Nucleon Recoil in the Pion-Nucleon Scattering

IL NUOVO CIMENTO VoL. IV, N. 6 1 o Dicembre 1956

Nucleon Recoil in the Pion-Nucleon Scattering.

L. FO:NDA und I . REINA

Istituto di Fisica dell' Universith - Trieste

(rieevuto il 6 Set tcmbre 1956)

S u m m a r y . - - The effect of nucleon recoil has been taken into account in the extended source theory of pion-nucleon s c a t t e r i n g with PS-PV interact ion Hamiltonian. We have compared the results on /)-waves with experimental da ta and with the results of the Ch~w's cut-off theory and discussed the influence bf nucleon recoil in the determinat ion of the rei'lormalized coupling constant . In order to discuss, on the same theo- retical line, the influence of nucleon recoil on S-phases, we have added tc the PS-PV interact ion Hamil tonian two terms, which are bilinear in the meson field, as can be obta ined from=a Dyson-Foldy transform- at ion applied to the PS-PS coupling. The effect of nucleon recoil is found to be remarkable.

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

R e c e n t l y CHEW (~) h~s d e v e l o p e d a m e t h o d of c a l c u l a t i o n b a s e d on t h e

a p p l i c a t i o n of t he T a m m - D a n c o f f f o r m a l i s m to t h e p i o n - n u c l e o n s ca t t e r i ng ,

u s i n g P S - P V i n t e r a c t i o n H a m i l t o n i a n w i t h cut-off . T h e r e su l t s he o b t a i n e d

,~t low ene rgy a re in s u b s t a n t i a l a g r e e m e n t w i t h t h e e x p e r i m e n t a l r e su l t s on

P - w a v e s ca t t e r i ng .

I n t h e w o r k of CHEW, aS wel l in those he w r o t e in c o l l a b o r a t i o n w i t h

L o w (2), t h e n u c l e o n is c o n s i d e r e d as a f ixed source of mesons a n d t h e r e f o r e

n o t s u b j e c t e d to a n y recoi l in a s c a t t e r i n g p rocess w i t h mesons .

I n th i s p a p e r we w a n t to look u p o n t h e ef fec t of nuc l e on recoi l a n d to show

in wh ich d i rec t ion CHEW'S resu l t s a re t o ' be mod i f i ed . I n t h e first s ec t ions

(1) G. F . Cunw: Phys. Rev., 95,~ 1669 (1954) . (2) G. F. CHEW and F. E. Low: Phys. Rev., 1.01, 1570, 1579 (1956).

14o) L. FONDA and I. REINA

we shall treat the P-wave scattering and point out how the value of the renor-

realized coupling constant is to be increased, compared with tha t of the fixed

source, in order to fit the experimental data. Taken into consideration two

terms, bilinear in the meson field, we shall discuss in the Sect. 4 the inter-

ference of these ones with nucleon recoil terms of the PS-PV Hamil tonian in

the S-wave scattering. I t will be seen how nucleon recoil acts on the behaviour

of the S-phases contributing a large amount to them. I t is to be thought

t ha t the qualitative aspects of these corrections will be kept also in a more

refined theory of pion-nucleon scattering.

2. - T h e E x t e n d e d S o u r c e T h e o r y .

We start with the Hamil tonian (3):

(1) H(t) = H o + H o + Nucleons pions

- % / ~ d4x~ d'x~5(t-- t~)5(t-- tt)/(x I - - x2)~(x,)rsy.'r ~ ~p(X,) , +# with

(2) ](x~--x2) = 1 ;v(k) exp [ ik . (X l - -x2 )Jd3k ( 2 ~ ) 3 j ,

where v(k) is a function of the modulus of k going to zero for high momenta ,

whose value is 1 for k - - 0 . Using the free-particle expansions for the field operators and eliminating

the small components of the Dirac fields in favour of the large ones, in the

non relativistic limit we obtain for the Hamil tonian the following form (*):

(3) " = ..... + . o f fd3k, d3 ,

�9 [yJ~(p'-- k ' )F(p' , k')(v + ' - - �9 o%)~o(p ) ~p+(p')F(p', k')('~.o'k,)~p(p'--k')],

with F(p ' , k'), given by :

(4) F(p', k') = (a. k ' ) - - - - f ~" ~ + Ep, a, ]]'

where the second term is referred to as (~ galileian ~) and the first is the only one

retained in the fixed source theory.

(*) We have omitted the exponential dependence from the Hamiltonian (3), since in the application of the Tamm-Dancoff formalism these terms disappear.

(3) p. BUDINI: NUOVO Cimento, 3, 835 (1956).

NUCLEON RECOIL IN THE P I O N - N U C L E O N SCATTERING 1401

3 . - P i o n - N u c l e o n S c a t t e r i n g i n t h e P - S t a t e .

We have to resolve the Schr6dinger equat ion for scattering processes of pions on nucleons at low energy:

(5) ( H - - E ) I P ) = 0 ,

where E, eigenvalue of the total Hamiltonian, is calculated on energy shell:

(6) E = Eko ~- coo �9

We develop now the state vector I P~ in eigenstates for the unper turbed I-Iamiltonian H o and retain the terms in one nucleon, neglecting those in three or more mesons. With the usual procedure (4) we obtain, in the center-of-mass

system, the following equation for the scattering ampli tude aq(k)~ae(--k, k):

]2 /" d3k , , . . . . . . (7) (Ek § co § S -- E)%(k) -- 4~/a~j ~ vt~)v(~ ).

[ k)F<o, k') F(- l �9 Tq~a M - - E - + ~ E k + k , + c o + c o - - E l '

where the term S arises as a result of a self-energy process and has the form:

al: F(-- k, k')F(-- k, k') (8) S = 4u~#2J o) v2(k') E,+k, + o) + co' -- E"

We drop such a te rm now; it will be taken properly into account in the re- normalizat ion of the equation.

The separation of the equation (7) for eigenstates of isotopic spin can be done directly; one obtains:

i (E~ + (o -- E)agkl -- 4~,l~,j V ~ 7 0(k)O(k').

[3F(O, k)F(O, k') F(-- k, k')f(-- k', k)l . . . .

(9) I "[-----M 7 - ~ + k - = + ~ - ~ 7 - - - ~ ] a � 8 9

(*) See for instance: F. J. D~'SON, M. Ross, E. E. SALmSTEI% 8. S. SCUWEBER. M. K. SUNDARESAN, W. M. VISSCttER and H. A. BETrIE: Phys. Rev., 95, 1644 (1954).

1402 L. FOND-A- and I. REINA

We shall consider now only the second of these equations and resolve it for P-s ta te and for the value j = ~ of the total angular momentum.

Inserting the explicit expressions of F ( - - k , k') and F ( - - k ' , k), the second of the equations (9) becomes:

r

. . . . i:__ ;. (10) (Ek + ~0 -- E)a~(k) 2~2#2j ~/o)o~' k'r

P

with

z,( o I,o') --fdO , +" k')(a, k) a~(k') Ek+k, + C

4g

Ek+k' 47t f (o + o/ (o" k')(~" k) a~(k')

(]1) I3(o)[r : d~k, 2E-~k+-~' Ek+k, + C ' 4~

(r k')] f : oJ~o' [(a" k) (r k) + I a ( r Ek .+Ek+k - - - - 7 Ek+k'l

4rt

.I(~.k') (~.k') (~.k)] ~+(k') [ Ek, + ~ + k , § E~+k-----TJE~+~----7-~C '

and

(12) C ---- ~o + ~ o ' - - E .

(13)

We develop [Ek+ k, -~ C] -1 and [Ek+k,] -1 in Legendre po lynomia l series (~):

[ [Ek+k, ~- C ] - 1 ~ Zn(OJ[(Df)-lDn(Ok') / [ [Ek+k,] -~ = ~ X,~(co[co')P,,(Ok,),

n=0

Z, and X, are given by :

(14) I 1-r x.(o~Io/) = z~

(~) See reference (4).

N U C L E O N R E C O I L I N T H E P I O N - N U C L E O N S C A T T E R I N G 1403

with Ek+k, + Ek-k' kk' C

- 2 , r = ~ - - ~ , c =~.

Let us determine now the first two coefficients and find approximate formulas~ more suitable in calculations, for them:

Zo(coico') _~ [E~+k, § ]-~ 2 + co § co ' - -E ,

z,(colco ' ) "-' 2kk ' [Ek+~. + F~_~, 1-~ - E~+~7 T:Ek_,<, [ 2 + co + co'- E ,

(15) Xo(co]co') "-~I E~'+~' + E~'-~"] -1

- - . 2

2 ~

all the others being negligible. We can evaluate now the integrals 11, Is, I3 and 14 at the right hand side

of the equation (10). Iqeglecting terms of order higher than k3Zo(#/M), for the P-s ta te , writing a(IkI, 0 ) = a(co), we obtain:

co

4] 2 f . , v(k)v(k')kco'k '~ (16) (co - - coo + E~ - E~;)a~3(w) =- 3~--~ 2 j ctco 7

._ . " % / C O C O <

# z

�9 co + o ' - coo + ~+k, +2 Ek_~, Ek~ 1 + Dk~, ~- N~_~,) ~(co ) '

This equation differs from the corresponding one in Chew's theory, both at the L.H.S. and at the R.H.S. In these members we have in fact respectively

Ek+k. § Ek-k. the terms Ek---E~ o and 2 --Eko which are not included in the

Chew's equation; moreover at the I~.I-I.S. appears the factor 1 § E k ~ + E~-k' due to the galfleian term.

The equation (16) can be renormalized by using the method proposed by C ~ w (6); ia the considered approximation we write

(17) (~o - - coo + E~ -- E~o)[1 +/[~(Oo - - CO)]a33(CO) = r

- 3 ~ co' V ~ [ 1 + A~(coo-- co - - co')]

g ~ �9 co + c o , co. + Ek+k' + Ek-k' Eko 1 § Eke' + E.~-.v 2 ~(co ) '

where ] is he rea f t e r t he r e n o r m a l i z e d coupl ing cons tan t .

(6) G. F. CHEW: Phys. Rev., 94, 1748, 1755 (1954).

1404 z. FONDA and I. RI~INA

A, has the following form:

co

= 3]a f~ xk ~a (18) d, (x) ,tt~o"v2(k ') .

= ~ ' J (co' -- x) o, '~ /*

The integral equat ion for the ampl i tude aa3 is thus :

K(w[o~') (19) a33((o) = 6(Ek q- co - - E) q- td(o'

t*

where the kernel K(a>lco' ) is given by:

a 3 3 ( ( , ) ' ) ,

( 2 0 ) K ( ~ Io , ' ) = - - 412 v(k)v(k ' )ko fk '2

37I# 2 ~ / ~ [ 1 -k d , ( ( ~ 0 - - o~ - - ~ ' ) ] [ 1 q- A , ( o 0 - - co)]

[Ek+~, § E~_~, ]-~( ~o +~o ' �9 2 + oJ 4- o>'-- E 1 5- Ek+k ' + Ek-k'] "

In order to solve the integral equat ion (19), one takes its F redho lm solutio~ and subst i tutes it buck in (19) (7), then, in the usual way, the ~33 phase shift, turns out to be

(1 (21) tgS~a 413 ~ §

-- 3# ~ k~ [ E2~. + M ~Oo § 2

with Ao((Oo) and zl~(Wo) given by

2~Oo ) v2(ko) E,k~ + M

/~.)[1 +A~(-- coo)]

I + do(~Oo)--Ll,(O,o) 1 - - ed, (o~o)

oa [ oo§ 2o~ (22) do(wo)---- 4~7 ~ fdco' 1 -+- 1 Jr k'3v2(k') �9

3~tt J E~.+k, + E~.-k'] E~k~ + M P

( c~176 E2~o q-2 M E,o)[1 § At(-- O)o)]

( ; o)'q- Ek.+k' § E~~ _ Ek~ [1 § d , ( - - co')]:(o/q- E~, - - E)[1 Jr zJ,(~oo -- o/)] 2

r

(22') d,(eOo) - , din' 1 Jr E,k, q- M

t*

1 "( 2co'2(ooq-(( E2k, + M)/2) - -Ek . ) [ l +Ar(a)o--2a)') ](m'--moq-E~;--Ekl)[ l q- d , (o)o--o/ )] "

(7) j . L. GAMMEL: Phys. Rev., 95, 209 (1954); P. BUDINI: NUOVO Cimento, 3, 1104 (1956).

N U C L E O N R E C O I L I N T I l E P I O N - N U C L E O N S C A T T E R I N G 1 4 0 5

T a b l e I shows t h e r e su l t s c a l c u l a t e d choos ing for v(k) t h e cut-off f u n c t i o n

of Chew, i.e. i t s v a l u e is one for 1 ~ oJ ~ 5.6, zero for o~ > 5.6.

TABLE I. -- ($33 phase shi]t (o~ . . . . = 5.6).

o) j

1.3 1.6 1.9 2.0

EI~ b (MeV)

57 122 190 215

633 (Exp)

7 . 6 ~

32 ~ 84 ~

105 ~

= ]ch~w = 0.058

fixed source (~33

8 . 0 ~

39.4 ~ 85.8 ~ 94.6 ~

with recoil

(~3a

5 . 0 ~

18.7 ~ 41.6 ~ 50.8 ~

] 2 = 0.0632

with recoil ($aa

9 . 2 ~

42.9 ~ 84.6 ~ 93.8 ~

The a b o v e Tab le I shows u n d e r co lumn f o u r t h t h e c a l c u l a t e d va lues of b3~

p h a s e - s h i f t f r om Chew's t h e o r y a n d u n d e r c o lumns f i f th a n d s ix th t h e r e su l t s

w h e n nuc l eon recoi l is t a k e n in to accoun t , for t h e two va lues of t h e r enor -

m a l i z e d coup l ing c o n s t a n t , /~ ----/ch, w 2 a n d 1 2 = 1.09]c*ho . W e o b t a i n good a g r e e m e n t w i th e x p e r i m e n t a l d a t a a n d w i th Chew's r e s u l t s

w i th t h e second v a l u e of t h e r e n o r m a l i z e d coup l ing c o n s t a n t (12= 0.0632).

W e obse rve t h a t (~a3-phase sh i f t is c o n s i d e r a b l y a f fec ted b y nuc l eon reco i l

e spec ia l ly a t h igh ene rgy . The inc rease of t h e r e n o r m a l i z e d coup l ing c o n s t a n t

in o r d e r to fit t h e e x p e r i m e n t a l d a t a , is, however , r a t h e r smal l , lower t h a n

1/10 (*) in r e spec t of Chew's choice.

4. - P i o n - N u c l e o n S c a t t e r i n g in t h e S - S t a t e .

I n t r o d u c i n g two t e r m s b i l i nea r in t h e meson field in t h e p rev ious H a m i l -

t o n i a n a n d neg l ec t i ng n u c l e o n recoi l , DRELL, FRIEDMAN a n d ZACHARIASEN (s)

ob t a in , w i t h t h e Chew-Low m e t h o d , good a g r e e m e n t w i t h e x p e r i m e n t a l d a t a on

S - w a v e s c a t t e r i n g of p ions on nucleons , as fa r as t h e i n c i d e n t m e s o n ene rgy

is less t h a n ~Oo = 1.25. W e now shal l a p p l y t h e T a m m - D a n c o f f f o r m a l i s m in o r d e r to ach ieve

r e su l t s s imi la r to t hose shown b y DRELL et al. (s) when nuc l eon recoi l is ne -

g lec ted . The co r r ec t i on due to th is effect is t h e n e v a l u a t e d .

= ]che~ = 0.058, we must increase the cut-off (*) We point out t ha t keeping ]2 2 energy to about ~Omax. = 6, in order to obtain results comparable with CHEW'S data .

(s) Work r eported by W~.ISSKOPF at the Rochester Conference (1956) ; see also : S. FU- BINI: .Nuovo Cimento, 5, 564 (1953); N. FUKUDA, S. GOTO, S. OKUBO and K. SAWADA: Prog. Theor. Phys., 12, 79 (1954); J. F. DYSON, M. Ross, E. E. SALP~.TER, S. S. SCHWE- BER, ~VL K. S U N D A R E S A N , W. M. VISSCHER and H. A. BETH]~: Phys. Rev., 95, 1644 1954); M. M. L~vY and R. E. MARSHAK: NUOVO Cimento., 4, 366 (1954).

1406 L. FONDA and I. REINA

We choose therefore for our Hamiltoniun the following one:

(23) / / - -Ho +H,~ +//1 +H~,

where H ~ is the usual PS-PV coupling and the form of //1 + H , is suggested from a Dyson-Foldy transformation applied to the PS-PS interaction Hamfl- tonian (9). In order to obtain a form factor in the PS-PV interaction Hamfl- tonian, we start from a correspondingly non-local PS-PS coupling, this gives for //1 and H~:

( ~ )

//1 = 20 I I Id4xt d~x2 d4x3/(x~ - - x 2 ) / ( x 1 - - x3 ) (~(t - - t l ) (~(t - - - t2) ( ] ( t - - t3)" d J J

. ~(x~)(~(x~).~(x~)) ~(~),

H-2 ~-= 2 I I I daXl d4x2 d4x3/(x~ - x2)](x~-- x3) 6 ( t - - t~) (~(t-- t~) ~ ( t - - t3)" J d d

�9 ~)+(X 1 )~'" ( ~ (X2) ~ I I ( ~ 3 ) ) ~0(Xl ) -

The coefficients 20 and 2 are given by :

(25)

G: 4zp 20 = l o ~ = l o ' 2 M #'--7-'

I (~ where l0 and l are positive parameters whose magnitude is left now open.

We develop the field operators in Fourier series, using eq. (2) we obtain (*).

(26)

t l _ m 2o ;ff?3pd3 ,d3kd k' (k)vIk,I 2 ( 2 = ) U J J J Vo~o' YJ+(P)fl~P(P')"

�9 [ ~ ( k ) ~ , ( k ' ) ~ ( p ' - - p + k + k') + ~ $ ( k ) ~ ~ ( k ' ) 5 ( p ' - - p - k - k') +

+ ~ ( # ) ~ ( k ' ) ~(p'-- p - - k + k') + ~ ( k ) ~ ( k ' ) ( 5 ( p ' - - p + k - - k')] ,

H2 -- 2T2~ -3 ~r ~o+(P)TlY'(P')" (t2a)J J J J

�9 { ( ~ o ' - . , ) [ ~ ( k ) ~ ( k ' ) ~ ( p ' - - p + k + # ' ) - - ~+ (k)~+(k ') ~ ( p ' - - p - - k - - k ' ) ] +

+ ( ~ ' + (o)[~;(k)~3(k')(~(p'--P+ k ' - - k ) - ~+(k')~2(k)(~(p'--p+k - - k')]}.

(9) F. J. DYSON: Phys. Rev., 73, 929 (1948) ; L. L. FOLDu Phys. Rev., 84, 168 (1951); S. D. DRELL and E. ~ . HENLEY: Phys. Rev., 88, 1053 (1952).

l ~ i s referred to cyclic permutation of the suffixes 1, 2, 3; (*) The summation C )

here too we have omitted the exponential time dependence.

N U C L E O N R E C O I L I N T I I E P I O N - N U C L E O N S C A T T E R I N G 1407

Let us resolve now the Schr6dinger equation (5) with the new Hamil- toni~m (23), in the second Tamm-Dancoff approximation, taking into account, however, only terms of order /2/#~. We obtain, in the center-of-mass system; the following equation for the amplitude a ( k ) :

(27) ,~o f d 3k' (Ek 4- ~ - - E)ae(k) -- (2~)~j ~ / ~ v(k)v(k')aq(k') - -

i~ t d3k ' 2(2~)3 j ~ / ~ , v(l~)v(k')(o~ + ~o')'vAct(k') +

]~ ; d~k'- v(k)v(k') ~ ( 0 ' k )T(O, k') ve~:aa~(k, ) 4- 4~2#2 j x / ~ M - - E

]2 ; d3k' 4- 4~2#2 j ,~/o~w ~ v(k)v(k')

4-

T ( - k, k')T (-- k', k) Ek+h, 4- ~o + ~ ' - - E ~Qa~(k ' ) ,

where we have neglected all self-energy terms. The equation (27) can be easily separated for states of definite isotopic

spin and angular momentum, for the S-state we obtain (*)

(o) - - ~Oo 4- E~ - - El,.o)al(e)) =

f = ~ / d w ' [K/(o)](o') + Kf,((~)l~o') + Kf,((,)leo')]a~(co'),

# j

(2s) ((o - - O~o + El~ - - E~o)a3(~o) =

ca

-= d~'[K~((o !~o') + Kf~(o~ le~' ) 4- K~,(eol~,')]a~(o'), / z j

where the kernels with superscript r represent the effect of nucleon recoil and their form is

(2,~)

- - k ' ~ o ' K[(wtw') - , ~ / ~ v(k)v(k')go(o)]~o').

. I[o~k'2 w'k 2 ]

+ E-;J K;(~o I(,f) = -- 2K/(o~ ](o'),

+ Xo((O ] ~') e)k '2 + ~o'k*~

2 J '

as can be easily obtained using eqs. (13) and (15) and neglecting terms of order higher than k~Zo(#/M).

(*) In this approach tile IIamiltonians (24) do not give any contribution to the scattering in the P-state.

9 0 - l l N u o v o C i m e n t o .

1 4 0 8 L. F O N D A a n d L R E I N A

The kernels signed with superscripts / /1 and H2 are due to the Hamiltonians //1 and H~ respectively; their form is:

2U~o' K ~ ( ( o l ( o ' ) = K~,(~o](o') -- _ _ v ( k ) v ( k ' ) ' 2 M l o ,

7~\/ o)0,)'

(30) KT,((o[~o') -- 2 k ' w ' 7~/o)o~ v ( k ) v ( k ' ) . (o9 + co')l ,

K~,((~ ] ~o') = - - �89 [(f,(o~ [ o)') .

We see from (29) and (30) tha t nucleon recoil will give a contr ibut ion to the

S-phases, opposite in sign to the experimental values, while the effect of the H~mfltonian H2 will have the same sign as the experimental phases. The l : [ami l tonian/ / i does not depend on isotopic spin, consequently its contr ibut ion to the S-phases is quite the same in both cases, i.e. for T = 1 and T - ~ .

We resolve the integral equation first in Born approximat ion (see Table I I )

T A B L E I I . - 61 and 6 a phase shi]ts (]2__ 0 . 0 8 2 , / 0 = 0 . 0 2 8 8 , 1 : 0 . 3 8 8 2 ) .

Born approximation

(DO

1 . 3

1 . 6

1 . 9 2 1

w i t h o u t w i t h

r e c o i l r e c o i l

61

4 . 1 ~ 3 . 4 ~

8 . 0 ~ 6 . 1 ~

1 3 . 0 ~ 9 . 0 ~



_ _ 5 . 8 ~

9 . 5 ~

__ 1 3 . 3 ~

63

__ 4 . 5 ~

__ 5 . 5 ~

__ 5 . 4 ~

D r e l l ' s v a l u e s (*)

61 ~a

7 .1 ~ __ 6 . 0 ~

1 2 . 2 ~ - - 1 0 . 2 ~

1 8 . 8 ~ 1 5 . 0 ~

(*) T h e s e v a l u e s h a v e b e e n t a k e n f r o m t h e d i a g r a m of WE]SSKOPF'S r e p o r t a t t h e I '~ochester C o n f e r e n c e (1956).

and we obtain, when nucleon recoil is left out, results fairly similar to those given by D R E L L , FRIED)/IAN and Z A C H A R I A S E N , choosing for lo and 1 the same values as in Drell 's paper and with the value of tile coupling constant ]2 =__ 0.082. Table I I I shows the phase shifts calculated more carefully (First

Fredholm approximation).

T A B L E I I I . - 61 and 63 phase shi]ts (]2_ 0 . 0 8 2 , 10 : 0 . 0 2 8 8 , 1 = 0 . 3 8 8 2 , (D . . . . : 5 . 6 ) .

First Fredholm approximation.




r e c o i l r e c o i l (D o

61 63

1 0 . 6 ~

2 3 . 4 ~

3 9 . 5 ~

6 . 0 ~

1 1 . 0 ~

1 6 . 6 ~

1 .3

1 . 6

1 . 9 2 1

_ _ 3 . 9 ~

_ _ 6 . 3 ~

__ 8 . 8 ~

_ _ 4 . 1 ~

_ _ 5 . 2 ~

__ 5 . 3 ~

D r e l l ' s v a l u e s (*)

7 . 1 ~ - - 6 . 0 ~

1 2 . 2 ~ __ 1 0 . 2 ~

1 8 . 8 ~ - - 1 5 . 0 :

(*) T h e s e v a l u e s h a v e b e e n t a k e n f r o m t h e d i a g r a m of WEISSKOPF'S r e p o r t a t t h e R o c h e s t e r Confe rence (1956).

N U C L E O N R E C O I L IN T H E P I O N - N U C L E O N S C A T T E R I N G 1409

We observe how the effect of nucleon recoil is rather large (~o) and how

it increases with increasing energy. The contribution from nucleon recoil is

increasing faster (with k 3) than the one due to the Hamiltoni~n //1 + H2

(wi~h k); furthermore it has opposite sign so that , at high energy, the S-phase

shifts could reverse their signs. In our case, with the chosen values for the

parameters lo and 1 the change in sign is obtained at too high ener~:y. Pro-

bably, with a proper choice of the parameters lo and l the experimental be-

haviour of the S-phase shifts could be better approached, but this would not,

make much sense if terms of order higher than ]~/tt 2 and renorm~dization

effects are not properly taken into account.

We believe however, tha t any theory which claims to reproduce ,~-wave

scattaring, must look upon the effect of nucleon recoil if it wants to expb~in

some remarkable experimental feature.

I t is a great pleasure to thank Professor P. BUDINI for having suggested

this work and for his continuous advice and encouragement.

(10) Set E. M. }{ENLEY and M. A. RUDERMAN: Phys. Rev., 90, 719 (1953).

R I A S S U N T O (*)

Nell'~stensione della teoria delle sorgenti dello scattering pione-nucleone con hamiltoniana d'interazione PS-PV si 8 tenuto conto dell'effetto del rinculo dei nucleoni. Abbiamo eonfronta~o i risultati sulle onde P coi dati sperimentali e eoi risultati del taglio di Chew e discusso l'influenza del ri:~enlo dei nucieoni su]la determinazione della co- stal~te d'accoppiamento rinormalizz,~ta. Per discu[ere suIle stesse basi teoriche l'influenza del rinculo dei nucleoni sulle fasi S, abbiamo aggiunto alla hamiltoniana d'interazione PS-PV due termini, bilineari nel campo mesonico, quali si possono ottenere da lma trasformazione di Dyson-Foldy applicata all'accol)piamento PS-PS. L'effetto del rinculo dei nucleoni risulta notevo]e.

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Nucleon Recoil in the Pion-Nucleon Scattering