First and second order hadron mass formulas in internal SU(6)

Pram~na, ¥ol. 16, No. 6, June 1981, pp, 493-510. © Printed in India.

First and second order hadron mass formulas in internal SU(6)

C P SINGH*

Department of Physics, Panjab University, Chandigaxh 160 014, India.

*Permanent address: Department of Physics, VSSD College, Kanpur, India.

MS received 17 January 1981 ; revised 18 April 1981

Abstract. We use broken SU(6) internal symmetry to derive the mass formulas amongst hadrons (112 +, 3/2% 0-, 1-) including second order mass contributions from symmetric 405 representation. Some hybrid mass relations are also obtained by relating second order parameters.

Keywords. Hadron masses; internal SU(6) symmetry; second order effects.

1. Introduction

In the last few years, high energy experiments have provided us with far-reaching developments. On the one side, the discovery of the J/~b particles (Aubert et al 1974;

Augustia et al 1974) and of charmed mesons (Goldhaber et al 1976) and baryons (Cazzoli et al 1975; Knapp et al 1976) has confirmed the theoretical model of charm (Glashow et al 1970). On the other side, the upsilon family 1", T' (Herb et al 1977;

Berger et al 1978) with masses 9.46 and 10.06 GeV discovered in the reaction, p + (Cu, P0 ~/z+ + / z - + anything,

could not be accommodated within the SU(4) framework which compelled people to think beyond charm. The new particles, like J/~ as cc, are interpreted as bound states (bb-) of a new quark flavour b (beauty), which suggest that a rich spectrum of new heavier particles might exist. Recently, Thorndike (1980) has reported that T(4S) is seen at CESR with mass 10.55 GeV and has a large width, suggesting that the mass of the b-quark meson may be between 5.16 ~< Db ~< 5"28 GeV, further con- firming the existence of the b-quark hadrons. Next, the discovery of a new heavy lepton ~(Perl et al 1975) accompanied by its own neutrino v¢ (Perl et al 1977) supports the expectation of a sixth quark t (taste) from the quark-lepton symmetry considerations (Harari 1975) and the unified gauge theories (Kobayashi and Maskawa 1973). Though there is no experimental evidence for the t quark hadrons until now, it is hoped that with improved experimental techniques at PETRA (Wiik 1980) the theoretical predictions for the quark t will become true. Many authors (Boal 1978; Aubrecht and Scott 1979; Camiz et al 1979; Misra and Sastry 1979;

Singh et al 1980; Khanna 1980; Singh 1981) have therefore started theoretical studies on their properties in different frameworks.

493

494 C P S i n g h

In this paper we consider the masses of the b and t quark hadrons in the internal SU(6) symmetry. Since the higher internal symmetries are badly broken due to the large mass differences amongst the quarks, contributions from the higher order perturbation terms may be significant. The higher order effects on the masses of charmed hadrons (Verma and Khanna 1978; Singh et al 1980) have already been studied and found to be important. In § 3 we derive the first order mass relations and then incorporate the second order effects which relate the discrepancies present in the first order mass relations. Higher order symmetry breaking parameters are also evaluated in terms of the masses of particles. § 4 deals with the discussion of the hybrid mass relations between the baryons and mesons.

2. Prdiminaries

Since baryons are supposed to be made up of three quarks, they are conveniently described by Young diagrams with three boxes generated in the direct product

6 ® 6 ® 6 = 56~ ® 20 A ® 70M® 70 M. (1)

Table 1. JP --- 1/2 + baryons

SU (6) SU (5) SU (4) SU (3) States

7%

3~c ( ~ f~c~)

6 A 3b (A/~ 7Rg)

lc,b (fl,eb)

4 3ha (~bb l'~ bb)

1 ~ (~bb)

10 A 6~ ~, (M ¢.;)

lcbt (L'~cbt) t

15 s lOs 6t (Z't ~t ~t)

3ct (~ct ~'~ct) lcct (~'~cct)

4 3bt (~bt fZbJ

l~t (ft~bt)

1 lbbt (l"lbbt)

5 4 3it (~tt ~'~tt)

lctt (~ctt)

1 lbtt (~-~btt)

Hadron mass formulas in SU(6) 495 f l ' = 1/2+ baryons are assigned in the 70 M representation and can be represented by the wave function

B[a/~] 5' satisfying B[a/~] 5 = - - BLSa] 5'

and BD/3] 5 + B[flS] a + B[sa]/3 : 0. (2)

The SU(3) decompositions of these is given in table 1. The totally symmetric representation 56s contain 3/2 + isobars and represented by the symmetric wave func- tion Da~5. The indices a, r, 8 refer to the flavour of the quarks and run from 1 to 6 representing u, d, s, c, b, t quarks. The SU(3) decomposition of the 3/2+ baryons is given in table Z

i i

Ill

Table 2. j e = 3/2+ baryons

SU (6) SU (5) SU (4) SU (3) States

56 s 35 s 20 s 10 (A 2:* ~* ~*)

6~ (2:. ~ , n*)

lcc~ (fi%~)

10 6 b (2:; ~ [~,)

3~b (~*~, fl%)

l**b (~*c~,)

4 3bb (~?,~ fZT, D

lcbb (~*cbb)

1 1 bbb ( ~ ' ~ b b )

15 s 10 s 6t (~'* ~* fi*)

3ct (~ct [~*t) lcct (~*cct)

4 3bt ('Q~t "~t)

lcbt (~'~*cbt)

1 l~t (fi~'bt)

5 4 3tt ('~'~t [~*t)

lctt (~'~*tt)

1 lbtt (~'~tt)

1 1 lttt (~'~ttt)

i

496 C P Singh

Table 3. Pseudoscalar mesons

SU (6) SU (5) SU (4) SU 0 ) States

1: 1

35: 5

1:

24:

1 1 [P0]

4 3 [D +, D O , F+I

1 16~]

1 1 [hr8

1 1 [Hil

71. 1 [G~

g [D;, r~, F/I

I 1 [,P.s]

4 3 [Db, D~, Fg]

[D~, D~, F~

] : 1 [P~,I

15: 3 [ D + c, O~, F+]

g [Ds, ric °, Fb- ]

1: [*'~d

[K+, ~o1 [e.l D+, ~o, ~-1 There will be 36 mesons generated in the direct produet

6®g= 1~35, (3)

which are represented in table 3. The tensor representations for 1/2 +, 3/2 + baryons, and mesons are gvien in appendix 1, 2 and 3 respectively.

To break the SU(6) symmetry, we extend the idea originally introduced by Gell- Mann (1962) and Okubo (1962) for SU(3) and later employed by Mathur et al (1975) for SU(4). We divide the Hamiltonian into two parts

H = He + HS.B. (4)

where He is invariant under SU(6) transformations and HS.B. breaks the symmetry.

The first order mass breaking operator (excluding electromagnetism) is taken to transform like

+ b q + c + (5)

Hadron mass formulas in SU(6) 497 component of 35. The second order mass-breaking Hamiltonian then would have an SU(6) transformation property dictated by the direct product

35 ® 35 = 1 ~ 35 ~ 35' ~ 189 ~ 280 ~ 280 ~ 405. (6) The representation 35 has already been taken in the first order breaking (equation (5)). We wish to consider the effects of totally symmetric 405 representation as second order mass contributions (Machacek and Tomozawa 1976). The general mass operator in SU(6) can be written as

,~ m o + a D/F T~ + b DIe T44 -a t- c D/F T# + d D/F T~,

} _,'r,(33) _,L. fT(44) _}_ oT(55) _~_ h T ( ~ ) .{_: ,'r(34) -l-- ;,'r(35} _~_ kT(36} .31_ IT(45)

z ( 3 4 ) ~ d z ( 3 5 ) " ' ~ ( 3 6 ) " - - ( 4 f i t '

_,v(4s) + .T(56) (7)

u~(46) " ' - - ( t i 6 ) "

3. Ha&on mass relations

In this section we first derive the first order mass relations. The second order contri- butions which relate the discrepancies present in the first order mass relatibns are then included. We use the particle symbol to denote its mass.

3.1 J P = 1/2 + baryons

The first and the second order mass contributions are obtained from the contraction

and

(8) (9)

respectively, where T~ and T ~ ) represent the first and the second order mass breaking spurion.

A. First order mass relations are 3 A + 2 7 = 2 E + 2 N

(4539.3 MoV) (4508.9 MoV), (10)

3A~ + Z'~ -- 2 E~c + 2N, (11)

3Ab + 27b = 2 ~bb + 2N, (12)

498 C P Singh

3 A , + 2 , = 2 ~t, + 2N, (13)

(f~tt ^-- ~,t) = (f~bb - - ~Ob) = (0,~ - - ~ ) = (27 - - N ) (252"9 MeV), (14) ( a ~ , - . % , ) = ( a ~ , - ~ c , ) = (.=, - 270 = ( a , - ~ , )

= ( a c ~ - ~ . ) = ( a . - ~ . ) = ( ~ . - 2 , ) = ( a ~ - F.~)

= (.~c - - 27c) ---- (½ (~ - - N ) (187.7 MeV), (15)

( f ~ , t - - ~n) = ( ~ b n - - ~ n ) = (2~ - - N ) (1560"5 MeV), (16)

(~2bn - - ~,,) = (2n - - N ) (17)

( a t , , - ~ , ) = ( . . . . - 2 0 = ( a ~ . - ~,~,) = ( a ~ c ~ - ~ )

= (E~b - - Zb) = ½ (gee - - N),

(~bb, - - Ebt) = (~bt - - 2 t ) = ½ ( ~ b - - N ) , 2 ~ + ~c - - 3~.'~ = 2(2~ - - 27) (2615 MeV), 2~bb + S~ - - 3 ~ = 2 ( 2 b - - 27),

2~bb + ~ - - 3 ~ b = 2(2b - - 27~), 2Nit + .=, - - 3N', = 2(27, - - 2 ) , 2.~tt + r a c , - - 3 ~ ; , = 2(E, - - 27~),

9 , = 2(27, - G ) , 2"~tt -~- "~bt - - 3-b,

~ , - - 3~'~. + 2~bb + 2(27 + 2c - - 27~,) = O,

~2c, - - 3 g ; , + 2Sb~ + 2(27 + Z'¢ - - 27b) = O, ab, - - 312;, + 2~,t + 2(2' + Z'~ - - Z',) = O,

~ c b t - -

T h e mass relations q u a r k sectors.

(18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28)

3~2'~b, + 2S,t + 2(2¢ + 27n - - Zt) = 0 (29)

(10) to (13) are the G e l l - M a n n - O k u b o mass formulas in different B. Inclusion o f the second order effects give following mass relations and the values o f the higher order parameters:

4(~cc - - flcc) - - 3( A'~ - - ~'c) = (~c - - 2c) - - 2(27 - - N), 0 o ) 4 ( s , , - % b ) - 3 ( A ~ - . ) = ( s , - 27,) - 2 ( 2 - N ) . ( 3 1 )

H a d r o n m a s s f o r m u l a s in S U ( 6 ) 499 4(~bb - - flcba) - - 3 ( A ~ - - ~'b) = (-~cb - - 27~) - - 2(27, - - N ) , ( 3 2 )

4(~tt - - f l t t ) - - 3 ( A ; - - E't) = ( E , - - Z',) - - 2(27 - - N ) , ( 3 3 ) 4 ( . ~ , - - n a t ) - - 3 ( A ; - - ~ ; ) = ( ~ a - - 27,) - - 2(27~ - - N ) . ( 3 4 ) 4 ( ~ , t - - o . b , ) - - 3 ( A ' , - - ~',) = ( ~ b , - - 27,) - - 2(27b - - N ) (35)

3(A~, - - fl'~b) + (flc~, - - 27b)

= 4(2.~bb - - ~2bb - - ~2~bb) + 2(~2,~ + 27, - - ~cc - - 2 N ) ( 3 6 )

3 ( A " - - 9 . ' a ) + ( o , - - 2:t)

= 4 ( 2 ~ , - - f t , - - O.ctt) -{- 2(~2~, + 2:~ - - .=,, - - 2 N ) (37) 3 ( A ; - ~ , ) + ( a ~ , - 2:,)

= 4 (2 ~ u - - f h , - - f ~ t , ) + 2 (Obb + Z~ - - "a~ - - 2 N ) (38) 3 (A', - - n ' ~ , ) + ( a ~ , - - 2:,)

= 4 (2 ~ , , - - n a t - - ~2bu) + 2 ( 9 . , ~ + 27a - - .=-~ - - 2 N ) (39) el/~ = (f~t + 27t - - 2 ~,) = ( f ~ + 2:~ - - 2 ~.~) = (fZ~ + 27e - - 2 ~-c)

= ½ [2 ~ + 2 N - - 27 - - 3 A ] ( - - 15.15 M e V ) ( 4 0 ) f l / a = (~2ca + 2:t - - 2 ~ct) = ( ~ c a , + 27b - - 2 ~,cb) (41)

= ½ 1 2 ~ + 2 N - - 2 7 ~ + - - 3 A A

gll~ = (~bbt + 27t - - 2 r~b, ) = ½ [2 "='bb + 2 N - - Z'b - - 3 A b ] , (42)

hl/z = ½ [2 r~,, + 2 N - - 2:, - - 3 A , ] , (43)

il/~ = (~** - % 9 - (27 - 2v), (44)

Ja/~ = (~2bb - - ~ b ) - - (27 - - N ) , " (45)

k~/~ = ( ~ 2 , - - Et,) - - (27 - - N ) , (46)

ll/9. = (O,~bb - - Ebb) - - (27~ - - N ) , (47)

ml/2 = (~2~,t - - .=,,) - -

(27~

N),

nx/~ = (fl~,, - - "~,,) - - ( 27~ - - N ) . (49)

500

C P Singh

3.2

JP

baryons

The first order mass contributions come from the contraction

(~amn Dflmn ) Z~,

whereas the second order f r o m

(~f[3m Ds~lm) T(~l)

A. The first order contribution gives the equal spacing rules as follows:

( s * - a ) = ( . * - s * ) = ( a * - v . * ) = (.~* - s ~ * )

(152 MeV) (149 MeV) (139 MeV)

= (fl~' - ~ * - c ) = ( a * c - = * 2:*

- ~ o ) = (-~* - b) = ( a * - .b)~*

= ( ~ * - = , • ~ ,

. o ~ ) = ( a ~ - .~*~) = ( . , - 2:*) = ( a * - ~ * ) - ( a ~ * ~ * . . , , ) = ( a * , - .=*,) * *

- - - - = ( ~ 2 b , - ~ b t ) ( 5 2 )

( a * - a * ) = (a~:~ - a * ) = (a**~ - ~ * ) = ( a , % - ~ * )

: ( a c % - a ~ * ) = ( a c * - a * ) : ( a t * , - ~ * ) = ( ~ , % , - a * )

(2:* - A) = (a~*~c - - -~*~) ---- =* ( . ¢ c - z * ) (53)

= (-cb ~ * _ _ Z b ) = (~c, - * 2:*,)

(27* -- A) = (N't, - - 27*) = (fl*b - - N*) (54).

= ( ~ , - 2 7 , )

( 2 : , - a ) = ( a , * - s * ) = ( a * , , - ~*,,) (55)

= ( ~ * . - - .~*,) = (~*`, - - .~* ) (56)

( 2 : * - 2:*) = ( a * , - a * ) = ( a L - a~*,)

= (.o*, - - ~,*) = (~*,, - - a*,) (57)

(27.` . 27*) . ( a ~ ' , , . a * ~ , ) = C a * , , . a , ~ , ) * (58)

.

P.---5

Hadron mass formulas in

SU(6)

S e c o n d order effects relates the disorepanoies in the equal spaoing rules and give

3 ( ~ * - Z * ) = ( a * - A) (59)

(447 M e V ) (443 M e V )

3 (so* ^- 2"*) = ( a , , c - * A) (60)

3 (~*. - - 2 * ) = (n**b - - A) (61)

3 ( S * - - Z * ) = (fi*tt - - A), (62)

(~c*,, - - 12") ---- 3 (fl*c - - n * ) , (63)

( f l * ~ - - f l * ) = 3 (n*~ - - n * ) , (64)

( a , . - a*) = 3 _{( a . - a 3,} (65)

( a * b ~ - a J * ---- 3 (a*~b - - fitch), * (66)

( a * , - nccc) * = 3 ( a c t , - - a.,,t), * * (67)

( f l * t - - fl*bb) = 3 ( O * t t - O*bt), (68)

½ea/2 = f l * + Z',* - - 2 ~ * = a * + z * - 2 ta*

- f l * + 2'* - - 2 = * = f l * + 2"* - - 2 ~ * (3 M e V ) , (69)

- - w e

fwa flc*t + Z~ -- 2.~c* = oct b + 2"2 - - 2Sa,

---- °c** + 2"* - - "-'-'cc'~=* --- c~*c + 2"* - - 2..,* , (70) ]ga/~ = n*~,, + 2"* - - 2 ~ * = fl*~,l,q- 2"~ - - 2~*t,, (71) ha/, = a~j., + 2"* - - 2 ~ * = fl~'. + 2"* - - 2 ~ (72) ti~/~ = ( ~ * - z ~ ) - (2" * * - - A) = ( a * - - -'c / =*~ - (E* - - 2"*), (73)

"~Ja/~ = (-=* - - 2"*) - - (2"* - - A) = (n~' - - ,~,*) - - (~,* - - L'*), (74) {k3/z = ( ~ * - -

Z*)

{/,,,

A)

2"*)

502 C P Singh

= - - -- •* ~* ( ~ * - 27*) (77)

~ / ~ ,-c,~=* - 27*) ( ~ * A) = ( ~ , , - . o , ) -

~n8/2

3.3 M e s o n s

Now there are 36 mesons (0- and 1- both) belonging to the 35 ~ 1 irreducible repre- sentation of SU(6). The first order mass contribution to these comes from the con- traction

(P~ P~ %- P~ P~) T~, (79)

whereas the second order contribution comes from

@ ⁽⁸⁰⁾

A. The following relations are obtained in the first order considerations.

(Ft - - D t ) = (Fa - - Db) = (Fc - - D , ) = ( K - - rt) (81) (174.5 i e V ) (357 i e V )

(677'6 MeV 2) (226.6 MeV ~)

(Gt - - Dr) = (Gb - - Db) = (Fc - - K ) = (De - - 7r) (8,2) (1533 MeV) (1733 MeV)

3G, - - 2 D , - - F t = 3Gb - - 2 D b - - F~ = F~ + 2D~ - - ~r - - 2 K (83)

a n , - - (G, + F , + 2 D , )

= (G~ + Fb + 2 D b ) - - ( F , + D , + K + zr ) (84)

3(Ps - ~) = 4 ( K - - ,,) (85)

(1230 MeV) (1432 MeV)

2 2

(849 MeV) (917 MeV)

6(P~5 - ' 0 = ( K - - ~ ) + 9 ( D , - - ~) (86)

10(P~4 - - 7r) ---- ( K - - 7r) + ( Dc - - ~ ) + 16 (D~ -- ~r) (87) 15(P3~ -- 7r) : (K -- ~r) + (D~ -- rr) + (Db -- ~') + 25 (Dr -- ,r) (88) 3(e 0 -- ,r) = ( K - - ~r) + (De -- 7r) + (Db -- ~r) + (D, -- rr ) (89)

Hadron mass formulas in SU(6) 503 The available mass values do not seem to satisfy these relations in the linear as well as in the quadratic form, indicating a large seoond order mass contributions.

B. The second order effects will give the following values of higher order parameters

io = (-Pc - - D e ) - - ( K - - rr) ( - - 1 8 3 M e V )

J o : ( F o - - D o ) - - ( K - - ¢r), k o = ( F , - D , ) - ( K - - . ) ,

!o =

(Go

n o = ( H , - - D , ) - - ( D a - - rr).

(90) (91) (92) (93) (94) (95) Similar relations follow for vector mesons by replacing corresponding vector particles for each pseudo-scalar mesons.

As the mass of the one b-quark meson (D~) is now known (Thorndike 1980), the masses of the others can be estimated and they come out to be

Do : 5"25 GeV, Fb : (5.61 + i ) GeV,

Gb

(6.97

l) GoV,

where j and I are the higher order parameters.

4. Hybrid mass formulas

Many authors (Eliezer and Singer 1973; Verma and Khanna 1978; Singh et al 1980) have studied the hybrid mass relations among the baryons and mesons upto SU(4), using different considerations. Assuming the universality of the ratio of higher order parameters (which are obtained in terms of the masses of the particles in § § 3.1, 3.2 and 3.3 for 1/2 +, 3/2 + and mesons respectively) various hybrid mass formulas can be obtained, such as

(nbb -- ~ ) -- (27 -- N ) (.~* -- 27~*) -- ( Z * -- N )

(ii-.-.[-a-~.,)--- ( & - - N ) ( ~ * - - Z * ) - - ( Z * - - A) (F~ - D J - - ( g - - ~)

( H , - - D , ) - - ( D . - - ~) (96)

504 C P Singh 5. Conclusions

In this paper we have extended the SU(3) broken symmetry scheme of Gell-Mann and Okubo to an SU(6) broken internal symmetry. First we have derived the first order mass relations and then by incorporating the second order effects, the discre- pancies present are related. The SU(6) symmetry will be badly broken so that mass contribution from the higher order will be significant as it is seen in charm sector (Verma and Khanna 1978; Singh et al 1980). We see that for 1/2 + baryon the analo- gous Gell Mann-Okubo mass formulas are found for all the sectors in the first order but the second order effects do not maintain these relations. For 3/2+ baryons the first order gives equal spacing rules which are not maintained under the second order considerations. At present because of the nonavailability of the experimental values of the masses of heavier hadrons our relation cannot be tested. Our studies give the definite test for the strength and pattern of symmetry-breaking mechanism when the masses are available.

Acknowledgements

The author is grateful to Dr M P Khanna and Dr R C Verma for helpful discussions and acknowledges the financial assistance from the University Grants Commission, New Delhi.

Appendix 1. Tensor representation for 1/2 + baryons ,B[12] 1 :p B[131 x : I+ == ~++ B[l~t]l c B[15] 1 ---,--- z~ + B[16] 1 .-~ z~ ++ B[~] 1 = ,,1._AO~, 1 lo x/6 V2 1 ,+ 1 1 ,So 1 1 St+

st~6~ = ~-~ A: + +

B[12] 2 : n B[1312 = -- "V~ Btx412 = -- A~ -- V--2

,o1

±u:

9113] 3 : -- ~/~ A 0 B[1313 : ~o

_(±z+ _{1 =,o 1 .~o) II I~+)

~.% ~/~

r~ L~ t~

__ I_Z~ + 1 ~+ __ 1 ~,++ 1 =++ __ 1 :,+ ~_.~+

-~cb

B[36]3 = ~o 1

D_,oo_f_.~_..~ D.,,+b

c~ ~'++ B[1414 ~cc B[1415 : -- '¢2/3 -~'+ a/97~ ~ ,++ B[14] 6 = -- v*./.., ~ct

(& ,+ 1 +)

B[,s] 5 = E°o B[15] 6 %/2"~"+ ~,++ B[1616 = ~tt Bt,,al, ~ = -- V'2/3 E'o

Bt~lo

~'+ J-,dcc B[241s V/2/3 =,o

BE2~1~ C273 ~'+ (~o,o , )

II io,

~+

+ Io

II II

~+

~+ ^~+

II II

I I

to

~+

I[

[0 ~+

I

$ I

[~] ~+

II

IO

II tl rl Ir

t

I I

g+

~='

H I

e..

g~

fl I

Ix]

I

[i|

II

~+

II II I[

I[

~+

~+

E

[I

<.

I0

li

[tl =+

LO~

(9)nS u

~ . svlmudof ssoul uo~pvH

508 C P Singh

Appendix 2. Tensor representation for 3/2 + baryons

= ~ I A +

Dll I : A++ Dn~ ~/~

1 Z , + D1~3 1 Z , 0

Dn3 -- ~/~ = ~-~

D ~ = ~ - 1 2 * - D ~ 3 = ~ * -

1 _ 1 2 , +

D144 = ~-~ ~ c c

_ 1 I , + _ 1 l , o

Dl15 V ~ DI~ -- ~-~

1 1 ,+

1 _ 1 f ~ , o

~,~.,=~x *+ D~.., = ~ *°

I 1

_ I D , +

l

DI~---~ A ° D2~---- A-

1 1 _~*0

D m - ~-~ Z*- D~3 = ~-~

9114 ~/3 D124 V 6 c D ~ - -

-v-~ "°

_ ~ n , o

= 1 ~2,+ D~4 = O *++

D~4 V~ ~ " ' ~

D~m = -~ I*- D135 ---- --~ .=.*b 0

1 =,o

1 Z , + +

_ 1 E , + D~..~ = - ~ E ~ °

D18s

"'

=In,+

= 1 ¢o,+

DI~ = -~2~ +÷ D~es ~

D . = ~ n:,+~ l D~. = o.*++ " ' t t t

Hadron mass formulas in SU(6)

Appendix 3. Tensor representations for JP = O- mesons (P[)

509

P~=

P; .+ K+ o2 n+ o o

~- p~ K0 D; D ° D7 K- Ko p.. F: ~ F/, D°o D~ + F: P1 G~ o ° D~- D O ~ G; p5 n~"

- - 0 D, D, F, G, Ht + P6 + + - - 0 6

Where

P~ = &I'V~ + PsI'V6 + .ed'V-fi + P~41v"~ + &d'V~ + PolV'~- P[

P31V'~ + Psi'V-6 + e~lv"ff + P,.jv, T6 + P~lV"J6 + PolV-~

P~ = - 2&l'v'6 + v,d ~ + Vulv/~ + &d v"J6 + PoIV-6 P1 = -- V312 P15 + e u l d ~ + & d ~ + PotV'-6

For vector mesons (1-) similar representations can be obtained.

References

Aubert J J e t al 1974 Phys. Rev. Lett. 33 1404

Aubrecht II G J and Scott D M 1979 SU(6) as an internal symmetry: Meson mass spoetra and decay widths, Ohio State Univ. Preprint No. COO-1545-240.

Aubrecht II G J and Scott D M 1979 Nuovo Cimento 3,50 241 Angustin J e t al 1974 Phys. Rev. Lett. 33 1406

Berger Ch et al 1978 Phys. Lett 1376 243 Boal D H 1978 Phys. Rev. D18 3446

Camiz P, Dattoli G, and Mignani R 1979 Nuovo Cimento 26 15 Cazzoli E G e t al 1975 Phys. Rev. Lett. 34 1125

Eliezer S and Singer P 1973 Phys. Rev. D8 2235 Gell-Mann M 1962 Phys. Rev. 125 1067

Glashow S L, Illiopoulos J and Maiani L 1970 Phys. Rev. I)2 1285 Goldhaber G e t al 1976 Phys. Rev. Lett. 37 255

Harari H 1975 Proc. Int. Symp. on Lepton and Photon interactions at higher energies, Stanford.

510 C P Singh

Herb S W e t a11977 Phys. Rev. Lett. 39 252 Khanna M P 1980 Lett. Nuovo Cimento 28 201 Knapp B e t al 1976 Phys. Rev. Lett. 37 882

Kobayashi M and Maskawa K 1973 Prog. Theor. Phys. 49 652

Machacek M and Tomozawa Y 1976 J. Math. Phys. 17 458 and ref. therein Mathur V S, Okubo S and Borchardt S 1975 Phys. Rev. D l l 2572

Misra D and Sastry C V 1979 Pramana 13 163 Okubo S 1962 Prog. Theor. Phys. 27 949 Pcrl M L et al 1975 Phys. Rev. Lett. 35 1489 Pod M L e t al 1977 Phys. Lett. 1370 487

Singh C P, Vcrma R C and Khanna M P 1980 Phys. Rev. D21 1388 Singh C P, Kanwar S and Khanna M P 1980 Pramana 14 433 Singh C P 1981 Phys. Rev. D23 (to appear)

Thorndike E H 1980 Proc. Int. Conf. on HEP, Madison, Wisconsin Verma R C and Khanna M P 1978 Phys. Rev. D18 828

Wilk B H 1980 DESY-80[129