IL NUOVO CIMENT0 VOL. IX, N. 2 16 Luglio 1958

On the ChaTge Conjugation of Baryons (*).

P. BUDINI, ~N. DALLAPORTA a nd L. FONDA

I s t i t u t i di F i s ica delle Universi td di Padova e Trieste I s t i tu to Naz ionale di F i s i ea Nucleare - Sezione di Padova

(ricevuto il 17 Maggio 1958)

S u m m a r y , - - If we consider the baryon as composed by a bare spinor without charge nor strangeness and the known meson fields (both n ~nd K) one may investigate the behaviour of the lagrangian describing this model under an operation which reverses the sign of charge and strangeness of the mesons wihout altering the. bare baryon (boson conjugation B) ~nd an operation which turns the bare baryon into its antiparticle without changing the meson fields (spinor conjugation S), the product of these two operations being the usual charge conjugation C. It is found, when we neglect mass differences between nucleon and hyperons, that the lagrangian of this system is invar iant for each of the three operations B, S and C separately. The same property is deduced also for the d 'Espagnat- Prentki lagrangian of an assembly of real nucleons, hyperons and mesons and for the interactions with the electromagnetic field of all these par- ticles. Finally it is suggested that the mass difference between nucleon and E could be due to an interaction term in the lagrangian which should be invar iant under charge conjugation but not under the B and S conju- gations separately and a simple proposal for such a term is made.

1. - I t is well k n o w n t h a t in pure e l ec t rodynamics , the opera t ion l ead ing

to charge c o n j u g a t i o n is ach ieved b y l i nk ing ti le t r a n s f o r m a t i o n which reverses

the s ign of the e l ec t romagne t i c field w i th the t r a n s f o r m a t i o n which conver t s

the par t ic les in to an t ipa r t i c l e s (~),

(*) This work in a preliminary form has been presented at the P a d u a - V e n i c e Conference ou Mesons and recently diseocered Particles, September 1957.

(1) A. I)Ms and R. JOST: Phys . Rev., 87, 871 (1952); L. WOLFENST~II~ and D. G. RAVENItaLL: Phys . Rev., 88, 279 (1952).

ON Ttt]~ C H A R G E C O N J U G A T I O N OF B A R Y O N S 317

Up to now, the same formalism has been assumed to hold also for the charge

conjugat ion of nucleons, and by implicit extension of all kind of baryons ;

t h a t is, it has been assumed tha t also for baryons, the reversal of the sign of the electric charge implies tile simultaneous t ransformat ion of the baryon into

the corresponding ant ibaryon. As the t ransformat ion of a baryon into an an t iba ryon should imply the change of sign of the baryon number, this means

tha t in the operation of charge conjugation, it is implicitly assumed tlm.t electric charge and b~ryon number are linked together in order tha t they both

s imul taneously reverse their sign when charge conjugation is applied. This however, does not seem to be a necessary condition, at least for some

possible ba ryon models; moreover, a number of different baryon states has been in these last years exper imental ly observed, and if, ao(,ording to tile sug- gestion of different authors, we disregard their m~ss differenees, these different

s tates may be tried to be a( 'eounted for as a consequence of general symmet ry propert ies applied to the baryons. In this line of thought one eould therefore

be t empted to consider wha t kind of general properties or symmetries should be expected and obtained when we define operations in whi('h either the sign

of the ele(,trie vharges is reversed without ( 'hanging the baryon nulnt)er or the baryon number is reversed without ( 'hanging the ele('tri(' charges; and

whether some known experimental baryon state would fit into these s(,hemes. In order to develop first these considerations on a well (|(,titLed case, we will

t r y in the present work to define and apply these possible new operations on

nucleons. I t will be found tha t the result of the first operation (change of charge wi thout changing ba ryon number) which will be called the boson

conjugat ion B transforms, when differences of ulass are disregarded, the nucleons into the ~.'s, while the result of the second operation ((;h'mge of ba ryon number without c h a n g i n g electric charges) which will be called the

spinor conjugat ion S t ransforms the nucleons into anti ~.'s, while of course

these two combined operations, which are equivalent to usual particle-anti- part icle conjugat ion C, t ransform the nucleons into autinucleons and ~?s into ant i ~ 's . The results so achieved will be extended to all baryons, and finally

the effect of possible reasons for the original mass difference will be briefly

discussed.

2 . - As a first point, we must stress our tha t our aim is not to look for just a ma themat ica l possibility, allowing to write formally the t ransformations

corresponding to our preceding definitions bu t to point out wh'Lt their physical

me-ruing could be as related to what we m ay call the s tructure of the nucleon. I n this sense, the easiest way to obtain some physical insight in the problem

we are looking for, m ay consist in assuming some model for the nucleon, ade- quate to underline the fundamenta l features of its constitution, in agreement

with our present knowledge. We shall adap t to this aim a model tha t has al-

318 P. BUDINI , N. DALLAPORTA and L. FONDA

ready been proposed (2), consisting of a bare spinor baryon Bo with no isospin

and no strangeness, which is the source of both the K and the pion field; the

real baryons are then obtained by clothing the bare Bo both with K's and :: 's .

This gives the possibility of constructing all the different states of electric

charge and strangeness tha t have been ohserved up to now. The nucleons

in part icular will then be obtained by clothing first Bo with the two K-mesons

of positive strangeness, K + and K0 obtaining thus two states N O (570 + and ~0 °)

and then letting pions being exchanged between these states in order to obtain

the real proton and neutron. In symbols we will have:

(1) 5Io = Bo + K ,

N -- 5To+~ ,

1"I0 ~ = Bo + K + ,

e = N i l + ~ + = ~ o + ~ ° ,

Iq ° = Bo + K °

= No + + ~ - = N ~ + ~ o .

I t may be observed that by strangeness we mean here what has been termed

by SCnWIr~GER (a) <(hypercharge }) and differs by one positive unit from

Gell-Mann'~ strangeness. Let us first remark that the nature of the K-meson field is such that for

mesons positive charge is always associated with positive strangeness, and

negative charge with negative strangeness, and never the reverse (i.e. only

K + and K - exist). Thus a reversal of the charge for all the mesons can only

happen if also there is a simultaneous reversal of the str 'mgeness. This means

that there is no physical meaning in a charge conjugation of the mesons

without a simultaneous strangeness conjugation. We are now in a position to give a physical definition to the operations

we are looking for:

1) We shall define as boson conjugation the simultaneous reversal of

the sign of electric charge and strangeness of the whole meson field without

touching anything concerning the bare baryon Bo: Then K transforms to K

and our formulas (1) become:

(2) E = B o + K ~ 0

- E + = 0 7~ 5

E o = Bo + K - ,

- - . ~ o + = _ - + = . M, = ~ 0 = '~ '0 '

~o = Bo + K ° ~ 0

The states in which the ~N o and N particles transform are here defined as

.% and ~ and it may be easily recognized, if we neglect mass differences, tha t

they in effect possess the right values for charge and strangeness in order to

identify them with tile experimental 7~'s.

(2 ) N . D A L L A P O R T A : Nuovo Cimento, 7, 200 (1958). (a) j . SCHWIN(+Er¢: Phys. Rev., i04, 1164 (1956).

ON T H E C H A R G E C O N J U G A T I O N OF B A R Y O N S 319

2) L e t us n e x t def ine as s p i n o r c o n j u g a t i o n t h e t r a n s f o r m a t i o n of t he

b a r e b a r y o n Bo in to i t s a n t i p a r t i c l e , w i t h o u t c h a n g i n g ~ n y t h i n g to t h e who le

o f i t s m e s o n field. B0 c h a n g e s t h e n in to lJo, a n d f o r m u l a s (1) b e c o m e :

X o = B o + K , {3) _

X = X o + r : ,

Xo + == B o + K + ,

X + =X00 + ~ : + = X o + + ~ o ,

X ° B o + K ° ,

X ° - X ~ + r c - = X o ° + r ~ ° .

T h e n e w ~mt is ta tes o b t a i n e d b y th is t r a n s f o r m a t i o n a re g iven t h e s y m b o l s

X ° a n d X . W e sha l l i d e n t i f y t h e m a l i t t l e fu r the r .

3) L e t us c o m b i n e n o w t h e t w o B a n d S c o n j u g a t i o n s , in o r d e r to r eve r se

s i m u l t a n e o u s l y t h e s igns of c h a r g e a n d s t r a n g e n e s s of t h e who le m e s o n field,

a n d to t r a n s f o r m t h e b a r e b a r y o n i n t o i ts a n t i p a r t i c l e . W h a t we o b t a i n t h e n ,

is o b v i o u s l y w h a t is u s u a l l y ca l l ed cha rge c o n j u g a t i o n a n d t h e t r a n s f o r m e d

s t a t e s for No a n d N a r e :

(4) N,, = B° + K ,

N = N o + ~ ,

N O B o + K ,

P - = N , , + = ° N I l + ~ - ,

Nil l~o + K o ,

No N,, + ~.+ = Nil + =o .

N,~ a n d N ~u'e now o b v i o u s l y t h e a n t i n u c l e o n s . Moreove r , t i le C B. ,S

t r a n s f o r m a t i o n w h i c h e n a b l e s us to go f r o m (1) to (4), if now :+pplied to (2)

wi l l g ive us t h e s t a t e s de f ined b y (3). This shows t h a t t he r e l a t i o n b e t w e e n

t h e X a n d ~2~ to t i l e ~ a n d Eo, is t h e s a m e as t h e r e l a t i o n b e t w e e n t i le an t i -

n u c l e o n s N~ a n d N to t h e n u c l e o n s No a n d N. This t h e r e f o r e enab le s us to

def ine t h e X~ a n d X as t h e a n t i p a r t i c l e s of ~o a n d ~ a n d a l lows us to g ive

t h e m t h e s y m b o l s X0 = Eo, X + = E+-

3. - W e shM1 now g ive t h e m a t h e m a t i e M e x p r e s s i o n for t h e t r a n s f o r m a t i o n

we a r e cons ide r ing . L e t is i n t r o d u c e t h e fo l lowing s y m b o l s for t h e p a r t i c l e

s t a t e s :

( 5 ) N o = ~ o , N = _ , K K o

a n d def ine

0) u = ± iv3

3 2 0 P. B U D I N I , N. D A L L A P O R T A a n d L. F O N D A

Then the Lagrangian of our model will be assumed as

L = L ° + L ' ,

(6) L ' = f lP' d x d y d z ,

~ ' = g~iN~5~No::~ + g~.N,,uK(F)Bo + h. c. + 8m~NN + 8mNoNo~ 0 .

The operation F means either iy5 or 1 according as we assume pseudo-

scalar or scalar coupling; we have chosen u representation for which ~ = 1;

g.~ and g~ are the pion and the interaction constants.

Then the transformations defining the B, the S and the C conjugation are given as follows:

(7)

B S C

~:i = , ( - - 1)' - - =i x i ( - - 1) '+1 K' - - K r - - K ]~r

B'o - - B o - - C - , B T c - , B T

N ' B ~ S -~ ~ C : ~ ~

where CyniC -~ ----y~,,-- r B = TT~°I~xl(}] , S ~ C a . ~[,q. : TT. ( . ~ 111) .

(s) C = C u , u = ('o-~), B . S = - - C .

One may easily verify tha t the Lagrangian (6) is invariant for all three

operations B, S, and C given according to the transformations (7).

4. - The previous model or any other of the same kind that distinguishes

the bare baryon from the mesonic field was necessary in order to allow us to

give a precise physical meaning to the different conjugation states of the

physical baryons tha t are obtained by applying the different operations we

have been considering. However, now that these new physical states have

been obtained according to the foregoing physical insight, we may use them

in order to extend our results and apply our operation not only to a given

baryon model, but to a real assembly of physical baryons of ~ and K mesons.

I f we assume the usually a.dopted interaction terms (~) for the different part- ides, we obtain ~ general interaction lagrangian:

L ' : f ~ ' d x d y d z ,

(9) /P' = g~iNysr~NT:~ + g,zA [A(F)n" Z + h. e.] --

(4) B. D'ESPAGNAT and J. PRENTKI: Nuclear Physics, t, 33 (1956).

ON THE Ct][ARGE CONJUGATION OF BARYONS 321

s ~ is the u n i t a r y a n t i s y m m e t r i c t e n s o r ; t he i n t e r a c t i o n cons t an t s , if ex-

p re s sed w i t h those def ined in the work of D'ESPAGNAT, PRENTKI a n d S~-

zA~I (~) a r e ;

g~ = gi = g4 , gA'-' = - - g2 , gx'z = - - g~, g,A - - g5 = g7 , g~r = g~ = ~ gs "

I t is n o w easy to show t h a t w i t h the a d d i t i o n to t ab l e (7) of some ne w oper-

a t i o n s c o n c e r n i n g the par t i c les t h a t were n o t cons idered "~s ye t t a b u l a t e d in

t a b l e (10)

B S C (10) A ' - - A - - C-~-X 'r C-1A 'r

Y]I ~ i ( - - 1) i+1 C - ' ~ : ' C -1 Z~ ' ( - - ]),+1

one can e x t e n d the resu l t s of Sect . 3 to th i s more genera l L a g r a n g i n n a n d ver i fy

t h a t i t is in f~ct i n v a r i a n t for the th ree k i nds of t r a n s f o r m a t i o n s (B, S a n d

C ( ' on juga t ion ) we h a v e cons idered . The conc lus ion is t h e n t h a t s t rong pion

a~nd K i n t e r a c t i o n s , which are the on ly ones i n c l u d e d in our L a g r ~ n g i a n are

i n w t r i a n t u n d e r the B a n d S ope ra t i on sepa ra t e ly , a n d as charge c o n j u g a t i o n

expresses the r e l a t i on b e t w e e n e + and e-, t he th ree B, S a n d C c o n j u g a t i o n s

m a y es tab l i sh the c o n n e c t i o n b e t w e e n the four s t a te 5 7 E E N .

5. - Le t us now i n t r o d u c e in our L a g r a n g i a n f u r t h e r t e rms co r re spond ing

to the i n t e r a c t i o n w i t h an e l e c t r o m a g n e t i c field: we have of course to select

o n l y cha rged c o m p o n e n t s a n d m o r e o v e r t he s ign of the E c o m p o n e n t s of N

is n o w reve r sed in r e spec t to the s ign of the n u c l e o n s ; therefore our p rev ious

s y m b o l s (5) c a n n o t be used a n y m o r e and we m u s t wr i te wi th usuul n o t a t i o n s :

(11)

~ ' = i e [ K + grad~ K - - - g r a d K + K -] -A~ +

+ ie(r:+ g r a d ~ - - - g r a d ~ + ~ : - ) A ~ - - e ~ (K+K - + T:+~-)A~ + ePy~P .A~ - -

L e t us i n v e s t i g a t e the i n v a r i a n c e p rope r t i e s of these t e rms wi th respect

to t h e s ame th ree k i n d s of t r a n s f o r m a t i o n s . I t c a n t h e n be verif ied as before

t h a t if we a s s u m e for A the fo l lowing t r a n s f o r m a t i o n proper t i es :

B S C (12) A' - - A A - - A

(5) B. D'EsPAGNAT, J. PRENTKI ~nd A. SALAM: Nuclear Physics, 3, 446 (1957).

3 2 2 P. BUDINI , N. DALLAPORTA 8 n d L. FONDA

also ~11 these new electromagnet ic interact ion terms remain invar ian t under

the three B, S and C t ransformat ions . We can therefore generalize our previous

s t a t emen t and conclude t ha t boson and spinor invar iance are valid for all the strong and electromagnet ic interact ion te rms generally considered in the usual

hamil tonians .

6. - All the previous derivat ions have been obtained according to the

assumpt ion t ha t the masses of all baryons are equal, and especially the masses

of nucleons and ~ 's . This of course is not true, and the very conspicuous

mass difference of these two particles indicates tha t there is a ra ther s t rong dis turbance which prevents the B and S invariance propert ies to be effective.

The reason for this d i ssymet ry has been generally a t t r ibu ted to a different

s t rength in the coupling for part icles of opposed strangeness. As an example in the first Schwinger (3) scheme, it was supposed tha t the pion field was

genera ted bo th b y nuclear charge and hypercharge , and as the pairs of these

q u a n t u m numbers were different for nucleons and 7~'s~ it tu rned out tha t the to ta l s t rength of the pion field genera ted by each of them, was different;

in the Gell-Mann (6) scheme, it was explicit ly supposed tha t the coupling

constant gxK¥ and gzKv are different. The consideration outlined in the present paper m a y perhaps suggest tha t

the reason of the mass differences of the baryons would be accounted for b y the introduct ion into the hamil tonian of one or more te rms which, in order

to preserve the equal i ty of the masses of particles and antipart icles as sup- por ted by the propert ies of antinucleons, should preserve invariance under charge

conjugation, bu t not under B and S conjugat ion separately. And as the mass spli t t ing is obviously connected with the strangeness

dependent interactions, it would be na tura l to postula te tha t such a t e rm should connect the baryons with the K mesons. A simples choice for it could be:

i~7~ (K bK b K K ) 5x# bx#

where by B we intend any kind of ba ryon (Bo, No, N for the Lagrangian (6), K ~

or :N, A, E, for Lagrangian (9) and now K = KO and K - ~ I K - K ° I .

I t m a y be shown tha t such an interact ion changes sign for bo th B and S conjugat ion and remains invar ian t for charge conjugation. According to it a ba ryon can emit and absorb in a single act two different K-mesons. I f we add

(6) M. GELL-MANN: Phys. Rev., 106, 1296 (1957).

O N T I I I ~ C I [ A R G I ~ " C O N , ) I J ( ; A T I O N O F B A R Y O N S 323

s u c h t e r m s t o o u r L ~ g r ~ n g i ~ n s , o n e m ~ y see t h a t se l f m a s s t e r m s c o r r e s p o n d i n g

t o d i a g r a m s s u c h ~s :

K K R R

Y Y N Y Y

Fig. 1.

c l u m g e s i g n w h e n w e t r a n s f o r m n u c l e o n s i n t o 7~'s o r ~mti ~ ' s , w h i l e t h e sitzn

rem~rins t h e s~,me if n u c l e o n s :~re t r ~ m s f o r m e d i n t o ~mt inuc l eons . T h e r e f o r e ,

o n e w o u l d e x p e c t tht~t t h e m ~ s s of b o t h =, a.nd wnti ~ s h o u l d be s h i f t e d in

r e s p e c t t o t h e m a s s e s of n u c l e o n s ~md : m t i n u c l e o n s , ,qs is in f i lc t o b s e r v e d at.

let~st f o r n u c l e o n s ~md 7~'s.

R 1 A S S I ~ N T ( )

Se eons ide r i amo i bar ioni come eoml)ost i da un bar ione nudo senza ear ica n5 spin e da i sol i t i c amp i mesonici (sia rc che K), si pus esaminare il c o m p o r t a m e n t o del laffran- g i ano che descr ive llil ta le s i s t ema nei r iguard i di una operazione the inve r t a i segni de l la ea r ica e del la s t r anezza dei mesoni senza toeeare il bar ione nudo (coniuffazione boson i ca B) e di una operaz ione che t ras formi il bar ione nudo nella propr ia an t ipar t i - ee l la senza tocea re i eampi mesonic i (eoniuffazione spinoria le S); il p rodo t to di tal l due ope raz ion i essendo e v i d e n t e m e n t e uguale al la so l i ta coniugazione di car ica C. Si t r o v a che q u a n d o si t r a s cu rano le differenze di massa t r a nucleone ed ipe ron i , ta le lagrangiano

i n v a r i a n t e pe r t u t t e e t re le operaz ioni B, S e C prese s epa ra t amen te . La stessa pro- p r i e t~ si pub pu re d i m o s t r a r e pe r il l agrang iano di d ' E s p a g n a t e P r en tk i di un ins ieme di nue leoni , iperoni e mesoni real i e per Ie in te raz ion i di ta l l par t ice l le coi campo e l e t t r o m a g n e t i c o . Inf ine v iene p r o s p e t t a t o c h e l a differenza di massa t ra nucleone o la E possa essere d o v u t a ad un t e rmine di i n t e raz ione che sia i nva r i an t e r i spe t to a C m a non r i spe t t o a B ed ~ s e p a r a t a m e n t e e si p ropone una semplice espressione che sod- disfi a ta l l p ropr ie th .