💼 Case: lets consider a family of particles which have the same Baryon $(B)$, stangeness $(S)$, charmness $(C)$, bottomness $(\tilde B)$ and spin parity ($J^P$)
For this we can define the hypercharge as
$$ Y=B+S+C+\tilde B $$
Which allows us to define the third component of the isospin
$$ I_3=Q-\frac{Y}{2} $$
where $Q$ is the charge
And the Isospin
$$ I=(I_3)_\text{max}=\text{maximum third isospin of the family} $$
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Isospin multiplet: Particles with the same $B,S,C,\tilde B$ and $J^P$ but different $Q$
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💃 Example:
Since the isospin appears to depend on the composition we can write the isospin of quarks
| Quark | $B$ | $Y$ | $Q$ | $I_3$ | $I$ |
|---|---|---|---|---|---|
| $d$ | $1/3$ | $1/3$ | $-1/3$ | $-1/2$ | $1/2$ |
| $u$ | $1/3$ | $1/3$ | $2/3$ | $1/2$ | $1/2$ |
| $s$ | $1/3$ | $-2/3$ | $-1/3$ | $0$ | $0$ |
| $c$ | $1/3$ | $4/3$ | $2/3$ | $0$ | $0$ |
| $b$ | $1/3$ | $-2/3$ | $-1/3$ | $0$ | $0$ |
| $t$ | $1/3$ | $4/3$ | $2/3$ | $0$ | $0$ |
💎 Conclusion: the up quark and down quark are an isospin doublet
We can thus write $I_3$ in terms of the quark quantities rather than the quark numbers
$$ I_3=\frac{1}{2}((N_u-N_{\overline u})-(N_d-N_{\overline d})) $$
where $N$ is the number of the specific quark or antiquark
⚙️ Properties: of Isospin a composite objects in term of constituents $a ,b$
Addition of Isospin
$$ \vec I = \vec I_a+\vec I_b $$
with allowed values $I_a+I_b,\,I_a+I_b-1\,\ldots ,|I_a-I_b|$
Addition of $I_3$
$$ I_3=I_3^a+I_3^b $$
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Supermultiples: particles with the same $B$ and $J^P$
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💼 Case: lets consider the the supermultiples of light hadrons ie $C=\tilde B =0$, thus the quarks will be composed of $u,d,s$
The hypercharge and third component isospin is
$$ Y=B+S \qquad I_3=Q-\frac{Y}{2} $$
Now lets consider the baryon case and the meson case