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IL NUOVO CIMENTO VOL. u X. 3 1 o Settembre 1957
Strange-Particle Effects in S-Wave Pion-Nucleon Scattering.
J. S. LA~aER
Department o/ Mathematical Physics - University of Birmingham
(ricevuto il 10 Giuvno 1957)
Summary. --- An attempt is made to explain the isotopic-spin splitting of the S-wave phase shifgs in pion-nucleon scattering in terms of virtual strange-particle effects. A very simple calculation is found to agree quite well with experiment.
Although pseudoscalar meson theory has achieved considerable success in reproducing the large P-wave phase shifts in meson-nucleon scattering, it has so far not been able to explain certain featm'es of the S-wave data. In part i- cular, the observed splitting of the two isotopic-spin states is not evident in any simple form of the present theory. I t has been suggested tha t this dis- crepancy ma y be explained in terms of virtuM strange-particle effects. The purpose of this paper is to investigate the possibility of including these effects in a scattering formalism, and to present a ten ta t ive ~nd very simple solution
which shows surprisingly good agreement with experiment. The familiar form of the S-wave interact ion resulting from a Foldy-Dyson
t ransformat ion of the pseudoscalar Hamil tonian includes the terms )~0~ 2 ~. + ) ~ . ~ • where ~o= g~/2m, ~ =g/2m and g is the pseudoscMar coupling
constant. Although the ~ te rm gives far too great a scattering in Born ap- proximation, it is well unders tood tha t it actually corresponds to a strong bu t very short-ranged effective potential , and tha t a more exact t r ea tmen t yields phase shifts of the correct order of magnitude. Nevertheless, a t e rm of this size will still dominate the low-energy scattering and can never give the correct isotopic spin dependence. Unless the scattering ma t r ix passes through zero or infinity at a total energy less than the meson rest mass, causing one of the phase shifts to change sign, it would seem that , for some reason, the ~ te rm plays almost no par t in the in te rac t ion . DI~ELL, FRIEDI~IAN and
STRANGE-PARTICLE EFFECTS IN ~-WAVE PION-NUCLEON SCATTERING 675
ZACt~AnlASn~ (~) have used the above interaction in a Chew-Low type of cal- culation, treating the ~'s us completely adjustable parameters. They find that they can fit the data quite well by choosing the (renormalized) 2(~) to be only ~ /2m times los originul value relative to ,~. The hypothesis of this paper is thut this reduction may be at least partly due to strange particle effects.
The interaction scheme that will be used here corresponds to the Lagrangian density :
[ -~ - - .~o - ~ ' , ~ ' o ' ~ v ~ - ~ + ~ g ~ - ~ x , . l . , , ~ . ~ -
[ - g ~ ( ~ r + @ ~ K ~ ' ~ : ~ ) ,
where ~o includes the usual expressions for the free fields. I t has been assumed that the ~-meson has no direct interaction with the K, that there is only one K-meson which has spin zero and negative parity relative to the E, that the E has spin �89 and that A and 7~ particles may be neglected . The vc-E inter- action has been chosen in complete analogy to the ~-~T interaction with g~/4-: = g~/~,": ~ 15. Throughout this paper, g3, the coupling constant cor- responding to production of strange particles, will be assumed to be consi- derably smaller than gl or g2. Except for the neglect of A and ~ particles, which is in no way essential to the argument, all of the above assumptions are in accord with the scheme suggested by G~,LT,-MA~ and SCKWIN~ER at the 1957 Rochester Conference. Using the technique devised by SCI~WlNGE~ (~)
one can derive a formidable set of covariant, coupled, integral equations which relate the various scatterings that can take place between the four particles. An outline of this derivation is given in the Appendix. The first of these equations may be written symbolically in the form:
(2)
+ v ~ ( ~ I~:) ~ (=) s (~)~ ' (~= I ~ ) + 1 ~-'(~,= [~;K) A (K) S(X)T~(X~: I ~ : ) +
+ V~(Nr: ] EK)A (K) S(E)T~(EK I ~7:) + ....
Here, the T's are the usual covariant scattering matrices describing transitions
between the sets of particles indicated inside the parentheses to the right and
left of the vertical line. 2 and S are the Boson and Fermion propagators
respectively. The superscripts are the Boson isotopic-spin indices. Several terms involving expressions of the form 3T/~(q~} have not been written out explicitly, and are indicated by + .... This equation, along with the definition
i
(1) S, D, DRELL, M. H . FRIEDMAN a n d F. ZACHAR1ASElq: Phys. Rev., 104, 236 (1956). (2) j . S. SenWlNG~:t~: Proc. Am. Acad. Sci., 37, 452. (1951).
676 J .s . LANGER
Of the V's~ is written in terms of Feynman diagrams in Fig. 1. equation will be used to provide an expression for T(EKI~T~ l .
(3)
The next
+ Vk~(Xx I Xx) A (7:) S(E)T~k(EK I Nx) + V~(EK I Nx) A (K) S(~T)T~(~K INK) +
+ V~a(EK I EK) S(E) ~ (K)T~(EK I N~) + ....
The third equation for T(~K I~K) will not be needed.
-- + ~ +
N N N N 4 N
' ~ a . +
Fig. 1.
I t is hardly necessary to say that is it impracticable to solve these equa- tions as they stand. One might first note that the last two terms in (3) are smaller than the preeeeding ones by a factor (g~/gl) ~, and may perhaps be neglected. There remains an unrenormalized integral equation for T ( E K I ~ ). A no-recoil approximation is out of the question because the kernel corresponds to intermediate states with two 7:'s and a K in the field at the same time. In the following ealeulation~ T(EK 15%~) will be approximated by the remaining
STRANGE-PARTICLE ]EFFECTS IN ~'-WAVE PION-NUCLEOIq SCATTERING 677
inhomogeneous terms in (3); t ha t is:
(3') ~ ( E K ] ~ ) ~_ V ~ r ( Z K l ~ ) + W ~ ( E K ] ~ ) A ( ~ ) S ( N ) T ~ ' ( ~ I ~ ) .
I t is not at all clear to what extent this approximation is valid. If the react ion 7 : - k l q - > K + E goes through an effective ~[-funetion potent ia l in the S-wave as does ~ +lq -+~ -klq, then the R.H.S. of (3') is probably too large. On the other hand, due to the assumed small value of g.~ relative to g~ and the large mass of the Y-meson, the two processes may not be str ict ly comparable. In any case, (3') can be expected to show the correct sign and isotopic spii~ behaviour.
On substi tuting (3') into (2), one arrives at the equation:
(4)
where
(5)
+ v ' ~ ( ~ z 12K 'A (K) S ( 2 ) W ' ( ~ K ] ~T=).
((I~ote tha t it is necessary to include some of the ~T / ~(~} terms in order to derive (4) exactly). U is just the lowest order Born approximation to pion- nucleon scattering plus its first radiat ive corrections due to heavy mesons only.
~ N = 4- 4-
~. N N N
~- terms of oraer ( g3/g 1 )2
Fig. 2.
Since only pions occur in the intermediate states in equation (4), a no-recoil
approximat ion for the nucleon should be reasonably accurate. Separating out
the S-wave part , and assuming tha t the integrat ion over meson m o m e n t a
cuts off at some value A of the order of the nucleon mass, m, one gets t h e simple algebraic relationship:
(6)
A
fT(co) = uT(co) + U?(o4 ~ - - ~o ~ + / ~ - - ie '-
~- crossed terms, o g ~ m ~
.678 a . s . LANGER
where ~o is the to ta l energy of the meson. U~'~(w) has been eva lua ted f rom (5)
b y covar iant in tegrat ion and then, i.e. af ter renormalizat ion, expanded in powers of ~o. Renormal iza t ion of ve r t ex par t s has been per formed in such a
w a y tha t radia t ive corrections vanish a t the unphysical point where the three par t ic les satisfy ip § = iO' + m = 0, (p - - p')~ +#5 = 0. This is consistent
wi th the definition of the P -wave coupling cons tant given b y CuEw and L o w (~) The result of this calculation is:
~7) u - ( ~ ) - m(2~)~\ ~J~.~ :t + G,~(2~)~ ~ - I - 4,~- / �9
t~ W~' - , \
= + N ~" N
N
"~- -F- ~. -~- . " ~ ,
+ crossed terms
Fig. 3.
'The numerical factors come f rom F e y n m a n integrals involving masses of N, E, K , 7:. For the choice g~/4= = 1.8, the t e rm propor t iona l to d~.~ vanishes. Th is cancellat ion is a direct result of the assumpt ion t ha t the K-meson has negat ive pa r i t y relat ive to the E particle. I f the opposite pa r i t y is assumed,
the cancellation does not occur. The remaining t e rm is small and strongly isotopic-spin dependent . I f i t is inserted into (6), i t is obvious t ha t T = U to
a good approximat ion . The resul t ing expression for the phase shift,
g~e~knT k [ -~2 for T = �89 (8) ~. = 2(2~)~m~ - - 0.05 - n r , n~ = - - /~ ~ - - 1 for T - - ~ ,
shows the correct isotopic-spin spli t t ing and is actual ly somewhat too small.
The essence of the above technique is the construct ion os an effective pion-
nucleon potent ia l U~(r which m a y be used in the Schr6dinger equat ion (6). I t seems fairly clear t h a t a potent ia l of the form derived here does produce
the correct S-wave scat ter ing a t low energies. There remains the question of
(3) G. F. C~Ew and F. E. Low: Phys. Re.v, 101, 1570 (1956).
S T R A N G E - P A R T I C L E ] E F F ] ~ C T S I N ~ - W A V ] ~ P I O N - N U C L E O N S C A T T E R I R G 679
whether this t r ea tment of strange-particle effects is consistent with the P-wave
da ta . In view of the fact t ha t the (3, 3) scattering is dominated by a pion resonance which cannot be expected to recur in the K-meson scattering at these energies, and tha t the other P phase shifts are so poorly known experi- mentally, it is felt t ha t there are no real objections here. A more serious a t t empt at fitting the data must wait unti l more is known about the strange particles. Meanwhile, one must consider i t possible tha t their effects are signi- ficant even at the lowest energies.
I should like to thank Professor R. E. PE~ERLS, Dr. S. F. EDWARDS and Dr. P: T. M~T~EWS for suggesting this problem and for m an y helpful discus- sions. I should Mso like to acknowledge the awurd of a scholarship from the
5~urshall Aid Commemorat ion Commission.
A P P E N D I X
Derivation of the integral equations.
Following Schwinger~s technique and notation, the (~ source ~> term, ~-~v~ q- ~- ~ § J~v= + K~v~ § ~ K are added to ~he Lagrangian (1). The equation for the true nucleon propagator, S~(x, x'), is
(A.1) [ ( )] ~ ( ~ , x ' ) = ~ ( ~ - x') - g ~ % ~ <~K~> = S; ~ + g~yS (~v~> - i ~j~
where S ; 1 = y,(~/~x~,) + m~, and all bu t isotopic-spin vector indices have been suppressed. The second te rm on the R.tI .S. m ay be rewri t ten using the re- lationships:
j - - ! , , ,\ J - - "/ <~(x)~<(~,z(x)w.,(x ))+> - i ~
~<~K(x)> _ ~'K(x, y ) , 3K(y)
~he K-meson propagator and the definition:
~A.2) t J ' ! \ s,:,(x~ x ) =- i<(~(x)~N(x'))+> - i<~w <~,.(x')~.
6 8 0 . S . ] . A N G E R
Substi tut ing these into (A.1) and dropping terms which vanish when ~1 -+ 0~ K--~ 0, we get:
< ~ - - i ~ j~ ~ ; ( x , x') = d ( x - - x') +
�9 [ '~ ~ ( ~ , ~') + ~g~y~ j/~K(x, y) ~<~dy): d~y.
A similar equation may be derived for S' EN*
(A.4)
= - - g ~ [ < ~ ( x , X ~ ) [
3S~(x, x') d~y ] - i fAK(x, y) 3<~,~(y)>
Returning to the simplified nota t ion used above (the part icle subscripts go, into parentheses)~ the relevunt T matrices 'are expressed as follows
(A.5) ~<G: ~" ~ G > (~'(~))-~ '
~"S'(Z~T)
~<G~ ~<G, ~ (~'(~))-~"
After operating to the left on (A.2) and (A.3) with S(N) and S(E) respectivelyr i terating once, applying (A.5), and combining terms, the resulting equation is:.
(A.6)
where
and
§ V~(Nu]EK)3 ' (K)S ' (E)T~(EKI~q~) § crossed terms ,'--
+ z~s ~ ( ~ ) ~ ( ~ ]N~:) + ZK~'(~)Tr~(~ ] ~ ) + ...},
E ~ = - - i g ~ y S S ( ~ ) L J ' ( ~ ) y s ~ j ,
are self energy parts. 3T/8(~> which hgve been neglected. To the order of this approximation,
(A.7)
Ei ( = - - i g~ys r ~_~(E)A'(K)yJ j ,
The (( -- ... ~ ugain stands for several terms involving
s'(~) ~ s(~ ~) § ( ~ 4- ~vij ' (~) �9
STRA:NGE-PARTICLE EFFECTS IN ~ - W A V E FION-NUCLEON SCATTERING 681
A little algebr~ then shows tha t the fuetor (S'(N))-IS(I~) exact ly cancels the self energy te rms in (A.6). Approx imat ing 5" and d ' b y S and A, one arrives ,ut equat ion (2). Equa t ion (3) m a y be derived in exac t ly the same manner .
R I A S S U N T 0 (*)
Si cerca di spiegare 1~ separazione degli spin isotopici degli sfasamenti dell'ond~ S hello scattering pione-nucleone in termini di effetti di p~rticelle strane virtuali. Un ca lcolo semplicissimo si accord~ assai bene con l'esperienz~.
( *) Tre~duz ione a eura del la R e d a z i o n e .
