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Scattering amplitudes that satisfy a mandelstam representation with one subtraction and unitarity
-.~
Nuclear Physics B l l (1969) 573-587. North-Holland Publ. Comp., A m s t e r d a m
SCATTERING AMPLITUDES THAT SATISFY A MANDELSTAM REPRESENTATION WITH ONE SUBTRACTION AND UNITARITY J. K U P S C H
CERN, Geneva Received 16 April 1969
A b s t r a c t : It is proved that there exist functions which satisfy a Mandelstam r e p r e s e n tation with one subtraction elastic unitarity and the inelastic unitarity bounds Im fl(s) >~ I fl(s)] 2 for all e n e r g i e s . We give solutions with positive and with o s c i l lating double s p e c t r a l functions.
i . INTRODUCTION
T h e s t u d y of t h e M a n d e l s t a m r e p r e s e n t a t i o n [1] to g i v e a c r o s s i n g s y m m e t r i c a n d u n i t a r y a m p l i t u d e h a s c o n s i d e r a b l y a d v a n c e d by t h e w o r k of A t k i n s o n [2-4]. A t k i n s o n p r o v e d , f o r t h e c a s e w i t h o u t s u b t r a c t i o n s , t h a t there exist functions which satisfy these properties. The inelastic unitarity i s t h e r e b y r e d u c e d to the i n e q u a l i t y w h i c h c a n b e w r i t t e n w i t h the t w o - p a r t i c l e s c a t t e r i n g a m p l i t u d e a l o n e . A n e x t e n s i o n of t h i s p r o o f to a M a n d e l s t a m r e p r e s e n t a t i o n w i t h a f i n i t e n u m b e r of s u b t r a c t i o n s h a s b e e n s u g g e s t e d a l s o by A t k i n s o n [4], b u t n o t e v e n the u n i t a r i t y b o u n d s 0 --< Im fl(s) --< 1 h a v e b e e n derived. T h a t t h e s e i n e q u a l i t i e s 0 < Im fl(s) < 1 f o r a l l l a r e i n d e e d r a t h e r r e s t r i c t i v e h a s b e e n s h o w n by G o e b e l [5] and M a r t i n [6]. If t h e d o u b l e s p e c t r a l f u n c t i o n i s p o s i t i v e then t h e t o t a l c r o s s s e c t i o n h a s to v a n i s h [5, 6] f o r s --* and the a m p l i t u d e c a n a l w a y s b e w r i t t e n in a s p e c i a l f o r m of a M a n d e l s t a m r e p r e s e n t a t i o n w i t h two s u b t r a c t i o n s [6]. In a p r e c e d i n g p a p e r [7] w e h a v e s t u d i e d the r e d u c e d p r o b l e m of a M a n delstam representation with only one subtraction. The double spectral funct i o n p(s, t) w a s n o t r e s t r i c t e d to b e p o s i t i v e b u t w e u s e d ( a m o n g o t h e r c o n s t r a i n t s ) b o u n d s f o r Ip(s,t) I s u g g e s t e d by r e f . [6]. T h e e l a s t i c u n i t a r i t y of s u c h a m p l i t u d e s w a s s h o w n , the i n e l a s t i c i n e q u a l i t i e s Im fl(s) > I fl(s)l 2 f o l l o w e d o n l y f o r a f i n i t e r a n g e of the e n e r g y . F o r h i g h e r e n e r g i e s we j u s t o b t a i n e d 0 ~< Im fl(s) < 1. T h e a i m of t h i s p a p e r i s to d e r i v e the b o u n d s Im fl(s) >i [fl(s)[ 2 f o r a l l e n e r g i e s . To be c o m p l e t e a n d to g i v e the n o t a t i o n s , w e i n d i c a t e the p r o o f of e l a s t i c u n i t a r i t y of r e f . [7] in s e c t . 2. T h e m a i n r e s t r i c t i o n s on the s p e c t r a l f u n c t i o n s a r e s t a t e d in s e c t . 3 and t h e i n e l a s t i c u n i t a r i t y Imfl(s)>1 Ifl (s)
574
J. KUPSCH
is obtained in sect. 4. The double s p e c t r a l function m a y o s c i l l a t e but in s e c t . 4 the m o d u l u s Ip(s, t) I still s a t i s f i e s the r e s t r i c t i o n s f o r p o s i t i v e double s p e c t r a l functions in ref. [6]. T h i s is no l o n g e r the c a s e f o r the e x a m p l e s we d i s c u s s in sect. 5. H e r e p(s, t) i n c r e a s e s s u c h that Ip(s, t)[ would r e q u i r e n s u b t r a c t i o n s (n = 1,2, 3 , . . . ) . But the o s c i l l a t i o n s a r e so s t r o n g that the u n i t a r i t y bounds a r e m a i n t a i n e d , m o r e o v e r the a m p l i t u d e m a y still be w r i t t e n with one s u b t r a c t i o n and h e n c e the total c r o s s s e c t i o n v a n i s h e s f o r s--" ~ .
2. T H E P R O O F O F E L A S T I C UNITARITY We c o n s i d e r the s c a t t e r i n g of two equal n e u t r a l (pseudo) s c a l a r p a r t i c l e s . The s c a t t e r i n g a m p l i t u d e is given by
s+t {. p(s',t')ds'dt' F(s, t) = g(s) + g(t) + g(u) + ~ - J ( s ; - ~ ) i F - T ( s ' ¥ t ' ) s+u
p(s',u')ds'du' t+u [. p_(t'!u')dt'du' -~-~+.,) +-~ J(t,-z)(.,-.)(t,+u,)'
+ -fi- f ( s , - ~ ,
(2.1)
with
g(s) = ~i f~,(s,) ds'. The r e s t r i c t i o n s on p(s, t) and q~(s)will be given below. To denote the functional d e p e n d e n c e on p and q~ we a l s o w r i t e F[p, ~]. The u n i t a r i t y i n t e g r a l d e f i n e s a function
¢(s, t) =- ¢[~, ~] = x(s)
$ +1 2~ 1 1 f dz' f dO F* (s, z')F (s, zz' - (1 - z2)~(1 - z ' 2 ) ~ c o s 0), -1
0
(~(s, z) = F(s, t)).
(2.2)
H e r e we have i n t r o d u c e d a c u t - o f f function X (s) with oo
k(s) e C
,
k(s) = 1
if
4 -- 1 6 ) ,
0 --
if
s X< s < s A < ~o,
k(s) = 0
if
s > s A.
T h e r e e x i s t uniquely a double s p e c t r a l function pel(s, t) and a function q~el(s) s u c h that Im F [pel, ~ el] = ~b[p, q~].
(2.3)
SCATTERING AMPLITUDES
575
The e x p l i c i t equation f o r p e l ( s , t) is [1]
pel(s, t) = k(s)
2 [s(s-4)]
-½
ttl t2 -½ . X fdtldt2[t2 +t2+t 2-2(ttl+tt2+tlt2) - 4 ~_4] D (sl, tl)D(Sl,t2) ,
(2.4)
with the r a n g e of i n t e g r a t i o n t 1 >~4, t 2 >/ 4 and
#
t2
tl~ ½ t (1.s_4j
_1
1
D(s, t) is the a b s o r p t i v e p a r t in the t - c h a n n e l D ( s , t ) = go(t) + -s+t f [~(s', t)ds' t + u F [~(t,u')du' (s'-s)(s'+t) +--Tr J(u;~')"
T h e function
(2.5)
T h e single d i s c o n t i n u i t y goel(s) has to be d e t e r m i n e d by the l = 0 p a r t i a l wave p r o j e c t i o n of eq. (2.3)
(s_~)
½
-
½goel(s) +
, o s+t fpel(s,t')dt, [s(s-4)] -~ f dt j~U~i(-~) 4-s 1
o
= ¼X(s)[[s(s -4)] -~ f dt F(s,t)[ 2. 4-s
(2.6)
If we take a s y m m e t r i c function co(s,t) with s u p p o r t s >/ 16, t >I 16 and a function X(S) with s u p p o r t s >/ 16, the mapping
p(s,t)-* p'(s,t) -- pel(s,t) + p e l ( t , s ) + ¢o(s,t), I g0(s)-~ q~'(s) = goel(s) + X(S), is well defined. Under c e r t a i n conditions on co(s,t) and ×(s) this mapping has a fixed point solution fi (s, t), ~(s) (refs. [2-4, 7]). The a m p l i t u d e F[fi, ~] s a t i s f i e s e l a s t i c u n i t a r i t y and, of c o u r s e , c r o s s i n g s y m m e t r y . Following Atkinson [2-4], we have defined in ref. [71 a B a n a c h s p a c e q~o of H61der continuous functions by lims~oo f(s, t) = limt_+oo f(s, t) = 0 and the no r E
IIf(s,t)llo -]f(s 2' t2 ) - f(s 1' t i ) [lo g2 ( ~ ) lo g2 (~ t-)l =suPd
. s 2. - s. l . . ~. ~-U . Is - - - ~ l
,-/~2-~-1-~-~--~ -
-['
(2.7)
J
with g = m i n ( s l , s 2 ) , [ = m i n ( t l , t 2 ) , 0 < /z < ½, ~ > 1. To d e s c r i b e i n c r e a s ing functions we have i n t r o d u c e d a s p a c e c~ :f(s,t) ~ 93 if and only if ( s + t ) -1 f(s,t) ~ ~ o , the n o r m is given by ][f(s,t)]] = ]] ( s + t ) -1 f(s,t)[[ o.
576
J. KUPSCH
T h e e x i s t e n c e of a f i x e d p o i n t of the m a p p i n g (2.7) w a s shown in r e f . [7] f o r s p e c t r a l f u n c t i o n s q~(s) and p ( s , t ) i n 9 3 [the s u b t r a c t i o n f u n c t i o n g(s) in eq. (2.1) w a s g i v e n by g(s) = d+ (s/n) - s ) ] d s ' ] . To d e r i v e the u n i t a r i t y b o u n d s 0 ~< Im fl(s) --< 1 w e n e e d e d a d d i t i o n a l r e s t r i c t i o n s on the s p e c t r a l f u n c t i o n s . To o b t a i n the s t r o n g e r i n e q u a l i t i e s I m f l ( s ) >i ]fl(s)[2 f o r a l l e n e r g i e s the f u n c t i o n s X(S) e 93 and ¢o(s,t) e 93 h a v e t h e r e f o r e to s a t i s f y s e v e r a l c o n s t r a i n t s w h i c h a r e d i s c u s s e d in the f o l l o w i n g s e c t i o n s . [ E s p e c i a l l y X(s) h a s to d e c r e a s e f o r s --* oo and we can t a k e g(s) = ( l / n ) f ( p ( s ' ) / ( s ' - s ) d s ' in eq. (2.1).]
f,(s,)/[s,(s,
3. R E S T R I C T I O N S ON THE S P E C T R A L F U N C T I O N S We a s s u m e t h a t ¢o(s,t) c a n be w r i t t e n a s
¢o(s,t) = tp(s,t) + ~V(t,s),
(3.1)
w h e r e ~ ( s , t) i s c h o s e n s u c h t h a t s - ~p(s, t) e 93 o r , e q u i v a l e n t l y , sis + t) -1 tp(s, t) e 93o" T h e f u n c t i o n ~V(s, t) i s t h e n a l s o an e l e m e n t of 93 w i t h ] l t p ( s , t ) [ I <½11 s . ~ v ( s , t ) ll. If the f u n c t i o n s . ~ ( s , t ) l i e s w i t h i n the s p h e r e ] I s . ~p(s,t) l[ --< B 1 then I]c0(s,t)]l ~< B 1 and we m a y s u b s t i t u t e the q u a n t i t y B in r e f . [7] by B 1. T h e s u p p o r t of ~V(s, t) h a s to be r e s t r i c t e d to s >/ 16, t >/ 16. T h e r e a r e s o m e r e a s o n s to t a k e a s u p p o r t of co(s, t) in a s m a l l e r r e g i o n than s >1 16, t >/ 16 b e c a u s e co(s,t) s h o u l d c o n t a i n (for 16 -< s --< s~, o r 16 -< t < s k ) o n l y t h e c o n t r i b u t i o n of t h o s e F e y n m a n g r a p h s w h i c h c a n n o t b e o b t a i n e d by i t e r a t i o n [3]. So one e x p e c t s a b o u n d a r y of the s u p p o r t s-4 t = 16 s ~ " T h i s c h o i c e w a s p o s s i b l e in r e f . [7] and d o e s t h e r e f o r e s t i l l a p p l y f o r s and t b e l o w the c u t - o f f e n e r g y s k [ s e e the a r g u m e n t s f o l l o w i n g r e f . [7], eq. (38)]. But in o u r p r e s e n t p r o o f w e o n l y r e a c h t = 1 6 + 6 4 ( s - 4 ) -1 if s > s x and s = 16 + 6 4 ( t - 4 ) -1 if t > s x (see the end of s e c t . 4). S i n c e * p e l ( s , t) v a n i s h e s f o r s > s A the f u n c t i o n s . p e l ( s , t) b e l o n g s to93 and t h e r e f o r e p(s, t) =pel(s, t) + p e l ( t , s) + co(s, t) h a s a l s o the s t r u c t u r e of eq. (3.1). A s i m p l e m e t h o d to o b t a i n eq. (3.1) i s by f a c t o r i z a t i o n . To t h i s end w e i n t r o d u c e the B a n a c h s p a c e 5Co of H S l d e r c o n t i n u o u s f u n c t i o n s of one v a r i a b l e w i t h the n o r m
lllJ
]f ( s 2 ) _ f ( s l ) Ilog2(~2~ ) Sl~S2t~ sup
4< S l , S 2 < oo
<3.2)
]s2_sllP
and lims__~oo f ( s ) = 0 [the c o n s t a n t s /~ and [2 a r e t a k e n a s in eq. (2.7)]. T h e i n c l u s i o n f r o x ~ o c 93o f o r p r o d u c t s of f u n c t i o n s i s o b v i o u s . Now l e t 5r_ 1 b e the s e t of a l l f u n c t i o n s f ( s ) s u c h t h a t s f ( s ) e ~ o and l e t 5r 1 c o n t a i n a l l f u n c t i o n s w h i c h s a t i s f y s - l f ( s ) e ~o. T h e n ~r o × 5rl and 5 r l x ~ o a r e s u b s e t s of 93, and if g(s) e ~ - 1 and h(t) e ~ 1 the a n s a t z ~p(s, t) = = g(s). h(t) e v e n f u l f i l l s s . ~ ( s , t) e 93 and m a y be u s e d in eq. (3.1). • In the following p(s, t), pel(s, t), ~p(s), q~el(s) denote the fixed point solution.
577
SCATTERING AMPLITUDES
]Is.
F r o m eq. (3.1) we get an e s t i m a t e of co(s,t), lco(s,t)] < @(s,t)]] x [2 + (s/t) + (t/s)] s-/z log -2 (~s) t - P log -2 (~t) w h i c h is s o m e w h a t m o r e r e s t r i c t i v e than condition (325 of ref. [7]. The f o r m (3.1) of a double s p e c t r a l function a l s o i m p l i e s s o m e bounds f o r d i s p e r s i o n i n t e g r a l s . The i n t e g r a l
Dl(s,t )
s+t f w(s',t) = - 7 ~ (s' - s)(s'+ t) d s ' ,
is e s t i m a t e d by (see a p p e n d i x A)
[Dl(s,t)l < C[l +
(i+ lslS-"log-2(a(l+ ] s l ) ) + ~ ] x t -/z
log-2(f~t),
(3.4)
f o r t > 16 and all s with C -< c o n s t B 1. T h e double i n t e g r a l
Fl(s't)=~
s + t f ds_: dt' w(s',t') "s'-s[;-t s'+t' '
can be m a j o r i z e d by
]Fl(s,t) l --< C[1
+
[tll-Pl°g-2(fZ(l+
Itl)) + ]sll-~log-2(n(l+ lsl))
1 + Isl
,
(3.55
1 + ]t]
f o r all s and t with C --< c o n s t B 1. T h e d i s c o n t i n u i t y X(S) e cB is a s s u m e d to v a n i s h at infinity, o r m o r e e x p l i c i t l y s-E X(S) has to b e l o n g to ~ o f o r s o m e ¢, 0 < 2e
4. T H E I N E L A S T I C U N I T A R I T Y BOUNDS Within the l i m i t a t i o n s of the p r e v i o u s s e c t i o n the f u n c t i o n s c0(s, t) and X(s) can be a r r a n g e d so that the a m p l i t u d e (2.1) a l s o s a t i s f i e s the i n e l a s t i c unitarity bounds
I
Im fl(s) >~ fz(s> ]2,
l = o, 2 , 4 , . . .
,
(4.1)
f o r all e n e r g i e s s > 16. T h i s p r o o f is divided in two p a r t s , the e a s e l = 0 and the e a s e l > 0.
578
J. KUPSCH
4.1. Angular m o m e n t u m l = 0 To obtain Im fo(S) >1 ]fo(S)] 2 we have to show [see eq. (2.6)] S
1
1
~~/\ T ]-4~'~ X(s) + [s(s - 4)] -2 f
0
dt s + t
4-s
4f•
pel(t', s) + co(s, t' )
(t'-t)(s+t')
dt'
>~¼(l-k(s))[s(s-4)]-I ]} F(s,t)dt[2. 4-s
(4.2)
The integral o v e r the amplitude (2.1) may be written as
f
o
4-s
F(s,t)dt = (s-4)g(s) + 2
o
f
4-s
s+t ( p(s',t')ds'dt' dt[g(t) + - 2 - J ( s ' - s ) ( t ' - t ) ( s ' + t ' ) ]
o t+u f p(t',u')dt'du' dt 2-½s ~- a(t'-t)(u'-u)(t'+u')"
+2 f
(4.3)
The second t e r m in eq. (4.2) d e c r e a s e s f a s t e r than s-/~ and the right-hand side f a s t e r than s 2 ( e - P ) ( s e e below). The function X(S)with Ills-E ×is)III ~ B1, a r e a p p r o p r i ately small. To get the r e q u e s t e d bounds we f i r s t consider those t e r m s in eqs. (4.2) and (4.3) containing the double s p e c t r a l function p(s, t) which can be e s t i mated with the help of eqs. (3.4) and (3.5). These inequalities a r e valid also for p ( s , t ) = pel(s, t) + pel(t, s)+ w ( s , t ) , only the constant C has to be r e placed by c o n s t . (B 1 + (A 1 + B1)2) to account for the contribution of pel(s, t) and p el (t, s) [compare the a r g u m e n t s following eq. (32) in ref. [7]]. The integrals dt in eqs (4 2) and (4 3) yield the bound s l - g, so all t e r m s in eq. (4.2) originated by the double s p e c t r a l function a r e m a j o r i z e d by const (B 1 + (A 1 + B1)2) • s- U. On the other hand we know f r o m Ills-~ X(S)]H < A1 that Ig(s) l < < c o n s t . AI(1 + I sl)£-~ t for high values of s." Since g(s) o r g(t) show up only in the quadratic t e r m s , eq. (4.2) follows with a p p r o p r i a t e values of A 1 and B 1•
fo
4 - $
"
"
"
4.2. Angular m o m e n t u m l >~2 The p a r t i a l wave amplitude is now given by the F r o i s s a r t - G r i b o v f o r mula 2 is(s_4)]-½ f fz(s) = ~-
QI(I+ s~4)D(s,t)dt.
4 To obtain eq. (4.1) we need s o m e positivity p r o p e r t i e s of w(s, t). As in
(4.4)
SCATTERING AMPLITUDES
579
ref. [7] we a s s u m e that w is, t) is p o s i t i v e along the boundary of its s u p p o r t and that this p a r t d o m i n a t e s in the s e n s e of eqs. (39)-(41) of ref. [7]. An a n s a t z which s a t i s f i e s t h e s e r e s t r i c t i o n s is
¢o (s, t) = COl(S , t) + ¢o2(s , t),
(4.5)
with Wl(S, t), w2(s , t) of the f o r m (3.1), ~l(S, t) = h(s)(t- 16) 1-/~ log-2(~2t)
= bl(S-16)/~,
if
16< s < 17,
h(s) e~7_l , h(s)=t > 0
if
17< s < 18,
=0
if
s > 18,
l
and I1 " VX(s,t)l]
<
IIs-
b 1> 0,
vl(S,t)ll.
In the following we a s s u m e for s i m p l i c i t y this a n s a t z with I I s . ~ 2 ( s , t ) l l - - < ½ ] ] S - ~ l ( s , t ) ] l . The n o r m o f w l ( s , t ) c a n b e changed by the p a r a m e t e r b 1. Under s o m e w h a t m o r e g e n e r a l conditions than those given h e r e we have d e r i v e d in ref. [7] the unitarity bounds (4.1) up to the e n e r g y sk, and 0 < Im fl(S) < 1 f o r all e n e r g i e s . The e n e r g y s k is d e t e r m i n e d by the c u t off •(s) in eq. (2.2) and is taken l a r g e r than 18. F r o m the d e t a i l s of this p r o o f we r e m e m b e r (see ref. [7], sect. 4.2)
]Im/z(s)l
(4.6)
½,
f o r all s, l = 2 , 4 , . . . if B 1 is chosen s m a l l enough (this bound is also i m plicitly contained in the following e s t i m a t e s ) and p e l ( s , t ) >/ 0 ,
(4.7)
f o r all t if 4 ~< s ~< 16. To obtain eq. (4.1), we have now to p r o v e
Im fl(s) >~ 21Re fl(s) l 2,
l=2,4,...,
s > s k.
(4.8)
F r o m eq. (4.7) we know
16 since p(s, t) = pel(t, s) f o r t < 16. The function pel(s, t) depends q u a d r a t i c a l l y on w(s, t) and X(S) t h e r e f o r e we get w i t h s m a l l n o r m s
Im fl(s) >~-~1[ s ( s - 4 ) ] - ½ f 16
Qz (1 + s2~t_4)COl(s,t)dt,
w h e r e we only take into a c c o u n t the d o m i n a n t p a r t COl(S , t) of eq. (4.5). F o r values of s > 18, this g i v e s
580
J. KUPSCH 18
1
Im
fl(s) >i171 [s(s - 4 ) ] - ~ ( s -
16) 1-~z log-2 (as)
f Ql(l + s2~t4)h(t)dt.
(4.9)
16
The r e a l p a r t of eq. (4.4) is divided in 1 17
R11)(=):~=(=-4)I -~ f Ql(l + s2-----~t4)Re D(s, t)d' , 4
and = ~-
~--2-~_ 4) Re
D(s,t)d,,
17
The p r o o f of eq. (4.8) is then reduced to
Imfl(s ) >I 4iR~l'2)(s)l 2
= > s~,
l : 2,4,
Let us f i r s t d i s c u s s R~2)(s). The contribution of 4v(') does not exceed
2[=(=_4)] -½ f~ Q,O+ .-~4)I~(')t d,
_
71"
17
oo
~< const- (AI÷ ( A I + B 1 ) 2 ) s
e-Az f Ql(l+x)x¢-Pdx V
with v = 34/(s - 4 ) , and for the f i r s t d i s p e r s i o n i n t e g r a l in eq. (2.5) the e s t i m a t e (3.4) gives
2I=(=-4)1 -½ f= oz(1 + 7-~-4) 2, Ial(=")l dt 1'7
oo
~< c o n s t .
(BI+(AI+B1)2){s -~
f Ql(l+x)x-~ dx V oo
+ s -~z log -2
(~s)Ql(1
+
34 [ t-l-p dt + s-/~ f Ql(1+ x ) x 1-/2 dx}. -s-~-4) 17
v
The second d i s p e r s i o n i n t e g r a l has the s a m e bound. The function iRl(2)(s)[2 d e c r e a s e s t h e r e f o r e at l e a s t as s 2 ( ~ - P ) if s goes to infinity. Since 0 ~ 2e < p this is dominated by eq. (4.9). The l b e h a v i o u r of the above e x p r e s s i o n s is d e t e r m i n e d by Ql(1 + 34/[s - 4 ] ) [we m a y w r i t e for all the i n t e g r a l s
SCATTERING AMPLITUDES
f
581
Ql(Z)f(z) dz = fQl(Z)Z2z-2f(z)dz ~
l+v
f
z-2f(z)dz,
l+v
b e c a u s e z2 Ql(Z) i s d e c r e a s i n g in z f o r l >/ 2]. H e n c e the / - d e p e n d e n c e of IR~2)(s) [ 2 is e s t i m a t e d by
+s4
and t h e r e i s no p r o b l e m to o b t a i n
I m f l ( s ) >I 4 IRl2)(s)] 2 , for s > sk, l = 2 , 4 , . . . if the values of A1 and B1 are small enough. More difficult arguments are needed for R~l)(s). We know Re D(s,t) ~ c(t-4) ~ s l-/z log-2(~2s), f o r 4 --< t ~< 17 with c < const. (AI+BI+ (AI+B1)2). This implies
IR~I)(s) I < const. (AI + B1+ (A1 + B1)2)2 s -2~ log-4(~2s)
× ]f
17
(t-4)"Ql(l+
_ )dtl 2 2t
4
T h e l a s t i n t e g r a l can be e s t i m a t e d by (see a p p e n d i x B) 17
If
17
(t-4) ~
Q/(I+
s-~4) dt[ 2 ~< c o n s t , l o g s -
4
f
(t-16) ~ O l ( l +
s2--~t4)dt.
16
But this g i v e s the w a n t e d r e s u l t I R~l)(s)] 2 --< ¼Im
fl (s),
f o r s m a l l v a l u e s of A 1 and B 1. T h e p r o o f in a p p e n d i x B s h o w s that we m a y s u b s t i t u t e the l o w e r i n t e g r a tion l i m i t t = 16 by t = 16+ 6 4 ( s - 4 ) -1. T h e r e f o r e the s u p p o r t of w(s, t) m a y be r e s t r i c t e d to t >I 16+ 6 4 ( s - 4 ) -1 if s > s x and s >/ 16+ 6 4 ( t - 4 ) -1 if t > s x and we still obtain eq. (4.1) (for 16 --< s,t ~
582 5. MORE
J. KUPSCH GENERAL
CASES
In this section, we give examples of crossing s y m m e t r i c and unitary amplitudes the double spectral functions of which do not lie in c/8. The r e p resentation (2.1) is still valid because of the strong oscillations of p(s, t), but Ip(s, t)] would require n subtractions. The ansatz for w(s, t) is
co(s,t) = Wl(S , t) + w2(s , t),
(5.1)
with Wl(S , t) as in eq. (4.5) and
w2(s,t) = b 2 0 ( s - 2 0 ) O ( t - 2 0 ) ( s - 2 0 ) n ( t - 2 0 ) n s i n
(~r[(s-20)(t-20)]n+l).
(5.2)
In appendix C, we study
D2(s, t)
1 f°°w2(s',t) -~ ~o s' - s
ds',
and obtain Re D2(s, t) = b 2 O(s - 20) O(t - 20)(s - 20) n (t - 20) n x cos (v[(s- 20)(t-20)] n+l) + g(s, t),
(5.3)
with a differentiable function g(s, t)
Ig(s,t) t --
< c o n s t - I b 2 ] ( l + ] s ] ) - 2 ( l + Itl) - 1 ,
~~g(s, l t)
~
-2.
The contribution of w2(s , t) to eqs. (2.1) and (2.5) may therefore be written in an unsubtracted form, i.e., ¢o2(s' , t ' ) d s ' d t ' F(s,t) =F[Pl,q~ ] + 1 f ~ - s ) - ~ - T ) + crossed terms ~2 where F[Pl, ~0] is the amplitude (2.1) with
Pl (S, t) = pel(s, t) + pel(t, s) + COl(S , t) , and as absorptive part we have
D(s,t) = D[Pl,q~] + D2(s , t) + D2(u ,t). With these definitions, the proof of elastic unitarity goes through without
SCATTERING AMPLITUDES
583
c o n s i d e r a b l e m o d i f i c a t i o n s . [The n o r m of c0 (s, t) h a s now to be s u b s t i t u t e d by ][Wl(S,t)[I + Ib21, o r I b l ] + Ib21. ] In this c a s e it is c o n v e n i e n t to c h o o s e the c u t - o f f e n e r g i e s s x = 18 and s A = 20 s u c h that the o s c i l l a t i n g p a r t of D2(s , t) d o e s not show up in eqs. (2.2)-(2.4). T h e c o n t r i b u t i o n of ¢o2(s, t) to the p a r t i a l w a v e s is given by
f 2)(s) =2
oo
2t f Ql (1 + ~Z-_~_4)(D2(s,t)+ D2(4-s-t,t))dt, 4 l = 0,2,4 .....
T h e s e f u n c t i o n s can be m a j o r i z e d by p a r t i a l w a v e s of the type c o n s i d e r e d in sect. 4. The only t r o u b l e could a r i s e f r o m the i n c r e a s i n g (and o s c i l l a t i n g ) p a r t of D2(s,t) , but the e s t i m a t e oo
i f Ql(1 + ~:-~_4/w2(s,t), 2t
--< Q / \ (1 -
20
+
4O
f
16+ s - I
]w2(s't)]dt
16
--< c o n s t " ] b 2 ] s -1
Ql(1 + s~4)
s h o w s that t h e r e is no difficulty. So we o b t a i n the i n e l a s t i c u n i t a r i t y bounds (4.1) f o r a s m a l l b2. In this e x a m p l e , the o s c i l l a t i o n s a r e so s t r o n g that we do not need s u b t r a c t i o n s f o r ¢o2(s,t) and the c o n t r i b u t i o n to the p a r t i a l w a v e s is m u c h s m a l l e r than a l l o w e d by the p r o o f . This is no l o n g e r the c a s e if we take w 2 is, t) = b 2 (s + t ) l + 6 ( s - 20) n (t - 20) n sin (~[(s - 20) (t - 2 0 ) I n + l ) , (0< 5< 1-~). H e r e we have to u s e the s u b t r a c t e d d i s p e r s i o n i n t e g r a l s of sect. 2. A c a l culation a s b e f o r e p r o v i d e s a u n i t a r y a m p l i t u d e f o r s m a l l b 2.
6. C O N C L U D I N G R E M A R K S We have o b t a i n e d a c l a s s of f u n c t i o n s w h i c h s a t i s f y a M a n d e l s t a m r e p r e s e n t a t i o n with one s u b t r a c t i o n e l a s t i c u n i t a r i t y and the i n e l a s t i c u n i t a r i t y bounds Ira flis) >i Ill (s)]2. T h e double s p e c t r a l f u n c t i o n s p(s, t) a r e p o s i t i v e f o r s --< 17 o r t < 17 and m a y o s c i l l a t e e l s e w h e r e . But this s t r i p w h e r e p(s, t) is p o s i t i v e still p r e v e n t s the a m p l i t u d e to d e v e l o p R e g g e a s y m p t o t i c s . The p r o o f i n c l u d e s p o l y n o m i a l l y i n c r e a s i n g double s p e c t r a l f u n c t i o n s with r a t h e r p a t h o l o g i c a l o s c i l l a t i o n s (their p h a s e v a r i e s with sn). F o r all s o l u t i o n s the total c r o s s s e c t i o n is bounded by c o n s t , s - ~ ( l o g s ) -1 and can be a r r a n g e d to b e h a v e like s - g ( l o g s ) - l . Since we m a y take ~ > 0 a r b i t r a r i l y s m a l l this r e p r e s e n t s one s t e p f u r t h e r to a r e a l i s t i c ~ a m p l i -
584
J. K U P S C H
tude. However, this proof does not allow ~ = 0, and still m o r e w o r k has to be done to obtain a non-vanishing total cross section. I a m indebted to Professor A. Martin for helpful suggestions and a critical reading of the manuscript.
APPENDIX A To derive the estimate (3.4) we consider
Jl(S, t) =
16
~(s', t)ds' (~-s-)~+})
J2(s, t) =
and
S ~(t, s')ds' 16 ( s ' - s ) ( s ' + t ) '
withs.~V(s,t) ~ 93, IIs'~V(s,t) ll --
S . J l ( S , t ) = ( ds' J s' - s
s ' ~ ( s ' , t ) + f d s ' ~V(s',t) s' + t'
s' + t
'
is a function in 930 and the second term belongs to g o which follows from
~ ( s ' t l ) l"
IJl(S,t) --4,
(A. 1)
t>4,
with C = const. B 1. The integral J2(s, t) is easily estimated:
]J2(s,t) l -< C. s -~ log-2(~2s)t -1-~ log-2(•t), s>4,
(A.2)
t>4.
These results can be extended to all real values of s (see ref. [2]), we have only to replace s by 1+ Is l and to take a somewhat l a r g e r C. Then eq. (3.4) follows. The same technique can be applied to the double integral ( ~h(s', t ' ) d s ' d t ' J3(s't) = J ( s ' - s ) ( t ' - t ) ( s ' + t ' ) '
s. J3(s,t ) = f s ' ~ ( s ' , t ' ) d s ' d t ' J(s'-s)(t'-t)(s'+t')
r~(s',t')ds'dt' +J (t'-t)(s'+t') "
The f i r s t t e r m of the right-hand side is a function in 930, the second t e r m
SCATTERING AMPLITUDES
585
a function in ~r o. Hence we get an e s t i m a t e of the type (A.1). If we add the c r o s s e d t e r m and extend the r e s u l t to all values of s and t the inequality (3.5) follows.
APPENDIX B In this appendix we p r o v e
If
17
(r(s)+l (t_4);
Ol(1 + s "2t~ ) d t [ 2 --< c o n s t ' l o g s f
4
(t_cr(s))~ Ql(1 + - ~2t4 4- ) dt
a(s)
f o r l = 0 , 1 , 2 . . . . a n d s > 4 , with any f u n c t i o n a ( s ) , 16
(t _ 4) ~ v l ( 1
+
2t _4)dt[ 2 ~(s)+l
~< c o n s t , l o g s
f or(s)
since ( t - 4)~ is bounded in 5 < t ~< 17 and As a f i r s t step we d e r i v e
2t
(t-or(s)) ~ Ql(1 + ~-:-~_4)d t ,
(B. 1)
Ql (1 + 2t/[s-417 is d e c r e a s i n g .
[Ql(Z)]2 ~<3z Qo(Z)Ql (2z 2 - 1),
(B.2)
for l=0,1,.., and 1< z < oo. If we apply S c h w a r z ' s inequality to the i n t e g r a l r e p r e s e n t a t i o n oo
Ql(Z) = f
(z + ~
coshO)-l-ldo,
0
we obtain
[Ql(Z)]2 ~
[Ql(Z)] 2 ~
Ql (z) ql+l (z)
Ql-1 (z) Ol(Z)
Qo (z) Q-~
3z,
and hence
Q2l(Z) ~<3z Q2/+l (z).
(B.4)
586
J. KUPSCH
The i n t e g r a l f o r Q21+l(Z) m a y be e s t i m a t e d a s CO
Q2I+I(Z) = f
1
[(z+ (z 2 - 1) i c o s h 0)2] - / - 1 dO
O
CO
=f
[u+ (u 2 - 1)½ c o s h 0 + (z 2 - 1 ) s i n h 2 o]-l-ldo
O CO
< f
1
(u+ (u 2 - 1) i c o s h o)-l-ldo = Ql(U),
O
w h e r e we have i n t r o d u c e d u = 2z 2 - 1. T a k i n g a c c o u n t of eqs. (B.3) and (B.4) this i m p l i e s the r e l a t i o n (B.2). F o r z = 1+ 8 / ( s - 4 ) we have 2z 2 - 1 = 1+ ( 2 / [ s - 4 ] ) ( 1 6 + 6 4 / [ s - 4 ] ) and therefore
Ol(1 + s - ~ - / j
-.< } I o g s .
+
Ql
f o r any a ( s ) w i t h 16 ~ 0. The e s t i m a t e (B. 1) follows now e a s i l y f r o m S c h w a r z ' s inequality
if
5
5 ( t _ 4 ) ~ Ql(1 + 2t
2t
4
4
and eq. (B.5).
APPENDIX C
F o r the function
f(x, y) = O(x) O(y) (xy) n-1 sin [= (xy)~ , n > 1 fixed, the H i l b e r t t r a n s f o r m m a y be defined by
h(x,y) =
lim
1 p ?/(x',y)
r-~ co n
o
-~ Z x
dx'.
(C.1)
T h i s definition g i v e s i m m e d i a t e l y h(x, y) = 0 if y < 0 and the functional r e lation
h(x,y) = h(xy, 1), if y > 0. L e t us c o n s i d e r the function
1i F(z)= lim ~X'*c°
O
~n-1 exp (i~ n) d~. ~-z
(C.2)
SCATTERING AMPLITUDES
587
T h i s i n t e g r a l i s r e l a t e d to h(x,y) by
h(x, 1) = ½Im [F(x+iE) + F(x- iE)].
(C.3)
W i t h i n t h e s e c t o r 0 < a r g ~ --< ~r/(2n) the f u n c t i o n e x p (i~ n) d e c r e a s e s z e r o if I~[ "-* ~o. W e m a y t h e r e f o r e u s e the C a u c h y t h e o r e m f o r
~J
~-z
to
d~
w i t h t h e i n t e g r a t i o n a l o n g 0 --* +oo -~ ~o. e x p [i~/(2n)] --* O. If small, we obtain
]Im z]
is rather
F(Z) = 0(Re z) 0(Im z) 2iz n-1 exp (iz n) + ~ (xv)n-1 exp (-x n) dx X7 Z o -
-ex, 0 F o r r e a l z t h e l a s t i n t e g r a l i s b o u n d e d by c o n s t (1 + ] z ] ) -1 and i t s d e r i v a t i v e d o e s n o t e x c e e d c o n s t (1 + I z ] ) -2. T h e r e l a t i o n s (C.2) and (C.3) y i e l d t h e n the r e s u l t
h(x, y)
=
O(x) O(y)(xy) n-1 c o s [TT(xy)n]
+
g(x, y) ,
with
]g(x,y)] -
1xi)-1(1+]yl) -1 ,
and
ag(x, y)
-< const.¢:+ Ix])-2(:+ [y])-:,
~g(x, y) --< c o n s t - ( 1 +
Ixl)-l(l+ lyl) -2,
w h i c h i s e q u i v a l e n t to eq. (5.3).
RE FERENCES [i] [2] [3] [4]
S. Mandelstam, Phys. Rev. 112 (1958) 1344. D. Atkinson, Nucl. Phys. B7 (1968) 375. D. Atkinson, Nucl. Phys. B8 (1968) 377. D. Atkinson, P r e l i m i n a r y version of l e c t u r e s to be delivered at the 11th Boulder Summer Institute for Theoretical P h y s i c s , University of Bonn p r e p r i n t (1968). [5] C. Goebel, P r o c . 1961 Midwest Conf. on theoretical physics, p. 34-38. [6] A. Martin, Nuovo Cimento 61A (1969) 56. [7] ft. Kupsch, CERN p r e p r i n t TH. 984 (1969), to be published in Nucl. Phys.
Nuclear Physics B l l (1969) 573-587. North-Holland Publ. Comp., A m s t e r d a m
SCATTERING AMPLITUDES THAT SATISFY A MANDELSTAM REPRESENTATION WITH ONE SUBTRACTION AND UNITARITY J. K U P S C H
CERN, Geneva Received 16 April 1969
A b s t r a c t : It is proved that there exist functions which satisfy a Mandelstam r e p r e s e n tation with one subtraction elastic unitarity and the inelastic unitarity bounds Im fl(s) >~ I fl(s)] 2 for all e n e r g i e s . We give solutions with positive and with o s c i l lating double s p e c t r a l functions.
i . INTRODUCTION
T h e s t u d y of t h e M a n d e l s t a m r e p r e s e n t a t i o n [1] to g i v e a c r o s s i n g s y m m e t r i c a n d u n i t a r y a m p l i t u d e h a s c o n s i d e r a b l y a d v a n c e d by t h e w o r k of A t k i n s o n [2-4]. A t k i n s o n p r o v e d , f o r t h e c a s e w i t h o u t s u b t r a c t i o n s , t h a t there exist functions which satisfy these properties. The inelastic unitarity i s t h e r e b y r e d u c e d to the i n e q u a l i t y w h i c h c a n b e w r i t t e n w i t h the t w o - p a r t i c l e s c a t t e r i n g a m p l i t u d e a l o n e . A n e x t e n s i o n of t h i s p r o o f to a M a n d e l s t a m r e p r e s e n t a t i o n w i t h a f i n i t e n u m b e r of s u b t r a c t i o n s h a s b e e n s u g g e s t e d a l s o by A t k i n s o n [4], b u t n o t e v e n the u n i t a r i t y b o u n d s 0 --< Im fl(s) --< 1 h a v e b e e n derived. T h a t t h e s e i n e q u a l i t i e s 0 < Im fl(s) < 1 f o r a l l l a r e i n d e e d r a t h e r r e s t r i c t i v e h a s b e e n s h o w n by G o e b e l [5] and M a r t i n [6]. If t h e d o u b l e s p e c t r a l f u n c t i o n i s p o s i t i v e then t h e t o t a l c r o s s s e c t i o n h a s to v a n i s h [5, 6] f o r s --* and the a m p l i t u d e c a n a l w a y s b e w r i t t e n in a s p e c i a l f o r m of a M a n d e l s t a m r e p r e s e n t a t i o n w i t h two s u b t r a c t i o n s [6]. In a p r e c e d i n g p a p e r [7] w e h a v e s t u d i e d the r e d u c e d p r o b l e m of a M a n delstam representation with only one subtraction. The double spectral funct i o n p(s, t) w a s n o t r e s t r i c t e d to b e p o s i t i v e b u t w e u s e d ( a m o n g o t h e r c o n s t r a i n t s ) b o u n d s f o r Ip(s,t) I s u g g e s t e d by r e f . [6]. T h e e l a s t i c u n i t a r i t y of s u c h a m p l i t u d e s w a s s h o w n , the i n e l a s t i c i n e q u a l i t i e s Im fl(s) > I fl(s)l 2 f o l l o w e d o n l y f o r a f i n i t e r a n g e of the e n e r g y . F o r h i g h e r e n e r g i e s we j u s t o b t a i n e d 0 ~< Im fl(s) < 1. T h e a i m of t h i s p a p e r i s to d e r i v e the b o u n d s Im fl(s) >i [fl(s)[ 2 f o r a l l e n e r g i e s . To be c o m p l e t e a n d to g i v e the n o t a t i o n s , w e i n d i c a t e the p r o o f of e l a s t i c u n i t a r i t y of r e f . [7] in s e c t . 2. T h e m a i n r e s t r i c t i o n s on the s p e c t r a l f u n c t i o n s a r e s t a t e d in s e c t . 3 and t h e i n e l a s t i c u n i t a r i t y Imfl(s)>1 Ifl (s)
574
J. KUPSCH
is obtained in sect. 4. The double s p e c t r a l function m a y o s c i l l a t e but in s e c t . 4 the m o d u l u s Ip(s, t) I still s a t i s f i e s the r e s t r i c t i o n s f o r p o s i t i v e double s p e c t r a l functions in ref. [6]. T h i s is no l o n g e r the c a s e f o r the e x a m p l e s we d i s c u s s in sect. 5. H e r e p(s, t) i n c r e a s e s s u c h that Ip(s, t)[ would r e q u i r e n s u b t r a c t i o n s (n = 1,2, 3 , . . . ) . But the o s c i l l a t i o n s a r e so s t r o n g that the u n i t a r i t y bounds a r e m a i n t a i n e d , m o r e o v e r the a m p l i t u d e m a y still be w r i t t e n with one s u b t r a c t i o n and h e n c e the total c r o s s s e c t i o n v a n i s h e s f o r s--" ~ .
2. T H E P R O O F O F E L A S T I C UNITARITY We c o n s i d e r the s c a t t e r i n g of two equal n e u t r a l (pseudo) s c a l a r p a r t i c l e s . The s c a t t e r i n g a m p l i t u d e is given by
s+t {. p(s',t')ds'dt' F(s, t) = g(s) + g(t) + g(u) + ~ - J ( s ; - ~ ) i F - T ( s ' ¥ t ' ) s+u
p(s',u')ds'du' t+u [. p_(t'!u')dt'du' -~-~+.,) +-~ J(t,-z)(.,-.)(t,+u,)'
+ -fi- f ( s , - ~ ,
(2.1)
with
g(s) = ~i f~,(s,) ds'. The r e s t r i c t i o n s on p(s, t) and q~(s)will be given below. To denote the functional d e p e n d e n c e on p and q~ we a l s o w r i t e F[p, ~]. The u n i t a r i t y i n t e g r a l d e f i n e s a function
¢(s, t) =- ¢[~, ~] = x(s)
$ +1 2~ 1 1 f dz' f dO F* (s, z')F (s, zz' - (1 - z2)~(1 - z ' 2 ) ~ c o s 0), -1
0
(~(s, z) = F(s, t)).
(2.2)
H e r e we have i n t r o d u c e d a c u t - o f f function X (s) with oo
k(s) e C
,
k(s) = 1
if
4 --
0 --
if
s X< s < s A < ~o,
k(s) = 0
if
s > s A.
T h e r e e x i s t uniquely a double s p e c t r a l function pel(s, t) and a function q~el(s) s u c h that Im F [pel, ~ el] = ~b[p, q~].
(2.3)
SCATTERING AMPLITUDES
575
The e x p l i c i t equation f o r p e l ( s , t) is [1]
pel(s, t) = k(s)
2 [s(s-4)]
-½
ttl t2 -½ . X fdtldt2[t2 +t2+t 2-2(ttl+tt2+tlt2) - 4 ~_4] D (sl, tl)D(Sl,t2) ,
(2.4)
with the r a n g e of i n t e g r a t i o n t 1 >~4, t 2 >/ 4 and
#
t2
tl~ ½ t (1.s_4j
_1
1
D(s, t) is the a b s o r p t i v e p a r t in the t - c h a n n e l D ( s , t ) = go(t) + -s+t f [~(s', t)ds' t + u F [~(t,u')du' (s'-s)(s'+t) +--Tr J(u;~')"
T h e function
(2.5)
T h e single d i s c o n t i n u i t y goel(s) has to be d e t e r m i n e d by the l = 0 p a r t i a l wave p r o j e c t i o n of eq. (2.3)
(s_~)
½
-
½goel(s) +
, o s+t fpel(s,t')dt, [s(s-4)] -~ f dt j~U~i(-~) 4-s 1
o
= ¼X(s)[[s(s -4)] -~ f dt F(s,t)[ 2. 4-s
(2.6)
If we take a s y m m e t r i c function co(s,t) with s u p p o r t s >/ 16, t >I 16 and a function X(S) with s u p p o r t s >/ 16, the mapping
p(s,t)-* p'(s,t) -- pel(s,t) + p e l ( t , s ) + ¢o(s,t), I g0(s)-~ q~'(s) = goel(s) + X(S), is well defined. Under c e r t a i n conditions on co(s,t) and ×(s) this mapping has a fixed point solution fi (s, t), ~(s) (refs. [2-4, 7]). The a m p l i t u d e F[fi, ~] s a t i s f i e s e l a s t i c u n i t a r i t y and, of c o u r s e , c r o s s i n g s y m m e t r y . Following Atkinson [2-4], we have defined in ref. [71 a B a n a c h s p a c e q~o of H61der continuous functions by lims~oo f(s, t) = limt_+oo f(s, t) = 0 and the no r E
IIf(s,t)llo -]f(s 2' t2 ) - f(s 1' t i ) [lo g2 ( ~ ) lo g2 (~ t-)l =suPd
. s 2. - s. l . . ~. ~-U . Is - - - ~ l
,-/~2-~-1-~-~--~ -
-['
(2.7)
J
with g = m i n ( s l , s 2 ) , [ = m i n ( t l , t 2 ) , 0 < /z < ½, ~ > 1. To d e s c r i b e i n c r e a s ing functions we have i n t r o d u c e d a s p a c e c~ :f(s,t) ~ 93 if and only if ( s + t ) -1 f(s,t) ~ ~ o , the n o r m is given by ][f(s,t)]] = ]] ( s + t ) -1 f(s,t)[[ o.
576
J. KUPSCH
T h e e x i s t e n c e of a f i x e d p o i n t of the m a p p i n g (2.7) w a s shown in r e f . [7] f o r s p e c t r a l f u n c t i o n s q~(s) and p ( s , t ) i n 9 3 [the s u b t r a c t i o n f u n c t i o n g(s) in eq. (2.1) w a s g i v e n by g(s) = d+ (s/n) - s ) ] d s ' ] . To d e r i v e the u n i t a r i t y b o u n d s 0 ~< Im fl(s) --< 1 w e n e e d e d a d d i t i o n a l r e s t r i c t i o n s on the s p e c t r a l f u n c t i o n s . To o b t a i n the s t r o n g e r i n e q u a l i t i e s I m f l ( s ) >i ]fl(s)[2 f o r a l l e n e r g i e s the f u n c t i o n s X(S) e 93 and ¢o(s,t) e 93 h a v e t h e r e f o r e to s a t i s f y s e v e r a l c o n s t r a i n t s w h i c h a r e d i s c u s s e d in the f o l l o w i n g s e c t i o n s . [ E s p e c i a l l y X(s) h a s to d e c r e a s e f o r s --* oo and we can t a k e g(s) = ( l / n ) f ( p ( s ' ) / ( s ' - s ) d s ' in eq. (2.1).]
f,(s,)/[s,(s,
3. R E S T R I C T I O N S ON THE S P E C T R A L F U N C T I O N S We a s s u m e t h a t ¢o(s,t) c a n be w r i t t e n a s
¢o(s,t) = tp(s,t) + ~V(t,s),
(3.1)
w h e r e ~ ( s , t) i s c h o s e n s u c h t h a t s - ~p(s, t) e 93 o r , e q u i v a l e n t l y , sis + t) -1 tp(s, t) e 93o" T h e f u n c t i o n ~V(s, t) i s t h e n a l s o an e l e m e n t of 93 w i t h ] l t p ( s , t ) [ I <½11 s . ~ v ( s , t ) ll. If the f u n c t i o n s . ~ ( s , t ) l i e s w i t h i n the s p h e r e ] I s . ~p(s,t) l[ --< B 1 then I]c0(s,t)]l ~< B 1 and we m a y s u b s t i t u t e the q u a n t i t y B in r e f . [7] by B 1. T h e s u p p o r t of ~V(s, t) h a s to be r e s t r i c t e d to s >/ 16, t >/ 16. T h e r e a r e s o m e r e a s o n s to t a k e a s u p p o r t of co(s, t) in a s m a l l e r r e g i o n than s >1 16, t >/ 16 b e c a u s e co(s,t) s h o u l d c o n t a i n (for 16 -< s --< s~, o r 16 -< t < s k ) o n l y t h e c o n t r i b u t i o n of t h o s e F e y n m a n g r a p h s w h i c h c a n n o t b e o b t a i n e d by i t e r a t i o n [3]. So one e x p e c t s a b o u n d a r y of the s u p p o r t s-4 t = 16 s ~ " T h i s c h o i c e w a s p o s s i b l e in r e f . [7] and d o e s t h e r e f o r e s t i l l a p p l y f o r s and t b e l o w the c u t - o f f e n e r g y s k [ s e e the a r g u m e n t s f o l l o w i n g r e f . [7], eq. (38)]. But in o u r p r e s e n t p r o o f w e o n l y r e a c h t = 1 6 + 6 4 ( s - 4 ) -1 if s > s x and s = 16 + 6 4 ( t - 4 ) -1 if t > s x (see the end of s e c t . 4). S i n c e * p e l ( s , t) v a n i s h e s f o r s > s A the f u n c t i o n s . p e l ( s , t) b e l o n g s to93 and t h e r e f o r e p(s, t) =pel(s, t) + p e l ( t , s) + co(s, t) h a s a l s o the s t r u c t u r e of eq. (3.1). A s i m p l e m e t h o d to o b t a i n eq. (3.1) i s by f a c t o r i z a t i o n . To t h i s end w e i n t r o d u c e the B a n a c h s p a c e 5Co of H S l d e r c o n t i n u o u s f u n c t i o n s of one v a r i a b l e w i t h the n o r m
lllJ
]f ( s 2 ) _ f ( s l ) Ilog2(~2~ ) Sl~S2t~ sup
4< S l , S 2 < oo
<3.2)
]s2_sllP
and lims__~oo f ( s ) = 0 [the c o n s t a n t s /~ and [2 a r e t a k e n a s in eq. (2.7)]. T h e i n c l u s i o n f r o x ~ o c 93o f o r p r o d u c t s of f u n c t i o n s i s o b v i o u s . Now l e t 5r_ 1 b e the s e t of a l l f u n c t i o n s f ( s ) s u c h t h a t s f ( s ) e ~ o and l e t 5r 1 c o n t a i n a l l f u n c t i o n s w h i c h s a t i s f y s - l f ( s ) e ~o. T h e n ~r o × 5rl and 5 r l x ~ o a r e s u b s e t s of 93, and if g(s) e ~ - 1 and h(t) e ~ 1 the a n s a t z ~p(s, t) = = g(s). h(t) e v e n f u l f i l l s s . ~ ( s , t) e 93 and m a y be u s e d in eq. (3.1). • In the following p(s, t), pel(s, t), ~p(s), q~el(s) denote the fixed point solution.
577
SCATTERING AMPLITUDES
]Is.
F r o m eq. (3.1) we get an e s t i m a t e of co(s,t), lco(s,t)] < @(s,t)]] x [2 + (s/t) + (t/s)] s-/z log -2 (~s) t - P log -2 (~t) w h i c h is s o m e w h a t m o r e r e s t r i c t i v e than condition (325 of ref. [7]. The f o r m (3.1) of a double s p e c t r a l function a l s o i m p l i e s s o m e bounds f o r d i s p e r s i o n i n t e g r a l s . The i n t e g r a l
Dl(s,t )
s+t f w(s',t) = - 7 ~ (s' - s)(s'+ t) d s ' ,
is e s t i m a t e d by (see a p p e n d i x A)
[Dl(s,t)l < C[l +
(i+ lslS-"log-2(a(l+ ] s l ) ) + ~ ] x t -/z
log-2(f~t),
(3.4)
f o r t > 16 and all s with C -< c o n s t B 1. T h e double i n t e g r a l
Fl(s't)=~
s + t f ds_: dt' w(s',t') "s'-s[;-t s'+t' '
can be m a j o r i z e d by
]Fl(s,t) l --< C[1
+
[tll-Pl°g-2(fZ(l+
Itl)) + ]sll-~log-2(n(l+ lsl))
1 + Isl
,
(3.55
1 + ]t]
f o r all s and t with C --< c o n s t B 1. T h e d i s c o n t i n u i t y X(S) e cB is a s s u m e d to v a n i s h at infinity, o r m o r e e x p l i c i t l y s-E X(S) has to b e l o n g to ~ o f o r s o m e ¢, 0 < 2e
4. T H E I N E L A S T I C U N I T A R I T Y BOUNDS Within the l i m i t a t i o n s of the p r e v i o u s s e c t i o n the f u n c t i o n s c0(s, t) and X(s) can be a r r a n g e d so that the a m p l i t u d e (2.1) a l s o s a t i s f i e s the i n e l a s t i c unitarity bounds
I
Im fl(s) >~ fz(s> ]2,
l = o, 2 , 4 , . . .
,
(4.1)
f o r all e n e r g i e s s > 16. T h i s p r o o f is divided in two p a r t s , the e a s e l = 0 and the e a s e l > 0.
578
J. KUPSCH
4.1. Angular m o m e n t u m l = 0 To obtain Im fo(S) >1 ]fo(S)] 2 we have to show [see eq. (2.6)] S
1
1
~~/\ T ]-4~'~ X(s) + [s(s - 4)] -2 f
0
dt s + t
4-s
4f•
pel(t', s) + co(s, t' )
(t'-t)(s+t')
dt'
>~¼(l-k(s))[s(s-4)]-I ]} F(s,t)dt[2. 4-s
(4.2)
The integral o v e r the amplitude (2.1) may be written as
f
o
4-s
F(s,t)dt = (s-4)g(s) + 2
o
f
4-s
s+t ( p(s',t')ds'dt' dt[g(t) + - 2 - J ( s ' - s ) ( t ' - t ) ( s ' + t ' ) ]
o t+u f p(t',u')dt'du' dt 2-½s ~- a(t'-t)(u'-u)(t'+u')"
+2 f
(4.3)
The second t e r m in eq. (4.2) d e c r e a s e s f a s t e r than s-/~ and the right-hand side f a s t e r than s 2 ( e - P ) ( s e e below). The function X(S)with Ills-E ×is)III ~
fo
4 - $
"
"
"
4.2. Angular m o m e n t u m l >~2 The p a r t i a l wave amplitude is now given by the F r o i s s a r t - G r i b o v f o r mula 2 is(s_4)]-½ f fz(s) = ~-
QI(I+ s~4)D(s,t)dt.
4 To obtain eq. (4.1) we need s o m e positivity p r o p e r t i e s of w(s, t). As in
(4.4)
SCATTERING AMPLITUDES
579
ref. [7] we a s s u m e that w is, t) is p o s i t i v e along the boundary of its s u p p o r t and that this p a r t d o m i n a t e s in the s e n s e of eqs. (39)-(41) of ref. [7]. An a n s a t z which s a t i s f i e s t h e s e r e s t r i c t i o n s is
¢o (s, t) = COl(S , t) + ¢o2(s , t),
(4.5)
with Wl(S, t), w2(s , t) of the f o r m (3.1), ~l(S, t) = h(s)(t- 16) 1-/~ log-2(~2t)
= bl(S-16)/~,
if
16< s < 17,
h(s) e~7_l , h(s)=t > 0
if
17< s < 18,
=0
if
s > 18,
l
and I1 " VX(s,t)l]
<
IIs-
b 1> 0,
vl(S,t)ll.
In the following we a s s u m e for s i m p l i c i t y this a n s a t z with I I s . ~ 2 ( s , t ) l l - - < ½ ] ] S - ~ l ( s , t ) ] l . The n o r m o f w l ( s , t ) c a n b e changed by the p a r a m e t e r b 1. Under s o m e w h a t m o r e g e n e r a l conditions than those given h e r e we have d e r i v e d in ref. [7] the unitarity bounds (4.1) up to the e n e r g y sk, and 0 < Im fl(S) < 1 f o r all e n e r g i e s . The e n e r g y s k is d e t e r m i n e d by the c u t off •(s) in eq. (2.2) and is taken l a r g e r than 18. F r o m the d e t a i l s of this p r o o f we r e m e m b e r (see ref. [7], sect. 4.2)
]Im/z(s)l
(4.6)
½,
f o r all s, l = 2 , 4 , . . . if B 1 is chosen s m a l l enough (this bound is also i m plicitly contained in the following e s t i m a t e s ) and p e l ( s , t ) >/ 0 ,
(4.7)
f o r all t if 4 ~< s ~< 16. To obtain eq. (4.1), we have now to p r o v e
Im fl(s) >~ 21Re fl(s) l 2,
l=2,4,...,
s > s k.
(4.8)
F r o m eq. (4.7) we know
16 since p(s, t) = pel(t, s) f o r t < 16. The function pel(s, t) depends q u a d r a t i c a l l y on w(s, t) and X(S) t h e r e f o r e we get w i t h s m a l l n o r m s
Im fl(s) >~-~1[ s ( s - 4 ) ] - ½ f 16
Qz (1 + s2~t_4)COl(s,t)dt,
w h e r e we only take into a c c o u n t the d o m i n a n t p a r t COl(S , t) of eq. (4.5). F o r values of s > 18, this g i v e s
580
J. KUPSCH 18
1
Im
fl(s) >i171 [s(s - 4 ) ] - ~ ( s -
16) 1-~z log-2 (as)
f Ql(l + s2~t4)h(t)dt.
(4.9)
16
The r e a l p a r t of eq. (4.4) is divided in 1 17
R11)(=):~=(=-4)I -~ f Ql(l + s2-----~t4)Re D(s, t)d' , 4
and = ~-
~--2-~_ 4) Re
D(s,t)d,,
17
The p r o o f of eq. (4.8) is then reduced to
Imfl(s ) >I 4iR~l'2)(s)l 2
= > s~,
l : 2,4,
Let us f i r s t d i s c u s s R~2)(s). The contribution of 4v(') does not exceed
2[=(=_4)] -½ f~ Q,O+ .-~4)I~(')t d,
_
71"
17
oo
~< const- (AI÷ ( A I + B 1 ) 2 ) s
e-Az f Ql(l+x)x¢-Pdx V
with v = 34/(s - 4 ) , and for the f i r s t d i s p e r s i o n i n t e g r a l in eq. (2.5) the e s t i m a t e (3.4) gives
2I=(=-4)1 -½ f= oz(1 + 7-~-4) 2, Ial(=")l dt 1'7
oo
~< c o n s t .
(BI+(AI+B1)2){s -~
f Ql(l+x)x-~ dx V oo
+ s -~z log -2
(~s)Ql(1
+
34 [ t-l-p dt + s-/~ f Ql(1+ x ) x 1-/2 dx}. -s-~-4) 17
v
The second d i s p e r s i o n i n t e g r a l has the s a m e bound. The function iRl(2)(s)[2 d e c r e a s e s t h e r e f o r e at l e a s t as s 2 ( ~ - P ) if s goes to infinity. Since 0 ~ 2e < p this is dominated by eq. (4.9). The l b e h a v i o u r of the above e x p r e s s i o n s is d e t e r m i n e d by Ql(1 + 34/[s - 4 ] ) [we m a y w r i t e for all the i n t e g r a l s
SCATTERING AMPLITUDES
f
581
Ql(Z)f(z) dz = fQl(Z)Z2z-2f(z)dz ~
l+v
f
z-2f(z)dz,
l+v
b e c a u s e z2 Ql(Z) i s d e c r e a s i n g in z f o r l >/ 2]. H e n c e the / - d e p e n d e n c e of IR~2)(s) [ 2 is e s t i m a t e d by
+s4
and t h e r e i s no p r o b l e m to o b t a i n
I m f l ( s ) >I 4 IRl2)(s)] 2 , for s > sk, l = 2 , 4 , . . . if the values of A1 and B1 are small enough. More difficult arguments are needed for R~l)(s). We know Re D(s,t) ~ c(t-4) ~ s l-/z log-2(~2s), f o r 4 --< t ~< 17 with c < const. (AI+BI+ (AI+B1)2). This implies
IR~I)(s) I < const. (AI + B1+ (A1 + B1)2)2 s -2~ log-4(~2s)
× ]f
17
(t-4)"Ql(l+
_ )dtl 2 2t
4
T h e l a s t i n t e g r a l can be e s t i m a t e d by (see a p p e n d i x B) 17
If
17
(t-4) ~
Q/(I+
s-~4) dt[ 2 ~< c o n s t , l o g s -
4
f
(t-16) ~ O l ( l +
s2--~t4)dt.
16
But this g i v e s the w a n t e d r e s u l t I R~l)(s)] 2 --< ¼Im
fl (s),
f o r s m a l l v a l u e s of A 1 and B 1. T h e p r o o f in a p p e n d i x B s h o w s that we m a y s u b s t i t u t e the l o w e r i n t e g r a tion l i m i t t = 16 by t = 16+ 6 4 ( s - 4 ) -1. T h e r e f o r e the s u p p o r t of w(s, t) m a y be r e s t r i c t e d to t >I 16+ 6 4 ( s - 4 ) -1 if s > s x and s >/ 16+ 6 4 ( t - 4 ) -1 if t > s x and we still obtain eq. (4.1) (for 16 --< s,t ~
582 5. MORE
J. KUPSCH GENERAL
CASES
In this section, we give examples of crossing s y m m e t r i c and unitary amplitudes the double spectral functions of which do not lie in c/8. The r e p resentation (2.1) is still valid because of the strong oscillations of p(s, t), but Ip(s, t)] would require n subtractions. The ansatz for w(s, t) is
co(s,t) = Wl(S , t) + w2(s , t),
(5.1)
with Wl(S , t) as in eq. (4.5) and
w2(s,t) = b 2 0 ( s - 2 0 ) O ( t - 2 0 ) ( s - 2 0 ) n ( t - 2 0 ) n s i n
(~r[(s-20)(t-20)]n+l).
(5.2)
In appendix C, we study
D2(s, t)
1 f°°w2(s',t) -~ ~o s' - s
ds',
and obtain Re D2(s, t) = b 2 O(s - 20) O(t - 20)(s - 20) n (t - 20) n x cos (v[(s- 20)(t-20)] n+l) + g(s, t),
(5.3)
with a differentiable function g(s, t)
Ig(s,t) t --
< c o n s t - I b 2 ] ( l + ] s ] ) - 2 ( l + Itl) - 1 ,
~~g(s, l t)
~
-2.
The contribution of w2(s , t) to eqs. (2.1) and (2.5) may therefore be written in an unsubtracted form, i.e., ¢o2(s' , t ' ) d s ' d t ' F(s,t) =F[Pl,q~ ] + 1 f ~ - s ) - ~ - T ) + crossed terms ~2 where F[Pl, ~0] is the amplitude (2.1) with
Pl (S, t) = pel(s, t) + pel(t, s) + COl(S , t) , and as absorptive part we have
D(s,t) = D[Pl,q~] + D2(s , t) + D2(u ,t). With these definitions, the proof of elastic unitarity goes through without
SCATTERING AMPLITUDES
583
c o n s i d e r a b l e m o d i f i c a t i o n s . [The n o r m of c0 (s, t) h a s now to be s u b s t i t u t e d by ][Wl(S,t)[I + Ib21, o r I b l ] + Ib21. ] In this c a s e it is c o n v e n i e n t to c h o o s e the c u t - o f f e n e r g i e s s x = 18 and s A = 20 s u c h that the o s c i l l a t i n g p a r t of D2(s , t) d o e s not show up in eqs. (2.2)-(2.4). T h e c o n t r i b u t i o n of ¢o2(s, t) to the p a r t i a l w a v e s is given by
f 2)(s) =2
oo
2t f Ql (1 + ~Z-_~_4)(D2(s,t)+ D2(4-s-t,t))dt, 4 l = 0,2,4 .....
T h e s e f u n c t i o n s can be m a j o r i z e d by p a r t i a l w a v e s of the type c o n s i d e r e d in sect. 4. The only t r o u b l e could a r i s e f r o m the i n c r e a s i n g (and o s c i l l a t i n g ) p a r t of D2(s,t) , but the e s t i m a t e oo
i f Ql(1 + ~:-~_4/w2(s,t), 2t
--< Q / \ (1 -
20
+
4O
f
16+ s - I
]w2(s't)]dt
16
--< c o n s t " ] b 2 ] s -1
Ql(1 + s~4)
s h o w s that t h e r e is no difficulty. So we o b t a i n the i n e l a s t i c u n i t a r i t y bounds (4.1) f o r a s m a l l b2. In this e x a m p l e , the o s c i l l a t i o n s a r e so s t r o n g that we do not need s u b t r a c t i o n s f o r ¢o2(s,t) and the c o n t r i b u t i o n to the p a r t i a l w a v e s is m u c h s m a l l e r than a l l o w e d by the p r o o f . This is no l o n g e r the c a s e if we take w 2 is, t) = b 2 (s + t ) l + 6 ( s - 20) n (t - 20) n sin (~[(s - 20) (t - 2 0 ) I n + l ) , (0< 5< 1-~). H e r e we have to u s e the s u b t r a c t e d d i s p e r s i o n i n t e g r a l s of sect. 2. A c a l culation a s b e f o r e p r o v i d e s a u n i t a r y a m p l i t u d e f o r s m a l l b 2.
6. C O N C L U D I N G R E M A R K S We have o b t a i n e d a c l a s s of f u n c t i o n s w h i c h s a t i s f y a M a n d e l s t a m r e p r e s e n t a t i o n with one s u b t r a c t i o n e l a s t i c u n i t a r i t y and the i n e l a s t i c u n i t a r i t y bounds Ira flis) >i Ill (s)]2. T h e double s p e c t r a l f u n c t i o n s p(s, t) a r e p o s i t i v e f o r s --< 17 o r t < 17 and m a y o s c i l l a t e e l s e w h e r e . But this s t r i p w h e r e p(s, t) is p o s i t i v e still p r e v e n t s the a m p l i t u d e to d e v e l o p R e g g e a s y m p t o t i c s . The p r o o f i n c l u d e s p o l y n o m i a l l y i n c r e a s i n g double s p e c t r a l f u n c t i o n s with r a t h e r p a t h o l o g i c a l o s c i l l a t i o n s (their p h a s e v a r i e s with sn). F o r all s o l u t i o n s the total c r o s s s e c t i o n is bounded by c o n s t , s - ~ ( l o g s ) -1 and can be a r r a n g e d to b e h a v e like s - g ( l o g s ) - l . Since we m a y take ~ > 0 a r b i t r a r i l y s m a l l this r e p r e s e n t s one s t e p f u r t h e r to a r e a l i s t i c ~ a m p l i -
584
J. K U P S C H
tude. However, this proof does not allow ~ = 0, and still m o r e w o r k has to be done to obtain a non-vanishing total cross section. I a m indebted to Professor A. Martin for helpful suggestions and a critical reading of the manuscript.
APPENDIX A To derive the estimate (3.4) we consider
Jl(S, t) =
16
~(s', t)ds' (~-s-)~+})
J2(s, t) =
and
S ~(t, s')ds' 16 ( s ' - s ) ( s ' + t ) '
withs.~V(s,t) ~ 93, IIs'~V(s,t) ll --
S . J l ( S , t ) = ( ds' J s' - s
s ' ~ ( s ' , t ) + f d s ' ~V(s',t) s' + t'
s' + t
'
is a function in 930 and the second term belongs to g o which follows from
~ ( s ' t l ) l"
IJl(S,t) --
(A. 1)
t>4,
with C = const. B 1. The integral J2(s, t) is easily estimated:
]J2(s,t) l -< C. s -~ log-2(~2s)t -1-~ log-2(•t), s>4,
(A.2)
t>4.
These results can be extended to all real values of s (see ref. [2]), we have only to replace s by 1+ Is l and to take a somewhat l a r g e r C. Then eq. (3.4) follows. The same technique can be applied to the double integral ( ~h(s', t ' ) d s ' d t ' J3(s't) = J ( s ' - s ) ( t ' - t ) ( s ' + t ' ) '
s. J3(s,t ) = f s ' ~ ( s ' , t ' ) d s ' d t ' J(s'-s)(t'-t)(s'+t')
r~(s',t')ds'dt' +J (t'-t)(s'+t') "
The f i r s t t e r m of the right-hand side is a function in 930, the second t e r m
SCATTERING AMPLITUDES
585
a function in ~r o. Hence we get an e s t i m a t e of the type (A.1). If we add the c r o s s e d t e r m and extend the r e s u l t to all values of s and t the inequality (3.5) follows.
APPENDIX B In this appendix we p r o v e
If
17
(r(s)+l (t_4);
Ol(1 + s "2t~ ) d t [ 2 --< c o n s t ' l o g s f
4
(t_cr(s))~ Ql(1 + - ~2t4 4- ) dt
a(s)
f o r l = 0 , 1 , 2 . . . . a n d s > 4 , with any f u n c t i o n a ( s ) , 16
(t _ 4) ~ v l ( 1
+
2t _4)dt[ 2 ~(s)+l
~< c o n s t , l o g s
f or(s)
since ( t - 4)~ is bounded in 5 < t ~< 17 and As a f i r s t step we d e r i v e
2t
(t-or(s)) ~ Ql(1 + ~-:-~_4)d t ,
(B. 1)
Ql (1 + 2t/[s-417 is d e c r e a s i n g .
[Ql(Z)]2 ~<3z Qo(Z)Ql (2z 2 - 1),
(B.2)
for l=0,1,.., and 1< z < oo. If we apply S c h w a r z ' s inequality to the i n t e g r a l r e p r e s e n t a t i o n oo
Ql(Z) = f
(z + ~
coshO)-l-ldo,
0
we obtain
[Ql(Z)]2 ~
[Ql(Z)] 2 ~
Ql (z) ql+l (z)
Ql-1 (z) Ol(Z)
Qo (z) Q-~
3z,
and hence
Q2l(Z) ~<3z Q2/+l (z).
(B.4)
586
J. KUPSCH
The i n t e g r a l f o r Q21+l(Z) m a y be e s t i m a t e d a s CO
Q2I+I(Z) = f
1
[(z+ (z 2 - 1) i c o s h 0)2] - / - 1 dO
O
CO
=f
[u+ (u 2 - 1)½ c o s h 0 + (z 2 - 1 ) s i n h 2 o]-l-ldo
O CO
< f
1
(u+ (u 2 - 1) i c o s h o)-l-ldo = Ql(U),
O
w h e r e we have i n t r o d u c e d u = 2z 2 - 1. T a k i n g a c c o u n t of eqs. (B.3) and (B.4) this i m p l i e s the r e l a t i o n (B.2). F o r z = 1+ 8 / ( s - 4 ) we have 2z 2 - 1 = 1+ ( 2 / [ s - 4 ] ) ( 1 6 + 6 4 / [ s - 4 ] ) and therefore
Ol(1 + s - ~ - / j
-.< } I o g s .
+
Ql
f o r any a ( s ) w i t h 16 ~
if
5
5 ( t _ 4 ) ~ Ql(1 + 2t
2t
4
4
and eq. (B.5).
APPENDIX C
F o r the function
f(x, y) = O(x) O(y) (xy) n-1 sin [= (xy)~ , n > 1 fixed, the H i l b e r t t r a n s f o r m m a y be defined by
h(x,y) =
lim
1 p ?/(x',y)
r-~ co n
o
-~ Z x
dx'.
(C.1)
T h i s definition g i v e s i m m e d i a t e l y h(x, y) = 0 if y < 0 and the functional r e lation
h(x,y) = h(xy, 1), if y > 0. L e t us c o n s i d e r the function
1i F(z)= lim ~X'*c°
O
~n-1 exp (i~ n) d~. ~-z
(C.2)
SCATTERING AMPLITUDES
587
T h i s i n t e g r a l i s r e l a t e d to h(x,y) by
h(x, 1) = ½Im [F(x+iE) + F(x- iE)].
(C.3)
W i t h i n t h e s e c t o r 0 < a r g ~ --< ~r/(2n) the f u n c t i o n e x p (i~ n) d e c r e a s e s z e r o if I~[ "-* ~o. W e m a y t h e r e f o r e u s e the C a u c h y t h e o r e m f o r
~J
~-z
to
d~
w i t h t h e i n t e g r a t i o n a l o n g 0 --* +oo -~ ~o. e x p [i~/(2n)] --* O. If small, we obtain
]Im z]
is rather
F(Z) = 0(Re z) 0(Im z) 2iz n-1 exp (iz n) + ~ (xv)n-1 exp (-x n) dx X7 Z o -
-ex, 0 F o r r e a l z t h e l a s t i n t e g r a l i s b o u n d e d by c o n s t (1 + ] z ] ) -1 and i t s d e r i v a t i v e d o e s n o t e x c e e d c o n s t (1 + I z ] ) -2. T h e r e l a t i o n s (C.2) and (C.3) y i e l d t h e n the r e s u l t
h(x, y)
=
O(x) O(y)(xy) n-1 c o s [TT(xy)n]
+
g(x, y) ,
with
]g(x,y)] -
1xi)-1(1+]yl) -1 ,
and
ag(x, y)
-< const.¢:+ Ix])-2(:+ [y])-:,
~g(x, y) --< c o n s t - ( 1 +
Ixl)-l(l+ lyl) -2,
w h i c h i s e q u i v a l e n t to eq. (5.3).
RE FERENCES [i] [2] [3] [4]
S. Mandelstam, Phys. Rev. 112 (1958) 1344. D. Atkinson, Nucl. Phys. B7 (1968) 375. D. Atkinson, Nucl. Phys. B8 (1968) 377. D. Atkinson, P r e l i m i n a r y version of l e c t u r e s to be delivered at the 11th Boulder Summer Institute for Theoretical P h y s i c s , University of Bonn p r e p r i n t (1968). [5] C. Goebel, P r o c . 1961 Midwest Conf. on theoretical physics, p. 34-38. [6] A. Martin, Nuovo Cimento 61A (1969) 56. [7] ft. Kupsch, CERN p r e p r i n t TH. 984 (1969), to be published in Nucl. Phys.