]L NUOVO CIMENTO

VOL. XXV, N. 3

1o Aprile 1962

Bound States and Elementary Particles. F. E. L o w Institute o] Physics, University o] Rome - Rome (*) (ricevuto il 19 Aprile 1962)

- - A simple theory is constructed within which one cannot distinguish which of two particles (here called a nucleon and a ~-hyperon) is elementary and which is a bound state of two other particles (here called a A and a K, or ]~, meson) or whether both are elementary. No condition need be imposed on the masses except those required to ensure stability, and to ensure that the center of mass of the ~ and the be greater than the mass of the A. Summary.

1. -

Introduction.

A m o n g the s t r o n g l y i n t e r a c t i n g particles t h e difficulty o f defining t h e dist i n c t i o n b e t w e e n an e l e m e n t a r y particle a n d a b o u n d state of t w o o t h e r particles is well k n o w n . Clearly, no p u r e l y e x p e r i m e n t a l distinction exists, since all k n o w n particles can be m a d e to t r a n s m u t e i n t o each other. Also, f r o m t h e p o i n t of view of dispersion t h e o r y a n d p o l o l o g y all particles give rise t o t h e same poles a n d cuts irrespective of their origin (1). I n a L a g r a n g i a n field t h e o r y , however, there is at least a f o r m a l difference b e t w e e n those particles which c o r r e s p o n d to the fields i n t r o d u c e d into t h e e q u a t i o n s a n d all others, w h i c h are b o u n d states whose masses a n d coupling c o n s t a n t s can be calculated f r o m those of the original ones. T h e idea t h a t m o s t particles can be a c c o u n t e d for as b o u n d states of a few f u n d a m e n t a l ones is v e r y appealing. On t h e o t h e r h a n d , a t h e o r y t h a t does this requires a criterion for w h i c h particles m u s t be p u t i n t o t h e equutions of m o t i o n a n d w h i c h d e r i v e d f r o m t h e m . W i t h i n t h e r e a l m of s t r o n g (') Permanent adress: Physics Department, Massaehussetts Institute of Technology, C~tmbridge, Mass. (~) K. NISHIJIMA: Progr. Theor. Phys., l i i , 995 (1958); R. HAAG: Phys. Rev., 112, 669 (1958); W. ZIMMERMAN: NUOVO Cimento, 10, 597 (1958); 13, 503 (1959).

BOUNI)

STATES

AND ELEMENTARY

PARTICLES

6~9

interactions there appears to be little e x p e r i m e n t a l guidance for the choice. E x p e r i m e n t a l l y a particle A looks like a bound state of particles B and C when m ~ = m B + m ~ , - - e , with ~ small c o m p a r e d to all three masses, and when the quasi-elastic scattering of a high energy projectile b y A is a p p r o x i m a t e l y equal, in the a p p r o p r i a t e region of phase space, to the scattering b y B (or C) alone. This is not the case a m o n g the e l e m e n t a r y particles. An a t t r a c t i v e idea which overcomes this difficulty is t h a t nature m a y be to a certain e x t e n t indifferent to the choice of L a g r a n g i a n (~). T h a t is, masses and coupling constants of the particles could be such that, starting f r o m a large n u m b e r of suitable subsets of particles, the others could always be recovered as b o u n d states, the scattering being identical for all choices of the initial set. There would then be no distinction between e l e m e n t a r y and comp o u n d particles. H o w wide this indifference could be is h a r d to say. E v e n if one considers only renormalizable interactions between known stable (with respect to the strong interactions) particles the r e q u i r e m e n t of complete indifference leads to m a n y more equations (between masses and coupling eonstants) t h a n unknowns. Thus unless there are some r e m a r k a b l e identities there can be no solutions. I n addition, even if one succeeds in satisfying the equations for the masses and coupling constants, it does not a u t o m a t i c a l l y follow t h a t the scattering will be identical. We h a v e investigated the trivial model described in the n e x t section in the hope of throwing some light on this question. T h a t is, do there exist consistent systems of the sort described, and are there identities reducing the n u m b e r of equations connecting theme. The answer to b o t h questions, for our model, is affirmative, although the existence of the identities m a y well be a consequence of the e x t r e m e s y m m e t r y introduced f r o m the beginning. Our model is physical in t h a t the states in it h a v e the same additive q u a n t u m n u m b e r s as real particles: the b a r y o n s of K-mesons. I t does not correspond to the real world~ since it neglects b a r y o n recoil in an unjustifiable way. Also it assumes even K A ~ and K A E parity, which does not a p p e a r to be the case in nature.

2.-

The model.

We consider three (3) alternative models: 1) A nucleon (of mass m~o), a A-hyperon (of mass ma) and a K - m e s o n (2) R. P. FEY~'~IAN: Talk at Aix-en-Provence Con]erence on Elementary Particles (1961); G. F. CHEW and S. C. FRAUTSCHI: Phys. Rev. Lett., 7, 354 (1961). (a) Models (1) and (2) were discussed in a preprint whose title was the same as that of the present paper. The possibility that model (3) might give the same result

680

1+. E. LOW

(of mass 1), coupled b y the interaction

(1)

r

_(0)~

H~ : y~ '~vx~vA~vK -[- h. c.,

where g~0~ is t h e unrenormalized coupling constant. I n this scheme the ~ ~h y p e r o n emerges as a b o u n d state of the s y s t e m (AK); its mass m~ and renormalized coupling constant g~ can be calculated from m ~ v , m A and g~ Scalar coupling in (1) makes the A K interaction a t t r a c t i v e in the J = ½ state and thus produces a ": with J = 1. A pseudo-scalar interaction would be attractive in the J = ~- state. 2) A E-hyperon (of mass m_-), a A - h y p e r o n and a K-meson, coupled b y the interaction (2)

H '2 = g~'~a~VAC~K ÷ h. e.,

where C is a charge conjugation matrix. I n this scheme the nucleon emerges as a b o u n d state of the system (AK); its mass m~v and coupling constant gl can be calculated from mA, m~ and g~. 3) B o t h a nucleon and a E-hyperon, coupled b y the interaction (3)

[

!

H a =

H 1 ÷

!

H 2 •

E a c h model, of course, has its own bare Hilbert space and its own bare coupling constant (s). W e wish to show t h a t for an appropriate choice of the renormalized coupling constants gl and g~ all three models m a y lead to the same A K and A K scattering (and hence to the same location of the Ap and 7= masses). A direct L a g r a n g i a n calculation which is sufficiently n o n - p e r t u r b a t i v e to produce binding leads to serious difficulty with renormalization effects. I n a n y case, the F o c k space equations for these models cannot be solved exactly, so t h a t one is in practice reduced to a t t e m p t i n g to sum a sub-set of the p e r t u r b a t i o n series. I n the static limit, which we take, one can sum t h a t sub-set of terms t h a t corresponds to a solution of the dispersion equations in the one meson approximation. One of course chooses t h a t solution whose power series expansion for small g2 satisfies the dispersion relations t e r m b y term. One can do some-

was suggested to the author by a theorem of Salam, for which the agreement of all three models provides a partial experimental check, although not all the steps leading to the final result can be verified within our model. The author thanks Professor SALAM for an interesting discussion of his theorem and for a very pleasant visit to Imperial College.

BOUND

STATES AND E L E M E N T A R Y

PARTICLES

681

what better 0), and obtain an exact consistency check, by lumping all the terms one cannot calculate into an assumed ratio of inelastic to elastic scattering cross-section, where this ratio is taken to be the same for all three theories. To be specific we assume

(4)

[m

1

= -- Ax(co),

]KA and 1 Im ]KA -- -- Ax(o)),

(5)

where o~ is the meson energy and ]KA and /~A are, respectively, the elastic K A and K A scattering amplitudes. One has At:= q + B~ , (6)

A~=

q÷B~,

where q = ~ / o ) "~- 1 ,

o)>l,

q~O,

(o<1 ,

and B K and B~ are also positive, and vanish for (7)

<,>4:- m A < mov + m.~

"rod (s)

respectively. We assume, of course, t h a t the ~ absorption threshold is smaller t h a n o ) = 1 . Otherwise, tile A's are zero for o ) < 1 . F o r ~ o > l each B is q times the ratio of inelastic to elastic scattering for t h a t particle. E v i d e n t l y the threshold (as well as the functions A) depend on the existence of both states (.N~ and 7~) so t h a t for model (1) and (2) our procedure is slightly back-handed, since we have not yet solved the elastic scattering problem to find the particles. Still, as an exact consistency check, we can imagine t h a t we have solved for the B's separately and exactly, and t h a t they are the same for the three models. We can then ask whether the elastic scattering is also the same. We now turn to t h a t l)roblem. (4) This possibility was pointed out to me by W. E. LA~tB.

682

F.E.

3. -

Dispersion

relations

LOW

and solutions.

The dispersion relations (always in the static limit) are

(9)

/i(A((o)

--

g~ co + ml

+

gl

-+ (o + rna

1 I

d(o' I m~f_TA

+ -

5~

(2} - -

(o/ ) + _l ( d o J ' I m ] ( ' 09

21;

.]

~ a ~0 )

O) - [ - ( 0

'

and

(lO)

/XA{~) =/xA(-- (o),

where m, = m A - ~'nx and ma-- m = - - m A. Evidently, for stability, and [m2]< 1. Of course, eq. (9) and (10) must be supplemented b y t a r i t y r e l a t i o n s (5) and (6). The solution of (5), (6), (9) and (10) corresponding to an e l e m e n t a r y i.e. t h a t solution which satisfies (5), (6), (9), and (10) term b y term as series in g~ for g22 0 is

[m~ [< 1 the uninucleon, a power

l ~ ( ( o ) -- - - g~/(o + m l 1 - - g~I~((o) '

(11) where

(12)

-

(o + m jd(o,I n

,

+

[((o + ~ 2 0 ' ) ( ( o ' - - m D 2 (

&(q)

° / H e})

provided AK and Ai~ are such t h a t the integrals converge at infinity, and provided (13)

g~Ii(--

m2) =

1.

Since A K and A~ are positive, m2-- ml must be positive, i.e. m = + m ~ v > 2 m A. Since 11((o) decreases monotonically in the interval where it is real, there is at most one bound state, whose coupling constant is given b y

(14)

g~ . . . . . . . . . . . . . .

l

, .....

- - (m,~-- m d L ( - -

rn~)

> O .

.(1) z \ The solution corresponding to an eleOf course IRate)) is given b y ]KA(--(O)" (1) m e n t a r y = is, analogously,

(15)

,,3,

]KA

.....

g~/(o + m~ ~

...........

1 - - gfl~((o)

,

BOUND STATES AND ELEMENTARY PARTICLES

683

where (16)

I~((o)

A~(~')

AK((~/)

+ m°fd(,'[

]

(~)'-- m2)~(o/ + ~o)~ ,

provided (17)

g212(--m,) = 1.

As before, g~ will be automatically positive. The simplest procedure to check the identity of ](" and f2> is to take m~ and m2 as given, and eliminate g~ and g~ b y eq. (13) and (17). We must then show t h a t

(18)

(o~ + m=)[I2(-- mr) - - Ia((o)] = - - (o, + m~)[Ii(-- m2) - - I1((~))] •

T h a t this is the case follows immediately upon substitution of (12) and (16) into (18). Finally, the solution corresponding to elementary ~V and ~ is

(19)

]~):~ =

--g[/o) + m ~ ( t - - g ~ I 2 ( - - ~V+I)) + g2/o) + m ~ ( 1 - - g ~ I ~ ( - - m 2 ) )

= Y - g~.q~r,( - m~)I~(- m~)- g~(1- gV2(- ~)) L ((o)- g~(~- g ~ ( - m~))I2(~) ' and

(2o)

(3)

/

x

KA (50) =

(3)

] K A ( - - (O) •

E v i d e n t l y ](3>XAis indeterminate at the physical value of g~ and g~. However, it has a unique limit at t h a t point, which is in fact equal to JKA t:(1) and ]KA'(2) To see this, following eq. (13) and (17), we let (2])

g22 i 2 ( - -

ml) :

1 -- x

and

(2'_))

gl2I i ( - - m~) = 1 - - ~ x ,

where x - ~ 0, ~ remains arbitrary. and denominator we have

(23)

• ~o

--

Keeping linear terms in x in n u m e r a t o r

g2/~ + m2 2 1 ~ ~ - - agxI~(~o ) - - g~23 1- - 2 ( ~ )

,

which appears to depend on ~ but in fact does not, provided g~ and g~ have their correct values as given b y eq. (13) and (17). In particular it is trivial to verify t h a t the equality of ]KA(1)and ]KA~)implies t h a t ]xA(3)(as given b y eq. (23)) has, independent of ~, the same value as at ~ - 0 (which is identically equal

684

F.E. LOW

to ]~)A)- Therefore, as stated, the K A and K A scattering amplitudes are identical for all three models. F u r t h e r m o r e , the n u m b e r of independent equations is two (rather t h a n four) since one m a y choose m~ and m2 arbitrarily and calculate g~ and g~ from them.

4. -

Example:

one-meson

approximation.

I n this limit, A K = A~-----q, and all the integrals are elementary. the results for completeness. The coupling constants are given b y g~

(24)

V1-

m~

g~

1 - - m 2 m l -~ %/1 - - m l2" V 1 . . .-. .-. m22

V 1 - m~

m ~ - m~

We list

I n the weak binding limit, m2-~1, g2~-+~/2(1-m2), in agreement with usual expression (the weakly b o u n d nucleon of course corresponds to the limit m~ --> - - 1 ) .

The scattering phase shift is given b y

(25)

- - g~

- - q ctg o)+ml

g~ ~KA =

1 ~

Vl--m~

] + mv~

e)+ml

and (26)

g~

g~

to - - ml q ctg 5KA ~ I

1-

ml~o

~ / 1 - - m~ co - - m~ '

in terms of the basic parameters of model (1). To translate from model (1) to model (2), g~-->--g~ and m i - ~ m 2 . Of course the two sets of expressions will be identical if eq. (24) is satisfied.

I should like to t h a n k the Guggenheim F o u n d a t i o n and the F u l b r i g h t Commission for support, and the University of R o m e and m y colleagues at the I n s t i t u t e of Physics for their kind hospitality.

RIASSUNTO

(*)

Si costruisce una semplice teoria nella quale non si pus distinguere se una delle due particelle (qui chiamate nucleone e iperone E) ~ elementare e l'altra 5 uno state legato di oltre due particelle (qui chiamate mesoni A e K) o se tutt'e due sono elementari. Non ~ necessario imporre alcuna condizione alle masse salvo quelle richieste per assicurare la stabilit~ e che il centre di massa de]l'2~ e del E sia maggiore della massa del A. (*) T r a d u z i o n c

a c u r a della R e d a z i o n e .