LETTERE AL NUOVO CIMENTO VOL. 14, N. 4 27 S e t t e m b r e 1975

Are there Four +-Particles?

C. S. KALMAN

Loyola Campus, Concordia University - Montreal

(r icevuto i l 29 Lugl io 1975)

I t has been sugges t ed (1) t h a t the recent d i s covery of ~-part ie les (2) is a mani - f e s ta t ion of the charm s y m m e t r y group S U t. W i t h i n the m i x e d 15 + 1 representa t ions of th i s group, ~ becomes assoc ia ted w i t h the J~)= 1- meson none t of usua l S U 3 s y m m e t r y to form a tenfo ld J~ = l - mul t ip l e t . I f so the sequence of three ~-part ic les should share some c o m m o n propert ies w i t h the corresponding observed none t mesons . In the SUI. l mode l (~) series of part ic les w i t h ident ica l spin, par i ty , s trangeness , ordinary and i so top ic spin are d i s t ingu i shed b y a s ingle a d d i t i v e q u a n t u m n u m b e r k. I t w i l l be a s sumed (4) tha t al l members of a t enfo ld J ~ - - 1- m u l t i p l e t h a v e a c o m m o n v a l u e of k.

In the m o d e l the mass of such a part ic le is g i v e n b y

(1) m a s s - - C o + CIc(K- -k + 1) .

(1) See e . g . S . BORCHARDT V. •. )/~ATHUR and S. OKUBO: Phys. Rev. Left . 34 38 (1975); D. LICHTE]g- BERG: Indiana U n i v e r s i t y preprint Coo-2009-92. (2) D i s c o v e r y of the 5 or ~(3100) b y J . J . AUBERT U. BECKER, P. J . BIGGS, J . BURGER, M. CI~EN, G. EVERHART, P. GOLDHAGEN, J . LEONG, T. MCCORRISTO~, W. G. RHOADES, M. t~OttDE, S. C. C. WING, S. L. WU and Y. u LEE: Phys. Rev. Le f t . , 33 , 1404 (1974); J . -E . AUGUSTIN, A. )/[. BOYARSKI, ~r BREI- DENBACH, F. BULOS, J . T. DAKIN, G. J . FELDSIAR', (J. E. FISCHER, D. FRYBERGER, G. ~IANSON, n . JEAN- MARIE, R. R. LARSEN, V. LUTH, H . L. LYNCH, D. LYON, C. C. MOREHOUSE, J . M. PATERSON, M. L. PERL, B. RICHTER, P. RAPIDIS, n . f . SCHWITTERS, W. 1VI. TANENBAUM, F. VAN~UCCI, G. S. ABRAMS, D. BRIGGS, W. CHINOWSKY, C. E. FRIEDBERG, C. GOLDHABER, ]~. J . HOLLEBEEK, J . A. I~ADYK, S . LULU, Y. PIERRE, G. H. TRILLING, J. S. WHITAKER, J. WISS and J. E. ZIPSE: Phys. Rev. Lel t . , 33, 1406 (1974); C. BACCI, R. BALDINI CELIO, M. BERNARDINI, G. CAPO]~ ~, 1~. DEL FABBRO, M. GRILLI, E . IAROCCI, L . JONES, C. MENCUCCINI, G . P . MURTAS, G. PENSO, G. SALVINI, M. SPINETTI, M. SPANO, B. STELLA, V. VALENTE, B. BARTOLI, D. BISELLO, B. ESPOSITO, ~. FELICETTI, ~). MONACELLI, M. NIGRO, L. PAOLUZZI, I . PERUZZI, G. PIANO MORTARI, M. PICCOLO, F. I~ONGA, F. SEBASTIANI, L. TRASATTI, F. VANOLI, G. BARBARINO, G. BARBIELLINI, C. BEMPORAD, R. BIANCASTELLI, Y. CEVENINI, ~ . CASTELLANO~ 1V~. CALVETTI, F. CO- STANTINI, P. LARICCIA, P. PARASCANDALO, E. SASSI, C. SPENCER, L. TORTORA, U. TROYA an d S. VI- TALE: Phys. Rev. Leit . , 33, 1408 (1974). Disc ove r y of the ~(3700) by G. S. ABRAMS, D. BRIGGS, Vr CIIINOWSKY, C. E. FRIEDBERG, G. GOLDIIABER, R. ~. ItOLLEEEEK, J . A. KADYK, A. LITKE, B. LULU, Y. PIERRE, n . SA])OULET, G. H. TRILLING, J . S. ~VHITAKER, J . ~VIss, J . E. ZIPSE, J . -E . AUGUSTIN, A. IV[. BOYARSI~I, M. BREIDENBACH, F. BI)'LOS, G. J . FELDMAN, G. E. FISCIIER, D. FRYBERGER, G. I tANSO~ B. JEAn-MARIE, R. R. LARSEN, V. LUTH, H . L . LYNCH, D. LYON, C. C. MOI~EHOUSE, J . M. ~ATERSON, M. L. PERL, B. I~ICHTER, P. RAI'IDIS, R . F. SCIIWITTERS, W. TANENBAUM and F. VANNUCCI: Phys. Rev. Lef t . , 33, 1453 (1974). (s) C . S . KALMAN and J . PATERA: Let t . Nuovo Cimento, 5, 78 (1972). (') The use of sequences of oc te t s rather than s ingle part i c l e s is m a d e to m a k e an ex tens iv e classifica- t i on of the baryons in C. S. KALMAN: Can. Journ. Phys . , 51, 1573 (1973); C. S. KALMAN: Par t i c l e s and N u c l e i , 5, 183 (1973).

115

116 c . s . KALMAN

Here C a n d Co are rea l c o n s t a n t s , K is a pos i t ive i n t ege r c o n s t a n t a n d k t akes on pos i t i ve i n t ege r va lues a n d is i n c r e m e n t e d b y one for succeed ing pa r t i c l e s in t h e sequences . Moreove r

(2) ~ > k ~ K + 1 .

F o r t h r e e successive par t ic les w i t h q u a n t u m n u m b e r s kl, k2, k3 a n d c o r r e s p o n d i n g masses 2t/1, M 2, M s, eq. (1) can be w r i t t e n in t h e fo rm

M~ - - M 1 2

M a - - M 2 2k a - - K

T h e v a l u e of R var ies w i t h i n n a r r o w l imi t s R ( K = 0, k = 1) = 2.0 a n d R ( K - > ~ ) = 1. I f K, k are e s t a b l i s h e d f rom a sequence of n o n e t mesons , t h e n t h e masses of t h e

two obse rved ~ ' s a t 3.095 a n d 3.685 GeV can be used to p r ed i c t f u r t h e r ~ 's . F o r t h i s p u r p o s e t h e sequence p(770), p '(1250), p"(1600) is used. T h e mass of p(1600) is in some d i spu t e p a r t i c u l a r l y because i t is s o m e w h a t b r o a d ; h o w e v e r t he lowes t q u o t e d va lues co r r e spond to a la rger v a l u e of R t h a n p e r m i t t e d in SUI. I t h e o r y (5). Also t h e l a rges t v a l u e of t h e mass t h a t ha s been o b t a i n e d ha s n o t been f o u n d accep tab l e (6). T h u s for t h e pu rpose of t h i s p a p e r M ( p ) = 770MeV, M(p')= (1256 • 10) MeV, M(p")

(1590 • 20) MeV. R t h e n lies be t w een 1.30 a n d 1.63. The va lues of R w i t h i n t h i s r a n g e t h a t co r r e spond to pos i t i ve i n t ege r va lues k, K sa t i s fy ing eq. (2) are R a R ~ 1.40, R = 1.50. T h e v a l u e R = ~ a l t h o u g h no t exc luded b y t h e d a t a is un l ike ly s ince t h e masses of b o t h t h e p' a n d p" wou ld h a v e to be nea r t h e e x t r e m e s p e r m i t t e d b y t h e er rors in t h e expe r i m en t s . Fo r R = 1.40 two so lu t ions of K , k are o b t a i n e d : K : 3 , k (p" or ~ " ) ~ 4 a n d K : I , k(p" or ~ " ) ~ 3 . Fo r : R : l . 5 0 t h e r e are also two poss ib le so lu t ions n a m e l y K = 2, k(p" or ~ " ) = 3 a n d K = 0, ]c(p" or ~ " ) ~ 2. T h e va lues of ]~ decrease w i t h inc reas ing mass a n d t h u s eq. (2) impl ies t h a t for K = 3 ( R = 1.40) a n d K = 2 ( ~ = 1.50) t h e r e are t h r e e pa r t i c les in t h e sequence, a n d for K ~ 1 (B = 1.40) a n d K = 0 (R = 1.50) t h e r e are four pa r t i c les in t h e sequence . T h e masses of t h e par t ic les are l i s t ed in T a b l e I . T he masses of those l i s t ed for t h e f i rs t t h r e e p 's are t h e e x p e r i m e n t a l va lues l i s t ed p rev ious ly r e s t r i c t e d b y t h e i n d i c a t e d choice of R. Those of t h e f irst two ~ 's are based on t h e l a t e s t e x p e r i m e n t a l f ind ings (7). T h e mass of t h e o the r ~( 's) can be c o m p a r e d w i t h t he r ecen t d i scovery (s) of s t r u c t u r e in t h e t o t a l c ross-sec t ion for h a d r o n p r o d u c t i o n b y e l ec t ron -pos i t ron a n n i h i l a t i o n in t h e 4.1 GeV region. T h e p r e s en t s t a t e of t h e d a t a m a k e s i t unc l ea r as to w h e t h e r t h i s s t r u c t u r e cons is t s of one or more resonances . I f i t is a s s u m e d to cons i s t of on ly one resonance , t he p e a k wou ld be cen t r ed a t 4.15 GeV, w h i c h wou ld t h e n s t rong ly f a v o u r R = 1.4, K - - 3. I f i t cons is t s of two peaks , t he d a t a sugges t t h a t t h e y wou ld b o t h be w i t h i n t h e r ange ( 4 . 0 - - 4 . 3 ) G e V , wh ich wou ld s t rong ly f a v o u r / ~ 1.5, K = 0.

(0 The value R = 2 is not altogether excluded, but if applied to the ~-mesons, this predicts a third at 3.98 GcV at variance with experimental findings (see ref. (8)). (~) V. CItALOIYPKA, C. BRICMAN, A. BARBARO-GALTIERI, D. ]~J~. CHEW, R . L. KELLY, T. A. LASI~SKI, ~4~. RITrENBERG, A. l{. t{OSENFELD, T. G. TRIPPE, F. UCHIYAMA, N. BARASIt-Scn~IIDT, P . SODING a n d M. R o o s : Phys. Left. , SOB, 1 (1974). (7) V. CI=I[ALOUPKA, C. BRICMAN, i . BARBARO'GALTIERI, D. M. CHEW, R . L. KELLY, T. A . LASINSKI, A.. RITTENBERG, A. H . ROSENFELD, T. G. TRIPPE, F. UCIUYAMA, G. P. YOST, N. BARASH-ScHMIDT a n d ~ . R o o s : Rcv. Mod. Phys. , 47, 535 (1975). (s) J . - E . AUGUSTIN, A. M. BOYARSKI, M. BREIDENBACH, F. BULOS, J . T. DAKIN, G. J . FELDMAN, G. E. FISCHER, D. FRYBERGER, G. HANSON, B. JEAN-MARIE, R . R . LARSEN, V. L(~TYI, i . L . L ~ C H , D. LYON, C. C. ~r J . M. PATERSON, M. I . PERL, B. RICHTER, R . F . SCH'WITTERS, f . VANNUCCI, G. S. ABRAMS, D. BRIGGS, ~V. CHINOWSKY, C. E. FRIEDBERG, G. GOLDHABER, R . J . ]:IOLLEBEEK, J . A . KADYK, G. H . TRILLINQ, J . S. ~VHITAKER a n d J . E. ZIPSE: Phys. Rev. Left., 34, 764 (1975).

AlOE T H E R E FOUI~ ~-PARTICL]~8

TABLE I . -- M a s s e s o] p, ~ a n d o~ m e s o n s ]or d i ] / e ren t ia l v a l u e s o] R a n d K .

117

R K /c Particle ~r (MeV)

1.4 3 6 p 770 input

5 p' 1246 -- 1260 modified input

4 p" 1586 --1610 modified input

6 ~ 3095 input

5 ~' 3685 input

4 ~" 4106 predicted

6 co 783 input

5 r 1298 4- 5 predicted

4 r 1666 4- 7 input

1 5 p 770 input

4 p' 1246.1260 modified input

3 p" 1586.1610 modified input

2 p" 1790 -- 1820 predicted

5 ~ 3095 input

4 ~' 3685 input

3 ~" 4106 predicted

2 ~" 4358 predicted

5 (o 783 input

4 o~' 1298 4- 5 predicted

3 co" 1666 4- 7 input

2 o "1 1887 4- 15 predicted

1.5 2 5 p 770 input

4 p' 1250.1266 modified input

3 p" 1570.1597 modified input

5 ~ 3095 input

4 ~' 3685 input

3 +" 4078 predicted

5 co 783 input

4 o ' 1313 -k 5 predicted

3 r 1666 ~= 7 input

0 4 p 770 input

3 p' 1250 --1266 modified input

2 p" 1570 --1597 modified input

1 p" 1730.1762 predicted

118

TABLE I (eonti~tued).

C. S. K A L M A N

R K k Particle Mass (MeV)

4 ~ 3095 input

3 ~/' 3085 input

2 ~" 4078 predicted

1 ~/" 4275 predicted

4 r 783 input

3 r 1315 ! 5 predicted

2 co" 1666• 7 input

1 co" 1843-~ 14 predicted

A difficulty with the lat ter possibility would be explaining the nonobservance of a fourth p near 1.75 GeV. Since such a p would be broader than tha t near 1.59 GeV, it is of course possible tha t i t would be very difficult to find. Finally since two co-mesons have been observed, the masses of the remaining ones can be calculated. Comparing masses of the co's with those of the p'ss uggests 0)=0)(783), ~"=co(1675). The masses of the other ~ ' s are given in Table I.

This paper was begun while the author was on a visit to Indiana Universi ty at Bloomington to give a seminar. The author would like to thank D. B. LmHTENB~.RG for valuable discussions.