A fundamental particle model (II)

Nuclear Phys,cs 3 4 (1962) 3 4 1 - - 3 6 6 , ~ ) North-Holland Publ,sh,ng Co, Amsterdam Not to be reproduced by photopnnt or microfilm without written pernnssmn from the publisher

A FUNDAMENTAL E

P A R T I C L E M O D E L (II)

VAN DER SPUY

Atom,c Energy Board, Pretorm, South Air*ca and

Oak R,dge Nat,onal Laboratory l, Oak R,dge, Tennessee Received 5 F e b r u a r y 1969. A b s t r a c t . T i n s p a p e r p u r s u e s t h e d i s c u s s i o n of a f u n d a m e n t a l p a r t i c l e m o d e l p r o p o s e d in a prev i o u s p a p e r T h e m o d e l for b a r y o n s a n d m e s o n s s t a r t s f r o m a f u n d a m e n t a l n o n - h n e a r e q u a t i o n of m o t i o n for t h e field o p e r a t o r in t h e H e l s e n b e r g r e p r e s e n t a t i o n , w i t h a f o r m a l l y s y m m e t r m self-Interaction E q u a t i o n s of m o t i o n a r e g e n e r a t e d for p a r t i c l e s a n d particle i n t e r a c t i o n s f r o m t h e e q u a t i o n of m o t i o n of t h e field b y u s i n g t h e Y-function f o r m a h s m T h e e q u a t i o n of m o t i o n of a single b a r y o n as discussed a n d it is a r g u e d t h a t , if o n e confines t h e a n a l y s i s to t h e m a s s shells a n d t h e a s y m p t o t i c d o m a i n , a s o l u t i o n is f o u n d w h i c h s h o u l d b e r a t h e r a c c u r a t e T i n s e q u a t i o n of m o t i o n is t h e n c o m p l e t e l y I n t e g r a t e d In m o m e n t u m s p a c e for p s e u d o s c a l a r , v e c t o r a n d p s e u d o v e c t o r self-interaction t e r m s A n e w m e t h o d is d i s c u s s e d for f i n d i n g t h e low e n e r g y p e r i p h e r a l i n t e r a c t i o n of t w o b a r y o n s f r o m t h e f u n d a m e n t a l e q u a t i o n , for all t h r e e t y p e s of self-interaction T h e v e c t o r a n d p s e u d o v e c t o r t y p e s h a v e e x p h c l t y v e l o c i t y d e p e n d e n t t e r m s e v e n in t h e lowest a p p r o x i m a t i o n Possible c o n f h c t s b e t w e e n t h e d e m a n d s of t h e low e n e r g y i n t e r a c t i o n of t w o b a r y o n s , a n d t h e b a r y o n s e l f - e n e r g y p r o b l e m a r e discussed I t a p p e a r s possible t h a t t h e o n g m of s e l f - e n e r g y h e s in a v e r y s h o r t r a n g e s e l f - i n t e r a c t i o n l ~ u m e n c a l c o m p u t a t i o n s of t h e b a r y o n s e l f - e n e r g y p r o b l e m a r e t h e n e x h i b i t e d for a n e x t e n s i v e r a n g e of m e s o n m a s s p a r a m e t e r s Possible cases of s e l f - i n t e r a c t i o n a r e d i s c u s s e d w i n c h w o u l d give t h e n u c l e o n a n d A - b a r y o n m a s s e s , w i t h t h e basic m a s s m 0 = 0 I n one of t h e s e cases it is s h o w n f r o m t h e n u m e r i c a l r e s u l t s t h a t t h e g h o s t m a s s a p p e a r i n g in t h e b a r y o n p r o p a g a t o r does n o t s a t i s f y t h e a s y m p t o t i c w a v e e q u a t i o n w h i c h t h e real m a s s e s do s a t i s f y I t is f u r t h e r s u g g e s t e d t h a t t h e l e p t o n s a n d t h e p h o t o n a r e a s s o c i a t e d w i t h t h e n o t i o n of s a t u r a t e d c o m p o u n d s in t h e s t r o n g coupling t h e o r y A s t r o n g c o u p h n g t h e o r y w o u l d t h e n h a v e t h e m o r e m a s s i v e s t r o n g l y i n t e r a c t i n g particles, a n d h a v e t h e s a t u r a t e d c o m p o u n d s as lighter, w e a k l y m t e r a c t m g p a r t i c l e s A t e n t a t i v e s k e t c h is g i v e n of a f u n d a m e n t a l t h e o r y b a s e d on t h r e e basic b a r y o n s , n, p, A in s t r o n g self-interaction, in w h i c h s t r o n g l y I n t e r a c t i n g h y p e r o n s a n d m e s o n s a p p e a r as well as w e a k l y I n t e r a c t i n g l e p t o n s a n d p h o t o n s

1. I n t r o d u c t i o n In the present paper the self-consistent cut-off baryon-meson model of a previous paper 1), now referred to as paper I, is discussed further The basic ideas of this model are discussed in paper I and are not repeated here in detail It is a model for fundamental particles, and was confined in paper I to the baryon-meson system with only strong pseudoscalar coupling The model alms at relating the ratios of the masses of particles of the system to the coupling constant, and at estabhshmg a fundamental calculation of the forces between the particles t Operated b y U n i o n Carbide Corporation for the U . S A t o m i c E n e r g y Commission June 1962

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To do this one starts from a fundamental non-hnear equation of motion for the baryon field operator ~v, in the Helsenberg representation Basic reqmrements and a heuristic approach are used to select what appears to be a useful fundamental equation It is suggested, however, that once this equation is selected, all results of the model shall be deduced from it To do this one uses it to derive equations of motion for z-functions 3) (and the associated Cfunctions) corresponding to the relevant physical situation This technique is essentially that employed by Helsenberg s) for another fundamental equation One uses such a discussion to discover the suitable form and content of the fundamental equation In the present fundamental equation ~vis a 12-component splnor, 4 components each for tile proton, neutron and A parts of the basic baryon field In the latter selection the model follows Sakata 4) Other strongly coupled particles have as wave-functions C-functions generated by the appropriate 1, 4) ~v products Binary products, for example, generate the meson wavefunctlons The fundamental equation is non-hnear and refers to the basic baryon fields in a formally symmetric self-interaction The dlscusslon suggests that the p, n, A fields hence share the same propagator To get a finite theory the baryon propagator must be cut off but the cut-off is introduced in a selfconsistent way Thus the real particle masses appearing in the propagator satisfy the equation of motion of a single baryon The different basic baryons distinguish themselves in this respect by propagating only on their relevant mass shells when they are free It was thus shown in paper I that a number of different masses is not inconsistent with a formally symmetric coupling, and that in fact each mass shell m a y even have a distract effective coupling constant For the discussion of the role of ghost masses in the propagators we refer to paper I The present discussion proceeds further on this basis Section 2 discusses a slmphflcatlon of the fundamental equation consequent on reassessment of the relative accuracies of the lowest approximations of the equation of motion of the single baryon and t h a t of the propagators The prospect of an exact solution of the baryon equation of motion is then discussed Section 3 deals with the baryon equation of motion for the case of a basic pseudovector and a vector, as well as the pseudoscalar interaction so far used It also lnchcates the momentum transform of this equation and its complete integration. Section 4 lndmates a new treatment of the baryon-baryon scattering problem leading to a full statement of the low-energy form of the wave-equation for this problem, for the types of interaction mentioned above Section 5 reports detailed numerical calculations concerning the baryon self-energy problem Section 6 discusses the application of the above techniques to the leptons, and introduces a tentative proposal for the leptons and photon as part of the baryon-meson model Section 7 discusses the present position and possible future research on this model

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The symbols of paper I are used again and we refer to paper I for their deflmtlon in general

2. Modified F u n d a m e n t a l Equation The fundamental equation used m paper I 1s

(1) where ~vIS the 12-component splnor already mentioned above, a IS the structure constant = G~/4z~hc, and ~ is an eight-component vector, each component being a 3 x 3 matrix 1) The function ~ ' r ( x ) was considered to be a dressed F e y n m a n propagator an paper I but in general the research is also aimed at fmdlng the most suitable form and content for this function The operator ~® ----- zV®--mo--(*Vz--m) was used in paper I but for reasons to be discussed below, it is for the present analysis taken as

¢~ = * ~ - m 0

(2)

Eqs (1), (2) define the fundamental equation for the present paper, except that in later sections we include other interaction types vector and pseudovector, in addition to the above pseudoscalar term To chscuss the equation of motion of a single baryon, one considers z~Z({x) = where [~o>IS a single baryon state Then d

=

X1 ~

F(x--xl)rS.ra.SP r

Pr

*,,p

(XllXlx)

(3)

One starts by noticing that eq (3) is exact The approximation comes in when one tries to reduce the third order ~-functlon in the lntegrand In the asymptotic domain for the ~v operators in Helsenberg representation one has the following exact relatmn by Wick's rule and the s y m m e t r y of the fundamental equation nktn ts,,a (x'llxl x)

=

<0IT

g,d'(xdv,,'~(xl)V,i"(x)l~o>

nkm

.,1. r = ¢,.p (XllXl~)+~{s ,,a(0)~.~p m ( I x ) - s t ~ p , ( z - z l ) ~ . . ~ ,

k

(1~1)},

(a)

where n/mr

~b~,# (XllZ1. ) ---- <0IN v~"(zl)vA*(xl)y~a~'(m)19>, the normal ordering N being defined for the asymptotic domain For the rateraction domain one m a y follow Freese 2) by using eq (4) as a definition of the $-function Then following Heisenberg s) the lowest order ~-function theory arises b y assuming nk, m ¢,.p (Zll~, ~) = 0

(5)

~4

~

VAN DER

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in eq (4) and inserting the resultant reduction of the third order z-function in eq (3) The baryon equation of motion then reduces to = ~

f d4Xl ~'F(X--Xl)y5 S'F(x--xl)Ts~(Ixl)

(6)

As shown in paper I, ff one assumes that the most important part of the antlcommutator {W~*(x), ~p(Xl) } is its c-number part, one readily obtains an equation such as (6), but in whmh ~([x) IS replaced b y S'(x) Furthermore, b y the above ~-functlon techmque, one can discuss the equatmn of motmn of the interesting T-function ~m 1 t *.B(ylx) ---- (0IT ~p~'(y)~pp'~(x)lO) -= -~,,,,S F,~,(x--y)

One gets eq (6) with v([x) replaced b y S'F(x), if one uses the approxlmatmn ~tmn ¢~,~,, (yx 1Ix 1x)

:

O,

(7)

where

C

k ~'ttn /

m the asymptotic domain, and is defined b y a relation analogous to that of eq (4) m the interactmn domain B y general arguments ~) the momentum transforms of S'~(x) and S'(x) differ only in contour, and thus apparently the above assumption m connection with the antlcommutator is in the present connection equivalent m effect to the assumptmn m eq (7) Only the latter assumptmn is referred to In connection w~th the equation of motion for the baryon propagators, from now on Now if one transforms eq (6) to momentum space one obtains

Z, (s) Z'(P) = ~

f c F d ' k (k,_~2), 1 r5 ( p - - h1- - m ) ' r~'

where the symbols have the same meaning as m paper I. The domain of/~ for which eq (8) should hold depends on the significance of the assumptions used m deriving eq (6) from the exact eq (3) If one assumes that the assumptions expressed b y eqs (5) and (7) have the same vahdlty, then eq (8) should hold for a continuous range of p* up to such values of p2 as are determined b y these assumptions This was the assumption of paper I, and having to satisfy eq (8) over a p2 range necessitated elaborating 0. to ¢® ~ ,~®--m o - (,~®--m)' This last term, which is important off the mass shells only, complicates the analysis of course Reconsideration of eqs (5) and (7) has suggested that assumption (5) can readily be used more accurately for our purposes than eq (7) This arises somewhat as follows

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Firstly, both eqs (5) and (7) are exact In the asymptotic domain where the ~v operators In the Helsenberg representation presumably are related to the free field operators with suitable normahzatlon Now the asymptotm domain is of great physical significance for the baryon equation of motion and intuitively corresponds to a "free" baryon, moving on the mass shell Thus the restriction of ]9> and eq (3) to the mass shell still leaves a most Important case Such a condition on [~) does not restrict single baryon final states but only intermediate states The present analysis shall not use a knowledge of intermediate states except in the most general terms Of course if one were to extend eq (5) into the Interaction domain as IS required to reduce eqs (3) to (6), an approximation IS involved because of the integration over 4-space This has the usual quahtatlve ]ustlficatlon of the Tamm-Dancoff method, in that the rhany-particle correction to a single particle problem is assumed small at not too high an energy This point and a possible exact analysis of eq (3), are discussed further below For the propagators the interaction domain is of essential importance and the qualitative ]ustlhcatlon on the line of the Tamm-Dancoff method for extending eq (7) to the Interaction domain IS not readily available for the many-particle nature of the dressed propagator In the present analysis eq (5) is thus assumed more generally and accurately valid than eq (7), and for this purpose ]~) defining z(]x) is required to be on the mass shell Hence eq (8) needs only be satisfied for # on the mass shell, and 0 x as defined by eq (2) suffices, at least for the present It Is, however, of the utmost Importance for assessing this model to Investigate whether the exact equations of motion of the v-functions can be solved exactly, or alternatively how good a given approximation is The present paper does not attempt such an analysis but rather sketches some features of the way in which it can perhaps be done All known exact wave-equations for the z-functions of the present model are similar to eq (3) Because of its basic importance and being the simplest case, the features of an exact solution of eq (3) will be specifically discussed Many of the remarks, however, apply to the case of more complex v-functions Consider eqs (3) and (4) with ]~) defined to be on the mass shell As already mentioned the asymptotic domain is here of great importance Here eq (5) applies exactly, giving an exact reduction of the relevant third order zfunction to the first order z-functions as for eq (4) together with eq (5) Thus this third order z-function depends hnearly on the first order z-function in the asymptotic domain Since this gives an extra exact condition one m a y ask whether it can be used with the exact eq (3) without extending its domain of validity Since in the present program one tries given functmns for ~'F(X), until one finds the best, this function can in this connectmn be taken as known The best approach to an exact solution of eq (3) m a y thus be to invert it as an

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integral equation for the tbard order z-function The result will be t~,# ~*'*(Xl[X1 x) as a functional of zpZ([x) The functional t a k e n in the a s y m p t o t i c domain, t o g e t h e r with eqs (4), (5), gives an exact solution T h e fact t h a t one o n l y w a n t s the a s y m p t o t i c form of the inversion of eq (3), and other features to be mentioned, In practice suggest a simpler, practical technique On quite general m v a r l a n c e a r g u m e n t s s) one notes t h a t 5

5

n k ^ Zm

7a.7~pp, v,

*zkm

Tt,p (Xl]XlX) = / = ~ ( x - - x l ) e-(P~ ~ =

%

where P , is the f o u r - m o m e n t u m of state ]9) a n d / z (x) is lust a general function, for the present, with two indices l, ~ F u r t h e r d~. ~pzp~([x) = Mz~(tx) where M = m - - m o, and m is the mass c o n s t a n t for the respective mass shell, on this mass shell, P , = m in eq (3) T h u s eq (3) reduces to

--M

(0)

f

Of course M depends on the scale of units I t IS not e x p e c t e d t h a t M changes r a p i d l y with ~ I t is e x p e c t e d r a t h e r t h a t the ratios of the various M values are functions of ~ A crude indication of the kind of dependence m a y be h a d from the first order p e r t u r b a t i o n t h e o r y with a cut-off

where Me denotes the cut-off mass This e q u a t i o n shows t h a t r a t h e r t h a n M h a v i n g a linear dependence on ~, it fixes M e / M as a more complicated function of ~: ~ - = exp

+

This relation lndmates a probable feature of more c o m p h c a t e d cut-offs, t h a t the mass ratios are smaller for strong coupling F o r large e the mass ratios have a slow dependence on e T h e present t h e o r y employs a more complicated cut-off, and does not use a p e r t u r b a t i o n expansion in powers of e, and the relation of the masses is hence considerably more complicated See section 3 in this connection The a p p r o x i m a t i o n is of the T a m m - D a n c o f f kind and it is the purpose of the present discussion to show t h a t as p e r f o r m e d in paper I and in the present paper, the a p p r o x i m a t i o n is likely to be good This discussion suggests t h a t the l e f t - h a n d side of eq (9) p r o b a b l y varies n e a r l y as I/a, and is hence small for v e r y strong coupling. The inversion of

A F U N D A M E N T A L PARTICLE MODEL (II)

347

eq (9) suggests/0~(X--Xl) a s a functional of (M/x)r~z([x) B y the non-hnearlty of the parent fundamental equation (1), this dependence wall in general be non-linear The non-hnear terms of nth order wall, however, be damped b y practically 1/,t ~" compared to the first order linear term The non-linear terms m a y hence be expected to be negligible for very strong couphng Furthermore, since they do not occur in the asymptotic domain b y eqs (4) and (5) they probably wall only affect a confined part of the interaction domain, anyway In general /~z(X--Xl) can have two other components not appearing in the asymptotic domain" (a) such functions ]~(x--xl) as are orthogonal to ~'F(x--xl) and hence at present independent of v(lx) which will presumably predominate in the limit of very strong or mflmte coupling, these functions, if any, must have a considerably shorter range than S'F(X--Xl) b y eqs (4) and (5), (b) other functions /a(x--xl) as coefficients of ,(Ix1) but again of much shorter range than S'F(X--Xl) Shorter range would mean deflmtlon b y larger masses than A Type (a) functions wall not affect the analysis leading to eq (6) at all Thus of them all only type (b) functions m a y mvahdate eq (6) Since it is beheved that eq (4) and eq (5) together are exact asymptotically, it is assumed that type (b) functions are of very short range and m a y probably be neglected In a good first approximation This discussion suggests that If one limits considerations to a 1~0) on the mass shell, eq (6) is probably a rather good approximation In further analysis one can try short range perturbations of S'F(X--Xl) to see the effect of components of type (b) Moreover, the above discussion m a y suggest techniques of inverting eq (3) more exactly Especially since the general solution/~Z(X--Xl) m @ clearly contain arbitrary proportions of type (a) components, this and other features of the solution will be determined b y boundary conditions for the third order v-function, in the interaction domain This is a complex problem because the boundary conditions and behavlour In the Interaction domain, which are the best for the success of the model, will depend on testing the principles involved also in problems other than the equation of motion of a single baryon Especially interesting may be the behavlour of other third order v-functions as afforded b y the suggested nature of the 2: and ~ hyperons, or b y the baryon-meson scattering problem. We shall thus use eq (8), as in previous calculations, restricting/~ to be on the mass shell for high accuracy For the same reason, 1~0) must, however, also be restricted to states possible in the asymptotic domain Thus eq (8) shall not be resisted on for/~ on the ghost mass shell On the contrary, if eq (8) is of good accuracy, as suggested above, and if in addition/~ on the ghost mass shell does not satisfy eq (8) but ~ on the real mass shell does, one has a sufhclent though perhaps not necessary indication that the ghost mass does not appear in the asymptotic domain The numerical calculations illustrate this posslblhty

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Because ghost masses are a problem, it Is the a p p r o a c h in the numerical work to see how far one can get with the m i n i m u m n u m b e r of ghost masses I n the present paper, discussing o n l y b a r y o n self-energy and b a r y o n - b a r y o n interaction in detail, only one ghost mass IS needed from the s t a r t I t is assumed to be t h a t in 1 / ( ~ - - m ) ' In the present work (as in p a p e r I), one t h u s uses t

1 (~--m)'

Ct Co Ck (~--Z) + ( ~ - - g ) + ( ~ - - k ) C~

(10)

=

T h e positive l, g, k are m ascenchng order of m a g n i t u d e T h u s g corresponds to the ghost mass and C o, is negative One has to discover the appropriate m e a n i n g of N'F@) and its m o m e n t u m transform, assumed = 1/(k~--#2) ' for the present F o r a particular set of b o u n d a r y conditions for the 3-function, correspondmg to meson motion, p a p e r I discussed some aspects of the t h e o r y in the case when the # is equal to or near the actual meson masses T h e numerical work in p a p e r I, however, suggested t h a t since (kS--#2) ' m a y contain k2--/~0 ~ which resembles a free field contribution,/~o being a basic mass, the # need not be equal to or near the actual meson masses Therefore m line with the m m to see w h a t m a y be obt a i n e d with the m i n i m u m n u m b e r of ghosts the following IS tried in this paper,

1

1 -

(k~--#~) '

k2--#~ '

(11)

where/~ need not be n e a r or equal to an actual meson mass This does not necessarily m e a n t h a t one expects one meson mass, as ~ ' r ( x ) m a y lust be a simple Green's function r a t h e r t h a n the fully dressed meson p r o p a g a t o r I n fact eqs (2) a n d (10) really force one to use an a p p r o a c h to the solution of the meson equation of m o t i o n different from t h a t e m p l o y e d in p a p e r I This m a y include a different b o u n d a r y condition for 3(0]0) a n d a different assessment of the relative i m p o r t a n c e of the t e r m s in the meson equation of m o t i o n Discussion of this a n d relevant numerical work is c o n t e m p l a t e d for a f u t u r e publication I n anticipation, it m a y be m e n t i o n e d t h a t a p r e h m l n a r y theoretical discussion of the case, with b o u n d a r y conchtlon 3(0[0) = 0, is not incompatible with eq (11) leading to a n u m b e r of meson masses In a n y case, the b a r y o n self-energy calculations are m a d e for a n u m b e r of #3 values and the more general choice of 1/(k2--#z) ' can thus be simply c o m p o u n d e d from its partial fractions B y a suitable choice of n o r m a l i z a t i o n of ~0, ~'F(X) and ,t (see eq (1)), one can always arrange to h a v e the normalization used in eq (11) and have Ct = 1 t No contnbutmn f r o m a c o n t i n u u m is a s s u m e d t h e c a l c n l a t m n s t h e r e ~s n o d e m a n d f o r i t

as yet because up to the present stage of

A FUNDAMENTAL PARTICLE MODEL (II)

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3. I n t e g r a t i o n of the B a r y o n E q u a t i o n of M o t i o n For this and the next two sections, one considers the fundamental equation extended to include pseudoscalar, pseudovector and vector interactions, using the respective suffixes ps, pv, v Thus eq (1) then reads

0. v2 (x) ---- -- 2~,:¢ps T f d* x 1~ F t' ( x - - x l ) ( ~ (xl)~5 ~p (x1)) ~,6~p (x) +2g,~vT

f dtXl~FP(X--Xl)

--2~,~,vTf d'x~F*'(X--Xl)

(~(xl)rI'~(Xl)) 7 t , ~ ( x )

(12)

(~(xl)y*ya~0(xl)) ygys~W(x ),

where ~. is defined b y eq (2), and #, p, a as indices on ~F(x) refer to the respective masses m the momentum transforms as in eq (11) All couplings are assumed either strong or not present Thus the ~ are either larger than 1, or = 0 Furthermore with C t = + 1, a positive a indicates that the mass in ~F(X) iS not a ghost, and ~ negative corresponds to this mass being a ghost Since ~F(x) is merely a Green's function for the present, the latter as not yet a statement about the physical meson masses B y the dlscusslon in the previous section the equation of motion for the single baryon, on the mass shell, now reads (~,~:(Ix) --~ ~ * % s

f d'Xl

~F~'(x--xx)(r6S'F(X--X~)ys)V(IXl)

-~- ~6-~7~$~pvf d4Xl ~.~Fo'(X--Xl)(r/~r 5 S'F(~g--Xl)rpr~)T(I*I) Transforming this equation to momentum space and using a single Feynman parametric integration y~elds /~ -- m0 = ~4- ~ C ~

{ f2d ( x_m)lnC, %s

~2~¢v f j d x ( , x - - 2 m ) l n

C,+2%v f l° dx(l~x*2m)ln C~},

(13)

where Cl, = --p2X(1--x)~-#2x~-m2(1--x) and Cp, C~ are the same but with p, a, respectively, substituted for/~ in Cg The Integration above over momentum space gives logarithmic divergences for the contribution of each particular m To remove these, as has already been done in the summation on the right-hand side of eq (13), a constraint is required on the C~, as follows

~, C,,, = 0 = ~_,mCm 9n

(14)

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This means that some of the C~ must be negative, corresponding to a ghost mass contribution Eq (10) can be consistent with eq (14) with Cg negative, as has been suggested Alternative ways of removing these divergences, which will not, however, be explored m the present paper, are offered b y suitably compounding contributions of dafferent meson masses or of different Interaction types The parametric Integrations are hence of two kinds

J ' d P L ~s, m s) = f l° dx J'2(P*, #s, m 2)

ln{--pSx(1--x)+#2x+mS(1--x)},

= f: ~vdxln{--pSx(1--x)+#Sx+m2(1--x)}

(15)

A number of cases arise depending on the relative values of p2, #2 m s Since only the baryon-meson case is reported here in full numerical detail, one notes that for this ps or m s only assumes the values l 2, gS, k 2, with extremes l s and k s corresponding to the nucleon and A masses squared Further, since # need not equal a meson mass, and we avoid a difficulty already discussed in paper I b y doing so, we only consider ps< (/z+l)2 B y what has already been said about p2, ls, this provides a lower limit for # of about 345 m e With these restrictions one has one kind of integral of eq (15) Define cos ~0 = (m2+#s--ps)/2#m, with 0 <~0< ~ Then

~

J ' l ( p 2,/z 2,m 2) = 1 n # 2 - - 2 + \

]ln

~-

t

+~-~osln~o

(16)

= In # 2 - - 2 + J ~ ( p s , / , 2 , mS), J'~(P2'#2'm2) = { l n # 2 - - ½ + t \ + \

2P 2

2-~

/\~-

./ - - ~

In (17)

~ sin , - - 1]

= ½1n/~s--½A-J2(p2, #s, m s) Due to (ln#2-- 2) and ~2/1-1n'~S--2Jl~being independent of m these parts disappear in the summation over m in eq (13) due to eq (14) Thus only 3¢1(p s, # s mS), J s ( f i s,/z s, m s) as defined above play a role in eq (13) Clearly the latter m a y now be written /~_mo = Kps{fij2(p2, # 2 ) _ j l ( p 2 ' #2)} +2Kv{/~d2(ib 2, p2)--2Jl(p2,

p2)}+2Kpv{l~J2(p2, a 2 ) + 2 J l ( p

2, a2)}, (18)

where 4

4

Kps= ~ccps, Kv = ~ v ,

4

Kpv-- 3~CCPv'

(19)

A FUNDAMENTAL PARTICLE MODEL

(II)

351

while J l ( p s, vs) = ~ mCmdfl(~ s, v s, mS),

(20)

J~(ib s, vs) ~- ~ C,n J2(ib s, vs, m2),

(21)

vs having the possible values #s, pS, as As has been mentioned, eq (18) should be a good approximation for/~, p9 on the mass shells In applying it one calculates the terms in the curly brackets on the mass shells for a n u m b e r of v2 values (/~s, pS, a~) Consider the mass shell n, or/~ = n, ps = n s Then the following notation will be used for the numerical work in section 5 J , s ( n , vs) = nJ2(n s, vs)--Jl(n s, vs), i v ( n , Vs) : nJs(n 2, v 2 ) - - 2 J l ( n s, vs),

Jpv(n,

= nJ (n s,

(22)

s)+2Jl(n s,

In paper I a normalization constant N was introduced in the equation of motion for b a r y o n - b a r y o n interaction This arises n a t u r a l l y from identifying a self-energy term m this equation of motion as leading to the baryons to move on their mass shells in the a s y m p t o t i c domain A little consideration Indicates N z as N on the mass shell l to be given b y the coefficient of/S in eq (18) when written as N~(~--I) = 0, or

Nt = l _ g p s j 2 ( / 2 , / ~ s ) _ 2 K v J 2 ( l s, pS)_2K, vJ2(/s, aS)

(23)

4. The Baryon-Baryon Interaction at Low Energies In paper I the equation of motion for the interaction of two baryons was deduced from eq (1) above, in the lowest order z-function t h e o r y The latter neglects the fourth order 6-function in the reduction of the relevant fourth order z-function A technique was t h e n d e m o n s t r a t e d for deducing the wave equation, at low colhslon energies and not too close an approach, in the form of the usual Schrodmger t y p e of equation In the present section an alternative technique for deriving the wave equation at low colhslon energies and not too close an approach, wAll be developed, which leads more reachly to a full exphclt s t a t e m e n t of the lowest approxlmatlon to the low energy interaction This technique is itself more exphclt and less lntmtlve a n d it is beheved it m a y be used to provide the velocity dependence corrections of the interaction effective at somewhat higher energies, b y a n a t u r a l extension One m a y pres u m a b l y also a p p l y it to calculate velocity-dependent Interaction effects, arising from the self-energy term nl the z-function equation of motion In the lowest approxlmatlon employed in paper I and in the present paper, this

~5~

]E VAN D~;R SPUY

term is taken to contribute only to the constant baryon mass l concerned and the normalization Nz As in paper I, z:,a(ixy) ks = (0ITy,~k(x)w/(y)]q~ ) is a suitable wave-function for the baryon-baryon Interaction problem, where ]~) is the relevant two-baryon state The lowest order in T-function theory for the wave-equation follows as in paper I, but here from the more general fundamental equation (12) Assuming for simplicity that the two baryons concerned have the same mass, l for definiteness, the self-energy term and the role of the p matrices m a y be simphfxed :), as follows

N/('V2--[)T(lxy) = :T~,8{05psf d ' ~ 1..~F~(y-xl.) (StF(~--Xl)rS)lr2 5

--iX v f d.'~l .,@FP(y--Xl)( StF (X--Xl)~#): ~'/t2

where s = ~4 , - - ~s when the wave-function v~p([xy) ~ IS respectively symmetric or antisymmetric in the baryon indices k, l Recognizing this simple dependence one m a y henceforward leave the baryon indices out as in this equation On account of the exclusion principle ~) these elgenvalues of e apply respectively to states antlsymmetrlc and symmetric under exchange of both space and spin together A lower index 1, 2 attached to the 7 matrices and the V operator signifies :) that the operation IS on the first and second v2 operator or its coordinates, respectively Hence =

7~8 - - a v f d a x I ~ F p(y--r:) { ( ~ z - - l ) S ' F (x --x:)~$}: ~2~2

+ ov j'

(24)

T

This expression can be used in the scattering formalism to be developed Let z°(xy) be the outgoing wave-function of two baryons both moving on the mass shell l Thus

(z@:--l)7:°(xy)

0

(zV,--/)v (xy)

(25)

Consider the two baryons in their centre of mass system and note the alternative coordinates z -----x - - y , Ye = ½ ( x + y ) (26)

A FUNDAMENTAL PARTICLE MODEL (II)

353

Quite generally, ~°(xy) = exp(--~Pt ye)~°(z) lust as T([xy) = exp(--~Pl" ye) × ¢(z), where e f -~ 0 One readily finds from eq (25) ~

(~10._~ ~20) / T ° ( y )

)

(27)

The scattering Into an outgoing state is determined by the scattenng amphtude 2) A,z : f dax --

f d~y~°t(xy)v([xY)

f d4x

f d4~]'T°(xy)($Vl--l)(gV2--1)7~([xy),

(28) i

--l(7~o + 72o)+ p0} v (]xy)

=

where one uses eqs (25), (27) and the orthogonahty of ,°(xy), ,([xy) in the remote past, together with suitable partial Integrations The scattering formalism up to now IS exact and relativistic From now on, however, the special case IS considered of low colhslon energy, for not too close an approach For this peripheral interaction one lntmtlvely expects a reasonable first approximation to correspond to what in the usual language would be called a single meson exchange On this basis one has to transform

Azz--

N,f d4x f d4Y~°(xY) 7~,8

× E~psf d4x~ 2F~(y--x~){(~Q~--I)S'F(X--XI)?'5}I)'25

"-~OCpvf d4"~1

~F°'(y--x,1){(1,~.--I)S'F(X,--Xl)~d~S}I(~#~)")2~ T([~ly )

One notes in general that since v°(xy) has both particles on the mass shell l, the curly brackets above will vanish except for the contribution of the corresponding (mass l) singularity of the propagator S"F(x--xl) The operator ( ~ . - - l ) m the above therefore lust selects the constant C z determining the effective coupling constants of baryons on this mass shell ¢ To effect the transformation most readily leading to the low energy peripheral interaction in the lowest approxlmatmn, consider v°(xy), v([xy) as the sums of products of Dlrac particle plane waves 3 ° (xy) = 2"o ulO (~)fl)e-"n

• (Ixy) = _ru

~ uz0 (Pf2)e-''

(ibil)e-"u " u 2(pt2)e-',.. "

", (20)

t Considerations (to be published) of final states in terms of an asymptotic condltmn shows C z = C~ Thus only N~ can produce different effective coupling constants for dafferent baryons See eq (32) and sect 5 in this respect

354

E

VAN D E R S P U Y

The u functions are Dlrac plane wave splnors of the relevant four-momentum The summation m a y be left quite arbitrary, the weighting being an arbitrary function of the momenta, not revolving the 7 matrices The latter means that the spin-orbit coupling is taken in the lowest approximation The end result shall then again be stated in terms of ~°(xy), ~([xy), without reference to the particular identification Using (29) and effectmg the space-time integrations one has

Air

--

~eC t (2=)s 27027 f d' k~*(k+pr~--p,1)~'(--k+prz--p,2) _ 4zt~N t _

m p ~ [~lo(prl)715Ul(Pil)][~220(pf2)r25u2(Pts)] (k2__#2)

X

~v _

_

(k s_p2)

%v ~') [a?(p~)rl.rl~m(p,~)][a2o(p~,)r.sr.~us(p~s)]} (kS--a The square brackets m a y be slmphfled to give the first approximation for the low energy peripheral interaction by using well-known properties e) of spmors and 7 matrices When this has been done the results may readily be written in terms of "c°@y), ~([xy)

(30)

A,, = - f d'~ f d,v~o(~v)o,s(x-v)~(l.v), where (see eq (26))

o,,(~)- ~--~sC~ fc d4ke*~ * I X

am (al" k)(as (k,_#2) 4l 2

k)

*iv 1 + (k2 p2) 4/z {4/2+k2--4Vz2-~-4~k

• Vz--2[(Oel--~a2)×k]

• V,

+ (a I • k)(a~" k) -- (al" a2)k 2}

~pv

1 { - a q a l . a~)+4(al-v,)ca~. V,)--Cal'k)(,7~-k)

+ (k2--a2~ 4/2

--2s(a 1 • V,)(a s • k ) - - 2 , ( q l • k ) ( a 2 • V,)}I.

A FUNDAMENTAL

PARTICLE M O D E L

(II)

385

Hence after integrating over the centre of mass coordinates, eqs (28), (30) lead to A~z = - - , (2:~)' ~' (Pr -- Pt) f d8 zco* (z) {*(a 1-- a2)" V~ --1 (7~°+ ~,2° ) + Pt°}¢ (z)

= --(2sr)'~5"(Pt--P,) f

d'z¢°*(z)~,, ° ~,2°012(z)¢(z)

= -- (2:~)' 0' ( P f - - Pl) f d s zco* (z)7, ° 72° 0x2(z)¢ (z), where 012(z ) IS the integral of Ox2(z) over the relative tnne coordinate In the centre of mass system, C°(z) does not depend on the relative time, and neither does $(z) for the peripheral interaction, when all the particles move on the mass shells all the time This relative tlme integration essentially removes this as a coordinate, and at the same time smoothes out oscillations of 012(z ) as a function of z ° which is not observable in the laboratory frame of reference Conservation of the centre of mass f o u r - m o m e n t u m being understood, we m a y thus define an equivalent equatmn of motion in the relative spatial coordinates

{$ (a 1 - a 2)" V z--I (~10-~-~20) + PI 0}¢ (z) :

-- Srl 0 ~22°012 (z) ¢ (z) = "~P12(z)¢ (z), (3 1 )

where ¢P12(z)is the equivalent interaction potential This equation can readily be converted to the Schrodmger form 1) B y simple transformation, the interaction potential m energy units turns out to be

~12(z)c~ -- --*yl ° 720 c~012(z) = N---~

i, c2

~-2 ~ - - ~ -

+mpC2 v p, "+-2

-]-

+812 (~-~+~-~+~'3

-1+4

;; r . v , 1

({71+O"3) [r X ~Vr] --}-~o'1 • o'2-- 512

e -~'r { 1 +m'C2%v-7, r- - - a l " a S + /i (al" V')(a2" V,) + ~

--2

x

as

. az+Sx2 " 1

~ - ~ --1-~

1

+ - -~rr +

{(0"3 r)(o'l" V , ) + ( e l " r)(e2" V,)}

,

356

E

VAN DER

SPUY

where m~ 1s the mass corresponding to # =

m~,c/~, for example,

and

3(a 1 • r)(a2-r)

S12 ~---

--0'1

},2

" (}'2

the tensor 1nteraction operator The usual vector r has been used m (32) instead of z and r is the corresponding magnitude The validity of the method of deriving equatmns (31), (32), restricts r not to be too small and the momentum (or the result of the operator V~) not to be too large For low energy interactmns one can, as a first approximation, neglect the terms lnvolwng V, which are only contributed b y the vector and pseudovector interactions These velocity-dependent terms can then be determined as perturbations They are equivalent to an effective change of mass with radius, and in some cases with spin At higher energies and closer approach, the interaction is fully velocity-dependent due, not only to the 7 matrices, but also to the integral nature of the primary equation of motion The vector Interaction gives a spin-orbit coupling The pseudovector interaction has velocity-dependent terms which are also a function of the spins, showing some analogy to the tensor interaction for its spin dependence These terms have a simple form for S conhguratmns of relative motion, when (al •

.

(al" r)(a2" V,) =

where one uses the

1•

( 2;

1 ~ Ca1 • r ) ( e 2 • r) r 3 r - -

,

r

3 (el

spherical s y m m e t r y of S states

a2)~r, and

hence

also

$12 = 0 Now C ~and N z are near 1 for the cases considered, a n d , = {, --3 respectively for states antlsymmetrlc and symmetric in exchange of both space and spin together Thus for *¢pspositive, the Interaction due to the pseudoscalar term is attractive for its central and tensor parts for the 3S1, 3D1 and ~S0 configurations of the very low energy nucleon, nucleon data, as it should be For ~v, ~pv positive, the signs of the Interaction due to these couphngs are unsatisfactory, if one neglects the explimtly velomty-dependent contributions This means that the vector and pseudovector contributions probably cannot occur alone as far as the interaction at low energy is concerned, and probably that p, a are rather large compared with /z so that these interactions only occur at very small rachl, if at all An actual test of these interactions m detail will be left to another occasion, when the demands of both the crucial barynn and meson self-energy problems have been mchcated

A FUNDAMENTAL PARTICLE MODEL (II)

357

5. N u m e r i c a l Results on the Baryon Self-energy P r o b l e m The technique of section 4 was applied to find the low energy interaction of two baryons corresponding to the numerical solution of the baryon selfenergy problem with a meson dipole 1) The interaction turned out to have the wrong sign at larger radn for the low energy 3S~, SDx, ~So configurations, but the magnitudes were of the right order Thus at least for this case of pseudoscalar self-Interaction there appears to be contrachctory claims of the low energy interaction and the ba r yon self-energy problems Intuitively the selfenergy problem requires repulsive effects for the masses of the nucleon and A to be above ~) m0 whilst the low energy Interaction IS attractive In the above configurations These contradictory reqmrements suggested t hat self-energy and low energy interactions m a y have chfferent sources The former m a y then be due to an Interaction of ver y short range, and even of a different t ype than pseudoscalar, while the low energy interactions m a y be due to a longer range effect of possibly pseudoscalar orlgan. If one establishes a statable method of calculating the higher order ,-function contributions and the full effect of velocity dependence, one m a y even find t hat the very short range interactions give rise to suitable long range low energy interactions This will, however, have to await further developments In the meantime it seems of Importance to calculate the self-energy problem for a num ber of interaction types and for a large range of meson mass constants In the ~'F(x) function It also seemed advisable to take the simplest possible form for this function, involving only one mass for a given interaction type, as in eq (11), and the m m l m u m number of ghosts, as in eq (10) In this w a y it m a y be easier to separate the operation of the different factors Essentially this section will therefore report a detailed set of calculations of the baryon self-energy problem, based on section 3 As In the numerical work in paper 1, consider l = 1837 385, k ~-- 2182 5 and g = ½(l+k) ~ 2009 9425, where the electron mass 1s the mass unit here and for the rest of this paper T h e n C z = 1 = C k a n d C g = - 2 , b y e q (14) Now one notes t ha t if b y an appropriate selection of interaction and coupling, one can arrange m 0 = 0, the basic mass will have been accounted for and in addition the t h e o r y m a y then be In a better position to take in low mass objects such as leptons and photons All mass will then arise from field energy We shall now investigate, amongst others, whether this can be done Referring to table 1 one notices the following If one relaxes the conditions m 0 = 0 and x > 0, there are m a n y ways of satlsfyang the conditions of the self-energy problem alone If one wants m 0 ---- 0 and c¢ > 0, however, one must at least use some pseudovector interaction, preferably for a > 2500, where Jpv(A, a 2) > Jpv(N, a 2) To get the A-nucleon mass difference, one must then use some vector Interaction in addatlon, preferably for p in the range 400 to 1200 Unless higher order z-function corrections contribute the correct forces, the lowest order forces for such a selection of

358

E VAN DER SPUY

fundamental interactions alone are probably wrong In this connection one has, however, still to ascertain the role of the velocity dependence A posslb l h t y m a y be xf one uses p, a as large as possible for the self-energy problem, and # small to give the correct low energy interaction for the larger radii, as due to a pseudoscalar fundamental interaction TABLE 1 The self-energy t e r m s p or n

v 400 700 1200 1837 385 2182 5 2500 2800

1837 1837 1837 1837 1837 1837 1837

385 385 385 385 385 385 385

400 700 1200 1837 385 2182 5 2500 2800

2182 2182 2182 2182 2182 2182 2182

5 5 5 5 5 5 5

700 1837 385

2009 9425 2009 9425

J l ( p s, v*) --9 16 26 26 24 23 22

J s ( p s, v*)

Jps(n, v s)

J r ( n , v~)

J p v ( n , v s)

3293 8725 6510 3392 9201 4652 0924

--0 --0 --0 --0 --0 --0 0

0292149 0126657 00404603 00097262 00034263 00006926 00016823

--44 3497 --40 1443 --34 0851 --28 1263 --255496 --23 5925 --21 7833

--35 --57 --60 --54 --50 --47 --43

0204 0168 7361 4655 4697 0577 8757

--72 10 45 50 49 46 44

3376 4732 8679 8913 2107 8031 4939

--275 859 --18 0880 22 0022 26 1188 25 1527 23 8371 22 4966

--0 --0 --0 --0 --0 --0 0

152752 0297163 00696414 00159649 00065637 00019164 00006286

--57 --46 --37 --29 --26 --24 --22

218 --28 --59 --55 --51 --48 --44

3375 6798 2036 7219 7379 0925 8560

--885 --101 28 48 48 47 45

098 0318 8052 7533 8729 2559 1304

5 9483 26 3004

--0 0183607 - - 0 00124047

5214 7678 2014 6031 5852 2554 3594

--42 8522 --28 7937

--48 8005 --55 0941

--25 0073 50 1075

If one does not demand ~ > 0, then m 0 = 0 can be satisfied b y the pseudoscalar interaction with/~ around 700 Then the lowest order low energy interaction of two baryons has the wrong sign The next order ~-functlon term may, however, have both the correct sign (If it depends on ~2) and be strong enough and of larger range To make a selection one will have to pursue the outstanding points, namely the higher order contributions to the low energy interactions of two baryons, and the bearing of the fundamental interaction on the meson self-energy problem Furthermore a large range of unexplored posslblhtles of different choices of g and of more masses in (l~--m)' m a y have to be tested We conclude this section with two examples of satisfying the conditions of the baryon self-energy problem alone (a) Consider, as already suggested, a = 1837 385 and p ---- 700, for example Then by eqs (18), (22) with m o = 0, and table I, n = 2182 5

yields

2182 5 = 2{Kv(--28 6798)+Kvv(48 7533)},

n = 1837 385

yields

1837 385 = 2{Kv(--57 0168)+Kpv(50 8913)}

A

FUNDAMENTAL PARTICL]~ MODEL (II)

359

Hence Kpv = 27 1712,

Kv =

8 1394,

or

:¢pv = 64 0206,

:iv ~ 19 1780

Now test w h e t h e r the self-energy e q u a t i o n IS satisfied on the ghost mass shell At p = g ---- 2009 9425 the self-energy is given b y 2{Kv(--48 8005)+Kpv(50 1075)} ---- 1928 55 This is off the ghost mass shell b y 81 39 me This m a y be sufficient to indicate t h a t the ghost mass does not a p p e a r on the mass shell in the a s y m p t o t i c d o m a i n B y eq. (23) it readily follows for the above case t h a t NN = 1 25904, N A = 1 57050 (b) Consider the case of pseudoscalar i n t e r a c t i o n with aDS < 0 b u t m 0 = 0 B y eqs (18), (22) for/~ = 400, p ---- 1837 385 is satisfied when 1837 385 = Kps × ( - - 4 4 3497), or Kps = --41 4295 F o r this, p ---- 2182 5 gives the self-energy ----K~s(--57.5214) = 2383 08 F o r /~----700, p - ~ 1837 385 IS satisfied when 1837 385 = Kps(--40 1443), or Kps = --45 7695 F o r this, ib ~- 2182 5 gives the self-energy = K p , ( - - 4 6 7678) ---- 2140 54 L i n e a r i n t e r p o l a t i o n suggests t h a t f o r / , ---- 648 10 the self-energy equation is satisfied b o t h on the nucleon a n d A mass shells, with K ~ = - - 4 5 0187, or Ctps ---- --106 0728 I t m a y be n o t e d t h a t the self-energy alone requires large coupling constants, a l t h o u g h in case (a), the larger range v e c t o r interaction has a coupling cons t a n t of the f a m l h a r order I n this case the v e c t o r Interaction IS chiefly Ins t r u m e n t a l In effectmg the A-nucleon mass difference

6. The Lepton P r o b l e m T h e n u m e r i c a l work on the b a r y o n - m e s o n s y s t e m encourages one to t r y similar w o r k for the leptons n e u t n n o , electron a n d m u o n One a t t e m p t s to a c c o u n t for the l a t t e r mass s p e c t r u m b y a self-interaction T w o different a p p r o a c h e s were tried (a) One assumes an electromagnetic t y p e of self-interaction alone This is the strongest k n o w n coupling to the electron a n d muon, a n d such an a p p r o a c h t h e n I m m e d i a t e l y excludes the u n c h a r g e d neutrino F u r t h e r m o r e one realizes at the outset t h a t for the usual p e r t u r b a t i o n t h e o r y the fine s t r u c t u r e c o n s t a n t o~ = e2/4a?~c m 1 - ~ is m u c h too small to give a self-energy equal to the electron mass, say, with a n y reasonable a n d simple cut-off Nevertheless, since the ghost masses m a y m a k e an essential difference, a n d to find the coupling required, it was decided to investigate thls possibility Also, the electromagnetic coupling to the massive p r o t o n m a y change the picture T h e n too, a large mass ratio such as in the l e p t o n s y s t e m m a y be associated n a t u r a l l y w~th a small

360

F. VAN DER SeUY

couphng constant It IS also clear that Y-function theory m a y provide a close and transparent connection between the zero mass of the photon, gauge mvanance, charge conservation and the equality of the low energy coupling constant Consider thus a fundamental equation (~z ~01 (X) = 27~CXv T f d 41~1~ F p (~--~I)(tP m (~1)~ 'p ~om (Xl)))"p ~l Ca), where the lepton index l (or m) m a y have two values corresponding to the electron and muon parts of the 8-component splnor ~o(x) One sums over the two values of m when repeated, as on the right-hand slde By considering the low energy peripheral interaction, one can readily show (as in section 4) t h a t for this to have the infinite Coulomb force range, p = 0 One can also readily extend the analysis to Include the electromagnetic field generated by proton currents It turns out, however, if one assumes (OJTeplO), (OlT#plO) to be zero, that in the lowest order z-function theory, which IS all we attempted numerically, this cross-coupling to the proton currents has no contribution to make to the equation of motion of a single lepton These z-functions will naturally vamsh as long as e,/z, p (electron, muon, proton) truly are elementary particle fields One notes, in this connection, t h a t strong and electronmagnetic couphng alone leaves these particles stable in the current view In view of the above, we considered only the noted equation in the electron-muon fields, leaving the couphng constants to be determined by the equation of motion of the leptons on the mass shells It turns out that with p = 0, one has the difficulty of Imaginary contributions to the self-energy equation In momentum space on the muon mass shell If one assumes that these m a y be cancelled by adding a suitable very short range self-interaction to the above equation, without changing the real selfenergy terms, one can proceed to analyze the latter on the two mass shells in the usual way Consider one example, in which (#--m)' has four factors with masses 1, I04, 206 86, 1022 in units of the electron mass, the second and fourth being two ghost masses and the third the muon mass It turns out that one must have two ghost masses, because only one ghost leads to the wrong sign for the electron Coulomb force. This is due to N z being negative 111 this case. For the given four-mass case, the equation analogous to eq (18) can be satisfied with m 0 = 216 4 By considering the low energy peripheral interaction of two leptons (a Coulomb force) in this case one finds that the structure constant ~v corresponds to ee2/4~?ic = 2 99 and e~,2/4~?ic = 12 34, for the cases of two electrons and two muons, respectively This should be compared with ,tm ~ The size of the effective couphng (and with lnflmte range) is the most serious criticism of this result The difference of ee2 and %z is also a problem but this may possibly be solved b y choosing different ghost masses The other difficulty with this approach is the occurrence of imaginary self-energy parts

A FUNDAMENTAL

PARTICLE

MODEL

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which can only be ehmlnated in the most artificial, and possibly unphyslcal way It appears t h a t this approach involving the vector self-interaction of infinite range is apparently not the explanation for the charged masses, that is also compatible with their known electromagnetic mteractlon To be absolutely sure one should really take account of the electromagnetic coupling to the proton in the higher order ,-function theory The lowest order is so far out that the comphcatlon seems unwarranted (b) The other approach offered itself by reahzlng that perhaps the strong interaction needed to explain the lepton masses was of very short range, and t h a t the short range is the reason for its not having been detected so far In this case a vector self-lnteractlon and a fundamental equation such as in eq (12), with %s = 0 = %v, was considered The field operator ~(x) then IS a 12-component splnor, four components for each of the neutrino, electron and muon fields The analogy 8) between the basic leptons and basic baryons makes such a form perhaps appropriate Since one already reahzes by the above t h a t the coupling will have to be strong, elementary considerations show that for such an interaction to be less than the Coulomb force at rachl of the order of the muon Compton wavelength, say, p must be of the order of the nucleon mass or larger Here p = 1837 385, the mean nucleon mass, was tried In this approach It is natural to view the electron and neutrino as having the same mass, namely zero, so that the actual electron-neutrmo mass difference arises from the electromagnetic field It was expected that m 0 m a y now be of the order of the nucleon mass, so that the proposed vector self-mteractlon provides the large depression of the actual lepton masses below m0, and that then the relatively small electron mass will naturally be comparable to the protonneutron mass difference It was hoped that if a vector self-mteraction of the same order of coupling formed the basis of the baryon-meson system, one m a y be led to a way of umfylng the lepton-baryon system With the above value of p, and following section 3, but with a posslblhty of four masses in the fermlon propagator in eq (10), calculations were made on the lepton equation of motion in momentum space The lepton masses used were m ---- 0, 103 43, 206 86, 1837 385, where the first'and second are for the electron (and neutrino) and the muon, and the other two are ghosts It turns out t h a t the self-energy terms J l ( p 2, p2), j2(p2, p2) have a very small change (of the order of a percent or even less) from the electron (p = 0) to the muon (p = 206 86) mass shells This means that the eq (18) can apparently only be fitted with unnaturally large couphng constants and m o values It m a y thus be briefly stated that this approach seems unpromising for explammg the lepton mass spectrum without invoking couphngs of such a strength as should already have been observed Such a statement also covers the first approach In the present case one would also have to discuss why the lepton system does not show boson type compounds, as the baryon system does

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The present section will now be concluded b y sketching a tentative proposal for dealing with the lepton problem When one considers the details of the above discussion closely, it seems probable that unless one changes to some completely chfferent approach for fundamental particles of the lepton family, one has to face a strong coupling of some kind to account for the lepton masses The real problem is thus how such a strong coupling IS compatible with the apparent weak interaction of leptons amongst themselves, and with all other particles, at such longer ranges as have been slgmflcantly tested If one has to countenance some strong coupling for the leptons anyway, then an obvious desire of umfymg the theory of all elementary particles naturally suggests that one should try to consider baryons, mesons and leptons in a unified picture The resolution of the interaction problem immediately suggests that one should look for some selection rules, presumably of a dynamical kind Now the present approach has used fermions as the basic fields, and they obey the exclusion principle (see paper I for the appearance of this pnnclple in the z-function theory) This suggest a notion that m a y be helpful It is the notion that we shall refer to as saturation To delineate the notion, consider a probably crude analogy from non-relativistic particle physics" a particular kind of fermlon which taken two at a time would strongly interact withm a certain range of one another Since they obey the exclusion principle, only one of them at a time can occupy a given state in a system of such fermlons If one such level is occupied and at an interval z] from the nearest available unoccupied level, then the particular fermion will tend to show a weak apparent interaction with an incoming fermlon of the same kind, if the bombarding energy is sensibly smaller than A In a certain sense the system m a y be called saturated in the given fermlon and for the given level and bombarding energy The saturation is even here a relative concept Consider thus the analogy of such saturated compounds in the fundamental particle model, which has been called the baryon-meson model, and which is essentially a strong couphng model It IS suggested that perhaps certain saturated entities (or eventually particles) exist within this model, especially when m o = 0 as we have shown it to be possible, which will have a small mass elgenvalue and which will have no (or only a weak) interaction with all particles of the model except at rather high colhslon energies and within very small ranges Since this is a tentative model which has still to be tested theoretically b y the y-function theory, in its full dynamical detail, and has to be compared carefully with experiments, these saturated entities will not be given the names of known particles They will be referred to b y symbols, operator symbols in fact, which b y correspondence with usual practice for the leptons and photons, will indicate possible analogies There are three basic held operators in the present model, p, n, A, which are all included in the 12-component spmor field operator ~o, of eq (12), say.

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These generate the wavefunctlons of actual particles by the z and $ functions To be saturated in a given baryon type. for our purposes four of the relevant operators must occur in the ,-function one each for the two possible spin orientations of both particle and antiparticle We consider z-functions such as ,(#) = (0IT0[ 9)where 0 is a sum of baryon operator products, in general, with an odd number of operators if [9) is a fermlon state, and an even number of operators if [9) is a boson state For these cases z(0). or rather the correspondlng 6(0) is the wave-function of a fermlon and boson, respectively Consider some of the generating operators 0. in descending order of saturation (a) 0 : A~ = (finfin~pOTt,pAAAA), a boson operator (b) 0 ---- e ---- (5nfinpp~)AAAA) ---- (P)h or (P)hole as a new notation Indicating that a p has been removed from the saturated 0 = (finfnDpF)pAAAA). 0 =

v~ =

0=v~=

(n)h,

(A).,

with antiparticle operators ~ = (P)h. Ve = (fi)~. ~a = (A)~ All these are fermlon operators Next should follow less saturated boson operators, but these will be left out This is not necessarily because the corresponchng model particles do not exist, but because the present author IS not aware of known analogies to explore (c) The fermlon operators next in order of saturation employ nine operators. or three holes To avoid • being unsaturated in a given baryon (leachng to the corresponding compound having strong interactions with the given baryon type), the most saturated operators of this class are presumably the following in a fundamental theory with formally symmetric, strong self-interaction 0 ----/~o ---- (npA)~.

0 =/~-1 ---- (fipA)h.

• ----/z1 = (npA),

with their antiparticles These correspond to charged fermlons If the fundamental strong coupling is not formally symmetric (say with respect to A). a neutral fermlon operator of perhaps greater saturation than /~0 m a y be 0

t(PP--fin)

Written in this way p_l, #ob, /zl has a formal correspondence with holes of /:+, /:o, Z'- We shall, however, for slmphclty only discuss the most saturated compounds in the symmetric theory Consider any given saturated operator 0 To get the lowest mass and the most saturated particle of the glvqen type one Imagines that • Is symmetric (as an S level m particle physics) in all relative coordinates of the individual operators This means that • really should include an appropriate integration over all such coordinates, leaving just the centre of gravity coordinate This integration

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and the summation over all equivalent combinations of operators and the appropriate 7 matrices with the baryon field operators, shall not be mentioned explicitly at present The symbol for d~ IS thus a formal one for the moment We sketch now in general terms how these saturated operators m a y generate particles of low mass and weak interaction Consider any 0 = (B 1B 2 B, Bn) where B, is a baryon field operator To get the mass of the corresponding particle, one studies the elgenvalue problem arising from _-

_-

with ¢ = ,~d~ = (BIB ~

B.

B~) = (BIB ~ ,

(E~BpBr) ,

Bn)

in a purely formal way by the fundamental equation The operator ~ has a baryon order higher by two than d~B.+1 To get the interaction of the given particle with another baryon one studies in effect the equation of relative motion of d~l = (BIB, B~B,+I) or d)l = (B1B~ .. 13~... B~+I) = (BIB ~.

(B~BaB~) ~

Bn+l)

which is 3 baryon orders higher than • Also since one sums over the possible values of the baryon Index, r, ~b and 0t are the sum of various terms Hence if 0 is one of the nearly saturated operators already mentioned, the present proposal hopefully expects with the higher order of ~, d)t that the exclusion principle will give a small expectation value for the latter operators, or a small mass and a weak interaction One notes that m o m a y be zero Thus the possibility of zero mass or of zero self-energy will depend sensitively on the proper conjunction of fields lmphed by 0 The constraint on interaction at not too high energies seems to be stronger than that on the mass One expects the interaction between such saturated compounds to be even less, and these compounds will be very confined systems with a hmlted surrounding cloud Clearly class (c), the # compounds, will presumably have a stronger interaction, at larger radii and lower energies, as well as a larger mass, than classes (b), (a), which are more saturated Since the constraint on the interaction is probably stronger than on the mass, class (c) may have a fairly high mass To decide the existence of the saturated compounds within the strong couphng model, one has to study these higher order functions The simplest m a y be that of class (a) If one can develop the proper technique for this more saturated class, the classes (b), (c) may perhaps be treated from the point of view of holes in the more saturated system It is the Idea of this proposal t h a t the existence of all these particles is already determined within the scope of a strong coupling theory Their Interactions with one another and with the baryons and mesons may however be determined especially at lower energies by certain additional or residual weak interactions which will not necessarily be symmetrical with respect to the particles of the model The electromagnetic

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interaction discriminates between p and n, A In this proposal the basic p held carries the charge of all the particles of the model, and since strong coupling determines the nature of all particles, it is perhaps natural that the effective charge for all these particles should be the same, when determined at low energies and at a suitable range Pending knowledge of the exact nature of tile fundamental self-interaction, especially as to symmetry, it is uncertain whether/~0b lS saturated enough to exist as a lepton In the model One anticipates that if a partially saturated compound is not saturated enough to exist as a lepton the corresponding conlunctlon of fields m a y not be able to exist as an elementary particle in any form, although it m a y form a hypernucleus At a certain degree of saturation self-interaction is effectively strong and the combination massive enough to break up into smaller units hyperons, nucleons and mesons The group e, %, va,/~0,/~-1, #1 m a y be called model leptons, if the present proposal has the anticipated dynamical properties They all have B = --1 The lepton number L = - - B here One m a y here refer to analogies noted by Marshak et al s) between the lepton and baryon systems One can assign lsospm and strangeness quantum numbers to the different ~ operators, but this is not helpful, because It is an Incomplete specification of #, which m a y be quite misleading if used carelessly without reference to d~ and the underlying dynamics One will have to know more about these model leptons, and also about the model hyperons and mesons to know whether the model suggests natural addatlonal weak interactions 7. D i s c u s s i o n The present model has so far used a formally symmetric fundamental selfInteraction This was done not only for slmphclty but also to show that it is not incompatible with a baryon mass spectrum If it should be found necessary one can discuss a model in which the fundamental Interaction is unsymmetrical with respect to the basic baryon helds, m an analogous way It is useful to realize that known words when used in a new context m a y have a rather special and new meaning Such is, for example, the word "compound" When one suggests in the present model that x - IS a compound generated by the basic field operators (On), it is well to realize that a totally new meaning is associated wath compound in this context It is not, for example, an antiproton bound to a neutron Even if one could talk of antlproton and neutron fields (or particles) in this context, one reahzes that they would propagate completely off their mass shells, and that they are presumably dressed differently from a single antlproton and neutron, respectively Moreover the dynamms must be completely relativistic, employing a fully covanant m a n y time

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formalism in the process This has no analogue in the classical non-relativistic framework m which the usual meaning of compound arises In view of the novelty of this domain great care is needed in the use of words which come naturally with the usual concept of compound, but m a y in the field of fundamental particles not only lead to error but m a y In fact be of little use and unnecessary Such words m a y be "excited states" and "intermediate states". This care is probably especially necessary in strong coupling theory These words need critical discussion To be able to state the other essential demands on the present model the next stage should attempt the full calculation of the simplest compound problem the meson mass elgenvalues, with numerical computatson This problem depends essentially on the boundary conditions for the more complex vfunctions needed After that one could study the higher order v-functions arising in the meson-baryon scattering problem, or with the compound baryons (2, g), or the existence of saturated compounds The present author should hke to thank the Oak Ridge National Laboratories for the hospitality of the Neutron Physics Division which he enjoyed during the performance of this research He also thanks the South African Atomic Energy Board for the opportunity to do this work on his overseas tour of duty References 1) E van der Spuy, Nuclear Physms 29 (1962) 400 2) E Freese, Zelt f Naturf 8a (1953) 776 3) H -P Durr, W I-Ielsenberg, H Mltter, S Schheder and K Yamazakl, Zelt f Naturf l g a (1959) 441, and numerous references mdmated m this paper 4) S Sakata, Prog Theor Phys (Kyoto) 16 (1956) 686 5) H Lehmann, iXiuovo Clm 11 (1954) 342 6) S S Schweber, H A Bethe and F de Hoffmann, Mesons and fields, Vol 1, Fmlds (Row, Peterson and C o , Evanston, 1955) p 35 7) L Rosenfeld, Nuclear forces (North-Holland Publ Co , Amsterdam, 1948) p 56 8) 1~ E Marshak and S Okubo, Nuovo Clm 19 (1961) 1226 and references mdmated m this paper