- 1 .

SODE INE%UALITIES FOR THE FORWARD SCATTZRIMG AMPLITUDE'

THE UNIVERSITY OF CHICAGO

THE ENRlCO . FERMl INSTITUTE FOR NUCLEAR STUDIES

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

DISCLAIMER

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SOME INEQUALITIES FOR THE FORWARD SCATTERING AMPLITUDE

A. P. Balachandran

The Enrico Fermi Institute for Nuclear Studies

The University of Chicago, Chicago, Illinois

L E G A L N O T I C E

October, 1963

Contract No. A~(l1-1) -264

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I .

1

/ A. P. Balachandran

The Enrico Fermi Institute for Nuclear Studies

The University of Chicago, Chicago, Illinois

ABSTRACT

We develop a class of inequalities involving the real parts

of the forward scattering amplitude at arbitrary energies and

certain finite integrals over total cross sections. These should

prove useful in the resolution of phase shift ambiguities in the

analyses of scattering data and are sufficiently flexlble to be

applied even in situations where, for instance, the magnitude of

the residue at a pole is not known.

* Work supported by the U. S. Atomic Energy Commission..

+On leave of absence from The Institute of Mathematical Sciences, Madras, India.

I . In t roduct ion

Since t h e f i r s t development of t h e sub jec t by Goldberger and

others , ' forward d i spe r s ion r e l a t i o n s have been extens ive ly used

3 4 i n t h e phenomenological a n a l y s i s of I J - N , ~ K-N and N-N s c a t t e r i n g

. d a t a . The technique has had a somewhat l i m i t e d success, however,

due t o t h e presence of t h r e e adverse f a c t o r s : a ) Ignorance of

t o t a l c ross s e c t i o n s i n t h e high energy region, b ) Ignorance of

t h e magnitude's of t h e r e s idues a t c e r t a i n .poles , and c ) The pre-

. sence of unphysical regions i n t h e d i spe r s ion i n t e g r a l s . The

d i spe r s ion r e l a t i o n s have h i t h e r t o been used a s i d e n t i t i e s f o r

t h e r e a l p a r t s of t h e forward s c a t t e r i n g amplitudes i n terms of . c e r t a i n i n t e g r a l s over t o t a l c ross s e c t i o n s . We wish t o demon-

s t r a t e i n t h i s paper t h a t i f t h e s e a r e converted i n t o s u i t a b l e

i n e q u a l i t i e s , t h e f a c t o r s a ) and b ) need no longer be problems

while t h e d i f f i c u l t i e s a s soc ia ted with t h e f a c t o r C) can be

rendered much l e s s severe . It is , however, t r u e t h a t i n t h i s

process, we l o s e some of t h e information which a r e i n p r i n c i p l e

contained i n t h e canonical i d e n t i t i e s .

Sec t ion 11 begins wi th a d e r i v a t i o n of t h e s e i n e q u a l i t i e s +

f o r IJ--p s c a t t e r i n g under very weak assumptions regarding t h e u.

asymptotic behav i .9 of t h e amplitude. These r e s u l t s a r e then

extended by observing t h a t t h e a--p t o t a l c r o s s s e c t i o n seems t o

+ be always somewhat l a r g e r than t he a -p t o t a l cross s e c t i o n be-

yond a c e r t a i n energy. S t i l l f u r t h e r i n e q u a l i t i e s can-be proved

i f we a r e w i l l i n g t o assume t h a t t h e r e a l p a r t of t h e amplitude

does not inc rease l i k e energy i t s e l f a t l a r g e energies . Evidence

f o r t h i s assumption i s somewhat ambiguous, but i t seems i n t u i t i v e l y

r a t h e r p l a u s i b l e .

+ Sect ion I11 i s concerned wi th t h e K--p system which i s a

t y p i c a l example where both unknown coupling cons tants and unphysi-

c a l regions occur. We show how i n e q u a l i t i e s can be w r i t t e n down

which do not involve coupling cons tants provided t h e i r s igns a r e

known. These s igns depend e s s e n t i a l l y only on t h e r e l a t i v e p a r i -

t i e s of t h e p a r t i c l e s involved and a r e r a t h e r wel l e s t a b l i s h e d f o r

t h i s system.' The s e c t i o n concludes with some remarks on t h e

unphysical region.

The appendix descr ibes how one can de r ive a whole c l a s s of

s impler i n e q u a l i t i e s from t h e previous r e s u l t s and follows t h e \

discuss ion of a . s i m i l a r problem by t h e author i n a d i f f e r e n t

context . b

The extension of t h e foregoing cons idera t ions t o o t h e r

s c a t t e r i n g processes i s f a i r l y t r i v i a l and i s not t h e r e f o r e con-

s ide red i n t h i s paper.

For t h e b e n e f i t of t h e reader i n t e r e s t e d p r imar i ly i n t h e

r e s u l t s , we mention t h a t t h e s e a r e contained i n Eqs. ( I I . ~ ) ,

(11 .16) , (11.23), ( 1 1 1 . 4 ) ~ ( I I I . ~ ) , (111.7)~ (111.8)~ ( I I I . ~ ) ,

(111.10) and t h e appendix. The p r i n c i p a l conventions about nota-

t i o n a r e explained a t t h e beginning of Sec t ions I1 and 111, i n

t h e t h ~ e e paragraphs which follow Eq. (11.8) and i n t h e ones

which inc lude Eqs. (11.11) and (111.7) .

11. The T-p Forward S c a t t e r i n g Amplitude:?

Let cd and .(1 denote t h e l abora to ry energy and momentum of t h e

pion and l e t mT and mN denote t h e pion and nucleonmasses . The

twice sub t rac ted forward d i spe r s ion r e l a t i o n i s then

- where f - (0) a r e t h e T + - ~ forward s c a t t e r i n g amplitudes, f 2 = 0.08

+ 2, " - w and aN - Jr/*wN For (11.1) t o be t r u e , i t i s s u f f i c i e n t t h a t

f - (w) be O(&@a) f o r 641 and f o r some a a s (3 + i n any

d i r e c t i o n . I f 6 ( W ) - a r e t h e t o t a l c ross s e c t i o n s , f - (a) a r e + +

normalized such t h a t

The symmetry r e l a t i o n ,

i s a l s o necessary t o de r ive (11.1)

The symbols 0 and & w i l l o f t e n be used i n t h i s paper i n

c e r t a i n well-known senses . Thus, f - (U) = ~ ( c r ) ) a s (A) ++ a means

t h a t if-(*)( < C la\ f o r some f i x e d e a s U ) + a w h i l e f - ( r ~ ) =

Qb W-40, &(o)~means t h a t ' 1 f - ( a ) ) ( h ( ~ 1 f o r any d > 0 a s w + a .

The l a t t e r i n p a r t i c u l a r can a l s o be w r i t t e n a s f j 0 -

a s fd j a .

We w i l l de r ive i n e q u a l i t i e s f o r t h e values of f - (d) only a t

energies with an immediate experimental s i g n i f i c a n c e . It i s easy

enough, however, t o extend t h e s e r e s u l t s i n t o Tnequa l i t i e s involv-

ing i t s value a t any r e a l o r complex Point by techniques s i m i l a r

t o those descr ibed i n Referenc'e 6.

Let Wi ( i = 1, 2, ... , p) be any p d i s t i n c t po in t s a t

which Re f ' - (d) i s known through, f o r ins tance , a phase s h i f t

a n a l y s i s and l e t U i ( i = p + 1, p + 2, . . . , n ) be t h e negat ives

of any n-p po in t s a t w h i c h Re f+(u) i s known. Further , l e t

E . ( i = 1, 2, . . . , n-3) be some r e a l v a r i a b l e s . Define 1

*'a TT cw- c,) v- (w) = &-(a) 3 ((w-Wc)

(11.4) 4 s 1

where t h e product i n t h e numerator i s t o be s e t equal t o one if

n = 3 We assume t h a t f - ((3) = o ( W 2+6~nau)) f o r s ( 1 and f o r

some a a s t ~ ) -+ 00 i n any d i r e c t i o n . For r e a l d i r e c t i o n s , t h i s

i s almost c e r t a i n l y t r u e , being much,weaker than t h e F r o i s s a r t

bound. The d i spe r s ion r e l a t i o n f o r g - (a) with t h e s e assumptions

reads

where t h e p r i n c i p l e value i s t o be taken a t each of t h e W i and

a t @ . The r e l a t i v e minus s i g n of t h e two terms i n t h e i n t e g r a l

has come about from t h e i d e n t i t y

n-3 JT caw?- E,) 8y~-3

(t: 9 - ,. d L = \ T [wi+ E L ) - w C

. Tr 1 b:\ 4 =a-

and i s important f o r oi l r an3.1 yei.a.

2 The F r o i s s a r t bound7 a l s o impl ies t h a t Re f (o)) =a&) ) a s - W + @ along t h e r e a l a x i s . Therefore, g-(u) = &(l/$tb ) o r t h e

c o e f f i c i e n t of 1 / ~ on t h e r i g h t s i d e of (11.5) must vanish as

. That is,

A t t h e negat ive 'u Re f (@.) can be e x p ~ e s s e d i n terms of jy J

Re f + ( - W . ) through t h e symmetry r e l a t i o n (11.3). The sum r u l e J (11.7) t hus involves only measurable q u a n t i t i e s . For s u f f i c i e n t l y

l a r g e w", each term of t h e in tegrand i s p o s i t i v e d e f i n i t e s i n c e

I m f - ( a 1 ) i s ' s o because.of ( 1 1 . 2 ) . Therefore ,cancel la t ion be- +

tween t h e s e two terms cannot be respons ib le f o r t h e convergence

of t h e i n t e g r a l s . B u t . s i n c e t h e i n t e g r a l s must n e c e s s a r i l y

e x i s t i f what we have assumed about Re f - (@) i s c o r r e c t , I m f - (w) - -

2+@ a +

must i n f a c t be&(a2/Rn W ) and not merely O(W n cr)) a s

+ along t h e r e a l a x i s . We w i l l o f t e n encounter t h i s type

of r e s u l t i n our subsequent a n a l y s i s . 8

Let us now def ine two r e a l energies Q,) and such t h a t

wr> max W h (i = 1, 2, . . . , p) and Wk > m a x I ~ ~ (

( i = p + 1 , p + 2, . . . , n ) . We can then r e w r i t e (11 .7) a s

06

i T(w' 2 ) i" I-(@'+ E' 1 - 1 dw ' !mi ( ,d l ) 4-1 dw' jm #p)') r(W~+C) - 7r - T ( w l - W , )

ud *4 (11.8)

It i s convenient t o in t roduce a few d e f i n i t i o n s a t t h i s

poin t : a ) For any r e a l a and b with a _L b, t E i ] E ~ ( a , b ) if

some p a i r s of E a r e equal and assume any r e a l value and t h e r e s t i

a a r e cons t ra ined by t h e inequality a ,C E ~ , < b e b) W & Ep((lQI) if

a,> maxWi ( i = q, q + 1, . . . , q + p ) i c ) For a r e a l M ,

Or E F p ( % , Y i f Wr > max W i ( i = q, q + 1, . . . , q + p ) o r

(,dr ) @ according a s whether max Wi > l(r o r > max

f o r i = q, q + 1, ... ,, q + p. I n t h i s case, i f maxWi =My

s h o u l d be > @ . The corresponding n o t a t i o n f o r w can be L i i l l u s t r a t e d by observing t h a t i n ( I I . ~ ) , 3 E . E ( I Wp + ) .

n-P d) FOP a r e a l w , [ ~ i ~ C R ( P ) i f a l l t h e Wi a r e ) w. e ) E u i )

6 K ( w ) i f a l l t h e L d i a r e > p f ) {wit 6 s(-p) i f a l l t h e U)

a r e ,( -IJ-.

The d e f i n i t i o n s a ) d) e ) and f ) w i l l a l s o be used f o r t h e

ind iv idua l ai. b U , ThusA i f only t h e f i r s t p of t h e LL)ils a r e IJ-, .

we w i l l w r i t e Wi E ~ ( p ) ( A = 1, 2, . . . , p) .

The contents of t h i s n o t a t i o n w i l l perhaps be c l a r i f i e d i f

i t i s observed t h a t i f a l l t h e Wi a r e 'JT--p po in t s , {ui\e R(mT) A

while i f only t h e f i r s t p of them a r e 'JT--p po in t s and t h e r e s t

4 negat ives of ' J T + - ~ po in t s , O i E R(mi,) f o r i = 1, 2, . . . p and

k). E S(-m ) f o r i = p + 1, p + 2, . . . ', n. 1 'JT

The domains ofWr and u) have b e e n s o defined i n (11.8).as e t o render t h e products TI-(*, -mi) and a(af +Wi) p o s i t i v e d e f i n i t e

OM. i t s r i g h t s i d e . Therefore, s i n c e I m f - ( W ) a r e a l s o p o s i t i v e +

d e f i n i t e , t h e r i g h t s i d e a s a whole i s so when ~ E J E D ( - W W,) . 4'

That is,

i f t h e f i r s t p of t h e W i t s E ~(rn~), t h e r e s t of t h e W i t s

G s ( - m T ) , E E p ( q ) > 9 € En-p(J

This i s t h e f i r s t of t h e promised i n e q u a l i t i e s and has t h e i n t e r -

+ e s t i n g f e a t u r e t h a t i t allows t h e use ot' TI---p and T -p d a t a

simultaneously. I n c i d e n t a l l y , i t i s not permit ted t o s e t both

0) and equal t o i n f i n i t y i n ( I I . ~ ) , f o r then., we ge t back r 4 (11 .7) . The i n e q u a l i t y of course g e t s s t r o n g e r wi th increas i .ng

'l'he reason.why we arranged f o r a minus s i g n between t h e

two terms of t h e i n t e g r a l i n (11.5) should now be c l e a r . . I f i t

had been a p lus , we could not have got any i n e q u a l i t y involving.

only f i n i t e i n t e g r a l s without e x t r a assumptions regarding t h e r e l a -

t i v e magnitudes of t h e two t o t a l c ross s e c t i o n s . This l a s t obser-

va t ion w i l l be developed f u r t h e r p r e s e n t l y . [Cf. Eq. (11.16)

with condi t ion a ) and E = - oO . ] n-3

It i s i n s t r u c t i v e t o r e w r i t e (11.9) i n terms of f + ( ~ ) :

The W (1 = 1, 2, . . . p) now correspond t o r+-p poin t s and t h e

r e s t of t h e O i correspond t o negat ives of t h e rr--p p o i n t s .

To proceed f u r t h e r , l e t us assume, a s i s ind ica ted by exper i -

ments, t h a t r (u ) ) - ) ~ ( 0 ) f o r W 3 W 0 where uo is t o be d e t e r -

mined empir ica l ly . Let

where - (u) >, 0 f o r W >,Wo. I n t h e f i r s t ins tance , l e t

<wi) 6 ~ ( m ~ ) , t h a t i s , l e t a l l of them be rr--p p o i n t s . We can

r e w r i t e (11.7) i n t h e form

*M TTCWl- E,) - 1 \ b u r g_ (wl) -

7r Tr PI- 4

where U), E Fn(WL, Wo) and W4 E ~ ~ ( 0 ~ ) . The f i r s t i n t e g r a l on

t h e r i g h t s i d e of (11.12) i s p o s i t i v e i f f E ~ ) E D ( - 0 0 , u p ) *

i f any one of t h e following condi t ions a r e s a t i s f i e d :

a ) A l l t h e Wi E R ( m T ) , W r Fn(ulJ No) , u4 E ~ ~ ( 0 ~ ) and E i 4 0 .

8. b ) The f i r s t p of t h e w i E R(m,), t h e r e s t of t h e w i E s ( - m T ) , I

t h e choice of t h e Wi i s such t h a t (11.13) holds f o r ~ 4 3 9 E F (LJ1, ao), Ut E E ~ ( I w ~ ( ) and Ei' 0 : c ) W r P E F ~ ( w ~ J U ) ~ )

and 9 ( ~ ~ ( 1 0 ~ 1 ) , and t E i f E D(-(AB J ? ) o r f E i 3 E D(-($.Wr)

according as whether %.Or o r Y >ar. The {oif here a r e -

t h e usual mixture of p o i n t s . This l a s t condi t ion follows

d i r e c t l y from (11.12) while b ) i s evident from t h e d e r i v a t i o n

leading t o (11.16).

It may be use fu l t o observe t h a t we do not r e a l l y r e q u i r e

9 - (u)) t o be p o s i t i v e d e f i n i t e f o r a l l W 3 LOo. It i s s u f f i c i e n t

f o r t h e appropr ia t e values of Ei, o r i n case i t i s negat ive, i f

t h e second i n t e g r a l i n (11.12) dominates t h e f i r s t f o r t h e s e Ei.

Eq. (11.17) looks r a t h e r p l a u s i b l e s i n c e a t t h e observed energies

> WOJ B-(w) is p o s i t i v e while t h e con t r ibu t ion from a . poss ib le

change of s i g n o f 9 - (w) a t h igher energies tends t o ge t suppressed

by th.e e x t r a f a c t o r i n t h e in tegrand. I f des i red , t h e high

energy c o n t r i b u t i o n can be suppressed f u r t h e r by reducing t h e

number of Ei t o n-3-2p f o r some i n t e g e r p . This w i l l change t h e

f a c t o r 1/wt3 t o 3+2p while l eav ing t h e i n e q u a l i t i e s u n a l t e r -

Observations seem to- reveal'that total cross sections

approach constants at high energies which implies that

Im f ( 0 ) ~ ~ 6 (a) for large 0. Due to the incoherence of many i 9T 7

phase shifts and to the dominance of inelastic channels, we would

also expect Re f - (~)/Irn f - (a) or equivalently, %f - ( w ) / u to + + +

tend to zero'with increasing energy. Slightly weaker assumptions

than these are sufficient for our purposes. Thus we shall assume

that'f - (0) = O(W '+'~n~w) for 1 and for some a ascl, + in any direction and Re f - j 0 as c.0 -+ 00 along the

1

real axis.' If we choose n-2, instead of n-3, Ei s with n 3 2,

(11.7) is seen to be replaced by

where the products in the numerators are as usual to be set equal

to one if n = 2. Write this equation in the form

- L dw' Cp ( w ' ) S -IT (w'- EL) - n T (w '- lo')

T r . .

With t h e s t a t e d assumptions, t h e two i n f i n i t e i n t e g r a l s i n (11.19)

w i l l e x i s t .8 Let ILL).) 1 E ~ ( m ~ ) and W

and

I f o r Ei 6 0 f o r e v e r y i, t h e l as t i n t e g r a l i s p o s i t i v e i n t h i s

range of Ei because of (11.13). I f we assume f u r t h e r t h a t

~ . ( i = 1, 2, . . . , n-3), t h e l e f t . s i d e of (11.21) i s l i n e a r i n 1

En-2 and w i l l t h e r e f o r e be p o s i t i v e f o r -a & E 6 0 i f i t 11-2

i s p o s i t i v e a t t h e po in t s En - - - - 00 and En - = 0 . The poin t

En-2 = - @ gives us back (11.16) while En - = 0 gives

when E ~ C 0 (1 = 1, 2, . . . - 3 ) We should now s e t E a l s o n-3

equal t o zero s i n c e E = - d) gives back (11.16) with i t s n-3

= 0 . Proceeding i n t h i s way, we f i n a l l y ge t

w-& - + w

'a-2 ~ w ' i w ' x

where {di] ~ ( m ~ ) , W r E F,(W~, Wo) and W E En(w,). This -4 equat ion i s a l s o t r u e i f t h e l a s t n-p s 'denote negat ives of

i + t h e rr -p po in t s provided E F ~ ( w ~ , Uo) , f% E E , ( I W ~ I ) and 4

t h e choice of t h e Wi i s such t h a t (11.13) holds f o r U)i&u) . 4'

I f q-(O) i s not p o s i t i v e d e f i n i t e f o r up w we requ i re / * 0)

i n s t e a d t h a t

I f i t a l s o t u r n s out t o b e negat ive, i t i s s u f f i c i e n t i f t h e

r i g h t s i d e of (11.19) i s p o s i t i v e when a l l t h e E a r e zero. i

111. The K - . D Forward S c a t t e r i n g A m ~ l i t u d e

The K--p forward d i spe r s ion r e l a t i o n reads

U ) - W * e w

W - w y 00

W 3

(111.1)

wi th

and

a 1 C -WlJ ' wl, ) A ' = (111.2)

The res idues f 2 and f 2 a t t h e A and poles have p o s i t i v e A Z s igns s i n c e t h e K- l\ and t h e K- r e l a t i v e p a r i t i e s a r e .odd. 5 '

I f f - (w) = O ( W 2 + 6 ~ n a u ) f o r some a and f o r r< 1 as

2 b2 + 00 i n any d i r e c t i o n and i f Re f - (a) =Md) a s :--)a , ,

along t h e r e a l a x i s , we can prove a sum r u l e , s i m i l a r t o (11.7):

The ranges' of t h e ind ices have been i n d i c a t e d i n (111.3) as a

reminder. The f i r s t p of t h e Oi a r e taken t o be K--p po in t s

(no t n e c e s s a r i l y ,111 t he physical reg ion) and t h e remaining LOi a r e

+ negat ives uP K -p po in t s . We know t h a t I m f - (w) w i l l vanish l i k e + a a s U) -+ rnK i f we exclude t h e improbable s i t u a t i o n where t h e

t o t a l c ross s e c t i o n becomes i n f i n i t e a t zero k i n e t i c ' e n e r g y . This

follows t r i v i a l l y fyom t h e op.Lical, theorem. However, s i n c e t h e r e

i s no such r e s t r i c t i o n a t t h e poin t k) = p, I m f (u) may not even - vanish ( l e t along vanish s u f f i c i e n t l y f a s t ) a s QJ .--) p.. The

p r i n c i p a l value i n t e g r a l may t h e r e f o r e diverge i f u i = p so t h a t

we should r e q u i r e Oi E R(p) f o r i = 1, 2, .. . , p. Choosing an

E Fp(ul, mK) and an O E E ) , we l e a r n from 4 n-P (111.3) t h a t

when {E$ E D(- y, up). The unphysical region has not been

removed from ( I I I . ~ ) ~ t h i s poin t w i l l be discussed l a t e r on. When Q1

t h e argument of f - (w) becomes negat ive, t h e analogue of t h e symme-

t r y (11.3) can be used t o c a l c u l a t e i t s value.

Since t h e magnitudes of f 2 and f 2 a r e but poorly laown, A L

we s h a l l - now proceed t o e l imina te them from (111.4) . Rewriting

i t i n t h e form

7 @e observe t h a t we have e s s e n t i a l l y t o search f o r ranges of Ei

. which w i l l nlakt! t h e r i g h t s i d e p o s i t i v e . Two cases a r e poss ib le :

a ) s(Uh -W i ) > 0, h = A ,z . I n t h i s case, t h e r i g h t s i d e

i s p o s i t i v e i f i) an odd number of Ei a r e >, LC) , ii) an even z number of t h e remaining Ei s a t i s f y W ~~h L3

A E: and iii) & &

t h e r e s t of Ei E D( -a , &,, ) . A b ) r(Oh -W i ) < 0, h = A , 2 . I n t h i s case, t h e r i g h t

s i d e i s p o s i t i v e i f i) an even number of Ei a r e ), W , ii) an 1

even number of t h e remaining Ei s a t i s f y W Ei g W and z iii) t h e r e s t of t h e Ei E D(-CO , b,,).

Combining t h e s e with t h e r e s t r i c t i o n 1 E ~ ) D( - , O r),

we f i n a l l y a r r i v e a t t h e i n e q u a l i t y 4

when W r E Eb (dl. mK). (dq E n - p ( l ~ p + l l ) and 1) an odd (even)

number of Ei a r e contained i n W Q Ei< Or, ii) an even number z of t h e remaining Ei a r e contained i n 4) Ei $ W

A z and

i i i i ) B t h e r e s t of t h e E ~ ' E D(-CtJ . t)h ) i f rr(wh -ai)) 0 (( 0 ) . 4 Here only t h e f i r s t p of t h e Wi need belong t o E(p,). Notice t h a t

~ ( ( 3 ~ - W i ) > 0 o r < 0 auuurding a s whether p i s even o r odd. I

Experimental evidence i n d i c a t e s t h a t r ( w ) - & 6+(~) f o r

every f i n i t e 0. This means t h a t I m f - (w) - I m f + @ ) =?-(a) 3 0

f o r a >, mK. The analogue of (11.16) i s t h e r e f o r e

- L = r -- > o Q ~ , . , !,:.,,T. . c . (m . 7)

. .

i f any one of t h e following condi t ions a r e s a t i s f i e d : a ) [ w ~ ~ ~ w ( w ) , E Fn ( ~ ~ , m ~ ) , 0, E En (w,) and Ei,C 0'. b) The f i r s t p of t h e

a. Is E K ( u ) , t h e r e s t E s(-%), t h e choice of t h e W i s such t h a t 1

(11.13) holds. f o r h~ a , o E F ~ ( L ~ ~ , , %) , % E ~ ~ ( 1 0 ~ 0 and

E i S 0.8 c ) W r € Fp (ul, %) and ic) 4 E En (lull ) 9 a n d p i )

E D ( - ~ J , @ ) o r { E ~ ~ E D ( - ~ , w,) according a s whether 0 < W r 4 1 Q

o r ' w > O r . Here too, only t h e f i r s t p Cdi need belong t o 8 ( p ) . .x We can e l imina te t h e f

2 h

n-3 terms e a s i l y . Thus

and E l & 0, o r b) The f i r s t p of t h e ai E R(w) where p i s odd,

W r e Fp 9 E En ((ul\ ), (11.13) i s t r u e f o r w*> ' W 4 a n d E i & O , o r e ) 0, E F P (9, % ) , ( . d e E En ( (u l \ ) and i) an

odd (even) number of Ei a r e contained i n a < E. ( min 2. 1.- i r . 9 3 ii) an even number of t h e remaining E i . . a r e contained i n IdA ,L E i 6 Wz

and iii) t h e r e s t of t h e Ei E D(-0 ,U ) i f r(uA -ai) > 0 ( 4 0 ) . e A

Fina l ly , i f f - (a) = 0 ($+9nag) f o r 6 ( 1 and. f o r some a as

C1) i n any d i r e c t i o n , Re f - (LO) =,&(a) a s cc) + oO along t h e

r e a l axis a.nd (W) >, 0 f o r UJ >/ mK, -

> o (111.9) if[Uli) c "(p) , U E Fn (ul, y) and O E En (Y) . If only t h e

4 f i r s t ) of t h e Wi E " ( w ) , we r e q u i r e i n s t e a d t h a t W, E F ~ ( G ~ ,

m ~ ) , E~ ( ) w ~ \ ) and (11.13) be t r u e f o r W ' ~ U a .

The elimination of the f2 terms is now trivial and the result h

in either case is

(111.10)

if an odd number of Wi E 8 ( p ) the rest of the W E S(-rnK) and

the restrictions on W are unaltered. r, ll

As we have already observed in the previous section, the demand

that $) - (a) be non-negative for 0 3 % can be relaxed a great deal

in the inequalities starting from (111.7).

All the equations of this section involve contributions from

the unphysical region )L % of the K--p scattering. It is

unfortunately true that unitarity in the unphysical region does

not impose any positive definiteness condition on Im f - (w) and this

is precisely what prevents us from getting rid of these terms. It

is, however, sufficient for our purposes if we can get an idea of

the signs of these integrals for an appropriate range of the Eils

and t h i s can o f t en be done, e s p e c i a l l y i n K--p s c a t t e r i n g , by

simple c a l c u l a t i o n s involving, f o r ins tance , t h e known resonance

con t r ibu t ions . Once t h i s i s done, i t i s usua l ly poss ib le t o

remove t h e s e terms by an appropr ia t e choice of t h e parameters of

2 t h e i n e q u a l i t i e s J u s t a s we removed t h e terms involving f 2 and f . A Z

We emphasize t h a t even some e r r o r i n t h e es t imates of t h e s e s igns

can be t o l e r a t e d . This i s because our i n e q u a l i t i e s have h i t h e r t o

been consequences of e q u a l i t i e s of t h e form (11.8) where t h e r i g h t

s i d e i s p o s i t i v e and non-zero. Therefore, when we b r ing o v e r

a d d i t i o n a l terms of ambiguous s igns from l e f t t o r i g h t , we only

requ i re t h a t t h e s e ambigui t ies do not overwhelm t h e q u a n t i t y which

was i n i t i a l l y on t h e r i g h t . It i s important t o note t h a t we

I commit ourselves much l e s s i n i n e q u a l i t i e s than i n e q u a l i t i e s and

this gi-eatly. l>eiluces the chances of mistakes. We a l s o need know r

much l e s s t o use them. We pay f o r t h e s e advantages by l o s i n g some

of t h e information contained i n t h e e q u a l i t i e s .

We remark i n conclusion t h a t i t i s poss ib le t o s tudy t h e

quest ions t r e a t e d i n t h i s paper and i n reference 6 i n terms of

c e r t a i n reduced moment problems.1° This observat ion w i l l be

developed f u r t h e r i n a &*publication.

The au thor i s g r a t e f u l t o P. P. Divakaran f o r a c a r e f u l read-

ing of t h e manuscript and f o r many suggest ions and t o C . R .

Schumacher f o r h e l p f u l d i scuss ions .

References

. 1. M . Gell-Mann, M . ' L . Goldberger and W . Thi r r ing , Phys. Rev.

95, 1612 (1954); M. L . Goldberger, Phys. Rev. 99, 979 (1955); - -

M. L . Goldberger, H. Miyazawa and R . Oehme, Phys. Rev. - 99,

2. H . L . Anderson, W. C . Davidon and U . E. Kruse, Phys. Rev. - 100,

339 (1955); U . Haber-Schaim, Phys. Rev. - 104, 1113 (1956);

T . Spearman, Nucl. Phys. - 16, 402 (1960); F. Salzman and

G . Salzman, Phys. Rev. - 120, 599 (1960). See G . Puppi, 1958

Annual I n t e r n a t i o n a l Conference a t CERN, p. 43 f o r f u r t h e r

r e fe rences .

I 3. See R . H. Da l i t z , 1958 Annual I n t e r n a t i o n a l Conference a t

CERN, p . 191 and R . KaPplus, L. Kerth and T . Kycia, Phys. 1

Rev. L e t t e r s - 2, 510 (1959).

4. Riazuddin, Phys. Rev. - 121, 1509 (1961).

5 . A summary of t h e evidence and d e t a i l e d references a r e t o be

found i n G . A . Snow, 1962 I n t e r n a t i o n a l Conference on High

Energy Physics, p . 795.

6 . A . P . Balachandran, Boundary Conditions f o r a Partial-wave

Amplitude, EFINS-63-67 and t o be publ ished.

7. M. F r o i s s a r t , Phys. Rev. - 123, 1053 (1961); A . Martin, 1962

I n t e r n a t i o n a l Conference on High Energy Physics, p. 566. I

8. See i n t h i s connection D. Amati, M . F i e r z and V. Glaser,

Phys. Rev. L e t t e r s - 4, 89 (1960); M. Sugawara and A . Kanazawa,

Phys. Rev. - 123, 1895 (1961); S. Weinberg, Phys. Rev. 124,

2049 (1961).

9. These are, however, stronger than what is required to prove

the Pomeranchuk theorem, I. Ia. Pomeranchuk, J. Exptl.

Theoret. Phys. (u.S.S.R.) - 34, 725 (1958) [Translation:

Soviet Phys. - JETP 7, 499 (1958)l. See also Reference 8.

10. J. A. Shohat and J. D. Tamarkin, The Problem of Moments

(American Mathematical Society, 1943) ,

Appendix

We descr ibe how t h e i n e q u a l i t i e s of t h e previous s e c t i o n s can

be s i m p l i f i e d and c o l l e c t a l l t h e re levant formulae i n t h i s appen-

d ix . Much of t h e d iscuss ion i s taken from t h e appendix of Refer-

ence 6 although t h e n o t a t i o n has been s l i g h t l y a l t e r e d t o a form

more s u i t a b l e f o r our purposes.

Af ter a few t r i v i a l changes of va r i ab les , an3 i n e q u a l i t y of

t h i s paper can be w r i t t e n a s

m-8 7r (0- E,)

m > o v

2 / = i

when 1 ~ ~ 3 l i e s i n some r e a l domain. The i n t e g r a t i o n i s always

over a r e a l range V which may poss ib ly c o n s i s t of s e v e r a l d i s t i n c t

p ieces . The s p e c t r a l funct ion W (a) conta ins 8-funct ions t o t ake

i n t o account t h e poles of h i . The subsc r ip t m of h i means t h a t

it i s a funct ion of El, E2, . . . , Em which.occur as t h e product m a (a' - Ei) i n t h e numerator of i t s in tegrand. The supersc r ip t j

i = l means t h a t i n t h i s product, t h e &hh j E i t s a r e t o be s e t equal

t o zero. This index i s a c t u a l l y zero i n t h e equat ions we have

h i t h e r t o e n c o ~ ~ n t e r e d .

The fol lowing i d e n t i t y i s o f t e n u s e f u l : '

where &(f,&) denotes t h e sum over a l l combinations of t h e p

va r i ab les E: : . ' : ,E ,.:* E . , taken a t a time and 4 ( p , o) m--#-p+*: ; m-9-b;aw* i s defined t o be equal f o one. For example,

It may have been observed t h a t p s tands f o r t h e number of E i l s

with respect t o which we have expanded t h e in tegrand and then

i n t e g r a t e d term by term.

We w r i t e down two more use fu l i d e n t i t i e s . Thus,

where

- if Ell-8-$imfd+dii = E w - g = E. Further ,

, - where

i f E =. -E - - . . . = -E - +

= E. Eqs. (A.4) and ( ~ . 6 ) m-j-~p+, * - a - 2 p + ~

can be proved by expanding t h e in tegrand i n a binomial s e r i e s .

1 n s t e a d . o f a t tempting t o reduce every i n e q u a l i t y we have

encountered i n t o ones with no E i t s a t a l l , we s h a l l d i scuss two 1

r e p r e s e n t a t i v e cases 1 ~ ~ 3 D(a,b) f o r some r e a l a and b and

. v .. E ~ $ 0 f o r every i. It i s s t r a igh t fo rward t o handle t h e o t h e r

equat ions along s i m i l a r l i n e s . Let us f i r s t s tudy t h e case where

. { ~~j E ~ ( a , b ) . With t . = 0, ( A . 1) then reads

when 1 ~ ~ ) ~ ( a , b) . I f we expand h: i n terms of t h e l a s t two

E . ' s , we have, 1

With t h e o t h e r v a r i a b l e s f ixed , we f i n d t h a t two cases a r e now

poss ib le : a ) Em-l = Em and assumes any r e a l value and b) a $ Em - I - '3 a Q Em& b. I n case a ) , ( A . 8) g ives

( A . 10)

f o r any r e a l Em. Therefore, s i n c e ( A . l O ) i s quadra t ic i n Em,

TO reduce case b ) , observe t h a t f o r f ixed Em, ( A . 9 ) i s l i n e a r i n

E m - l and w i l l , t h e r e f o r e , be pos,dtive f o r a 4 Em-l< b i f i t is.

Po$&tive a t t h e end po in t s E = a and Em - = b . That is , we m - 1

r e q u i r e

But t h e s e a r e i n t u r n l i n e a r i n Em and should theyefore be ~ o ~ i t i v e

f o r Em = a a n d Em = b. only t h e combination Em #'Em-1 gives ' a

r e su lb d i f f e r e n t from ( ~ . 1 0 ) and t h i s reads

( A . 13)

/ I n b o t h (~.11) and ( ~ . 1 3 ) , E. 1 ~ ( a , b ) f o r i = 1, 2, .... , m - 3.

Note t h e s i m p l i f i c a t i o n when a o r be g e t s unbounded. Thus i f a=-00,

t , ( A . 1 3 ) becomes

( A . 14 )

We have not succeeded i n . s i m p l i f y i n g ( A . 1 1 ) and ( A . 1 3 ) s t i l l - f u r t h e r i n t o a s e t of i n e q u a l i t i e s which completely exhaust them

and a l s o conta in no E i l s a t a l l . It i s however easy t o w r i t e down

t h e i n e q u a l i t i e s implied by t h e s e equat ions w h e n a s Ei& b f o r

every i. To f i n d t h e s e equations, we have t o s e t each of t h e Ei

equal t o a and b i n t u r n i n ( A . 1 1 ) and ( A . 1 3 ) . This w i l l genera te

a c l a s s of funct ions wi th no E i l s . We denote any member of t h i s ad * (1.2) 0-

c l a s s by#:. These can be c a l c u l a t e d by repeated a p p l i c a t i o n of A t h e i d e n t i t y ( A . 4) . It may then be shown t h a t ( A . 11) and ( A . 13)

iinply t h e fol lowing equat ions:

where p = 0, 1, ... , i n t e g e r ( 3 . The cor re -

. .spending members of t h e c l a s s should of course be i n s e r t e d i n t h e

f i r s t and l as t l i n e s of ( A . 1 5 ) . I f t h e maximum value of p i s

m - ) , t h e r e w i l l be one Ei l e f t over' i n ( A . 15) which i s then

d t o be s e t equal t o a and b i n t u r n . I f a = b t h e l a s t equat ion of

( A . 1 5 ) i s unnecessary, being a l r eady implied by t h e f i r s t . t h r e e .

This i s c l e a r ' from t h e d i s c u s s i o n l e a d i n g t o ( A . 1 3 ) .

I n c i d e n t a l l y , ( ~ . 1 5 ) conta ins r a t h e r more information than

t h a t implied by t h e v a r i a t i o n of every Ei between a and b i n

( A ) and ( A . Thus, when t h e v a r i a b l e p i s increased by one

u n i t , we have e f f e c t i v e l y s e t a p a i r of Ei equal i n (~.11) and

( A . 13) and made them a r b i t r a r i l y l a r g e .

The case where E i < 0 f o r every i i s e a s i e r t o d i scuss .

Since

" E +-oo and m-p

t h e technique leading t o ( ~ . 1 3 ) shows t h a t t h e necessary and

s u f f i c i e n t condi t ions f o r ( ~ . 8 ) t o be t r u e when Ei,C 0 a r e t h e i n -

e q u a l i t i e s

The reader f o r whose purposes t h e equat ions of t h i s appen-

d i x a r e too d e t a i l e d may be a b l e t o f i n d more convenient ones by

g iv ing t h e E i l s s u i t a b l e values i n t h e appropr ia t e ranges i n t h e

i n e q u a l i t i e s given i n t h e body of t h e paper .