IL NUOVO CIMENTO VoL. V, N. 1 1 ° Gcnnaio 1957

Pion-Nucleon Interaction and Strange Particles.

P. BUDINI and L. FONDA

I s t i t u to di F i s i c a dell' Un iver s i th - Tr ies te I s t i t u to N a z i o n a l e di .Fisica N u c l e are - Gruppo di Tr ies te

(ricevuto il 26 Novembre 1956)

In a previous paper (1) we discussed the possibility of ascribing the non loca- lizability of the pion-nucleon interaction to the existence of an intermediate field. In particular a simple model was presented and applied to the problem of pion- nucleon scattering obtaining results equivalent to the well known ones of the extended source theory (*). The model had only a mathematical character and the quanta of the internal field had no connection with reality. The purpose of the present note is to propose a more realistic theory in which the internal quanta can be iden- tified with existing particles. At low energy pion-nucleon interactions these part- icles must be present only in virtual states, thus their rest mass must be higher than the one of the pion. At high energy pion-nuclcon collisions they must be apt to be created in real states. We know that K-mesons can be created in such collisions, thus the simplest hypothesis is to take them as quanta of the internal field.

As it is now well known, experimental evidence concerning all phenomena connected with K-mesons indicate that these particles are isospinors (~), while the most reasonable value for their spin seems to be zero. So if we wish an interaction lagrangian linear in the pion field u, it has to be at least quadratic in the K field. Owing to the fact that the pion is pseudoscalar, it is necessary to postulate different parities for the two K wave functions in the interaction term. This possibility seems not to be excluded by reality, as it appears now probable that both parities, + and --, are to be connected with K-mesons (~). Therefore, according to these ideas, the simplest invariant interaction lagrangian between pion and K field is:

(1) La~ , = - - g~EgK~*'cKp, + ~ K * ' c K . ]

where K. and KD. represent two isospinor fields of the first kind (4), scalar and pseudo-

(') P. BUDINI: N'ttovo Cimento, 3, 1104 (1956). (*) I t can be shown t h a t such a model is in fact equiva len t wi th some restrietinnR to & non-

local theory of the Pais-Uhlenbeck type (A. PAre and G. E. UHLENBECK: Phys. Rev., 79, 145 (19ti0D. (") /~I. GELL-MANN: Phys. Rev., 92, 833 (1953) (s) T. D. LEE and C. N. YANG: Phys. Rev., 102, 290 (1956). ~a) B. D'ESP~GNAT and J. Pm~NTKI: Nuclear Physics, | , 33 (1956).

PION-NUCLEON INTERACTION AND STRANGE PARTICLES 30~

scalar respectively. These fields in tu rn should in terac t direct ly wi th the nucleon field and, for the corresponding lagrangian, we choose:

(2)

(3)

L~ ~' = q.[,7~,AK. + R*~A~'~]

where N refers to nucleons and A to isoscalar hyperons (*). The to ta l in teract ion lagrangian, sum of these, is obviously invar ian t with respect to the Lorentz group, rotat ions and reflections in the isobaric space.

The field equat ions of t he K and ~ fields, supposing equal the masses of the K-particles, are the following:

(4) ( D - - m ~ ) K p . - - igr.,)A75,1; ~ + g=rcTK.

(l~ - . m ~ ) ~ = g=K*~ r K p . + g r : K ~ ' v K , .

The system can be easily quant ized with the usual methods. From these equat ions one can el iminate the K fields obtaining an integral

equat ion for the ~: field. This equat ion is ra ther complicated and non linear. If we consider only weak in teract iug fields, we are justified in e l iminat ing the non linear terms from these equat ions , which is equ iva len t to neglect ing the react ion of the 7: field on the K fields and of these on t im A-N field. Considering besides the ~.-fieid only the nucleon field present in the l imit t ~+ ~: ~-, we obtain:

(5) ([-3 m ~ ) ~ ~gr:qsqt, 1 x d 4 x " . A t z ( x - - x ') e ,, - . . . . . . . . - - = h~(x - x ) ~ ( x )[75,,~A(x x ) ] + ~ ( x )

where A~ and ~ are thc causal Green funct ions of the K and A fields. This equat ion can be thought of as derived from the lagrangian densi ty:

(6)

with

L ( x ) = L ° i g ~ q . q v . J J d a x ' d % '~ F(x, x ' , - - , X h') ~/',~(Xl)g(X) ,

F ( x , x ' , x") = - - 2MAA~(x x ' ) A ~ ( x " - x)A~A(X ' X") .

Thus the l inear approx imat ion of this theory is a non-local covar iant form factor theory (5).

Taking this lagrangian as s ta r t ing point, the self-energy of the nucleon, the nucleon vacuum polarizat ion as well as the o ther pr imi t ive divergent S-mat r ix

(*) Toghe te r w i t h an isoscalar hyperon , one can t a k e an i sovector and an isospinor one (Z, ~), the results , a p a r t f r o m numer ica l factors , are the s a m e ; one ob ta in thus a connect ion be tween the K - h y p e r o n and pion-nucleon in te rac t ions and a dependence of the corresponding coupl ing cons tants .

(1) p. KRISTENSEN a a d C. MOLLE~: Dan. Ma t . Fys . Medd. , 27, no. 7 (1952).

308 P. B U D I N I and L. FONDA

e lements of t he local theory , can b e c a l c u l a t e d w i t h t he Y a n g - F e l d m a n m e t h o d (6) and resu l t finite.

This model could be t e s t ed on t he ca lcu la t ion of known e x p e r i m e n t a l d a t a as those on p ion-nucleon a n d K-nuc leon sca t te r ing , magne t i c m o m e n t a n d charge d i s t r ibu t ion of t he nucleon, etc. I t implies in p a r t i c u l a r an inf luence of t h e pre- sence of K-mesons in these processes, and should p red ic t t he K - p r o d u c t i o n in h igh energy p lea -nuc leon collisions.

In order to compare t he s ta t ic a p p r o x i m a t i o n of th i s t h e o r y w i th t he pseudo- vec tor ex t ended source theory , one has to s u b s t i t u t e 75 wi th YsY~ a n d K~. w i th V~I(p~ in Z~ a) and to inse r t ~$-functions for t h e sources of t he K-fields ( this of course will in- t roduce divergences in to t he theory) . W e ob ta in t he following equa t i on for t he r: field:

1 (7) (V 2 --~n~)~ i = - - (].nq.%.ri~3 • V ) - - exp [ - - 2 m K r ~

• 16,her ~ ,

which can be der ived f rom an e x t e n d e d source p scudovec to r lagrangia l l w i th .Fourier t r ans fo rm of t he fo rm-fac to r :

2m K k (8) v(lc) .... k arc tg 2m-----K "

This form-fac tor has a knee a t k ~ 2m K = 1.05M~ which is of t he order of m a g n i t u d e of the cut-offs used in the p ion-nuc leon sca t t e r ing theor ies (7). I t is to be expec ted t h a t t he r -1 s ingu la r i ty of t he K fields, which appears a t t he R .H.S . of (7), will be s m o o t h e d down in a dynamica l theory , a n y w a y t h e knee of the form fac to r (8), which will p r e sen t a more rap id convergence for k--> ~ , will r ema in a t ,~ 2m K.

Some po in t s of th is model will be fu r the r anal ized in a f o r t h c o m i n g paper .

W e t h a n k Prof. N. DA•LAeOaTA for an in te res t ing disoussion on t h i s sub j ec t .

(6) C. N. Y/t~G aad D. FELDMAN: Phys. Rev., 79, 972 (1950). (~) G. C. WICK: Rev. Mod, Phys., 27, 339 (1955).