Abstract
Full Text
Yu. M. MALYUTA
MASS FORMULAS IN THE SUPERMULTIPLET MODEL
(Presented by Academician N. N. Bogolyubov, January 12, 1965)
- Recently, Gürsey, Radicati, and Pais proposed a model of strongly interacting particles \((^{1,2})\), based on the group \(SU(6)\). This model systematizes hadrons according to \(F\)-spin–spin supermultiplets. In papers \((^{3-5})\), mass formulas were found for meson, baryon, and selected supermultiplets.
Fig. 1
In the present paper we shall give a formulation of this model in the language of quarks \((^{6})\). The advantage of such an approach is that it makes it possible to use diagrammatic techniques for obtaining mass formulas.
- Let us introduce into consideration a basic sextet of quarks \((q_{11}, q_{12}, q_{21}, q_{22}, q_{31}, q_{32})\) and represent it in the form
\[ q_{ij}=u_i v_j, \tag{1} \]
where \((u_1,u_2,u_3)\) is a unitary triplet, and \((v_1,v_2)\) is a spin doublet. Symbolically, relation (1) may be written as follows: \(6=(3,2)\). The quantum numbers of the particles \(u_i\) and \(v_j\) are given in Table 1 (\(B\) is baryon number, \(I\) is isospin, \(Y\) is hypercharge, \(J\) is spin).
Table 1
| \(B\) | \(I\) | \(Y\) | \(J\) | |
|---|---|---|---|---|
| \(u_1\) | \(1/3\) | \(1/2\) | \(1/3\) | \(0\) |
| \(u_2\) | \(1/3\) | \(1/2\) | \(1/3\) | \(0\) |
| \(u_3\) | \(1/3\) | \(0\) | \(-2/3\) | \(0\) |
| \(v_1\) | \(0\) | \(0\) | \(0\) | \(1/2\) |
| \(v_2\) | \(0\) | \(0\) | \(0\) | \(1/2\) |
Table 2
| \(3\) | \(v_1v_1\) | \(2\) | \((\{v_1v_2\}v_1-2v_1v_1v_2)/\sqrt{6}\) |
| \(3\) | \(\{v_1v_2\}/\sqrt{2}\) | \(2\) | \((2v_2v_2v_1-\{v_1v_2\}v_2)/\sqrt{6}\) |
| \(3\) | \(v_2v_2\) | \(4\) | \(v_1v_1v_1\) |
| \(1\) | \([v_1v_2]/\sqrt{2}\) | \(4\) | \((\{v_1v_2\}v_1+v_1v_1v_2)/\sqrt{3}\) |
| \(2\) | \([v_1v_2]v_1/\sqrt{2}\) | \(4\) | \((v_2v_2v_1+\{v_1v_2\}v_2)/\sqrt{3}\) |
| \(2\) | \([v_1v_2]v_2/\sqrt{2}\) | \(4\) | \(v_2v_2v_2\) |
Table 3
| 8 | $\bar u_2u_1$ $(\bar u_2u_1-\bar u_2u_2)/\sqrt{2}$ $\bar u_1u_2$ $\bar u_3u_1$ $\bar u_3u_2$ $\bar u_2u_3$ $\bar u_1u_3$ $(\bar u_1u_1+\bar u_2u_2-2\bar u_3u_3)/\sqrt{6}$ |
8 | $[u_3u_1]\,u_1/\sqrt{2}$ $([u_2u_3]\,u_1-[u_3u_1]\,u_2)/2$ $[u_2u_3]\,u_2/\sqrt{2}$ $[u_1u_2]\,u_1/\sqrt{2}$ $[u_1u_2]\,u_2/\sqrt{2}$ $[u_3u_1]\,u_3/\sqrt{2}$ $[u_2u_3]\,u_3/\sqrt{2}$ $[u_1u_2]\,u_3/\sqrt{2}$ |
10 | $u_1u_1u_1$ $\{u_1u_1u_2\}/\sqrt{3}$ $\{u_1u_2u_2\}/\sqrt{3}$ $u_2u_2u_2$ $\{u_1u_1u_3\}/\sqrt{3}$ $\{u_1u_2u_3\}/\sqrt{6}$ $\{u_2u_2u_3\}/\sqrt{3}$ $\{u_1u_3u_3\}/\sqrt{3}$ $\{u_2u_3u_3\}/\sqrt{3}$ $u_3u_3u_3$ |
| 1 | $(\bar u_1u_1+\bar u_2u_2+\bar u_3u_3)/\sqrt{3}$ | 1 | $[u_1u_2\,u_3]/\sqrt{6}$ |
Following the standard method (7), we construct from quarks the supermultiplets of higher dimensionality:
\[ \bar 6\times 6=35+1 \quad \text{(mesons),} \]
\[ 6\times 6\times 6=70+70+56+20 \quad \text{(baryons and isobars),} \]
where
\[ 35=(8,3)+(8,1)+(1,3)+(8,2)+(1,2), \]
\[ 1=(1,1), \]
\[ 70=(10,2)+(8,4)+(10,2), \]
\[ 56=(10,4)+(8,2), \qquad 20=(8,2)+(1,4). \]
The $F$-spin–spin multiplets $(n,m)$ are given in Tables 2 and 3 (for an explanation of the notation $\{ab\}$, $[ab]$, $\{aab\}$, $\{abc\}$, $[abc]$ see Ref. (8)).
Fig. 2
3. Using the data of Tables 2 and 3, it is not difficult to construct self-energy diagrams for mesons, baryons, and isobars, taking into account the breaking of the $S\tilde U$ (6)-symmetry in first order in the interaction
\[ L=\bar q_{31}q_{31}+\bar q_{32}q_{32}. \]
For the 35-plet and 56-plet these diagrams are shown in Figs. 1 and 2 (wavy lines denote quarks $q_{11}$, $q_{12}$, $q_{21}$, $q_{22}$,
straight lines are quarks \(q_{31}, q_{32}\), black vertices are the interaction \(L\).
The diagrams presented immediately lead to the mass formulas.
35-plet (the particle symbols denote the squares of their masses):
\[ \omega=\rho,\qquad \varphi+\rho=2K^*;\qquad K=\frac14(3\eta+\pi);\qquad K^*-\rho=K-\pi . \]
56-plet (the particle symbols denote their masses):
\[ \Omega-\Xi^*=\Xi^*-Y_1^*=Y_1^*-N^*; \]
\[ \Xi-\Sigma=\Sigma-N,\qquad \Lambda=\Sigma;\qquad \Xi^*-Y_1^*=\Xi-\Sigma . \]
- In an analogous way, the mass formulas for the 70-plet and the 20-plet are obtained.
70-plet:
\[ \Omega_0^{1/2}-\Xi_{1/2}^{*1/2} =\Xi_{1/2}^{*1/2}-Y_1^{*1/2} =Y_1^{*1/2}-N_{3/2}^{*1/2}; \]
\[ \Xi_{1/2}^{3/2}-Y_1^{3/2} =Y_1^{3/2}-N_{1/2}^{3/2},\qquad Y_0^{3/2}=Y_1^{3/2}; \]
\[ \Xi_{1/2}^{1/2}-Y_1^{1/2} =Y_1^{1/2}-N_{1/2}^{1/2},\qquad Y_0^{1/2}+Y_0^*=2Y_1^{1/2},\qquad Y_0^{1/2}Y_0^*=\frac23\,Y_1^{1/2}Y_1^{1/2}; \]
\[ \Xi_{1/2}^{*1/2}-Y_1^{*1/2} =\Xi_{1/2}^{3/2}-Y_1^{3/2} =\Xi_{1/2}^{1/2}-Y_1^{1/2}. \]
20-plet:
\[ \Xi_{1/2}^{\prime 1/2}-Y_1^{\prime 1/2} =Y_1^{\prime 1/2}-N_{1/2}^{\prime 1/2},\qquad Y_0^{\prime 1/2}=Y_1^{\prime 1/2}. \]
The formulas we have found for the 70-plet differ from the formulas given in works \((^3,^5)\).
Institute of Physics
Academy of Sciences of the Ukrainian SSR
Received
7 I 1965
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