P H Y S I C A L R E V I E W D V O L U M E 11, N U M B E R 5 1 M A R C H 1 9 7 5

Real parts of forward elastic hadronic amplitudes*

Deepinder P. Sidhut and Uday P. Sukhatmef: Brookhaven National Laboratory, Upton, New York 11973

(Received 27 November 1974)

We compute the real parts of forward elastic hadronic scattering amplitudes using derivative analyticity relations and recent measurements of total cross sections at the Fermi National Accelerator Laboratory. The results on the ratios of real to imaginary parts, p(s), for p p , p p , r ' p , and K = p are compared with the available experimental data. The agreement with the data is reasonably good. The real parts are predicted to change signs at energies s -- 180, 430, 120, 250, < 20, and 145 GeV2 for p p , p p , T - p , r ' p , K p , and K + p , respectively.

The calculation of the r e a l p a r t s of forward e las t i c hadronic amplitudes using forward dis- persion relations is a well-known technique. The method i s based on analyticity and crossing prop- e r t i e s of the scat ter ing amplitude, and the agree- ment of the calculated rea l p a r t s with the experi- mental data h a s d i rec t bearing on the validity of the underlying assumptions. Although this i s t rue i n principle, an accurate evaluation of the real p a r t s through fixed-t dispersion relat ions i s a l - ways very difficult because they a r e nonlocal and require knowledge of the total c r o s s section over the complete energy range, which i s obviously not available. However, i t has recently been shown that a t high energies the real p a r t s a r e giv- e n by quasilocal derivative analyticity relations,',' which greatly simplifies their computation. We presen t h e r e a n evaluation of the real p a r t s of forward elast ic hadronic amplitudes using deriva- tive analyticity relations and recent measurements of total c r o s s sections a t F e r m i National Acceler- a t o r Laboratory ( F e ~ - m i l a b ) . ~ Direct measure- ments of these rea l p a r t s f rom Coulomb interfer- ence measurements should be forthcoming soon.

Our procedure for obtaining real p a r t s is straightforward. We choose the normalization of scat ter ing amplitudes such that the optical theo- r e m h a s the form

Even- and odd-signature amplitudes a r e defined by

where X = lr, K, N, e tc . The sum and difference of par t ic le and antiparticle total c r o s s sect ions a r e defined by o$ -u,(x-p) i u , ( X + p ) . At t = 0 , deriva- tive analyticity relations give the r e a l par t in t e r m s of derivat ives of total c r o s s sect ions with respec t to Ins (Ref. 2):

sY -'lr +- 2 s e c z / z ) L- j?) +. . . 2 d l n s sY-'

It should be emphasized that these relat ions a r e based on solid, model-independent foundations. The only ingredients used in their derivation a r e analyticity and sufficient smoothness of total c r o s s sect ions to allow a Taylor - se r ies expan- s i ~ n . ~ The f r e e p a r a m e t e r s @ and y a r e chosen such that o$ /sa - ' and a;/sY-' have minimal ener - gy dependence. Hence, keeping only the f i r s t de r i - vative in Eqs. (3) and (4) gives a good evaluation of rea l par ts . It i s important to real ize that these equations a r e essentially qrcusilocal, and R e F a t a given energy is mainly controlled by the value of the f i r s t derivative, du,/dlns, a t the same energy. Different assumptions about the asymptotic be- havior of u, do not affect r e a l p a r t s in the F e r m i - l a b energy range. The application of Eqs. (3) and (4) to arzy good parameter izat ion of total c r o s s section data yields reliable resu l t s fo r the r e a l par t s . We chose to parameter ize the high-energy c r o s s section data [s i 15 ( ~ e v / c ) ' ] with functional

1352 R E A L P A R T S O F F O R W A R D E L A S T I C H A D R O N I C . . .

PP PP

-0.2 a AMALDI et al. 0 BARTENEV et al. v BEZNOGIKH et al. 0 FOLEY et al.

BELLETTlNl et al.

Q - 0.1

- 0 .2 FOLEY et al. APOKIN et 01.

K-P CAMPBELL et al.

A ANP COLLABORATION

FIG. 1. Calculation of p (s) = ReF (s, t = O)/ImE (s,t = 0 ) for various reactions using derivative analyticity rela- tions. (a) pp andpp; data points from Ref. 6. (b) 9 p ; data points from Ref. 7. (c) ~ * p ; data points from Ref. 8.

f o r m s partially motivated by Regge theory

and

This parameter izat ion gave excellent f i t s to the c r o s s data. The extrapolated curve for u,(pp) i s in agreement with the CERN ISR mea- s u r e m e n t ~ . ~ The corresponding r e a l p a r t s [using Eqs. (3) and (4)] a r e given by

and

P lo t s of p(X") - R ~ F ( X * ~ ) / I ~ F ( X * P ) a t t = 0 a r e shown in Fig. 1 and available experimental data a r e a l so The agreement with well-mea- sured pp scat ter ing i s very good. Experimental data on rea l p a r t s f o r nip and Kip a r e sparse ; th i s makes it difficult to draw definite conclu- s i o n ~ . ~ Note that the separat ion between the curves fo r p(X p) and p(X-p) is due to the odd- signature contribution ReF- [ s e e Eq. (2)]. This separation d e c r e a s e s a s the energy increases . At very high energies , only the even-signature con- tribution ReF, survives, and a good choice fo r the parameter p in Eq. (3) i s unity, which corresponds to the Pomeranchuk trajectory. This gives

7: R e F -- -

2 d lns '

and i t follows that r is ing c r o s s sections a t high energ ies must necessar i ly be accompanied by positive real par t s . Using the Serpukhov and r e - cent Fermi lab total c r o s s section measurements , we predict that p (s ) will change sign a t s - 180 GeV2 f o r Fp, s - 430 Gev2 f o r pp, s - 120 GeV2 f o r n-p, s - 250 GeV2 f o r nip, s < 20 G ~ V ~ for K-p, and s - 145 G ~ V ' fo r ~ ' p . ' "

The hospitality of the high-energy physics group a t Brookhaven National Laboratory i s gratefully acknowledged.

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

tpermanent Address: Department of Physics, Rutgers, The State University, New Brunswick, N. J. 08903.

$Permanent Address: Department of Physics, Univer- sity of Michigan, Ann Arbor, Mich. 48104.

i ~ . B. Bronzan, in proceedings of the Argonne Sympo- sium on the Pomeron, ANL Report KO. ANL/HEP- 7327, 1973 (unpublished). The leading term was pre- viously derived in the Regge representation by V . N .

Gribov and A. A. Illigdal, Yad. Fiz . 8, 1002 (1968) [Sov. J. Nucl. Phys. 8, 702 (1969)l.

'J. B. Bronzan, G. L. Kane, and U. P. Sukhatme, Phys. Lett. e, 272 (1974).

3 ~ . S. Carroll et a l . (BNL-Fermilab-Rockefeller Colla- boration), paper submitted to the XVII International Conference on High Energy Physics, London, 1974 (un- published).

4 ~ . P. Denisov et a1 ., Nucl. Phys. E, 1 (1973); W. Galbraith et a1 ., Phys. Rev. 138, B913 (1965).

11 - D . P . S I D H U A N D

5 ~ . Amaldi e ta l . , Phys. Lett. e, 231 (1973); S. R. Amendolia et a1 ., ibid. e, 119 (1972).

'u. Amaldi et a1 ., Phys. Lett. G, 231 (1973); V. Bar- tenev e ta l . , Phys. Rev. Lett. 3 l , 1367 (1973); G. Bez- nogikh et a1 ., Phys. Lett. E, 411 (1972); K. J. Foley et u l . , Phys. Rev. Lett. 2, 857 (1967); G. Bellettini et al., Phys. Lett. 14, 164 (1965).

'K. J. Foley e t a l . , Phys. Rev. 181, 1775 (1969); V. D. Apokin et a1 ., paper submitted to Second International Conference on Elementary Particles, Aix-en-Provence, 1973 (unpublished).

'J. R . Campbell e tu l . , Nucl. Phys. E, 1 (1973); Amsterdam-Nijmegen-Paris Collaboration, paper

U. P. S U K H A T M E 1353

presented at the Second International Conference on Elementary Particles, Aix-en-Provence, 1973 (unpub- lished).

or calculation of the real par ts in the Fermilab energy range using conventional dispersion relations, see R . E. Hendrick and B . Lautrup, Phys. Rev. D 11, 529 (1975).

'O~ecent measurements of ReFb-p) at Serpukhov ener- gies strongly suggest a zero in the vicinity of s -120 G ~ v ~ ; see V . D. Apokin et al.. paper submitted to the XVII International Conference on High Energy Physics, London, 1974 (unpublished).