PX156 - F5 - quark combinations

• all non-lepton particles are formed by combining quarks and anti-quarks
• quarks are never found alone
• the principle of confinement states that quarks always combine into larger compounds, which is a property of the strong force

rules

• every combination must have zero or integer values of electric charge
• every combination must be colourless
  • colour + anti-colour, eg: ($r\overline{r}$)
  • add ($rgb$) or $(\overline{r}\overline{g}\overline{b})$

hadrons

• compound states of quarks

mesons

• combinations of a quark and an antiquark $(q_1{\overline{q}}_2)$

• must be of the same colour so that they are colourless $(r\overline{r})$

• can be different flavours such that the net charge $\in \mathbb{Z}$

• generally, $Q_{meson}=0,\pm 1$

• eg:

  • ${\pi }^{+}:(u\overline{d})$
    • it is the lightest meson
    • $Q_{{\pi }^{+}}=+\frac{2}{3}e+\frac{1}{3}e=+1e$
  • $K^{+}:(u\overline{s})$
    • $Q_{K^{+}}=+\frac{2}{3}e+\frac{1}{3}e=+1e$
  • $D^0:(c\overline{u})$
    • $Q_{D^0}=+\frac{2}{3}e-\frac{2}{3}e=0$
  • $B^{-}:(b\overline{u})$
    • $Q_{B^{-}}=-\frac{1}{3}e-\frac{2}{3}e=-1e$

baryons

• combination of three quarks or three anti-quarks

• can be different flavours such that the net charge $\in \mathbb{Z}$

• all quarks must have different colours

• eg:

  • proton $p:(uud)$
    • $Q_p=2\times \frac{2}{3}e+(-\frac{1}{3}e)=+1e$
  • ${\mathrm{\Sigma }}^0:(uds)$
    • $Q_{\mathrm{\Sigma }}=\frac{2}{3}e-\frac{1}{3}e-\frac{1}{3}e=0$
  • ${\overline{\mathrm{\Lambda }}}_c^{-}:(\overline{u}\overline{d}\overline{c})$
    • $Q_{\mathrm{\Sigma }}=-\frac{2}{3}e+\frac{1}{3}e-\frac{2}{3}e=-1e$

baryon number

• similar to the global lepton number

• quarks have a baryon number of $+\frac{1}{3}$

• anti-quarks have a baryon number of $-\frac{1}{3}$

• conserved by all forces

• its violation violates the conservation of mass

• baryons have a baryon number of 1

• eg: $B(p)=+\frac{1}{3}(u)+\frac{1}{3}(u)+\frac{1}{3}(d)=+1$

• mesons have a baryon number of 0

• eg: $B({\pi }^{+})=\frac{+1}{3}(u)-\frac{1}{3}(\overline{d})=0$

others

• tetraquark $(q_1{\overline{q}}_2{q}_3{\overline{q}}_4)$ with $(r\overline{r}g\overline{g})$
• pentaquark $(q_1{q}_2{q}_3{q}_4{\overline{q}}_5)$ with $(rgbg\overline{g})$
• rarely seen, only in colliders
• highly unstable due to high mass

quantum numbers

• knowing quantum numbers allows to work out if an interaction can happen or not and also gives the quantum numbers of the compound particles
• eg: ${\mathrm{\Omega }}^{-}:(sss)$
  • $Q=3\times (-\frac{1}{3}e)=-1e$
  • $B=3\times (+\frac{1}{3})=+1$
  • $u=3\times (0)=0$
  • $s=3\times (-1)=+3$