LETTERE AL NUOVO CIMI~NTO VOL. 21, ~. 6 11 Febbra io 1978

Masses of 27-plet Mesons. Possible Existence of Low-Mass Long-Lived Exotics (*).

C. S. KALMA~

Physics Department, Ind iana University - Bloomington, Ind iana 47401, U S A

Physics Department Loyola Campus, Coneordia University - Montreal, P . Q . , Canada H 4 B 1R6

(ricevuto il 9 N ovembre 1977)

W i t h the success of the quark model , it has been a s sumed t h a t exo t ic -meson s t a t e s like q ~ q and q q q qqq would even tua l ly be seen (i). Such exot ic combina t ions in addi- t ion to t he oc te t and singlet SU3 r ep resen ta t ions con ta in S U 3 27-plets. Recen t ly i t has been sugges ted (2.~) t h a t a n u m b e r of observed meson s ta tes are of th is type . I n s ta tes of t y p e q ~ q , s ta t i s t ics will u n d o u b t e d l y cut down the n u m b e r of avai lable mesons, b u t nonetheless , pseudoscalar , vec tor and t ensor par t ic les ought to occur. These will each be separa te ly considered in th is paper . In a separa te pape r (5), sum rules for 27-plet mesons are given based on the mass formulae also p resen ted here . In th is pape r some addi t iona l phys ica l a s sumpt ions are m a d e in order to connec t such mesons to ex is t ing none t s and t h e n to p red ic t the ac tua l masses of the exot ic 27-plet mesons. These masses are ca lcu la ted using an S U 4 dynamica l g roup scheme based on a s t rong coupl ing model for quark in te rac t ions . If a l inear mass fo rmula is used, one can p red ic t low-mass pseudosca lar mesons wi th s t rangeness 0, -4- 1, -4- 2, which a re long l ived (3 ~ 10 -s s). The charge s ta tes of such mesons inc lude -4- 2. If a quadra t i c mass fo rmula is used, only the pseudosca la r mesons wi th s t rangeness -4- 1 and • 2 are p red ic t ed to be long l ived. The expe r imen ta l ev idence (s) ne i ther confirms no r denies t he ex is tence of such par t ic les .

The usual S Ua mass opera to r is t he p roduc t of an S U 3 t r ip le t or quark opera to r w i t h a 3 or a n t i q u a r k opera tor . Le t qk and qk, k ~ 1, 2, 3 be such quark and a n t i q u a r k

(*) Work supported in part by the National Research Council (Canada) and in part by the U. S. Energy Research and Development Administration. (1) J . J . J . KOKKEDEE: The Quark 3lodel (New York, N.Y., 1969). (2) D. WEINGARTEN and S. OKUBO: Phys. Rev. Left., 34, 1201 (1975). (3) M. BANDER, G. L. S~AW, P. THOMAS and S. MESHKOV: Phys. Rev. Left,, 36, 695 (1976). (a) •. L. JAFFE and K. JOHNSON: M.I.T. Preprint 508 (1975). (5) C. S. ]~:A~fAN: Left. Nuovo Cimento, 18, 201 (1977). (~) PARTICLE DATA GROUP: Rev. 2~lod. Phys., 48, 51 (1976).

201

2 0 2 c . s . KALMAN

opera to r s . T h e n if A~i; i , j = 1, 2, 3 are t he g e n e r a t o r s of SUa,

(1) [Ai~, Akin] = ~i.~Ak~ - - (5jkAim ,

(2) ~ .Aii= O, i=1

(3) [Air qk] = bikqi,

(4) [Ai~, qk] = - - ~jkqi,

i , j , k = l , 2 , 3 ,

i , j , k = 1 , 2 , 3 .

I n a s t r o n g coupl ing mode l pos ing t h e cond i t ion

(5)

(7), a closed Lie a lgebra of A ' s a n d q's is o b t a i n e d b y im-

[qi, qJ] = O ( ~ i ~ A 4 ~ - - A r , i , j = 1 , 2 , 3 ,

where 0 = + 1 cor responds to t h e Lie a lgebra of S U4,

= - 1 to t h a t of S U1,3,

= 0 to t h a t of T 7 | 3,

a n d A44 a (( d iagona l ~) g e n e r a t o r in add i t i on to A , , i = 1, 2, 3 needed to comple t e t h e f irst two of t he above Lie a lgebras . This d y n a m i c a l c o n d i t i o n l imi t s (s) t h e q u a r k r a d i a l field v a r i a b l e to a fixed (( r ad ius ~) a n d th i s co r re sponds to t h e n o n o b s e r v a n c e of free quarks . The group T 7 @ S U a is ru led out because b y i t s use, one c a n n o t p r ed i c t t r a n - s i t ions b e t w e e n e l e m e n t a r y par t i c les . The a lgeb ra of S UI,a will be t r e a t e d e l sewhere a n d t h e a lgebra of S U4 is e x a m i n e d here .

Calcu la t ions are p e r f o r m e d us ing a Ge l ' l and bas is (9)

(6) ]m> = I Tt~14 Tfb24 /34 /44~

!13 /rb23 m33 ~ .

~12 m22 /

mll /

To satisfy the unimodular condition

(7) m44 = 0 .

The r e m a i n i n g p a r a m e t e r s are pos i t ive in tegers s a t i s fy ing t h e cond i t ions

(8) mi~ ~ mi+i, i , mi~> mi , i_ i~ mi+ld, j = 2, 3, 4, i = 1, 2, 3, 4 .

(7) J. t~URIY.~N and E. G. G. SUDARSHAIr Phys. Rev., 162, 1650 (1967). (8) Y. DOTHAN and Y. NE'EMAN: A. E. C. Research and Development Report CALT-68-41 (1965). (~) G. E. BAIRD and L. C. BIEDENItARN: Journ. Math. Phys., 4, 1449 (1963).

MASSES OF 27-PLET MESONS ETC.

Iden t i f i c a t i on w i t h phys ica l q u a n t i t i e s is as fol lows:

P ' q I ~

(9) r , I , Is/

m12 m13 - - m23,

~- m 2 2 - - ~(m13 -[- m23 ~ m33),

T h u s for S U 3 oc te t r e p r e s e n t a t i o n s

(lO) m13 = m23 -~- 1 = m33 --~ 2

for s ing le t r e p r e s e n t a t i o n s

(ii) m 1 3 : m23 = m33

for 27-ple t r e p r e s e n t a t i o n s

(12) m13 = m2a ~- 2 ~ ma3 ~- 4 .

TABLE I. -- 27-ple t meson states.

m23 - - m33

�89 - - m22),

2 0 3

mlj " - - �89 -~- m22)> "

Particle name ~ I

P 0 2

R 1

R --1

S 2 I

~:, p, A S (27) 0 1

S - - 2 1

K, K* (27) 1 �89

K, K* (27) - - 1 �89

~, (I), f (27) 0 0

A comple t e desc r ip t ion of mesons in t h e 27-plet r e p r e s e n t a t i o n b a s e d on eq. (9) is g i v e n in t a b l e I. T h e c o m b i n a t i o n of eqs. (8) a n d (10) r e s t r i c t s t h e poss ible va lues of m2a a n d m34 to four poss ib le cases:

( 1 3 a ) m2a = m13 , m34 : m 1 3 - - 1 ,

(13b) m2a = m13 - - 1 , m34 ~ m~3 - - 2 ,

(13c) m24 : m13 , m34 ~ m13-- 2 ,

(13d) m24 = m 1 3 - 1 , m34 = m 1 3 - 1 .

E q u a t i o n s (13a), (13b) co r re spond to t h e same S U 4 r e p r e s e n t a t i o n . Such a r ep re sen t a - t ion in gene ra l c o n t a i n s two octe ts , two 27-piers, a decup le t a n d a n a n t i d e c u p l e t re-

2 0 4 c . S . KALMAN

presenta t ion. Thus this representa t ion is sui table for a descript ion of baryons. An S/74 representa t ion corresponding to eq. (13c) conta ins a single octet , one or more 27-plets, a decuple t and an an t idecuple t representa t ion and is sui table for a descr ipt ion of mixed s ta tes of baryons and an t ibaryons . F ina l ly the S U4 representa t ion corresponding to eq. (13d) contains a single 27-plet, an octet and a singlet representa t ion and is thus sui table for a descript ion of mesons.

I n first-order t ime- independen t theory, the mass of each meson is g iven by

(14) 4

M(m) = Co -{- <ms[ ~ C~r162 =

= C o + Ci(mls~lm> + Ce<m[u~lm> + Ca<mid(Tim),

M(m) can be replaced by M2(m) for quadra t ic mass formulae C2 : C a ~= C~ is requi red by isospin invar iance. L e t n be the value of m13 for octet mesons. F r o m eqs. (9) and (13d), the only free parameters in the ma t r ix e lement <m]qi~ti]m) are n and m,4. However , n is re la ted to m14 by the condi t ion

(15) n : (m,4 + 3)/2

in order t ha t all mesons wi th same value isospin bu t s trangeness of opposi te sign have the same mass, Note t h a t the smallest va lue of m,4 consistent wi th bo th eqs. (8) and (151 is m14 = 3. The only zero t r i a l i ty SUa-plets conta ined in this SU4 representa t ion are the singlet and octet . (Such representa t ions mus t be consistent wi th eqs. (7), (8), (13d) and are defined by eq. (9).) Fo r m,a = 5, the zero t r i a l i ty SU3-plets are the singlet, oc te t and 27-plet. Fo r m14 = 7, the zero t r i a l i ty s tates are the singlet, octet , 27-plet and 64-plet. In general an S U 4 representa t ion defined by eqs. (7), (13d), and (15) con- ta ins the t r ia l i ty zero S U s ple t representa t ions g iven by ( p , q ) = (0,0) , (1, 1), (2, 2) . . . . , (n - - 2, n - - 2). The s i tuat ion summar ized in table I I is consis tent w i th each

TABLE II . - Towers o] SU4 representations.

m14 Zero t r i a l i ty S U3 representa t ions

Dimension (p, q)

3 1 @ s (o, o), (1, 1)

5 1 Q 8 0 2 7 (0,0), (1,1), (2,2)

2 n - - 3 1 0 8 e 2 7 0 . . . (0,0), (1,1), (2, 2) . . . . ( n - - 2 , n - - 2 )

such S U4 representa t ion consist ing of n - - 2 products of q~. I t wil l thus be assumed t h a t the usual meson nonet corresponds to the case n = 3. Exot ic mesons of t ype q ~ q wil l be assumed to correspond to n = 4 and consist of an exotic nonet and an exotic 27-plet. In ei ther case, f rom eq. (14) the masses of singlet , octet and 27-plet mesons are as follows:

(16) 1

M(Po, Vo, To) : C0 ~- -: (n 2 - - 1)(C1 + 2C~), 3

1 (17) .z]l(rc, p, _A_2) : C O -}- (n 2 - - 4)(Ci q- 2C~'), g

MASSV.S OF 27-PLET MESONS ETC. 205

1 (18) M(K, K*) = C O + -: (n 2 - 4)(2Cx + 3C'2),

O

(19) 1

M(Ps, Vs, Ts) = Co + ~ (n 2 - 4)(7Cx + 8Cry), IO

1 (20) M(P) = Co + ~ (n 2 - - 9)(3C1 + 16C~'),

1 (21) M(R) = C O + • (n 2 - 9)(6C 1 + 13C~'),

z l

1 (22) M(S) = C O + ~ (n ~ - 9)(9C x + lOCk),

z l

(23) 1

M(rc, p, As; 27) = C o + ~ (n 2 - - 9)(39C1 + 56C~),

(24) 1

M(K, K * ; 27) = Co + (n ~ - - 9)(48C1 + 47C~2), 105

(25) 1

M(~,, r f ; 27) = C o + 105 (n2 - - 9)(51C1 + 44C~).

Tile group theore t ica l methods used give no in format ion about the spins and pari t ies of the mesons. The connect ion be tween our expressions for the masses in eqs. (16)-(25) and the masses of physical mesons is made by the assumpt ion t h a t the coefficients Co, C1 and Cr~ are the same for al l mesons of an SU4 tower p rov ided t h e y all have the same spin and par i ty . The usual nonet t hen corresponds to n = 3 and the masses correspond to eqs. (16)-(19). The constants C o, C x and C~ for the ent i re tower are obta ined by the subs t i tu t ing of the physical masses M(T:, p, As) in eq. (17), M(K, K*) in eq. (18) and the masses M(Ps, Vs, Ts), M(Po, V o, To) in eqs. (19) and (16), respect ively. These last two are obta ined f rom the physical masses as follows. E q u a t i o n s (17), (18) and (19) are re la ted by the mass sum rule

(26) 3[M(Ps, Vs, Ts) - - M(K, K*)] = M(K, K*) - - M(r:, p, A~)

which is s imply the Gel l -Mann-Okubo formula. The va lue of M(P o, V o, To) then fol- lows f rom the re la t ion be tween mixed and unmixed meson masses (lo)

(27) M(Po, Vo, To) + M(Ps, Vs, Ts) M(~, o~, f) + M(-,,)', O, f ' ) .

The resul t ing values of Co, C1, C'~ are noted in table I I I . Subs t i tu t ing these values and the va lue n = 4 corresponding to q q q q combinat ions in eqs. (20)-(25) yields the values of the masses of the mesons in the SUa 27-plet representa t ion g iven in table IV. Sub- s t i tu t ing in eqs. (16)-(19) yields the masses described in t ab le V of an exot ic octe t and singlet composed of two qua rk -an t iqua rk pairs.

Note t h a t for the psendoscalar mesons, the masses of t he charge states of • 2 for s trangeness 0, • 1, ~: 2 obta ined by use of a l inear mass formula are respect ively below the 3~, ur:K and r~KK thresholds. F o r the quadra t ic formula , the strangeness -}- 1, ~: 2

(10) B. FELD." Models o/ Elementary Particles ( W a l t h a m , ~Iass., 1969).

206

TABLE I I I . - Values of C o, C1, C~.

C. S. K A L M A N

P a r a m e t e r L inea r re la t ion (MeV) Quadrat ic re la t ion (GeV) ~

Spin Spin

0 1 2 0 1 2

C o - - 4 8 4 822 1481 - -1 .04 0.637 2.15

C1 412 87.2 54.6 0.393 0.154 0.153

C~ 52 - -34 .8 - - 5 6 . 4 0.166 - -0 .049 - -0 .145

TABLE IV. - Masses of 27-plet mesons.

Par t ic le L inea r re la t ion (MeV) Quadrat ic re la t ion (MeV)

Spin Spin

0 1 2 0 1 2

P 205 724 1234 491 726 1234

R 565 846 1346 684 855 135I

S 925 967 1457 877 983 1467

r:, p, A e (27) 781 919 1412 777 923 1416

K, K* (27) 997 992 1479 860 987 1478

~, (I), f (27) 1067 1016 1501 888 1008 1499

TABLE V. - Masses of octet and singlet composed of q q q q combinations.

Par t i c le L inea r re la t ion (MeV) Quadra t ic re la t ion (MeV)

Spin Spin

0 1 2 0 1 2

~ ,~ , A~ 1004 697 1071 1225 730 1055

K , K * 1868 990 1337 1430 1010 1354

~, r 2156 1087 1426 1492 1087 1440

~ , ~,f~ 2096 910 1190 1609 955 1205

s ta tes are below the r:r:K and = K K thresholds. No s t rong decays of such par t ic les is possible. Indeed in the case of the l inear mass formula , since the strangeness 0, ~: 1 par- t icles are below the ~7: and ~ K thresholds t h e y should be par t i cu la r ly long-l ived. The fas tes t decay possible for the s trangeness 0-meson is

(28) P++-~ e+~+v-.

One ant ic ipates a l i fe t ime grea te r than or equal to the 7: l i fe t ime of ~ 10 -s s, for the P++. The exper imenta l evidence ne i ther confirms nor denies the existence of such particles.

MASSES OF 2 7 - P L E T MESONS ETC. 207

The Pa r t i c l e D a t a Group (~) l is ts four references of searches for s t ab le par t ic les , b u t none a re in t he mass r ange of t a b l e I I I i n d i c a t e d b y t h e i r ca lcula t ions .

S ta t e s w i t h n o r m a l q u a n t u m n u m b e r s are also c o n t a i n e d in t he 27-plet r e p r e s e n t a t i o n . F o r e x a m p l e in t h e q u a d r a t i c r e l a t i o n case those w i t h q u a n t u m n u m b e r s iden t i ca l w i t h t h e ~, p, A 2 are l i s ted a t 777, 923 a n d 1416 MeV. Since these , un l ike t he usua l par t ic les w i t h these q u a n t u m n u m b e r s , a re composed of a q ~ q c o m b i n a t i o n , t h e i r p r o d u c t i o n cross-sect ions are l ike ly to be small . Moreover s ince t h e y h a v e ample decay channe ls , t h e y m a y t u r n ou t to be too wide to be seen. B o t h l i nea r a n d q u a d r a t i c for- mu lae give t he pseudosca la r d o u b l y s t r ange S-meson wel l be low t h e r :KK th resho ld . T h u s i t shou ld be long- l ived ( ~ 10 -s s). Typ ica l r eac t ions are

r:+p --~ S++Eo,

z:-p ---)- SoE o, S+E - ,

~ z - n ----> S o E - ,

K+p---> S++Zo, S++A,

K+n___> S++E - '

K~ ~ S++E -, S+E o '

KOn ---> S+E -, SOEO.

Al l these e v e n t s a re d i s t inc t ive a n d o b s e r v a t i o n of a s ingle e v e n t shou ld be sufficien~ for iden t i f i ca t ion . P r o b a b l y t he bes t b e t would be

~ -p .__> S+E - 1---~ r:+K o

I n conclus ion, us ing e i t he r a q u a d r a t i c or a l i nea r mass fo rmu la , a t leas t one long- l ived exot ic m e s o n is p red ic ted .

The a u t h o r wou ld l ike to t h a n k D. LICHTENBERG for he lp fu l c o m m e n t s a n d sug- gest ions.