Abidi, S T H

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PROPERTIES OF 35-PLET OF ONE BARYON AND TWO-BARYON SYSTEMSS. T. H. ABIDID e p a k t m k n t o f P h y s i c s , I n d i a n I n s t it u t e of T e i h n o l o o y , K h a r a g p u rI n d i a(Received October 30, 1907, Reaubmitted June 17, 1908)ABSTRACT. The states bolougiiig to (4,1) irreducible roprescutations of SU3 have boon rlassilleiJ by their idiargo and t7-spin. 'J'his clasRifi<*ation has then been used to derive mass rolatitais. The probable value of spin J ol‘ u baryon-baryoii system belonging to (4, 1) repre- st'iilntion lias been sliowa to bo 1 . It has also boon found that (J^ * - -- 5/2 ‘ 1 ) baryon states can only belong to (4, 0, 0, 0, 1) irreducible ropresentation of SIIq. Seine branching ratios in the decays 3 5 8 ( ^ 2 7 are calculated.51I N T R O D U C T I O NKbcently Dashon and Sliarj) (1965) liavo used tho statin model of Abers' ef (d (1964) in conjunction with /S-inatrix perturbation theory of Dashen and Frautschi (1964) to discuss tlie masses and dc^ cay of 35-plet of baryons — 5/2 ^into (i) an octet of baryons and an octet of mesons and (ii) an octet of mesons and a diicimeut of baryons. Zimennau and Fagundes (1966) liavo calculated certain sum-rules among the coupling constants in the decay of 35-plot into octet of 0 ' mesons and doeimont of 3/2 »- baryons. In the present paper the states belonging to (4, 1) irreducible representations of SU3 have been classified accjording to their cliarge and ?7-spin (introduced by Lipkin, Mesbkov and Levinson 1963). This classification has been found useful in deriving mass relations among the different members of the multiplot. The relations obtained are in perfect agreement with the masses calculated by Daahon and Sharp. The possible irriducible represen­tation of SU ^into which this (4,1) multiplot can be accomodated has been shown to be (4 ,0,0 , 0, 1).The results obtained by this analysis may he utilized with advantage to study tile properties of the 35-plot of dibaryons. Some attempts (Oakes 1963, Gorstein 1964 and Lipkin 1965) have already been made to classify the multi-baryonic states according to irreducible representations of SIT3 with limited success. Oakes (1963), for example, has shown that deuteron is a member of a lO-multi- plot. The other members of this multiplet may, possibly, be produced in baryon- baryon interactions. The possible multiplets that may arise in baryon-baryon interactions are an octet, a decimot, a 27-plot and a 35-plet. Some properties of4654 6 6 a. T. H. Abidithis 35-plet have been studied by the method presented in this paper. The pos­sible spin values aie specified and some interesting brandling ratios in the decay of the 35-plet into an octet and a 27-plot have been calculated.Cr.SPIN E I G E N - S T A T E S O F 35-P L E TIn SU3 multipletfc like (3//, 0) and (0, S/i) where /ti is any integer, since there is never more than one state at a given point on tlio /g— Y diagram the transfor­mation from I-spin classification to i7-spin classification of states obtainod simply by a rotation of /g— Y diagram by an angle of 120°. In the multiplot (4,1), however, one finds that there are twenty points on I>^—Y diagram where two states occur at the same point. Consequently, in 35-plot the eigenfunctions of are not the eigen-functions of E7^ . The f/-spin oigen-statos can, however, be expressed as linear combinations of isor pin eigon-statos. In table 1 the ?7-fepin eigen-states, along with their charges, as linear combinations of ieospin states are listed. The coefficients in these combinations have been determined with the help of commutation relations between raising and lowering operators U_and / . . For the isospin oigen-states notations of Lipkin and Harari (1964) have been used.Table 1I/ -^spin eigen-statesCharge Q U Ua Notation3 1 /2 + 1 /2 *3 11-2 - 1 /2 2^ 5* H-+2 1 + 1 /■*22 1 0 " ( I ) * " ' * " + ( ! ■ ) * " • '2 1 — 12 0 01 3/2 + 3/2 l2^1 3/2 + 1 /2 ■ ( r ) * " • • '+ ( 4 ) * w *1 3/2 - 1 /2 Fa*‘* =1 3/2 - 3 /2 Sa*+^5-Plet of BaryomsTable ] (contd.)4 6 7Charge Q u u.3 Ifotation1 1 / 2 -f-1 / 2 - a ) -1 1 / 2 - 1 / 2' ( - 2- ) ‘0 2 + 20 2 + 1 A^ 6*°•“ ( H i)0 2 0 1 -2*00 2 - 1 S3*“ ' ( 2 ( 4 ) ‘ s . -0 2 — 2 Hi*®0 1 H 1 IV.., *0 .0 1 0 5-,*o = ( i0 1 — 1 S i* “ ~ f-J— 1 6 /2 -1-6 /2 2~~ 1 5/2 H-3/2 A -,*- =- ( y ' - . -— 1 6 /2 4 1 / 2 =■ (l-‘ ( ? „ ) * > ' . -— 1 6 /2 - 1 / 2 Sa*" -— 1 6 /2 S j 2 h i* - = ( 1 ) * -— 1 6 /2 — 6 /2 -aTi-- 1 3/2 + 3/2 1^3*- =4 6 8 S. T . H . A bid iTable 1 (contd.)Change Q U U3 Notation— 1 3/2 ^■l/2 r'l*- =— 1 3/2 -1 /2 s . -— 1 3/2 -3 /2 ilo* ' =- 2 2 -1 2 N^*--—•2 2 + 1_2 2 0 Ha*_2 2 — 1_2 2 _2 x r -IM A S S - R E L A T I O N SThe above TZ-spin classification may be used to find relations amc ng tlie masses of (4, 1) multiplet. I f the strong interactions were fully invariant under SU3 all mombors of a given multiplet would have the same mass provided that the small mass splitting duo to electromagnetic interaction be neglected. Since the masses of different particles classified in the same multiplot are difloront, the strong interaction Hamiltonian operator may bo written asH = H o+H , 0 )wliero is SU3 invariant and is not. / / i , nevertheless, being a strong inter­action Hamiltonian must not violate liypercharge an 1 isospin conservation. Tlic simplest assumption, usually made, is that mass breaking operator transforms a* ly-meson i.c.,T ji- \ xjo2 2(2)where is a ?7-spin vector operator and is scalar operator. From eqs. (1) and (2) one obtains the follo’wing matrix elements of mass breaking operator :< / , + + + ! ! / , + + + > = - i <iv,*+++iff^|iv,*+++> = iII r/‘ I I I> - 1 < iII IIi >< 1 II t7 i|| 1 > _ J < 1 Iu;.|| <^6*^ == — <1 (I t/0 j|l>Z6~Plet o f Baryona< n - + | / / , | r , .+ * > „ _ V S „ p . II j II II 1 >I V > - < f II f'- II » > - i < s II t;. II {?. >> — ^ < t II ^ M l g > - J < § II / 7 0 I I , >I f / + > - - ^ 1 < ^ II II I > - i < 5 II UO II « < H 3 * + | ^ r , | H 3 * ^ > ^ < f I I C 7 M I I > - i < s II r70|| 3 >I / 2« > ^ < 2 II r / i | | 2 > _ J < 2 II no II 2 >//, |i^5-> ^ <-> II II 1>> - J < 2 II no II 2>< V ® I 1 v ® > =- ~ i < 2 II n o II 2 >< Es*®! / / , 1 S,,*«> ~ < 2 II f / ' II 2 > - - J < 2 II 7/0 II 2 > = - v /3 < 2 II 7 /1 II 2 > - ^ < 2 II 7/0 II 2 >< A “ | / A I A - > = < § II C ^ M U > - i < f I I 770 II s >_ 3V 3\/5< II II ! > -- i < S II f/o II I >I - i < f I! 770|| I > - V | < | | | 7 7 i | | $ > - J < | l | r / o | | | >4 6 94 7 0 S. T. H. Abidi< A * --<xr\ih\x, -> - ^ r. II 771?7* I -i; >< 0 II f/" II § >< 5 II II« >< N , * - 1H, I N ^ *-> =- V3 < 2 II ?/’ II 2 > - i < 2 II | 2 >< Fa— |//J 7^ * - > - ' " 'I < 2 II II 2 > - J < 2 II f/® 2 | >< Ss— I / /1IH3* i < 2 II r/o !| 2<Oi»— 1 //, IQ,*—> =■ < 2 II r;* II2 > - I < 2 II ?/«II2 >< X i - | 7 7 i| A V - > - - A / 3 < 2 1 |rn ||2 >-^< 2 i|r7 "||2 > ... (:5)whore <n | ?71| a > is the reduced matrix element between states with U s])in equal to n. Those matrix eleraonts load to th(i following relations among tlu! masses of different particlcis :M (1,)+M {Y*) J J l f (A V )+ J J W )M(r,)+M(E,*) = ^3f(.V,*)+*3/(Ar3*)+Jilf(7,*)+4J/(74*)M{I,)+M{a*) - Jil/(7s.*)+Jilf(7i*)M (Y^)+M (il*) = M (N *)^M (X,) ^ 2M(S3*) (4)Those relations are found to be in perfect agreement with the masses given by Dashen and Sharp (1965).SUa C I > A S S I F I C A T I O N A N D P R O B A B L E V A L D E S OF S P I NIn this section we find out the irreducible representations of SUe to w hich the 35-plet of SUg m ay belong. Let us first consider the baryon states having baryon number B = \. These states will appear as resonances in meson- baryon interactions at high energies. They will, therefore, by contained in^5-Plet of Baryons mor more, of the irreducible representations of SCTg that are obtained in tlio reduo- lion of the direct i>roduct 56<g>35. This redu(‘tion is given by Itzykson and Mauenborg (1966)56® 35 = 7OO01134©7O©56Noa\' il wo exam ine the (SU 3, S U 2) contents o f all tlie irreducible representations occuring on the righ t side o f equation (5) w c find tliat tlien^ a m (i) tw o 35-plets o f SU3 with spins J — 5 /2 and J — 3 /2 respectively in the 700-(liiuensional repre- scataiion o f SUg and (ii) tw o 35-plets Avith J == 3/2 and J I j2 in 1134 dim ensional ?'(‘]>njs(»ntatioa o f SUg. N o 35-plet occurs in 70-dijnonsioiial and 56-diinonsional iv]>rosentation. W o, tlioreforo, find that baryons can on ly bo accom o-datjul in 70-dim ensionaI representation o f SUg.Th(' h aryon -baryon 35-plots ( U 2) will, sim ilarly, be fo u n l in sonu'. irrodu- ribl<' represemtations o f tliat are obtaim ul in the nMluction : ([tzyk son and Nauenberg 1966)5()®56 10500490©! 1340462 . (6 )TJic generalized Pauli principle allows only the tAVo auti-SA i^iimetric representations 490 and 1050 to bo realized in nature. The 35-plets of SU^ octuir in both tlu s^o r(ii)resoutations. In 490-dimensional irreducible representation only those 35-plct of jSILj occur AA'hich have spin J 1 while in 1050-dimensional representation of there are throe 35-plets Avith spin values J ^ 1, 2 and 3. Thus the low- lying baryon-baryon states will have spin J = 1.H H A X C H I N G K A T I O S I N T H E D E 0 A Y O F 35-P L E T1'lic decays of a 35-plet o f single baryons 5/2' into (i) a meson octet andbaryon octet and into (ii) a meson octet and a baryou diNcimeai have been discussiul by Dashen and Sharp (1965) and by Zimmerman and Fagundes (1966). Their I'csults could be used for di-baryon states also if avo replace the meson octet by baryou octet. We want to discuss the decay of 35 di-baryon states into an octet of meson and a 27-plet of di-baryons. We assume that for higher dimensional resonances the violation of unitary symmetry primarily affects the phase space aufl form factors in the decay widths. The branching ratios may then bo predicted irom the following wave-functions :I /*++> = _ ( i ) i / 2 1 I I+{*)»/*I V > = - ( i ) * '* I - (i)»/® I K\,^X\,^>+(1)1/* 17T+iX_i*>+ ( ! ) ! « I 7tW > + (^ ) i /* 17r-,X ,*>I - (A )" '* 1 K\,^XK^,,> - ( i ) i /^ | ^ -3 /,J X i_ ,„> + ( i ) i /* I 7T«iiC*_i>+(i)i/*|7r-iJTo‘*> •■ (7)where . -2C stand for a mombor of 27-plot having quantum numbers Y = 1'',* -*31 = 1 ' and /'s. The other thirty-one wavt^-funetions can be similarly written dovra using the Clobsch-Gordon eoefficients of 8U3. The branching ratios are then given by (Gasiorowics 1964) :4 7 ^ 3 . T. B , Abidi\P~m J P'M.f+A^ («)Hero Ml and Mg are the inenibors of meson octet; and are members of 27-plet of baryon-baryon states; PMi and PM2 »ro the momenta in the rospec;- tivo decays, I is the orbital angular momentum in the decay and A reflects the source size. Thus, using wave-functions (7) we obtain :p = K++„^X\i,) _ 1• M(7g+++-».jr^-]-,AV) '2P _ > ^■’■ + 3/2X*]/g)_ 3* M (V+->7T++ iA,*)'^ 4R = L® 7r«4-iX|) 4„ _ M (7g+-.A++3/g^_,) _ 3* M(7j+^Tr^-f7A*_,)” ‘ 2‘ M (/,+-fzrO-fiA„*)' ~ 8p -^^+a/a^^-3/2) 1• M(7g®-> 7T<>4qz5r,r" “ 4R _ A®4-3/S^*-1/2) 3The branching ratios in the decays of other particles o f this plot can als > be shni- larly found out.35“P/e< o f Baryons 473ACKNOW LEDGM ENTI am greatly indebted to Dr. G Bandopadliyay for his suggestions and encouragement.R E F E B E N C E SAbers, E. S., Balazs, L. A. P. and Kara, Y., 1964, Phyt^Rev., 186B, 1382.Dasheii, R. F. iind Sharp, 1). If., 1965, Phyfi, R ev,, 188B, 223.DaHlieii, R. F. and FraiiiflG.lii, S. C., 1964, Phy^. R ev., 186, B 1J90. Gaworowoz, S., 1963, Phys, R ev ., 181, 2808.Itzyksori, C. and Nauonborg, M„ 1966, R ev. M o d . P h ye ., 88, 95.Lipkin, H. J,, Moshkov, S. and Lovuisoji, V. A., 1903, Phys. R ev. Lett., 10,361. Lipkiu, H. J., and Harari, H., 1964, Phyft. Rev. L ett., 18, 345.Zimmerman, A. H. and Fagundes, K. V., 1960, II Sunvo Ciw^rtto 48, 233.

Properties of 35-Plet of One Baryon and Two-Baryon Systems

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Properties of 35-Plet of One Baryon and Two-Baryon Systems

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