🧠 Remember:
| Hadrons | Composition |
|---|---|
| Meson | $q\overline q$ |
| Baryons | $qqq$ |
| Anti-Baryons | $\overline q\overline q\overline q$ |
<aside> <img src="attachment:f7f818dd-2aa9-4b90-bbcc-5c3743a11e9e:charge.png" alt="attachment:f7f818dd-2aa9-4b90-bbcc-5c3743a11e9e:charge.png" width="40px" />
Charge:
$$ Q_\text{hadron}=\sum_i q_i $$
where $q_i$ is the charge of a constituent
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<aside> <img src="attachment:54f43de8-f443-46b9-bf8f-070a6cb2ebd3:baryon_number.png" alt="attachment:54f43de8-f443-46b9-bf8f-070a6cb2ebd3:baryon_number.png" width="40px" />
Baryon Number:
$$ B=\tfrac 13[N_q-N_{\overline q}] \qquad $$
Mesons: $B=0$ ; baryons: $B=1$ ; anti-baryons: $B=-1$
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<aside> <img src="attachment:a2fc46f0-c3f1-40e5-8bfe-981638cb4e79:strangeness.gif" alt="attachment:a2fc46f0-c3f1-40e5-8bfe-981638cb4e79:strangeness.gif" width="40px" />
Strangeness:
$$ S=-[N_s-N_{\overline s}] $$
where $N_s$ and $N_{\overline s}$ is the number of strange and anti strange quarks respectively
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<aside> <img src="attachment:967ed0a8-54bc-41fc-9cf7-dd7b6c2c8691:charmness.gif" alt="attachment:967ed0a8-54bc-41fc-9cf7-dd7b6c2c8691:charmness.gif" width="40px" />
Charmness:
$$ C=[N_c-N_{\overline c}] $$
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<aside> <img src="attachment:3f74897b-e2ed-4bc7-b6ab-ff43c46dd3de:bottomness.png" alt="attachment:3f74897b-e2ed-4bc7-b6ab-ff43c46dd3de:bottomness.png" width="40px" />
Bottomness:
$$ \tilde B=-[N_b-N_{\overline b}] $$
🗒️ Note: the tilde ~ is to distinguish between Baryon number and Bottomness
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💃 Examples:
| Particle | Composition | $Q$ | $S$ | $C$ | $\tilde B$ |
|---|---|---|---|---|---|
| $p$ | $uud$ | $1$ | $0$ | $0$ | $0$ |
| $n$ | $udd$ | $0$ | $0$ | $0$ | $0$ |
| $\Lambda$ | $uds$ | $0$ | $-1$ | $0$ | $0$ |
| $\Lambda_c$ | $udc$ | $1$ | $0$ | $1$ | $0$ |
| $\Lambda_b$ | $udb$ | $0$ | $0$ | $0$ | $-1$ |
| $\pi^+$ | $u\overline d$ | $1$ | $0$ | $0$ | $0$ |
| $K^-$ | $s \overline u$ | $-1$ | $-1$ | $0$ | $0$ |
| $D^-$ | $d\overline c$ | $-1$ | $0$ | $-1$ | $0$ |
| $D^+_s$ | $c \overline s$ | $1$ | $1$ | $1$ | $0$ |
| $B^-$ | $d \overline u$ | $-1$ | $0$ | $0$ | $-1$ |
| $\Upsilon$ | $b \overline b$ | $0$ | $0$ | $0$ | $0$ |
🗒️ Note: blue is baryons and green is mesons
🧠 Remember: The spin of hadron is the sum of the total angular momentum constituents
$$ {\text{Spin of bound state } \hat {\bold S}B = \hat {\bold J}\text{constituents} \text{ total angular momentum of the constituents}} $$
This is equal to $(\hat {\bold L} + \hat {\bold S})_\text{constituents}$
Spectroscopic notation: $\boxed{^{2S+1}L_J}$ where we use $S,P,D,F$ for $L=0,1,2,3$
🗒️ Note: the lowest energy state is usually $L=0$
⚙️ Properties:
- Addition of spins: $\hat {\bold S}= \hat {\bold S}_1+ \hat {\bold S}_2$ where $s\in\{ s_1+s_2,s_1+s_2-1,\ldots |s_1-s_2|\}$
- Addition of $\hat {\bold J}$: $\hat {\bold S}_\text{hadron}=\hat {\bold L}+\hat {\bold S}=\hat {\bold J}$ with $j\in\{ l+s, l+s-1,\ldots |l-s|\}$
💼 Case: a meson, quark antiquark and angular momentum $l$
The spin configurations are
$$ \hat {\bold S}_\text{M}= \hat {\bold S}q+ \hat {\bold S}{\overline q} $$
which allowed values $s\in\{0,1\}$
Lets find the total angular momentum for $l=0$
$$ j=l+s=0+(1\text{ or } 0)=1\text{ or } 0 $$
which we can write as the following states $^{2S+1}L_{J}=\ ^1S_0$ and $^3S_1$ for spin-$0,1$ respectively
For $l=1$ we have
$$ j=\overbrace{|j-s|}^{1-1},\overbrace{j+s-1}^{1+1-1},\overbrace{j+s}^{1+1}=2,1,0 $$
which gives $^1P_1, \ ^3P_2, \ ^3P_1, \ ^3 P_0$ where the first is spin-$0$ and the others spin-$1$