AN U P P E R B O U N D TO A S C A T T E R I N G C R O S S S E C T I O N

IN T H E R E G I O N O F T H E F O R W A R D A N D B A C K W A R D

D I F F R A C T I O N P E A K S

G. G. A r u s h a n o v a n d I, I . P i r m a t o v UDC 530.145

In the study of hadron scat ter ing at high energ ies it is convenient to write the scat ter ing amplitude in the Hankel integral representa t ion (the impact p a r a m e t e r representat ion) (cf. , e . g . , [1])

o o

F (t) = i V ~ ~,~(t) (p) ]~, (PI t 11'2) # p , (1 ) b

where the spec t ra l function ~(t){p) can be expressed as a sum of par t ia l waves in the usual angular moment- um representa t ion by the formula

oo

/ = 0

The total e last ic sca t te r ing c ross section in the representa t ion (1) has the fo rm o ~

o~ = 2= [ I ~(t) (P) I : pap. (3)

This resul t is most eas i ly seen by immediate substitution of (2) into (3) then integrat ion with use of the formula

~.? il, J2~ .:~ (;.' :-~,,,-:~ (~) ~-~ d~ = ~,n 2 (2t ~- 1). (4)

Representa t ion (1 ) is especiaUy convenient in the case of diffraction scat ter ing (small momentum transfer) . At maximum momentum t rans fe r (for sca t ter ing angles close to 180 ~ it is convenient to use the angle | = 7r--| which is small, in place of the sca t te r ing angle | and instead of (1) to write

F ( . ) = i I /Y. (~p~") (p) J,., f~ t u" I "'2) :.d:., u' = ~ - - ~,~. ( 1 ' ) b

where

~t = -- 2~: (1 § cos O) + ,,, u~ = (m~ -- :,-~ s -I, (5)

m and/~ are the masse s of the colliding par t ic les . In the above formulas s, t, and u are the usual re la t iv- i s t ica l ly invariant Mandelstam variables. The spec t ra l function ~p(U)(p) is expressed in t e rms of al by means of (2), in which it is n e c e s s a r y only to replace a I by (--1)~a/. The total elast ic c ro s s section can be writ ten in t e rms of ~(U)(p) in a form completely analogous to (3).

Using the Bunyakovsky--Cauchy--Schwartz inequality, we obtain f rom (1), for example

at ~ V E F~ (0) i ~ ) (P) dg (PI t l ~ -') :-dP - - F R (0) t v~'~ (:~) J3 (;.I L I ] z)p~p (6)

(where R and I denote the rea l and imaginary parts) . An ex t remal value of the r ight side of (6) was obtained in [3, 4] for given total e las t ic c ro s s section a e l and forward sca t te r ing amplitude F(0). This value is equal to [4]

o , o 2) ~:2) F~(O) i J ~ ( R t l t l - ) - F J i ( R t t t i ~ -: ~e(J '~(Rl t l t I +J '~ (R~ t l t i 12 ) ) ] , (7)

V. I. Lenin Tashkent State Universi ty. Trans la ted f rom Izvest iya VU Z, Pizika, No. 3, pp. 109-112, March, 1973. Original ar t ic le submitted December 24, 1971.

�9 19 75 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 1001 t. No part o f this publication may be reproduced, stored in a retriepal system, or transmitted, in any form or by any means, electronic, meehanical, photocopyb~g, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15. O&

379

1 ~ ~ " ~ : , , . . . . . . . . . . . .

, \ , I ,u~u li- m,,~C, evlc

0,6 i " o- a, gGevlc

' t , l ' l l l ~ i i ,1

: e ' i ~ �9 6 ' 9 ~

Fig. i

d6 /,o' ~)- ] a e

~-: 9,SGev/c 0.8 '~I , x ~, ~- #,ec,~/~

' ~)~, x 'p O-7,eGe,dc g.4 l ~ - Y.OOev/c

g 4 ~ d 32

Fig. 3

/,0

o,5

~2

3

dO

Z'# l *- :o, sGev:< L i - a , SCevi~

r g,.f Gev/ c -~-P �9 *-/o,8c~+7c

- 'r~_/yOe~] c

5 6 9 X ~

Fig. 2

de l #e

t iI, oo.

.~- r GeV/c �9 _ L . i , . 1 t I , - - ' 1 . . . . !

2 4 , a m ~'~2 Fig. 4

where 5 is the ra t io of the r ea l and imaginary p a r t s of the amplitude at t --- 0, and the p a r a m e t e r s R t and Rlt have the physica l significance of in terac t ion radi i and a re re la ted by the express ion

4F~ (m - ,~; + ,~--] (8)

We shall find a conditional e x t r e m u m of (7) with the r e s t r i c t i on (8). We shall solve the p rob lem by the method of undetermined Lagrange mult ipl iers . A n e c e s s a r y condition for an ex t r emum is

RI21 g' I .Vl (t:~t I t I | 2) ~- ;k ~- J l (RI t } t } li2) Rft, I t } (9)

(where k is the Lagrange mult ipl ier) , i . e . , f r o m (8) we find

Thus,

R t --'= R l t -~ "~el " -~ t~O

the e x t r e ma l value of (7) is equal to

dtd~: t=o (J~ (Rt ] t ] I;2) + j~ (Rt l t I a,'2) ),

(zo)

(11)

where R t is de termined by (10). Fo r a sufficient condition for an ex t remum, we find that the sign of the second der ivat ive must coincide with the signal of the function

- - ( R t I t l I!~)-I j~ (R~I t I ''~) do (Rt I t 11:2), (12)

which is negative for Rtlt l t/2 <- 2.4, which co r responds to smal l momentum t r an s f e r Itl < 0.12 (Gev/c) 2. The re fo re the ex t r emum found is a m a ~ m u m at small momentum t rans fe r s . For sca t te r ing through angles c lose to 180 ~ (small ]ul) we can also obtain a r e su l t analogous to (11) (where the~backward c ros s sect ion appears ins tead of the forward). Exper iment shows, however, that the main contribution to the total e las t ic sca t te r ing c r o s s sect ion ee l comes f rom the in te rva l of smal l angles, and the backward dif ferent ia l c ro s s

380

sect ion da/dulu,= o dec rea se s rapidly with increas ing energy. There fo re the resu l t (11), with Rt de te rmined by (10), turns out to be much too large for sca t te r ing into the backward half Sphere. In the case of s ca t t e r - ing with small momentum t r ans f e r in the exchange channel (small lul), we can write the total c ros s section Oel back into the backward half sphere in (10) ins tead of the to ta le las t i c c ro s s section ~e/back. Infact , this quantity has the fo rm (3), where by ~p(u)ip) we mean that pa r t of the spec t ra l function which dominates the c ros s sect ion for sca t te r ing into the backward half sphere. Thus, instead of (11) we have in this case

a~ ,,'=o" (J~ (R~ I ~zr 11 -,) - j~ (R~, I ~' 1 i~2)) (i3)

with

R~= 4 .~'~ u'=~J (14) :etback da

There are quanti t ies which can be d i rec t ly m easu red by exper iment in the ex t r ema l values (11) and (13): the different ial c ros s sect ions for forward and backward e las t ic sca t te r ing and the total e las t ic c ross sect ions for sca t te r ing into the forward and backward half spheres . There fo re they can be subjected to d i rec t exper imenta l verif icat ion. F igures 1-3 give the exper imenta l data fo r the normal ized different ial c ros s sect ion do/dt [da/dLIt=o as a function of the var iable x~ = R~ltl for pp- and pp - sca t t e r ing (Fig. 1), ~+p and u-p (Fig. 2), and K+p and K-p (Fig. 3) at different energies . Fig. 4 gives the exper imenta l data fo r the different ial c ross section d~/dulda/dulu,= o for e las t ic ~r+p- and v -p - sca t t e r i ng at different energ ies as a

2 = R 2 [ U , ]. The exper imenta l data for fo rward sca t te r ing a re taken f rom [5] for function of the p a r a m e t e r x u the pp case, [6] for pp and K• and [7] for v• and for backward sca t te r ing f ro m [7]. In each figure the theore t ica l curve is the graph of J~(x) + J~(x) as a function of x 2. We see f rom these f igures that in the l imits of exper imenta l e r r o r (which are smal l for fo rward sca t te r ing and large for backward scat ter ing, espec ia l ly for the ~-p case; hence we have also given the horizontal e r r o r ia Fig. 4) the theore t ica l curve l ies dist inct ly above the exper imenta l data; for small Itl and lul the upper l imit obtained is v e r y close to the exper imenta l resu l t s for all p r o c e s s e s and energies .

The authors would like to thank Prof . S. A. Azimov for d i scuss ions of this problem.

1.

2.

3. 4. 5.

6. 7.

LITERATURE CITED

G. G. Arushanov, in: Phys ic s of E l emen ta ry P a r t i c l e s and Cosmic Rays [in Russian], Izd. "Fan , " Taskent , UzSSR (1969). I. S. Gradshtein arid I. M. Ryzhik, Tables of Integrals , Sums, Ser ies , and Different ia ls [in Russian], GIFML, Moscow (1962). G. G. Arushanov, Izv. VUZ SSSR, Fizika, No. 12, 86 (1967). G. G. Arushanov, Dokl. Akad. NaukUzSSR, No. 3, 21 (1969). G. C. Fox and C. Quigg, Compilation of Elas t ic Scat ter ing Data, Univers i ty of California, UCRL- 20001 (1970). IL J. Foley, S. J. Lindenbaum, W. A. Love, e t a l . , Phys. Rev. Le t t . , 1_!, 503 (1963). G. Gaicomelli , P. Pin/ , and S. Stagni, Compilation of Pion--Nucleon Scat ter ing Data, CERN-HERA 69-1 (1969).

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