S U ( 8 ) m a s s relations a m o n g JP = 1/2 + a n d 3/2 + baryons R A M E S H C VERMA and M P K H A N N A
Department of Physics, Panjab University, Chandigarh 160014 MS received 5 February 1977; in revised form 23 March 1977
Abstract. Mass relations among charmed and uncharmed baryons belonging to 20 and 20-- multiplets of SU(4) are derived in the framework of SU (8) symmetry, Spin singlet mass breaking interaction is found to give unsatisfactory results. Second order effects and spin triplet mass breaking interactions are studied to improve the situation.
Keywords. SU (8) ; hadron masses ; baryons and baryon resonances.
1. Introduction
SU (4) charm scheme proposed to understand ~k resonances predicts a spectium of large number of charmed particles. Subsequent experiments (Cazzoli et al 1975; K n a p p e t a l 1976) have piovided some evidence for the existence of these particles. In view of these, it is useful to study the properties of charmed paIti- clcs in the framework of higher symmetries, as it helps tb.e experimental physicist in his search for these particles. Moreover tl=e study of the properties of these particles will provide tests for the validity of higher symmetries.
Mass relations among these particles have been obtained by several authors.
Some of these have made explicit use of the quaik models (Hendry and Lichtenberg 1975; Franklin 1975) while others have assumed a certain transformation property of the mass operator in SU(4) (Kobayashi 1972; Okubo 1975; Moffat 1975;
Boal 1975; Gupta 1976). In this paper we discuss the masses of the charmed and uncharmed baryons and isobars in the framework of SU (8). We assume the interaction responsible for breaking the symmetry to transform like:
H ' = axTx 1 + a2T3 s + aaT44 (1)
member of adjoint representation 15 in the SU (4) subgroup of SU (8). The electro- magnetic (em) mass operator is assumed to transform as T1 x + T44+ fl T~ ~ component o f 15 @_1 in SU (4). Hence in the symmetry breaking Hamiltonian (1) em mass breaking is also introduced. In this case masses of the baryons can be expressed in terms of only four parameters. We have calculated the masses of 1/2 + baryons in table 1 (column one) using three octet masses and Zx ° (2.5 GeV) as input.
524
S U (8) mass relations in baryons Table 1. Masses of the charmed baryons JP (1/2+).
Mass breaking Hamiltonian.
Spin structure Spin singlet Spin singlet Spin singlet + Spin triplet
Number of parameters Four Seven Seven
SU(3) multiplet and particles (first order) (first order + secord (first order)
(GeV) order) (GeV) (GeV)
2"1"+ 2" 494 2" 494 2" 494
I 2~1+ 2"497 2"497 2"497
B (6) / 271° 2" 500 (input) 2" 500 (input) 2- 500 (input)
~1+ 2.619 2-684 2.684
| ~t ° 2" 622 2" 687 2" 687
~.~dj 0 2" 745 2" 874 2" 874
fA J '+ 2.497 2- 260 (input) 2" 260 (input)
B(3)* ~ l '+ 2-619 2-315 2-503
t ~1' o 2" 622 2" 508 2" 508
{ ~++ 3" 924 3" 793 3. 793
B(3) ( B~+ 3"927 3"796 3"799
(~'22 + 4" 049 4" 054 4" 043
tmixing masses)
ma~* 0 --0"067 --0"0015
m a : + ~,~, 0 - - O" 2 0 8 - - 0 - 0 0 1 5
m~a'o=~,o 0 -- O" 112 -- 0"0367
m ~ / + ~ + 0 - - O" 3 1 3 - - 0 " 0 3 8 2
We find t h a t if mass b r e a k i n g i n t e r a c t i o n is taken t o be spin singlet, the masses o f all the f o r t y b a I y o n s o b e y equal spacing rule. Mass relations f o r 1/2' baryons d o not agree with experimental values. F u r t h e r m o r e the states mixed due to SU (2) a n d S U ( 3 ) breaking [i.e.: A - X ° a n d B ( 3 * ) - B(6)] are predicted to be mass degenerate a n d mixing angles c a n n o t be fixed.
Second o r d e r mass b r e a k i n g is considered to improve the situation. We take second o r d e r mass breaking H a m i l t o n i a n to transform like
H " = b~ r ~ + b~ T ~ + b a T I , " + b, ~ + b~ r,~, ' + b. T h" (2) component of 20~ and 84 representations at the SU (4~ level. H" is also assumed to be spin singlet. Resulting mass relations are given in section (2.1). Agree- m e n t o t these relations with experiment is good. Second order effects also remove the degeneracy between the mixed states, but predicts very large mixing angles, i.e.,
Oa270 = 0A'1271 = 081g~ = 300 (3)
while these mixing angl~s a t e expected to be o f the o r d e r o f , ~ 1 °.
T o remove the discrepancies in the e m mass relations o b t a i n e d in S U (6), K u o a n d Yao (1965) suggested the inclusion o f the em mass splitting due t o magnetic interactions (Liehtenberg 1975) which t r a n s f o r m as (8_ 3) c o m p o n e n t o f 35: SU (3) breaking in the spin singlet interaction is a b o u t 100 MeV in the S U (6) symmetry a n d SU (6) breaking difference o f the spin singlet and triplet interactions is also
f o u n d to be a t least t h a t much (Franklin 1968).
P ~ 5
In view of these arguments, we include the spin triplet mass breaking contributions to the masses of the baryons. Wc obtain the well satisfied Cokman- Glashow relation and electromagnetically modified G M O mass sum rule for the uncharmed baryons. Various mass relations aie given in section (2.2). We have used the particle symbol to denote its mass. Mass values of different charmed baryons and isobars are given in tables 1 and 2 respectively. Introduction of spin triplet mass breaking does not disturb the equal spacing rule for 3/2 + baryons.
We have also calculated the mixing angles between the mixed states in section (2.3).
2. Mass relations
We assume the baryons ( j r = 1/2 + and 3/2 +) to belong to the symmetric representa- tion 120-~ (20', 2) + (20, 4) in SU(8). When mass operator is assumed to trans- form like (15, 1) component of 63, we gct the various mass values of baryons given in table 1. Mixed states arc mass degenerate and mixing anglcs cannot be fixed.
2.1. Second order effects
Second order contributions are obtained from the following contractions:
1~ as'c' BAsc M~, M~,
(4)
where mass breaking Hamiltonian corresponds to 1232 representation of SU (8).
Mass breaking Hamiltonians are taken to transform like (20", 1) and ( ~ , 1) compo- nents of 1232. It is found that (20 ~, 1) and (84__, 1) components do not contri- bute to 3/2 + and 1/2 + baryons respectively. Thus the masses of 1/2 + and 3/2 + baryons are expressed in terms of seven and ten parameters respectively. We obtain ten mass relations (5, a-d) and (6, a-e) for 3/2 + isobars. Thirteen mass relations for 1/2 + baryons are giwn in (7, a-g) and (8, a-e). With five octet masses, A ' + ( 2 . 2 6 G-eV) and ~0 (2.5 GeV) as input we get different parameters for 1/2 + baryons:
Table 2.
Misses of the charmed isobars JP (3/2 +) SU (3) multiplets and Equal spacing ruleParticles (four parameters) (GeV)
[ 271"++ 2" 494
I ~ *+
D (6) ~ ?:
* 0 2 " 2 " 500497
E1 *+ 2" 642
/ S * o 2" 645
D11,o
2- 790f B, *++ 3.760
D (3) ~ Sa*+ 3- 763
3.908
D (1) {Qs *++ 5 "026
i
(input)
527 me = 0.941 GeV, al = - 1.95 MeV, as ° = 0-186 G e V ,
az = 2 - 3 4 G e V , b4 = 2.01 MeV, b5 = 0.405 GeV, be = 0 . 4 2 2 GeV.
Masses o f 1/2 + b a r y o n s are given in table 1 (column two).
Second o r d e r c o n t r i b u t i o n s d o n o t predict equal s p a c i n g rule f o r the 3/2 + iso- bars, r a t h e r relate t h e disclepancies present in this rule in the f o l l o w i n g m a n n e r
A + + - - A - = 3 ( A + - - A o) ( 5 a)
Z *+ + Z * - - - 2 Z * ° : A + + A - - - 2A° : Z 1 *++ + E l * ° - - 2)71 *+ (5b)
( ~ , 0 _ s * - ) + ( A o - - a ~) = 2 ( X *0 - - X * - ) (5 c )
(S~* + - - 3 z * , ) + ( a ° - - a - ) : 2 (El*+ - - X~ *°) (5 d) ( t a ~ * + - - ~ * + ) + ( f l - - - g * - ) = 2 ( t a , * o - - ~ , o ) ( 6 a)
£~1 *° + Y.1 * ° - 2 ~ 1 *° = f ~ - + E * - - 2 3 " - ( 6 b )
( f ~ - - - A - ) = 3 ( 3 " - - Z * - ) ( 6 c)
(f~3 *+ + - - f~-) = 3 (f~2 *+ - - f~l *°) (6 d)
~z *+ + Z * - - - 2 ~ 1 *° ---- $2 *~ + A - - - 2Xx *° ( 6 e ) R e l a t i o n (5 a) has been o b t a i n e d in SU ( 3 ) s y m m e t r y c o n s i d e r a t i o n ( I w a o 1965).
We are u n a b l e t o determine all the tell parameters f o r 3/2 + b a r y o n s as very less i n f o r m a t i o n exists c o n c e r n i n g c h a r m e d isobars. Because o f this masses o f all the 3/2 + isobars c a n n o t be calculated. I n the a b s e n c e o f e l e c t r o m a g n e t i c interactions, mass relations a m o n g various 3/2 + isomultiplets c a n be o b t a i n e d simply by r e m o v i n g the c h a r g e s in (6, a-e).
I n case o f 112 + b a r y o n s , second order effects predict the f o l l o w i n g r e l a t i o n s :
3 ( A + Z °) = 2 ( Z - + P + S O ) ( 7 a )
3(A'1 + + ~"~1 +) = 2{E1 ° + P + 32 ++) ( 7 b )
~ n + _ ~ + ) + ( ~ - _ x - ) = 2 ( s ~ o - x~o) ( 7 c)
~'~10 - - ~-.,~10 = .~.=,~10 - - '~I 0 ( 7 d )
2 ( E l ° - Z - ) ---- 2 S ~ + + $1 ° - 3 S ' 1 ° ( 7 e )
S - - - P --- 2 ( Z - - - N ) ( 7 f )
Ss + - - P : 2 ( Z t ° - - N ) ( 7 g )
~ ° - - 3 - - - Z + + Z - : N - - P (8a)
S~++ - - $ 2 + - - Z 1 ++ + Z l ° : N - - P ( 8 b ) Zl ++ - - Zi + ---- Z1 + - Z~ ° : $1+ - - 810 ( 8 c )
Z + - - Z ° : Z ° - Z - (8 d)
3 ( S ' a + - - 3'~ °) : 2 ( N - - P) + 2(Z ° - - Z - ) + 3 (h'~ + - Z1 °) (8 e)
In the absence of electromagnetic mass breaking we get:
( n ~ - 3~) + ( z - x ) = 2 t z ~ - x D
~ i -- g l = g l -- X l
2 (X; 1 -- X) = 2 g2 + g , - - 3 g , ' 3 + N = 2 Z
g,~ + N = 2Xl A', =X;x and A = Z
(9 a) (9b) (9 c) (9 d) t9 e) ( 9 f ) Relations (9 a) to (9 e) are already obtained in charmed quark models (Hendry and Lichtenberg 1975). (9 f ) and (9 g) are collapsed forms of GMO relation (12 b) and its charmed analog (12 c). This happens because of relation (g f ) which is unsatisfied experimentally. The relation (9 f ) exists, because in the absence of electromagnetic interactions, second order effects in SU(8) are unable to remove degeneracy between A',E 1 and A-X. This fact is also reflected in (14) where mA;+2:x+ = mA,r, o i.e., A'i-Ei + mixing is purely electromagnetic in origin.
Relation (8 a) is the well known Coleman-Glashow relation and (8 a) is GMO mass sum rule modified electromagnetically. Second order effects also remove the mass degeneracy between the mixed states. Mixing masses are calculated
to be
maze : ~/~/2(A -- X;°); mA,l+2:÷ = a/3/2(A'l + -- Xt+); m s l ' s , = \/3/2 (3'1 -- ~ ) so that
0A27o = OA~+Za+ :- 0 ~ , ~ : 30 ° (10)
which are too large.
2.2. (Spin singlet G spin triplet) mass breaking interaction
Spin triplet mass breaking Hamiltonian transforms like (15, 3) components of 63. In SU(4) sub group, it is taken as a'xT, x + a'~T3 s + a'zT44 member of 15.
H--ore the masses of 1/2 + baryons are expressed in terms of seven palameters which are evaluated to be
m 0 = 0.930 GeV, a x ---- -- 8 6 M~:V az = 0. 345 GeV, aa = 2. 250 GeV a t ' = 5 " 3 MeV, a' = 2 --0"127 GeV, a' = 3 -- 0" 262 GeV.
With both the spin singlet and spin triplet mass breakings the follov/ing results are obtained,
(1) Relations (7, a-e), (8, a-d) are maintained.
(2) In addition we get:
~ i + - - Zx ° = Y~+-- Z ° (11 a)
S ~ + + - - 3 a + = 3 0 - ~ - (11 b)
3 ( S ' 1 + - - ~1 'o) -~ 5 ( g ° - ~ - ) + ( P - - N) (11 c)
2 ( ~ o _ Z o) = ( g o _ N). (11 d)
Note t h a t the C o l e m a n - G l a s h o w relation and the modified G M O sum rule are again obtained. In the absence o f em interactions, relations (9 a) to (9 c) are repeated. I n a d d i t i o n we g e t :
2 ( S t - - X 1) = ( g - - N). ( 12 a)
3A + Z = 2 ( N + 8 ) ( 1 2 b )
3A' 1 + Xx = 2 ( N + E~). (12 c)
Inclusion o f spin triplet interaction does not disturb equal spacing rule for 3/2 + isobars because spin triplet mass breaking contributions are n o t i n d e p e n d e n t o f that o f spin singlet breaking. Masses o f 20 isobars are expressed in terms o f four parameters. Effective parameters are evaluated to be:
too---1.232 GeV, A 1 = 9 MeV. A ~ = 0 . 4 3 5 GeV, Aa = 3"790 GeV.
various mass relations are
f ~ s * - - f ~ * = f~2* - - ~ 1 " = f~x* - - f~. ( 1 3 a)
f ~ - - g * = g * - - Z * = Z * - - A (13 b)
= S 1 " - - E l * = ~'~1" - - $ 1 - * = ~"~2" - - " ~ 2 "
A + + - - A + = A + - A ° = A ° - A - = x * + - Z * °
= Z * ° - X * - = ~ , 0 _ _ 4 * - = Xx *++ - - Xx*+
= Xt *÷ - - Xl *° = $ 1 " + - - ~ t *° = 3~ *++ - - 32"+ (13 c) Because mass relations are identical with or without the spin triplet mass breaking.
Hence f o r b o t h the cases we give the masses of 3/2 + isobars in a single c o l u m n o f table 2.
2.3. Mixing angles
In the 20' multiplet o f S U (4) A a n d Z ° o f octet are mixed due t o S U (2) breaking a n d B ( 3 * ) is mixed with the states o f B ( 6 ) via S U ( 3 ) breaking. S U ( 2 ) mixing is expected t o be small ( o f the order o f e m breaking), while S U (3) mixing m a y be larger. These mixing effects are expected to be negligible therefore we have not considered t h e m in determining mass relations. But f o r the sake o f complete- ness we have calculated the mixing angles. Considering mass breaking due to b o t h the spin siuglet a n d spilt triplet interactions, we get t h e following mixing
m a s s e s :
1 2 - S O )
mAZ, O = mA,i+2~1-. = ~ - ~ ( P - - N + - - = 1" 53 MeV.
1 ( S O - N - - 2 ( Y - + - - P ) ) = - - 36" 71 MeV.
m g , l o ~ o =
1 g o ) )
mg,+gx÷ -~ ~ ( P - ~ - - - 2 (Z + - - --- - - 38-24 MeV (14)
using the mass values given in column 3 of table 1 we get:
OAsO -- 1°,4'; 0A~+221. : 0 °, 22'; Ogl'g 1 ~ 10 ° 6' (15) 01127o = I °, 4' has been obtained in the SU (3) symmetry framework (Carruthers 1966).
3. Discussions and conclusions
We see that S U ( 8 ) symmetry does not predict satisfactory results if mass break- ing is attributed to spin singlet interaction alone. Relations for 1/2 + baryons are too strong to be true. Second order effects definitely seem to improve the results but predict very large mixing angles of the mixed states. When mass break- ing due to spin triplet interaction is included in first order, SU (8) predicts well satisfied relations and small mixing angles. H e n c e w e may expect mass relations among charmed baryons also to be true. Any discrepancies preEent in the rela- tion can be removed by further considerations of second order effects. Know- ledge of the mass of one charmed baryons in 20 and of the two in case of 20' multiplet will provide the masses of all the other charmed baryons and isobars.
Recently a charmed antibaryon state has been observed (Knapp 1976) at 2.26 GeV decaying to ~, n - z c-r c+. Another state decaying to the first one and a positive pion has also been observed at 2.5 GeV. Assuming first state to be A~' + and second one to be either ~t *° (3/2 +) or f l ° (1/2+)we have calculated masses of charmed baryons given in tables 1 and 2.
Quark models (Hendry and Lichtenberg 1975; Franklin 1975) do not give well known Gcll-Man Okubo octet mass sum rule and equal spacing rule for decimet without additional assumptions. Both the relations follow, when the two body interaction is assumed to satisfy:
Dsp = ½ ( D s s + Dpp) (16)
where D,j is the two body Interaction energy between the i and j quark in triplet state. Extension of this assumption to
D,, = ½(D,, + Dj,) i , j = p , n , s , e (17) leads to further relations among charmed baryons, e.g., equal spacing rule for 20 multiplet and analog of GMO relation (12 c). No particular leason can be given to justify this assumption, except that it represents a simple form of symmetry break- ing. In SU (8) symmetry framework, we have derived relations similar to those obtained i n quark models, only by assuming a certain transformationc haract¢ r of the mass breaking interaction. Our relations are modified electromagnetically since we have introduced em bleaking also. If em breaking is neglected, the relations (9 a), (9 b), (9 c), (12 b) and (12c), among isomultiplets of 20' have been already obtained within the framework of quark model in broken SU (8) symmetry. The relation (12) is new. However this relation has been derived in quark-diquark model (Lichtenberg 1975b),
In case of 20" baryons, inclusion of second order effects also gives all the rela- tions obtained in quark models. If em breaking is neglected it gives two relations:
S + N = 2 E
~ + N : 2E 1 (18)
which are collapsed form of G M O relation (12 b) and its charmed analog (12 e). For 20-multiplet, second order effects relate the violations of the equal spacing rule. But inclusion of spin triplet interaction does not disturb equal spacing rule for 20 isobars.
If the mass breaking operator M~ A transforming as 63 of SU (8) is considered up to all orders, then it is seen that for arbitrary powers of M ~ , we can at most have the following three distinct baryons contractions.
These correspond to 63, 1232 and 13104 representations of SU(8), presentin the direct product:
120" ® 120 = 1 • 63 (+3 1232 @ 13104 (19)
We have not considered the contributions of 13104 as it will lead to the intro- duction of large number of additional parameters in the Hamiltonian. This deft- nitely does not lead to any useful mass relations to compare with experiment.
Like em mass splitting (Sakita 1964), if we assume general mass breaking Hamiltonian to be of the current @ current form, which transforms like
6...33 Q 63 = 1 O 63..s @ 63..._a O 720 Q 945 O 945* O 123 2 t20) Then representations common in the two direct products (19) and (20) will contribute to the mass splitting terms of the baryons. Natural candidates then are 1, 63 and 1232. Hence 13104 is neglected.
In summary we have extended the l~sual SU (6) (Pais 1966) to SU (8), thereby incorporating new quantum number charm. We see that mass relations obtained in the framework of SU (8) are well satisfied even though the SU (8) is very badly broken symmetry. Yet one may feel that breaking mechanism is systematic and there may be some deeper reason which explains why perturbation theory works well in such a badly broken higher symmetries (Hendry and Lichtenberg 1975).
If a few charmed baryons are found, mass relations obtained can be employed to confirm whether SU (8) symmetry is a worthwhile symmetry scheme.
Acknowledgement
One of us (RCV) would like to thank CSIR, New Delhi, for the financial support.
References
Boal D /-I et al 1975 Phys. Rev. Lett. 34 54
Carrutbers P A 1966 Introduction to Unitary Symmetry (N.Y: John Wiley and Sons) p. 157 Cazzoli E G et al 1975 Phys. Rev. Lett. 34 1125
Franklin J 1968 Phys. Rev. 172 1807 Franklin J 1975 Phys. Rev. D12 2077
Galliard M K, Lee B W and Rosner J L 1975 Rev. Mod. Phys. 47 277
Gupta V 1976 Prama.na 7 277; III HEP Symposium, Bhubaneshwar, 1976 Vol. II 29 I-Iendry A W and Lichtenberg D B 1975 Phys. Rev. D I 2 2756
Iwao S 1965 Nucl. Phys. 68 632
Knapp B e t al 1976 Phys. Rev. Lett. 37 882
Kobztyashi M, Nakgawa M and Nitro I-I 1972 Prog. Theor. Phys. 47 982 K u o T K and Yao T 1965 Phys. Rev. Lett. 14 79
Lichtenberg D B 1975a Phys. Rev. D12 3760 Lichtenberg D B 1975b Nuovo Cimento A28 563
Moffat J W 1975 Phys. Rev. DI2 286 Okubo S 1975 Phys. Rev. D U 3261
Okubo S, Mathur V S and Borchardt S 1 9 7 5 Phys. Rev. Left. 34 38 and 236 Pals A 1966 Rev. Mod. Phys. 38 215
Sakita B 1964 Phys. Rev. Lett. 13 643
