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Unconventional asymptotics
V o l u m e 3 l B . n u m b e r 10
PHYSICS
LETTERS
UNCONVENTIONAL
11 May 1970
ASYMPTOTICS*
V. B A R G E R Deparlmcnl qlPh3'sics, l:nirersily of Wisco~sin, Madiso*~. [Viscot~si~t 53706
and R. J, N, P H I L L I P S Rtdhe~Ttbrd High Entry3' Laboralory. Chillo~z. Didcol. Be;'kshi;'c. En14/a~zd
Received 6 April 1970
We c o n s i d e r w h e t h e r the high e n e r g y rrN qnd KN s c a t t e r i n g a m p l i l u d e s m a y c o n t a i n u n c o n v e n t i o n a l t e r m s l e a d i n g to v i o l a t i o n of the P o m e r a n e h u k t h e o r e m a n d / o r l o g a r i t h m i c a l l y i n c r e a s i n g c r o s s s e c t i o n s . The total c r o s s s e c t i o n and f o r w a r d r e a l p a r t data. with d i s p e r s i o n r e l a t i o n e ( m s t r a i n t s , a r e c o n s i s t e n t with a p p r e c i a b l e t e r n s of t h e s e k i n d s .
It is c o m m o n l y a s s u m e d t h e s e d a y s that h a d r o n i c t o t a l c r o s s s e c t i o n s t e n d to c o n s t a n t s at i n f i n i t e e n e r g y , and a l s o that p a r t i c l e and a r t i p a r t i c l e c r o s s s e c t i o n s on the s a m e t a r g e t t e n d to e q u a l i t y {the P o m e r a n c h u k t h e o r e m ) . T h e usual R e g g e p o l e and eut m o d e l s a l l h a v e t h e s e c o n ventional properties. H o w e v e r , the l a t e s t r e s u l t s f r o m S e r p u k h o v [11 w i t h ~-, K - and ~ b e a m s in the 25-65 G e V / c r a n g e , s u g g e s t to an u n p r e j u d i c e d e y e that e f T ( K - p ) and crT(K+p) m a y not be t e n d i n g to the s a m e l i m i t a f t e r all, and a l l o w one to b e l i e v e that crT0r-p) and CrT(rr+p) m a y not h a v e a c o m m o n l i m i t e i t h e r ° A l t h o u g h t h e s e d a t a can in f a c t be f i t t e d without v i o l a t i n g the P o m e r a n e h u k t h e o r e m , by i n v o k i n g s t r o n g R e g g e c u t s [2], it r e m a i n s i m p o r t a n t to e x p l o r e a l s o s o m e of the u n e o n v e n t i o n a l p o s s i b i l i t i e s s u g g e s t e d by t h e s e r e s u l t s . In the p r e s e n t w o r k , we show that s u b s t a n t i a l v i o l a t i o n s of the P o m e r a n c h u k t h e o r e m 131 and logarithmically increasing total cross sections a r e both c o m p a t i b l e w i t h t o t a l c r o s s s e c t i o n d a t a [1, 4, 5] and with d i s p e r s i o n r e l a t i o n c a l c u l a t i o n s of f o r w a r d r e a l p a r t s ° S p e c i f i c a l l y we c o n s i d e r the p o s s i b i l i t y that the s y m m e t r i z e d 7rp and Kp a m p l i t u d e s A':L(rrp) = ½/A' (rr-p) ± A ' 0 r + p ) . a n d A':i:(Kp) = = ½ [ A ' ( K - p ) + A ' i K ÷ p ) ] contain t e r m s at high e n e r g y a n d ! = 0 o f the f o r m s * W o r k s u p p o r t e d in p a r t by the U n i v e r s i t y of W i s c o n s i n
Research Committee with funds granted by the Wisconsin A l u m i n i R e s e a r c h Foundation. and in part by the U.S. Atomic Energy Commission under contract AT(111)-881. C00-262.
+
A'+ = i ? D : ' [ l n V - b D - ½ i ~ l
(1)
- )s
A'-=~D~[ln~-bD-½ivl
(2)
w h e r e v is the total m e s o n lab e n e r g y , T h e s e r e p r e s e n t the s i m p l e s t v i o l a t i o n s of c o n v e n t i o n a l a s y m p t o t i c s , p r e s e r v i n g the u s u a l a n a l y c i t y and c r o s s i n g p r o p e r t i e s . S i n c e crT = I m A ' ( / = 0 ) / p , w h e r e p is the lab m o m e n t u m , p r e s e n c e of T f) v i o l a t e s the P o m e r a n c h u k t h e o r e m , and the p r e s e n c e of ~1~ i m p l i e s that ~zT i n c r e a s e s l o g a r i t h m i c a l l y . The F r o i s s a r t bound c~T < c(ln t,) 2 is obeyed. In R e g g e l a n g u a g e the l e a d i n g t e r m s in eq, (2) a r e " d i p o l e s " at a = 1, and can be d e r i v e d by t a k i n g the d e r i v a t i v e s d / d R of n o r m a l R e g g e p o l e t e r m s * * . The a d d i t i v e c o n s t a n t Ys' not i n c l u d e d in the dipole, r e p r e s e n t s an a d d i t i o n a l a r b i t r a r i n e s s in A ~+, F o r c o m p a r i s o n , the n o r m a l R e g g e p o l e e x pansions are A,+
~_ = k
e~k 1i 7k v e x p ( - ~ ~'ak)
,
A ' - = ~'-~ig v cvk exp (-½ 7r~ k) k k where
Otk(t) is
the trajectory
and
(3) (4)
7k(/)
is related
**To be precise, eq. (2) comes from the derivative of a R e g g e pole with no s i n g u l a r i t y at ~ 1: it i s thus an e x t i n c t d i p o l e . R e g g e d i p o l e s ha ve p r e v i o u s l y b e e n i n t r o d u c e d , f o r d i f f e r e n t r e a s o n s [61.
643
Volume
3lb.
number
10
PHYSICS
to the r e s i d u e of the t - c h a n n e l R e g g e p o l e k. A '+ a n d A ' - c o n t a i n only e v e n a n d o d d - s i g n a t u r e p o l e s . respectively: W e e s t i m a t e t h e p o s s i b l e s i z e of t h e new t e r m s in e % (1) - (2), by p a r a m e t r i c f i t t i n g of h i g h e n e r g y aT data above 5 GeV/c. We then use dispersion r e l a t i o n s to c h e c k t h a t R e A '+ a r e c o n s i s t e n t with experiment. Although dispersion calculations r e p r o d u c e t h e R e A '+ f r o m e q s . ( 1 ) - ( 4 ) at h i g h energies, deviations may occur at lower energies. U s u a l l y t h e c h o i c e of d i s p e r s i o n r e l a t i o n s f o r A'd: d e p e n d s c r i t i c a l l y on t h e d e g r e e of c o n v e r g e n c e a t infinity° H o w e v e r , s i n c e t h e h i g h e n e r g y i n t e g r a l s w i l l in a n y c a s e b e p e r f o r m e d u s i n g s o m e a n a l y t i c p a r a m e t r i c f o r m , we g e t a s i m p l e r a n d m o r e u n i f i e d t r e a t m e n t by c o m p l e t i n g t h e h i g h e n e r g y i n t e g r a l s on a f i n i t e c o n t o u r r a t h e r t h a t on s e m i c i r c l e s a t infinity~ T h i s a v o i d s a l l c o n v e r g e n c e p r o b l e m s . T h e p h i l o s o p h y is t h e same as for finite energy sum rules*o Our finite contour dispersion relations can then have the " u n s u b t r a c t e d " f o r m b o t h A '+ a n d A ' - , r e g a r d l e s s of a s y m p t o t i c c o n v e r g e n c e * * R e A ' + ( ~ o) = p o l e t e r m s +
LETTERS
i i May 1970
28
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I I
II
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I I I II
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finite-contour
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Fig. I. Fit to~-N a n d K N
total cross seetiondata with
the dipoles plus pole model described in the text. k(a--~+)d °
+
"o f, d'-d_~
c u2-u2o
where the contour c is the upper semicircle from v=N+i~ to u = - N + i ~ , andN> I~io Analytic a l l y - c o n t i n u a b l e p a r a m e t r i c f o r m s U f o r A'd: a r e needed for the semicircle integrals. S i n c e t h e i n p u t to e q s . (5) a n d (6) i s j u s t I m A '+ on t h e r e a l a x i s , R e A ' + ( v ) c a n o n l y b e determined within an additive polynomial, even f o r A '+, odd f o r A ' - . If t h e a m p l i t u d e s a r e b o u n d e d b y ~ln~, t h e r e i s a n a r b i t r a r y c o n s t a n t to b e a d d e d to R e A '+ a n d a n a r b i t r a r y m u l t i p l e of ~ to b e a d d e d to R e A ' - , w h i c h a r e a l r e a d y r e p r e s e n t e d by t h e p a r a m e t e r s 7 s a n d b D in e q s . (1) a n d (2). T h e s e p a r a m e t e r s p l a y t h e s a m e r o l e a s s u b t r a c t i o n c o n s t a n t s do i n o r d i n a r y d i s p e r s i o n relations° * The
I i I I
Plab (6e Vlc)
R e A ' - ( ~ o) = p o l e t e r m s +
+
I
approach
to finite
energy
sum
rules was f i r s t s t r e s s e d by Meshcheriakov et a|. [7]. ** Eqs. (5)-(6) may be derived by considering the integrals of pA"/(P2-~ '2) and A'-/(J2-~ '2) on a finite contour, along the real axis from ~ , N t o l / -N and closed around the upper s e m i c i r c l e . The antis y m m e t r y of A ' - and tJA' in z/ is used. 644
O u r r e s u l t s a r e t h e f o l l o w i n g . W e f i t aT(VN ) d a t a [ 1 , 4 ] b y i n t r o d u c i n g d i p o l e s a s i n e q s . (1) a n d (2) p l u s n o r m a l R e g g e p o l e s P ' a n d p a s i n eqso (3) a n d (4), w i t h (~p, = 0.4 a n d a o = 0.5. p ' ,. The Regge pole is already included in the dipole f o r m u l a t h r o u g h b ; . A t y p i c a l good f i t i s g i v e n by 7~)=2o9, 7p,=103, yD=-0.65,
(7) Tp = 6,
b ; = - 15
o
The p a r a m e t e r s bD and 7's r e m a i n free, not entering a T directly° The r e s u l t s of this fit a r e i l l u s t r a t e d in fig° I. To compare the ~N predictions for ReA '+ with those of our conventional p a r a m e t r i z a t i o n s , we have evaluated the d i s p e r s i o n i n t e g r a l s in eqs. (5) and (6). The p a r a m e t e r s bD and Ts a r e d e t e r mined from a best fit to the ReA d: and forward ~r-p~ir°n m e a s u r e m e n t s [8, 9]:~. We find bD = 5.0 At each rnomentum the~-p--~1~°ndata point with the smallest /-value was used as the forward differential cross section.
Volume 31B. number 10
PHYSICS
IO0
LETTERS
11 May 1970
F o r KN s c a t t e r i n g , t h e d i p o l e q u a n t u m n u m bers need further specification, since A'+(Kp) has contributions from both P-and A2-type exchanges, while A'-(Kp) has both w- and p-type e x c h a n g e s . F o r s i m p l i c i t y , we r e s t r i c t o u r s e l v e s to P - a n d w - t y p e d i p o l e s (with c~ = 1), p l u s n o r m a l P ' . w. p a n d A 2 R e g g e p o l e s w i t h i n t e r c e p t s c~p, = c~w = 0°4 a n d c ~ p = C~A,2 = 0~5: A t y p i c a l
> L
Plob(GeWc)
good fit to cTT d a t a [1, 4] a b o v e 5 G e V / c f o r K+N i s g i v e n by T
4-
-I00
+
?D = 2.9,
b D = -12,
Tw = 22,
7 p , = 6,
? p , = 53, 7A 2
7D = - 0 " 8 '
5 .
÷
-200
5
I0
15
20
25
Plab (GeWc)
u
I.
t cca. *l E
.~
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i|111
5
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Pla b(GeV/c ) Fig. 2. a) Dispersion relation r e s u l t s for the real p a r t s of the /rN forward amplitudes. The solid curves a r e based on the double-dipoles plus pole model d e s c r i b e d in the text. The dashed c u r v e s illustrate r e s u l t s from a conventional power law p a r a m e t e r i z a t i o n for total c r o s s section from ref, ISl. b) P r o j e c t i o n s of forward d(J/dt (Tr-p .#°n) from the p dipole plus pole model. Data from ref. [9]. a n d 7 s = - 4 6 . Fig. 2 s h o w s R e A '+ f r o m t h i s model, compared with results from a convent i o n a l p o w e r law p a r a m e t e r i z a t i o n of t o t a l c r o s s s e c t i o n s by t h e L i n d e n b a u m g r o u p [8]. T h e s e c o m p a r i s o n s show t h a t o u r d i p o l e p r e d i c t i o n s a r e n o t d i s t i n g u i s h a b l e f r o m p r e v i o u s s e c c e s s f u l f i t s to R e A '+ b e l o w 20 G e V ~ . ~/ln fig, 1. the difference between the dipole and Lindenbaum curves above 20 G e V / c reflects in part the failure of the Lindenbaum fft parameteriz'~tion to des c r i b e the Serpukhov data. Horn [1()1 has independently checked the consistency of ReA'-(~'N) d i s p e r s i o n relations, making the s t r o n g e r assumption that fft(~'-p)-fft(Tr 'p) is a constant from 30 GeV to infinity.
A s b e f o r e ~ bD a n d T s r e m a i n f r e e . T h i s p a r a m e t e r i z a t i o n of t h e K:ep d a t a i s s h o w n in fig. 1~ I n t e r e s t i n g l y the d i p o l e m o d e l a l s o a c c o u n t s f o r t h e u p w a r d c u r v a t u r e o b s e r v e d r e c e n t l y i n c~t(K+p) d a t a b e l o w 3.3 G e V / c [11]~ D i s p e r s i o n r e l a t i o n r e s u l t s f o r R e A '+ f r o m t h i s d i p o l e m o d e l a r e s i m i l a r to t h o s e f o r c o n v e n t i o n a l p o l e m o d e l s b e l o w 20 G e V I12, 13]~ Our remarks and conclusions may be summarized as follows: (i) O u r m o t i v a t i o n f o r c o n s i d e r i n g R e g g e dipoles at all is that they provide the simplest parametric forms violating conventional asympt o t i c s , T h e n a t u r e of t h e S e r p u k h o v d a t a e n c o u r a g e s u s to c o n s i d e r u n c c n v e n t i o n a l f o r m s . W e do n o t p r o p o s e to a s s o c i a t e t h e s e R e g g e dipoles with known physical particles° (ii) T h e A ' + d i p o l e e x p l a i n s t h e l e v e l i n g out of CrT(+)"" t h r o u g h a g r o w i n g In v c o n t r i b u t i o n t h a t eventually overcomes the decreasing Regge pole t e r m s ° In c o n t r a s t , a R e g g e - c u t e x p l a n a t i o n 12] g i v e s cr ~ a + b / ( l n v ) k r i s e to a f i n i t e a s y m p t o t i c total cross section° (iii) W i t h a A ' - d i p o l e , t h e d i f f e r e n c e of p a r ticle and antiparticle total cross sections tends to a f i n i t e n o n - z e r o a s y m p t o t i c limit~ (iv) D i s p e r s i o n r e l a t i o n c a c u l a t i o n s of R e A '+ w i t h o u r d i p o l e a s y m p t o t i c s a r e not d i s t i n g u i s h a b l e f r o m r e s u l t s w i t h c o n v e n t i o n a l p o w e r law a s y m p t o t i c s (and h e n c e f r o m e x p e r i m e n t ) b e l o w 20 GeV, (v) O u r f i n i t e c o n t o u r d i s p e r s i o n r e l a t i o n s r e s u i t s show t h a t d i s p e r s i o n r e l a t i o n s c h e c k a n a l y t ±city b u t s a y l i t t l e a b o u t a s y m p t o t i c b e h a v i o r , In o r d e r to f i n d out w h a t h a p p e n s a t h i g h e n e r g i e s , w e h a v e to go t h e r e , (vi) R e g g e d i p o l e s c a n r e a d i l y b e i n c l u d e d i n finite energy sum rule analyses, as can cuts, a l b e i t t h e i r e f f e c t s c a n b e f a k e d by p o l e s in finite regions, (vii) If d i p o l e s a r e p r e s e n t in b o t h A '+ t h e n 645
V o l u m e 31B. n u m h e r 10
PHYSICS
t h e r a t i o s c~ = R e A ' / l m A ' f o r 7r+p a n d 7r-p t e n d to f i n i t e c o n s t a n t s of o p p o s i t e s i g n ° S i m i l a r l y f o r K+p and K-p. In our examples these asymptotic values are a ( u - p ) : -(~(Tr+p) : - 0 , 2 ,
a ( K - p ) = -c~ (K+p) = - 0 . 3
( v i i i ) T h e d i p o l e c o n t r i b u t i o n s to d ( r / d ! e l a s t i c and charge exchange must shrimk at least logarithmically with increasing energy for the int e g r a t e d c r o s s s e c t i o n s to b e c o m p a t i b l e w i t h ~rT ~ in v. (ix) T h e e x i s t e n c e of u n c o n v e n t i o n a l a s y m p t o t i c t e r m s i n A '+ a n d A ' - c a n b e s e p a r a t e l y examined~ We have in fact made fits with vlnv t e r m s p r e s e n t in o n l y o n e o r t h e o t h e r * (x) A d i r e c t m e a s u r e of A ' ' ( r r p ) i s g i v e n b y dry/d! (Tr-p- ~Tr°n). W e h a v e c h e c k e d t h a t o u r d i p o l e model agrees with existing forward data on this reaction° Projections of d~/d/(n'-p- ~ 7;°n) r e a c h a minimum value at 300 GeV/c, followed by a (lnv) 2 rise at higher momenta, as illustrated i n f i g . 2~ T h e r e g e n e r a t i o n cross section K2P-~KlP provides a similar measure of A ' - ( K ° p ) t2,151o (xi) D i p o l e s c o u l d a p p e a r i n o t h e r p r o c e s s e s beside elastic scattering (e.g. P-type dipole in diffraction processes; c o - t y p e d i p o l e in ) p - ,~Op)o
the total c r o s s s e c t i o n i s bounded by a c o n s t a n t , a A ' - ,It /=0 r('(lilil'es a n"i()F(~ (.:onlplie'tted J-plane s t r u c t u r e for l ¢0 than that of a llcggc dipole [141 . I1
vIl]vdcl)cndcllcC f o r
646
LETTERS
Ii May 1970
O n e of u s (R. P . ) i s g r a t e f u l to B. R. M a r t i n a n d R. G. M o o r h o u s e f o r h e l p f u l d i s c u s s i o n s ,
l, J . V . A i l a b 5 et ul.. P h y s . L e t t e r s 30B (1969) 49~. 2. V. B a r g e r and R.J.N,Phillips. Regge c u t s P o m e r a n chuk t h e o r e m , and the S e r p u k h o v total c r o s s s e c t i o n data. U n i v e r s i t y of W i s c o n s i n r e p o r t ~ C00-260. O c t o b e r . 1969. 3. 1. Ya. P o m e r a n c h u k . Soviel P h y s i c s J E T P 7 (1958) 499: A . M a r t i n , Nuovo C i m e n t o 39 (1965) 704. Some r e c e n t g e n e r a l rcmt~rks on p o s s i b l e violation o[" the P o m e r a n c h u k t h e n r e m a r c given by R . J . Eden. P h y s . Rex'. dn p r e s s ) : A. M a r t i n . CERN r e p o r t Th. 1075. talk g i v e n "it Stony B r o o k Conl'ercncc (1969). 4. K . J . F o l e y et ul.. P h y s . Rev. L e t t e r s 19 (1967) 330. 5. W. G a l b r a i t h c,t ill.. P h y s . Rev. 138 (1965) B913. 6. G. F r y e . P h y s . Rex'. 129 (1963) 1453: R. K r e p s and J. Moffat, P h y s . Re','. 175 (1958) 1942: Y. T o m o z a w a , U n i v e r s i t y of M i c h i g a n p r e p r i n t (19691. 7. V . A . M e s h e h e r i a k o v et al.. P h y s . L e t t e r s 25B (1967) 3tl. K . J , F o l e y et al.. P h y s . Rc~v. 181 0969) 1775. 9. O . G u i s a n . p r i v a t e c o m m u n i a t i o n . P. S o n d e r e g g e r el :it.. P h y s . L e t t e r s 20 (1966) 75. 10. D. Horn. Cal. T e c h n . r e p o r i C A L T - 6 8 - 2 2 8 . 11. R . J . A b r a m s . R . L . Cool, G, G i a e o m e l l i , T , F . K y c i t t . B . A , L e o n t i e . K . K . Li and D.N. M i c h a e l . B r o o k h a v e n National L a b o r a t o r y r e p o r t 11046 (1969), 12. V. B a r g e r , M . O l s s o n und D. R c e d e r . N u c l e a r P h y s . B5 (]96S~ . i l l . 13. G.V. D a s s . C. Michael and R . J , N . P h i l l i p s . N u c l e u r Phj, s, B9 (1969) 549. 14. J. F i n k e l s t c i n . Stanford U n i v e r s i t y R e p o r t I T P - 3 5 2 (1969). 15. K, K l e i n k n e c h t . P h y s . L e t t e r s 30B 0969) 514.
PHYSICS
LETTERS
UNCONVENTIONAL
11 May 1970
ASYMPTOTICS*
V. B A R G E R Deparlmcnl qlPh3'sics, l:nirersily of Wisco~sin, Madiso*~. [Viscot~si~t 53706
and R. J, N, P H I L L I P S Rtdhe~Ttbrd High Entry3' Laboralory. Chillo~z. Didcol. Be;'kshi;'c. En14/a~zd
Received 6 April 1970
We c o n s i d e r w h e t h e r the high e n e r g y rrN qnd KN s c a t t e r i n g a m p l i l u d e s m a y c o n t a i n u n c o n v e n t i o n a l t e r m s l e a d i n g to v i o l a t i o n of the P o m e r a n e h u k t h e o r e m a n d / o r l o g a r i t h m i c a l l y i n c r e a s i n g c r o s s s e c t i o n s . The total c r o s s s e c t i o n and f o r w a r d r e a l p a r t data. with d i s p e r s i o n r e l a t i o n e ( m s t r a i n t s , a r e c o n s i s t e n t with a p p r e c i a b l e t e r n s of t h e s e k i n d s .
It is c o m m o n l y a s s u m e d t h e s e d a y s that h a d r o n i c t o t a l c r o s s s e c t i o n s t e n d to c o n s t a n t s at i n f i n i t e e n e r g y , and a l s o that p a r t i c l e and a r t i p a r t i c l e c r o s s s e c t i o n s on the s a m e t a r g e t t e n d to e q u a l i t y {the P o m e r a n c h u k t h e o r e m ) . T h e usual R e g g e p o l e and eut m o d e l s a l l h a v e t h e s e c o n ventional properties. H o w e v e r , the l a t e s t r e s u l t s f r o m S e r p u k h o v [11 w i t h ~-, K - and ~ b e a m s in the 25-65 G e V / c r a n g e , s u g g e s t to an u n p r e j u d i c e d e y e that e f T ( K - p ) and crT(K+p) m a y not be t e n d i n g to the s a m e l i m i t a f t e r all, and a l l o w one to b e l i e v e that crT0r-p) and CrT(rr+p) m a y not h a v e a c o m m o n l i m i t e i t h e r ° A l t h o u g h t h e s e d a t a can in f a c t be f i t t e d without v i o l a t i n g the P o m e r a n e h u k t h e o r e m , by i n v o k i n g s t r o n g R e g g e c u t s [2], it r e m a i n s i m p o r t a n t to e x p l o r e a l s o s o m e of the u n e o n v e n t i o n a l p o s s i b i l i t i e s s u g g e s t e d by t h e s e r e s u l t s . In the p r e s e n t w o r k , we show that s u b s t a n t i a l v i o l a t i o n s of the P o m e r a n c h u k t h e o r e m 131 and logarithmically increasing total cross sections a r e both c o m p a t i b l e w i t h t o t a l c r o s s s e c t i o n d a t a [1, 4, 5] and with d i s p e r s i o n r e l a t i o n c a l c u l a t i o n s of f o r w a r d r e a l p a r t s ° S p e c i f i c a l l y we c o n s i d e r the p o s s i b i l i t y that the s y m m e t r i z e d 7rp and Kp a m p l i t u d e s A':L(rrp) = ½/A' (rr-p) ± A ' 0 r + p ) . a n d A':i:(Kp) = = ½ [ A ' ( K - p ) + A ' i K ÷ p ) ] contain t e r m s at high e n e r g y a n d ! = 0 o f the f o r m s * W o r k s u p p o r t e d in p a r t by the U n i v e r s i t y of W i s c o n s i n
Research Committee with funds granted by the Wisconsin A l u m i n i R e s e a r c h Foundation. and in part by the U.S. Atomic Energy Commission under contract AT(111)-881. C00-262.
+
A'+ = i ? D : ' [ l n V - b D - ½ i ~ l
(1)
- )s
A'-=~D~[ln~-bD-½ivl
(2)
w h e r e v is the total m e s o n lab e n e r g y , T h e s e r e p r e s e n t the s i m p l e s t v i o l a t i o n s of c o n v e n t i o n a l a s y m p t o t i c s , p r e s e r v i n g the u s u a l a n a l y c i t y and c r o s s i n g p r o p e r t i e s . S i n c e crT = I m A ' ( / = 0 ) / p , w h e r e p is the lab m o m e n t u m , p r e s e n c e of T f) v i o l a t e s the P o m e r a n c h u k t h e o r e m , and the p r e s e n c e of ~1~ i m p l i e s that ~zT i n c r e a s e s l o g a r i t h m i c a l l y . The F r o i s s a r t bound c~T < c(ln t,) 2 is obeyed. In R e g g e l a n g u a g e the l e a d i n g t e r m s in eq, (2) a r e " d i p o l e s " at a = 1, and can be d e r i v e d by t a k i n g the d e r i v a t i v e s d / d R of n o r m a l R e g g e p o l e t e r m s * * . The a d d i t i v e c o n s t a n t Ys' not i n c l u d e d in the dipole, r e p r e s e n t s an a d d i t i o n a l a r b i t r a r i n e s s in A ~+, F o r c o m p a r i s o n , the n o r m a l R e g g e p o l e e x pansions are A,+
~_ = k
e~k 1i 7k v e x p ( - ~ ~'ak)
,
A ' - = ~'-~ig v cvk exp (-½ 7r~ k) k k where
Otk(t) is
the trajectory
and
(3) (4)
7k(/)
is related
**To be precise, eq. (2) comes from the derivative of a R e g g e pole with no s i n g u l a r i t y at ~ 1: it i s thus an e x t i n c t d i p o l e . R e g g e d i p o l e s ha ve p r e v i o u s l y b e e n i n t r o d u c e d , f o r d i f f e r e n t r e a s o n s [61.
643
Volume
3lb.
number
10
PHYSICS
to the r e s i d u e of the t - c h a n n e l R e g g e p o l e k. A '+ a n d A ' - c o n t a i n only e v e n a n d o d d - s i g n a t u r e p o l e s . respectively: W e e s t i m a t e t h e p o s s i b l e s i z e of t h e new t e r m s in e % (1) - (2), by p a r a m e t r i c f i t t i n g of h i g h e n e r g y aT data above 5 GeV/c. We then use dispersion r e l a t i o n s to c h e c k t h a t R e A '+ a r e c o n s i s t e n t with experiment. Although dispersion calculations r e p r o d u c e t h e R e A '+ f r o m e q s . ( 1 ) - ( 4 ) at h i g h energies, deviations may occur at lower energies. U s u a l l y t h e c h o i c e of d i s p e r s i o n r e l a t i o n s f o r A'd: d e p e n d s c r i t i c a l l y on t h e d e g r e e of c o n v e r g e n c e a t infinity° H o w e v e r , s i n c e t h e h i g h e n e r g y i n t e g r a l s w i l l in a n y c a s e b e p e r f o r m e d u s i n g s o m e a n a l y t i c p a r a m e t r i c f o r m , we g e t a s i m p l e r a n d m o r e u n i f i e d t r e a t m e n t by c o m p l e t i n g t h e h i g h e n e r g y i n t e g r a l s on a f i n i t e c o n t o u r r a t h e r t h a t on s e m i c i r c l e s a t infinity~ T h i s a v o i d s a l l c o n v e r g e n c e p r o b l e m s . T h e p h i l o s o p h y is t h e same as for finite energy sum rules*o Our finite contour dispersion relations can then have the " u n s u b t r a c t e d " f o r m b o t h A '+ a n d A ' - , r e g a r d l e s s of a s y m p t o t i c c o n v e r g e n c e * * R e A ' + ( ~ o) = p o l e t e r m s +
LETTERS
i i May 1970
28
i
I I
II
I
I
I
I I I Ill
I
I
I
I I I II
26
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24
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+
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(6)
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finite-contour
I I I]
I0
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I00
I000
Fig. I. Fit to~-N a n d K N
total cross seetiondata with
the dipoles plus pole model described in the text. k(a--~+)d °
+
"o f, d'-d_~
c u2-u2o
where the contour c is the upper semicircle from v=N+i~ to u = - N + i ~ , andN> I~io Analytic a l l y - c o n t i n u a b l e p a r a m e t r i c f o r m s U f o r A'd: a r e needed for the semicircle integrals. S i n c e t h e i n p u t to e q s . (5) a n d (6) i s j u s t I m A '+ on t h e r e a l a x i s , R e A ' + ( v ) c a n o n l y b e determined within an additive polynomial, even f o r A '+, odd f o r A ' - . If t h e a m p l i t u d e s a r e b o u n d e d b y ~ln~, t h e r e i s a n a r b i t r a r y c o n s t a n t to b e a d d e d to R e A '+ a n d a n a r b i t r a r y m u l t i p l e of ~ to b e a d d e d to R e A ' - , w h i c h a r e a l r e a d y r e p r e s e n t e d by t h e p a r a m e t e r s 7 s a n d b D in e q s . (1) a n d (2). T h e s e p a r a m e t e r s p l a y t h e s a m e r o l e a s s u b t r a c t i o n c o n s t a n t s do i n o r d i n a r y d i s p e r s i o n relations° * The
I i I I
Plab (6e Vlc)
R e A ' - ( ~ o) = p o l e t e r m s +
+
I
approach
to finite
energy
sum
rules was f i r s t s t r e s s e d by Meshcheriakov et a|. [7]. ** Eqs. (5)-(6) may be derived by considering the integrals of pA"/(P2-~ '2) and A'-/(J2-~ '2) on a finite contour, along the real axis from ~ , N t o l / -N and closed around the upper s e m i c i r c l e . The antis y m m e t r y of A ' - and tJA' in z/ is used. 644
O u r r e s u l t s a r e t h e f o l l o w i n g . W e f i t aT(VN ) d a t a [ 1 , 4 ] b y i n t r o d u c i n g d i p o l e s a s i n e q s . (1) a n d (2) p l u s n o r m a l R e g g e p o l e s P ' a n d p a s i n eqso (3) a n d (4), w i t h (~p, = 0.4 a n d a o = 0.5. p ' ,. The Regge pole is already included in the dipole f o r m u l a t h r o u g h b ; . A t y p i c a l good f i t i s g i v e n by 7~)=2o9, 7p,=103, yD=-0.65,
(7) Tp = 6,
b ; = - 15
o
The p a r a m e t e r s bD and 7's r e m a i n free, not entering a T directly° The r e s u l t s of this fit a r e i l l u s t r a t e d in fig° I. To compare the ~N predictions for ReA '+ with those of our conventional p a r a m e t r i z a t i o n s , we have evaluated the d i s p e r s i o n i n t e g r a l s in eqs. (5) and (6). The p a r a m e t e r s bD and Ts a r e d e t e r mined from a best fit to the ReA d: and forward ~r-p~ir°n m e a s u r e m e n t s [8, 9]:~. We find bD = 5.0 At each rnomentum the~-p--~1~°ndata point with the smallest /-value was used as the forward differential cross section.
Volume 31B. number 10
PHYSICS
IO0
LETTERS
11 May 1970
F o r KN s c a t t e r i n g , t h e d i p o l e q u a n t u m n u m bers need further specification, since A'+(Kp) has contributions from both P-and A2-type exchanges, while A'-(Kp) has both w- and p-type e x c h a n g e s . F o r s i m p l i c i t y , we r e s t r i c t o u r s e l v e s to P - a n d w - t y p e d i p o l e s (with c~ = 1), p l u s n o r m a l P ' . w. p a n d A 2 R e g g e p o l e s w i t h i n t e r c e p t s c~p, = c~w = 0°4 a n d c ~ p = C~A,2 = 0~5: A t y p i c a l
> L
Plob(GeWc)
good fit to cTT d a t a [1, 4] a b o v e 5 G e V / c f o r K+N i s g i v e n by T
4-
-I00
+
?D = 2.9,
b D = -12,
Tw = 22,
7 p , = 6,
? p , = 53, 7A 2
7D = - 0 " 8 '
5 .
÷
-200
5
I0
15
20
25
Plab (GeWc)
u
I.
t cca. *l E
.~
,0~
i|111
5
IO
i
1
i
I |11111
I00
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| |Jill
I000
Pla b(GeV/c ) Fig. 2. a) Dispersion relation r e s u l t s for the real p a r t s of the /rN forward amplitudes. The solid curves a r e based on the double-dipoles plus pole model d e s c r i b e d in the text. The dashed c u r v e s illustrate r e s u l t s from a conventional power law p a r a m e t e r i z a t i o n for total c r o s s section from ref, ISl. b) P r o j e c t i o n s of forward d(J/dt (Tr-p .#°n) from the p dipole plus pole model. Data from ref. [9]. a n d 7 s = - 4 6 . Fig. 2 s h o w s R e A '+ f r o m t h i s model, compared with results from a convent i o n a l p o w e r law p a r a m e t e r i z a t i o n of t o t a l c r o s s s e c t i o n s by t h e L i n d e n b a u m g r o u p [8]. T h e s e c o m p a r i s o n s show t h a t o u r d i p o l e p r e d i c t i o n s a r e n o t d i s t i n g u i s h a b l e f r o m p r e v i o u s s e c c e s s f u l f i t s to R e A '+ b e l o w 20 G e V ~ . ~/ln fig, 1. the difference between the dipole and Lindenbaum curves above 20 G e V / c reflects in part the failure of the Lindenbaum fft parameteriz'~tion to des c r i b e the Serpukhov data. Horn [1()1 has independently checked the consistency of ReA'-(~'N) d i s p e r s i o n relations, making the s t r o n g e r assumption that fft(~'-p)-fft(Tr 'p) is a constant from 30 GeV to infinity.
A s b e f o r e ~ bD a n d T s r e m a i n f r e e . T h i s p a r a m e t e r i z a t i o n of t h e K:ep d a t a i s s h o w n in fig. 1~ I n t e r e s t i n g l y the d i p o l e m o d e l a l s o a c c o u n t s f o r t h e u p w a r d c u r v a t u r e o b s e r v e d r e c e n t l y i n c~t(K+p) d a t a b e l o w 3.3 G e V / c [11]~ D i s p e r s i o n r e l a t i o n r e s u l t s f o r R e A '+ f r o m t h i s d i p o l e m o d e l a r e s i m i l a r to t h o s e f o r c o n v e n t i o n a l p o l e m o d e l s b e l o w 20 G e V I12, 13]~ Our remarks and conclusions may be summarized as follows: (i) O u r m o t i v a t i o n f o r c o n s i d e r i n g R e g g e dipoles at all is that they provide the simplest parametric forms violating conventional asympt o t i c s , T h e n a t u r e of t h e S e r p u k h o v d a t a e n c o u r a g e s u s to c o n s i d e r u n c c n v e n t i o n a l f o r m s . W e do n o t p r o p o s e to a s s o c i a t e t h e s e R e g g e dipoles with known physical particles° (ii) T h e A ' + d i p o l e e x p l a i n s t h e l e v e l i n g out of CrT(+)"" t h r o u g h a g r o w i n g In v c o n t r i b u t i o n t h a t eventually overcomes the decreasing Regge pole t e r m s ° In c o n t r a s t , a R e g g e - c u t e x p l a n a t i o n 12] g i v e s cr ~ a + b / ( l n v ) k r i s e to a f i n i t e a s y m p t o t i c total cross section° (iii) W i t h a A ' - d i p o l e , t h e d i f f e r e n c e of p a r ticle and antiparticle total cross sections tends to a f i n i t e n o n - z e r o a s y m p t o t i c limit~ (iv) D i s p e r s i o n r e l a t i o n c a c u l a t i o n s of R e A '+ w i t h o u r d i p o l e a s y m p t o t i c s a r e not d i s t i n g u i s h a b l e f r o m r e s u l t s w i t h c o n v e n t i o n a l p o w e r law a s y m p t o t i c s (and h e n c e f r o m e x p e r i m e n t ) b e l o w 20 GeV, (v) O u r f i n i t e c o n t o u r d i s p e r s i o n r e l a t i o n s r e s u i t s show t h a t d i s p e r s i o n r e l a t i o n s c h e c k a n a l y t ±city b u t s a y l i t t l e a b o u t a s y m p t o t i c b e h a v i o r , In o r d e r to f i n d out w h a t h a p p e n s a t h i g h e n e r g i e s , w e h a v e to go t h e r e , (vi) R e g g e d i p o l e s c a n r e a d i l y b e i n c l u d e d i n finite energy sum rule analyses, as can cuts, a l b e i t t h e i r e f f e c t s c a n b e f a k e d by p o l e s in finite regions, (vii) If d i p o l e s a r e p r e s e n t in b o t h A '+ t h e n 645
V o l u m e 31B. n u m h e r 10
PHYSICS
t h e r a t i o s c~ = R e A ' / l m A ' f o r 7r+p a n d 7r-p t e n d to f i n i t e c o n s t a n t s of o p p o s i t e s i g n ° S i m i l a r l y f o r K+p and K-p. In our examples these asymptotic values are a ( u - p ) : -(~(Tr+p) : - 0 , 2 ,
a ( K - p ) = -c~ (K+p) = - 0 . 3
( v i i i ) T h e d i p o l e c o n t r i b u t i o n s to d ( r / d ! e l a s t i c and charge exchange must shrimk at least logarithmically with increasing energy for the int e g r a t e d c r o s s s e c t i o n s to b e c o m p a t i b l e w i t h ~rT ~ in v. (ix) T h e e x i s t e n c e of u n c o n v e n t i o n a l a s y m p t o t i c t e r m s i n A '+ a n d A ' - c a n b e s e p a r a t e l y examined~ We have in fact made fits with vlnv t e r m s p r e s e n t in o n l y o n e o r t h e o t h e r * (x) A d i r e c t m e a s u r e of A ' ' ( r r p ) i s g i v e n b y dry/d! (Tr-p- ~Tr°n). W e h a v e c h e c k e d t h a t o u r d i p o l e model agrees with existing forward data on this reaction° Projections of d~/d/(n'-p- ~ 7;°n) r e a c h a minimum value at 300 GeV/c, followed by a (lnv) 2 rise at higher momenta, as illustrated i n f i g . 2~ T h e r e g e n e r a t i o n cross section K2P-~KlP provides a similar measure of A ' - ( K ° p ) t2,151o (xi) D i p o l e s c o u l d a p p e a r i n o t h e r p r o c e s s e s beside elastic scattering (e.g. P-type dipole in diffraction processes; c o - t y p e d i p o l e in ) p - ,~Op)o
the total c r o s s s e c t i o n i s bounded by a c o n s t a n t , a A ' - ,It /=0 r('(lilil'es a n"i()F(~ (.:onlplie'tted J-plane s t r u c t u r e for l ¢0 than that of a llcggc dipole [141 . I1
vIl]vdcl)cndcllcC f o r
646
LETTERS
Ii May 1970
O n e of u s (R. P . ) i s g r a t e f u l to B. R. M a r t i n a n d R. G. M o o r h o u s e f o r h e l p f u l d i s c u s s i o n s ,
l, J . V . A i l a b 5 et ul.. P h y s . L e t t e r s 30B (1969) 49~. 2. V. B a r g e r and R.J.N,Phillips. Regge c u t s P o m e r a n chuk t h e o r e m , and the S e r p u k h o v total c r o s s s e c t i o n data. U n i v e r s i t y of W i s c o n s i n r e p o r t ~ C00-260. O c t o b e r . 1969. 3. 1. Ya. P o m e r a n c h u k . Soviel P h y s i c s J E T P 7 (1958) 499: A . M a r t i n , Nuovo C i m e n t o 39 (1965) 704. Some r e c e n t g e n e r a l rcmt~rks on p o s s i b l e violation o[" the P o m e r a n c h u k t h e n r e m a r c given by R . J . Eden. P h y s . Rex'. dn p r e s s ) : A. M a r t i n . CERN r e p o r t Th. 1075. talk g i v e n "it Stony B r o o k Conl'ercncc (1969). 4. K . J . F o l e y et ul.. P h y s . Rev. L e t t e r s 19 (1967) 330. 5. W. G a l b r a i t h c,t ill.. P h y s . Rev. 138 (1965) B913. 6. G. F r y e . P h y s . Rex'. 129 (1963) 1453: R. K r e p s and J. Moffat, P h y s . Re','. 175 (1958) 1942: Y. T o m o z a w a , U n i v e r s i t y of M i c h i g a n p r e p r i n t (19691. 7. V . A . M e s h e h e r i a k o v et al.. P h y s . L e t t e r s 25B (1967) 3tl. K . J , F o l e y et al.. P h y s . Rc~v. 181 0969) 1775. 9. O . G u i s a n . p r i v a t e c o m m u n i a t i o n . P. S o n d e r e g g e r el :it.. P h y s . L e t t e r s 20 (1966) 75. 10. D. Horn. Cal. T e c h n . r e p o r i C A L T - 6 8 - 2 2 8 . 11. R . J . A b r a m s . R . L . Cool, G, G i a e o m e l l i , T , F . K y c i t t . B . A , L e o n t i e . K . K . Li and D.N. M i c h a e l . B r o o k h a v e n National L a b o r a t o r y r e p o r t 11046 (1969), 12. V. B a r g e r , M . O l s s o n und D. R c e d e r . N u c l e a r P h y s . B5 (]96S~ . i l l . 13. G.V. D a s s . C. Michael and R . J , N . P h i l l i p s . N u c l e u r Phj, s, B9 (1969) 549. 14. J. F i n k e l s t c i n . Stanford U n i v e r s i t y R e p o r t I T P - 3 5 2 (1969). 15. K, K l e i n k n e c h t . P h y s . L e t t e r s 30B 0969) 514.