a v e i s o s p i n 0 enhancement a s c e r t a i n e d t h r o u g h the a n a l y s i s of low e n e r g y e l a s t i c p~.on nucleon data, In 1996 a c r u c i a l new event tool~: p l a c e when W e i n b e r g showe¢i that the c o m b i n e d use of the partially c o n s e r v e d c h ~ r a c t e r of the weak axial current, the lir..k be~we~, the n0n-r.ero d . i v e r g e n c e o~ th~.s c u r r e n t and ~he effects; assoc~'~tecl w~th pi~r~ e x

158

J . L . PETERSEN

change, and the b r o k e n s y m m e t r y p r o p e r t i e s of h a d r o n i c i n t e r a c t i o n s r e f e r r e d to a s c u r r e n t a l g e b r a (sub~ection 3.1) lead to a lew e n e r ~ r b e h a v i o u r of the ~n amplitude which w a s f u n d a m e u ~ l l y d i f f e r e n t f r o m the o n e s p r e v i o u s l y considered. Working f r o m an unphysical l i m i t w h e r e the m e s o n m o m e n t u m g o e s to zero, or, a5 c o m m o n l y s a i d when the pion i s "soft", this a p p r o a c h is m o s t f r u i t f u l and i n t e r e s t i n g for low e n e r g y pionic a m p l i t u d e s . N e v e r t h e l e s s , the m o d e l turned out to have the f u r t h e r advantage t h a t r e a l i s t i c r e s o n a n c e s could r e a d i l y be incorpor a t e d into it ( s u b s e c t i o n 3.2). At the s a m e t i m e , generali:~ation to o t h e r mesonmeson c h a n n e l s was p o s s i b l e and a : t r a c t i v e . All of that is v e r y c o n v e n i e n t l y t r a n s l a t e d in t e r m s of ;eneziano models which s u m m a r i z e what may now be c o n s i d e r e d the o r t h o d o x pict a r e of m e s o n - m e s o n s c a t t e r i n g ( s u b s e c t i o n 3.3). T h e m a i n i n g r e d i e n t s which ~vill be l a t e r discussed are" (1) soft m e s o n theory of low e n e r g y p a r a m e t e r s ; (2) SU 3 nonet p a t t e r n s for n a t u r a l spin p ~ r i t y m e s o n s (those a r e the ones that couple to two p s r u d o s c a l a r m e s o n s ) ; (31 a b s e n c e of so-call~.d exotic r e s o n a n c e s - that is m e s o n - m e s o n resonances of i s o s p i n 2 ff n o n - s t r a n g e or of i s o s p i n 3 / 2 if s t r a n g e , the absolute vahe of the s t r a n g e n e s s being l i m i t e d to 1 - and f i n a l l y , (4) the l i n e a r i t y of the Regge t r a j e c t o r i e s a:ssociated with the p r o m i n e n t m e s o n r e s o n a n c e s such a s the p and ! m e s o n s . Thrc, ughout the p a p e r we shall s t r e s s t h i s p i c t a r e s u g g e s t i n g that it is sufficiently a t t r a c t i v e that it can be used a s a r e f e r e n c e f r a m e f r o m which deviations can be conveniently m e a s u r e d . This is yet an a p p r o x i m a t i o n and d e p a r t i n g f r o m it s e v e r a l a t t e m p t s have been made to p r o d u c e quantitative e s t i m a t e s . They u s e a v a r i e t y of d e v i c e s - hard meson or effective Lagrangian techniques, breaking s c h e m e s .for hadronic syn metries, etc. In m o s t e a s e s , no g e n e r a l c o n s e n s u s f o r the i m p o r t a n c e o r even dir e c t i o n of the~e c o r r e c t i o n s h a v e been r e a c h e d and we shall deal with those topics to a l e s s e r extent. The Weinbe rg and the V e n e z i a n o mod.~ls f o r m a l l y ignore u n i t a r i t y effects comi tetely and t h e r e has been a g r e a t confugion a s to the i m p o r t a n c e of that. Recently, however, a c o n s i d e r a b l e c l a r i f i c a t i o n a p p e a r s to have taken p l a c e thanks in p a r t to a r e v i v a l of the d i s p e r s i o n theory m o d e l s (and r e l a t e d ones), and a cons i s t e n t p i c t u r e of the 7rTr i n t e r a c t i o n below 900 MeW e m e r g e s which it is topical to review. The m a i n p r o b l e m s left o v e r a r e thosc a s s o c i a t e d with i n e l a s t i c i t y and in p a r t i c u l a r the n e e d for c o n s i d e r i n g o t h e r m e s o n - m e s o n c h a n n e l s . Section 2 should be r e a d in s o m e d e t a i l s i n c e it contains all the conventions and definitions and in p a r t i c u l a r g i v e s a d i s c u s s i o n of the v e r y i m p o r t a n t crossing p r o p e r t i e s f o r the different m e s o n - m e s o n a m p l i t u d e s (subsection 2.4). R e a d e r s i n t e r e s t e d in the soft m e s o n a p p r o a c h to the low e n e r g y amplitudes should a f t e r r e a d i n g section 2 c o n s u l t s u b s e c t i o n 3.1. The low e n e r g y region is then c o n n e c t e d with the r e s o n a n c e region via the K S F R r e l a t i o n s (subsection 3.2) and finally the p i c t u r e is c o m p l e t e d by w r i t i n g down Veneziano m o d e l s (subsection 3.3) the m o s t i n t e r e s t i n g and p r o m i n e n t p r o p e r t i e s of which a r e discussed. On the o t h e r hand, r e a d e r s m a i n l y i n t e r e s t e d in the p r e s e n t s t a t u s of the d:.sperslon t h e o r y m o d e l s couid d i r e c t l y r e a d s u b s e c t i o n 3.4. T h r o u g h o u t the p a p e r the e m p h a s i z is on a d e t a i l e d d i a c u s s i o n of assumptions and r e s u l t s w h e r e a s the s t e p s leading from one to the cti:er a r e m a i n l y sketched. The o r i g i n a l l i t e r a t u r e r e f e r r e d to will have to b e t r a c e d for a m o r e detailed der cation which the r e a d e r m i g h t like to s e . s p e l l e d out. The p r i n c i p a l b a s i s for e x p e r i m e n t a l m e s o n - m e s o n studies was l a i d very early by the C h e w .t,')w-Goebel s u g g e s t i o n for a n a l y z i n g high energy, p e m p h e r a l pro-

MESON-MESON SCATTERING

159

duction p r o c e s s e s w h e r e o n e - p i o n e x c h a n g e (OPE) is a l l o w e d b e t w e e n the i n c o m ing meson and the t a r g e t n u c l e o n , s o that th,e m e s o n - m e s o n s c a t t e r i n g a m p l i t u d e can be factored out in tb~ m e s o n t w o - m e s o n t r a n s i t i o n a m p l i t u d e off a nucleo:~. Strictly s p e a k i n g , t h i s fact~,ri,,g out is p o s s i b l e only f o r the r e s i d u e of the OPE pole which is s i t u a t e d at an e n p h y s i c a l point, but the a n a l ) t i c p r o p e r t i e s ot the re:..ction cross-section makes it possible in principle to obtai~ thi~ pole residue Iro~ the data in the physical region through analytic continuatmn, provided the, experimental accuracy is sufficiently high. Over the p..~st decade the statistics ,:,f available data has had a big !increa~o so that the goal of a model independent pole extrapolation has come ever closer. So far, however, it has been necessary tc make a number of additional assumptions on the production mechanism" dominance of OPE, neglect of off-shell corrections, etc. These have recently been much (riticized. In sect.'on 4, after an outline of the basic formalism needed to describe these processes, ve give a user oriented discussion of the relative practical imporlance of various points that have been raised in the literature, and provide a tool for experimentalists based on various ,~p to date mo,:'Lels. -fhi'~ review of tI,c techniques used to extract information on meson-meson scattex ing from meson production data could bc read independently of section 3. N, ,or h,! less, rc, method is free from theoretical prejudice and powerful methods ra~her be..~efit from them. Reference to the more theoretical section 3 is therefore very useful though not a

prerequisite. The existing experimental situation is reviewed in subsections 4.6-4.8. Valuc:-~ 0f masses and widths for decay of natural spin parity resonances into two pseudo-. ~calar mesons are constantly updated l)y the Particle Data Group Tar)los ~nd we shall not be concerned with those to any large extent. Rather we shall define experimental meson-meson ,,;cattering as; the stage at which bump hunting changes into p/,.a.~e shift p h e n ~ : n c n o l o g y . D e s p i l e the g eat diffic,,l~i~-~ ~n .n~rt~""i~" ................ .!~_ tic nr, phase s h i f t s now a r e a v a i l a b l e with b e l i e v a b l e d e t a i l s a s s o c i a t e d with thew. and wtriou~, a m b i g u i t i e ~ a r e g e t t i n g res:)lved. A l s o , b r a v e a t t e m p t s to e x t r a c t K: phase sh:~ts a r e t a k i n g place altho.lgh the d i f f i c u l t i e s a r e much g r e a t e r , r o d tf'l~ field is n~uch less matured. As for the KK system, c~ ~)l}ngs. to inelasIic chan-nels only can as yet be studied and little is known out~.~e i~he prominent resonance bumps (f, f', g, p, A2). One exception is the S* enhancement which promises to play a major: cole in sorting out the combind ,~v and KR behaviour of the s wa~,es near the KF, ti reshold. Fo" KK or for processes involving r~'s, next to nothing that we shall be concerned xith in this paper ~s known exp~.rimen[ally. [/high energy periphera! di-meson prc
-_

:

.

-

2. BASIC FORMALISM }

i In this s e c t i o n we c o m p i l e s e . r e r a l f o c m u l a s of g e n e r a l validity ~c~t~'-~,~i~,~', ~o i meson-meson s c a t t e r i n g .

"_60

J . L . PETERSEN

2.1. Amplitude normalizations. F o r the t r a n s i t i o n

C r o s s sections

the S rna,trix element Sfi and the invariant Fe?mman amplitude c~. i are thus rel~ted: ,,

c~fi

Sfl = 5fi - i ( 2 ~ ) 4 54(P~ - P i ) ~

.

(2.1)

1 We use u n i t s ~ = c = 1 t h r o u g h o u t . State v e c t o r s a r e n o n - i m , ariantt:¢ n o r m a i i z e d , (i[i) = (fir> = 1, s p e c i f i c a l l y f o r o n e - p a r t i c l e s t a t e s

= ( ~ ) z ~ ( p , .~) _. v ~pp, -. 9 p , w h e r e V-= 1 is the q u a n t i z a t i o n v o l u m e , P i ( P f ) i s the s u m oi all I o u r - n ~ o m e ~ t a in the i n i t i a l (final) s t a t e and U] is the e n e r g y of the j t h p a r t i c l e , j r u n n i r . g over both i n i t i a l and f i n a l p a r t i c l e s . F o r the b i n a r y r e a c t i o n

o+b'-'c+d we define s t a n d a r d invariantt:~ s =- ( p a +

pb )2

S + l

= ?rl~l

,

2+.,2+2

+ U

p5) 2 '

t =- ( l , d m C

u -= ( P c -

Pb )2'

+ ~

( m e t r i c " P~t q~ = Po qo - P " q ) " Many d i l f e r e n t n o r m a ' i z a t i o n s a r e iT, c o m m o n use variant amplitude

No define for e a c

,V tt'e ,r,-

5-.~ lco~

N

, r~

T h e n the d i f f e r e n t i a l c r o s s - s e c t i o n in tke c e n t r e - o f - m a s s dcr

4 N2 qf

2

"

dcr

-

"

h"2 a v

2

:

-

v

s y s t e m ( c . m . ) beccm~:s d~ a a

"I

w h c r e qi(qf) is the m a g n i t u J e of the inittaI (final) c.~n. ti r e e - m o m e n t v . m . The total e l a s t i c c r o s s - s e c t i o n is CreI - ~f

J

4v

d ~ dg d--~"

H e r e we h a v e d e f i n e d for l a t e r c o n v e n i e n c e a two-~)articie chameel l~(iex ¢f

=

i i io,.- two d i f f e r e n t p a r t i c l e s nt If) 1 } ~ f o r two i d e n t i c a l p a r t i c l e s in l i ) . I

2.2. Unitarity. Partial wave expcnsions The unitarity condition SS t = !I tg.kes the form

I'.:,,

MNSON-MESON

2 (2~) 4 ~4(pi

-

imC~fi

t;-t -.-~::?. ..........7~:.>-:-~

SCA~ TElliNg..,

:

) ~ tf,t) #

9gt- n

pf_ .p,)

(2,)8 2 4(

.....

j ~ (f,n) F o r i i) - t'-') t h i s g i v e s

the optical

-

~9 .7)

.S~ (i,n)

tl-:eorem

qv s lm NAi= f = ~ atotWe d e f i n e p : t r t i a l

wave

n

(u)

y •"~ f lt l S, )

amplitudes

'%~lfi(s, c o s O) = ~ ( 2 l where 0 is the c.m. For two-particle

(2.8) part~cle,~ oy

+ 1) ~4~, (.~.) P l ( c a s O?

scattering angle. tntermedtate states

(2.) 3

for spinlers

i ~. ~,) .,,~. \

o n l y (~°)

(2.) 3 ( ~'

St)

2N

i'

E

lm NA i = , ~ •

(~.)

.

. --sI., I ~x Ors " qu(AA~,4)

("'A , i ) '

Here ~ t s t h e ~.--,c h a a n ~ ~1 t h r e , ~ ; h o l d ,-~ lh~. m . ~ ' . , n i t u d e of th ,~ ;, chapr~c.-:i c r,,~ mon-,t~n~' aln a n d ¢ ~, i s d e f i n e d m eq . (. 2 . 6 ) . g~!~,lion ( 2 . 1 0 ) g ~ a V b e ~un~ter ( e r t a , ~ a .'~mdiei(:m:,.s i , :'.:,:i{:..:...,~,{ i,~.~ .: .... g~,'e r

l

tCAf~ I !m Ii,,, ........ ".... :;>t ~.,iq!!'-

a

.

:,

"-

"

F N a [ , * r N a l -t 2 ! + 1 1 ...... !~,:_~_i I .........5",! .... I z_j O ( s - s ) < v q V , L . k. aJ . .~Z.l

air v, =

TE

to g i v e

where 4.~ ',s t b e r<,al ph<:~se s h i f t . ,~,

: ZCO. & S

I

,' w~ ize~ ir~ ;~ar~i,. tta,~:

!r,~ ,,.~41.! = ~]=~21\ca! ' NAI'I; 2

,~,:,.,,

•q

(t..~

v h i c h is o n l y e q u i ~ , a i e n t to e q . t,, <'~..~n) fr:r s si: sf. Fo,' v i a . s : i c s c a t t e r i n g in a s t a t e of w e . l l - d e f i n e d i s o s p i a belc, w the f i r s t .i~,l'>~*~c . . . . . . ,. t h r e s h o l d

which '".~ay b e s o l v e d

"

Above

t h e lxrs.: ~r~etaski,.

ki,x,.::::,;,.,.<., ......~

~

I~$

J, L. PETERSEN

NAhI =._k_l 4-sr'~ e 2 i o { " 1 Are 2 2iq

(2.I4)

w h e r e ~{ i s s t i l l the ::eal p h a s e shift (not the p h a s e of elasticity,

NAI'I'

~nd ~ i s t,,e real in-

0 ~<~ < I.

(2.15)

Then we get f o r the total, the e l a s t i c and the i n e l a s t i c c r o s s - s e c t i o n s in a state with a n g a l a r m o m e n t u m l and isospin I

l,l

2'{r " ( 2q' + 1 )

w/-

~1)'

(rl'l el y(.2l (q,6-+I)[ 171e2i~ l,l

inel =

~(2~+ I)

11 2

(I - (r,{)2)

(2.1B)

(symbols (, q, etc., always pertain to initial state). F o r t w o - p a r t i c l e channels only, S m a t r i x e l e m e n t s may be thus defined:

sl,l

4s

for t , f ,

These a r e i n d e p e n d e n f the n o r m a l i z a t i o n , N. The unttarity condition StS =: l b e c o m e s a m a t r i x equation, h o w e v e r , only open chap nels a r e s u m m e d over. As an ilK~stration we w r i t e down the solution f o r the 7rTr -- n~, KK -" Kg and g~ --* KK s y s t e m assuming that no oth~.r channels couple, q :=rid k a r e c.m. threemomenta of pions and kaons r e s p e e t i v o l y and ~/(Tr~--. KK) ;s defined to be the phase of the amplitude NAl, l(rllr--.Kg). Then

(2.18) 2.3. Isospin conventions [ 8-10] Let [I, m, a) denote a set of p a r t i c l e s t a t e s t r a n s f o r m i n g a c c o r d i n g to isopia ! (13 -=--m ) ,

~I,~ and let I(I, m)*,a)denote the corresponding set of antiparticle states =

=

lo>.

Then the s e t !T m, •> defined by

[L m, a> - Cmm, ](I, m')*, a>

(2.!9)

will have the s a m e isospin t r a n s f o r m a t i o n prc~perties a~ [I, m, a) provided [8]

N E S O N - M ESON SCATTERING

Cram, = r ~ 6 m , . _ m

(_)l+m ,

I~?51 : 1,

t G3

i n d e p e n d e n t of m, m'

(2.20)

We :~hall t a k e

~

_

(_)'~J~-la

(2.21)

where Y5 is the hype~charge of ~. This choice is different from that of C~rruthers and Krish (71 ~ -I) [10] and of Neville (~I = I) [8! but conforms to De Swart's SU3 ionveniion [9] and has the advantage that for the sell-conjug-~te particl~3 ~ and ~,

~,,,> ,~- 1~, m~, [ ~, o> = I~, o>. Thus o u r c o n v e n t i o n f o r p s e u d o s c a l a r

Ix,½>

t,,"~

mesons becomes

= ix*>,

i~,.~>

-

I~°>

=

[,~o>,

ix,-½> = IKO>,

Iff,-½> = -IK-','

I',-~>--

l~'>

I,,°>

(I~' , 0> :

=

in>,

'tr~ ) ) .

With t h e s e c o n , , e n t i o n B w e l i s t t h e i s o s p i n d e c o m p o s i t i o n A ( g + ~ + - . ~t+~t+) = A ( c l - n - -" ~r-Tt-) = A (2)

(2.22)

f o : a few r e a c t i o n s

] (2.23)

:~ (. ~'rto -.. n+~,o~ - A ( , ~ - r r o - rr-Tto) = ½(A (2) + A (1)) A(n+u - _. ~ o ~ o )

= ~

(A(0)_ A , ~ ) )

i

l

A(7~+v- -. ~o~o) = ~ (2A(0) + 3A(1) + A(2))

A(K+~~ -'K+. +) = A(K-~--' K-u-) = A(3/2) A ( K * , - -' K+~ -) = A(K-Tr + --" K -rr+) = ~ (A:3/'/) + 2A(1/2)) 1~ A(K+~ - -' K°,~ ° ) = A(K -~+ -' I~°~°) = .7 . ,0) A(~-+~--~ K+K -) = ½ A(1)• + ~/ - ~~,~

t

(2.24)

(A(312) - A t l /~)) l

l

t2 25'~ ~.

,

a ( ~ + . ..... K ° l < °) : ~' . ~(~) - 4 ~ A ( ° ) 2.4. Isospin c r o s s i n g m a t r i c e s For p s e u d o s c a l a r m e s o n - m e s o n scribe s c a t t e r i n L of Y e 0 p a r t i c l e s by schannel'a

+b

s c a t t e r i n g w e s h a l l d e f i n e t h e s c h a n n e l to d e o n l y . T h e n tl-,e s, t a n d u c h a n n e l s a r e d e f i n e d

--'c + d

t c h a n n e l • d + b -" c + (7

(note particle order)

{2o2(~

u c h a n n e l " c + b - - ' a + d. The s c h a n n e l a m p l i t u d e A' (ab-~ cd) is s u p p o s e d to be an an'~}.:¢[ic [UL~CLIUt~ O~ (s, l: u) h a v i n g s ~ n g u l a r i t i e s g i v e n f o r ex~an:pie by ~ne L,a n u a u r u ~ , ~ ~ ~i ........... " .... tinued to the t o r u c h a n n e l p h y s i c d regiom~ it d e s c r i b e s a l i , ~ r crumb nati~':" o~ / or u c h a a n e l i s o s p i n a m p l i t u d e s , ..he c o e f f i c i e a t s of w h i c h a r e ~Zivea b~¢ ~h~: ~ rc~s5 lag m a t r i c e s . F o r a g e n e r a l d i s c a s s i o n of a l l t h e s u b t l e t i e s i n v o l v e d in finding that. see N e v . l l e [8]. H e r e w e l u s t s u m m a r i z e t h e r u l e s . Coe.sider t~'~ c t - ~ n n e l s 1 a n d 2 w i t h i s o s p i n s 11, I2 a n d w h e r e a e a t t e ~ t i o n is a.~ vet paid t~ p a r t i c l e o r d e r "

i1~

J . L . PETERSEN

channel 1 '

(ax) -.

(by), 11

channel 2" (ay) -- (bM), 12 . Th~n

(2,rt) where the crossing matrix is given by

'~x~ exZ(Xl,XZ) = (')ll+12+Ix-IY

lla ix l'-I (2/2 + 1) ~o

ib ly 12 "

Here ~ , ~y a r e the phases (el. (2.20)) pertaining to the anti~rticle (we call ~ . states with Y~ 0) of the p a i r (x, ~), (y, y). The last term is a 6 - j symbol [!2]. is a phase factor constructed accordin~ to the r u l e s of table 1. Then we h~ve for example

cles,

(2.~9)

~$u(Is,Iu)= (21u+l)(.)la+21b-lc-~(Y~t+Y~ ) IIb la Is1 Idiot

u

"

Table 1 Rules for constructing the phase factor tp of eq (2.28). F o r each of the four partmles, x,y, ~, ~', the two appropriate factors from the corresponding line in the table a r e selected. All of these eight factors are then mtdtiplied togetl'er to give ~ . Note that ~9 depends on ~-hether the p a r t i c l e s participating in the c r o s s i n g p r o c e s s a r e . 1s t . (beam in initial sta t e. forward in final state) or "2ndn (tar ~7t in initial state, bsckward in rival state). 1st

J'

2nd

Particle

Anttpart?cle

(Y >~0)

(Y < 0)

(-)~ +zrlz

I

(.)Ix

1

z

t_)Ib+~-ll

(_)ly

1

1

1

(-~

{-)/a +~ -I2

Using; similar expressions for Gut, ~ts, (~us, 6_'~ tu, the consistency m a y be checked by the relations

Qts = Q-1 st'

Cut = Qus" Qst

etc.

(2.30}

fin t~. (2.29} the convention el. (2.21) hae been used whereas the right-ha~d side of ~ . (2.28} is convention independent. In addition tc, '~sospln Rmplitude~, covariant amplitudes are often used. These are defined m terms of the covari~mt states. Thus for plons, I ~a), a = 1, 2, 3, ~e defined by

MESON-,Vi ESON ' ICATT E RING

l~*> = 2-x/2 For kaons

]~o>

(1~I> * ilx2>),

{(0)

1c 5

[.,r3>"

::

(2.31)

IK

+,K

for

K

[ K°,K° (2.32) K + K-

x c - Iv2 xK =

for

K = K°,K °

XT denotes tt,e h-aasposed spinor (row) a n d

are the Pauli matrices. Below we llst for each pseudoscalar binary reaction the crossing properties of the amplitudes in c o m m o n use as well as symmetry propertier following from Bose statistics and charge conjugation invariance. The lacking crossing matrices m a y be obtained from eq. (2.30). r~II~Tr

s c h a n n e l : rrn --, n n ; t c h a n n e l : 7rn ..... n n ; u c h a r m e l : n,'r ..... 7rn

C'st =

3 -5 ; -3 1

(3

su

=-

6

-2 2

3 3

( d . c [ T i b , a ) = A~ab 6cd + B6ac 5bd + C~r,dSbc

(2.33)

[lzl

(2,34)

(2.35) Sl

0

1

A(s,t,u) =B(t,s,u)

= C(u,t.s)

(2,36)

even u n d e r u ,-* t .

K~K~ 9===iMa~i

I/¢g ~'~ClTlKr~a'

1

1-1

4

= X~, {6ac A(+) + ½ [ * c , ~ ' a ] A ( - ) } ×K

,2.37) (2.38)

I66.

J . L . ~ETERSEN

A(+) = 6 - 1 / 2 A ~ 0) - ~t~s~ ,,~ a (3/2) + .4(1/2)s ) = ~-~° a (x3' -/'2u ) ~ A(_ )

+ Au(1/2))

(2,~0)

, ~.4(1/2) = ½A(t0) =~,"s - A s(3/2) ) =~(au(3/2) "Au(1/2) )

(2,40)

A(±)(s,t,u) = ±.a(.*)(u,t,s). KK~K s channel" ICg -. g g ; t c h a n n e l " ~ ' g -* g R ; u c h a n n e l " RK -" ~ g 1-1

Ao -_ A(O),

3

(2.4l)

(~.42)

A I - A(1)

$

S

(2.43)

A (Ol (s, t, u) = =~A (O)(s, u, t) .

~r~7~r77 s channel: ~

Qst

=

--* vq; t c h a n n e l : ~1~ -" ~,~; u c h a n n e l : ~q --' ~

- 1/~;

~su

=

(2.44~

1 (1)

<.~,c [ T{ ~,va> = 5ca A ,

(2.45)

A -z ,i s

(2.46~,

A ( s , t, u) = A ( u , t, s) .

s c h a n n e l : K~ --" K,3; t c h a n n e l : ,yr --" KK; u c h a n n e l : I ~ -~ ~rl

C s t = _ -]~: ~-6 ;

(2.47)

(?-su = -1

<,rlK, I TIT,,aK, , , , , = ~T, TaXK A ,

(2.48)

A - 3 - i / 2 A ( 1"s/ 2 ) .

(2.49)

A(s, t, u) = A(u, t s) w

KT/K~ s c h a n n e l " K~ -" Kr/ t c h a n n e l " Tr~ -- I ~ ;

Qst : - 1/'Ur2";

~-su~ T

0/K'i rlwK> = XK,XKA,

u c h a n n e l : K~ --* ~r/

(2.501

= 1 A = A(s b 2)

:2.51)

(2.5~)

A ( s , t, u) :- A ( u , t, s) rlTmrl ,,=,,m,..,=....m~

(?-st = C s u = 1. T h e (one) s c a t t e r i n g a m l : l i t u d e i s c o m p l e t e l y s y m m e t r i c a l

(2.53) in s, t, u.

I~ESON-M ESON SCATTERING

167

Replacing one o r m o r e ~ p a r t i c l e s with an ~7' l e a v e s the c r o s s i n g m a t r i c e s un.changed, h o w e v e r , s o m e of the s y m m e t r y p r o g e r t i e s get a l t e r e d . Our c o n v e n tions kere differ s l i g h t l y f r o m t h o s e of Canning [14] and Osborn [15].

3. THEORIES O F M E S O N - M E S O N

SCATTERING

In this s e c t i o n w e s h a l l t r y to s e t up a c o h e r e n t p i c u t r e of m e s o h = m e s o n s c a t tering as it m a y b e viewed at t h i s t i m e . We shall not c l a i m that ,t d e s c r i b e s t,~e truth in every r e s p e c t , but we s h a l l s u g g e s t that it a c c o m m o d a t e ~ a s u f f i c i e n t number ot ~ r a c t t v e i d e a s that d i s p r o v i n g it would be j u s t as i n t e r e s t i n g as having it conftrm,?d in d e t a i l . M o r e l i k e l y , s e v e r a l e s s e n t i a l p a r t s of the p~cture will survive and s e v e r a l will need m o d i f i c a t i o n s . We s h a l l m a k e c o m m e n t s on which features a p p e a r to be m o r e b e l i e v a b l e than o t h e r s . Those p r o c e s s e s involving q p a r t i c l e s will be f i r s t ignored and only in s u b section 3.5 s h a l l we c o m m e n t on s o m e of the d i f f i c u l t i e s that a p p e a r to be p e c u liar to that particle or else m a y indicate a m o r e fund:tmental difficulty with the approach as such. 3.1. 8 o f f - m e s o n t h e o r y o f l o w e n e r g y p a r a m e t e r s This p a r t i c u l a r t r e a t m e n t of low e n e r g y i n t e r a c t i o n s due to W e i n b e r g p r o c e e d s in several s t e p s [I 5, 16, 17]. 1) For the c o v a r i a n t a m p l i t u d e s of s u b s e c t i o n 12.~, one w r i t e s down the mos'~ . . . . ,,. ~i2 (th e v i r t u a l m sa s general o f f - s h e l l l i n e a r expansion in the v a r i a b l e,s s •, ~, squared of the ith m e s o n , i = 1 , . . . ,4). Imposing s e v e r a l c o n s t r a i n t s such as s + t + u = >,~J;q~ a n d s y m m e t r y p r o p e r t i e s coming f,'om Bose s t a t i s t i c s and C invariance, one-- z~,educes the n u m b e r of independent v a r i a b l e s and f r e e p a r a m e t e r s . We n;ake a few ¢ o m m e n t s on the s i g n i f i c a n c e of this !i.n_ear expansion. (i) It is the s i m p l e s t form c o m p a t i b l e with the r e q u i r e m e n t s of P C A C and c u r re~t a l g e b r a D e v i a t i o n s a r e c e r t a i n to e x i s t but t h e i r e v a l u a t i o n is m u c h m o r e model dependent. (ii)It can only be interpreted as the trunc~:t~on of a complete polynomial expansion of the amplitude iv a singularity free region in the space of all the var-iables. On-shell, this means in the triangle s ~So,

t
u ~
(3.1)

where So, t o and u o a r e the l o w e s t t h r e s h o l d s ( p h y s i c a l or p ,eudophysicaI) in the corresponding c h a n n e l s . The d e v i a t i o n f r o m the t r u e amplitude in the s channel P h y s i c a l r e g i o n w i l l be of the o r d e r ,'~

1 /~;

(So)~(s-sol''' ~(ao)3 ( s - s o)

for

IrnA

for

Re A

(3.2)

where a o is the s wave s c a t t e r i n g l e n g t h in the s c h a n n e l (possibly z e r o by Bose statistics - then the a c c u r a c y i s i m p r o v e d and d e t e r m i n e d by the p wave). C o n s e quently for not too l a r g e s c a t t e r i n g l e n g t h s the l i n e a r a p p r o x i m a t i o n may well be a sensible f i r s t a p p r o x i m a t i G n f o r the ~ e a l p a r t below the r e s o n a n c e r e g i o n . {iii) It i n f o r c e s " s m o o t h n e s s ~ in the f o r m of a b s e n c e of d aud h i g h e r waves. Then for the o n - s h e l l amplit~de the condition s + t + u = m~. = c o n s t a n t leads to

168

J.L. PETERSEN

a further reduction on the number of parameters thereby in general implying m m ru/~s for the scattering lengths. These are at least easy and convenient checks,~ models ignoring higher waves. 2) In the second st~p one employs the L S Z formalism to contract two m e ~ous of the total four-point function, using for the interpolating field the divergence of the axial current having the r~ght quantum numbers. This defines the one p~tleufar off-shell.,~orm for the amplitude which is relevant for imposing current a I ~ bra iuform~,.tion. It further involves introducing a P C A C constant F i normalized

by

where i is an SU3 index. For i = 7r this m a y be related to the 7r-g decay:

r

--

o 20c

Is. l

where G is the weak interaction coupling constant, 0 c is the Cabibbo angle and and ~ are m u o n and pion mass. Experimentally [18] this gives F~ "~ 0.66p. ~ 93 M e V .

m~

(3.51

The G o l d b e r g e r - T r e i m a n relation on the other hand says

Mg A

F~ = - - ~ r

~ 0.61g ~ 85 MeV

(3.6)

where M is the nucleon m a s s , gA -~ 1.23 and G2/4~ -~ 14.6. When qucttng theoretical p r e d i c t i o n s of low energy p a r a m e t e r s we shall give a " t h e o r e t i c a l uncertairAy" c o r r e s p o r d i n g to F~ = (89 + 4) MeV.

(3.7) 2 q~ = 0:

2

2

•

•

At the "Adler points" q~ = m~ ~ ~z), the m e s o n - m e s o n amplitudes vani~m when the particular off-shell'contihuation defined by PCAC is used. The Adler condition tikes the cormecticn between the on-shell and the off-shell beh~tviour ~jf the amplitude. However, taken alone ~,t provides n~ it,.fcrma Vion whatever on the on-shell behaviour. 4) At the " c u r r e n t algebra points" (CA points), q~ = q~.,/= 0, q2 = ink, k , ~,} current a l g e b r a s provide low energy t h e o r e m s that ~nay ~ i n t e r p r e t e d ott-shell using the Adler condition ~,,) t~ and the smoothness assumption (here linearity). T0 order qi" qj the amplitude at t h e s e points contains two termg invoi'dng the commutators 3~

The first t e r m is given by the usual SO3 x SU3 Cell-Mann c u r r e n t algebra. It is related to the vector c u r r e n t

fijk Vk Provided one car~ find an e,y~al.-mass channel for which this is one of the conserved c u r r e n t s , Vi, V2, ,'~|, 1:8, e derivative of the ~mplitude i s known exactly

MESON-MESON SCATT ERING

169

at th ~~ CA point. F o r p r o c e s s e s involving the rt, however-, this is not: p o s s i b l e and further a s s u m p t i o n s on the n a t u r e of $~U3 × $U 3 s y m m e t r y breaki,~g m u s t be made. The second t e r m a l s o r e q u i r e s knowledge on this b r e a k i n g , h~,~,ever, when no 7} partlcte is p r e s e n t it p r o v e s sufficient to a s s u m e n o n - e x o t i c i t y of the b r e a k i n g to flxthe (linear) a m p l i t u d e c o m p l e t e l y . Once a c o m p l e t e model fox" file o f f - s h e l l amplitude ( l i n e a r or not) is given, a number of m a t r i x e l e m e n t s of c o m m u t a t o r s of the f o r m (3.8) a r e fixed although they are mflmown a p r i o r i from the Gell-Mann a l g e b r a . This information m a y be used to throw f u r t h e r light on the n a t u r e of symmetry, b r e a k i n g which the model implies. It h a s b e e n fashionable to a n a l y z e this i n f o r m a t i o n according to the GellMann-Oakes-Renner (GOR) s c h e m e [19] a c c o r d i n g to which the Hamiltonian d e n s i ty is wrltte~ ~(*) = ~ o ( x ) - uO(x) - cuB(x).

(3.9)

,

Here, u', i-- 0 . . . . , 8 a r e s c a l a r SU 3 singlet and octet m e m b e r s oi a,: i r r e d u c i b l e (3,])$ (3,3) r e p r e s e n t a t i o n of SU 3 x SU 3 and p a r i t y . In fl,ls model, m a t r i x elements of ~ t e r m s ( l a s t t e r m in (3.8)) and m a t r i x element~ of n o n - c o n s e r v e d vector currents a r e e x p r e s s i b l e as l i n e a r c o m b i n a t i o n s of t~e quantities [:5]

where i and k a r e SU 3 labels for the m e s o n sta*:es. T h e coefficients of these depend on the p a r a m e t e r c of eq. (3.9). Analyzing a m p l i t u d e s for pions and kaons this why Osborn [15] obtained a c o n s i s t e n t value of m 2 ij~2 K 2m 2 + U2 _

c = -2J-2

(3.10)

which is the value also obtained by GOR [19], The fact that c ~ - ~ r a t h e r than c -~ 0 may be i n t e r p r e t e d as the s t a t e m e n t that SU 2 x SU 2 is a b e t t e r s y m m e t r y than SU3 (if o f f - s h e l l b e h a v i o u r s a r e roughly D.near). That point of view has r e cently been questioned on v a r i o u s g r o u n d s [20, 21], however, we shall not aiscuss this further h e r e . Below we shall give the r e s c l t s of the s o f t - m e s o n t r e a t m e n t of b i n a r y p r o cesses for pithS and kaons in what m a y be c o n s i d e r e d i n c r e a s i n g o r d e r of model dependence: (i) on-shell c o n s e q u e n c e s of l i n e a r i t y ; (if) consequences of the A d l e r conditions (PCAC), i.e., no new o n - s h e l l i n f o r m a tion; (iii) low-energy t h e o r e m s depending only on the GeH-Mann algebra: (iv) consequences of the n o n - e x o t i c i t y a s s u m p t i o n We take N -1 = 16~ in eq. (2.3). S c a t t e r i n g h:ngths and effective r a n g e s a r e d~ freed by

q~ +I cot 6 I= (a~)°I + ½ r~/q2 d O(q4).

(3.1t)

The :o.mection to the p a r t i a l wave a m p l i t u d e s )f eq. (" 7 4) is

2hr¢

q21+l cot

(:~.12)

1~

J.L. PETERSEN

where the f i r s t t e r m on the right f r o m unitartty is expected to be holomorphic in a region around the physical threshold.

~ e m o s t g e n e r a l linear f o r m of A (eq. (2.34)) tias no explicit dependence o n ~

(i=a,b,c,~:

A(~ ti=,q~•)

~,s.

= a ,~(t.=).

(i) on-shell this implies the important rule 2a ° (ll) P C A C

.

= 18~2a

(:,i:)

.

gives

A(/~2,/~2,/~2) = 0,

i.e.,

ot + 2~t2~ + ),•2 = 0.

(Ill) C A implies

a-~A(~2*~'0'~2"x)

F~

_-l_-

~-~

or

L --- -~( 2 a ° - 5a'~o) = 3p2a I = __E_ = (0.10÷=0.01)

8.~

.-1

(3.14)

wilere L is Welnberg' s mdversal length. (iv) non-exoticity of crterm:

A(U 2, 0,/~2) = C(t~2, 0, U2) = 0

er

~o/a2o= A°o( s = , 2 / 2 )

= o;

~o° -

A(s,t,u) =3-(s-~2) F~

-712

~z,=

(o.t7,o.o2)~-t (3.15)

A2o(~=~, 9-) =o; , oo = ( - T s .

= -½ L = o.'1) ~-1 ;

(-o.oso, c~.00s)~-1

r2o = (6.0 ± 0.6) t~-1 .

W e note that the well-known old s wave dominance theory of ~ Chew and Maudelstam [13,22] had

_~o/.~ ~ + 5/2

scattering by

(_~16)

correspo~Iding to an essentially c:onstant value for A (~-~),-~0). This apparently simple behaviour, however, rcqt=xres a very complic~tted off-shell extrapolation to cope with the current algebra informat. :n. Note in (3.!5) that not only are s wave scattering lengths small, but also effective ranges are large. K ~ - . Kn

For the amplitudes of eq. (2.38) one writes

M E S O N - M E S O N

SCATTERING

17i

A (') = A ' ( s - , , , )

(3.17)

where Pl, P2 are kaon momenta. (i) on-shell this gives the sum rule ta 1/2

,!I2.a3ol2

6mKl.l., 1 _~/~)

=

(3.18)

(it) PCAC implies

A+ , *

K: o,

+c

z)

(3.19)

Lo

(3.20)

(ill) PCAC + CA~"

'

-"

(iv) PCAC + n o n - e x o t i c i t y of a t e r m s

inK+ix

( a b s e n c e of I = 3/2)

D = 0 (i.e., P l , / ) 2 dependence redundant)

A(+)= ~ - ~1

"!

(s+u+22-2m2 K-2p

all2 /,,3/2

a~/2 = 0,

0

t '~0

=

2) ,

A ( - ) - 4 F "2 ( s - u )

- 2 ( s - w a v e d o m i n a n c e : a~/2/ao/2

(3.21)

-

1)

112 mK a 0 : 2 i n K + S t L = (0.':6 -. 0.02) ~ - 1

(3.22) .31'2

mK

_

N0

a~12

inK+IX L :-

r 0112 =

( - 0 . 0 7 8 ~ 0 , 0 0 8 ) t x -1

=

(0.011 ± 0.001) V.-3

(_5.0+0.5)~-I ;

r

~12= (17± 2) p. -I .

KK-+ KK (el. eq. (2.42))

A i = A + B(u+t) ÷ Cs

A° = A ' ( u - t ) ,

(i) No sum rule involving only physical scattering lengths exists.

(ii) PCACA+2Bm

+Cm

(3.23)

=0

(iii) PCAC + CAA' : 0 ,

a

=0,

~ - _

(3.24)

172

J . L . PF,TERSEN

(iv) PCAC + non-exoticity of a t e r m s AO_0 '

A1 = ~-~tt 1 , ÷ u - 2m 2 ),

a 1 -- - . ~m- K L = ( - 0 . 1 8 ~ 0 . 0 2 ) u -1 .

(3.25)~

'We s u m m a r i z e this s e c t i o n by s t r e s s i n g the i m p o r t a n c e of the two different p i e c e s of information p r o v i d e d by c u r r e n t a l g e b r a and the n o n - e x o t i c i t y assumption (bolLh i n t e r p r e t e d by P C A C ) : (;) G e l l - M a n n c u r r e n t a l g e b r a d e t e r m i n e s a d e r i v a t i v e of the a m p l i t u d e and fixe~ low e n e r g y p waves uniquely; (ii) the s t r u c t u r e of t~e a t e r m (i.e., the a m o u n t ol exotics) d e t e r m i n e s the zero patte:.m below t h r e s h o l d - in p a r t i c u l a r ratios of s wave s c a t t e r i n g lengths. This information is not obtained by the A d l e r conditions alone. In fact, several. a u t h o r s have c o n s i d e r e d the effect of allowing exotic co~atributions to the t e r m while of c o u r s e keeping the A d l e r z e r o [23-27]. T h e s e two types of i n f o r m a t i o n a r e c r u c i a l f o r fixing the a m p l i t u d e and any theory :must provide for t h e s e two basic types of input. In the following, we shall c o n s i d e r e x a m p l e s on how this c o m e s about. 3.2. l~esonance coupiings and KSFR relations MOving away from the low e n e r g y region we s t a r t this sectio~a by summarizing for e a s y referenc,~ the v e c t o r and t e n s o r m e s o n couplings to a p a i r of pseudos c a l a r s . Then as a f i r s t step t o w a r d s an u n d e r s t a n d i n g of the reLltion between the low e n e r g y and the r e s o n a n c e r e g i o n s we c o m m e n t on the d y n a m i c s behind the often d i s c u s s e d KSFR relation. F o r the v e c t o r - p s e u d o s c a l a r - p s e u d o s c a l a r (VPP) coupling we w r i t e

2Vp p = igOKK @;~(K½ r ~ K ) + gp== ~;~(~×~;~.~)+ igg,K=~Ur [K ~ 1 *

-

*

r K ~ ) + igK,K (-K/ [K p 0 ] +

+ [~uK]

+

(gwKKCO

+ gq~KK q~bt[1 ~ ~ g])

(3.2fi)

Here K = KO

,

K = (K-K°),

'~± =

(~1 ~ i~2),

~o = ;;3

(3.27}

and s i m i l a r l y for p and K*. The name of the field irfdicates the p a r t i c l e which is destroyed. Note that d e s F i t e the apparent difference: the pion convention is cons i s t e n t with that of eq. (2.31). Within an SU 3 f r a m e w o r k an w-~ mixing angle o c c u r s . In g e n e r a l , we define for physical fields X, X' X=X 8 sin0+X 1 cos0 X' =X 8cos0

-X 1 sin0

m ( x ' ) > re(x).

(3.28!

Then in a b r o k e n SU 3 s c h e m e [28, 18], w e have

gpTTTr = gpKK = 2g(1 + crl) ;

gK*K~ :: gK*K~?/v~ = g ( 1 - a 1 / 2 )

(3.2g)

F r o m the e x p e r i m e n t a l p and K* widths ~1 ~ -0.14

(3.30)

MESON-MESON SCATTERING

173

whereas in the limit of SU 3 s y m m e t r y c~1 : 0.

,qaJKK = ~f-33g stn t} ;

g:2KK = ~

g

cos

O.

(3.31)

Experimentally the co-go mixing angle is 0¢o~ ~ 40 °

(3.32)

whether a l i n e a r or a q u a d r a t i c m a s s formula is used. This is close to the ideal nonet value (SU 6 or quark model [29-31]) tg~ 0 = I / 2 .

(3.33)

With this value the ideal nonet coupling limit is elegantly wrltte.n for PV'v and TVV (T" tensor mesons, jPC = 2++) [31].

.~ : g2V T r { ( P 8 ~ . P 8 )

VII + gT[ 2 Tr {(/)8 *"3 ~ ""3 vP 8) Tt~ 9v}

+2xTr{(p8~ ~uP1)rgu}+y Tr{(P1~ ivP1)T9v}] with

1V4

J.L. P E T E R S E N

ei l,in

~1~ ..

_

q

=

"~

:i - m 2 + i~' q 3 '

3_ = 2

mp

rp/qp -~- 4n

,.

1

(3.37)

mp

and extrapolating to the threshold region }gnoring finitie width effects w e get (Z~-z = ZOo)

A ~z)

9~ rp

~ - " ~ ( t - , , ) = ....

= m--~--p

m3

(t-u).

Comparing with the c u r r e n t a l g e b r a prediction eq. (3.14), the KSFR relation (3.36) results,. : The ~alidity of this relation has been i n t e r p r e t e d as a s u c c e s s for p dominan~ and finite width corrections have b~=en considered to improve a g r e e m e n t with experiment [37, 36]. As d i s c u s s e d by Kamal [35], however, the dynamics behind app e a r s to be nowhere near p dominance in a straightforward meaning of the word. The point is that c r o s s e d ch~,nel p exchange gives a contribution to a~ at thresh. old about the Same magnitude as the direct channel r~monance. Interestingly enough, however, the kind of c (or e) particle predicted by Veneziano theory ( ~ he,It subsection) or the algebraic realization of chiral s y m m e t r y [ 38, 39] always "ccmspires" with the p to give back p r e c i s e l y eq. (3.36) in the absence of other coutributions. Consider for example an I s = 1, s = 0 d i s p e r s i o n relation (of. subsection 5.5). Saturating with narrow width p and ~ mesons gives =

=

+

-

+

g~ n~ m 2 . u

(3.38~ where

2

an~ the Lagrangian _P = g¢ zTr4..~r. ~ have been used. Expanding n e a r t = u = 2~ 2 anti comparing with the c u r r e n t algebra prediction, we get

1__ F 2=

~2n____~op 4 ~2 m2p +

m4

(3,.391

Ti2s r e p r o d u c e s eq. (3.36) p r e c i s e l y ff

"c = rap,

r~/rp = 9/2.

¢~.40i

This kind of c state f u r t h e r has the eff,~ct of exactly cancelling the crossed p contr~ibution to I s = 1, t = 0 dispersion relation, thereby partly justifying the above derivation. However, ~.lthough we thus have a suggestive link between eelS. (3.36) arid (3.40) none of the two can be derived at this level. Note that inconsistencies occur ff one t r i e s to account for c r o s s e d channel contributions a s in perturbation theory [35], r a t h e r than by using dispersion relations to avoid double counting.

MESON-MESON SCATT ERING

175

Equation (3.39) is the A d l e r s u m rule in its m o s t simple form [40] (cf. eqs. (3.14), {5.40)). In a phenomer;ological application of c o u r s e the high e n e r g y cor,.tribution (Reggeized p e×cha.~ge) should not be ignorecI (indications a r e that it is about 2 ~ [108]). A t this point we comment on an ofte.a cited p a p e r by Bowcock and John [4!1. The idea of t h e s e a u t h o r s is to c o n s i d e r the I = 1 s channel f o r w a r d amplitude supposed to be given by the p w a v e for 4~t2 ~< s <~ 1 GeV 2. In p r i n c i p l e one might perform an analytic continuation f r o m that region o~,.to the left-hand cut thereby obtaining in p a r t i c u l a r the i m a g i n a r y p a r t of the combined u channel I = 0, 1,2 contributions (el. (2.33)). I g n o r i n g I = 2 and taking the I = 1 p a r t f r o m the input, idorm.xtion on I = ~J r e s u l t s . In ref. [41], a s i m p l e par',.metrization of the p bump in terms 'ff known m a s s and width values is used. Using an ingenious p r e s c r i p tion for ca~'rylng out the analytic continuation, the left-hand cut [~ r e a c h e d a l though it is r e a l i z e d that due to the p r e s e n c e of e x p e r i m e n t a l e r r o r s the r e s u l t 0nly makes s e n s e on the a v e r a g e (see also s u b s e c t t o r s 4.4 and 5.2). However, the analytic form chosen did have an e s s e n t i a l l y z e r o im:tginary p a r t on the left-hand cut, so from the above we know that something like e¢;. (3.40) m u s t e m e r g e . We conclude that to p r o v e or d i s p r o v e the existence of a strongly coupled ~ state by this kind of a n a l y t i c continuation it is insufficient to exploit ~he presence of a p bump. In fact the object to be continued is the e x p e r i m e n t a l devialion of the shape from a pure B r e i t - W l g n e r f o r m (i.e., something with z e r o left-hand cut). We finish this s u b s e c t i o n by noticing that r e l a t i o n s s i m i l a r to eq. (3.36) may be der!ved in other r e a c t i o n s . T h u s , a s s u m i n g K* dominance o; the I = 1/2, K:7 p wave g~ves in a s i m i l a r way (g_pnn ~2

m2_4~ 2 - 4 m2z,

-

(3.41)

In the $U~ limit, inK. : rap. m K : ~, this is c o n s i s t e n t with ~ . (3.29) with al -- 0. However, the c o r r e c t i o n to SU 3 r e p r e s e n t e d by eq. (3..41) is . 3 0 ~ w h e r e a s the pheaomenological value of c~1 eq. (3.30) gi~,es a -35~ c o r r e c t i o n . T h e r e f o r e it does not seem p a r t i c u l a r l y useful to p u r s u e this g a m e much f,~rther. 3.3. Venezia,o theo~; In this s u b s e c t i o n we s u m m a r i z e and d i s c u s s propertie.~; of con~-cntional Veneztano models for b i n a r y p r o c e s s e s involving peons and kao:ts and postpone to subsection 3.5 any r e m a r k s c o n c e r n i n g the t r o u b l e s introduced by going outside this framework. L o v e l a c e [42] in nis fundamental p a p e r pointed to a p o s s i b l e deep connection between the dual t h e o r y of s t r o n g i n t e r a c t i o n s and c u r r e n t a l g e b r a s or other s y m m e t r y s c h e m e s : At thi_~ mnm-,nt it - p p e a r s quite u n c l e a r whether this dream will be r e a l i z e d in any c o m p l e t e sense. H o w e v e r , it has been d e m o n s t r a t e d that the most s i m p l e t r e z t m e n t in a very convenient way p r o v i d e s model a m p l i tudes thlt a c c o m m o d a t e rath-~r consister, tly: (i) the low e n e r g y a m p l i t u d e s of subsection 3.1; (ii} natural spin pa.rity m e s o n s in exchange d e g e n e r a t e SU 3 honer p a t t e r r s i~ reasonable a g r e e m e n t with e x p e r i m e n t . This acldevement in itself is i m p r e s s i v e , but r e c e n t l y the d i s c u s s i o n has centred rather oa the e x i s t i n g l i m i t a t i o n s p r e v e n t i n g f u r t h e r p r o g r e : ; s : (iii) unrealistic lower d a u g h t e r s , i.e., p r e s e n c e of ghosts and ~ack of general fac torization;

J.L. P E T E R S E N

176

[iLv) unreal~e:tic ~ f - s h e l l extrapolations to safely impose c u r r e n t algebra con. s~nts c r study s y m m e t r y patterns implied oy the model; Iv) lack of unRaribj and d~ffractive contributions. The new prediction.q of the model are(~iL) relations between m a s s e s and~ couplings of p a r t i c l e s with different (arbitrary) ~ i n at. the ~ r e n t level; (vili) existence of daughter states, the coupling of which one should not take too seriously; (v i~ii)s p e c i f i c behaviours of low .partial waves in exotic channels. ~,Ve now l t s t t h e Lovelace-Veneziano type models for ~T, rK and IC~ scattering. N~.~comprehensive derivation can be given, but we shall make brief ~omments on sewera! potnts and r e f e r t o ~ e o r l ~ n a l l i t e r a t u r e [42-46, 14] for m~ re detail as weir as to e ~ s t i n g reviews [6,47-50]. l)e~ing

~(s,t)

= -

~ 1 - <~x(s))F (1 - Cey(t)) F(1 -~x(S) - ~ . (t)) - = .,,(1 - o~(s) - o~y(t))B(1 -

C~x(S), 1. O~y(t)) (3.42)

we write (N "1 --16~) f o ~ ~ ~t.~ "~ ~

A(s.t,U~=f~Ti(Vpp(S,t)+ Vpp(S,u)

- Vpp(t,-))

(3.43)

for K~ -~ K~

AO:)(s, t, u) = ½.f~Tr(VK.p(S, t) ~ vK,O0,, t))

(3.44)

for KK -~ KK

A°(s, t, ~,) = f 2 ( V ¢ t

t, ,,) -- ffKX(V

, (t, u) - V~op(U, t))

,p(t. ,,) +

t)).

(s.45t

Several typas of input have been provided at this level. Thus ~ is Lhe degenerate p - f - ~ - A 2 trajectory, a,~ is ~le degenerate K*(890)-K*~(1420) t r a j e c t o r y , c~ is the degenerate ~(1019)-f'(1500) t r a j e c t o r y . These must be linear functions with equal slopes [48] from very general duMity requirements. The degeneracy stems Arom the following SU 3 or quark model type information (supported by experiment): absence of I -- 2 7rTrresonances implies p-f ~legeneracy, and presence of only one p state implies decoupling of f' from ~ Ab~;ence of I = ~KTr resonances implies K*-K** degeneracy. Absence of Y -- 2 KK resonances implies s e p a r a t e degeneracy of P-A2, ~r and cp-f' traJecLories. Figure 1 shows the Chew-FrauLschi plot for the natural spin parity mesons indic;tting that exchange degeneracy works well; also shown a r e daughter states (names f r o m ref. [47]), s o m e of which ~ v e e ~ e r i m e n t a l support ~'olack circles) and most of which have not (open circles). Equations (3.43-3.45) exhibit correct spi~ s t r u c t u r e s of the (real) resonance poles and the normalization constants may be determined from the observed p and K* widths. Also factorization of paren~ states is satisfied provided

MESON-MESON SCATT ERING

177

4

/

/

/ /

/ / I / l::p" / ,./ / .,/// /¢/// I'

A2

. /

../ / / /

/ /

0

¢

/

~

l

2

3

(Moss) 2 . a"

Fig. i. Cp, O~K, and ot(p as determined from eq. (3.47). Ex:,erimental states are indical:ed as M~2± 3./F in units {c~')-1. Dashed lines are first daughter ~. Circles indica:e prcdictecl resoaanc~,s lacking conclusive experimental evidence (black) or any at all (opon). I'~an~es from ref. [47].

2 2 /4K~ = / ~ J K K "

(3.46)

Lovelace f u r t h e r a s s u m e d t h a t t h e f o r m s (3.43)-(3.45) not only p r o v . d e d good ap.proximations to the t r u e a m p l i t u d e o n - s h e l l , but a l s o that going o f f - s h e l l they' cht~ose lo appx , x i r n a t e well the p a r t i c u l a r f o r m a e f i n e d by PCAC. With this bold assumption the A d l e r c o n d i t i o n g e t s p r e d i c t i v e p o w e r and a n u m b e r of s u r p r i s i n g results ~ o l I o ~ ~.

I) T r a j e c t o r y i n t e r c e p t s b e c o m e q u a n t i z e d a n d g

i

s-

with

m2 _~2 2(m2_ ~2)"

~"

,

C~K,(S ) : c~p(s) - A ,

~,o(s)

= ,:~K,(S) - La

~3.4'1)

178

J.L. P E T E R S E N

Those are the ones plotted in fig. 1 directly. Equation (3.47) follows only if each term in eqs. ~3.43)-(3.45) is required to satisfy the Adler condition separately, otlqerwise terms w~.th .. I~, PP and E ~_,~. in eq. (3.45) could occur [14]. These ~md satelIRe t e r m s a r e disc=rded ad hoc I42]. F r o m e l . (3.47) SU 3 ncnet m a s s formulae with ideal mixing angle (eq. (3.33)) ~ol3low. Thus for the JPC= 1"" and 2++ states [14,43,44]

.

A2

'

A2

in l~air a g r e e m e n t with experiment [30, 31, 51]. In addition of c o u r s e the linearlty gives r i s e to an infinity of further mass relations not contained in simple symmetry s c h e m e s , such as

The degeneracy of trajectories and the ideal slaglet octet mixing follows from day,lily and absence of exotics alone without using the specific forms (3.43)(3.45) [52]. A d e m o U o et al. [53] have used forms like el. (.~.42)in conj~nction with pion P C A C only, to suggest an even more general quantlzcd structure of intercepts. 2) The Euler function B(1 - cvx,1 - ~y) in eq. (3.42) is m u c h m o r e slowly varyinL: than Yxy itself within the triangle (3.1). Replacing it by a con~,tant, all the low ener~.T amplitudes of subsection 3.1 are precisely r~produced up to t ormalization. Th,.~ normalization is given by the low energy theorems, provided [45,46]

O

and or, allowing for the variation of the beta function,

! This m a y be considered a modified K S F R relation, however, in noticeably better agreement with experiment than eq. (3.36). Also the p versus c contribution ambiguity described in subsection 3.2 is here fixed and a model for the remaining contributions available. 3) Parent states factorize in general aria the condition that first daughters factorize is precisely the Adler condition eq. (3.47) [54]. Lower daughters do not factorize, however, for finite A (eq. 13.47)) and their predicted widths are t~rly meaningless. Examples on parent ana daughter width relaticns are (elastic widths always)

MESON-MESON £(, ATT ERING re,/r o = 9/2

(cf. eq. (3.40)),

rg,/r ° = 154-5/'112,

ro,/r °

F~¢/FK. = 6.6,

179

Ff,/rp = 9,/3/20

(3.52)

: ~/2

FK*~,/FK* = 1.0,

F K , , / F K , = 2.7.

For further d e t a i l s and c o m p a r i s o n with e x p e r i m e n t , see table 2 below. 4) Definite multiplet coupling p a t t e r n s a r i s e . Tbus for the s i g n a t u r i z e d a m p l i tudes v~v(s,t.u)

-

vxr(s,t)

•

(3.53)

vxy(~, ~)

the pole r e s i d u e at CZx{S) = J h a s the following spin s t r u c t u r e [14]" (l) at .'igM signature points

- 2(2c~' qtqf) a'(J- 1)' (2,/+ 1) C j

for

spin J

for

spin J

for

spin J - 1 .

-

1 .

(3.54)

(it) at wrong signature points

- 2(2ot , qi qf) J - l ' J a ' ( J - 1)1(2J- 1)

A

CJ_I

Here qi(qf) is initial (final) c . m . t h r e e - m o m e n t u m ~.~- . . . . . . . . . ~ ""~ ~". . . . . . .

zJ ~r (~) : c~ + " " A -~ -~[a'Em i2 ~ ~x(o) + 2% (o: - 11 B=- ~t

(3.5s)

a

For elastic s c a t t e r i n g the p a r t i a l wage ~.mplitude (~/j in the n a r r o w resf)nance a p p r o x i m a t i o n

R° Ij = F ' m j / q j

( F ' p a r t i a l width).

e 2ihtJ - l ) / 2 i q has a r e s ! d u e

(3..56)

In general it is r e l a t e d to the p r o d u c t of r e s o n a n c e coupling c o n s t a n t s to the initial {3.57~

RI(i --f) ~ gi(3)gf(J)(qiqf)d/mJ el. J Comparing eq$. (2.14), (3.54) anal (3.57) it is s e e n tha. the coupling cor, s~an s ~o,. the leading t r a j e c t o r y s t a t e s a r e independent of i n t e r c e p t s and m a s s e s and only depend on J and the quantum n u m b e r contents. As mentioned above they then just become the SU 3 couplings eq. (3.34) with ideaI mixing. This r e s u l t is certainly in--

180

J.L. P E T E R S E N

t e r e s t t n g although it is c l e a r that a f a i r amount of SU 3 and q u a r k model informa. tion has been conveyed to the formulae. A convenient way of visualizing the way this c o m e s about is given by the H a r a r i - R o s n e r d i a g r a m s [56]. At tile f i r s t daughter level mass splittings explicitly e n t e r the couplint':s via A and B e l . (3.55) and SU 3 is b r o k e n for these couplings in a p a r t i c u l a r w a y . Since still l o w e r d a u g h t e r s h a v e c o m p l e t e l y u n r e a l i s ~ c couplings, it is u n c l e a r what the significance of that i s , h o w e v e r ; Having established the principal s u c c e s s e s of the theory, we now c:~mment o~ some of the troubles that have recently been much discussed. I) Siv~e no Pomeron or diffractive contributions a r e p r e s e n t hz the theory it is not c l e a r whether the a g r e e m e n t of the low en¢,rgy amplitudes with the soft meson ones (or with experiment f o r that m a t t e r ) s h o u l d b e c o n s i d e r e d a s u c c e s s or a difficulty. B r o o k e r and T a y l e r [57] t r i e d (for #~) to add an ad hoc Pomeron f o r e (fixed at high energies by factorization in ~P, P P and PP) and estimated that corrections a s large as 500~ might occur. However, in a Regge absorption model, diffractive effects a r e introduced by multipiying partial wave amplitudes by a slowly varying factor giving a negligible low energy effect [47]. Also Kugler [58] noticed that (in ~ ) the amplitudes A~x=l (x = s, t, t~) at low e n e r g i e s satisfy unsub, t r a c t e d fixed x dispersion relatsons with no high energy P o m e r o u contribution (cf. also subsection 5.5). Since then all amplitudes can be e x p r e s s e d as a linear combination of these, the form (3.43) is justified and adding something would be double counting. The argume,~t of course ov.ly works at low ene~-gies and only when exotic channels a r e p r e s e n t . 2) A iarge amount of work has been devoted to studies of what the underlying breaking mechanism for hadron symmetries implied by the model is. It seems cle~tr by now that the model is not sufficiently well defined to m a k e such a study meaningful. W e shall briefly see at which ~evel the bre~kdown takes place. In ~ , the ~ term is given by A(0,~2,/~ 2) = -2 g 2

r(1 - ~o(0))r(1 - ~a(v2)) r(l- ap(0) - ap(g2)) = O(~ 2c~')

(3.58)

when C~p is given by eq. (3.47). This led Lovelace [42] and Yellin [59] to specul~,te that the deviation from SU 2 × SU2 chiral symmetry was sx~all and determined by (eq. (3.47))

ap(O)- I/2

= O(~ 2 a')

(3.,;9)

(in itself an impressive prediction) so that an analysis according to the G O R scheme [19] its suggested. Cronin and Kang [60], ho~,ever, noticed that in K~ -"K~ this is impossible with the form eq. (3.44). The point is that in addition to the two usual low energy theorems obtained by taking either the two p~oas or the two kaons. .off-~ho_ll, . . . . . . . . . . f;c~R pt.t, d l r t ~h~" th.,, ht,,,~ ¢,,,,,,.,t~,.,,,,-...... .. . ..,~.,-~...,~,.tI,. . . . ~_~..~ . . . . . . . ticular K~r amplitude to the two lines c o r r e s p o n d i n g to the one-pion soft and onekaon soft r e s p e c t i v e l y , have s i m i l a r s dependences. That is impossible .~or e l . (3.44) with m K - ~ z ~ 0. At f i r s t sight this may be cured s i m p l y by multiply ° ing eqs. (3.43)-(3.45) with off-shell f o m f a c t o r s , one for each p a r t i c l e [61,46 ], Xi(q 2) i = a, b, c, d with Xi(m 2) = I. I n t e r e s t i n g l y , these a r e then r e q u i ~ to l ~ e poles at the positions e x p e c t ~ l from ref. [53]. H o w e v e r , F a ~ a z u d d i n [62] and Ellis and Osborn [62] showed that whichever the form f a c t o r s chosen an inconsistency p e r s i s t s ~tlready with the Gell-Mann algebra. One c o n s i d e r s the difference between the values of the amplitudes at the. two off-shell points

MESON-MESON

(s. t. u) = (0, la2, m2K) ;

:~ :~

SCATTERING

(s. t, u) = (0, rn 2 , ~2 ).

By PCAC this is r e l a t e d to the m a t r i x e l e m e n t o~ a difference between twe c~ terms (3.8). Using the H e i s e n b e r g equations of motion and the Jacobi identity, that is given by the C e l l - M a n n c h a r g e ? l g e b r a . Using the low e n e r g y t h e o r e m s , Fn, F K and all form f a c t o r s m a y be e l i m i n a t e d to give

2 × { ~s vz*~ (s =,,K, 0) ~~ VK:~,p (s : 2 , 0)}- 1/2

(~.6o)

where .f+(0) is the well-known Kl3 form f a c t o r (see, e.g., ref. [83] and subsection 5.4). 2 his is an explicit function of the m a s s e s , x ;v (m2 K - t~2) c¢' cc SU 3 _ b r e a k i n g ,

], _= p.2c¢, cr SU 2 × SU 2 _ b r e a k i n g

and so

/+(0) = I +o(xy2) in contradiction with the A d e m o l l o - G a t t o t h e o r e m [6,..',]

£(o) -- 1 + o(x2). In KK ~ KK the i n c o n s i s t e n c y is s i m p l e r . With eq. (3.45) the low energy theorems read

1

3 V

2F2K - Ou

/O , , - - 2 )

r2

q~p, ," . . . . r~ , K K

(3.G:)

however, the two r i g h t - h a n d s i d e s a r e in the rat~.o 1.8- 1 ~nd this will continue to be true for any form f a c t o r m o d i f i c a t i o n ' both get multiplied by XK(0)XK(rn2:) [46]. What is *.he lesson to be l e a r n e d from t h e s e t r o u b l e s ? As we have noted, p r e s ent day Yeneziano t h e o r y even o n - s h e l l and in the t r e e - a p p r o x i m a t i o n is unable in a comprehensive form [65J to a c c o m m o d a t e p a r t i c l e s with r e a l i s t i c quantum n u m bers and t r a j e c t o r y i n t e r c e p t s . F u r t h e r , the o f f - s h e l l e x t r a p o l a t i o n s defined by eqs. (;t.43)-(3.45) whether with form f a c t o r s or not have u n r e t l i s t i c singularity structures [66] and a f u n d a m e n t a l l y sew a p p r o a c h is needed to i n c o r p o r a t e c u r rents in the theory. In view of that it is not s u r p r i s i n g that i n c o n s i s t e n c i e s occur. Perhaps it is m o r e s u r p r i s i n g that one has effectively t~ go to second o r d e r d e rivatives (as above} b e f o r e they become obvious. By patching with s a t e l l i t e s , the troubles may be moved a bit f u r t h e r away [67-69]. but nothing v e r y useful s e e m s to be l~arned thereby. 3) The m c d e l ' s lack of a n i t a r i t y was an obvious trouble from the very beginni~:~ and when c o m p a r i s o ~ with exper:me~-t t~ wanted at the phase shift le ~el, s o m e thing has to ~)e done about it. The dual multileop p r o g r a m has r e c e n t l y had very impressive f o r m a l p r o g r e s s but since it so f a r only applies to a ~rery u~Lrealistic model world, It has little p r a c t i c a l value as yet (see for example ref. [ ,0], and references therein). Since unit;,,rlty r e q u i r e s Regge t r a j e c t o r i e s to beco~ae c o m plex above t h r e s h o l d , the s i m p l e s t phenomenotogical p r e s c r i p t i o n has been to add to the ~'s of eq. (3.42), i m a g i n a . r / p a r t s above t h r e s h o l d and t<.ave :hem unchauged below. However, not only does that i n t r o d u c e a n c e s t o r s . ~,iolate c r o s s i n g

1~2

J . L . PETERSEN

L 120

9~

80

3O

,,,i

0

I

,

I

....

I

,

__|

._ ~

._

II-

-3C _

1

C,3

_

!

(~L,

t

I

Q5

Q8

~

1

1

(17

Q8 M.. (Gev)

I

Q9

~__.)=

tO

11

"12

Fig. 2. nn phase shifts 8o~ and 8o2 predicted by various models: LW - single channel uw~t~;.'ized Veueziano theory [47,247]; L - double channel mdtarized Venezisno theory [47]; ~r- :~d. [106], subsection 3.4; P - ref. [107], subsection 3.4; GMN - ref. [108], subsection 3. t. and a n a l y t i c i t y , but it still violates unitarity badly in that res:gnances like p a~d get equal widths although they have different co~iplings. L o v e l a c e ' s K matrix c evice o v e r c o m e s Rartly t h e s e troubles. F o r an e l a s t i c p a r t i a l wave amplitude, i t , n o r m a l i z e d a s ( e 2 i 8 1 - 1 ) [ 2 i q and for the c o r r e s ~ n d i n g p a r t i a l wave projectiot~ Of the Veneziano formula, IT/, one w r i t e s [47]

Fl(s) = Vt(s)/(1 +p(s) Vl(s)). This f o r m s a t i s f i e s e l a s t i c unitarity exactly provided p(s) h a s an imaginary !)art Im p(s) = -q

above t h r e s h o l d , Then infinities a s s o c i a t e d with r e a l axis p o l e s have been r e moved .and sensible p h a s e s h i f t s may be dr~wn (figs. 2 and 3). Since no genuin, ~ new i n f o r m a t i o n has been convp_yp.d tn th~ th~nry it iA nut tn h~ ~ t , t ~ l that r~r a m e t e r s like r e s o n a n c e m a s s e s and widths a t e improved o v e r the ones h~ the simple v e r s i o n . I~ p a r t i c u l a r , low e n e r g y c r o s s i n g is not well s a t i s f i e d sihc~ ~ )(s) has been taken independent of L T h e r e f o r e the [~tirly l a r g e p r e d i c t e d corrections to s c a t t e r i n g lengths neeci not i~e take~ s e r i o u s l y . I n / a c t s e v e r a l people have cons i d e r e d the effect of p a r t l y r e s t o r i n g c r o s s i n g by m ~ r e c o m p l i c a t e d Pl(S) functions t h e r e b y giving evidence that unitarity c o r r e c t i o n s to s c a t t e r i n g lengths ar~ pro~ably not m u c h m o r e than ~.0% [71-75]. A l s o (in ~ ) the A d l e r c o n s t r a i n t ma) reafily be r e - i n t r o d u c e d by putting pl(/~2) = 0, if d e s i r e d .

MESON-MESON SCATTERING

]

~3

'It

~z12

-r'

0

-~,14-

~- ~ 0,8

....

.~ 10

. . . . . . .

m.,,

~. 1.;

. L ~ ~

1.4

(GeV)

Fig. 3. Kff phase shifts predicted by tmi~arize~ ~'eneziano theory [47]. The K m a t r i x m e t h o d i s e a s i l y e x t e n d e d to t a k e i~lo a c c o u n t s e v e r a l two-bociy channels. L o v e l a c e thus c o n s i d e r e d the c o m b i n e d n7, -. =Tr, nn -. K K , KI{ -* KK system (cf. e q s . (2.17), (2.18)). T h i s h a s no u o t i c e a b h ; effect o~ the (Try) pha.'.e shifts below ~. 850 MeV, h o w e v e r , the p r e s e n c e of th.e S* (fi~. l~ f#hl~, 2~ :~r-: the i ~ threshold a t -.- 995 M e V c r u c i a l l y i n f l u e n c e s the ~ e h a v i o u r of 5 ° above that energy (see a l s o s u b s e c t i o n s 4 . 6 , 4.7). Table 2 is a ( s l i g h t l y u p - d a t e d ) r e p r o d u c t i o n of L , ~ r ¢ ] a c e ' s [4~zl with p a r e n t state p a r a m e t e r s l i s t e d f i r s t . T h e s e m a y s e r v e a s u.,eful r e f e r e n c e gaid~::~. As ~0r daughter p a r & m e t e r s the i m p o r t a n t c l a i m of the l h e o r y is that sucr~ s t a t e s should e x i s t . H o w e v e r , m a s s e s a n d c o u p l i n g s a r e l i k e l y to be s e v e r e l y di~;tort,~d. In par*:,cular p h a s e s h i f t s n e e d not r e s e m b l e B r e i t - W i g n e r ' s at a ' l . In fig,,;. 2 a~d 3, r e s o n a t i n g s w a v e s a r e t h u s l i k e l y to b e s t r o n g l y :.afluenced by (unreaiL~;t~c) lower d a u g h t e r s t a t e s &bore t h e r e s o n a n c e . T h e f o r n of exotic s w a v e s a~~d p wave,, is b, ;s m o d e l d~-pendent a l t h o u g h the a b o v e - r m . n t [ o n e d v i o l a t i o n of h w ertergy c r o s s i n g would r r e a n t h a t ( a b s o l u t e ) v a l u e s a r e :oo l a r g e , m a y be by ~¢ m u c t as 3(~0. F,~r f u r t h e r d e t a i l s on t h e e x p e r i m e n t a l situ~ lion s e e s e c t i o n s 4 a ~d 5 ns well as the P a r t i c l e D a t a G r o u p l i s t : n g s . 3.4. General p r i n c i p l e t h e o r i e s on low e n e r g y nn inte ~action wen o v e r h a l f the total t h e o r e t i c a l a c t i v i t y on me~, o n - m e s o n s c a t t e r i n g . . . . ',~s to be d e v o t e d to c o n s i d e r a t i o n s on low e n e r g y .~n ~catterir~g i~,. whi~ h th~ nuti0ns of c r o s s i n g , a n a l y t i c J t y a n d u a i t a = , t y c o n s t i t u t e (or a r e c l a i m e d ~y tl,~c ~uthors to c o n s t i t u t e ) t h e m o s t i m p o r t a n t g ~ r t of t h e input. Roughly h a l f o! th is ~ct ivit? takes the fo=m of m o d e l b u i l d i n g e m p l o y i n g t h e ab,~,,: c o n s t r a i n t s . ~ an a p proximate w a y , w h e r e a s t h e r e m a i n i n g half i s devote~l ~o w o r k i n g out r i g o : o u s consequences of t h e s e . On the m o d e l .¢tde w e f u r t h e r d i v i d e the d i s c u s s i o r a c c o r d i n g to the loll ~win~ ~ two s o m e w h a t r e c i p r o c a l p r o b l e m s "

184

J.L.

PETERSEN

Table 2 Updated versic~a of ruble 1, ref. [47], using p a r t i c l e d~ta group t a b l e s ~ kpril, 1971). Fo-f~rt h e r i n f o r m a t i o n On p a r t i c u l a r on t,', ~ and S*), see s e c t i o n s -. 5 arid 6. Name

J 1)

IC

blass (MeV)

Decay mode

.....................................

Theory

Expt

P a r t i a l width (McV) Theor)

Expt

128

125

~ 20

98 10

123 7

:~ 20 .~

p

1-

1+

749

765 • 10

a)

1-

0-

807

784

f

2+

0+

1289

126t e 1 0

Kl{

A2

2+

i-

1292

<1310)

KK

g

3-

1+

1671

1660 ± 20

Ka~

38 11

K*

1-

1/2

895

8q3

Err

3~

50

K**

2+

1/2

1381

1408

K;;

36

61

K***

3-

1/',',

1737

-

K~

24

(p

I-

0-

1009

1019

f'

2+

0+

1455

1514 •

£

0+

0+

59'o

5

0+

1-

752

962 -~ 5

-

p'

1-

14

1288

--

r,,-r KK

105 .~,;

~),

1

0-

1..,. 0

_

KK

22

f"

2+

0+

1661

-

7rTr KK

63 2O

,..: "

3-

0

1 ~: ; 3

-

KK

11

~C

0+

1/2

S99

-

K,'r

214

K*'

1-

1/2

1.380

-

K~

94

K**'

2+

1/2

1737

-

:~Tr

49

S*

0+

0+

911

1070 ± 30

~

15

-

¢~q

rr,=

-

KK 5

,~75J

K~

~7

7.2

1. ~.2

4 "~ 200 £. 14

.* 10

3.2 ~ 0.3

0 27

< 14

407

~.100

-.

< 150

(i) w h a t i s t h e m i n i m u m i n p u t n e e d e d in a d d i t i o n to t h e a b o v e g e n e r a l p r i n c i p l e s to c o r . , p l e t e l y d e t e r m i n e t h e low e n e r g y a m p l i t u d e ? {ii) a r e ct t r e n t a l g e b r a ( o r V e n e z i a n o ) t y p e a m p l i t u d e s c o n s i s t e n t w i t h unitari~y at a l l , a ,d if s o , w h a t i s t q e e x t r a p o l a t i o n t o t h e r e s o n a n c e r e g i o n (< i Ge':~ ? A n en~ r m o u s n u m b e r of a t t e m p t s h a s b e e n m a d e to s o l v e t h e s e p r o b l e m s , am ~,--~ .... , - o ~ , s s = o ~ , , , u ~ w = , ~ ua~ n u t b e e n l e s s e n o r m o u s . R e c e n t l y a con:sme r a b l ~ c l a r i f i c a t i o n ha~ t a k e n p l a c e , h o w e v e r , a n d it s e e m s tE~.;: r e s u l t s of g~nera! ph- s i c a t i n t e r e s t n o t c o n t a i n e d in t h e p r e v i o u s s u b s e c t i o n s , h a v e e m e r g e d ~h'~ i~ :;ue i n p a r t to t h e e q u a l l y l a r g e e f f o r t m a d e on t h e r i g o r o u s s i d e of w o r k i n g 0u: c o n s t r a i n t s a c c o r d i n g to t h e M a r t i n t , : o g r a m w h i c h ,re d i s c , A s s f i r s t ( f o r reviews, see refs. [76-79]). O n e c o n s i d e r s t h e f o l l o w i n g p r o b l e m . F i n d a n d d e s c r i b e in t h e m o s t cen~'(nien; w a y t h e c l a s s of f u n c t i o n s A ( s , t, u) s a t i s f y i n g t h e c o n d i t i e n s of c r o s s i n g , ar.alytiza n d D ) s i t i v i t y p e r t a i n i n g t o a rr~ a m ~ , l i t u d e (eq. (2.34)).

try

M ESON-M ESON SCATTE RTN(;

~:.;

The c r o s s i n g p ~ ' o p e r t i e s w e r e d e s c r i b e d in s u b s ~ , c t i o n 2.4. The anc.r~C~;city p r o p e r t i e s r e q u l r ~ . d a r e a s u b s e t o~ t h o s e fot.lowir~g f r o m I},~-. ,~iandels~am r e p r e s e n t a t i c n (for an e t a . ~ o r a t e l~s{ of r e s u l t s p;-oved :.,: a.-;~o:,.-:~t{~. fieldtheorv, s e e , e . g . , r e f . 178i): (.) for 0 s - 4/z 2 the ampl~.tude is r e a l a n a l y t i c in the [ p l a n e ca~. aJ.oai~, {.....:, -.,, and (4g 2, ~ ) , a n d s i r o i l a r l y (by c r o s s i n g ) f o r o t h e r p a i r s of (s, l, u ) ; in t h i s in terval t h e a m p l i t u d e .~zatJsfies a fixed s d i s p e r s i o n r e l a t i o n w i t h at m o s t two ,,,ubtr a c t i o n s ; (ii) ,or s ". 4g, 2 t h e a m p l i t u d e i s a n a l y t i c in s u c h a way th:~t p a r t i a l w a v e e x p a n sions c o n v e r g e w i t h i n t h e L e h m a n n e l l i p s e : l o c i a~ ~ -: 0 m~d t : - ( s - 4 ~ 2 1 a n d s e m i - m a j o r a x i s a(s) = J (s .~ 4/a2). Most o:f t h i s ~s c o n v e n i e n t l y s t ~ m m a r ~ z e d iu t h e w e t ~ - k n o w n M;_mdeis~a.m din... gram ~or , ~ s c a t t e r i n g (fig. 4). An s c h a n n e l p a r t i a l w a v e a m p l i t u d e is define~i ;,5 aa i n t e g r a t i o n of ,the a p p r o p r i a t e L e g e n d r e p o l y n o m i a ! m~,!liplied by t h e i avariarxt amplitude f o r {ixed s b e t w e e n t h e l i n e s x s == -1 a n d x s = +1 (x:; =~cos 0s), i . e . , ~ :: -{s-4~2~ a n d t = 0. F o r [ l x e d ~ :.- - 4 p a t h e n e ~ . r e s t s i n g u l a r i t i e s ~n c o s Os a r e ti-~: rand ~ c h a n n e l n o r m a l t h r e s h o l d s , t -: 4/z 2 (u :4tJ,2}. Ver~ m a n y p r o p e r t i e s of partial w a v e a m p l i t u d e s { p w a ' s ) b e c o m e o b v i o ~ s ',vl~en l o o k e d at t h i s way. F o r c a . ample s c h a n n e l p w a ' s a r e r e : , l a n a l y t i c f u n c t i o n s in t h e s p l a n e c u t a l o n g ( ....~., 0) and ( 4 ~ 2 ~). N e a r s = 4r~ 2

[m .Q(s) cc q4~+1 (?: ~" >',

and n e a r s ;, 0

~m f/{s)~: (-s) a/z !he coef~.c~en~ of the l a s t b e i n g ea'.~ily r e l a t e d to t h e cc~.,ssed c , : ~ e ! s,';:~:,~,ri::~le,,gths • ~. By t,.'.,siti?'itv is n-~.~nt t h a t ~'o~" pwa'.~ v ith '.leii~it<,' is~}si.,i~: ~,t:~.. t~:, {;. .): \

~

e

"N

....

~:

...........

g

7'

v,= a~"

ii?>, .......... ' q =: (.)

7

J

•

J . L . PETERS~TN

186

l m NAl'l(s) >~ 0,

s > 4 2

(3,~,4)

which i s a c o n s i 0 e r ~ b l y w e a k e r condition than the full u a i t a r i t v r e q u i r e m e n t (el. eq. (2.14)) vs w h e r e e = 1 / 2 f o r ~r~ --. rra a n d R

l°l/~I,l

- ~rtot'

el ~ 1,

s ~ 16g 2.

(3.6~)

T h a t a m p l i t u d e s do e x i s t w h i c h natisfy t h e s e r e q u i r e m e n t s e x a c t l y even with full Mana~ l s t a m a n a l y t i c i t y a n d with e q s . (3.65), (3.66) w a s s h o w n by Atkinson ([80] and e a r l i e r r e f e r e n c e s c i t e d : h e r e , s e e a l s ~ K u p s c h [81]). H o w e v e r , little's known a b o u t the low e n e r g y p a r a m e t e r s c o r r e s p o n ling to t h e s e m o ~ e l s and to wh, ! e::tent they e x h a u s t the p o s s i b i l i t i e s . By f a r t h e m a j o r i t y of the w o r k on the l ~ a r t l n p r o g r a m h a s t a k e n the f o r m of w o r k i n g out c o n s t r a i n t s a m o n g p w a ' s in the u n p h y s i c a l r e g i o n 0 .<. s ~ 4/a2 (for exc e p t i o n s , s e e R o s k i e s [82] and W a n d e r s [83]). T h ~ s e t a k e on a l t ~,orts of complic a t e d f o r m s and the s i g n i f i c a n c e of m a n y is not p r o p e r l y u n d e r s t o o d . Neverthel e s s , we s h a l l try to c o m m , . n t on the d i f f e r e n t k i n d s ~[ i n f o r m a t t ~ m obtaia,:d.

A. Constraints f r o m cross#~g s y m m e t , w aicme [ 8 4 - 8 ? ] N e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r a c a n d i d a t e set ,' Fa~rtla! wave ~.mpittudes Lo b e l o n g to a c r o s s i n g s y m m e t r i c a m p l i t u d e w e r e o b t a i n e d by Bal~chandran and NuTts [84] and given a v e r y s i m p l e i n t e r p r e t a t i o n b7 ! ~ ) s k i e s [85] z,nd Pasder a n t e t a ! . [86]. F o r the c o m p l e t e l y s y m m e t r i c a m p l i t u d e v','s, t, u) for ~o~o . ~0~0 for e x a m r , l e , one has with i n t e g e r N a n d P :.\,1 P !d::;~ :, ......,'* b'(',, /. e.') t

{}

e"

w h e r e the. i n t e g r a t i o n is o v e r ,he M a n a e i s t a m t r t a a g t e ~, . u O . . [ zce~r , ~n~aag (,e i n t e g r a t i o n s o v e r x s = c o s Os = 1 + 2 U ( s - 4 ~ 2) a n d s, r,-~!ations b e t w e e n poly.~om i a l m o m e n t s of the p w a ' s , f ~ ° ( s ) with g-~ 2N + I + P " e s u t t . T h e c o m p l e t e n e s s of t h e s e follow f r o m the com~.,leteness of the p o l y n o m , a . l s (in two v a r i a b l e s ) used o v e r the M a n d e l e t a m t r i a n g l e . T h e relatic, ns i n v o l v i n g s w a v e s o n t y , a r e 4k~ 2 f (s-4k2)(2fo°(S) o

4 2(

5f ( s ) ) d s : O,

s-4~2)(.~(s)

+ 2f;(.s)}ds

0. (a.a8

0

F o r f i n i t e L the n u m o e r of r e l a t i o n s i n v o l v i n g f / w i t h I ~ L o n l y . i n c r e a s e s with L t¢.._~.,u~L - 1 ,Lustre . . . . . a. .r.e. ~i~ej. . . . . ~ n e s e - R v s k i e s c o n s t r a i n t s " have b e e n p r o v e d pa~-tieul a r l y u s e f u l f o r b u i l d i n g and t e s t i n g low e n e r ~ , v,~ m o d e l s . A n o t h e r lnLeres~n.,, c o ~ e q u e n c e of c r o s s i n g alone is ~ a ~ fe~ g i v e n , wa:.:-s, d ~ a v e s (and h i g h e r ) f o l l o x s u r p r i s m g l y u n i q u e l y {87]. ~ the m a n y ~:omp~,°~.~ea r e l a t i o n s e n s u r i n g that we q u o t e a s e . ~ m p t e s (for r,o~ro -~,o~,o) •

(]o~0.5'-~2) - f : , , 1 . 2 9 ~ ) 2 . 0 5 0

..

": f2(0.572} -:: (fo(0.572

"~

| ' o "

- fo(i.087~} '1.43:':

'

!%IES( ,N-,M[qS()N S(?A'I'Tb;~:~Nt;

. .. t h e p o i n t i'~ ~h . ~ c~ w a v e : : r i s e s [ r o m a hi.,4 q u a l i w ~~[v, ~ .at a oLa .. c u r \ : A : a r e :,~ . .::," i,< .. t h e t - u d k , ' e c t i o n w h i c h b v . ( , r o~s s r. ~ ~ ir:~i.qies a 1>i,,-. ,t~-~,.....,~-,t~,",~., .. ~ the .,:,:ir~,ii,,~ $ a show up in t h e S w a v e s~m',,: ~hat i s j u s { [he ~ t v e r a g e ~,:,~ , (.<:,r i t:¢~:,{ .,, .

B, C o t z d i t i o n s f ( , l l o w i n g . ~ ' o m

~ b e s i t i v ; t y a l o r e [ S g , 8 9 , 761 Also h e r e n e c e s s a r y and s ~ , . t ] t c i e a t c o n d i t i o n s h a v e b e e n o b t a i n e d [ 8 8 , 891, h o w ever, t h e c o n s t r a i n t s have a ,nuch moze compliceted f o r m in t h a t e i t h e r iuJfinitel7 many p ~ ' s a r e i n v o l v e d o r e l s e o n l y i:,teqt4-e::it.ies c a n b~ o b t a i n e d . B a s i c a l l y t w o ~ p e s of i n f o r m a t i o n a r i s e : (i) sigr.,s on d e r i v a t i v e s a r e * ' i x e d , it, ~act t h e s h a p e o; the :s w a v e amplitade f~°(s) is determined i n O < s " 4 ~ 2 a~.m,-~st u n i q u e l y

~°r,o " ~o~o u<) t(; t.~.r.~:c,~ ~

mattons fOO + b , l o:ou ._. a-~o

a.-

0,

~. 3. . . 7(;.~

b rea.l

and t h e c u r v a t u r e i s p o s i u v e i76l; (t~ l ~ s l f i v i t y i m p r ~ , v e s t h e s r n , . J o t h n e a s t i , f c r m ; ~ i o i , a°°

,°

<

'~

I+2

'

,,i ,:uml).'~c.~.~ ; ~!~',.,> 17(.;, tJ~!;i h~r

L <-2

(~. <'',l

.

wi~erea~ ~ ° a r e t h e ~O~Oelastic scatLerlnglengths; [ r o m fig. 4, ~ ~s -~1~:,:-, ~ : thit ( 3 . q l ) h o l d s f o r s u f f t e i e n t l y l a r g e l. h o w e v e r , a n a l v t | c ~ t v a l u m : , i~-: ,~nablc, !:, v.y l o t h o ~ l a r g e I. T o p.,~ove r e l a t i o n (3.'111~ o a c ~s~:: '!'; I ::~=.:,."-,~". :.;~:.:.,:, '.::. r.'.,.£a

[/,.r l • 2

r"~°¢.s)

-

4 ~'

rr(s - 4 a 2 5

<~ . - 4

<~ /

i.~'

2'

~,.itk .

,

~.

2r .

the v a l i d i t y of t,,,~ t c k l o ] l o u ; ~ f r o m analTL~SitV, t~ ~, (ii) a~Jc.,,.e. T L e b a s i c new infor" ' b ; t ( t h a t f o l l o w s sinc.~. , m yL :' 0 [or / matiep, v ~ e l d i n ~ 3 . 7 1 ) i s t h e p o s ~ t i v i t y of " "~ 0,1,2imo!i.~s[m .f?o..-,0 0:. , al,.~o ~ . f. o r ~¢,.. ~ ! 4 ~ x 2 a n d ~ ..4,,?.. , .... .,~. ~ ~, :..,.:~~, i.,,~ Pi(cos ~t) " I " 0). Similar relations

f o r a n y ! m a y b e deriv+-,i.

two caoes" nc s u S . f i c i e n t ,:,e~." .of c o n s t r a i n t s

have been

obf.aine¢i.

";'i~. ~. b:.n,,,v:~ :-:,i<,,; ;.,~:

~anspa, ,.,~ f o r m . i

.~ l s ..v~.h 1 < L o n l y , ~ u~kno',.vn It is g....... ,ess"~bL+. <~ t h a t gi+'en tt.~e n e e d e d e f f o ~ t <~. m,,r~ " ::.,",?~,~'." ' ~ : " . : . rained. As f a r a s t h e cc, n s t r a ~ n f . ~ o~ c r o s s ; ' e a n a ~ a . ¢ c i t v a~d, o..,-:sifi~ , . .-"',~r~, ~,' is i m p o r t a n t ~o, r e a l i z e t h a t t h e s e a r e ir'.,,.~ i~,,nt ,.,,..~ri..<,.r. ~:~,. ;~ ~ : - : . • ",

-..

,.

'

~.

-'~)

0 s

C z.~ ( , s .

~" ,'~ i

<.

.

.

.

! . , ~ : .' " :

188

J.L.

PRTERSEN

T h i s m e a n s t h a t t h e y a d m i t low e n e r g y s a n d p w a v e s w i t h v i r t u a l l y a n y scattering l e n ~ h s a n d e f f e c t i v e , r a n g e s , s o t h a t t a k e n a l o n e t h e s e c o n s t r a i n t s ,vould s e e m to b e i n s u f f i c i e n t to r e s o l v e p h a s e s h i f t a m b i g u i t i e s i o r e x a m p l e . C n c .- s and p waves a r e g i v e n t h e c o n s t r a i n t s a r c a..,le t,) fix a l m o s t u n i q u e l y t h e c n , . p l e t e .'mDii~ucle, however.

T o g e t a b s o l u t e bour, fis on t h e l o w e r p a r t i a l w a v e s ,

it is t h e r e f o r e n e c e s s a r y to e x p l o i t a t l e a s t t h e n o n - l i n e a r c o n t e n t of u n i t a r i t y . B o u n d s c a n t h e b e o b t a i n e d ( h o w e v e r , o n l y b e c a u s e t h e a d d i t i o n a l p h e n o m e n o l o g i c a l a s s u m p t i o n of no s*~.te v.~ s t a t e i s m a d e ) { 9 5 - 1 0 0 ] . Thu.~ , o r tim C h e w - M a n d e l s t a m c o u p l i n g c o n s t , nt

- _ 2 A ( 4 9 2 / 3 , 4 6 2 , ' 3 . 41a2/3) and for the scattering lengths, one obtains - 2 . 6 " a " 17

no2:: - 7 . 1 ~ When c o m p a r e d

1,

aoO~. _ 7 . 5 U - 1 ,

with the c u r r e n t

all °

- 2 . 3 0 ~ -I

algebra predictions

cq. (2.15) a n d

u 2 -.

..........

96rrb,2

(0.0083,0.0008,

,

a ~ '~

'i ~

2

1;~,.~,4>

•

:~ ,ao:~

(0. 025 .. 0.0031 ~ -1 (,~. ~5i

t h e y a p p e a r to h a v e m a i n l y a c a d e m i c I n t e r e s t , a l t h o u g h t h e y dc~ r e p r e s e n ~ a ,'ons i d e r a b l e i m p r o , . , e m m ~ t '.96. 100] o~,e, l h e orig~r~al v e r s i o n 19~.~i. We now d i s c u s s p r o b e m (ii) a b : w e : a r e c u r r e n t a l g e b r a t y p e a m p l i t u d e s consistent w:th unitaritv? T h e quef.~.ion i.~ v e r y r e l e v a n ~ . We s a w in s u b s t , c~icm5 3.2 a n d 3.3 t'~at a,:,, sta~e with m~ :: rap. F~ :-: 9 2 F a w a s e i t h e r i n d i c a t e d o," .~red~ ted. If :::,w the sar:~, :-.i:,~t~l:,-:::i!::{<,~i :-,:,as:,," ~: w h i c h led t,~ tlt~' ]ZSFI{ '.'{'l::t:~m is :~.ppli~'~l. '.~,e {:~°~ o~ ao

i'e /~ - I m~

.9. . .I'p fa - I --:- 0.7 ~.L-~ 2 ~e,.,

,3.,C

w h i c h is f o u r t i m e s the c u r r e n t a l g e b r a v a l u e s o t h a t a m a j o r i n c o n s i s t e n c y i.~ conceivable. v e r y l a r g e t m i t a r . t v c o r r e c t i o n s to S e v e r a l a u t h o r s h a v e c l a i m e d that in ,,,¢:t ' W e i n b e r g ' s s c a t t e r i n g l e n g t h s w e r e n e c e s s a r y a n d ' o r t h a t 5° ~< 30 ° b e l o w i ,';eV ~.~ required i101,34,102,103]. S e ~ , e r a l of ~ n e s e t r e a t m e n t s i g n o r e p a r t i y or c,,.. ~p l e t e l y c r o s s i n g c o n s : r a i n t s a n ~ s o m e L a v e b e e n s h o w n [85] to b e e x p l i c i t l y :ace:,s i s t e n t . O n e is :eft with t h e i m p r e s s i o n t h a t t h e m o r e c a r e f u l l y , cre..,ssin~' is t r e a t e d t h e s m a r t e r a r e t h e u n i t a r i t y c o r r e c t i o n s a n d the l a r g e r i s t h e : w}d,h so that in f a c t Pe,.;Fp - 9 / 2 m a y w e l t be c o n s i s t e n t ! ! 0 4 : ~05]. At a n y r a t e it ca~'~ ae'~: . ~ thai at)::,roxim:-te be s a i d w i t h c o n s i d e r a b I e c o n f i d e n c e that m o d e l .~ ~ do e x :".~' , : ' :<~se!~ W e : n b e r g ' s a m p l i t u d e , s h o w a b r o a d e r o s o n a n c e and s a t i s f y u n i t a r i : v a n , 4 , ' ~ , s s o lng to a h i g h d e g r e e ol a p p r o x i m a t i o n l t O b - i 0 8 ~ . P e r ~ a p s t h e m o s t e l e g a n t a p p r o a c h is t h a t b a s e d on a P a d e a p p r o x i m a n t ' :-eatm e n t o[ e f f e c t i v e ( , a g ~ ' a a g l a , ~ ~ a t l s f v : n g curre~:~- a l g e b r a w i t h n o n - e x v t i c .~ :, :-ms e x p l i c i : , y [ 1 0 6 , 1 0 7 1 . O n e e x p l o i t s t h e ar~alytie p r o p e r t i e s of t h e p e r t a r b a t : o : : s e r i e s ~ x p a n s i o n t e t r a s e ? t o n l y a s f u n c t i o n s of ., and l but a : s o ~f t h e ca~p! '.'2: c o n s t a n t . T h e p' n : l o s o p a' y " is "~,,~ ' ~ ,,,,.. ":~ i~ff~.,rmation is e n o u g h to c a l c u l a t e the ,..,-

M F,S

~ N - h,l~,TSt ~N N C A

:'a~: i,~. ::<~,~] i~ ~ f u n d a m c , n t a l wa;,.

F I ' I':} e t N t ',

to build a !ladronic

,models h a v e s o m e e s s e n t i a l drawbacks" the ratio aot r i g h t , [ ~ : a j e c t o r i e s behave in an unsatisfacN~ry

t h e o . r y ks l i o t c t c , , i .

/5xi~il~E

of the p width and the f width is w a y a n d ,~x;)tic r e s o n a n c e s o,,--

curat higher energies. However, as Jar as the low energ!; region goes it has been shown t h a t b a s i c a l l y the same unitary crossing ,~ymmetric current algebra-like amplitude emerges whether one starts w i t h t h e r m o d ~ ~] I t 0 6 ] a p r l p r e d i c t s the p {.x c ~ r v ( ' s in f i g . 2) o r o n e s t a r t s with an eleme:'~arv Ya~i:-Milln O fiehi coupled ~o a c~,n:.~erved c u r r e r d

[ 1 0 7 t, a n d

predicts

a .r (p c v t r v e s

iq f i g . 2).

N , ~ ~ ~h.~ ~'~"

0 dominance dynamics only makes s e n s e i n t h e w e a k c o u p l i n g l ~ m l f (~:1. su{~,,-;v,.~:0n 3.g), b u t c o n ~ i s t e n t t y renormalizes as the caupling is :ncreas(,d. We i ; n a l l y d i s c . , . s s t h e fi~-nt q u e s t i o n raised tn this subsection" uni~luo~-~ess ai ~he low. ~neigh, ,'~, ~r,-r amplitude. It is ;m i m p o r t a n t f a c t tha, t W e ~ , ~ b e tmttaritv. However, on-shell o a 2 ) a n d w e w, o uld like and a e

Ft.~~' S &

mpl{tud(,

i s c o , ~ s i ~. t e n (.

WeinberI: m - e d i c t s tee', p a r a n , , c i ~ , , r s { s a y ,
ty c o n s i s t e n t w i t h u n i t a r i t y so that a two-dimensional tim C h e w - M a n d e l s t a m tr~a'm~nt [ 2 2 ] it i s i m m e d i a t e l y since

a~ l e st <5 - ' 2 - d i m e n s i o n a l

wi~!: . .l,)'x ~,,,~, ,',, , . . ,'

for a large range ol decade of confusion

a m b i , < u i t v pr~,v;t{1 ~. ~:'r,-,,~, c l ~ , : ~ r t h , ~ :1,,. i~ .,,~.~,,,~. -. . . . . . . . . . ::m~o:~ "'-: *~, }"~'

a'

there

now

l..mts ,~ha~. in a , ' e r t a i ~ s m ~ s e ',h,_: a m b i f : u ~ t v i~ ,,,~.~,...:,~!a~!v ,..:~!'_ .,~':~ '! ..... . . . . . ~ . . . . . . . . . ,"wre , : r e e v e n i n t e r e s t i n } t ( t , i t m s a s t~ {he n a t ~ r e ~f i¢, One t"~tt.'~, tt~ b u i l d t h e a ~ n p [ ~ t u d e ~ r ~ n a p w a ' s : , a r a m c t r , . : c c i ~ s~, :~,- ",.vav ,,:.-.,{i,, ,. ,,'nOS~ : ,:-~:mt}:: I h i n g

t() d o

Eti.'..;b e e : :

t:~ a s s u n ~ : ?

{It~, p

v,.~:v:, i ,= ~.',, ,:iv<.:: : :,. : ;:i-::

: "

"R0sk,,s-constraints and o.he:s of t h e a b o v e k i n d s e e m s largeij ~(, b e a m : : t t c r of fast:.. In : h e l a t t e r c a s e , part of the posltiv;,ty inlo:'mation i s s a j ; < r ~ , : : :t~:.~:,::-:::{ :ea!ly by t h e c h o s e n u n i t a r y p a r a m e t r i z a t i o n . A careful t r : , a . t n : e n i a l t c v , i n , ~ : ~'(:~: ... ~.,.t

sider::~,~e ~ l e x i b t l ~ t v id Wav~",

,:re

t s t h a t of 1.0e G u i l l o u

noSier:ted

in

m:~s{

.,:t a l . [ 1 0 8 I .

,e~:.",:fr::,~":ts)

a:'o

-~:~> ',: ' ! , : ' : - ,

i.,.l:-l:a:~,.~

: 2 ] : : D o l"3 . "" : ~ . ~l v '

1

}FO&I.C,'i

t'"',:~'~'}? ',, : " : ; : ,, •

:na~tn:.: s , ~ m e of t i l e p a r a m e t e r s . M(~:',.~ :~r{' ~:,,:e(! ~'~' ,:~}}~sl~:v, .r}<:~,.'.,..,,~.'-' e.,~:: ::':::'-', i::the )Jaadelstar:: triant:le. T h e , r:,s,~tl,-. ;:: . . . . .:. ,.:.-:: .. e, i;::,:,~ .: :' :. e: all,>:,~,d s o l u t i t ; n s For th,

.F.v.

>~,tu~ion ~,f i~}t. 2.. ~I,,

.:l~a,,,.

.e

2,"

c,,rresptmds

-

~ ,. :

5re 2

,-,~rrected

t:~ e l 5 ~-~ a t ~h~' ~ . ; ~ ' : _

:-, ' :

•

~-':.-' : . . . . . . .

-

-' ..

.

1at.t"

a I

-

.

~

w-.,rsio:~ t.~ t h e i ~ r m u { a

:

t~ r e ~ .

;~ ~

~,

:a c ? n t ; . : ~ t ~ c t : o n w i t i h t h e a s s u m p t i o , q s . ~: fa<"~ ::~'..," ~ .--#. :~,~i .,.-:.t: ,1 .... ,. s~sten~ a v t h e s e s o l u t i o n s w e r { : : ' e , ~ ' e n r l v #,l~,>,:.::~r,,,~" : : :' >-:<-". The " . - . : u t ~ o n s of I . e G u i l l , ' , u e.* , : i . [ 1 0 ~ ~ :!:'c,. :2 , ; : , : , . , :.~ ; : :

' -'

: " ' ' '

"

'

190

J.L. PETERSEN

with the possible existence of an e meson. Although 5° barely passes througk 900 and if so at a very high energy, a second sheet ~ pole is found, ha~'ing a very stable mass of me -~ 420 M e V with a very large but less stable width c ~ F~ 270-520 M e V . Also, corresponding to the solution of fig. 2, the
4" /q

the partial wave normalizedas

Rpole -- 14.7 e i56° ~2

for

(e 16° s i n 8~)) i s [118]

(me, I'~) = (420 MeV, 380 MeV)

(s.'~8)

whereas a narrow resonance approximation would give

RN.R. = - I t ,

m ~ t ~fmm~- 4~ l : - 22. 0~ 2 .

(3,70)

This is a show-case for demonstrating how little a resonating please shift n~ed resemble a Brelt-Wigner one, and how large the penalties are likely to be if narr0~, resonance formulae are used in phenomenologlcal work on the e. Below threshold the ~Tr amplitude of ref. [108] can no doubt be fitted very well by real axls poles {it is close to either Welnberg's or Lovelace-Venezlano's forms) bu' Jbtalned ~ rameters need ,have littleto do wlth the dy~tmlc~ fly slgniflcant rei~Id~e eq. (3.78). This is also part of the reason why the e,~timate (3.76) is lalr'y meaningless, the other part of the reason being that the derivatlon ignores the #re sence of a below threshold zero (eqs. (3.15)) that strongly influences tie shape of the I = 0 s wa~e. Results of the above models are further illuminated by the :york of Dilley [llgI, He considers the class of amplitudes A(s,t,u) (first used by lll.opoulos [120})that can be represented by polyaomials in the variables

,!!

-Qi

11ttll t'

i Ill

ol

. . . . . . .

Q2/ |

_

a~ .

.

-00 -QIO~ '/

/! Fig. 5. Regions in " the (ao ,a~!) plane allowed by low energ', e r o ~ s i r g and ~mitarity a, ,m)~,~w, by Dilley [119]. Branch I (in'~icating author's artistic view) was fou~'~d also by Chew ~ ai [22]. CA i. Weinberg~s current algebra prediction, eq. (3.t5).

MESON- MES()N S C A T T E R I N G

{ :~ ;

C r o s s i n g s y m m e t ~ ' y and c o r r e c t t h r e s h o . , d b e h a v i o u r is r e a d i l y imposec~ and elastic u n i t a r i t y can be a p p r o x i m a t e l y e n f o r c e d by a n u m e r i c a l m i n i m i s a t i o u p r o cedure. Fig. 5 a h o w s what p a r i s of the (a oo,~ r2° ) p~ot a r e f, o p u l a t e d by solution',. Near zero, they can be described in a first appro:~mati~,,~ as two (:)ne-dim:ms,onai)

L.les_havtng abo/a~o ~ 5 / 2 (I) ( t h e s w a v e d o m i n a n t s o l u t i , m s of C h e w eL al, [22]) a n d aO/a_2 ~ - 7 / 2 (II) ( W e i n b e r g ' s n o n - e x o t i c or t e r m v a l u e ) , ff a I is a s s u m e d f r o m the p°angd w a v e s a r e i g n o r e d , t h e l i n e ~ o _ 5Co 2 _ 1 8 # 2 a ~ (eq. ' ( 3 . 1 3 ) ) i n t e r s e c t s : branch It c l o s e to the CA point. T h i s is c o m p l e t e l y c o n s i s t e n t with the a f o r e mentioned m o d e l s . If it i s t r u e it s u g g e s t s t h e f o l l o w i n g exce~din{,dy i n t e r e s t i n g "theorem" ( i g n o r i n g the b e n d - a w a y of b r a n c h H)' For not too '~-:-,;c v a l u e s of t h e s c a t t e r i n g l e n g t h s , the r e q u i r e m e n t s ~,f c r o s s h;,% a n a l y t l c i t y a n d u a i t a : , i t y ^ i m p l y e i t h e r (I) t , ' # a v e d o m i n a n c e , a oo/a : zn - _~ 5/ 2 , o r (2) no~,-exottclty, a°o/az . - - - ~ / 2 . Thus if the V e n e z l a n o m o d e l h a d had ,70 ,:,2 a p p r e c i a b l y diffe~ ent f r o m -7 '2 it woald have b e e n i n c o n s i s t e n t w t t h low e n e r g y u n i t a r i t y . N3w, h o w e v e r , it is so consistent that u n t t a r l t y (at l e a s t tn t t s low e n e r g y e l a s t i c f o r m ) is u n a b l e to fix the o v e r - a l l ec a s t a n t (this s t a t e m e n t is c o n s i s t e n t w i t h , but d o e s not follow f r o m . the multtloop p e r t u r b a t t ~ e p o i n t of v i e w ) . There is c , ~ , , s t d e r a b l e r e a s o n to b e l i e v e t h a t t h i s t h e o r e m is not l i t e r a l l y true. E x i s t i n g w o r k l a o w e v e r , w o u l d s u g g e s t t h a t s o m e p e n a l t i e s l i k e ~" .,~r~¢, d waves c r u n r e a l i s t i c high e n e r g y c o n t r i b u t i o n s w o u l d be a s s o c i a t e d with m o d e l s correspor.dlng to p o i n t s o u t s i d e t h e r e g i o n s of fig. 5. What is next n e e ~ M i~ a wJv ~[ describing a n d m e a s u r i n g w h a t " l a r g e " and " u n r e a l i s t i c " m e a n s ( s e e also. s u h sectto~ 5 5). Tae re.,ult, a s. . .it. . . s t a n d ~ , r~~,,._., p a r t l y ex~Iain vchv so m a n y w¢~rl(.~ in tho n:.,~, . . . . . ,~,:_ suming s( m u c h a!~out the sfn ~ l a r i t y s t r u c t o r e {even r e s o n a n e e . n m a : ; s e s ) b.ave beer, so u n s u c c e s s f u l . The p r o s p e c t s for a c o m b i n e d low e n e r g y and high energy:!, bootstrap ts h a r d e r to a s s e s s . T h e l a t e s t s t r i p a p p r o x i mat i(~ ~r¢,: it me~t,s b a s e d ..;~ P, P'anci p t r a j e , : t o r i e s ~mve l(,w enert4v a m p l i t u d e s s i m i l a r t(~ fib,i. 2. ~,.,t ~,~.,~,~-:: Ab~.r te,~tures a r e u n s a t i s f a c t t ~ r y o r aot c l e a r l y ande.rstu,~d i l 2 1 i ~.

3.5. The ~ p r o b l e m 7ae s i m p l e s t V e n e z i a a o f o r m u ] a f o r nr/ • ~zt s c a t t e r ' r , g h a s ~he f o r m [14, 54,

122,123] (cf. eq. ¢3.42)) A{,~,:,u) :~ VAzj-¢s,:) + V A 2 , 1 2 ( s , , , ) +

r

t:fA2(t,u )

(3.8])

which e n s u r e s c o r r e c t s i g n a t u r e s fo~' A.2 a n d f ( d e g e n e r a t e with p and ,'.' - s¢;e s~.~i~section 3.3}. I m p o s i n g the A d l e r zer.~ on e a c h t e r m in the u s u a i w a y ,:.ires Iwhett',-

"Note added i n / ) r o o f : Recent v ; ~ r k s o f M o f f a t a n d c o l l a b o r a t o r ~ (,I~ W. Mc, f f a t , P h y s . t O.,v. t)3 ~1 :~" I, i 'o',~.i'..:'.~ ,~t~,~ ~: C u r r y e t a l . , P h y s . Re,..'. D3 ~ ] 9 7 ] ) l?.:~:.~} nr,, n o t : ' . ~ ' . T t h 3 I,,~ d,.-i~ ~'~,.:~,~,i . . . . i :: ,!~:~,~,: treatment ~f ~ a n d K r, i n ~ e . r a c t l o n : ~ w h i c h w i l l , m, d o u b t , b e c h a r ; ~ c t ~ . r i s ~ i c oi ! t ~ t u - c , , ~ :

mille hetd. Their model appears to be based on a mixtur,', of two-c~,r.por~,~ t ,~u:*litv an~i kt~erfere:ce model ideas. High energy behavJour (Reg~e and

Pomer:.,.nch . : ; ~

~,-,., ~ .... ~:..

algebra forn~s are 'mposed at the input Iew.~l and th,, autbor~ not:,.:~' ~ ,n~ this is r',~:,-. sis~ent ~,~th approxi,;,ate Mandelstam analyticity. However, severe difflculti, ,~ at,' er~-countered in the tntc_maediate energy region: a n c e s t o r s are pr~sent, ghost I~ro ~f,.rat.. u~,less S~tc!~es aze included ( r u i , i n g much of tbe predictive p~w~:~) and ,~:,,tarity is l~re~.:.:,.~ !,. alarge araount above the rho--mass {~fl r~,,-). Ynr th,'.n~ r e a s o n s i t is ,:~n¢'!, ,~" .,'~at ',h..: i,: :.;.~', t0be rem,mbered should be. Some of the troubles r.~ay he liabl,~ to ~.,~ ,_:s ~J. W. Mof[a~,

current

192

J.L. PETERSEN

er in the soft pion limit (s=m~ I, t= ~2 1 - Up(~ 2) _

ap(m 2)

= C,

u= m,2)) o r in the soft 77 limit (s =/~2

i.e.,

(eq. (3.47))

roT/ = / a .

(3.82)

Much disc,assion has beeT, aevote~l to this m i s e r a b l e p r e d i c t i o n and many cures have been p r o p o s e d , but ',othing very s~.tisfactory h a s s o I a r e m e r g e d . B a a c k e et a l . [ 122] po.nt ou,~ that w h e n a p p l i e d to the r e a c t i o n 7r~' - v,q (releva~l to 7' -" 2 ~ d e c a y - one of t h e r ~ r e c a s e s w h e r e e x p e r i m e n t a l t n f e r m a t i o n on a 0 " 0 - -. 0 - 0 - amplitude involving tl,~. W is conceivable), the f o r m ( 3 . 8 1 ) is a n ~ y

inconsistent with phenomenolcgy in ~hat a l a r g e 5 coupling is p r e d i c t e d in disa g r e e m e n t with the o b s e r v e d s m a l l 8 vidth. If one then feels f o r c e d to modify eq. (3.81~ by adding s a t e l l i t e s it is ear y enough to achieve an A d l e r z e r o without eq. (3.82) via a cancellation mechani m. That, however, is a g a i n s t the rules which gave predictive power to the Adler c o n s t r a i n t in s u b s e c t i o n 3.3. Osborn notices that a l r e a d y at the soIt m e s o n level a m p l i t u d e s involving ~'s have a qualitatively different off-shell extrapc~ation from the o n e s of subsection 3.1 [15]. T h u s in the GOR model, the uW - ~71 amplitude is ,

m2_.

where q ' s and k ' s a r e pion and r / m o m e n t a . Th is explicit dependence ;)n the b,'s a r e indicated and if this is r e p l a c e d by the o r - ~hell value the A d l e r z e r o is ,oi p r e s e n t (i.e., this new o f f - s h e l l form is i r r , : , e w ~t for CA). Maybe therefore the off-shell extrapolation of eq. (3.81) is q ' , a l i t a t i v e l y different f r o m that for eqs. (3.43)-(3.45). Similar points have been m,,de [124-126] a r g u i n g f r o m simple Lagrangian m o d e l s satisfying the GOR s c h e m e [19]. The form eq. (3.83) gi~,es on~ sbmll a constant amplitude correspondi~Lg to a s c a t t e r : m ,~ length aol ~ 0.008~L_l

(~

:

F,).

(3.~4

The t r e a t m e n t is u n r e a l i s t i c , however, in that the coupling oI 77' to the dlvcr!~ettce O~A~ has to be ignor ,~. Brout ct al. [127] us.ng v a r i o u s types of c h i r a l and dual input achieve the Adler condition through a special cancellation s c h e m e (,ot using Veneziano formsl. thereby obtaining the a t t r a c t i v e ruleJ (together with many m o r e ) at/(0) : -1/4,

~r/,(0) = -3,;4.

(3.8~

H o w e v e r , even though one may be able to get around the A d l e r trouble at this level, the prediction (3.82) a p p e a r s to be a m o r e fundamental f e a t u r e of the whole dual s c h e m e . v~'e have seen in 0-0" -" 0 " 0 - that SU 3 quantum n u m b e r s o( e~.~er~ai and i n t e r n a l states fixes m a s s e s and couplings of the intvrnal (V, T, etc.) s~ates at the q u ~ ' k mode: nonet v a l u e s ; for example m w = m o and e~f- ~A? is predi.:ted. S i m i l a r l y by c o n s i d e r i n g b i n a r y p r o c e s s e s with external v e c t o r m e s o n s so that p s e u d o ~ c a l a r s can couple i n t e r n a i l y , the r e s u l t (3.82) i., e x t r e m e l y hat5 to ~v~i~. [128-130]. M o r e r e c e n t l y , F i n k e l s t o i n [131] h a s given a v e r y si~nple but gene~l a r g u m e n t for relieving that dual m e a o n - m e s o n sca~tering a m p l i t u d e s saturated with n a r r o w r e s o n a n c e s on l i n e a r l y r i s i n g h ' a j e c t o r i e s with u n i v e r s a l slopes ~lt e i t h e r have exotic s t a t e s o r e l s e coat,tin p a r i t y doublets (eq. (3.82)). Maybe, then, the title of thib subsection should have been the pseudosc~lar meson p r o b l e m . We shall not t r y to ,~ssess /he f-~ture p r o s p e c t s of dual the(~ries, but take the p r a g m a t i c point o; view that no s t r o n g suggestion f o r amplitudes per-

MES{)N -MES()N SCATT E R NG taining (o pl=ocesses with the ,~ is at aand. Howeve~ iMorm.:tion is av,ailable, a n y w a y .

e s s e n t i a l l y no ,~xp (•-..:i:u

; :~,:!, e n ta

i

3.6. ('2 , s i , g .,,-tinny'ks St , e r a l f i e l d s on the f r i n g e of m e s o n - m e s o n p h y s i c s have not b e e n t r e a t e d , These include h a r d m e s o n o r e f f e c t i v e L a g r a n g i a n t r e a t m e n t s [132-1341, c a l c d l a tions of 5o°(u~t)by c o n s i d e r a t i o n of the f o r m fac: , (~1~] n ) I 1 3 5 - 1 3 8 1 anti t h e o r i e s 0a some s y m m e t r y b r e a k i n g [la a, 1,10]. Wh~ie ~he.,e have b e a r i n g s on s e v e r a l a s pects of high e n e r g y p h y s i c s t h e g s e e m to p r o v i d e little i n f o r m a t i o n on m e s o n meson a m p l i t u d e s , not a l r e a d y c o n t a i a e d in the p r e v i o u s s u b s e c t i o n s . Pcfore we go to the e x p e r i m e n t a l o~rI we s t r e s s [he m o s t i m p o r t a n t t h e o r e t i c ~ l

pr,:,aettons. {:) ~ow energie$ (subsection 3.1): scattering lengU~s are snutll and effective r:mges a r e l a r g e ; s i g n s f a v o u r r e s o n a n c e s in n o n - e x o t i c c h a n n e l s and d i s f a your r e s o n a n c e s in e x o t i c c h a n n e l s . (li) Medium high energl,~s: p h a s e s h i f t s ({lgs. 2, 3) a r e s a g g e s t e d for n,~ and K,~, Much i n t e r e s t c o n c e r n s /}~(r,~r). W h e r e a s t h e o r i e s r e a s o n a b l y a g r e e bel,)w the p (~o(500 MeV) ~ 45 ° ~ 10o) V e n e z i a u o t h e o r y p r e f e r s a v a l u e considera',dy higher than 90o abovo the p w h i l e the r e m a i n i n g c u r v e s a r e at o r bel:)w J0 <' ~n fig. 2. H o w e v e r , the f i r s t is b i a s e d in f a v o u r of l a r g e v a l u e s by the I~r~:.~'~,( ,' of higa energy, low da ught e r S state with p o s s i b l y u n r e a l i s t i c c:mp~in~:, w h e r e a s the l a t t e r a r e b i a s e d a g a i n s t l a r g e v a l u e s by the p a r a m e t r i z a t i o n chosen and n e g l e c t of c g m p l i c a t l o n s f r o m the KK t h r e s h o l d 5.* r e g i o n . ~:; f~,r 5J2(K~I (fig. 31 a ~ s t a t e (table 2) is s t r o n g l y i n d i c a t e d but the p h a s e shift could d i f f e r v e r y much m o r e fro n a B r e i t - W i g n e r . Exotic r~hases~..~ a r e rath~,r uatquely p r e d i c t e d (in p a r t i c u l a r the n e a r z e r o : ?.!',_:e :~'f ~'~/z (K.:)). flit) For h i g h e r energn] r e s o n a n c e s ,~ee table 2 and comm~mts in st:bsecti~m 3.3. Hanttng~a for t r a c e s (if t¢~wer ( l a u g h t e r s and exotic~ will be fc~()wod wi~b. ~y;~:... ti,'ular n t e r e s t . 4. INbOHaMATION FROM PEI{IPItEtLAL DI-MESL)N I'IL(JDUCI",O;, The a b s e n c e , for the t i m e b e i n g , of m e s o n i c t a r g e t s i m p l i e s that e x p e r i m e n t a l information oe m e s o n - r a e s o n s c a t t e r i n g is p o s s i b l e only in highly indirec¢ ways° By far the m a j o r i t y of o u r k n o w l e d g e d e r i v e s f r o m high e n e r g y p e r i p h e r a l m e s o n production. N~:w .b.gh s t a , : s t i c s e x p e r i m e n t s now being planned o r a n a l y z e d e n s u r e that t~'.~s wilt :o:,.inue to be b,. ~or a n u m b e r of y e a r s . R e v i e w s : r e f s . [4, 5, ~41-~451. 4,t, .~',~eno~,~e.tological f~'amewort, Consider a p r o c e s s (fig. 6) for ~'hich pion e x c h a n g e is :~ll~)wed. }I~r(' ~ 0 me:~0 r t ( f o u r - m o m e n t u m q, m a s s m) s, a t t e r s (>n a t a r g e t n u c l e ~ (f¢~ut'-x~,~m~'n~ur~)'; mass .'~ri to p r o d u c e a t h r e e - b o d y ', inal s t a t e of a b a r y o n and two 0- m~stms ".ri!i~ ba.ryo~ will be , ~ t h e r a nvcte~m ~)r a A~,(123(;~ r~'~,,~:,, ~"

..- (g'?a')

(q,mF-- ~.....

(P.m) 7

t 1""

J.L. PETER:BEN

194

total c.m. energy s q u a r e d s = (q' +q,,'~2 52 :

effective d i - m e s o n m a s s squared lnv.~riant ,~quared momentum t r a n s f e r to baryon (> 0 in physical region)

_(p_2,',2

u = (q _q,)2

t = (q _¢,)2, M2M,m,

=

(4.~)

M2Apm,

(q, +p,)2,

=

(q,_p,)2.

Momentum conservation and m a s s - s h e l l conditions imply

s~- M2Mm,+ , M2M,m.= W2+M'2+

,n'2+ m ''2

(Dalitzplot)

s + t +u =m 2 + m ' 2 + m''2 - A 2.

(4,2)

For e x p e r i m e n t a l studies, the following v a r i a b l e s a r e p r e f e r r e d :

W2

(fixed for given beam momentum)

( s . a 2)

(4.3!

(Chew-l,ow plot)

(0,¢)

polar ~,~,les of the outgoing m e s o n m' in the dl-meson c.m.

An infinity of co-or ~' :hate s y s t e m s can define (0, ~). In most studies either the (GJ) [146] or the helicity ~rame (H) is used. In both the y ~ i s is taken o-'thogonal to the production plane (containing the beam and the oatgotng baryon). Further in the GJ, z is the direction of the incoming 0- m e s o n (i.e., q as seen in the dlmeson centre-of-mass) and in the H frame, z is opposite to the direction of the outgoing baryon (i.e., -p' as seen in the di-meson centre-of-mass). In e~'ery case x is specified by the right-hand rule. Goebel [14'/] and C h e w and L o w [148] suggested to au,dyze data according to the for,nulae [ 149]

cs

dcr(W2~ s. 52, 0,
a2

ds dA 2 d~

(.~

( , ~ ( d ~ t))

¢4.4a

+/J.,

for final baryon = nucleon, and d olW 2, s,

,,2, e ,¢_)

_

4-f

~ -5 2

8,3 p2 W2 q(

ds dA 2 d ~ d M ~

1

•Ides.

~h

)(~2+ ,2)-2 0 ( ' 5 2 ) M'2 cr(M')(--d~ ~']mesz (4.4b,

ior final baryon = 533(1236). p is the initial centre-of-mass three-momentum, ~ i~ the pion mass and •

--

,,

=

!Q(_52))2_

~ S t -

,

.

.

.

.

.

.

,

1 [,W2 _ (M+ ~f-A-2)2]{M'2 - (M- ~-Fi-~2}2] 45p2

f4.5

?,f is the proton m a s s , G2/4.~ = 2 x 14.6 for n "p -, n- ~+ n, (r(M') is the ~N crosssection in the 5 3 q region and the above c r o s s - s e c t i o n s have been summed over final and averag¢¢~ over imtial baryon polarizations. ( d ~ / d ~ ) m e s o n is tne on-shell m e s o n - m e s o n scatf er!ng c r o s s - s e c t i o n corresponding to the upper vertex o~ fig.ft.

MESON-M

ES()N

SCATT

r; I I I N (

! !:':

The w m i s h t n g ol cq. (4.4a) at A 2 : 0 is a p e c u l i a r ( e a t u r e of p s e u d o s c a l a r c o u pling to the N"N v e r t e x ( e v a s i o n ) . The point b e h i n d e q s . ( 4 . 4 ~ i s that ac('('(~rdin,~ Io it., " e r a l ~) belief ( s t u d i e s ()1" Fevrman graphs) the a m p l i t u d e of fig. 6 ha~ . t.,~de :~ A . . . . /a'-. and th.,, ~'ross-s(,,'iii,t.. has ;t double p o l e , the r e s i d u e of which is given e x a c A y by e q s . ( 4 . 4 ) . T h e r e f o r e , one attempts to suitably e x t r a p o l a t e the l e f t - h a n d s i d e s to the u n p h y s i c a l r e g i o n , ~II = . g 2 t h e r e b y o b t a i n i n g (dcr,/d~) m e s o n . Then tt~e p r o b l e m of finding m e s o n meson phase s h i f t s is just a~ e a s y (and just a s d i f f t c u l ~ as for ' o r d i n a r y ' p~:ocesses. H o w e v e r , it t u r n s out t h a t eqs.(4.4} a r e s a t i s f i e d so p o o r l y ie the p h y s i c a l region that m o d i f i c a t i o n s a r e n e e d e d . The m o s t g e n e r a l f o r m for the i n v a r i a u t a m p l i t u d e ;or the p r o c e s . ' b,

,

0"(m) a~(M)'* - O - ( m ' ) O ' ( m " ) , : +(,~r~' is [ 1 .~-.,,

?~' (w 2.

s, a 2 , 0, ~,! ,

' A2~5(q

u(fi){AlrS(q'y~,

7) ' A a 7 5 ( q

,,

"'r') ' A4 ~,~>tv,r

).,a

q

t.t ,~e q

4. (;~

gq. (4.4a) c o r r e s p o n d s

to

A(S_,2)_ trg u(f)~,5u(P) a2 + ~2 c

"~

~'

._

where A ( s . t ) is the i n v a r t a n t m e s o n - m e ~ , , n s c a t t e r i n g a m p l i t u d e ~,~,t C i~ tl~,, !,~.~r~ m~¢:le(.n coy.piing c o n s t a n t . E v l d e c t l y th( : i y n a m i c a i s()ph~st~ca~':)~ (~i ti~at ~:~,,~ti~:; ;:-. exceedingly low c o n , p a r e d with the m o s t g e n e r a l p o s s i b i l i t y . For p h e n o m e n o l o g l c a l s t u d i e s , one n o r m a l l y e m p l o y s heli('ity amplitu,:!es rather than i n v a r i a n t ones 11511. T h e s e ~nay i;e ana:lyze.~i ~t~ (~'r~,~ (,i ,i~ ....;,,,:c()7~ :::.: gmar m o m e n t u m . A2

I/

t{

:, i~

8ere ).(,~') is the initial (final) b a r y o n he!.icity in the o v e r - a l l c e n t r e - o f - m a s s , ), a;,, s the a m p l i t u d e for prociucing the dipion s y s t e m in a s t a t e of a n g u l a r m o fl, mestum l a n d h e l i c i t y ~t. Eq. (4.8) a p p l i e s d i r e c t l y to the h e l i e i t y f r a m e . The dif ferential c r o s s - s e c t i o n is ( p ' is the final b a r y o n c e n t r e - o f - m a s s t h r e e - m ( ) m e n t u m " ............

i

i 2

';dA 2d f~H ' 2 .~p ~ , t i [

"7'

( , ll'

'

'

,)ii

'

• ] ' ,,( ~," i C i~ i~ " ' ; ' i " ' ,ji( ,..p,,

- l !' b~ t.t'

?

i

•

,'

.... 9 (~ % rt i2~-

with dr clA 2

;~.'

~ ; , \ J 2~'~' a

The hel~city a m p l i t a d e s s a t i s f y

, ~;-~

~(_) ~ i

, :]5;:

,

196

J.L. PETERSEN

where ~ is a notation dependent phase independent of ~ (but dependent on parities and k,k') [151,152]. Equations (4.9) and (4.10) imply

( p .ll'~ , _

it' H ~,)H= (.)~ -bt' 1o~,~,)

d~'g,)H* = (r)! q ~H ,,- ~,u-

(4.i~)

ll' ,)I'I= Tr{pH) = 1. ll ' g g ' As pointed out by Gottfried and Jackson [146], eqs. (4.8)=(4.11) hold in frame as well provided the following changes are made

/tO, tp) H -* (8,tP)GJ; "rk'~t;k

f../(c) ~ k'k;m ;

c}~,/.t; k -.9tt~kc)~;m , ll'

,H -.

(P~,~'/

,

It'

,GJ

(Prom '/

(4.11)

; m , . ^ and -'k'k;m f/(c) a r e helicity amplitudes for the c r o s s channel procen,, Here c~C).

B(M) B(M')-. m, ~rlt

,

r?/~

analytically continued on to the phy~ tcal region for fig. 6. The density matrices tn the two f r a m e s are connected by

t.,ll'

,GJ = (_)I.~"U'

~ I~ P" )

~.

mm

ll' .H-l( ,~pmm,) a X)m ~d l' (X)m' ~,

(4.13}

where the crossing angle X is given by [152] E _ A 2 + ~ . , . 2 ) ( v,,2 + s - M '2) - 2.~8]:,

COS X =

(4.14)

with

6-=s+itf-M

'z - m 2.

Given the density matrix elements, the angular distribution is fixed, however, the converse is not true. Writing in either frame

w(o, ~,) = ~ LM

( ) ,.M,, , , y M ( o ' ¢) ""

(4.15)

""

with

w(e, ~) _=

da d~ I dsdA 2df~ d s d A 2

the moments (or statisticaltensors) (Y~) have been defined (~yO,~ ~ 1/Vr~,). These are the directly measurable quantities. One has (see, for example, Rose [153], eq. (4.34))

MES()N-MESON

SCATTERING

197

V~-(2~ 1)(2I' + 1 ) ( - ) m ( i , l ' , M + r n , - m l l l ' L M

)

ll'm

× (ll'O0]ll'LO)(1+ (_)l+l'-L)R e P_If' M+m, ~'/~

(4.16)

For fixed b e a m momentum (i.e., W2) they depend on the point (s, A2) in the "Chew-Low p l a n e " bounded by the "Chew-Low boundary"" I) l~e line

s = (m' + m") 2

~) the curve

-

1 [w4 _

+ (M2-m2l(M'~-s) ± 4 W~pp']. (4.17)

To p e r f o r m a "Chew-Low extrapolation" f r o m this physical region to the pion pole, some knowledge on the singularity s t r u c t u r e of th~ e x t r a p o l a t e d object is necessary. I) The i~variant amplitudes A 1 , . . . ,A 4 (eq. (4.6)) have d y n a m i c a l singularities only [150]. F o r each of the ten t w o - p a r t i c l e channels, below t h r e s h o l d poles ( c o r responding to the existence of a stable pa. ticle with the quantum n u m b e r s of the channel) and n o r m a l t h r e s h o l d b r a n c h points o c c u r in the invariant associated with the channel. Thus for fixed p h y s i c a l values of W2, s, t and M~i,m,, the A i ' s a r e analytic in the A2 plane cut along (-~,A~) and (A~, ~o) possibly w,:th a pole at ~2 :.~2 Here A2=

91L2

for

uN--. 2~N'

and

x2 = 4~ ~'

~'-

""

'~'-~'

9 9 whereas A~ depends on s, t. M~M,m , and is a s s o c i a t e d with a threslxold in one of the remaining nine channels. Little is known about higher ordey singularities as given by the Landau rules [ 11]. Generally they a r e supposed to be further away irom the physical region. H o w e v e r , for B' = A33(1236) the d i a g r a m s of fig. 7 give rise to anomalous t h r e s h o l d s Z~ ~ - 5~t2(t + i) + 2 X 2 ( g / M + i) ~ (-3.6 + i 4.6) U2

(a)

Aa2 ~ - 2 ~ 2 ( l +i) + ½X2{~z/M +i) ~. (-1.6+ i2.8) ~2

(b)

with X2 = M 2 _ M 2 _ U2 A 33

a)

b)

Fig. 7. Triangle graphs giving r~se to anc,malous ,5,2 thresholds, eqs. (4.}~).

(4.18)

198

J.L. PETERSE N

and associated cuts exteadtng from ~I- In the absence of any information about the dtsconti~taity a c r o s s these cuts, one must suspect that the one-plon exchange (OPE) pole is considerably m o r e difficult to separate from background singularities than for p r o c e s s e s with B ' = N (see, however, the r e m a r k at the end of subsection 4.3). 2) The partial wave helicity amplitudes f / , ~;k (DCtlA) andy~,Cx);,.. (CCHA) have two t:v-pes of additional singularities. (i) ~e'~ng pwa's they a r e obtained from the lnvarlant amplitudes b~ lntegraUonS over t and M~/,m,. That produces In g e n e r a l complex sing~.larittes of complicated "orms [154], which gives a s m a l l e r effective value .or A~. (i~) Being helitity amplihutes they contain kinematical slng,~larlttes which a r e ~ . ceedlngly complicated to describe for a five-point function [ISS, 156]. A stm. plified idea about them can be obtained by viewing the outgoing di-meson ~/stern as an effective particle [q'~,/] (which is reasonable at least at a resor.ance) and r e s o r t to the extensive t r e a t m e n t s for four-point functions {152, 157-159]. Then: a) on the boundary of the physical region (in particular (4.17)) ~quare root rio. gularttles may be p r e s e n t via the half-angle ~'actors:

isin (el2)] I

llcos

,z(c) =[s n (ec/2)] I m- X'+ l [co8 (ec/2)] JYt'~t;m

t7,<,,;,< I

{4.1 1

;l L/(c) '* " r e r e g u l a r a c r o s s the boundawhere the reduced amplitudes Y~'tt,~, J)~'~.m ry. O (Oc) is the direct channel (c~rossed ci~annel) c e n t r e - o f - m a ~ s scattering angle of the outgoing, baryon; a c r o s s th,~ boundary sin (0/2) vanishes as a square root of the i~variants ( c o s ( e / 2 ) ~ 1); thi~ is true for Oc as well

~(W A, A2) = _4A2Pc2 pC2 sin Oc

(4.20)

where Pc(P~') is the initial (final) c e n t r e - o f - m a s s t h r e e - m o m e n t u m for the p r o c e s s B]~~; -'0-(m) [~/'s,/], and ~ ( W ' , A2) is the Kibble function [160], a polynomial in the invariants with first o r d e r z e r o s at the kinematical boundar i e s , positive in physical regions and negative outside; b) at thresholds and pseudothresholds [W2 = (Mr m) 2, W2 = (M ~± .,Fs)2 for DCHA's and A2 = -(M+ 214")2, A2 = -(m ± q'~)2 for CCHA's] and at W2 = 0 for DCHA's, A2 = 0 for CCHA's, square root singclartties in positive and negative powers may be uret~ent. These a r e i r r e l e v a n t for DCHA:s as far as Chew-Low extrapolation goes but could be of m~jor importance for CCHA's, in particular toc M = 2if' but see below. 3) Density mairix eie,aents and moments have: (a) a subset of the singularities of the .fl's; s e v e r a l singularities between the physical region and the OPE pole drop out, whi,gh is e x t r e m e l y fortunate Thus, from eqs. (4.9) and (4.1t;)we see that ( ~ ; x , ) ' s with m = m' and (I~)'s'" with M = 0 (those a r e the ones c< atainivg the OPE double pole) have no kinematical boundary singularities [ (2, ii, a)above]. Also, the a l a r m i n g pseudothreshold singularity at A~= 0 (for M = M') disappears from (pl~m,)C~3due to constraints among the ~l(e) 's as is evident from_eqs. (4.13), (4.14); (b) additional poles a r e ~,J~'X:m present whenever (dcr/dsdA2)is zero; the positions com-

MESON-MESON SCATTERING

199

pletely depend on dynamical details and a r e unknown apric, ri; a.; we shat: see in subsection 4.3 one is a l m o s t certainly p r e s e n t betwee, a 2 = 0 and ~2 = _ p 2 which constitutes a m a j o r w o r r y for Chew-Low extrapol::tions of norma'.ized quantities. The GJ f=.ame [146] was introduced as a m e a n s of discriminating between v a r t0u~ exchange n~echanisrns. T h u s Io" normal parity exchange

(Pm, lI'

- m'

)GJ

=

.(_)m',~ Pll' m,

m

,)GJ

for abnormal parity exchange ~Pm, " ll' -m ,)GJ = (_)m',~Pm, ll' m ,)GJ

(4.21)

More specifically, for pion-exchange'only

Prom in

= 5m°

8m'°f(l'l')

(4.22)

particular , M) =0, ~Yi:

(~.23}

M#0

which is just the ~ell known T r e i m a n - Y a n g test for spin zero exchange: the dipion an~lar dtstrlb,~tion is independent of the T r e i m a n - Y a n g angle, rpGJ. The rules (4.21). however, a r e approximately valid in the helicity frame as well •ell, the correction being of o r d e , ~,2~ W2 I161]. ~ u a t t o n s (4.21)-(4.23) constitute tests for OPE. Others a r e specific oredic• ~ns of cxts.(i.t.,): v~ntshing of (a) at &9. = 0. specific tx2 dependence and specific ~2 depen~tence (Pith). In the following s u b s e c t i o n ~. . . w.~ . ~l~a~l . . .u ~.u u :. ~ . . ~.o w. t.o t t t : , t I ~tth deviations from these as suggested by theory or experiment. 4.2.

Off-shell effects

One very POl~alar way of ~iewtng di-meso~, produ,'tiou is to consider a modified 0PE mechanism by which the graph in fig. 6 is suppose::l to r e p r e s e n t scattering of a m'.~son on a virtual pion. In such a picture the Lorentz s t r u c t u r e is still that d ~ . ,4.7) (for B' =N) but one allows for the generalization ~)/~= ~(P')Y5u(P)

A(s, t. A2) c

N~.2~l

~2 + ~t2

where A(". t. ~x2) is ~:alled an off-shell meson-meson scattering amplitude. Two modffie,,~tlol,s have been m,~ch covsidered"

(1} meson-rues ~n pwa's, a l, a r e moi_:f~ed off-shell by real multipl~cativefo'r~;~ f',ctco'~ d ~ n ~ n d i n ~ ' nT, ~ a n d A 2 only; a particular way suggested by PCAC.

Fon~ factors

.In ihe rtial wave orojeetion pc -formed to obtain fl,~ [a common name for (f~, _)HI'and 0 ~ l ")i3J] the integration limits deDendi~dn zx2. This effect m t r o tl',~;a., a..~...,,:,}~.~..;~t A2~l ~n f r if that was the only non-pole ~Z dependence ve would have

200

O. I,. PETERSEN

'

= \ ~'(pt2) :

'

-~

(4.25'

l ,, 2 , ~t2 )] tAa=.l.t I where [f(v)(s, A")(A 2 = al(:~)fl(G,(v)) and the quantities/3(G, (v i)ma,: be eva!uated e.xactly (see, for exit mple, ref. [145]). If al(s)is dominated by a narr o w r e s o n a n c e this is just what eovartant F e y n m a n rules would give, howe~rer, the A2 behaviour is in g r o s s d i s a g r e e m e n t with experiment. Diirr and Pilkuhn (DP) [162] and B e n e c k e and ERtrr (BD) [163] showed that the replacement (~(-A2)/~'(p2))/-, F I ( - A 2) with

F / ( - A 2) ~ (~(-AZ)/~(p.2)) / for

_A2 ~ (4-~.¢ m)2 14.2~

FI(U2) = 1 ]constant (DP) I for A2 . oo F / ( - A 2 ) -" IIA2{-1

(BD)

a r o s e n a t u r a l l y in n o n - r e l a t i v i s t i c models with force range R (DP), or tn ladder app~'oximation t r e a t m e n t s of B e t h e - S a l p e t e r equation soluti~ms for exchange f o r t e s f r o m a particle with m a s s = R -1 (BD). F/t-A2) is a "pion form factor" [or ti:e v e r t e x " ~ ( - a 2 ) " --' 0-(m) + [ ~ ; / ] e x p r e s s i b l e in t e r m s of s p h e r i c a l B e s s e l and Neumann functions (DP) or Legendre functions of the second kind (BD). Specil!cally ,~9 P (-A 2) :-: 1,

P~I P ( - a 2) ~ N 1

/¢ q("a2)

where N 1 is determined by eq. (4.26) and w h e r e R is a p a r a m e t e l F r o m the first eq. (4.27), DP suggest the name, k i n e m a t i c a l form f a c t o r s . Wolf [164] in a s e r i e s of papers fitted a !,~rff,e numbec of differential crosss e c t i o n s for reactions supposed to be dominated by OPE. In addition to the aL<~ve f o r m f a c t o r s he also supplied the ones for the lower vertex and the propaffator o,' fig. 6. The relatively nice a p p e a r a n c e el t h e s e tits made Mi,~g Ma et al. [1651 ~s'ee also Schlein [142,143] and subsequently many other w o r k e r s to consider form factor modified Chew-Low extrapolations. We m a k e , few g e n e r a l comments o~ this approach. (i) Modified with form f a c t o r s , eqs. (4.4) continue to decoupie the W2 and ~i~e52 dependence in alcparent a g r e e m e n t with ex'per~,nent [164]. This is a su,¢ ess of OPP. only if the pion t r a j e c t o r y has z e r o siope but indicates failure ~! OP~ if o¢,~ has univers0~l slope. (it) The ability of f o r m - f a c t o r - m o d i f i e d OPE fom,~s to fit (d~/ds dA 2) in c.~ses w h e r e Pn.m(n or m ~0) is kno~.n not to be small (o 1. and ~,10 in ~-n -. "~,-~ • . , "~'1 e'10 ~ ' • ' s e e t o r example ref. [ 145]) is a warning that these do not provide ~trir,~em t e s t s for OPE; tilt) In perturbation ¢.heori(.s of weak a~d e l e c t r o m a g n e t i c int,?ractions..fc~(:,:riza-

M ESON-M£SON

~,C,4 T T E K I N ( ;

Z{) 1

bh' form f a c t o r s may be d e f i n e d . In h a d r o n t c p r o c e s s e s this is only p o s s i b l e ir~ ~n o b v i o u s way for p o l e rcsid,,,cs. T l , e r e f o r e , although the idea of m i n i miziaf, the n u m b e r of f r e e p a r a ) n e t e r s ~) a C h e w - L o w e x t r a p o l a t i o ~ by im:~.lefronting f o r m f a c t o r s o b t a i n e d f r o m fit,,,: to e t h e r r e a c t i o n s ~-, at~rae ~; e, i,: can h a r d l y be j u s t i i i e d . T h i s is so in p a , t i c u i a r when ~i~. approact~ is m o d i fied to a l l o w for n o n - e v a s i v e t e r m s [165]. {iv) DP or BD f o r m f a c t o r s a l l o w f o r a d d i t m n a l ¢ i n g u l a r i t i e s in the A 2 channel s wave. S i n c e , h o w e v e r , the c r o s s e d c h a n n e l p a r t i a l wa~e e x p a n s i o n anyway d i v e r g e s in the r e g i o n of i n t e r e s t the s i g n i f i c a n c e of thi s is u n c l e a r . The d i vergence t r o u b l e is w e l l k n o w n to be o v e r c o m e by Reggeiz:ttion. Then the L r r e n t z content is m o r e c o m p l i c a t e d than tn e q s . (4.7), ¢4.24) (whether oe~ is conspiring or e v a d i n g - s e e next subsec~i(,n). F a r t h e r , the r e s i d u e functi{m ,~ does f a c t o r i z e (bein~ a j p l a n e pole r e s i d u e ) . U n f o r t u n a t e t y tim a t r a j e c t a ~ is e x t r e m e l y poorl3 known at p r e s e n t . {v) F o r m f a c t o r s a r e c l e a r l 3 to be used w h e r e they belong but n o w h e r e e l s e . B e low we s h a l l c o n s i d e r c a s e s w h e r e blind f o r m f a c t o r fi~s to low s t a t i s t i c s data will p r o d u c e s y s t e m a t i c e r r o r s .

Correlation effects Models i n c o r p o r a t i n g PCAC s u g g e s t a p a r t i c u l a r behavi~)ur of the o f f - s h e l l pwa's that cannot be d e s c r i b e d by f a c t o r i z i n g the ¢ and A 2 ~,lepende~ ~.~s like

al{~, A 2} : at{s)Fl{A2}.

(4.28) have

Thus %r .~ s c a t t e r i n g (eq. ( 3 1 5)) 4(0)

1 (2,~ A 2 - 2 t ~ 2 ) ,

t
A

{2~

I .2

(t:2 - ~x2 - s )

(,!. 29

describing a p p r o x i m a t e l y pur~: s, p and s w a v e s r e s p e { ' t i v e i v ( t h i s is exaci
This is a

::-p " 7.-7, sn), ¢'q. (4.29)

ewes for the s wave (4.30~

A o - { , ' a2} = 2;22_(.: _ A 2 . ~2}. Making an a v e r a g e fit of that to the f o r m (4.28) ()~'('r tLe iat(:,rval {) ~'ill giw, in an a v e r a g e ~ e n s e ~.$

C()~-ip~wu

i,~ t ~

~F ~

,--:

S2 • ,q ;2

"-'~-

,r,

1

Wa~,,,v [166] a n a l y z e d t h i s effect using the K m a t r i x un!tarize{l I~{,',.'],,;'e Venezia~ , m o d e l for the u p p e r v e r t e x fig. 6. I n s t e a d of BD or DP f,<>rr~: fa~'~:,:':: ".~:'

202

J.L. P E T E R S E N

(~,2)l t r o u b l e was dealt w~th by an o v e r a l l m u l t i p l i c a t i v e exponential e -A~,2 roughIy analogous to Reggeization at one e n e r g y . The s wave was found to d e c r e a s e for positive ~2 as eq. (4.30) for low e n e r g i e s (<~ 500 MeV). F o r h i g h e r e n e r g i e s only, the p ~ a v e s s h o w e d a :slight ( ~ 2 0 ~ ) d e c r e a s e (at b~ =9p@), w h e r e a s the s wa~e was l o w e r e d b y a s much a s 7505 to 6 0 ~ - I o r ~ b e t w e e n 500 a n d 900 MeV; ~o2 ~ a ~ffected by l e s s than 3005. Gutay et al. [167], L o o s et al. [168] and Scl-~rengutvei et a.l. [27] w ~ t tl~e ~ r way a n d a t t e m p t e d to d e t e r m i n e ~ p a r a m e t e r s f r o m t h e o b s e r v e d co/'relat~ons. I~ the l i n e a r a p p r o x i m a t i o n one has with P C A C i m p o s e d ( s u b s e c t i o n 3.1)

(4.321 and

a~a2o = -

(4~ -

3
H.331

It is a r g u e d that the m o s t m o ( i e l - i n d e r e n d e n t f e a t u r e of the e£fect is the behavior of the point, &2 = &~(s) w h e r e the f o r w a r d b a c k w a r d a s y m m e t r y vanishes, a~ a function of s. That is obtained by putting eq. (4.32) z e r o , i.e., 3~

=

+

2~

"

E x p e r i m e n t a l l y a s t r a i g h t line was indeed found for ,]-s <~c_,G0 MaY, and )~/~ was equal to the CA value, -1, to within 40%~ giving by eq. (4.33) the imp.)rt~nt ~sult

ze/...2

o' ¢=o -

-

3.3

0.6 - 1.1

+

(4.35i

~p a g r e e m e n t with W e i n b e r g ' s ,,'alue of - 7 / 2 (eq. (3.15)). Fi~.lre 8 shows ~{s} v.=rsus s [168] /or ~/s :~ 1 GeV. The tow energ~ straight line behaviour s e e m s to b,s modified in the right d i r e c t i o n by the V¢~,aziano ansatz.

--- S, PD.

- - - S.PDF

i _!11 '

0

8

~6

```%` "

2a

32

40

48

..~

Fig. 8. A2 versus s. ref. [168]. Straight l:ne f~ts and Veneziano model predictions are shown. Dashed and d o t t ~ lines include d a,"~d f waves.

MESON-MESOt~ JCATT ER1NG

203

It is not q u i t e c~,ear what the s i g n i f i c a n c e of t h e s e nice r e s u l t s is. When the lower ~ertex ia fig. 6 is w e a k (lepton t a r g e t ) the A d l e r condition is a cot~sequence 0f the c a r r e n t c u r r e n t n a t u r e of the I n t e r a c t i o n . In the p r e s e n t h a d r o n i c context, however, t h e r e ts no reason to believe in i t s v a l i d i t y and w i t h o u t it, the r e s u l t ~ . (4.35) d o e s not follow. At l e a s t , one m o r e c h a r g e m o d e n e e d s to be c o n s i d ered. Rowever, the i m p o r t a n t p o i n t to r e m e m b e r i s that c o r r e l a t i o n e f f e c t s of the above-raen~oned o r d e r of m a g n i t u d e follow j u s t f r o m s y m m e t r y c o n s i d e r a t i o n s of the off-~helI a m p l i t u d e without impo~ tng P C A C . One s i m p l e way of d e a l i n g w*th them ts to m a k e the r e p l a c e m e n t

A2+~2

A2~2

(4.36}

+b

where A ~ ( s . t) ~.a the o n - s h e l l ~r;; s c a t t e r i n g a m p l i t u d e and b is a. r e l a t i v e l y Ieatureles~ b a c k g r o u n d d e p e n d i n g on {s, t, h2) but constant in the l i n e a r approxi-. m a t t ~ and d ~ e n d l n g on t ~ v ~ p i n c h a n n e l s In a k n o w n way (cf. s u b s e c t i o n 2.4). Mt~t'ds {4.38) m a y b e f u r t h e r m o d i f i e d by f o r m f a c t o r s . C o r r e l a t i o n ~ f e c t $ ~)f a c o m p l e t e l y d i i f e r e n t o r i g i n have bem~ d i s c u s s e d . Th._~s Bl~ats ,~t al. {169] ~ u g g e s t e d to u s e C o u l o m b i n t e r f e r e n c e to .J,bta!.n ~o',~, er, e r g y ph~e shlftg. F o r ~ v a t t e r l n g of c h a r g e d p i o n s , o n e - p h o t o n e x c h a n g e s p r o d u c e poles a r t : 0 ( u = 0 ) . F o r , ~ 2 = _ ~ t 2 t h e s e o c c u r a t 0 : 0(~;) w h e ~ e a s l o r ~2 0they are outset,de the p h y s i c a l r e g i o n a n d c o n t r i b u t e a f o r w a r d b a c k v , a r d a s y m m e t r y t*epending m s and A 2 A l t h o u g h that does nat s e e m to be the m o s t powerful method a~ailabl,., for s t u d y i n g low e n e r ~ , p h a s e sl',!fts, it c o n s t i t u t e s a ~ alculab~e effect ~h!eh should not be c o n f u s e d with o t h e r ones.

4.'. A b s o r p t i v e e f f e c t s 0ne~,f the m o s t e x t , . n s i v e l v s t u d i e d r e a c t i o n s fr,~m the p,~ip,~ ~ v~ew ~f (;:~e~ ,0~. e x t r a p o l a t i o n s i:-

• hlch would s e e m to be a p a r t i c u l a r l y a t t r a c t i v e p u r e OPI:'. c a n d i d a t e . ~..~.,{ ttowever, (1) M a r a t e c k et a l . [170] found that e x t r a p o l a t i n g the. ~o~ 1 ~a cr,~sssection using the e ~ a s i c e c o n s t r a ~ - * _d~___=0 ds d,~ 2

at

M. e~t. ~";.4a)}o ,~ the r e s u l t s mass. F o r a n o n - e v a s i v e acc~acy, A l s o (2) if s and an~lar d~:~tribtat~on (~n the 0O b~*

A2 = 0 w e r e c l e a r l y below the p w~v~ ~ n i t a r i t y ~]mi! :~t ih~:: e x t r a p o l a t i o n d a t a w e r e i n s u f f i c i e n t to ?,{old a ,..,sc!:~ p w a v e s only '~re c~.~~sidered, the ts(~r,~t~i ' tect~ t.,~ ~.,:, ~ 1 r'~ GJ ~ . a m e ; is gtve:~ by (~e4. (4 . 9;)

11 !I (201! * P-I,-1 }.

For OPE only, ehe l a s t t e r m i s z e r o and the i s o t r o p i c p a r t is a m a a s u r ~ ,~i ,ti.: ' s~'ave. H o w e v e r , doing a g a i n an evasi~'e e .. . r., , ~~, , ~, a p o l •a ,4- m a , the r e s u l t this i i ~ , <~ceeds eva u n i t a r y l i m i t £er ..= w a r e s ~v-. .~. ........ . y ! a r g._o amo~n~ " • ' [ l a.5 , 1~1 ~' ~. Bath ~-f t h e s e o b s e r v a t i o n s s t r o n g l y ~=u,,,~,,~ .. ~ . tha.* . . . . . .,=omo ..... , -,~, .~:~ri{v.e-:-

204

J.L. PETERSEN

changes axeipresent s o that ( ~ / d s d & 2 ) ~ 0 for &2 = 0 and depolarized O's domt~ hate the Isotropic distribution. ~.~ Below we briefly 0iscuss predictions of these effects in various theoretical models (vector meson dominance (VMD), absorption, conspiracy) and try to ~ . mate~f:rt)m the models w h a t t h e cures wttl be in practical Chew-Low extrarolatl~ Studies. ~!~ Much! interest has !focussed on the p o u i b l e connection ( v i a VMD)b~.~,w.~ -~,~ t r a n s v e r s e n e u t ~ p production and (inverse) isovector photoprodu~lon

7vn.-p~+.

'=~'P ~ D~r n,

(4i.)

For *.he f i r s t the GJ frame cross-section vanishes in a pure OPE model, whereas experimentally, the second shows a forward peak (within A2 ,,/~2) [172]. Th~ ,uvartaut amplitude for the reactions (4.39) may be decomFosed as [175]

oNE=l~(P')y5{Bl~ # , 2B2X÷ 2~(c.q) ÷ 2B4(.k

Here k = q' + q" is the vector particle moment~,% Also R = j ( P ÷ P'). B4,7 do=or contribute to the physical c r o s s - s e c t i o n but play a role in dis=ussinl| possible Mnematical singularitiee (B i, i = 1 , . . . , 8 do not have any [173]). The conditions p be coupled to a conserved current a r e

m= st. P

- v')

(a= ÷ m=p÷

s s ÷ z=p284 = 0

- ~.B~ • ½ (w2 - ~ s 6 , 2m~ a 7 , (a= , m2p , ~=) as = o

(~.41)

where U = (q-/~)2. For m2O = 0 this is just the requirement of gauge invarlanee [174-m6]. For OPE only, G'f

=

.2B4"

ei-

o,

,s.4

where G2/4~ = 14.6 and f = g n ~ or e for the two processtm (4.39). Thus OPli Is not a gauge invariant descrip~i0U for m~ = 0 (Reggeized p~.on exchange is [174]), This ts well known in perturbation theory n which in g e n e r a / a l l diagrams of given order must be taken together. Doing this in the electric Born model (no magnetic pNl~ coupling) one has from the nucleon l~.le t e r m s [1/(W2-M2), 1/([/-M2)] the further specific prediction for B 1 (&2/W 2 s m~_l) [177 ] (t oge ~her i with a s i m i l a r one for B2) ~

B l ~ 2 ~ G . / W -2

in~~t

tn-~'

,.,'2 .

(4,~)

A priori there is no reason whatever to expect that this model makes sense for high energy periphera~ produ, tion, but as we shall see, that value of B 1 arLses in more complicated models as well [178]. Cross-sections for production ~ t r a n s v e r s e and longitudinal p ' s may be written down from the expressions fo. the heltci.ty amplitudes. Thus in the channel one has fo~" A2/Wr2 sr~all [176] (notation. M~tN,;X(O,7,), ;iN, ~k~*) are parity co~tserving amplitudes; eq. ( , 4 1 ) has been used to replace B2,B 6 by B4, B7):

205

(4.443

4

+) =M

+-I

-;I+ + M-;-It

= JK;~+ - K;,l,

1 = (-W2B1 t 2MRg) ax := (-+

B1+ 2A2Bg+ 2MB5)-7&

;;r theelectric Burn mo&l the heltclty frame cross-sections

are

(4.45)

Th@if& (tzxbnsoeree) cro88-section exh,lbits the non-zero forward peak in (inMel@ phot0producU0n. 8imikr 4~~#~~~etons may be written down for A2 channel helicity amplitudes &ndpolarfzed cfu#b-sections in the GJ frame [ 1741. Here the pion pole (BQ)does tMcontrfbwteto non-zero h&city states (q. (4.22)). ThereL FE:&_jd~2)~~ exh&w rtoforward structure for p praductfon. This has bet+nused as an argllment ndttVMD [ I?$]. However, kinematical, singularities at A2 = (JQ f ,$2 (see ctfon 4.1) introduce a strong ?tz2d endence in the pGJ ‘is, whereas the cor(MM@ in pH ‘Tas .little effect on high mspondingsingularfttsa at W 2 = (Mfp)~ forwati structures. Therefore, I&D prediction%?zould be straightforward interpret VMD as independence of the Ball (this ts unfortunately not an unambiguousase two eqst, (4.41) are actually four equations and 8 generally oq3ressed in u&on. Using the pseudomodel of ode1 based on absorbed Reggeizaand t,qnrrrverse cross-sections for p production may ts from Hyams et al. [181]). = 0 (t = 0 in the figures). Howis featureless. Parm8F, (dcr,/d&2)H hai a ftxwaxd peak whereas (da,/dA2)GJ ticufarlyintetesttng are the very forward 6? lstructures in P~J (ti in fig. 9b). m&e predfction~ should soon be tested. The forms corresponflo a value of JIsj& muchas in eq. (4.43). ?%@connection between this value and absorption is particularly evident in the m&Hof WiINams (1821. Far trmsverse p production in the hellcity frame, the U.*ngle farctops 43~. (4.19) suggest that 0nlJrnucleon flip terms matter fox &* sm;rll (*I in eq. (4.44)). Pure OPE eq. (4.42) gives to

J.L. P E T E R S E N

~"~6

1 ) 3 ~.~- -

- . . . . . .

I'""-

. . . . . . .

I

. . . . .

-

~'1

02 !

-0.2

-04 ~ 10~

,,~ - - - - -

---[--,

t

s ~d o ~

21~s ~do/dr 2~)s ~ do/dt (s ~ doklt) w

~

°' o -Ot~

(a' 0

....

t

_,

{tOt

....

|

-

O00

Or~

lit tr-~21c ~)

0

oot lit

ixm

~

~ V ~ ~)

Yig. 9. Predictions of cross-sections ~a) and density matrix elements (b). for the procou lr-p --. ~'+~r"n at 11.2 GeV/c using photoproductto,~ data and VMD |177}. The symbols s,/0|~. correspor~t to W2, - A 2 and (G T) in the text.

~4"

A2 )ccA2+~3-I

~2 ~2+U2"

(4'@)

H e r e the constant f i r s t t e r m contributes an s wave that v i o l a t e s unltarity enor~ m o u s l y a t high e n e r g i e s . W i l l . m s a r g u e s that the main h e l i c i t y dependent effects of a b s o r p t i o n [183] is to complete)y r e m o v e such t e r m s , By eq. (4.~A) th~s devlm is seen to produce p r e c i s e l y eq. (4.43'~ [178]. Regge conspiracy fits have a s t m i h t r i m p o r t a n t 81 [184,185] c o r r e s l ~ n d ~ to an t s o s c a l a r ] plane pole ~ o r r e l a t e d to the pion at A z = 0, which, however, 1 ~ to well-known troubles with factorization in o t h e r p r o c e s s e s [ 186}. The r e l e ~ point here, ~ . . . . . _.. . . . . . . . . . . . . . . . . ishing B 1 of the type e l . (4.43). The Smportant consequence of this for

is easily seen f r o m eq. (4.45). T h e O P E

zero at A 2 = 0 is n o w shifted to a value

2()7

M E S O N - M E S O N SCATTERING

-%

Q~

.Q2

.~

~

0

1

t

2

3

~

S

0 -2

' -I

2

3

4

S

Pig. 10. Mode| predicUons (fur| curves) of o H - P~'lU (a) and the f o r w a r d b a c k w a r d ; s y m m e try (b} ~n ~'p -' ;;'+;;'-n ugtng a ~impltfiod absorption calculation [1821, Dashed lines indicate "smooth" extrapolations. 42 between 0 and -Vt2. H o w e v e r , neither d % / d A 2 nor dcr,/dA 2 has a zero at that ~me place. T h e r e f o r e the n a r m a t ~ e d density m a t r i x elements and moments will Mve a pole at that point and possibly a z e r o n e a r b y as emphasized by Eane and Ross [144,187]. Williams e s t i m a t e d the consequence of this for Chew-Low extrapolations in his p a r t i c u l a r model. ¢ l g u r e 10a shows that ignoring the effect can p r ~ u c e s y s t e m a t i c e r r o r s of ~ 20% and s u g g e s t s an explanation for why the unitary limit at the p was not r e a c h e d in ref. [170] (see above). F i g u r e 10b indicates that th~ f o r w a r d b a c k w a r d a s y m m e t r y m a y be affected to a l e s s e r extent by the pole z e r o anomaly (provided ver~" p e r i p h e r a l events a r e ~gnored C)). s i m i l a r calculations w e r e c a r r i e d out by Ktmel who a l s o provided a s c h e m e for doing a b sorl~on c o r r e c t i o n s at s u c c e s s i v e levels of a c c u r a c y [188]. Due to the absence of OPE c o n s t r a i n t s for B' = A33(1236) at a 2 = 0 (eq. (4.4b)} we may expect that a c o r r e s p o n d i n g effect is a b s e n t fox' these p r o c e s s e s . This In the following two s u b s e c t i o n s we d i s c u s s two different w~ys of dealing with M| these effects in p r a c t i c a l Chew-Low extrapolation ~,~ork, which may be denoted I) cross-section e x t r a p o l a t i o n , and 2) amp]Rud~ p a r a m e t r i z a t ' o n .

4.4. Cross-section exb'apol~ This method f~-nD,hes in principle a completely model independent analysis. H~ever, vezT high statistics is re~luired and physical information difficult to implemeat

$08

J.L. P E T E R S E N

To avoid k i n e m a t i c a l s i n g u l a r i t i e s between the p h y s i c a l r e g i o n and the pi~n pole, A2 = _~2, we c o n s i d e r the ,_~.e y M ) G J ' s with M = 0. T h e s e q u a n t i t i e s have a double pole a t A2 = -~2 but unlike d~/dA-2 d s d l 2 they have no s i n g l e pole (subsection 4.1). T o avoid the p o l e s at p l a c e s w e r e dc,/ds dA2 = 0 (see p r e c e d i n g subsection) it is ~urther a d v i s a b l e tc c o n s i d e r the u n n o r m a l i z e d o b j e c t s

dc

dA2 ds (Re

>

(4.47)

•

In g e n e r a l the absolute r,o r m a l i z a t i o n is e x p e r i m e n t a l l y p o o r l y known but to the extent that only one o v e r - a l l c o n s t a n t can a l l o w f o r the i g n o r a n c e , that may be det e r m i n e d by r e q u i r i n g e x t r a p o l a t e d m e s o n - l n e s o n p w a ' s to be u n i t a r y . (~le then m u l t i p l i e s I',4.47) by (A 2 + ~t2) 2 or s o m e m o r e s u i t a b l e quantity (including if w a n t e d the i n v e r s e of a D P or BD type f o r m f a c t o r [ 165], h o w e v e r , c~-~ should be e x e r c t z e d that it c o n t a i n s no c o m p l e x p o l e s or o t h e r unwanted ne sing u l a r i t i e s ; c a l l the object t h u s obtained f(A ) f o r each value of s. Fitting f(A2) to a p o l y n o m i a l in A2 the r e s i d u e ts obtained by e v a l u a t i n g the p o l y n o m i a l at ~2

= _~2.

"

"

Several people (Cutskosky and Dee [189], Ciulll [190], see also I'isut [!91]~, however have suggested that improved accuracy, better error estimates and m o r e eccmomlcal use of the data Is achieved if these operations arc perf',rmed in s o m e conformally tra~sformed variable, z, rather than in A 2. Ideally ~he conformal t r a n s f o r m a t i o n st ould m a p the p a r t of the A 2 plane o u t s i d e the ~ t s pcrtain!n~ to (4.47), onto an e l l i t s e in the new z p l a n e s u c h that the i n t e r v a l ~,n the real positive A 2 a x i s on which ~Jata a r e to be used, is mapped onto the i n t e r v a l between the loci of the z plane e l l i p s e ; i . e . , without l o s s of g e n e r a l i t y , -1 s z 1. in practice one may a v o i d the c o m p l i c a t e d c o m p l e x cuts of (4.~7~ (cf. subsecti¢~a 4.1) by cut. ling away a c e r t a i n s e g m e n t of the A2 place. Exptictt e x p r e s s i o n s ',(or the needed t r a n s f o r m a t i o n a r e c o n v e n i e n t l y s u m m a r i z e d in ref. [189] and a r e ,,ery easy to i n c o r p o r a t e ~.n a c o m p u t e r p r o g r a m . T h e p o i n t s about t l ~ c o n f o r m a l transformation a r c a s lollows. (i) I g n o r i n g f i r s t the effec~ of experimeat:~l e r r ~ r s , l e a s t s q u a r e polynomial fits to the d a t a on -1 -.< z < ~ will coi~,er~,c throu~hou~ the z p l a n e r~llipse a:~,t t h e r e f o r e throughout the cut A2 platte. T h i s is to t)e c.)rnpared with A 2 polyn o m i a l fits that only c o n v e r g e ~n a s m a l l finite p a r t of the A2 plane (an ellipse wi~h l o c i at the end p o i n t s of the i n t e r v a l containing u s e d d a t a , and bounded A2 p l a n e singularitie,,~ - no! a c i r c l e a s often stated). A c c o r d i n g to general t h e o r e m s t h e r e f o r e [ t89-191] f e w e r t e r m s a r e needed f o r given apprc~ir~Rt~on a c c u r a c y in the z v a r i a b l e than ~.n the h~. c a r i a b l e . (i i) Low o r d e r p o l y n o m i a ' s in z c o r r e s p o m t to c o m p l i c a t e d " f o r m ~actors" :a h 2. A l s o . the pole p o s i t i o n is r e l a t i v e l y n e a r e r to the d a t a i n t e r v a l m the z plane than in the A2 plane. Therefore, use of the z v a r i a b l e m e a n s tt~at a ¢=mo~,~he:.' o b j e c t has to be e x t r a p o l a t e d a s h o r t e r d i s t a n c e . (iii) W h e n experiment~l errors are v--~---.~""~.~nly...._ the_ ......... lnwest order pol_vnom:al g ring fit to t~he data C~_rlbe used in a justifiable way to perform the extrap.,~lat~on. ]i~oth (i) and (ii) then argue ~n fa,'ot~r of z over A 2. The extrapolati~m error is a s u m of the usual statistical error and a truncattcrn error as~,,elated with neglect o~ higher terms in the polynomial e:_pans~cn. The [,~ter can be e s t i m a t e d using the fact ~hat at the pol~ p o s i t i o n , z =: Zp, the ra~c of c o n v e r g e n c e of the p o l y n o m i a l e x p a n s i o n is like that of a g e o m e t r i c s e r i e s with quotient

MESON-MESON SCATTERIN 3

y = r/R

':'.0! ~

= (a ~/3)/'(a + b ) .

Here a a n d b a r e the s e m i - a x e s of the z p l a n e e l l i p s e a n d a a n d fl a r e the s e m i - a . x e s of a new e l l i p s e w i t h [oc! at i l a n d a :: [z.. I. T h e t r u n c a t m n e r r o r is often n e g l e c t e d in e x T e r i m e n t a l Cl,~,v;,-L,c,w ~.'xt :apo~ation w o r k w h ~ ' h r e sults in p l o t s w h e r e t h e e r r o r b a r on the e x t r a p o ated v a l u e is s a ~ a l l e r tl,a~ the e r r o r b a r s on t h e d a t a p o i n t s . T h a t ' s highly_ u n r e a l i s t i c . Of c o u r s e the t r o u b l e i s only grea.'.er w h e n o n e w o r k s in t h e A~'. v a r i a b l e . S c h e m e s t h a t a r e t e c h n i c a l l y m o r e e l e g a n t n a v e b e e n w o r k e d out [1921 and in p a r t i c u l a r t h e d e l i c a t e ¢ . r r o r e s t i m a t e h a s b e e n g~ven a m o r e s a t i s f a c t o r y t r e a t meat. The u s e of q u a n t i t i e s l i k e ¢4..~7) anly ' : u a . ~ t i s f a c t : ~ r y in t k a t a c o n s i d e r a b l e amount of e x p e r i m e n t a l i n f o r m a t i o n i3 no' reed (.(~r[?f¢~)). Al,';o it i s not o b v i o u s that (4.47) i s t h e m o s t p r a c t i c a l q u a a t i W .o e x t r a p o l a t e . Chin C.han~, Sbih [~93] .vc ce~;tly s u g g e s t e d to w o r k f a t h e r w i t h c i , : , s s - s e c t i o n s with we: l - c l e f ! n e d v a l u e s ,of the : : w a r i a n t s (4.1) I n s t e a d of ~ v e l l - d e i l n e d v a l u e s of d i - m e s ,n a n g u l a r m o m e n t u m . For fixed W2 t h e q u a n t i t y

,~~ ¢ , n , " t. a 2 ) ~ (A2 , bt2) ( 1,:/'}/''[ 2,

F(s,

.

,1 4 t-!'~

.

is c o n s i d e r e d ,

w h e r e <[c)~[2> is t h e s p i n - a v e r a g e d l n v a r | a n t a m p l i t u d e squat,, ,! {eel. (4.6)) w h i c h m a y b e o b t a i n e d f r o m t h e d a t a . T h e n : - m a y ue s p l i i a s F(s.

/ ~ m " t ' a 2) = FB{,;.

' m " t ' a 2) , R ( s , t ) " { a 2 +

p2,

~

.

9

where the r e s i d u e R(s. t) is i n d e p e n d e n t of A 2 a n d '~./lI'm a.~(~ ,s f e i n t e d ~ ,~ known way to t h e ~ n v a r ~ a n t m e s o n - m e s o n s c a t , e r i n g a m p l i t a l o ' s 1 : a : ~ i l . n : ' . : : , ! ] ~ raises of s a n d a,~ . . . . . . . . F,~ a n d R a r e o a r a m e t r i z e d by s u i t s ~te ' "" c ~. ann t r , ' s p e c t i v e l y [ u s i n g p o l y n o m i a l s in ,:'on~orma~l:y t r a n s : , ~ toga ":ar~ab~es :~,,. ~.bo',e or s o m e t h i n g m o r e o r l e s s e q u i v a l e n ¢ [ 1 9 2 , 1 9 3 ] ) . Aa .:nbiase.d opIln~a! e ~ mat~ of R(.~. t~ i s oDtained f o r t h e f~t with t h e m o s t M~wiv v~ ~-via< ,:[~ (th;s c a n }w. pre, ise|y d e f i n e d [189,192]~. T h e l : a r a n ~ e t r i z a t i o n ': ,v (t A~! the act that s ~ n g u l a r i t i e ~ a r ,~ p r e s , : a t aot (miv f o r fi ~ v:, [ ar~;l i~.~ ~, for fixed v a l u e s of o t h e r i n v a r i a n t s l i n e a r l y d e p e n d e n t on t a r d A2 (ef. s u b s e c t i o n 4.1). T h e v e r y n o t e w o r t h y p o i n t a b o u t k e e p i n g M~¢, m , f i x e d r a t h e r than l ( m ' , m ') is t e e h a i c a K l y t h a t the h o r r i b l e c o m p l e x s i n g u l a r i t ; e s and kir, e w a t i c a l s i n g u ! a r i ties of the q u a n t i t i e s f{{y) a r e a v o i d e d a n d f u l l e r u s e of the a p . a ! y t i e i t v informa!ir,~: can be m~t¢|e. P h y s i c a l l y it m e a v ~ that the a a t a a r e a n a l y z e d f o r f i x e d p o i n t s ~n the ~ l } t z p l o t a n d one is a b l e to s t a y a w a y f r o m c o m p l i c a t e d N* r e s o n a n c e r~,{i0ns. The r a o s t i m p o r t a n t d r a w b a c k a p p e a r s to b e that t h e r e q u i r e m e n t of :~ (.,./) being i : , d e p e n d e n t of M~/, m , h a s not b e e n buii~ ~nto the m e t h o d . Th~s would agF t0be n e c e s s a r y with a n y f o r e s e e a b l e s t a t i s t i , s . ,,.~,

.

.

.

.

.

.

.

r

.

.

.

.

.

.

.

r

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Doin< the p a c a m e t r i z a t i o n at the amp!itud,, level is i~ principl,:' a rn,~c} v;~

~0werf,, w a y of m a k i n g u s e of t h e d a t a , a n d . ~so physi(::~l }n~orrnati~m .~':.~T~} . . ,

~"

,,

veyed t,., ' h e a n a l y s i s . To the extent that a s i n g l e theore i c a l model such as a ]~v~......

ooiat 'Je'aez~.ano a m p l i t u d e w a s a b l e to fit t h e d a t a , one rr:~,~ht.,:, f~,,~.,. ~:~ti~fi:':~.. ........ ~,~ :,,~ :,: least :~.,asistency with theory, h a d b e e n e s t a b l i s h e d . At p r e s e n t , h,.~wever. ~5 Pn~'n0mm~o!ogy i s f a r s h o r t of d e a l i n g with a l l t h e e f f e c t s of the p r e c e d i n ~ s,~h.~e::tions. T h e r e f o r e ,me h a s t9 r e s o r t to c o m p r o m i s e s and b u i l d m : ; d e l ~ with po,~re~

210

J.L. PETERSEN

predictive power but l a r g e r flexibility. If the latter is .emphasized the scheme ts likelyto be tecl,~cally quite complicated. So far no really satisfactory proposal has been given, tmwever, we shall comment on some of the ones available. Schlein [194] suggested to exploit the factorization assumption made for example in all absorption calculations f("v)(S, 4 2 ) = f(/v)(S, ~2) al(s)

(4.~0)

whe~-e al(s) is the-on-shell meson-meson pwa and the reduced amplitudes ~ / ~ are (rei~tively) real. To the extent that fie1 is slowly varying a r a t h e r e f f i c i e n t ~ r t , metrization is achieved which is able'tb accommodate generalized absorption elfects. Note theft eq. (4.50) is assumed also for those hellcity amplitudes to which the p o l e d o e s not contribute. This implies that the method r e l i e s on a certain flail state interaction philosophy more than on dominance of the pton pple. Froggat and Merga~. [ i95] noticed that the reduced amplitudes ~ ) cannot be slowly w r y i n g wtt~ Az due to the presence of kinematical singulax:~t~es. Instead (for ~N-" 2~N) the following par~mctrtz~.tton is suggested for 1 ffi 0, 1 (s and p waves)

½;0~, 4~) =az(s ) 42 +p2 ÷

y/(c)(s

=

(4.S])

f l ( c ) ~¢s

,-~;,1 (s' A2) = al(s) For l = 1 the expressions Rre obte.ined from formulae s i m i l a r to eq. (4.44), however, written down in the A2 channel [ 174,196]. Then the factortzation assumption eq. (4.50) is made on the Ball amplitudbs for each s

(4.S~) For i ~ 3 (4), Bi is assumed independent ~ A2 and s ( r ] is related to B 1 and FI0~ B 8) and for i = 3 the,p~n p o ~ is inserted. Also some of the ~ ' ~ do not contri~¢~ to leading order in q 4 z in ~ and ohhers a r e assumed small (near vanishing cl (P111 1)G~). In a completely s i m i l a r way for s wave production, fo~c)(s, 4 2) are ' . (I; p ~ r a m e t r i z ~ l in terms of (two) Invarlant amplitudes for the proc~ss ~N -, (rN, where ~ i s a scalar particle [196]. In a fit to data from s e v e r a l coll~boratlons on ~ °p -. ~:%-n Scharengulvel ct ~. [197] showed that with available statistics (which is poor in Me crucial region • A2 thi~ ~ : e l adequately describes the data. For B1 (i.e., F 1) a valu~ ~t - , -. . . . . . -- . . . . . . . s ~ m a t e eq. (4.43) results, whereaJs for B8 no consistent value ~ s ob~inr~d. The model illustrates the basic advantage o~ amplitude parametriz~h0u. As we saw in ~ubsechon 4.3, the degree of non-evasiveness determines the pole ~.ero anoma2y o~ normalized moments and prevent tho~e from eztrapol~ttng smoothly. However, due ~o the large steep dip in d~/ds d~ 2, the :-Mue

£~2~

(d /d dA21 _0)

MESON-MESON

SCATTERING

211

at A2 = 0 iS h a r d to d ~ t e r m m e .

In the model oi eq. (4.5i) on the o t h e r hand, the crucial p a r a m e t e r may be m o r e readily m e a s u r e d by looking at those quantities { ~ f ~ 0 ) f o r example) not containing the pion pole and ignored in n o m e n t extrapolation (cf. p r e c e d i n g s u b s e c t i o n ) . In fact in such s c h e m e s no extrapolaL~n is done at all (computattonally); the p a r : t m e t e r s a r e d e t e r m i n e d by fittings in the ph}'sical region and the pole r e s i d u e simply obtained by evaluating the p a r a rnetrized I o t a s at the pion pole. Equations (4.515 a r e c o m p l i c a t e d by the u n p l e a s a n t singularity at A2 : 0. However, (subsection 4,1) this i s a s o m e w h a t s p u r i o u s trouble as f a r a s o b s e r v a b l e s go. It is avoided by working in the helicity f r a m e and using the d i r e c t channel e x pressions eqs, (4.44). T h i s ts in e s s e n c e what Kane and Roos [187] do, allowing slightly m o r e g e n e r a l than c o n s t a n t behaviour for the /~i's (eq. (4.52)) but n e glecting all but nucleon flip t e r m s (as may be justified for t r a n s v e r s e p p r o d u c -

(Fi)

tlo.). As they stand, these s c h e m e s ignore (s. ~2) c o r r e l a t i o n s (subsectio~ 4.2). This m~? be p~trtly c u r e d by the d e v i c e eq. (4.36). At h i g h e r e n e r g i e s w h e r e d waves and v~.t higher p w a ' s a r e p r e s e n t , analogous p r o c e d u r e s may be quite laborious to wor~: out. M o r e i m p o r t a n t , h o w e v e r , is the c o m p l e t e ne~.lect o" competing processes such as ~ N -" ~ N *

or (related to that) diffractive d i s s o c i a t i o n . To allow in a c n s i s t e n t way for all possible e t f e e t s it would s e e m to be Inevitable to b a s e the r ~ t r a m e t r i z a t i o n on the amplitudes A i ( ' = 1 , . . . , 4), eq. (4.65. 4.6. E x p e r i m e n t a l r e s u l t s on ~

scattering The r e a c t i o n s mainly u s e d in p e r i p h e r a l pton p r ~ perlmental r e v i e w , see Schlein [143]):

;z-p - zr-=Op

(a)

7r+n --" 7,+~ -p

(e)

;r-p --" 7r-n'+n

(b)

r~+p --" rr+n+n

(~)

~-p -. ~o~on

(c)

;~+p - ~'~;;-A++

(g)

~+n --" nO~T:>p

(d)

n - p -~ n - ~ - a 4 +

(h~

:clio ~ a - e (for a recent ex-

4 . T~ .in)

Apart from (a) they a r e all chaxge exchange r e a c t i o n s so that I :: 0 exchange to the nucleon (in p a r t i c u l a r vacuum exchange) is forbidden. In (a) c l e a r w like contributions are s e e n ( T r e t m a n - Y a n g d i s t r i b u t i o n [198] cr(~-p--, pOn)/cr(n-p-, p-p) ¢ 2 Consider f i r s t the region below the KK t h r e s h o l d region (~ 1 CeV) where ~ scattering is o v e r w h e l m i n g l y d o m i n a t e d by the well stud~ed p m e s o n . Important s waves a r e p r e ~ e n t in addition and most of r e c e n t activity has fc cused on 5°. Figure 11 shows the new r e s u l t s of Baton et al. [200], u~tng r e a c t i o n (4.53a) to first obtain 61 and 80 (P1 and $2). The advantage of that is that n~ I - 0 is p r e s e n t . Chew-Low e x t r a p o l a t i o n using c o n l o r m a l m a p p i n g techniques was applied ( s u b s e c 27n 4.4) which is supposed to deal a u t o m a t i c a l l y with the ¢o. No violations of the evasive c o n s t r a i n t was observed._ We shall come back to other w a v e s in fig. :. i. Figure 12 shows r e s u l t s for 52 frc:m Colton et al. [201] using t~e modified n,~n evasive form f a c t o r method [165]"to e x t r a p o l a t e r e a c t ; .o n .(e, 53h) . .to the . pole Several older r e s u l t s and two t h e o r e # , c a l e s t i m a t e s a r e shown as well.

212

J . L . PETER~EN ~ m

~0~

IO0

50

Do -------.~:48~, ~ ~ ,,J,

O8

Q4

,,

1.0

06 M,. (G~V) 12

O0

10

t2

F~g. 11. Phase shifts u I u~, v2, ~o, uo {P1, DO, D2. t0, $2) [rom B~ton et at. [200|. D o ~ curves correspond to zero inelasticaUy and the b a ~ s result when Inelasticity is taken into account.

IG

_

®"r~ex~rime~ ('i~e I} o T h i s ~ ~ t (Tc~eIf)

Q2

0.4

Q6

Q8

tO

Fig. 12. Results for ~ from r~., [201] and several earlier determinations. C l e a r l y ~2 is n e g a t i v e , slowly d e c r e a s i n g in the r e g i o n (600 MeV < - ~ £ 850 MeV). At the p m a s s ~ w d u e of 52o = -15 ° , 5° is likely. Simihrly there is not m u c h discussion ~tbout 51 in this region (fig. 11). Most i~ziorrnation on 5° iI the D region at present c o m e s f r o m study of s-p in-

M ESON-MESON SCATT ~.WRING

213

160 +

"

o,,g'

..........

060 Q64 ll~

.

+

if'f2 a76 O.BO (184 080 M,, ("~V)

Fig. 13. Results for'8~ deduced from the forward backward asymmetry, ref. [202]. terference in reaction (4.53b}. Figure 13 shows the results of Scharenguivel et al.

[I02]. More s~ciflcally these authors study the forward backward asymmetry and extrapolate to the plon pole. F r o m the precedir,g sections we expect such a treatment to accommodate form factor dependences and correlation elfects and to be 0nly sh~htly affected by non-evaslve contributions, although this last point relying on fig.10b is m o d e l d e p e n d e n t . F i g u r e 13 e x h i b i t s the we",,-~,~,,~' . . . . . . . . . .,.~. .,.,. . ~.~. dow:: ~ m b i g aity [203 , 204 ]. T h i s i s i n h e r e n t to tI,e study of s - p i n t e r f e r e n c e only, in which (he ....

glecting 82) one measures sin 6} sin 5°o cos (51 - 5°), so.that the ambiguity

results D i f t e r e n t t r e a t m e n t s ( d i f f e r e n t v a l u e s f o r 82, use of f o r w a r d b a c k w a r d asyml, e t r y o r 70 ~ n o r < 20o, a n d the f a i r l y m o d e l ~ t d e p e n d e n t t h e o r e t i c a l p r e d i c t i o n s (fig, 2) holds up.

Cline et al. [205] proposed a simple way of measur°ig ~,cattering ~engths. Using ~ t a on r e a c t i o n s (4.53b, d f) t h e y po:n: out t h a t in fo~m .-,~u~,~ . . . . . . . . . u,,,~,~,~"~ ~-~c'~u'~ m0dets, t h e a p p r o x i m a t i o ~ c o s (5 ° . o "~2o) ~ I at low ~nergiP,~, y i e l d s 1

¢(~+,~+) =

Jr2 + 1 -

where r ~ s i n ~}°o/sin 82 .

,

c~(,~+~-) : 2 L r 2 + ~ + r

j

Z14

J.L.

PETERSEFi

/

80[- + ~ ~ ' , n A

,~E

/

EERt~E'~'COLC~iX~,

,TOF~tNTO.WIS~'~.'.'SIN

OERN

....

_

j

¢ ,

.

J l

~

,

o

t

03O

Fig.

1~.

i ~sults

for ~o from

O38

f.~rward

Or.6

t

L _ _

O~Z.

b a , , k w a r ~ ~, a s y m m ~ r y ,

O~

r-

. 127].

~>ata ~ l n t ~

nr,. l r-m

other earlier d~termlnation~.

Consistency yields

r -~ a 0° / -(~O 2

=

-(3.1 :t 1 " 1) ~-1

i'~"" 55i•

in nice agreement with Weinberg. PossibLe non-OPE forces and neglect :~f effective r a n g e s r e n d e r the r e s u l t uncertail~, blot the ~,~ethocl i l l u s t r a t e s the pm~,e~0o~ c o m b i n i n g dat~ Irom differen~ i n t e r a c t i o n s ~n the a n a l y s i s . S c h a r e n g u i v e l et al. [27] p u r s u e the ~o:'ward b a c k w a r d a s y m m e t r y me h¢:d. They m a k e a ,!oint l i n e a r s and ,x 2 p~ r a m e t r i z a t m n to allow t o t P c A c tyl,, ~',~,r~el a t i o n s (cf. s u b s e c t i o n 4.2). Knowledge on low e n e r g y 8{1 and ( "*. is obtained by ext r a p o l a t i n g f r o m h i g h e r e n e r g i e s by e f f e c t i v e r a n g e f o r m u l a e . . , m b i g u i t i , ~ s in this e x t r a p o l a t i o n a r e m a i n l y r e s p o , . s t b l e for the width in t h e i r band of solutions (fig. 14). In reaction (4.53b), the obvious way to cesolve the ambiguity is to get gced values for the extrapolated isotropic part of the ~+;r- distrit,ution since in that, onl}~ e v e n I w a v e s c o n t r i b u t e to the pole residL~e. A s we have s e e n , h o w e v e r , de'2~larized p':s c o m p l e t e l y d o m i n a t e the isotropic' p a r t in the ph'¢sical r e g i o n {subsection 4.3). M o s t r e c e n t attempt.s [o circ,:n: ,.tent t h i s difficulty a g r e e with figs. 14. 11 and 2 1 2 0 6 . 2 0 7 . 1 7 t . 2 0 0 ] . The ~oso c h m m e l s (4.53c, d) a r e fr~e ~ f d e p o h t r i z e : l ;~ t r o u b l e s but ex::erimen tally e : : t r e m e l y difficult. High s t a t i s t i c s : : x p e = i m e n t s ave n e v e r t h e l e s s n>w ~,n p r o g r e s s a n d a r e hoped .re p],ay an i m p o r t a n : r o t e in finally s e t t l i n g the q.~e~ion So f a r , how-eve:, no g e n e r a l a g r e e m e n t he ~ b e e n a r r i v e d at f r o m t h e s e re, act~.eae n e i t h e r below nor ak, ove the p. F i g s . . 5 I:0~] and 16 [209] r e p r e s e n t the :wc~ d::f e r e n t ~schools ~f thcught. C o m i n g to the above p r e g i o n and in p a z t i c : : ! a r the r e g i o n a b o v e I GeV, i: is generally realized tA~ ,,~• al!c'v?oace for }ne!~sticities and higher waves ace hexes-

MESON-,,~,.,~.'
"~

15

R~ /

l

/ N

¢J

(it,

Q6

0.5

Og m <" (Oe~/~)

a7 i

1

from up and down [ o r m s of 5¢c~, re[. 12~),~i. sarv. B:~t.o~ et a l . ~2001, fif~. 17, f i n d p o s s i b l e i n e l a . s t i c i l i e ~
~ 5 - 7 1 ~ , - ~ {it 3 . 9 O e V c

usin~

t h e ~}on-ev,_t.{i,'e

scbemv i1. <'~' servect W i t h i r ~ t h e f i r s t reach th,~, ~ l i . t ; ~ r 7 l i m i t .

thresh,,l.~ s u m m a r i z e

30 M e V a b o v e t~e K K tl,.:e.~i~,,i<;!, :" r, ~. :.i)~:',~ "T h ( . [olicJ,:viu!.~; ~.v,<~ra~:{.-s ,-:~.: t<,. !, :..~:I ]el,;} 7st,"; .,{,
the {ic~din~ls.

10

............

T ..........

T .............

08 %

i l~

(~ 04

-....

}7

T ...........

i" . . . . . . . .

1

............

l

!

,-I )

i ~:'

#.i

1

Fig. 15

l , J r m l a c ' t o r r.,~,,iifi(:.d O P E H~o i n a l a s t i c i l i e s ,v,,r~:, (>b

~,i,

L!e,,<~m':~.

216

J . L . PET~RSEN

..... '

=

~

~

0.5- ~o~

tO

0.5 ~P

1:0

o!

10

05

0

117

08

09

tO

U

't2

t4.. (C,e#)

Fig. 17. Inelasticil~y parameters pertaining to fig. 11, ref. {200}. <~(~-Tr+ -* K-K +) = (2.2 ~ 0.5) mb

~(~-~+ -. ~-~+~-~+) = (0.7 ± 0 . 2 ) m b ~(~-~+ -~ all inelastic) ~ (2.8 ~ 0.5) mb

(4.S or

~o < 0.45 +0.13

~0~4 3

The ~irst two lines should a g r e e for p u r e t = 0 s wave, h o w e v e r , important corr e c t i o n s could a r i s e f r o m neglected n, ~n-OPE f o r c e s . The s t r o n g onset o5 | ~e~sticity at the KK threshold (~ 995 MeV) is h ~ d to explain away in t e r m s of r r ~ tion m e c h a n i s m , however, and i3 likely to ~ a c r u c i a l a s p e c t of the mesonm e s o n s c a t t e r i n g . Also, the complete lack of e v e n t s below 995 MeV s~ong~y s ~ g e s t s that 5 ° is nearly e l a s t i c in this r e g i o n [211,212,198] and s u g g e s t s tha~ Jig. 17 represents spurious effects.

Oh et al. [198] anR1yze reactions (4.$3b) using an absorption modified OPE forrealism. Fig. 18 shows their results a~in with important but less marked iae~ticities.

MESON-MESON

I

-

SC A T T E RING

217

r---'--'r

~

-20°

= ' / a S T

OB

t0

~2

~

Fig. !8. Phase shifts and inelasticitics from ref. [198].

Evidence tltat the t- 0 s wave is appreciable in the fo region is clear in the angulax distribution of the decay fo-. ~+Tr- [213, 214] ~,~ qbown (reac~.ion (4°53g)) in fig.19. A strong Z = 2 component is evident, but at cos 0 = 0 there is a depletion 0[ events as c o m ~ e d to ct pure (P2(cos 0)~2 (dashed curve). The p wave, how.. ever, m~Ist be small causing little asymmetry. Figure 20 shows the A B C collaboration r e s u l ~ [214]. The genera.1 a g r e e m e n t with fig. t8 I198] a p p e a r s to be satis~ctory. ~ e r e ~s, however, considerable reasons to believe that the suggested constant behavtour o[ o y e a r f r o m " " ^ r~ev r~t, The Doiut ~s that an s wave KK tr~ela~tieitv will make w° d e c r ~ a s e above 2m~ as, ; ( ~ s . (2.16)-(2.18)). By ~nalyi~city this implies a [argo a c c e l e:~ti~ of 80 bd,~)w 995 MeV which by unttarit~" m u s t be in the p o s i t i v e direction. I~is eff<~ct ( r e m i n i s c e n t of the B a l l - F r a z e r m e c h a n i s m ) vary turn out te be c r u c~l for d i s c l o s i n g the b e h a v i o u r of 8 ° above the p, It is c l e a r l y b e s t analyzed ia ccu]unct~on with a KI~ production a n a l y s i s (see next subsection). Fine s t r u c t u r e s around 1100 MeV have been obse: veal by s e v e r a l people I215, ~IB,~00,198]. F i g u r e s !1 and 18 ~how examples° T l e i n t e r p r e t a t i o n is unclear, and it is also not c l e a r w n e t h e r in fact it is the s a m e e[fect. Baton e t a l . I200} and et alo I198] s u g g e s t a p' type i n t e r p r e ~ t i o n (dotte, i c u r v e s in fig. la). However, d,IJIJ~Ib

.....

215

J.L. P E T E R S E N

i 40¸

I I

\ \

0~ -I

~.~------i 0

I

cosE} Fig. 19. fo _. w÷~- decay angular distribution, ref. [214]. Miller et at. [216] ru~e out I = 1 from lack of appearance in the ~-~o channel. S* -. 2~ is generally discarded (the structure ts too narrow for S* and too stro~ for an s wave) whereas an I = 2 interpretation might work (full curve in fig. 18). Split structures Inthe fo (an SU3 companion of the A 2) has b ~ u looked for and

both claimed [217] and disclaimed [218]. Phase shift analysis in the g m e s o n region has so far not been possible and it is quite unclear whether one or more (possibly narrow) resonances are requtr~ (for a review, see ref. [219]). However, the e.xPected spin 3 a s s i ~ e n t appears by now to have been proven (reaction (4.53b)) [220]. Finally, fig. 21 shows s ~ s t l o n s for "-~",u-~.~,~,~,^-~ ,~-~o~..."'~-~;'~ ~+~" cr,~s-sections b~ Case et al. [221] from 1.5 to 3.0 GeV/c 2. The total cross-sectlons were obmln~ from forward elastic differential cross-sectlons assuming puxely imaginary amplitudes and. usl~lg~the opt;cal t h ~ r e m . The results are c o m ~ t t b l e w i ~ a ~ p -

totic estimates from ~N, NN and ~N using e a r l i e r quark and Regge .m~els: ~ t : 14 mb [222], .tot -~rw. __ 16 mb [223 I,

MESON-MESON

SCATTERING

2 t ~.}

1 I( 05 i 0

J,

J..........

,L___--~J

i Q5bt0 ~ .......... QB

1

1.0

1.2

1.4

(.~"~')EFI=MASS (OeV) Fig. 20. ~)l~ase shifCs and i n e l a c t i c i t t e s f r o m ref. [214]. 1

30-

T--~

T /

E

0

T

;

]

?

+~~+~,~%+++~+~+,~ T

. . . . . . . . . . . . .

tO

|.__

15

~,

ZO

Fi~:;. 21. T o t a l and e l a s t i c v+~7- c r o s s - s e c t i o n s

2.5

@l

30

trGm ref. [22l l, ( Je~)"

220

J.L. PETERSEN

4.7. ~vpe fimental information j~om I ~ producti(m The study of elastic KI~ scattering by means of a Chew-Low type technique is unlikely to become feasible in the near future: c r o s s - s e c t i o n s for KN ~- KKY are 100 times s m a l l e r than ~hose f o r ~N-*2~N and the K exchange pole is ~ 12 times further away from A2 -_ 0 than the OPE hole. Pton lnitiat on KK couplings to v~r° ious inelastic ~ t u s df experimental sophistication is f~r short of making a phase shift analysis meaningful. Available knowledge ts most conveniently summarized in t e r m s of partial decay modes for a variety of bumps f, f', g, S*

(natural spin parity)

~, A2

(unnaturalspln parity)

and ~tzdated constantly by ~rtlcle Data Group (see also table 2). For summaries on bump-hunting, see for example Lal [224] and Astier [2~5]. For a review on the ~ssibly split A 2 see Barbmro-Galtleri [226]. At pr,rse~t the situ~ttion appears m o r e e o u f u s ~ than ever by the fact that recent high s~tts. tics experiments have given no evidence for splitting [~27].

Pion initiatedS* production with subsequent decay into K ~ ',s~.n exception to the above general situation and a reaction which is likely to platya crucial role (combined_ withanaiysis of ~N -2~rN') in disentangling s waw~s in # - " ~r,, ~ - " k'K and KK -. KK. In the approximation that no other channels (4~) than ~ z,.J Ki; couple, eq. (2.17) gives for the S matrix for fixed ] and I [228}

(S0r~r)

$(~-.KI~)I_ (~}e 2i~(~)

~S(~-. KI~)

S(KR)/

i ~

tv~l_ ~ e i/{(;''r-"K~}

ei~(~-" K~) 7{e

y

with 6(n.--zR) + O(KR). Beusch [2281 fitted the Kl°i<~ mass spectrum [2101 to B r e i t - W i g n e r ' s for S*, In and A2 plus background to get e s t i m a t e s for the t ~ c r o s s - s e c t i o n s for S* and fo production. These were assumed to be p r o d u c ~ pion exchange for which there Is some evidence (whereas for A 2 ~ course OPE is forbidden and n a t u ~ l {rarity exchange a p p e a r s to dominate). With these, the s-d interfer~mce term 0"I~) may be interpreted to yield a value for cos ( 6 ~ ( ~ - . ~R)O "'~ -{~2(n~-KK). Assuming the d wave to be given by an inelastic fo Brett-Wi~er, b~(alr-, t~.~) is fixed mod~ (see fig. 22). Next, parametrizing S ~ ) by a complex effective range formula

S(KK) -- q-cot A + iq q cot A - iq '

1 q cot ~ -- a÷ib ....

~a,~.q2

(4.~,

0 are parametrized and t~e~oc an inelastic resonance :orm, both d~(KI~) and 17o fore (eq. (4.57)) {S(7rTr-. KK){2. As is w:AI known, the S* may be fired eitherway [229,230]. Parameters depend much on background assumptions. For the resonant solution Beusch gives

1040 MeV ~ m s , ~< 1100 MeV;

150 MeV .-~ FS, < 300 MeV

(4.50)

and for the form, eq. (4.58) a + ib = (1.3+i0.35)

fro"

r =

1.4 fro.

(4.6o)

M E S O N - M E S O N SCATTERING

22t

| ......

360

f

>__

_

~.

t:

)2

a)

,°0T

t3

)

u

t2

t3

b)

Fig. 22, (a) P h a s e s d e d , ced (torn s - d i n t e r f e r e n c e and d wave phase shift e x p e c t e d from the fo; (b) s wave r,~" "-" KK p h a s e s and ;r,~ -" ?rff p h a s e s (ref. [228]). may be obtained fron, the With e i t h e r p a r a m e t r i z a t i o n di°(KK) i s f i x e d a n d 5o,~nrr) o. value of 8°(zrr -~ KK). F i g u r e 2 3 s h o w s p r e d i c t e d ~r. s w a v e A r g a n d d i a g r a m s a s suming ~otr~-~.o~,., = 9 0 0 a t t h e KK t h r e s h o l d ; r o t a t e d d i a g r a m s wil~ r e s u l t f o r o t h e r values.

!

Fig. 23. ~[<:,de| A r g a n d d i a g - a m f o r ! = 0 s ~,~ave try anaplitude. Full c u r v e : S* r e s o n a n c e .... dotted c~rve- a c a t t e : i n g le~lgth with e t f e c t i v e range ( r e l . i2281~.

222

J.L. PETERSEN

As emphasized by Morgan [231] considerably m o r e Interesting results emerge if below threshold 7z~ information is conveyed to the analysis and If analyticity ts pro!mrly implbmented. Due then to the (infinite) acceleration of 6 ° ( ~ ) just below the ~ threshold (see preceding subsection), the ~ l u e of this phase at thresl'~Id i is a v e r y Inconvenient p a r a m e t e r . Instead, M o r g a n (using a K m a t r i x f o r m a l l ~ starting from e l . ( 4 . 5 7 ) ) p r ~ a c e s solutions for ~o{n~)andw°n that jotn tV either the down or the up b~nCfi ~Of ~ t o n e t a l . (flg~ 11) at 800 MeV, and f o r which the amount of accelerattot~:iS d e t e r m l n ~ by the Beusch s wave p a r t of the Kl°K° pro~ duction (neglecting 4~ inelasttcitles). It is found in either case ~ a t the I = d s wave has a zero near below the ~ threshold. This is In qualI "tative agreement with the prediction of Lovelace [47], fig. 2, who has, however, the zero at a low. er energy due to the lower rmass of the S* (it also appears to be on a different sheet~,. In either case, specific predictions for #+#- production in reactions (4.53b, g) follow. Thus for the solution joining onto the down branch at 8 ~ MeV, the #+~- re~tss spectrum will show a sharp drop just below the ~ threshold (this is impossible for the up solution, since that stays ~ear hhe bottom of the Argand diagram all the time [212]). Also, (Y~) will drop ~t t h r e s h o ~ in either case, however, for the down solution a moderate r i s e Just below will show up. These state~. ments a r e easily verified by looking at the b e ~ v i o u r s of the s ~,nd p ~ v e s ot~ the Argand diagram. Prelimir~zy data [21.2,232] do indeed seem t¢ p r e f e r the d~wn branch solution below 850 MeV. We finally emphasize again that with these kinds o ' c and S* resonances inure that el. (4.58) also has a pair of (KI~)second sheet poles, h~wever, below thresh. old) conventional Breit-Wigner phenor~enology is both Insufficient and misleading. 4.8. Experimental results on KTr scattering The following reactions have been mostly used: K+p --.K + ~ - A ++ (a) K%°a

~+n

++

(b)

K+Tr+n

(c)

K+r'+A O

(d)

---*K'~'-p

(a)

KO Op

(b)

(4.61}

(4.621

(c)

K - 7 - h ++

(a)

K "Tr+n

n.~ ~uj

g-, +tx°

(c)

i

(d)

%oao

K ' n -- K "Tr-p

(a) (b)

(4.62)

14.41

For a discussion of the different experimental peculiarities with these, see Schlein [233,143]. Experimentally, non.charge exchange reactions a r e d o m i n a t ~ b- natural ~ n

22:3

M E S O N - M ESON SCATTERING

parity exchange, which is l e s s d o m i n a n t (but n e v e r forbidden) in the c h a r g e e x change r e a c t i o n s (4.61)-(4.64). C o m p a r e d with the analogous s i t u a t i o n in nN ~ 2 r N ' almost e v e r y t h i n g a c t s a g a i n s t you. Data a r e f e w e r and the p r o d u c t i o n m e c h a n i m s m o r e c o m p l i c a t e d . A l s o B o s e s t a t i s t i c s do not help to cut down on the a u r a ber Or[ p a r t i a l w a v e s and no G p a r i t y s e l e c t i o n r u l e s help cut down on low e n e r g y inehstictty. F o r t h e s e r e a s o n s , m o r e c o u r a g e is n e e d e d to u n d e r t a k e a ChewLow analysis and even m o r e r e s e r v a t i o n f o r e v a l u a t i n g the r e s u l t s . One advantage, however, i s tltat KO's a r e c o n s i d e r a b l y e a s i e r to o b s e r v e than nO,s. Trtppe e t a ! . [234] used r e a c t i o n (4.61a) and found s o m e v e r y interesting" e v i dence for a b r o a d K s t a t e ( a s s u m i n g OPE d o m i n a n c e ) from the K'O(890) decay a s ; ~ m e t r y . T h a t r e a c t i o n s t i l l a p p e a r s to be one of the m o r e a t t r a c t i v e candi~ t e a (or obtaining Kn p h a s e s h i f t s , but c l e a r e v i d e n c e s for n o n - O P E c o n t r i b u tions m~ist,

Figure 24 shows density matrix e l e m e n t s for K*O production at 5.0 GeV/c [335], Although the T r e p a n - g a n g distribution is quite fle,t in this reaction, large de~atlons from O P E ( ~ I) are seen. Curves are based on absorption O P E c~,lc~tions but in ~tlcular fall to reproduce the large w.lue of Re PI0. Also the ~gnltude a ~ shape of ~ e differentlal cross-sectlon is unsatisfactory. Reaction (4.64b) should l~tvledifferential cross-sectlon equal to that of (4.61a) if O P E domi~ t e s . Actually R i s s m a l l e r by a f a c t o r 4 a t c o s 0 K , = 0.98 w h e r e a s the two b e coe::e equal a t c o s ~ , = 0.87. W e r n e r at al. [236] showed that this d i f f e r e n c e lC C9 C8

C~ (15

i

Q2

0t

a. 0 -(11-

"l

-Q2

Q. G ~

0

i

i

Q1

Q2

....

03

Fig 24. Density matrix elements ~or K~°(890) pr(~d~(:tion ~n rea(:tion (4 .(~;la) (ref. 12:i~i,t/.

224

J . L . PETERSEN

~)01--

~.o)

e)

÷ 295tt2

-

23,6-*&Z

§ ~,~. ~ O ~ G S 4,4z0S

_

o

Q2

~

~6

o

FAg, 25. The It'l distribution (,~2 in toxt) for various kinematical cuts, Numbers ~c!atod with exponential fits (straight lines) are exponent coefficients (re(. t23s)).

c}

120

L.

K~

-j _

o

!

I_~~

It'l~Q1~ev~.):

I

40

0 0

I

-1

0

1

cos9 Fig. 26. cos t9 distributions correst~ndlug to various kinerr, aticat cuts in K*° (890) decays (ref. [238]).

MESON-MESON SCATTERIFG

225

cannot be a c c o u n t e d for t y a b s o r b e d OPE so c l e a r l y o t h e r n a t u r a l p a r i t y exchanges with both c h a r g e c o n j u g a t i o n s m u s t be p r e s e n t in these r e a c t i o n s ( p o s s i bly A 2~p; d e n s i t y m a t r i x e l e m e n t s for the txvo a r e r a t h e r s i m i l a r ) . A s b n i l a r trouble is well-known in n e v e r a l o t h e r p a i r s of reactio.~s (cf., for e x a m p l e , Zane [23'71). Fu et al. [238] ~tudy in s o m e d e t a i l r e a c t i o n (4.61a) in an a t t e m p t to isolate OPE signals. Of p a r t i c u l a r c o m p e t i n g i m p o r t a n c e they c o n s i d e r a z-A++ enhar c e ment around 1.58 GeY which m u s t have I = , / 2 ( a b s e n c e of 7teA++ e n h a n c e m e n t in reaction (4.61b)). T h i s effect c o n t r i b u t e s p a r t of ' h e Kn a s y m m e t r y and should not be confused with K~ s c a t t e r i n g . It c o r r e s p o n d s to eve~ats with K+ very forwz* :'d in the GJ f r a m e and l a r g e K~-rr" e f f e c t i v e m a s s . F i g u r e 25 shows a t t e m p t s to e l i m i nate these e v e n t s by the cuts C-s--" 1.54 GeV and c o s 0 GJ < 0.5. T h e s e leave a v e r y forwar.~ peak ( ~ 2 < 3~t2, I t ' l in fig. 25e) which is a b s e n t in (c) and (d) and is e v i dence fo~ a p l e a pole effect. S i m i l a r l y fig. 26 shrews decay a n g u l a r d i s t r i b u t i o n s for K'(890) and K**(1420), F o r A 2 < 5/.t2 t h e s e a r e produced in a p o l a r i z e d f a s h ion typical of O P E , d i s t o r t e d p o s s i b l y by l o w e r K1; waves (a), (c), w h e r e a s for ~2 > 5~t2 a high d e g r e e of d e p o l a r i z a t i o n is evident Col, (d). F o r the K i n t e r p r e tation {i.e., l a r g e 8I/2) of the K*(890) a s y m m e t r y it is p e r h a p s i~,,'tunate that it seems to be a rrmtn'ly p e r i p h e r a l phenomenon. A l s o in favour of K is the fact that the large n u m b e r of e v e n t s b e t w e e n K*(890) and K**(1420) in the Krr . n a s s s p e c trum are m a i n l y ~,ery p e r i p h e r a l o n e s . A large d i f f e r e n c e b e t w e e n the f o r w a r d b a c k w a r d a~symmetries in r e a c t i o n s (4.61a.b} h~ts been known f o r s o m e t i m e to e x i s t above ~ 1 GeV [141]. If one re-fusc~ ~t, ~,[ ~rpret this in t e r m s of a n o n - O P E 1 = 1/2 Art c o r r e l a t i o n 1238}, a huge ~alue for the exotic .; = ~2K,-r c r o s s - s e c t i o n is o b t a i n e d . Thus De B : t e r e et al. [239] obtained a value ~ 3 / 2 ) 2 . 80 m b using r e a c t i o ~ (4.61d). B a k k e r et al. [239], h0weveE, pointed out that this l a s t r e a c t i o n is o~terw:~elmed by b a c k g r o u n d I rom 14.61a) and using r e a c t i o n (4.64a) they give ~3 '2) ~. (2 ~ 1)mb

(4.65)

0ver the r e g i o n f r o m t h r e s h o l d to 1.2 GeV in a g r e e m e n t wit~ ,Cl-,o e~ al. [239] usir~g the same r e a c t i o n up to 2 GeV and with e s t i m a t e s f r o m the c u r r e ~ t a l g e b r a s c a t tering length ( s u b s e c t i o n 3.1). , /2 ~ • Figure 27 s h o w s the l a t e s t r,~sults for 5/o f r o m the n t e r n a h o n a l K+ c o l l a b o r ation [240] (see a l s o ref. [241] I . R e a c t i o n s ( 4 . 6 2 a , b ) were used and mainly the polynomially p o l e e x t r a p o l a t e d s - p i n t e r f e r e n c e (I~°} used. The K*(890) was p~rametrized by an e l a s t i c Brett-Wi, g n e r (cf. fig. 27) an~. the t a i l of the K'*(1420) by a d ~'ave B r e i t - W i g n e r with 50% i n e l a s t i c i t y . T h e 5o3 2 can be obtained from the dillerent a s y m m e t r y in Kn e l a s t i ~ and c h a r g e e x c h a n g e s c a t t e r i n g . As noticed, however, this can be influenced t~e~vily by c o r r e l a t i o n s but using in addition the smallness of the K - z - c r o s s - s e c t i o n (4.65) l a r g e v a l u e s w e r e e x c l u d e d . Assumin~ • 3/2 ~ , .2/9_ . . . . . . . . . . ,.~ . . . . . . . . . ,.4 ~an ,,.,,,,,,,.,., ~ - h , = , u r l - , n l ~ v,r~ry';,~,~ ~1 ~o t~e n e g i i g i b i e , Oo'- c o m c ~ ,Jut n,~rn.,.~ ,x, ~u,,,., -,,..,- . . . . . . . . . . . . . . . . . . . ~ ..... Upto 1 GeV but having s e v e r a l unexplained 2-3 s t a n d a r d deviation fluctuations of •-20°. The n e a r z e r o of 5~/'2 is c o n f i r m e d by s t u d y i n g the a n g u l a r K - ~ - d i s t r i b u tiort ar,d may a l s o be obtained in (4.61a) f r o m the obtained p wave deviation from a pure K*([~90) (here taken as B r e i t - W i g n e r ) which g i v e s a negative v a l u e : -10 °. This is all in a g r e e m e n t with soft m e s o n o r V e n e z i a n o model e x p e c t a t i o n s (fiu. ~,). The i n t e r e s t i n g 51/2(g) is s e e n to exhibit an up-down a m b i g u i t y coming from •

Pre|iminary results wet- given by the Johns Hopkins group (R. Mercer et al.. Johns itopk'ins Uni',,ersity report (1970). unpublished).

t~?,6

J.L. I:'~TERSEN

'I'

"

}

I

"

I

....

K~ pl'mse JNfts 6', . S w (0.89L Q0,'S0) o ~ ,1.8 mb

°

,

~

-

, ,

" u p ( m ~ ~k~#~" _

.

.

.

.

.

.

.

.

m

-

- -

-

lbOo <). "Up"

+ 90 • -o-

"t~p t ~ l ~lown" SIP

30~

(y

i

l

m~

,~__

.......... l , _ _ .....

R,eV~ )

Fig. 27. Values for 5o/21(K~) (~o~in the figure) from the i~ternationRlK* collaboration[~40} (see also ref. [241]). measurement of s-p interference only. The down branch is rising considerably slower than in fig. 3, apparently not reaching 90o below 1100 M e V . The up bran~ passes through 90 ° at 8~5 M e V like in fig. 3, but corresponds to a roach narrov~r resonance (I"< 40 M e V ) that can always be hidden under experimental rez~lutiom. Regarding the existenc~ of the K meson it s e e m s that either the width has to be m u c h smaller than In table 2 (214 M e V ) or else it h~s to be m u c h larger. The down solution is sufficiently similar to what w e have become used to call an ( 8° in ~w that it cannot be considered evidence against g, however, next to nothi~ could be said about.,^the corresponding m a s s value. It might be very different f ~ the point w h e r e 8~/z p a s s e s t h r o u g h 90(' (if it d o e s - it does not have to, for r e s o n a n c e pole to be there). I n e l a s t i c i t i e s iu K~ a r e not cut down by G p a r t l y as in ~ , h o w e v e r , the lowest one m u s t b e p wave f r o m B o s e s t a t i s t i c s . One stud), [141] s e e m s to indicate t ~ they a r e s m a l l below I I 0 0 MeV. In the K**(1420) region the i n e l a s t i c i t y is well k~lown to be ~ 50% [242]. Whereas earlier measurements on the relative branching ratio

r(K* were in agreement with the S U 3 value of 30% (50%) for an unsplit (splR) A 2, tater measurements seemed to prefer a value close to 100% [242]. A small value f44~) was rece~,tly reported again, however [243].

927

MESOL'-MESON SCATTER,~G

G)

30

¢..,,

.lO

.0.5

10~ "

0

f15

...........

l.O~o-tO

-G5

0

0.5

1.0

04

06

08

10~E

= e)

d)

=

0

02

O~,

O6

0 0.8 -t ( ~ W c ) z

02

Fig. 28. Ks" doeay angular distributions and do'/dsdA2 in the K~ ma~s ba:-.ds 1.3-1.4 GeV (a,c) and 1.4--1.5 C,e.V {b,d) rcs!~etivcly, (ref. [244]). Partial w a v e s in ttae 1 3 0 0 - 1 5 0 0 MeV r e g i o n w e r e r e c e n t l y d i s c u s s e d by F i r e stone et al. [244] b a s e d on d a t a on r e a c t i o n (4.62a) at 12 GeV/c. F i g u r e 28 s h o w s K~rdecay a n g u l a r d i s t r i b u t i o n s a n d dcr/ds d ~ 2 in the Kn m a s s b a n d s 1 . 3 - 1 . 4 GeV (a),(c) and 1 . 4 - 1 . 5 Gt:V {b), (d) r e s p e c t i v e l y . F i g u r e 28b s h o w s the e x p e c t e d d i s tribution for a 2 ÷ o b j e c t (K**(1425)) p r o d u c e d by O P E . E v i d e n c e for apprecJ.abl~ s wave b a c k g r o u n d (depletion o:f e v e n t s n e a r c o s 0 : 0; ef. a l s o fig. 19) f,,;,und in reaction (4.61a) ~t 9 GeV/c [238] (fig. 26) is s t i l l c l e a r w h e r e a s no a p p r e c i a b l e p wave a s y m m e t r y is o i : s e r v e d . However-, the a n g u l a r d i s t r i b u t i o n in the l o w e r m u s band i s m a r k e d l y d i f f e r e n t e v e n though in the K~ m a s s s p e c t r u m the two t o gether s p e m to c o n t r i b u t e one p e a k . T h e a u t h o r s i n t e r p r e t t h i s a s e v i d e n c e for a neWKN(1370) r e s o n a n c e wtth r < 150 MeV. T h e s p i n a s s i g n m e n t is l i n k e d to the assumed p r o d u c t i o n m e c h a n i s m . Spin 2 + with p w a v e b a c k g r o u n d i.~ p o s s i b l e only with a ) p r e e i a b l e v e c t o r e x c h a n g e , w h e r e a s p i o n - e x c h a n g e ( e x p e r i m e n t a l T r e i man-Yang d i s t r i b u t i o n is fiat) s u g g e s t s 0 + with a d w a v e b a c k g r o u n d c o n s } s t e n t with the tail of a K * * ( 1 4 2 0 ) o f w i d t h 100 MeV. T h a t f i t s with an I = 1 ' 2 s wave passing 900 a b o v e 1100 MeV zoooing arnur'cj tc~ 9.7¢1o s t 1 ':tTn ~ " V onA ~,,~11 ~.,~...~.away from the b o t t o m of the A r g a n d diatz~-am at 149,0 MeV. It will be interes,'!n~:! tr, seei! the effect h o l d s up both in c o n n e c t i o n with q u e s t i o n s on a s p l i t v e r s u s ~nsplit 2÷ honer (a p r e c i o u s s e a r c h f o r a sp~.~'. K * * ( J 4 2 0 ) h a s b e e n n e g a t i v e [245]) and tn connection w t t h V e n e z t a n o - t y p e s e c o n d d a u g h t e r s t r u c t u r e s .

J . L. P E T E R S E N

5. I N F O R M A T I O N

FROM O T H E R

SOURCES

5.1. Low vnergy inelastic pion-nucleon interaction Single pion production in 7rN collisions in the region just above threshold and below the ~N* threshold (TTr ~ 4 1 0 MeV) can provide information on the lowest ~ scattering and thus supplement the high energy studies of the s a m e process in a convenient way. The production mechanism and analysis philosophy, however, ~s completely different. The di-pton mass distribution has been kno~Ta f o r a long time to deviate from phase sp~ce in a way that ]~ts led to suggestion,,~ for a low energy narrow I - 0 ~ resonance [246]. As emphasized by Roberts and Wagner [247] the observed effect is actually just what is exp~cted from a ~ I = J = 0 wave with small positive s c a t t e r i n g length and l a r g e negative effecti se range like predicted by W e l n b e r g or Veneziano theory (section 3). Frota elastic 7rN phase shift analysis [248] one knows that the P11 wav~, i.sthe only one showing important inelastlclties in the energy region in question. It foll o w s that the final di-pion state is either In an I : J = 0 relative state being s vntve with r e s p e c t to the nucleon o r , in an I = J = 1 s:ate being p wave with respect to the nucleon and thus being considerably d a m p e d by centrifugal ~arrier elfects (or forbidden as in the new ~rOTro data at 378 M e V [249]). In that picture the production amplitude takes the form A(~N-" 2~N) =Nx (phase-space) x f g ( s )

(5.11

where fg(s) is the on-shell n~ i' = J :-. 0 pwa and N is an energy dependent normalization constant that may be obtained either from the ~.N inelasticity qPI 1 or frozr the data (thus checking the elastic phase shift results). In the single channel K m a t r i x unitarized Veneziano model with a

"

,

(5.2t

=

(compare with eq. (3.15)) a good fit to available low energy ~+~- data was obtained [ 2 4 7 ] . Botke in an analysis incorporating the ~o~o data allowed a ° and ~o to be free parameters and also took into account small inelasticitles in other r,N waves (Sll and Dll). A s far as the ~ aspect of the work goes, the following results were 0btainted [250]: o (0.2 +0"08" - I ao= -0.1 )p'

0 ao

= 0.2

4

o : (-6.5,0.5) ~-I = 0.15 implies ~o

Implies ~) = (-5.2 +0.2. -I o -0.5 ) ~

(:,31

In-the linear model for the below threshold ~ amplitude, these may be translated into information on ~o and a2o:

2=

, 3aOO % u 2]

(5.41

The r e s u l t s thus obtained a r e plotted in fig. 29. ALso shown a r e the (aCcl,ao2) points corresponding to eq. (3.15) and ref. [247]. The indicated r e s u l t s o! Maung et al. [249] (~o~o experiment) were obtained by p a r a m e t r i z i n g f ~ ( s ) by the s wave

M E S O N - M NSON SCA'I'i" E RING

'2 2~)

/... 010

(105

-

,o,,

t_ I

.

t.....

- ( 1 t 5 ............... !=

0

01

_1 I t

L ~

C2

ORW

l

03

04

05

a~.~

Fig. 29. S u m m a r y o; r e s u l t s o b t a i n e d f r o m I o w - e n e r K v i n e l a s t i c ?rN s c a t t e r i n g : Maung et ~1. [249}, Botke 1250l r a s l n g eq. 15.4)) ( ) l s o n and T u r n e r [2(;) (()T', ;:;.: ' .... . . . , . ~ ,., ) ,~.... Roberts and W e g ~ r {2471 l a w ) (not a f i t to ~he d a t a ) , W e i n b e r g ' s p r e d i c t i o n is shown t"~r comparison,

dominant C h e w - M a n d e l s t a m f o r m s [22), i . e . , ha.¢ing a ° , a, 2 ~ 5 / 2 o r a o~o o o ~ 1. "the authors notice that their a~ a g r e e s with Weinbe:g, but the relatively inconsistent nature of the fRs (and large uncertainty) might be an indication that the Darametrlzatton in fact di~gre~ s with Weinberg. Neve.LneLess, the difference between the results obtained by the various groups (fro-,29) together with the fact that they all were able to m o r e or less fit the data, in, cute that these experiments m e a s u r e a certain combination of low e~ergy par',tmeter~ m u c h better than a ° a.nd ro individually. A similar p h e n o m e n o n is k n o ~ n in the current algebra treatment of the s a m e proce:~s. H e r e the soft pion techniques relate the amplitude for nN ....2zrN to that for ~N -. r.N at threshold in a wa.y that depends on the s a m e cT c o m m : ~ Itor that en~crs the r~ treatment, i.e., on a°/a 2. Olsson and Turner [26] tbus obtained rcs~tts tha.. m a y b e s t u m m a r i z e d a s (x'eI. [ 39]) o

°2

and plotted in :Cig.2q a s w e l l ( O T ) . A n a l y z i n g t h e D a l i t z d i s t r i b u t i o n f o r "n-p - r,-n*n at74 MeV by G r i b o v ' s f o r m u l a ( e f f e c t i v e r a n g e s i g n o r e d ) , r e f . [ 2 5 1 ] , B l a i r et el. [252] o b t a i n , ~ o a2 -1 ao " ' o = (0.42 :~0.10) I~

J.L. PETERSEN

230

5.2. Low energy elastic pion-nucleon interaction R e s u l t s of a n a l y s i s of t h i s p r o c e s s w e r e a m o n g the vecy f i r s t evidence~ that the I = J == 0 n,'r wave at low e n e r g i e s had a p r e d o m i n a n ' l y p o s i t i v e p h a s e shift 6~ ~f a p p r e c ; a b l e m a g n i t u d e [253]. T h e i d e a i s to take the a c c u r a t e i n f o r m a t i o n on ~rN s c a t t e r i n g in the p h y s k ~ r e g i o n s >~ '.M + ~)2, _4q2 .< t -< 0, and f r o m t h a t d e d u c e p r o p e r t i e s on the amplitade in the. p s e u d o p h y s i c a l r a n g e , 4 ~ 2 ~ t a 4 M 2 f o r the p r o c e s s ,'r~ --' hlN. In thl, r e g i o n the x r a p l i t u d e m a y be p a r t i a l wave e x p a n d e d and the p w a ' s t o t even (odd) a n g u l a r m o m e n t u m J and I = 0 (1) s a t i s f y the g e n e r a l i z e d u n i t a r i t y r u l e

~:gtfd~(t)) ~ ~ICt)(mod ,r),

t -
H e r e 5/ i s the ~r~rp h a s e s h i f t f o r a n g u l a r m o m e n t u m J and isospLn I = 0, 1, the *__s a b s c r i p t is a nucleon h e l i c i t y label, tin i~ the ( f f e , - , ' , e ) i n e l a s t i c threshold. A m o n g the s e v e r a l p r o p o s a l s for p e r f o r m i n g t! e above a n a l y t i c extrapolation, the m e t h o d u s i n g b a c k w a r d d i s p e r s i o n r e l a t i o n s it, the e a s i e s t to d e s c r i b e . Dcmn a c h i e et al. [254] used that m e t h o d to find e v i d e n c e for a J = I - 0 r e s o n a n c e . Details in the t r e a t m e n t have m e a n w h i l e been c r i t i c i z e d on v a r i o u s point~ and the c a l c u l a t i o n s r e c e n t l y r e d o n e by N i e l s e n et al. [255]. We s h a l l c o m m e n t briefly those p o i n t s and e s t i m a t e how the p r o c e s s can c o n t i n u e to s u p p l e m e n t i, fformatto~ from other sources. The m o s t i n t e r e s t i n g r e s u l t s c o m e f r o m s t u d y of f~+(t). Co~ s i d e r the specific rrN b a c k w a r d a m p l i t u d e

F(+)(v) = ,:,r, - - fE s (f~+} ( v e o s O = - 1 1 -

f(+)(v

,

cos 0 =-1))

(5,7: _ 1 8n

~

t J + l 2)(ptq:)Jf~.f{t)

J~ Pt2 Jeven The (~.) i n d i c a t e s that I : 0 in ,he t channel (r,r, -NNt has been s e l e c t e d ; [." ,s the s c h a n n e l c . m . nucleon e n e r g y , v ~ q2 the s c h a n n e l ¢.m. t h r e e - m o m e n t u : n s q u a r e d a n d v :-- -,it at c o s 0 = -1. M is the n u e i e o n m a s s and Pt and qt a r e the t c h a n n e l c.ra. nucleon and pion t h r e e - m o m e n t a . F(+)(v) i s a r e a l a n a l y t i c f u n c t i o n d e f i n e d in the .u p l a n e cut a t o n g (0, ~) (physical 7rN--. 7rN r e g i o n ) and (-co, _g2) (the p s e u d o p h y s i c a l , 4 g 2 ~ t ~ 4M 2, and vhysical, 4M 2 -< t -< ~o, 7r~--. bin r e g i o n s ) and h a v i n g a B o r n pole at , = _u2 + ~t4/4M 2 with knowm r e s i d u e . Using the f i r s t line in (5.7) F(+)(v) m a y be calctflated for v ~0 using kncwn ,'rN p h a s e sh*fts (for v-'~o b a r y o n Regge m o d e l s may" be (and have been) u s e d , h o w e v e r , t h i s r e g i o n is i r r e l e v a n t to the p u r p o s e at hard). E r r o r s in hetps to r e d u c e ti~em. P r o v i d e d F(+~(vl is kno~,~ua ,a ith stffticient ~ c c u r a c y on t~e r i g h t - L a n d cut, li :'nay be c o n t i n u e d onto the left-~. ,nd cut wher~ the last l~ne in eq. (5.7) is valid. T r u n c a t i n g that e x p a n s i o n in sot~e way o r o t h e r and separating the f i r s t t e r m l y ( t ) , eq. (5.6) can be ised to obtain 5~(t) ior t.._< ira. In all treatmen*s tin is ~ k e u of the o r d e r of 1 GeV 2 f o r w h i c h t h e r e ~s ucw good evidence ( s u b s e c t i o n 4.6) Above t h i s value f°(t~ even if it could be o b t a i n e d is not relaV'~ in any s i m p l e way to ~ ~ c a t t e r t u g . C o m p a r i n g this metiaoci with the C h e w - L o w m e t h o d ( ~ c t i o n 4) we make t~e fel-

MESON-MESON SCATT ERINt~

2.

lowing c o m m e n t s : both r e q u i r e a n e x t r a p o l a t i o n ; bow,.~ver, w h e r e a s i~, t.,. , i a t t e r we only n e e d t o e x t r a p o l a t e to an i s o l a t e d p o i n t (tl'~:~ p~o~ p~]~ ~ in ~he ho]om,.:~'Di~y region, in t h e f o r m e r e x t r a p o l a t i o n s onto t h e b o u a d a ~ 9 (the i e ~ i - h a ~ d ,., c,.,~t ~s " ~ quired: the m a t h e m a t i c a l e x t r a p o l a t i o n t h e o r y [ 1 8 9 - ! ~ ) I , 255] t e l l s a s t h a t tim p ~ , a r i e s f o r t h i s a r e t e r r i b l e . T h u s . iI ~ is a m e a s u r e of the e x p e r i m e n t a l e r r o r , the e x t r a p o l a t i o n e r r o r in t h e C h e w - L o w c a s e wil1 be ~CI., < e p

with .p s l i g h t l y s m a l l e r

whereas in t h e p r e s e n t

application

(a.8)

than 1

it will b e

(5.9)

( b a c k w '~: [ log ( I - 1

The effect of t h i s !s t h a t in t h e r~N b a c k w a r d ( a s e e x t r e m e l y p o o r r e s o l a t i o n of struct.,tres ~n f o r e s u l t s a n d o n l y a v e r a g e s o v e r ,the w h o l e r e g i o n up to 1 GeV is obtainable w t t h a u s e f u l a c c u r a c y ( ~ 50%). I m p r o v i n g rrN p h a s e s h i f t by an o r d e r of magaaitude at tow e n e r g i e s , a n o n - t r i v i a l c o m b i n a t i o n of a °c a n d r °o c o u l d be m e s s ured with g ~ a c c u r a c y , b u t f i n e s t r u c t u r e s ia t h e p r e g i o n a r e u n l i k e l y to be c,I~~lnable. T h e m a i n r e a s o n why t h i s k i n d of a n a l y s i s m a k e s s e n s e a t a l l is t b a t h.,energy, e l a s t i c 7rN a m p l t t a d e s a r e s o f a n t a s t i c a l l y m u c h s t m p l c ~ to Oescxibe, and better k n o w n t h a n h i g h e n e r g y i n e l a s * i c o n e s . The t r e a t m e n t h a s t h e a l m o s t u n i q u e a d v a n t a g e t h a t I :: 0 m a y b e isc~lated :~,.~:~ the sigm of b~o i s u n a m b i g u o u s l y o b t a i n e d a s p o s i t i v e , ruli,~g out s e v e r a l su~:~e~,.-,tions for the o p p o s i t e . The methc:ds of Donnachie et al. i2541 have been ci~I~cized ¢~n thc ioliowL~

points. (t) T o a c h i e v e t h e a n a l y t i c c o n t i n u a t i o n in ~ c o n v e n i e n t w a y , a c o n f o r m a l t : a n s f o r m a t i o n r o a c h l i k e t h e o n e d e s c r i b e d i~ suase,:~x,,n ,~.4 w a s a p p l i e d , h~w:c-v • e~, a c i r c l e r a t h e r .*han an e l l i p s e wa~: u s e d w h i c h is u n j u s t i l l a b l e i ! 9 0 [ . [¢~ p r a c t i c e t h e t w o d i f f e r by at m o s t a few p e r c e ~ , t it, this c:t.~, :,~,d the, ~,Ifc.,~' .:',~:~ be ign,~refl w i t h p r e s e n t a c c u r a "y [2[)5!. tit}In , d ) t a i n i n g f /) fr~:,m /,'(')i,,i f,~r t 4 . : . 2 ! : ' . . . . . ; ,~,~:a~..: v. . . . . . ' [ecttvely i~ored. T h i s h a s b e e n s h o w n to p r o d u c e s i g n i [ i c a , t s y s t e m a t i c e r r o r s {255] d u e to t h e p e c u l i a r I : e h a v i o u r of t h e s a n d u c h a n n e l B o r n p o l e s . T h e effect, h o w e v e r , is r e a d i l y t a k ~a ,nto a c c o u n t by e x p l i c i t t r e a t m e n t of the Born p o l e f o r nil J ' s in e q . (5."),. and n e g l e c t i n g t h e non-~>ole, p a w ~}f /'}~!~: f,,.~. al~ ..,~'s "- 2 c a n b e j u s t i f i e d . Figu "e 30 s h o w s t h e r e s u l t s f o r t ° ( t ) [ 2 5 5 } . S e v e r a l w a y s (,f p~,ri¢,rmir~g ~h<.... :~.trapola~ton w e r e f o u n d to a g r e e w i t h i n e r r o r s . C u r v e B in ~i~,*. 30,~ i ~. l.hr r e b u t ! ~,f the un}<,sti[ied t r u n c a t i o n of e q . i 5 . 7 , w n e r , > a s t h e mor.o ( ,-~rrec~ c ~ r , e B;~ was ~,t,tal~ed :, s d e s c r i b e d a b o v e . "[:he o l l o w i n g c o n c l u s i o n s c a n b e m a d e ' 5oo is p r e d o m i r m a t l y p~,a~iv{~ ,,.,1~,',~. average s h o u l d be r i g h t to a b o u t 50~{. T h u s it a p ~ a r s imp, e,ssi~;le ' ~,,r ~b~.. !~;' ;"~ " haven ~:lea.r c u t zer{~ n e a r t~~e o a n d end~n~ wit~: a la.:'<~ at}~,~a~iv~ '.'.~}:~<'::: .,,.~. : entirely p o s s , b l e in r e f . [2541, cf. c u r v e B'~ a n d t ~ u s the down b r a n , ' h in ~h~' r..-'.,r:,,'-: above the p ia ~ r e f e r r e d o f o r 6 °. If v a l u e s f o r 5~ a r e kr~cm,n f r o m at}..,'~ :-..:~.:r::: '. similar a n a l y s i s c a n p r o v i d e m o r e r e l i a b l e v a l u e s f o r t h e . , ].N c(mplip.!. 125 ,~',' ~f. however, n a r r o w r e s o n a n c e a ~ p r o x i m a t i o n s a r e u s e d I257! we e x p e c t frc, r~, " " I3.,8,, ~...,.~ v ~ t h a t s y s t e m a t i c ~ . r r o r s of a f a c t o r 2 - 3 ar<' lik~>lv ~,, ..~ ~ :~!.ea~ 14},

.'

~32

J.L. PETERSEN

!Re[.o

,m L a)

//B2 b)

/

40 30

30

20

2t2

I0

~0

0

81

--L

~0

. . . .

'

. . . . . . .

,~,"'

L

..........

30

0

....

L--.--

40

fire"

""-----

B

Fig. 30. R e s u l t s for f o obtained i r o m backwatal r~N disl,t.,r~:,,o,~,~ reiatiot, s ~"~,,~." In tb~ c u r ~ L is the r e s u l t of the direc~ t r u n c a t i o n of eq. (5.7} ~ h e r ~ m s curve B~ is tat,. r e s u t t of th.c moe~ c o r r e c t p r e s c r i p t i o n tsee t~ xt). Cuvve~ t~ anti Bo ropr,~.ucnt equal:y |Ik~:ly .~)ssibilitf,..,,s ,~,~.e, ref. [ 2 ~ 1)5.3. ' E x p e r i m e n t a l i n f o r ; n a t i o n f r o m K e 4 dc cm, K e 4 d e c a y w a s s h o w n by P a i s a n d T z e i m a n t2581 t,~ p r o v i d e a r e a c t t o n w h e r e low e n e r t . w ;,'rr s c a t t e r i n ~ c a n b e s t u d i e d in an e x t r e m i s t # c l e a r w a y l a b other' had r o n s in t h e f i n a l s t a t e ) u s i n g o n l y a m i n i n u m c" a~,;~umptttm~. F t w t h e r the 'Midity of t h e s e m a y b e t e s t e d o n t h e d a t a in a c . o m p . e t e l y model ~r,d e g e m , e~, way. Unfortunately. t h e b r a n c h i n g r a t i o i o r t h e proces.,.: is ,~,lv 3 • ]0 -5 and ;~v:r i n v a r i a n t a m p l i t u d e = , a n d f i v e i n v a r i a n t v ~ . r i a b l e s a r e n e e d e d to a c o m p l e t e d e . scription. So far only 338 fully determim.~d events [25q. 2601 bare been pu~,l.~hec ands more -)del-dependent a n a l , ; s ~ s m L ~ ~,~ .n~ade. } t o w c v e r . t r e ~,>tai n u m b e r of available events is e~'~pected to be soon ~.i
. :,~::-e+t,

:5.I~

has been studied. We shall assume tim:.-zeversal i n v a r i a n c e a n d t h e [&f I 2 r u l e b u t t h e s e m a y b e te~,ted g i v e n the n e e d e d s t a t i s t i c s . Tt.en o n l y ! = 0, ~ - : s c a t t e r i n g c o n t r i b u t e . , The k i n e m a t i c a l d i - p i o n m a s s r a n g e i s 2~., ::: 2 8 0 M e f -

~ ~=~~ < m K

,!94 M e V .

However, ex~erimentatly v i r t u a l l y no e v e n t s a r e p r e s e n t a b o v e 4 0 0 M e V . T~e:~fore an effective range parametrization of s a n d p v ' a v e s s h o u l d s , f f f i c e :o d . scribe t h e data. T h e q u e s t i o n s we m i d l i k e ~o a~,~"- a r c

(I) can current-algebr~-type 7,~ amplitudes describe the data'? L

.

.

.

.

.

.

.

.

. ~

~,<~; ff ~*.,, tO . . . . . . . . . . . . . . . . . . . . . . . (3) if n o t SO, w L a t g o e ~ w r o n g ? In a d d i t i o n , it is of g ~ e a t ~he,ereti_cal u:~,,r~,:~, t~; .:~.,,: ~i.d.F~r s e v e r , . 1 ciues:~ .,~ c o n n e c t e d w i t h th." s o f t p i o n r e l a t i o n s b e t w e e n K e 4 a n d K,. 3 [ 2{3t, 2 6 2 , :~3i. A s f o r a n s w e r s to ( 1 ) - { 3 ) t h e pre,,,.?,nt s t a t u s is" f r o m t h e V e n e : ~ a n o n~,.,,:, fin by R a b e r t . ~ arid W a g ~ e r [ ? $ 3 ] t h e a: s w e r to ( i ) i s : def~n~.tely yes'. , f r o m t h e m o r e p~-~e,7 ,~e: N.,.,t~' a~'d~': irt p r o o f at :h,c en,t of sL~bs°.,,.t~on .~.3.

M [-18<)N- M ~:;8(* ,1 8~.'A'I'T I.'I ~.',1,:;

......

n o m e n o l o g i c a f i t s of E l y e t a l . [ 2 6.0 t ~nct . l.~ e r e n. d s e¢ :'t {2,,.,.'. t h e a n s w e : ; ' ~ is. n,:,f ~", a n y ~ p p r e c ! . a b l e e::~e~.,.~ .~ n i t (se~' ,~l:.:~ K a . l m u s [ 2 l ) . To i~,wes't ~ r , ! e r i~.~ :v~,ak tui~,r'.::.' :,,,. t ! . ,t~ . ~ l : + , ~ : l < , (,,~ ,"~:~.l{}) !,~ ""

:V

' "¢P+~:-(t'.~':

,,~A{~

{e,. {t,

:;~¢c~,"

',',

¢9,, ....

,_.2,

with 1

[FPa

~ (;~.'x + R k x !

.~

H - ~::a,+:rtekt):,.,Q, ~

(5.12 ~, ana

.P+P+ + P _ ,

14 ::P+ - P _ .

T',.e I n v : : r i a n t a x i a l f , ~ r m f a c t : , r s F, (.'. R a ,- d t h e v e c t , ~ r f o r m f a c t o r ! I d e p ~ v l on Mrs: ' d i - m e < ~ o n m a s . ~," a.I~, -:- d i - l e p t o n m a s s , ¢),: o:ngl<. ¢~f ,.~ ~ i~t d l - - p i c ~ ~¢ .~0,, system w.r.t, the di-lep~n l i n e ,,f f l i g h t in : h e K ~ re.~{ s k s ~ ¢ , m . T h e ~vv:~ ;.~:ldi~.i~m.a:! ~-artables a r e tak~.-a a s

Of ~ a n g l e of e + in t h e a l l - - l e p t o n r e s t s y s t e m w . r . ~ . , t h e d i - [ e ~ ! o n li~.' :ff f l i g h t in t h e K + r e s t . s y s t e m ; ., :~ngt~ of t h e d i - l e p ~ o n p l a n e w . r ' t h e d i - p i ~ , n i~la:~, in ~}~,, ~' ' te~

Tt, c [.~rm f a c t o r R is of p a r ! t c u l a r !ween K~3 a n d K e 4 b u t g e t s m u l t i p l i e d :k'no:'eA f ~ r t h e r e s t . I¢)rm

!'~:k,' .

.

to s~,'~).

.

.

.

,

,,

, : h e r e , a,, ~ o n e w r i t e s

.~=.... 3.I !

impo~t:~:,,'~, t,,~. ~)~... ,,i: ~ ,.,~: , . . . . : : . : : : ~ ; :.:,. by ,,,2 in : ~ e ~::.~:~v ~t~s~rit.~,.~:: :,n(t ,.vil~ ~ ,

) : /s(M-

# -

-~265!

.~It~,)e ~0~ja~;'~

,51i;~?_. :,5: ' '

i

/:~(.,~! ..... r

+

,I-I/,.

1

284

J.L.

PETERSEN

t(Mzr,,,7,Mlv, Ozr, Ol,gO) = I 1 + 12 cos Ol + 13 s i n 2 0 l c o s 2¢p + I 4 s i n 2 0 l cose~ + ! 5 sin Ol , o s ( p + l 6 c o s Ol + t 7 sin Olsincc,+18 s i n 2 0 t s A n ~ . l 9 sin2 Ol'sin2o

w h e r e the Ii's m a y be given a,.., c o m p l i c a t e d e x p r e s s i o n s i n v o l v i n g kin~w~ ~-~.~~ac t o r s and F, G and H, but a r e d e p e n d i n g only on MTr,r, Mtu a n d ~ and m a n principle a l l be obtained f r o m the d a t a I n t e g r a t i n g o v e r Mlv and c o s 0~ to ob~ ~ (I/)'s, e q s . (5.14) l m p l y tar, (5°(M,~r) - 6~(Mzrzt)):: (17)/2(14)

T h i s is an e x t r e m e l y m o d e l - i n d e p e n d e n t m e t h o d for o b t a i n i n g (6 ° - 6t) as a function of M~r~ but it d e p e n d s on c o r r e l a t i o n s in the (01, (p) d i s t r i b u t i o n that are not c l e a r l y v i s i b l e in the m e a g r e data. T h e r e f o r e m a i n l y o n e - d i m e n s i o n a l distrib u t i o n s h a v e been s t u d i e d [ 2 6 0 , 2 6 4 , 2 6 6 ] . T h e f ~ r m f a c t o r s f s , - .- ,ht~ a r e a s s u m e d c o n s t a n t and 5° - 5~ p a r a m e t r i z e , t by a C h e w - M a n d e l s ~ m e f f e c t i v e r a ~ f o r m u l a n e g l e c t i n g 5~. T h e m o s t v a l u a b l e i n f o r m a t i o n c o m e ~ f r o m the ,~ d~-:t.-:~,,tion. W r i t i n g dF d~pdMr2r .

.

.

.

.

.

.

.

.

.

.

.

.

--,

~

a + b sincp + c cos~o

(~.18~

a and c c o n t r i b u t e s y m r e e t r i c a l l y a r o u n d o : 0 and a r e a l m o s t independent o~ ~o ~.~1 Fo~ 5 on the o t h e r hand [264] we h a v e "O - ~ 1" r . . . . . . . . . .

T ..............

~ ......

ii 1

i i

I 02 F

t 1

' 1

o~

i i

ol -~80

d

i

........ I

0

1~. 3 ]

.~. . . .~. .i.~. .s.'.~ ' t

~ "-

in V.e~. d e c , ~ ' ; ,

Do~cd

l~n~,~ a~'e ~ r o m

Lhe R o ~ e r t ~ - \ V a ~ n ~ o :

~ t ~f~i,,~

M E S()N- M E SON S C AT T V tl IN(;

:,

:

.!;~.,

i a K.f s gp sin (%-"° ',5~)

( 5.1 '3

where K is a p o s i t i v e d e f i n i t e k i ~ ) e m a t i c a l fun('ti(~n. F i g u r e 31 .~):,ow~ ~i~(, v..:p,::'i-. mental d i s t r t D u t i o n a n d f r o m t h e a s y m m e t r y , (b) i s s e e n to b e p o s i t i v e . T h u s t w o solutions a r e f o u n d ' ,'r~ .... 0: (I) ,%,'.s

a oo ,

~pLfs <

a oo

(il)

0,

0 <

o.

(5.20

Solution (I) a g r e e s in s l g m s w i t h W e i n b e r ( . : ter,.~ined f r o m t h e o t h e r d i s t r i b u t i o n s and dtstn-~ution, b u t s t a t i s t i c s d o n o t a l l o w to fits a r e s a t i s f a c t o r y a n d no d i s a g r e e m e n t implied by t h e C h e w - M a n d , : l s t a m form.

a~(t {Ii) d o e s not. F o r m f : c t o r " a~'~ d~-,.. tlae a m b i g u i t y m a y b e r e s : ) l v e d l.,y the "'i do t h a t . T a b l e 3 s h o w s t h e resul~ s. T h e is f o u n d w i t h t h e c o n s t r a i n t a ° r ~' > 1

,Tablt~ 3 lle,'.~[l,~ of Borerld$ ot al. [.2641 and Ely et al. [260l f o r the two s,)lutions eq. t5.20). ]'rc,~:ticttons of W e i n b e r g [261.16t a r e shown for (,o n p a r i s o n .

a0{~o 1) gp t :¢r.~

(-'t P

¢g

tap/t

B e r e n d s (I)

Ely (I)

0.71 ~ 0.37

1.26 + 0.6H

-().5"

~ ()

2.t)0

I . q.'~ . ~ 0. . t 7.

-'>.0D

, i).,~2

-l.iil

+ () 1 5

, i~.:~0

t'2.5

~ t .2

:

-2.~I

, ().'~;

-0.34 -4.70.,

~ 0.90 ,~ 0 . 3 7 1,96

! 1

-3.I

t

-!.()9

.~ 0._~1

-

B e r e a d s (II)

1 95 .

.... t , ' ~ ( J

I.';6

El"

:~g~

(~ ~,.~,i , ~ i

).:".i

',,v'.,

, )+i

~, ,

!,, • r.,

,,;L

',-.

!

~ ~) ) ( :

.~e~.twe e f f e c t i v e r a n g e . C o n s i d e r a t i o n s on q u a n t u m , ~ u m b e r a n d a n g u i a r tum c o n t e n t s of p o l e s in t h e v a r i o u s p o s s i b l e c h a n n e l s s u g g e s t to w r i t e F ~ G a: H a: B(1 -Crp(S), 1 - C , K , ( e ) )

momen-

!,~i.2a)

where s = : ~ a a n d t i s t h e m o m e n t u m t r a n s f e r b e t w e e n the ieptone, am( u a c pi~J~a. S0it pioa t h e o r e m s m a y b e i m p o s e d a n d ~Le f o r m (5.21) i f m a t r i x m . ~a:eiz, ed. g q u a tiort {5.!5) i s a p p r o x i m a t e l y satisfied and the only free parameter is t h e r e l a t i v e , ~ o r m a l t z a t i o n b e t w e e n !2 a n d (F. G). The ~lt h a s n o ' : e r y s t r o n g b e a r i n g on V.~aezia~ao t}~e.~rv a5 ~:v<'h, ,~:)~! :~ ct,.~:~5; , . , ~ exists_rig da~a c a n b e d e s c r i b e d bv a ~ e ~ n b e r g {vr)e 77 fectiy &..,eat f o r m r a t t l e r s . Vote ad.D,d in , ~ o o f :

~bmttte,] to the A m s ~ e r d . m (.'onfereace oa [-ilerne~tar~ ['art~:l~:~ .~uly 197.,. :~ee ',~:~,.; '..::.~ rap~rteur ~'llk by C. S~:hmid, CERN prepr'ir~t TH-~ i03 ¢19~,1)) a)l:.v.~:! !".: '.!:" !:::: :i: : a~.n~t~~i,, ~-',.'~-Tre~man method eq~. {5. i6), (::,. ~7~ and s~<:
i 5.4. Mi.~ ¢ ' . t l a n e ~ ¢ s a d d i t i o n a l r e a c t i o n s i K° • 2r. d e c a y s d e p e n d to s o m e e x t e n t on t k e o h a g e d}ff~r:m.,# , :,'~", .... : 0 gl~e s~.l ~s is l a r g e l y u n c h a n g e d s i n c e the r e v i e w of Kalm,a~. (2 ::.

~:!;ii,

J.L. P E T E R S E N

236

F o r K ~ -" 2~ the usual CP violating a m p l i t u d e s n+_ and 770o s a t i s f y [26"~] 77+- - ~oo oc i exp[i(82(mK ) - 6°(inK))]. (5.22) However, at this time, e x p e r i m e n t s [269] a r e in a g r e e m e n t with the "superwea': diction [ 268]

1'/+_ = 7'~00 in which c a s e eq. (5.22) is empty. F o r K ° -" 27r the ratio

R : r (K~-' ~+Tr-)/r(K ° -" ~o~ro) satisfies X - 2 = 6~f~ R e (A2/Ao) c o s ( ~ - 5°) + 2f (S,|3} where Ao(real ) and A 2 are the amplitudes for di-plon I = 0, 2. R - k m 0 if the IA:[ = 1/2 rule was exactly valid. Im A 2 ~ 0 is found experimentally and assumed. A 2 is measured from K + -- ~+Iro, however, lack of knowledge on A/-- 5/2, contributes a theoretical uncertainty of ~ 0.02 azld the s a m e i:; true with modeldependent electromagnetic corrections (these effects are lurrpea together under fin eq. (5.23)). Altogether one obtains [2,270]

~$oO(mK) _ 52(inK) = + ( 4 0 0 +_ 20oi 150~ rood (180o)

(5.24)

Due to the t h e o r e t i c a l u n c e r t a i n t i e s the e r r o r s cannot be dra:stically decreased and thus this piece of information is u n f o r t u n a t e l y unable to r e s o l v e the up-down ambiguity below the p. K/3 d e c a y s have been used by many people in a t t e m p t s to e x t r a c t K~ phase shifts. At this time tt~e t h e o r e t i c a l u n d e r s t a n d i n g of the e x p e r i m e n t a l results is as confused a s e v e r and s ~ , ~ e n t s on K~ i n t e r a c t i o n can hardly be made [63,262 I. A s s u m i n g pure v e c t o r t r a n s i t i o n , I A/1 = 1/2 rule and CP conservation, the amplitude i~ determined by

(2 )3

<,+(q)t

_-(k+ q),/+(t)+ (k-q), :_(t)

(S.2S)

with t - (k _q)2. P a r a m e t e r s ~%, ~o and ~ a r e defined by

f±(t) ~- f.. (0)(1 + X+ t:/~, 2) t

f (t) = f+(0)(1 + ~'o t/I ~2)

(5.261

= f_t0),%(0). 'Ihc C a l l a n - Y r e i m a n r e l a t i o n [271] s u g g e s t s (5.2;) E x p e r i m e n t a l l y , a s s u m i n g Cabibbo theory with one angle [272]

](o) - A(o) : FK/F t .28 0.061-1 whereupon

(S.281

'~:~'~

MESON-MESON SCATT b:RING ~o ~ 0"28~a2/'m2

~',,. :79~,

rc,~ults. D i s p e r s i o n r e l a t i o n s f o r .~(f) and fir! a r e d e t e r m i n e d r e s p e c t i v e l y by spin 1 and 0 i n t e r m e d i a t e s t a t e s . A s s u m i n g f o r t(/) an u n s u b t r a c t e d f,~rm d o m i nares by a ~ m e s o n :

f(tl, = .f+(O) ,,~ _ t

or

m K

1.06 GeV

(5."?)

obtains. T h i s a p p e a r s to be in f a i r l y nice a g l e e m e n t both with e x p e r i m e n t (subsection 4.8) and V e n e z i a n o t h e o r y ( h e r e m~ = i n K . ( 8 9 0 ) , h o w e v e r , b e i n g a wide oh]ect the: K m a y e a s i l y be f i t t e d b e l o w t h r e s h o l d by p o l e s at e i t h e r 900 MeV or IIOC MeV). T h e t r o u b l e is that by now d i r e c t e x p e r i m e n t a l m e a s u r e m e n t s of ~o appecu- tc j r e f e r a negative v a l u e ( e r r c r s a r e l a r g e ) [262]. Should that be taken as evidence for a negatl('e 6~/'2(Kn)? We m a k e the f o l l o w i n g c o m m e n t s s u g g e s t i n g the an~wer is no: (t) F o r f+(t) t h e r e is not m u c h q u e s t i o n about 51/2(KTr) but E*(890) d o m i n a n c e (h+ ~ 0 . 0 2 . ~ =0) is in e q u a l l y bad a g r e e m e n t with e x p e r i m e n t ( ~ ~ 0.05 ~ 0.02. = -o.9 ± 0.3).

(ii) E v e r y t h i n g m a y be s a v e d by allowing m o r e compliczAed '~ehavi~mrs for f. _~ and m o : l e l s e x i s t tllat s u g g e s t t h i s<) to be the c a s e ; h o w e v e r , ti~e ~heore~ica~. u n d e r s t a n d i n g is so p o o r t h a t 8~/~(K~)'s c a n n o t be obtained. (|~t} A m o d e l - i n d e p e n d e n t e x t r a p o l a t i ' o n of the data in the decay r e g i o n ,m~o the K~: cut would give the low e.nergy p h a s e s . In p r i n c i p l e t h i s m e t h o d is very s i m i lar to the a p p l i c a t i o n o J b a c k w a r d 7rN s c a t t e r i n g to obtain r e f o r m a t i o n ~n 6t(~). S i n c e the data a r e so e n o r m o u s l y m u c h p o o r e r , any a t t e m p t would be meaningless, however. K37r dec~y was t r e a t e d a m o n g o t h e r s by L o v e l a c e [42] in t e r m s of fig. 32a with X = K and the s q u a r e r e p r e s e n t i n g weak i n t e r a c t i o n s and I o~ ~!~e b~ob using a Veneziano f o r m with "one l~eavy p i o n ' . Jr- such a f r a m e w o r k the d~::~.:a:~d~str~buti.,v'., ~0uld m e a s u r e ~,n s c a t t e r i n g . T h e m o d e l h a s s u b s e q u e n t l y bceu shay ~,. ~,~ lea~d ~<, severe d f f I i c u l t t e s [7,'/3,274] p a r t l y h a v i n g to do with i n c l u s i o n of the ,:i~agram, ~ig, 32b and p a r t l y with o f f - s h e l l v a r i a t i o n s that a r e e.,cperimentally r e q u i r e d to be ~w,reasonably l a r g e . S i m i l a r t r e a t m e n t s h a v e b e e n t r i e d with 7?-~ 3zr but troubl ~ appear to be e v e n g r e a t e r . P o s s i b l e c u r e s h:~ve been c o n c e i v e d I27fi. 2q61. We c o n s i d e r t h e s e r e a c t i o n s at p r e s e n t to y i e l d no useful i n f o r m a t i o n on meson-meson s c a t t e r i n g . A n y w a y t h e i r u s e is d~st~nctly d i f f e r e n t f z o m that of ali other so f a r d e s c r i b e d p r o c e s s e s in that e v e r in p r i n c i p l e , exper~men+-al i n f o r m a tion a~one, h o w e v e r a c c u r a t e , wil'. be m s u f f i , : i e n t to solve the d i [ f i c u l t i e s : a n o t e or less c o m p l e t e theory for r e l a t i n g t~'.e a m p l i t u d e at the one e x p e r , rn,~n~al # o ~ ,

a)

b)

"~

Fig. 32. Model diagrams for X .... 3rr de~,a~:.

J.L. P E T E ~ E N

~8

m X = m K (or m~) to the ~r~ s c a t t e r i n g point m X = ~ is needed, w h e r e a s prevloUly the amplitude could be m e a s u r e d in a non-vanishing reg/on and the principle of analytic continuation be invoked. Similar r e m a r k s apply to the now famous Veneztano ~,tudies of ~N annihila~ons [42] to t h r e e mesons (X - N N with JP = 0- in ftg.32a). H e r e , using the m o d e L ~ e has the exciting possibility of m e a s u r i n g m e s o n - m e s o n s c a t t e r i n g in the " d ~ s p e c t r a l function region". Of course, the m a s s extrapolation penalties a r e uncontrollable, but this kind of study is no doubt of the g r e a t e s t i n t e r e s t tu ~ more gene r a l hadrodynamical (:lual) content. New s t u d i e s [277] of ~p -- 1?~+~r- [W 1~ acceding t o t h e Veneziano model for ~ s c a t t e r i n g [ 122] indicate qualitative a~ ,ement wLth:theory (provided 8 deccuples [122]). s i m i l a r but much m o r e complicated Veneziano fits [279] to the proces: .'ga|, at rest)

may be of g r e a t e r intera.st. T.~e isolation of the mode (5.31) f r o m the dat~ ,~

is ambiguous ( s e v e r ~ l (a~)O's with m a s s ~ m~o ~an co-exlst) and the to p~;~?~l e seen only above a very l a r g e t~ckground. F o r these r e a s o n s a g e n u I ~ q~antila~ tire a n a i y s l s has formldable difficulties. T h e fits indicate dominance of ~ o t 7 6 ~ ~ f(1260) and B(1230). However, more i n t e r e s t i n g for our p u r p o s e i~ an ,,'~served cliscrep~ncy in the ~r m a s s spectrum around the 1 GeV regic,n: ~mhanc, rnen~s at 850 and 1:115 MeV ax~ s e e n , s e p a r a t e d by a non-fitted dip at 940 MeV a~d t)d~ ~ s t r u c t u r e Is blamed on ~he /}~0r~). Similar conclusions w e r e obtained in final state r e s o n a n c e model fits [280] and in a s i m i l a r a n a l y s i s of ~p--* ~+~r-~.+~Ta s s u m i n g the initial 381 state to go predominantly to O plus an I = 0 dt-pton [251]. This is just what is expected from the combined ~ + S* + KI~ threshold anomaly ~cf. subsections 4.6 and 4.7) and the effect was qualitatively fitted Vy Morgan [~1} in these t e r m s . When taken together with t h e s e other p i e c e s of information {subsection 4;'7) this s e e m s m o r e s a t i s f a c t o r y than a fit using two s i m p l e elastic ~ n a n c e s [ 279, 281] . . . . ~: The annihilation of colliding e+e - b e a m s gives of c o u r s e valuable informattan, on the 0 p a r a m e t e r s [282,283] but has so f a r mainly been used to test VMD p ~ dictions. Several suggestions for using the p r o c e s s [284,285~ e+e" -~ ~+r"

~5,3~) •

:.,:

for studying ~ ( , ~ ) haCe been made and c r o s s - s e c t i o n s e s t i m a t e d using the mode! (in the ~, region) e~e- -. ~ -~e + (~) -* ~+~-~,

(5.~)

However, in conformal invariant (scale invarlant) theories the coupling is forbidden ]:l gauge in,:ari~tnce and current conservation when one ~9 is on-shel] and the other has zero m a s s [286], This is true for epp and e cow couplings as well. In

dual t h e o r i e s e ~ , 9 would be forbidc ~,n in t r e e - d i a g r a m s a u t o m a t i c a l l y by the H a r a r i - R o s n e r rules [56] but not cpp or ~ co~o [28~]. Di-pion photoproduction has been studied for some tinge in the P region andrecent e x p e r i m e n t s on nuclei produced some weak but i n t e r e s t i n g evidence for an

MESON-MESON

S C A T T El:rING

239

enhancement b e t w e e n 1300 and 1800 MeV [287] with a p r o d u c t i o n c r o s s - s e c t i o n , however, s e v e r a l h u n d r e d t i m e s s m a l l e r than f o r p. T,!:,1¢~, : , o w e v e r , is p e r f e c t l y compatible with the p' r e q u i r e d in the two-compone,~['~'ual p i c t u r e [288]. Finally, e x p e r i m e n t s of the kind p + d --* He 3 + 2~

[289]

(5.34)

N + N --' d + 2~r

[290]

(5.35)

looking for m i s s i n g m a s s to the final n u c l e o n s , Weinbe::'g f o r a °,

give much h i g h e r v a l u e s than

a°o ~ 1 to 2 p - 1 . [,~ the s p i r i t oI this p a p e r , we s h a l l t e n t a t i v e l y c o n s i d e r the o b s e r v e d effect to be oI ~:uclear p h y s i c s o r i g i n , f o r which t h e r e is s o m e e x c u s e in the fact that huge complex a n o m a l o u s t h r e s h o l d c u t s a r e p r e s e n t in the MTrTr v a r i a b l e . 5.5. Dispersion relation phenomenology D i s p e r s i o n r e l a t i o n s m a y be u s e d to s h a r p e n r e s u l t s c o m i n g out of the e x p e r i mental a n a l y s i s Just a s Is w e l l krzown in low e n e r g y m e s o n - n u c l e o n s c a t t e r i p : For m e s o n - m e s o n s c a t t e r i n g t h i s has so f a r only been done to a n y a p p r e c i a b l e extent for 7r~r s c a t t e r i n g . Soon, h o w e v e r , t w o - c h a n n e l t r e a t m e n t of ~ • ~ , r:r -" K~ and KTr -" KTr s h o u l d b e useful. In r,Tr s c a t t e r i n g [5] f o r w a r d d i s p e r s i o n r e l a t i o n s have in r e c e n t time been p r e ferred to p a r t i a l wave o n e s d u e to the m o r e p o w e r f u l way c r o s s i n g s y m m e t r y e a ters and due to the fact that unitar~:y is a m a n a g e a b l e p r o b l e m at low e n e r g i e s since ~o few p a r t i a l w a v e s a r e i m p o r t a n t . In addition to tl=e u s u a l (s, u, t) v a r i a b l e s the f e l l o w i n g a r e u s e f u l : z -: ( s - u)/4!~ 2

v =q2/~2 = ( s - 4 p 2 ) / 4 ~

2 (5.36)

w = ~ : --=pion lab e n e r g y (t = 0) /¢2 = o~,2 _ p 2 . Then [or a f o r w a r d (t = 0) s c h a n n e l a m p l i t u d e subtracted d i s p e r s i o n , r e l a t i o n r e a d 3 1 r°° lm F l ( z ' + i¢) dz' v(.z) = =?Tt

.- -

1

z'-z

1/o . . . . .

F~z)

of d e f i n i t e i s o s p i n ], an un-

Im F l ( - z '

J

Z

;7 1

~

+

ie) dz'

(5.3~;)

+z

with Im F l ( - z ' + i e ) = - Im ~ ' l ( - z ' - i e ) = - ~ ¢ u ( l , l ' ) IV

lm FI'(z ' +i¢)

----

(cf. eq. (2.33)). With n , r m a l i z a t i o n N :- 2 (eq. (2.3)), s u b t r a c t e d f o r m a a r e (see also refs. [4,291])

(.o

Al(v)-~ a~

+ Jl(u) + -~ J(u)

for

I --

(0) 21

(5.38)

240

J.L. P E T E R S E N

where v(g + 1)_foo

crI(v')_____

2v' + 1

(S.Sg) oO

( dr'

J ( v ) = -~

Im

[ ] A ° ( v ') + A l ( v ') - }A2(v')]

v'+ 1) ~"

The following sum r u l e s a r e important: ,(1)

evalaating At1 (the amplitude with t channel isospin I~ = 1) at t = 0, .s = 4~ 2 gives

2~-

3a~=12; 1

12;

dt~ , 0s Im {,~A t¢2"1

dk

+

.t ~A s1

~Ags~

1

='-~- 0 °-:'~lm A t 3

~o

?r+Tr

ff+~/+"

d v a t ° t "(v) - a~ot

•

~v)

(sAc)

1

(2) evaluating R e A~(vl/v at v = 0 gives 1 .0 x - 1 ~. 2 2 ; dw l m A s + Im(~ "is " ~ s " s A s l

8 ['°dk Im 2 1

=~

~-~

2jl d~ lm 1 s +~ (~+1)2 At

1 oodv (2v+ ..... 1) al(v) 1 ~' -4~ 2f [ v ( v + l ) ] 3/2+ -- v dv

(5.41) t ~vj" zr+zr" - ato ~+~+" Jv(v+l) v+l

['il~ twi~e ~ubtr~;t,~d forr~ ~= Qo~ ~c known to rigorously exist (cf. subsection 3.4). F i - ~ a Regge pole picture, the following asyu~p~.., *,,~ic relations are expected for forward amplitudes for s -~ ~o A0

. .. A / ~

-. i~, {e/e_1 + -,, -I

-..~ , ~~- ~ r

~ {.,lit

t

~. ~T

r i - e o t ( ~ - ~ p , ( O D l ( s / s , ~ ) aI~(O) t-

-

-

,~,

A l1 -" yp[i +tan( ~~ ~p(0))l(s/So) ad0)

. . . . .

v

(5.421

A~ = Y2~i -cot (~-~rr~eff))~S/So)~eff i

where a/~O) = 1 has been used and where ~p:(O) ~ 0.5 and ~eH is expected to be negative. Similariy, for example, relations for deriva, ives may be written down.

M E S O N - M ESON SC A T T E RIN G

241

Goebel and Shaw [292] h a v e m a d e i m p o r t a n t u s e of d i s p e r s i o n r e l a t i o u s for i n verse a m p l i t u d e s . F o r a f o r w a r d a m p l i t u d e d e s c r i b i n g s c a t t e r i n g of p h y s i c a l p a r ticles CF÷I the l e f t - h a n d cut d e s c r i b e s the c o r r e s p o n d i n g a n t i - p a r t i c l e r e a c t i o n (F') so that the i m a g i n a r y p a r t on both is p o s i t i v e from the o p t i c a l t h e o r e m . Then the sign of Im ( F "a) is fixed (negative). T h e o b s e r v a t i o n is m a d e that for r e a s o n a ble asymptotic b e h a v t o u r the n u m b e r of z e r o s of F+(z) is d e t e r m i n e d from the values a ~ ~ F(±/a).

For a ± both negative (no zeros) the following relation is suggested

(a±)_ 1

l

a±'z)

1 " ~'1

+ z-+i J

kz-1

o,

(5.4oj

wlth R~Im

(F -I)

q

at°t

-

it_ o

i.e., positive definite. T h u s any knowledge on the c r ~ , s s - s e c t i o n s puts a negative l~wer bound on a *. The s a m e is t r l v t , l l l y s a t i s f i e d if a ~ a r e not both ,~egative. With information a v a i l a b l e at the t i m e . the, r e s u l t o : , _0.5

aO

-1

(5.44)

was obtained thus ruling out s e v e r a l s a g g e s t i o n s for l a r g e n e g a t i v e values of 5~ (refs. cited in ref. [292]). E q u a t i o n (5.44) s w,~ch s t r o n g e r tnan the r i g o r o u s bounds eq.

(3.74).

Olsson [291] a r g u e d that a s a m e a n s of :letermir~ing a , eq. (5.41' is r a t h e r insensitive to asst" ned s w a v e s . He o b t a i n e , the value a 11 = ( o . 6 4 o , 0.O0b) u -3

(5.45)

is compared to W e i v b e r g ' s p r e d i c t i o n s (eq. (3.14)) for which

L

ma°-S.o21 --(0 10

0.0a). "1

and 1 a 1 = (0.033 ± 0.003) ~ -3

(5.46~

are equivalent. The most r e c e n t work of M o r g a n and Slaaw a p p e a r s to pro ve that at least ~,or ',, a value can be obtained which i s indeed v e r y ~ndependent of ;~,ssumed s waves a t , ' high energy b e h a v i o u r s [293]. T h e g e n e r a l o b j e c t i v e is to take a v a i l a b l e informa-tion on ~[, ~2 o and 8~ in the e n e r g y region 500 MeV < Vrs < 850 M e V

(5.47)

and deduce low e n e r g y e x t r a p o l a t , o n s and d w a v e s . In the f o r w a r d dispersi(,n r e latioas, the i n t e g r a t i o n r a n g e wa," divided into t h r e e pieces" low, i n t e r m e d i a t e

242

J.L. PETERSEN

o~ -03

.02

.0.1

0

01

0.2

03

~

0.5

Fig. 33. Ptot of sotuttons obt.~Jned in ref. [293] for varloua input forms for ~°o (U: up. D: do-,~, "i: between). Curves are for m~7 = 900 M e V (dashed llne), m r = 765 M e V (solidline} and m O = 600 M o V {dotted line). LV corresponds to curv,~ L W o{ tlg. 2.

(5.=~'~, j ,,,,~'"'~ ..,~,,..~;"~In " "~,,,,~fLzst two regions s, p and d wave~ "~'ere p~rametrized and s o m e parameters fixed by the input. The contribution in the low energy region from the high energy region was treated in a near model-inde~ndent way and parRmet~'ized by polynomi~tls in z (5.36). W e first consld,~.r results for the s wave scattering tengths a ° and a2o (fig. 33). For each a s s u m e d 5° in the intermediate region (5.47) (classi'~ed according to up and down ;orms defined by fig. !3 - "between ~ - forms were allowed as well in the below p region), the parameter m ~ is the e n e r ~ at which 5° passes through 900 and for the broad resonant solutions need have no strong relation to the resonance pole parameters. The striking point about fig. 33 is that despit the ~ery large range of allowed input forms of 5° and the high energy contributions (fixed by requiring unitarity of the dispersion relation output), the q~mtlty L is ~ccurately determine(,. -

-

+ 0.0:

(~.48)

T h l s r e s t t l t a p p e a r s to depend on the f~ct that only 5° f o r m s p a s s i n g through the up and down bands w e r e allowed [294 ]. A....,h....~..~.... ~ ; , , , - a h ~ a , - , , ~ t ; a n ;,= that rill 50 Ralutinn~ nroduce r,~ amplitudes having small d waves avd satisfying several crossing and posltlvity tests in the M a n d e l s t a m triangle. H o w e v e r the solutions D U I D U If, D D with aO/a 2 very dirferent from the Isoscalar ~ te r m' prediction, -'I/2,' need high energy ocon~ribuU0,s that ceuid not be explained in ter~ ~ of k n o w n high m a s s resonances (f,g) or ~ exchanges. This is in agreement with the m o r e theoretical qu~sl-theorem ol subsection 3.4, that the p mesot~ together with c.-ossing and unttarlty appears "reasonably" to constrain low energy scattering to the Weinb rg form.

p,

Q

(degmes) J,

~3

C~Z'O

m !

g) ol

r~ w

0

-\

L~

,,...,,,

t'O v

l

~

_

t.

-

[.

~ _

L--..-_.--

[

,/

~

i.........

J

. _

t

J

_

__

244

J . L . PETERSEN

Excluding the unsatisfactory solutk~ns or imposing the "experimental" deter. o o 2 (eqs. (4.35), (4.55)) 5°o is essenti~tlly t,niquely determb~ed below minations of ao/a the 0. Above, the down branch was preferred from comparison with the ~o~o mass spectrum of Deinet et al. [208] about which, however, there is no 'onger universal agreement (subsection 4.~), but as we have seen a number of other ~,rguments ~xist for believing in that (~ubsectiou 4.7, 5.2). Then the final fo:'ms for 60, r~ ~O, 51 and ~ result (figs. 34, 35) (~] was consistent with zero). The r e s u l t ~ low energy parameters were =

--

in remarkable a~'eement with Weinberg, eqs. (5.46), (3.1~). Further

(cf. eq. (5.24)). C~/~stoldi's[295] suggestion for a ImSittve a2o despite the negative values found for ~ above 500 MeV appears then to result frt~m Incomplete imposition of crossing (only the ! = 2 channel was considered). Pi~dt's [296] suggestion for a "turn-over" 5~ (a°< 0) partly suffers the same criticism and in ref. [293], i.tis possible only for DD, DUI~ "DUll solutions (where it is moreover required) which were considered unde~hrable for other re~sons.

s. C O N C L U S I O N In this section we attempt a qualitative and semi-quantitative description in words of those meson-meson partial waves for which this makes sense. ?'hatwill partly represent the conclusion of some of the material collected in the l~iper~t to a greater if not extreme extent will it represent the a u t h o r ' s prejudice, and warnings against that could not be ~,Ter-emphasized. For a m o r e reasonable eealnation of uncertainties and alternatives, reference must be made to the appropriate portions of the text.

At low energies, no strong evidence exists against Wetnberg's amplttt.de (subsection 3.1, eqs. (3.14),(3.15)). Theoretically it is not only consistent wi:h unltarry but may "zlmost" be required by unitarity and the existence of the p (sub* section 3.5, fig.5).Experimentally, several pieces of evidence in its favour exist 5o2 has a value of -15° k, .20o at the p position (figs. 2. II, 12, 35) and m~y level off in the 1 Gee region (fig. 11) above which it is unkno,. ~. ~ ~ 1 throughout. leve811Iis of course dominated by the p (figs. 11,34). Uncertai~tles at the 20 Me¥ continue to exist. Above 1 C_~V, it is surprisingly poorly known. Sugge~ms for moderate (figs. 20,18) and large (fig. 17) inelastic~.tlesexist. A p' signal may have been seen in photoproduction (subsection 5.4) bu~ almost not in Won pr~lucti~n (subsection 4.6) and the pwa is smal' at the fo(12(i0) (fig. 19).

MESON-MESt)],~ SCATT E RING

245

G° Below the p d~erc is increasing agreement about bet.ween forms boiL, from theory (fig. o), experiment (figs. 11, 14) and dispersion relation pheao~,enology {fig. 35, eq. (5.50)). Ke4 experiments are so far unable to confirm that (table 3) but indicate no disagreement (fig.31)t. Above the p the down branc~ is barely preferred up to 850 M e V (subsections 4.6, 4.7, 5.2, 5.5). M ~ s s and width values for a possible ~ res( nance can hardly be inferred from available data alone and Brelt-Wigner ph,~nomenology would be grossly misleading. For semitheo, ,-~cal suggestions, :f. table 2 and in particular eqs. (3.78), (3.79). Above 850 MeV, a large zccelerr*ion due to the S* inelasticity above the K K threshold (995 MeV) Is likely to shoot up ~ to around : 90o at threshold. Above i GeV, hg. 23 (rotated 180 o) gives qualitative Ide~.s about the behavio.~r depending on details of S*. q~o seems remarkably close to I below_ 1 G e V (subsection 4.6) but 6:'ops quickly above (eq. (4.56)) mainly due to K K inelasticity but 47~'s account for ~ S,~creaslngly large fraction, possibly as m u c h as 30% at 1100 MeV. At the fo(12~0) the pwa is s o m e w h e r e in the upper half of the Argand p~ot (fig. 19). The sero in the pwa near 1 G e V h,ay account for a dip seen in N N annihilation at rest (subsection 5.4, ci. ~Iso fig. 2, carve JJ. t~ is dominated by the near elastl "°(1260) and is non-negligible at the p po~}~tim )~ 2o, figs. 11, 34). ~ l s negligible below I G e V aml m~.y be neg~ttive above (fig. 11). -- K R (sub,~ectlon 4.7) Only for the I --0 s wave does the phase shift L:ngua{:e have meaning (fig° 22) trotambiguities are still too large ff'r numbers to be selected. Combined analysis wlth s~ -- ~n is compulsory but 4~ inelasticttles m a y not be negligible (eq. (4.56)). There appears good evidence that the S" corresponds to a pai~ of complex second sheet poles, but whet:I.ar to the right (resonance) or the left (scattering length) of the KK threshold is not known - it m a y even be considered of secondary importance. However, they are probably on the second K ~ sheet rather t},.:,on the sec'~nd n n sheet. XI -" K~

Again no evidence exists against the soft m e s o n low energy form (subsection 3.1,eqs. (3.18), (3.20)-(3.22)), but unlike in ~ no good quantitative results strongly support it. At m e d i u m energies, however, the pwa's are sufficiently ~logous to ~ ones that one rna7 feel confident about the low energy ~tmplj~'udes we

.

is consistent with z e r e (subsection 4.~) as expected (eq. (3.22), fig. ~). ~o/2 is negative, -I ~ ± 15 ° , up to 1050 MeV (subsection 4.8) in agrecment with expectation (el. (o.22), fig. 3). ~1/2 is dominated bel)w 1050 MeV by the elastic K*(890). Vhe pwa seems to t~,e ~-mMi.in the K**(i420) Tegion (figs. ~6,28). ~/2 (fig. 27) is different from fig. 3 whether the up or down solution is preferred. However, it is positive and of appreciable magnitude as expected ceq. (],]2)). In analogy wlth ~" one might expect the dov,nn branch to be the easier to get 0nwith, but no dispersion relatiov ,,henomenology exists so far. In that case a br0m/x state (in qualitative agre .ment with table 2) i~, ia,hcated but iar from esblished K/3 data (subs,.~ction G.4) are unfo~-tunately ~-.-nclusive on the ~. AbQve 1100 M e V Inelasticlties presumably b e c o m e ir~p~zmnc, but no good values t

'

'~ .~

I"See,ho -ever: Note added ~n proof at the end of subsection 5.3.

246

J . L . VE'~ c;RSEN

e x i s t (s, b s e c t i o n 4.8). The pwa is cf c o n s i d e r a b l e s i z e at '.:he K**(1420) (figs. 26, 28) and may even r e s o n a t e at 1379 MeV (fig. 28). 51t,[2 is d o m i n a t e d by the 50~ inelastic K**(1420).

ACKNOWLEDGEMEI~/r I a m v e r y much indebted to a large n u m b e r of people In and outside CERN fc,r s t i m u l a t i n g d i s c u s s i o n s on v a r i o u s t o p i c s of the p r e s e n t ' p a p e r . In particular thanks a r e due to Y. G o l d s e h m i d t - C l e r m o n t , D. Morgan, J. Pigdt and M. Roos for giving m e a c c e s s to their unpublished m ~ t e r i a l . The" hospitaltlty of the T h e o r e t i c a l Studies Division at CERN Is much appre. ciated. .RE FE RENCES

111 M e s o n Spectroscopy, Sds. C. Baltay and A. If. Roqenfeld (Ben}az ,in. N e w York, 1968). [21 Prec. Argonne Conference on ,~ and K~ Interact.ann, M a y 191:.4,Eds. F. Loeffler at~d E. Malamud.

131 Experimental Meson Spectroscopy, Eds. C. Balta~ anti A. II. R( senfeld (Columbia Univ e r s i t y P r e s s , 19 70).

I4l J. L. Basdevant and J . Reignier, ttereeg-Novi Lectur(, v - : .... ~1:' 70~. [51 D, Morgan and J. Pi~ifit, Springer Tracf.s in Modern ,'hy:,~*cs. V(,l. 55 {1970) p. t. ISl M.Jaeob, Hereeg-Novi Lecture Notes (1970). [71 S. Mandelstam, Phys. Rev. Letters 4 (1960) ,~4. Is] D.E. Neville, Pnys. Rev. 160 (1967) 1375. [91 J.J. De Swart, Rev. Mod. Phys. 35 (1963) 916.

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~ f E S ( ) N - M L S ( ) N S C A T YE ]t{tNt: I:~5{ A , ' . l i a m a ~ . P h y s . R e v . l S 0 (13{;.0~ 14.54.

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