IL NUOVO CIMENTO VOL. 11 A, N. 4 21 Ottobre 1972

A New Method for the Determination of 7~-S Scattering

Amplitudes (*).

M. GIFF0~ and R. LAVEn~I~RE

Institut de Physique Nuel~aire, Universitg Claude Bernard - Villeurbanne

(ricevuto il 17 Gennaio 1972)

S u m m a r y . - - Using the recent measurements of the /~ rotation para- meters, we propose a simple method to determine the various T:-JT scattering amplitudes. We compare the results with those obtained in a previous paper and those of the Regge model arid FESR.

1. - I n t r o d u c t i o n .

In a previous paper (1), hereaf ter quoted as I , we have shown that one

can reasonably hope to solve the problem of the construct ion of the various

r:-3~ amplitudes for the near forward direction. To this aim, we used the fol-

lowing experimental data: elastic and charge-exchange cross-sections and

polarizations and the rotat ion parameter _R-(w-p). We also tested the influence

of the phase variat ion on the determinat ions of the amplitudes.

We emphasized the difficulty to obtain precise results due to large errors

about some dat% part icular ly the charge-exchange cross-section and polariza-

tion. I n order to reduce the uncer ta in ty on the results due to these errors,

we present here a new method to construct the physical quanti t ies without

reference to the charge-exchange data. The method consists in determining

separately the amplitudes for the 7:+p and 7:-p processes using recent experi-

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (1) M. GIFFO~: Nuovo Cimento, 7A, 705 (1972).

917

918 M. GIFFON and It. LAV]~I~RI~R~

menta l measu remen t s abou t the _R + and ~ - ro ta t ion pa ramete r s . Unfor tu-

nate ly , for the t ime being, this me thod can only be used a t the l abora to ry m o m e n t u m p = 6 GeV/c because da ta a t o ther m o m e n t a are not still available.

The main in ten t of our new m e t hod is t ha t we choose unambiguous ly

the solution f rom m a t h e m a t i c a l and physical considerations. We give also

the ana ly t ic expression of the var ious ampl i tudes in a ve ry simple fo rm con-

ta in ing only the exper imen ta l dat:~ and one phase. The s impl ic i ty of our

de te rmina t ion allows us to reduce grea t ly the numer ica l calculus. F r o m the

e las t ic-ampli tude de te rmina t ion we deduce the various quant i t ies of physical

in te res t in order to give their t ransfer dependence a t the considered energy.

In this work we use only exper imen ta l da ta for the elastic processes

7~±p (~), more precise ly we consider da¢~/d[2, P~'~) and /~(~, where m-----4-

respec t ive ly for the 7~+p and = - p processes. We fit the exper imenta l da ta and

we give an evaluat ion of the error for each quan t i t y studied. We confine our inves t iga t ion be tween t = 0 and t = - - 0 . 6 (GeV/e) ~. Due to the l imi ta t ion of

values for the R~-o pa ramete r s , we ex t rapo la te t hem in the forward direct ion

and near the ex t reme poin t t = - - 0 . 6 (GeV/e) 2.

I n the nex t Section we explain the new method , giving analy t ic expres-

sions of the ampl i tudes in t e rms of exper imenta l pa ramete r s . We discuss our

resul ts in Sect. 3. We make an a t t e m p t to compare t h e m wi th those given

b y the Regge model and F E S R and we conclude in Sect. 4.

2. - Expression of the 7:-2V amplitudes in terms of the experimental data.

Among the following data : the differential cross-sections, the polariza-

t ions P of the recoil nucleon and the 1~ pa ramete r s , we can select a set of independent me-~surements now avai lable for the scat ter ing of pions on nucleons. We bui ld this set wi th da(~)/d/2]~ and the P(~) and /~(~) pa rame te r s , where

m = =[= refers to the ~ p - - ~ ± p processes, i.e. we use only the elastic-scat-

t e r ing data .

(2) K. J. FOLEY et al.: BNL preprint 13102 (1968) referred to in Compilation o] pion- nucleon scattering data; G. GIACOMELLI, ]9. PINI and S. STAGNI: CERN-HERA 69-] (November 1969); M. BOI~GHIM, L. DICK, L. DI LELLA, A. NAVARRO, J. C. OLIVIER, K. REIBEL, C. COIGNET, D. CI~ONENBERGER, G. GREGOIRE, K. KIJRODA, A. MICHALOWICZ, M. POULET, D. SILLOU, C. BELLETTINI, P. L. BRACCINI, W. DEL PRETE, L. FOA, G. SAN- GUINETTI and l~I. VALDATA: Phys. Lett., 31 B, 405 (1970); A. DE LESQUEN, ]3. AMBLARD, R. BEURTEY, G. BYSTRICKY, G. COZZIKA, J. DEREGEL, Y. DUCROS, J. M. FONTAINE, A. GAIDOT, M. HANSROUL, F. LEHAR, J. ]9. MERLO, S. MIYASHITA, J. MOVCtIET and L. VAN ROSSUM: Measurement o] spin rotation parameter in r~±p scattering at 6 GeV/e, Communication at the Amsterdam International Con]erenee o/ Elementary Particles (July 1971); J. P. MERLO: Measurement o] A and R parameters in ~p scattering at high energies, Seminars given at Daresbury, Rutherford Laboratory, June 1971.

A N E W METHOD FOR D E T E R M I N A T I O N OF r~-~ ~ SCATTERING A M P L I T U D E S 919

The c o r r e s p o n d i n g equa t ions in t e r m s of the rea l and i m a g i n a r y p a r t s of

the a m p l i t u d e s can be w r i t t e n

d~2 ----Ig(m)]2 + h e r e a f t e r quo t ed as 8 (m),

( d ~ ) ~ ) = 2 I m [ g ( ~ ) h * ( ~ ) ] , (1) e ~

R ~-~ = (Ig(m)] : - [h(~)I :) cos a + 2 sin a R e [g('n)h*(~'],

where g(~) a n d h~-0 d e n o t e the spin-nonf l ip and spin-fl ip 7:-3g' amp l i t udes . The

angle a is the angle b e t w e e n the m o m e n t a of the final nuc leon in the l a b o r a t o r y

s y s t e m a n d in t he c .m. sy s t em.

W e t r a n s f o r m the s y s t e m (1) b y s t a t i n g

g(~ = @(~) exp [i~(~)] a n d h (~) r (m) exp [i/5( ~)] .

Since we h a n d l e iden t i ca l ly t he two cases m = ~ we drop , for the t i m e be ing ,

t he i ndex m. W i t h t he a b o v e p a r a m e t r i z ~ t i o n , we ge t the s y s t e m

(2) 2@r sin (~ - - f l ) - - P S ,

(e 2 - r ~) cos a + 2@r cos (~ - - f l ) sin a - - ~ S .

W e no te theft the u n k n o w n q u a n t i t i e s a re now @, r~ c~ and/3. A f t e r some m a n i p - u la t ions we express @ a n d r in t e r m s of ~ - - f l and f inal ly we o b t a i n the fol lowing e q u a t i o n :

(3) P2e tg : (~ - - f l ) - -2PRs inac tg (~ - - f l ) t- P2cos~a + R2--cos2a--=O,

the roo t s of which are

(4) c tg(c~--f l ) - - (/? s i n a ~ A eosa)/P,

where the A - p a r a m e t e r is r e l a t ed to the o thers b y

A -- ( 1 - P~--_R~) ½ .

F r o m (2) a n d (4), we de r i ve the two o the r u n k n o w n s :

{ 20~- S(1 ~ R ( , o s a ~ A s i n a ) ,

(5) 2r" S ( 1 - - R c o s a I A s i n a ) .

920 M. GIFFON and R. LAVERRI~RE

We should like to drop the sign inde te rminacy to find the physical solution. Near the forward direct ion we have a ~_ g/2 so tha t

2e ~ ~ S(1 =~ A) and 2r ~ _~ S(:1 ± A ) .

We know tha t in this region A ~ 1 ; as physical ly @ is always larger than r, we must choose the lower sign in (4).

F rom the last equat ion (2) and f rom (4) and (5) we can derive fl in terms

of a and the exper imenta l data S, P and R. So to obtain the g(m) and h (m) func- tions, the only unknowns are the phases ~(~). To determine these phases we separate in the following manner , as in I : we allow them to va ry between

ctg a (~) = - 0.3 to ctg a ( ~ ) = - 0.1 and we choose the values leading to results in good agreement with the expected physical results:

i) ]ctg~+] > Ictg~-[ at t = O,

ii) ]h(m)]->O for t - > 0 ,

iii) Img(~)(t = O) giving the value in agreement with the optical point.

We find tha t these conditions are well verified if we choose

ctg ~+ = - - 0.25 and ctg ~- = - - 0.15.

Then we get the following expressions for g(~) and h('):

l~e

Im

Re

Im

This allows us

for the hel ici ty

.,m,} (8,o,y {cos g(m) = - 2 [ I + R (m) c o s a + A (m) sina]½ sin,(m)',

h('~) I I,~(m,\~ { h(") J = i T ) [ 1 - - t ~ ( m ) c ° s a - - A ( m ) s i n a ] ½ sinC°Sfl(m) 'fl(-),

to obtain simple expressions for the invar iant amplitudes and

amplitudes direct ly in terms of the exper imental quantit ies.

3. - D i scuss ion o f the results.

The first in teres t ing result is tha t we simplify our previous method be- cause the fact tha t the exper imenta l measurements of the R ± parameters are now available allows us to solve the problem symmetr ical ly in the two cases

7:+p. We can get simple expressions in terms of the physical parameters for the ampli tudes.

The recent measnrements of /~+ seem to show tha t this pa ramete r can be posi t ive in the domain 0 < - - t < 0.25 (GeV/cp cont rary to the results obtained

A NEW METItOD FOR DETERMINATION OF ~-~(~ SCATTERING AMPLITUDES 9 2 1

in our previous work. I n the following we examine the two poss ib i l i t i es /~+> 0

and R + < 0 in order to t es t the influence of the R sign over the various ampli- tudes in this domain.

We have eva lua ted the errors for afl the ampl i tudes . The errors for the quant i t ies re la ted wi th only one process are highly correlated, but those for

the quant i t ies re la ted to the two processes, as A'~ and B -+, are not. The errors

are larger in the forward direct ion; this is due to the fac t tha t the errors on the leading quant i t ies , i.e. the differential cross-sections, are b~rge in this domain.

Among all the results we e:dculate, we give on Tables only the quant i t ies of in te res t for compar ison with oar previous results and those obta ined by

o ther authors . The Tables f o r / ~ + < 0 are s topped ,q.t t = - - 0 . 2 8 (GeV/c) 2 be-

cause the following values are the same than those for /~+> 0.

3"1. The e~astic scattering. - The resul ts abou t the spin-nonflip ampl i tudes

are the same as in I , wha teve r sign of R+; we note a nonnegligible m o m e n t u m

t rans fe r dependence; the slopes of the imag ina ry par t s are ve ry close and the real pa r t s are comparab ly equal for the two processes.

Fo r the spin-flip ampl i tudes , the real pa r t s are similar to those obta ined in I wi th some m o m e n t u m dependence near the forward direction: Re h + is

always pos i t ive and Re h- is negat ive unt i l t ~ - - 0 . 4 (GeV/c) ~ where there is

a zero followed b y small oscillations. As for the imag ina ry par ts , there are two cases (Tables I and I I ) :

i) I m h- is the same as in I , wha teve r the sign o f / t+ ; i t exhibits a b u m p

near t ~ 0 and o ther s t ruc tures for - - t > 0 . 3 (GeV/c) ~, bu t remains always negat ive .

it) I m h + has a fair ly pronounced t ransfer m o m e n t u m dependence near the forward direction. I f / ~ + ~ 0, i t keeps the same shape than in I bu t with possibi l i ty of zeros near t - - - - 0 . 2 (GeV/c) 2. I f R + ~ O , i t becomes posi t ive

be tween t ~ 0 and t - - - - 0 . 2 6 (GeV/c) 2 where i t lms a zero, then i t presents m i n i m u m near t ~ - 0.33 (GeV/c) ~ and seems to have some other s t ructures

when t increases wi th possibi l i ty of zeros near t ~ - - 0 . 4 and - -0 .52 (GeV/c) 2.

Now let us examine the hel ici ty ampl i tudes in the s-channel. As quoted

in I , the s-channel ampl i tudes do not disappear . The spin-nonflip ampl i tudes

p resen t the same behav iour and the same magni tude than in I. The only

change for the spin-flip ampl i tudes concerns I m _F+_(~+p) in the forward direc-

t ion i f / ~ + ~ 0 (Tables I H ~nd IV). The main consequence of this modification is the occurrence of a zero for t ~ - 0.2 (GeV/c) ~ for this ampl i tude.

Now we inspect the inva r i an t ampl i tudes A and B (Tables I I I and IV):

i) I f R + ~ 0, a b u m p seems to appear near the forward direction for all the ampl i tudes ; this is the only modification compared to I.

922 M. GIFFON and R. LAVERRII~3RE

TABLE I. -- Moduli in (mb)½ o / I m h +, Im h- and various ~-J~ charge-exchange amplitudes with R + > 0 The variable - - t is always expressed in (GeV/c)%

- - t I m h + I m h - R e gO h n gO R e h ° I m h °

0.020 0.34=t=0.25 --0 .45- /0 .25 --0.32£=0.03 - -0 .27±0.02 0.294-0.04 0.55 ±0.35

0.060 0.69±0.25 --0 .84- /0 .22 - -0 .30±0.03 --0.10=]=-0.02 0.40-/0.03 1.07 ±0.30

0.090 0.79±0.22 --0.89-/0.19 - -0 .28+0.02 0.024-0.02 0.334-0.03 1.19 ±0.30

0.125 0.744-0.17 --0.824-0.14 --0.264-0.02 0.10=]=0.02 0.23=]=0.03 1.11 4-0.30

0.150 0.63-/0.17 --0.744-0.12 - - 0.23 ~0.02 0.06-/0.02 0.214-0.03 0.98 ±0.25

0.175 0.51-/0.14 --0.644-0.12 - - 0.20 =L0.02 0.02±0.02 0.20±0.03 0.81 =]=0.25

0.200 0.36=]=0.13 --0.51=]=0.10 - - 0.18±0.01 0.00±0.02 0.20±0.02 0.61 =]=0.20

0.230 0.17 -/0.12 --0.36=]=0.09 --0.164-0.01 - -0 .02±0.02 0.21=]=0.02 0.36 -/0.15

0.280 - -0 .07±0.12 - - 0.13-/0.09 --0.14-/0.01 0.05±0.02 0.19=]=0.02 0.04 -/0.15

0.320 --0 .13- /0 .12 --0.05=]=0.09 --0.13-/0.01 0.07±0.01 0.18-/0.02 --0.07 ±0.15

0.380 --0 .08- /0 .08 --0.194-0.07 - - 0.11 -/0.01 0.08=]=0.01 0.10±0.02 0.08 ±0.12

0.440 - - 0.05 ~0.05 - -0 .04±0.07 --0.09 0.09±0.01 0.07±0.02 0.00 =]=0.09

0.480 --0.03=]=0.05 - - 0 .05i0 .05 --0.08 0.05±0.01 0.03±0.02 0.01 ±0.07

0.540 - - 0.01 ±0.04 - - 0.09 -/0.04 - - 0.06 0.03 0.01 =]=0.01 0.056 ±0.06

0.575 - - 0.03 4-0.03 - - 0.08 =]=0.04 - - 0.05 - - 0.01 0.01 0.033 -/0.05

0.625 - - 0.06 4-0.03 - - 0.03 =]=0.03 - - 0.04 - - 0.01 0.01 - - 0.015 4-0.04

TABI~E II . - Moduli in (mb)½ o] Im h +, Im h-" and various r~-~ charge-exchange amplitudes with R + < O.

- - t I m h + h n It-- R e go I m gO R e h ° I m h °

0.02 - -0 .20±0.25 --0 .45- /0 .25 - -0 .34±0.03 - -0 .14±0.02 0.39-/0.04 0.16±0.35

0.06 - -0 .47- /0 .25 - -0 .84±0.22 - -0 .30±0.03 - - 0.17-/0.02 0.60-/0.03 0.26=t=0.30

0.09 --0.49=t=0.22 --0.894-0.19 - - 0 . 2 7 i 0 . 0 2 - -0 .04±0.02 0.56-/0.03 0.30=]=0.30

0.125 --0.354-0.17 - -0 .82- /0 .14 - -0 .25±0 .02 --0 .09- /0 .02 0.42-/0.03 0.33-/0.30

0.150 - - 0.18:J=0.17 --0.74 4-0.12 - -0 .22±0 .02 0.054-0.02 0.34-/0.03 0.40±0.25

0.175 - - 0.06-/0.14 --0.644-0.12 --0.204-0.02 0.04-/0.02 0.29-/0.03 0.41 -/0.25

0.200 0.00±0.13 --0.51 4-0.10 - - 0.18-/0.01 0.024-0.02 0.274-0.02 0.37-/0.20

0.230 0.10±0.12 --0.364-0.09 - -0 .16±0.01 - -0 .02±0.02 0.23-/0.02 0.26-/0.15

0.280 - -0 .08±0.12 - - 0.13 ~=0.09 --0.14=t=0.01 0.04-/0.02 0.20±0.02 0.03~:0.15

ii) I f R + > 0, t h e s ign of I m A(~z+p) a n d ImB(rc+p) has b e e n r e v e r s e d

c o m p a r e d to I a n d c o n s e q u e n t l y ~ s u p p l e m e n t a r y zero ~ppe~rs n e a r t - -

= - - 0 . 2 2 (GeV/c) ~ for t h e two ~mpl i t udes .

k N E W METHOD FOR DETERMINATION- OF ~ - ~ 8C&TTERING AMPLITUDES 9 2 3

TABLE I I I . - Helieity Im F+_(T:+p) and invariant I m A(r:+p) and Im B(rz+p) amplitudes respectively in (rob)½ /or I m F +_ and I m A and (rob)½ (GeV/c) -1 /or Im B with R+> O.

- - t Im F+_(u+p) Im A(~+p) Im B(~x~p)

0.02 - -0 .11 4- 0.26 - - 108 -~ 88 - - 4 7 4- 38

0.06 - -0 .35 4- 0.26 - - 137 4- 70 - - 6 1 4- 28

0.09 - -0 .42 4- 0.23 - - 131 :J: 46 - - 5 8 4- 20

0.125 - -0 .37 4- 0.22 - - 117 4- 46 - - 4 6 4- 16

0.150 - -0 .26 4- 0.20 - - 82 4- 42 - - 3 4 4- 16

0.175 - -0 .14 4- 0.14 - - 51 4- 30 - - 2 4 :J: 12

0.200 0.0 4- 0.14 - - 17 4- 20 - - 12 4- 10

0.230 0.17 4- 0.13 14 4- 20 0 :~ 8

0.280 0.38 4- 0.13 50 4- 16 12 4- 8

0.320 0.42 4- 0.13 53 4- 16 14 4- 8

0.380 0.34 4- 0.06 37 4- 10 10 4- 4

0.440 0.28 4- 0.05 29 4- 6 7 4- 3

0.480 0.24 4- 0.04 23 4- 6 6 4- 2

0.540 0.19 4- 0.03 16 4- 4 4 4- 1

0.575 0.19 4- 0.03 17 4- 4 4 4- 1

0.625 0.19 4- 0.02 17 4- 4 5 ~: 1

T A B L E I V . - Helieity Im F+_(rc+p) and invariant Im A(=+p) and Im B(r:+p) amplitudes respectively in (mb)½ ]or Im F+_ and Im A and (rob)½ (GeV/c) -1 ]or In] B with R + < 0.

- - t Im/~+_(~+p) Im A(r:+p) l m B(r:+p)

0.02 0.42 ± 0.26 226 4- 88 65 -4- 38

0.06 0.80 ± 0.26 255 ± 70 75 -4- 28

0.09 0.86 4- 0.23 220 4- 46 64 ± 20

0.125 0.71 4- 0.22 151 ± 46 43 ± 16

0.150 0.55 4- 0.20 101 4- 42 27 4- 16

0.175 0.42 4- 0.14 67 ± 30 17 4- 12

0.200 0.35 4- 0.14 50 ± 20 l l 4- 10

0.230 0.33 :J_ 0.13 43 ± 20 9 4- 8

0.280 0.38 4- 0.13 50 4- 16 12 :j_ 8

L e t us c o n s i d e r t h e i s o s p i n e v e n a n d o d d d e c o m p o s i t i o n A ' ± ,~nd B±

( T a b l e s V a n d V I ) . I n t h e e a s e _R+< O, t h e o n l y c h a n g e c o m p a r e d t o I is

t h e a p p e a r a n c e of a b u m p n e a r t = 0 fo r t h e a m p l i t u d e s A ' a n d I m B +. I n

924 M. GIFFON and i~. LAVERRI]~RE

TABLE V. - Isospin even and odd decompositions A" in (mb)½ and B in (rob)½ (GeV/c) -1 /or elastic

- - t R e A '+ I m A '+ R e A ' - I m A ' -

0 . 0 2 0 - - 161 ± 34 338 ± 70 394 ± 34 772 4- 70

0.060 - - 91 4- 20 286 4- 70 312 4- 20 810 4- 70

0.090 - - 74 4- 15 236 ± 45 209 ~= 15 732 4- 45

0.125 - - 64 4- 10 180 4- 35 125 4- 10 570 4- 35

0.150 - - 544- 6 1754-35 1034- 6 450~=35

0.175 - - 454- 6 169 4- 25 904- 6 3 5 0 4 - 2 5

0.200 - - 454- 4 1584-20 854- 4 2 4 0 4 - 2 0

0.230 - - 464- 4 1494-20 814- 4 1504-20

0.280 - - 44 4- 4 1304-16 684- 4 124-16

0.320 - - 444-t= 4 1084-16 584- 4 - - 3 0 4 - 1 6

0.380 - - 324- 4 1094- 9 154- 4 204- 9

0.440 - - 244- 2 62 4- 7 204- 2 - - 44 - 7

0.480 - - 23 4- 2 54 4- 5 84- 2 + 24- 5

0.540 - - 184- 1 484- 3 44- 1 -]- 24- 3

0.575 - - 184- 1 4 5 ± 3 34- 1 + 44 - 3

0.625 - - 94- 0.5 364- 2 34- 0.5 - - 14- 2

TABL~ VI. - Isospin even and odd decompositions A" in (mb)½ and B in (rob)½ (GeV/c) -1 ]or elastic

- - t R e A '+ I m A "+ R e A ' - I m A ' -

0 . 0 2 - - 2 8 5 4- 34 808 4- 70 530 4- 34 216 4- 70

0.06 - - 245 4- 20 896 4- 70 464 4- 20 200 4- 70

0.09 - -211 4- 15 798 4- 45 350 4- 15 172 4- 45

0.125 - - 162 4- 10 586 4- 35 224 :J- 10 168 4- 35

0.150 - - 121 4- 6 504 4- 35 168 4- 6 185 4- 35

0.175 - - 88:k 6 344 4- 25 132 4- 6 1754-25

0.200 - - 68 4- 4 260 4- 20 112=h 4 1 4 7 ~ 2 0

0.230 - - 564- 4 192 4- 20 90 4- 4 96 4- 20

0.280 - - 444- 4 130 j_16 704- 4 1 2 4 - 1 6

t h e c a s e R + > 0, t h e g e n e r a l f e a t u r e s a r e t h e s a m e b u t t h e s t r u c t u r e s a r e

m o r e p r o n o u n c e d f o r a l l t h e a m p l i t u d e s .

F i n a l l y w e s t u d y t h e h e l i c i t y a m p l i t u d e s i n t h e c r o s s e d c h a n n e l w h i c h w e

d i d n o t c o n s i d e r i n I ( T a b l e s V I I a n d V I I I ) :

A NEW METHOD FOR DETERMINATION OF 7~-~ ° SCATTERING AMPLITUDES

scat ter ing w i t h R + ~ O.

9 2 5

Re B + hn B + Re B- Im B-

- - 1 7 . l ~ 4 34.6:~:40 4 3 . 4 ± 4 81 ~-40

- - 9 . 5 ± 2 28.7zL20 3 4 . 8 ~ 2 90 ~L20

- - 7 . 7 i l 22 ± 17 23 ± 1 80 :J:17

- - 6 . 6 ~ 1 1 8 . 2 ± 1 2 t4 ~ 1 64 ~ t 2

- - 5.8~_0.8 17.8:~12 11 .5~0 .8 51 ± 1 2

- - 4 . 7 ± 0 . 8 1 6 . 8 : ~ 9 10 ~ 0 . 8 39 ± 9

- - ~ 4 . 6 ~ 0 . 5 16.2~ 8 9.5~_0.5 28 ± [ 8

- - 5.0~=0.5 1 5 . 6 ± 7 9 . 0 ± 0 . 5 16 ~- 7

- - 4 . 9 - L 0 . 5 1 3 . 8 + 7 8 . 0 ~ 0 . 5 1.5=~ 7

- - 4 . 9 ± 0 . 5 12.2-~ 6 6 . 8 ~ 0 . 5 - - 3 ~ 6

- - 3 . 7 ± 0 . 5 12 .0~ 4 2 . 0 ± 0 . 5 2.5~- 4

- - 2 . 9 ± 0 . 4 6 . 8 ± 3 2.4-~0.4 0 . 5 ± ~ 3

- - 2 . 6 i 0 . 4 5.8:~ 3 1 .0=L0.4 0 . 2 ± 3

- - 1 . 9 ~ 0 . 2 5.3 [: l 0 . 5 ± 0 . 2 1 . 5 ± 1

- - 2 . 1 ~ : 0 . 2 5.0:[~ 1 0 . 4 ± 0 . 2 1.0~- 1

- - 0 . 9 ± 0 . 1 4.l :L 1 0.4=-'_0.l - - 0 . 4 ± 1

scat tering w i t h R + ~ O.

Re B + Im B" Rc B- hn B -

- - 3 2 ~ : 4 89 : t :40 59 ~ 4 25 j _ 4 0

- - 2 7 ~ 2 9 6 . 4 ± 2 0 51 ± 2 22 ± 2 0

2 3 ± 1 84 ± 17 38 ~:1 20 ~:17

- - 1 7 : L l 61 :J: 12 24 ± l 18 :~ 12

13~ 0.8 47 ± 1 2 18 :L0.8 20 -El2

- - 9 ± 0 . 8 36 ~ 9 15 ± 0 . 8 t9 -[= 9

- - 7 ± 0 . 5

- - 6 ± 0 . 5

- - 5 ~: 0.5

28 ± 8 13 ~_0.5 17 -~ 8

2l J: 7 10.6-~0.5 11 ~- 7

14 ~ 7 8 ~:0.5 ] .5~: 7

i) I f R + < 0, I m Mo is a s m o o t h d e c r e a s i n g f u n c t i o n and I m No seems

to p r e s e n t a b u m p n e a r t he fo rwa rd d i r e c t i o n ; R e Mo is a n e g a t i v e and

s m o o t h l y i n c r e a s i n g f u n c t i o n , b u t Re No seems to h~ve a s t r u c t u r e n e a r t he

f o r w a r d d i r e c t i o n a n d a zero n e a r t = - - 0 . 3 9 (GeV/c)". F o r t he i m a g i n a r y

926 1~. GII~I~ON and R. LAVERRI]~RE

TABLe, VII . - Helicity amplitudes in isospin-O and -1 crossed channel in (rob)½ with R + > 0.

- - t R e M~ I m M~ Re 2~1 Im N~

0.02 0.23 ± 0.04 0.08 ± 0.12 0.23 ± 0.04 0.32 2= 0.25

0.06 0.19 i 0.04 --0.01 ± 0.12 0.34 i 0.04 0.59 4- 0.25

0.09 0.17 4- 0.04 --0.05 i 0.12 0.30 i 0.03 0.65 ± 0.25

0.125 0.16 i 0.04 --0.14 i 0.12 0.22 ± 0.03 0.58 ! 0.20

0.150 0.14 ± 0.03 --0.12 + 0.10 0.20 + 0.03 0.52 i 0.20

0.175 0.12 ± 0.03 --0.09 ± 0.08 0.18 4- 0.03 0.44 4- 0.14

0.200 0.11 ± 0.03 - - 0.06 ± 0.06 0.18 ± 0.03 0.35 + 0.14

0.230 0.09 ± 0.02 --0.02 ± 0.06 0.17 ± 0.03 0.23 ± 0.13

0.280 0.08 =L 0.02 --0.03 4- 0.06 0.15 4- 0.02 0.02 ± 0.12

0.320 0.07 ± 0.02 --0.04 4- 0.06 0.14 4- 0.02 --0.06 i 0.12

0.380 0.06 -4- 0.01 - - 0.06 ± 0.06 0.08 i 0.02 0.04 ± 0.07

0.440 0.05 2= 0.01 --0.06 + 0.04 0.06 4- 0.02 --0.02 i 0.05

0.480 0.05 ± 0.01 --0.03 -2= 0.04 0.03 + 0.01 --0.04 + 0.03

0.540 0.04 i 0.01 - - 0.03 4- 0.04 0.02 ± 0.01 0.03 ± 0.03

0.575 0.03 + 0.01 --0.00 ± 0.02 0.01 ± 0.01 0.02 ± 0.03

0.625 0.02 ± 0.01 0.01 ~= 0.02 0.01 i 0.01 --0.01 + 0.02

TABLE VIII . - Helicity amplitudes in isospin-O and -1 crossed channel in (mb)½ with R + < 0.

- - t R e M 1 I m M 1 Re N1 Im 2V 1

0.02 0.23 ± 0.04 0.09 ± 0.12 0.28 i 0.04 0.12 4- 0.25

0.06 0.18 ± 0.04 0.04 4- 0.12 0.44 ± 0.04 0.19 ± 0.25

0.09 0.16 + 0.04 0.01 ± 0.12 0.41 4- 0.03 0.21 ± 0.25

0.125 0.15 ± 0.04 --0.09 ± 0.12 0.31 + 0.03 0.23 4- 0.20

0.150 0.13 4- 0.03 --0.07 ± 0.10 0.26 ~- 0.03 0.27 ± 0.20

0.175 0.12 i 0.03 --0.07 ± 0.08 0.22 ± 0.03 0.28 i 0.14

0.200 0.10 4- 0.03 --0.05 + 0.06 0.20 ± 0.03 0.26 ± 0.44

0.230 0.09 ± 0.02 --0.01 4- 0.06 0.18 4- 0.03 0.18 4- 0.13

p a r t s of t he M1 a m p l i t u d e , we n o t e a zero n e a r t = - - 0 . 1 (GeV/c) ~ corre-

s p o n d i n g to t h e c ros s -ove r a n d t h e n i t r e m a i n s n e g a t i v e w i t h some f luc tua-

t ions . F o r t h e 2¢1 a m p l i t u d e , t h e i m a g i n a r y p a r t p r e s e n t s a m a x i m u m n e a r

t = - - 0 . 1 8 (GeV/c) ~ a n d a zero n e a r t = - - 0 . 3 (GeV/c) 2 fo l lowed b y f luc tua-

t ions . The R e M1 is a s m o o t h dec reas ing f u n c t i o n a n d R e N1 p r e s e n t s a

m a x i m u m n e a r t = - - 0 . 6 (GeV/c) ~ a n d t h e n decreases s lowly.

A N E W M E T H O D F O R D E T E R M I N A T I O N O F 7 ~ - J ~ ~ S C A T T E R I N G A M P L I T U D E S 9 2 7

Re Mo Im M o l le h% I m No

- - 1 . 0 3 ± 0 . 0 4 5 . 1 7 + 0.12 - - 0 . 1 2 ~= 0.04 0.33 ~_ 0.25

- - 0.89 ~: 0.04 4.5 ± 0.12 - - 0.16 ± 0.04 0.58 -L 0.25

- - 0.79 :L 0.04 3.98 :~ 0.12 - - 0.16 ~ 0.03 0.43 i 0.25

- - 0 . 6 5 j : 0.04 3.27 ± 0.12 -- 0.16 j : 0.03 0.59 -L 0.20

- - 0.60 ± 0A)3 3.00 ~_ 0.10 -- 0.15 ~ 0.03 0.57 ± 0.20

- - 0.54 ± 0.03 2.73 ± 0.08 -- 0.13 4 0.03 0.55 ± 0.14

- - 0.50 ± 0.03 2.50 ~_ 0.06 .... 0.13 ~ 0.03 0.52 ~_ 0.14

- - 0.44 ± 0.03 2.23 ~: 0.06 - 0.12 ~ 0.(t3 0.46 ~: 0.13

- - 0.37 ± 0.02 1.85 ± 0.06 - 0.12 j : 0.02 0.41 ± 0.12

- - 0.31 4_ 0.02 1.57 ~ 0.06 - 0.13 ~ 0.02 0.38 :[: 0.12

- - 0.24 4= 0.02 1.2:{ ~ 0.06 - - 0.10 ~ ().01 0.38 ± 0.07

- - 0.20 _k 0.01 1.02 ± 0.04 - 0.09 k 0.01 0.27 ~: 0.05

- - 0.17 ~: 0.0[ 0.9o ± 0.04 - 0.08 t- 0.01 0.25 ± 0.03

- - 0.14 ± 0.01 0.74 ± 0.04 0.07 ]: 0.0[ 0.23 ± 0.03

- - 0 . 1 2 ~ 0 . 0 1 0 . 6 6 ~ 0 . 0 2 0.07 L0.01 0 . 2 1 ± 0 . 0 3

- - 0.11 ± 0.01 0.53 ~ 0.02 0.04 ± 0.01 0.18 ± 0.02

Re 310 h n 310 l¢~e ~o h n ~V o ,,,ti

- - 1 . 0 2 LO.04 5.15 ~0.12 - - 0 . 1 8 ± 0 . 0 4 0 . 5 4 i 0 . 2 5

- - 0.88 ~ 0.04 4.45 :~ 0.12 0.2{i ± 0.04 0.99 ~ 0.25

- - 0.78 -]: 0.04 3.91 ~ 0.12 -- 0.27 [_ 0.03 1.07 ± 0.25

- - 0.64 ~: 0.04 3.2;{ [: 0. [2 - - 0.25 ± 0.03 0.94 ± 0.20

- - 0.58 ± 0.03 2.93 ± 0. [0 .... 0.21 ~ 0.03 0.82 ± 0.20

- - 0.54 4= 0.03 2.71 ~ 0.08 - - 0.17 ~: 0.03 0.70 ± 0.14

- - 0.50 ~: 0.03 2.49 ~ 0.06 - - 0.15 A: 0.03 0.61 -~ 0.14

- - 0.44 -L 0.02 2.22 ~ 0.06 --- 0.14 ~ 0.03 0.51 _~ 0.13

ii) I f R + > 0, t h e q u a l i t a t i v e f e a t u r e s f o r t h e i s o s p i n - 0 a n d -1 a m p l i -

t u d e s ~ re u n m o d i f i e d .

3"2. The charge-exchange scattering. - T h e r e s u l t s a b o u t t h e a m p l i t u d e s gO

a n d h ° ( T a b l e s I a n d I I ) p r e s e n t t h e s a m e f e a t u r e s t h a n in I , t h e c r o s s - o v e r

o c c u r r i n g n e a r t ~ - - 0 . 1 (GeV/c ) 2 i f R + ~ 0. I f / ~ + > 0, t h e a m p l i t u d e b e h a v -

928 M. GIFFON and R. LAV]ERRII~RE

iour is the same bu t more pronounced. The only change is t h a t now the dom- inan t cont r ibu t ion to the sca t te r ing arises f rom ] m h ° ins tead of Re h ° and Re gO.

The inva r i an t ampl i tudes r emain unchanged for /~+< 0. I f ~ ÷ > 0, the only change is the appearance of a b u m p near the forward direct ion for

I m A0 and h n Bo. The Ao, Po and /~o p a r a m e t e r s are ob ta ined b y difference of quant i t ies

which are small, of the same magn i tude and which suppor t large errors ; this

implies a large ins tab i l i ty on the resul ts which p reven t s us f rom tak ing t hem

into account .

4. - C o n c l u d i n g r e m a r k s .

The exper imenta l m e a s u r e m e n t of the ~+ p a r a m e t e r s allows us to deter-

mine separate ly , for the two elastic processes ~+p, the energ T and m o m e n t u m

dependence of the var ious ampl i tudes wi thou t reference to the charge-exchange

exper imen ta l data . The pr inc ipa l in te res t of our new me thod is to get unam-

biguously the physical solution only f rom the equat ion defining the observables

and physica l considerat ions. I n this pape r we t e s t ed the influence of the /~+ da ta near the forward direc-

t ion over the physica l ampl i tudes . To make predic t ions abou t the /~+ sign we recons t ruc t f rom our resul ts the charge-exchange cross-section in order to compare i t wi th the exper imen ta l data . I t seems tha t the bes t ag reemen t

is ob ta ined f rom /~+< 0. This leads us to th ink t h a t / ~ + < 0 near the forward direct ion as predic ted in I. The pr incipal difference be tween ~ + > 0 and ~ + < 0 abou t the ampl i tudes is to change the phases of h +, /~+_(7:+p), A'(r:+p) and B(7:+p), this leads to the appea rance of a zero for the imag ina ry pa r t s of

these ampl i tudes near t = - - - 0 . 2 2 (Ge¥/c) 2. Le t us now compare our resul ts wi th those ob ta ined b y other authors :

1) About the 7:- 1) process our resul ts are ve ry similar to those of BA~GER

and PmLLIPS except for the l ight b u m p near t = 0 for Re B- (a ) .

2) Concerning even and odd isospin decomposi t ions A '~ and J ~ , t hey

are somewhat different f rom those of H o ~ ] ~ et al. (~). I n fac t we find a zero for I m B - and I m A ' - near t = - - 0 . 3 (GeV/c) ~, followed b y small oscillations,

no zero for Re B - and Re A ' - , whereas HOLDER found zeros near t ~ - - 0 . 2

and - - 0.5 (GeV/e) ~.

(8) V. BA:aGER and R. J. N. PHILLIPS: Phys. Rev., 187, 2210 (1969). (4) G. HOHLER, J. BAACKE and G. ]~ISENBEIS8: Phis. Zett., 22, 203 (1966).

A NEW METHOD FOg DETERMINATION OF 7~-J~ SCATTERING AMPLITUDES 929

3) The results about the helici ty ampli tudes in the crossed channel are

only in par t ia l agreement with those of H~ZE~- and MICKAEL (5). We are in

good agreement for the results concerning M ° and N 1 bu t not for N ° and M i ;

perhaps this discrepancy arises f rom a different choice of exper imental data and phases. Bu t the main difference arises f rom the fact tha t we do not use the charge-exchange data; we verified this assumption b y comparing the present

results with those obtained by using these charge-exchange data with the method presen ted in I.

4) For the charge-exchange scattering, our results eoncerning I m B ° and Re B ° are in good agreement with those of ]3At¢GEI~ and PKILLIPS (3).

5) We also calculated /~,~--Ih±/g ± sinSsl, R 2 = ]F+_/F++ s i n O s I and R 3 = = IA'+/vB+I where v = ( s - - u ) / 4 M ; these calculations lead to the following re-

I n a r k s :

i) I f R + > 0 , in the in terva l 0 . 0 2 < - - t < 0 . 6 2 5 (GeV/c) 2 for the 7:+1 a

process, /~ decreases from :[1.5={-1 to 2.1~-0.5, R2 lies between 0.45:J:0.1 and 0 .9~0 .1 and R:3 between 1.56J:0.5 and 1 .84~0.5 . For the r:-p process, R~ is the same, R2 lies be tween 0.6 and 1.76 and R3 remains ve ry close to the previous one.

ii) I f /~+< 0, the only change concerns /~ which lies now between 0 .70~0 .1 and 1.55:L0.1.

Compared to I~ the results about R~ are unchanged~ contrary to/~2 and R3; this resul t m ay derive from the fact tha t we do not use the charge-exchange

data. We see tha t R~ and R~ :~re not compatible wi th 0 in ~greement with a conclusion drawn by AMnLA]~D (") and R3 is different f rom i as suggested by BARGEI¢ and PI-IILLIPS (3).

We see, as a l ready quoted in I, tha t the details of the interact ion between pions and nucleons ~re still impor tan t at the present energy. As emphasized by several authors (5,~), the helici ty ampli tudes in the s-channel do not disap- pear and the hel ici ty conservat ion of the pomcron does not seem to be exact. At this energy, we can see tha t our results do not seem in agreement with those of the Regge model and those given by the FESR.

We are ve ry grateful to Prof. E. PRED&ZZI for useful discussions and warm

hospi ta l i ty ~t the Tur in Ins t i tu te of Theoret ical Physics.

(5) l 0. HALZEN a n d C. MICHAEL: P]~y8. Let t . , 36 B, 367 (1971). (6) B. AMBLARD: Thes is of P&ris U n i v e r s i t y , No. CNRS A.O. 5421 (Mars 1971).

59 - II Nuovo Cimento A.

930 M. GIFFON &nd 1~. LAVERRI]~R]~

• R I A S S U N T O (')

Faccndo uso delle recent i misure dei pa rame t r i di rotazione R% si propone un metodo semplice per de te rmina te le var ie ampiezze dello scat ter ing r:-d~'. Si confrontano questi r i sul ta t i con quelli o t t enu t i in un articolo precedente e con quelli del modello di Regge e delle F E S R .

(*) Traduzione a cura della Redazione.

HoBblfi MeTO~I onpe,~e.~enu~t aMn~InTy~ ~-.~' paceeaHtm.

P e 3 m M e ( * ) . - I/IcrioJib3y~ ne~IaBane n3MeperInn 1~± poTauuomfbiX napaMeTpoB, MbI npeJxnaraeM npocTog~ MeTO~ JXmt onpe~eneHHa pa3o~a~Hbix aMnnaxy~x n-d~ pacceannm 1-Io.ny~eHHbie peay~-mTa'r~I cpaB:qriBamTc~ c pe3yab'raTaMa npeff/,I,ayn~e~ pa6oT~t ~ pe- 3y.rISTaraMn Mo,ae,nrI Pe,a:~e n I~ESR.

(*) Hepeeec)eno peOamtue&